Subgroups — C₃₁×#60819 #21517578

Order 7936.

2058 conjugacy classes of subgroups.

The subgroup lattice has 924 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3510	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3500	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3225	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3224	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3212	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3211	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3153	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#495	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#3140	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#3131	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#3044	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#3043	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#3042	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#3041	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#3011	C₂	1	1	yes	maximal, characteristic, complemented
16	C₃₁×#3010	C₂	1	1	yes	maximal, characteristic, complemented
17	C₃₁×#3009	C₂	1	1	yes	maximal, characteristic, complemented
18	C₃₁×#3008	C₂	1	1	yes	maximal, characteristic, complemented
19	C₃₁×#2922	C₂	1	1	yes	maximal, characteristic, complemented
20	C₂×C₃₁×2₋¹⁺⁴	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#137	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#137	C₂²	1	1	yes	characteristic, complemented
24	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic, complemented
26	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
27	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
28	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
29	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
30	C₃₁×#559	—	2	2	no	—
31	C₃₁×#556	—	2	2	no	—
32	C₃₁×#551	—	2	4	no	—
33	C₃₁×#548	—	2	2	no	—
34	C₃₁×#546	—	2	4	no	—
35	C₃₁×#546	—	2	2	no	—
36	C₃₁×#542	C₂²	1	4	yes	complemented
37	C₃₁×#536	C₂²	1	1	yes	characteristic
38	C₃₁×#536	C₂²	1	1	yes	characteristic
39	C₃₁×#536	C₂²	1	2	yes	complemented
40	C₃₁×#535	C₂²	1	1	yes	characteristic
41	C₃₁×#535	C₂²	1	1	yes	characteristic
42	C₃₁×#535	C₂²	1	2	yes	complemented
43	C₃₁×#533	C₂²	1	1	yes	characteristic, complemented
44	C₃₁×#532	C₂²	1	1	yes	characteristic
45	C₃₁×#532	C₂²	1	2	yes	complemented
46	C₃₁×#529	C₂²	1	4	yes	complemented
47	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
48	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic, complemented
49	C₂×C₃₁×#124	C₂²	1	2	yes	complemented
50	C₃₁×#519	C₂²	1	1	yes	characteristic, complemented
51	C₃₁×#518	C₂²	1	1	yes	characteristic, complemented
52	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
53	C₃₁×#497	C₂²	1	2	yes	complemented
54	C₃₁×#496	C₂²	1	2	yes	complemented
55	C₃₁×#495	C₂²	1	4	yes	complemented
56	C₃₁×#495	C₂²	1	4	yes	complemented
57	C₃₁×#495	C₂²	1	2	yes	complemented
58	C₃₁×#495	C₂²	1	2	yes	complemented
59	C₃₁×#489	C₂²	1	4	yes	complemented
60	C₃₁×#488	C₂²	1	2	yes	complemented
61	C₃₁×#488	C₂²	1	2	yes	complemented
62	C₃₁×#488	C₂²	1	4	yes	complemented
63	C₃₁×#487	C₂²	1	2	yes	complemented
64	C₃₁×#486	C₂²	1	2	yes	complemented
65	C₃₁×#464	C₂²	1	4	yes	complemented
66	C₃₁×#464	C₂²	1	4	yes	complemented
67	C₃₁×#463	C₂²	1	4	yes	complemented
68	C₃₁×#463	C₂²	1	4	yes	complemented
69	C₃₁×#462	C₂²	1	4	yes	complemented
70	C₃₁×#461	C₂²	1	4	yes	complemented
71	C₃₁×#460	C₂²	1	4	yes	complemented
72	C₃₁×#459	C₂²	1	4	yes	complemented
73	C₃₁×#452	C₂²	1	4	yes	complemented
74	C₃₁×#452	C₂²	1	4	yes	complemented
75	C₃₁×#451	C₂²	1	4	yes	complemented
76	C₃₁×#450	C₂²	1	4	yes	complemented
77	C₃₁×#449	C₂²	1	4	yes	—
78	C₃₁×#449	C₂²	1	4	yes	complemented
79	C₃₁×#448	C₂²	1	4	yes	complemented
80	C₃₁×#447	C₂²	1	4	yes	—
81	C₃₁×#433	C₂²	1	4	yes	complemented
82	C₃₁×#431	C₂²	1	4	yes	complemented
83	C₃₁×#424	C₂²	1	1	yes	characteristic, complemented
84	C₃₁×#424	C₂²	1	1	yes	characteristic, complemented
85	C₃₁×#419	C₂²	1	1	yes	characteristic, complemented
86	C₃₁×#419	C₂²	1	1	yes	characteristic, complemented
87	C₃₁×#418	C₂²	1	1	yes	characteristic
88	C₃₁×#418	C₂²	1	1	yes	characteristic
89	C₃₁×#416	C₂²	1	1	yes	characteristic
90	C₃₁×#416	C₂²	1	1	yes	characteristic
91	C₃₁×#416	C₂²	1	1	yes	characteristic, complemented
92	C₃₁×#416	C₂²	1	1	yes	characteristic, complemented
93	C₃₁×#408	C₂²	1	1	yes	characteristic
94	C₃₁×#408	C₂²	1	1	yes	characteristic
95	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
96	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
97	C₃₁×2₋¹⁺⁴	—	4	2	no	—
98	C₃₁×2₋¹⁺⁴	—	4	2	no	—
99	C₃₁×H(5,𝔽₂)	—	4	2	no	—
100	C₃₁×H(5,𝔽₂)	—	4	2	no	—
101	C₂×#41×C₃₁	—	2	1	no	—
102	C₂×#41×C₃₁	—	2	2	no	—
103	C₂×#41×C₃₁	—	2	1	no	—
104	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
105	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
106	C₂×#41×C₃₁	C₂³	1	2	yes	—
107	C₂×#41×C₃₁	C₂³	1	2	yes	—
108	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
109	C₂²×Q₈×C₃₁	—	2	1	no	—
110	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
111	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
112	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
113	C₂²×D₄×C₃₁	—	2	2	no	—
114	C₂²×D₄×C₃₁	—	2	1	no	—
115	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
116	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
117	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
118	C₃₁×#137	—	4	2	no	—
119	C₃₁×#137	—	4	2	no	—
120	C₃₁×AGL(1,ℤ/8)	—	4	2	no	—
121	C₃₁×AGL(1,ℤ/8)	—	4	2	no	—
122	C₂×Q₁₆×C₃₁	—	4	2	no	—
123	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
124	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
125	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
126	C₂×SD₁₆×C₃₁	—	4	2	no	—
127	C₂×SD₁₆×C₃₁	—	4	2	no	—
128	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
129	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
130	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
131	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
132	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
133	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
134	C₂×D₈×C₃₁	—	4	2	no	—
135	C₂×D₈×C₃₁	C₂³	1	2	yes	—
136	C₂×D₈×C₃₁	C₂³	1	4	yes	—
137	C₂×D₈×C₃₁	C₂³	1	2	yes	—
138	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
139	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
140	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
141	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
142	C₃₁×C₄⋊Q₈	—	2	1	no	—
143	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
144	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
145	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
146	C₃₁×#127	—	2	1	no	—
147	C₃₁×#127	C₂³	1	2	yes	—
148	C₃₁×#127	C₂³	1	1	yes	characteristic
149	C₃₁×#126	—	2	4	no	—
150	C₃₁×#126	—	2	4	no	—
151	C₃₁×#125	C₂³	1	2	yes	—
152	C₃₁×#124	—	2	1	no	—
153	C₃₁×#124	—	2	1	no	—
154	C₃₁×#124	—	2	4	no	—
155	C₃₁×#124	—	2	2	no	—
156	C₃₁×#124	—	2	4	no	—
157	C₃₁×#124	—	2	2	no	—
158	C₃₁×#124	C₂³	1	2	yes	—
159	C₃₁×#124	C₂³	1	2	yes	—
160	C₃₁×#124	C₂³	1	2	yes	—
161	C₃₁×#124	C₂³	1	1	yes	characteristic
162	C₃₁×#124	C₂³	1	1	yes	characteristic
163	C₃₁×#124	C₂³	1	2	yes	—
164	C₃₁×#124	C₂³	1	2	yes	—
165	C₃₁×#124	C₂³	1	2	yes	—
166	C₃₁×#123	—	2	2	no	—
167	C₃₁×#123	—	2	4	no	—
168	C₃₁×#123	—	2	2	no	—
169	C₃₁×#123	—	2	4	no	—
170	C₃₁×#123	—	2	4	no	—
171	C₃₁×C₂²⋊Q₈	—	2	2	no	—
172	C₃₁×C₂²⋊Q₈	—	2	2	no	—
173	C₃₁×C₂²⋊Q₈	—	2	4	no	—
174	C₃₁×C₂²⋊Q₈	—	2	4	no	—
175	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
176	C₃₁×#121	—	2	2	no	—
177	C₃₁×#121	—	2	4	no	—
178	C₃₁×#121	—	2	2	no	—
179	C₃₁×#121	—	2	4	no	—
180	C₃₁×#121	C₂³	1	4	yes	—
181	C₃₁×#120	—	2	2	no	—
182	C₃₁×#120	—	2	2	no	—
183	C₃₁×#120	—	2	4	no	—
184	C₄×Q₈×C₃₁	—	2	2	no	—
185	C₄×Q₈×C₃₁	—	2	2	no	—
186	C₄×Q₈×C₃₁	D₄	1	2	yes	—
187	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
188	C₄×D₄×C₃₁	—	2	4	no	—
189	C₄×D₄×C₃₁	—	2	2	no	—
190	C₄×D₄×C₃₁	—	2	2	no	—
191	C₄×D₄×C₃₁	—	2	4	no	—
192	C₄×D₄×C₃₁	D₄	1	4	yes	—
193	C₄×D₄×C₃₁	D₄	1	2	yes	—
194	C₄×D₄×C₃₁	C₂³	1	4	yes	—
195	C₃₁×#117	—	2	2	no	—
196	C₃₁×#117	—	2	4	no	—
197	C₃₁×#117	C₂³	1	1	yes	characteristic
198	C₃₁×#117	C₂³	1	1	yes	characteristic
199	C₃₁×#117	C₂³	1	1	yes	characteristic
200	C₃₁×#117	C₂³	1	2	yes	—
201	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
202	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
203	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
204	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
205	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
206	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
207	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
208	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
209	C₃₁×#103	C₂³	1	2	yes	—
210	C₃₁×#103	C₂³	1	2	yes	—
211	C₃₁×#103	C₂³	1	2	yes	—
212	C₃₁×#103	C₂³	1	2	yes	—
213	C₃₁×#103	C₂³	1	4	yes	—
214	C₃₁×#103	C₂³	1	4	yes	—
215	C₃₁×#102	C₂³	1	2	yes	—
216	C₃₁×#102	C₂³	1	2	yes	—
217	C₃₁×#102	C₂³	1	2	yes	—
218	C₃₁×#102	C₂³	1	2	yes	—
219	C₃₁×#102	C₂³	1	4	yes	—
220	C₃₁×#102	C₂³	1	4	yes	—
221	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
222	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
223	C₃₁×#97	C₂³	1	2	yes	—
224	C₄×C₈×C₃₁	C₂³	1	2	yes	—
225	C₄×C₈×C₃₁	C₂³	1	2	yes	—
226	C₄×C₈×C₃₁	C₂³	1	2	yes	—
227	C₂⁴×C₃₁	—	2	1	no	—
228	C₂⁴×C₃₁	—	2	2	no	—
229	C₂⁴×C₃₁	—	2	1	no	—
230	#41×C₃₁	—	2	2	no	—
231	#41×C₃₁	—	2	2	no	—
232	#41×C₃₁	—	2	2	no	—
233	#41×C₃₁	—	2	2	no	—
234	#41×C₃₁	—	2	2	no	—
235	#41×C₃₁	—	2	2	no	—
236	#41×C₃₁	—	4	2	no	—
237	#41×C₃₁	—	4	2	no	—
238	#41×C₃₁	—	4	2	no	—
239	#41×C₃₁	—	4	2	no	—
240	#41×C₃₁	—	4	8	no	—
241	#41×C₃₁	—	4	4	no	—
242	#41×C₃₁	—	4	4	no	—
243	C₂×Q₈×C₃₁	—	2	1	no	—
244	C₂×Q₈×C₃₁	—	2	2	no	—
245	C₂×Q₈×C₃₁	—	2	1	no	—
246	C₂×Q₈×C₃₁	—	2	2	no	—
247	C₂×Q₈×C₃₁	—	2	2	no	—
248	C₂×Q₈×C₃₁	—	2	2	no	—
249	C₂×Q₈×C₃₁	—	2	2	no	—
250	C₂×Q₈×C₃₁	—	2	2	no	—
251	C₂×Q₈×C₃₁	—	4	2	no	—
252	C₂×Q₈×C₃₁	—	4	2	no	—
253	C₂×Q₈×C₃₁	—	4	4	no	—
254	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
255	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
256	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
257	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
258	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
259	C₂×D₄×C₃₁	—	2	2	no	—
260	C₂×D₄×C₃₁	—	2	1	no	—
261	C₂×D₄×C₃₁	—	2	2	no	—
262	C₂×D₄×C₃₁	—	2	1	no	—
263	C₂×D₄×C₃₁	—	2	4	no	—
264	C₂×D₄×C₃₁	—	2	1	no	—
265	C₂×D₄×C₃₁	—	2	2	no	—
266	C₂×D₄×C₃₁	—	2	1	no	—
267	C₂×D₄×C₃₁	—	2	4	no	—
268	C₂×D₄×C₃₁	—	2	2	no	—
269	C₂×D₄×C₃₁	—	2	2	no	—
270	C₂×D₄×C₃₁	—	2	2	no	—
271	C₂×D₄×C₃₁	—	2	2	no	—
272	C₂×D₄×C₃₁	—	2	2	no	—
273	C₂×D₄×C₃₁	—	4	2	no	—
274	C₂×D₄×C₃₁	—	4	2	no	—
275	C₂×D₄×C₃₁	—	4	8	no	—
276	C₂×D₄×C₃₁	—	4	4	no	—
277	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
278	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
279	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
280	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
281	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
282	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
283	C₂²×C₄×C₃₁	—	2	1	no	—
284	C₂²×C₄×C₃₁	—	2	2	no	—
285	C₂²×C₄×C₃₁	—	2	1	no	—
286	C₂²×C₄×C₃₁	—	2	2	no	—
287	C₂²×C₄×C₃₁	—	2	1	no	—
288	C₂²×C₄×C₃₁	—	2	2	no	—
289	C₂²×C₄×C₃₁	—	2	1	no	—
290	C₂²×C₄×C₃₁	—	2	4	no	—
291	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
292	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
293	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
294	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
295	Q₁₆×C₃₁	—	4	4	no	—
296	Q₁₆×C₃₁	—	4	2	no	—
297	Q₁₆×C₃₁	—	4	2	no	—
298	SD₁₆×C₃₁	—	4	4	no	—
299	SD₁₆×C₃₁	—	4	4	no	—
300	SD₁₆×C₃₁	—	4	2	no	—
301	SD₁₆×C₃₁	—	4	2	no	—
302	SD₁₆×C₃₁	—	4	2	no	—
303	SD₁₆×C₃₁	—	4	2	no	—
304	D₈×C₃₁	—	4	4	no	—
305	D₈×C₃₁	—	4	2	no	—
306	D₈×C₃₁	—	4	2	no	—
307	M₁₆×C₃₁	—	2	2	no	—
308	M₁₆×C₃₁	—	2	2	no	—
309	C₂×C₈×C₃₁	—	2	2	no	—
310	C₂×C₈×C₃₁	—	2	2	no	—
311	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
312	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
313	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
314	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
315	C₄⋊C₄×C₃₁	—	2	2	no	—
316	C₄⋊C₄×C₃₁	—	2	4	no	—
317	C₄⋊C₄×C₃₁	—	2	4	no	—
318	C₄⋊C₄×C₃₁	—	2	2	no	—
319	C₄⋊C₄×C₃₁	—	2	4	no	—
320	C₄⋊C₄×C₃₁	—	2	2	no	—
321	C₄⋊C₄×C₃₁	—	2	2	no	—
322	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
323	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
324	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
325	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
326	C₂²⋊C₄×C₃₁	—	2	1	no	—
327	C₂²⋊C₄×C₃₁	—	2	1	no	—
328	C₂²⋊C₄×C₃₁	—	2	4	no	—
329	C₂²⋊C₄×C₃₁	—	2	4	no	—
330	C₂²⋊C₄×C₃₁	—	2	2	no	—
331	C₂²⋊C₄×C₃₁	—	2	4	no	—
332	C₂²⋊C₄×C₃₁	—	2	4	no	—
333	C₂²⋊C₄×C₃₁	—	2	4	no	—
334	C₂²⋊C₄×C₃₁	—	2	4	no	—
335	C₂²⋊C₄×C₃₁	—	2	4	no	—
336	C₂²⋊C₄×C₃₁	—	2	4	no	—
337	C₂²⋊C₄×C₃₁	—	2	4	no	—
338	C₂²⋊C₄×C₃₁	—	2	2	no	—
339	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
340	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
341	C₄²×C₃₁	—	2	4	no	—
342	C₄²×C₃₁	—	2	1	no	—
343	C₄²×C₃₁	—	2	1	no	—
344	C₄²×C₃₁	C₂×D₄	1	4	yes	—
345	C₄²×C₃₁	C₂×D₄	1	2	yes	—
346	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
347	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
348	C₄²×C₃₁	C₂⁴	1	2	yes	—
349	#60819	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
350	C₂³×C₃₁	—	2	2	no	—
351	C₂³×C₃₁	—	2	2	no	—
352	C₂³×C₃₁	—	2	1	no	—
353	C₂³×C₃₁	—	2	1	no	—
354	C₂³×C₃₁	—	4	4	no	—
355	C₂³×C₃₁	—	4	2	no	—
356	C₂³×C₃₁	—	4	2	no	—
357	C₂³×C₃₁	—	8	2	no	—
358	C₂³×C₃₁	—	8	2	no	—
359	C₂³×C₃₁	—	8	4	no	—
360	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
361	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
362	Q₈×C₃₁	—	2	2	no	—
363	Q₈×C₃₁	—	2	2	no	—
364	Q₈×C₃₁	—	2	2	no	—
365	Q₈×C₃₁	—	2	1	no	—
366	Q₈×C₃₁	—	2	1	no	—
367	Q₈×C₃₁	—	2	2	no	—
368	Q₈×C₃₁	—	2	2	no	—
369	Q₈×C₃₁	—	2	1	no	—
370	Q₈×C₃₁	—	2	1	no	—
371	Q₈×C₃₁	—	4	4	no	—
372	Q₈×C₃₁	—	4	2	no	—
373	Q₈×C₃₁	—	4	2	no	—
374	Q₈×C₃₁	—	8	2	no	—
375	D₄×C₃₁	—	2	4	no	—
376	D₄×C₃₁	—	2	2	no	—
377	D₄×C₃₁	—	2	2	no	—
378	D₄×C₃₁	—	2	2	no	—
379	D₄×C₃₁	—	2	1	no	—
380	D₄×C₃₁	—	2	1	no	—
381	D₄×C₃₁	—	2	2	no	—
382	D₄×C₃₁	—	2	2	no	—
383	D₄×C₃₁	—	2	1	no	—
384	D₄×C₃₁	—	2	1	no	—
385	D₄×C₃₁	—	4	4	no	—
386	D₄×C₃₁	—	4	4	no	—
387	D₄×C₃₁	—	4	4	no	—
388	D₄×C₃₁	—	4	2	no	—
389	D₄×C₃₁	—	4	2	no	—
390	D₄×C₃₁	—	4	2	no	—
391	D₄×C₃₁	—	4	2	no	—
392	D₄×C₃₁	—	8	4	no	—
393	D₄×C₃₁	—	8	2	no	—
394	D₄×C₃₁	—	8	4	no	—
395	C₂×C₄×C₃₁	—	2	2	no	—
396	C₂×C₄×C₃₁	—	2	2	no	—
397	C₂×C₄×C₃₁	—	2	2	no	—
398	C₂×C₄×C₃₁	—	2	2	no	—
399	C₂×C₄×C₃₁	—	2	2	no	—
400	C₂×C₄×C₃₁	—	2	2	no	—
401	C₂×C₄×C₃₁	—	2	1	no	—
402	C₂×C₄×C₃₁	—	2	1	no	—
403	C₂×C₄×C₃₁	—	4	4	no	—
404	C₂×C₄×C₃₁	—	4	2	no	—
405	C₂×C₄×C₃₁	—	4	4	no	—
406	C₂×C₄×C₃₁	—	4	4	no	—
407	C₂×C₄×C₃₁	—	4	4	no	—
408	C₂×C₄×C₃₁	—	4	2	no	—
409	C₂×C₄×C₃₁	—	4	2	no	—
410	C₂×C₄×C₃₁	—	4	2	no	—
411	C₂×C₄×C₃₁	—	4	2	no	—
412	C₂×C₄×C₃₁	—	8	4	no	—
413	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
414	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
415	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
416	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
417	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
418	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
419	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
420	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
421	C₈×C₃₁	—	2	2	no	—
422	C₈×C₃₁	—	2	2	no	—
423	C₈×C₃₁	—	2	2	no	—
424	C₈×C₃₁	—	2	2	no	—
425	#3510	C₂×C₃₁	1	1	yes	characteristic, complemented
426	#3500	C₂×C₃₁	1	1	yes	characteristic, complemented
427	#3225	C₂×C₃₁	1	4	yes	complemented
428	#3224	C₂×C₃₁	1	4	yes	complemented
429	#3212	C₂×C₃₁	1	4	yes	complemented
430	#3211	C₂×C₃₁	1	4	yes	complemented
431	#3153	C₂×C₃₁	1	1	yes	characteristic, complemented
432	C₂×#495	C₂×C₃₁	1	1	yes	characteristic, complemented
433	#3140	C₂×C₃₁	1	1	yes	characteristic, complemented
434	#3131	C₂×C₃₁	1	1	yes	characteristic, complemented
435	#3044	C₂×C₃₁	1	1	yes	characteristic, complemented
436	#3043	C₂×C₃₁	1	1	yes	characteristic, complemented
437	#3042	C₂×C₃₁	1	1	yes	characteristic, complemented
438	#3041	C₂×C₃₁	1	1	yes	characteristic, complemented
439	#3011	C₂×C₃₁	1	1	yes	characteristic, complemented
440	#3010	C₂×C₃₁	1	1	yes	characteristic, complemented
441	#3009	C₂×C₃₁	1	1	yes	characteristic, complemented
442	#3008	C₂×C₃₁	1	1	yes	characteristic, complemented
443	#2922	C₂×C₃₁	1	1	yes	characteristic, complemented
444	C₂²×C₃₁	—	2	2	no	—
445	C₂²×C₃₁	—	2	4	no	—
446	C₂²×C₃₁	—	2	2	no	—
447	C₂²×C₃₁	—	4	4	no	—
448	C₂²×C₃₁	—	4	2	no	—
449	C₂²×C₃₁	—	4	4	no	—
450	C₂²×C₃₁	—	4	2	no	—
451	C₂²×C₃₁	—	4	1	no	—
452	C₂²×C₃₁	—	4	2	no	—
453	C₂²×C₃₁	—	4	1	no	—
454	C₂²×C₃₁	—	4	2	no	—
455	C₂²×C₃₁	—	16	2	no	—
456	C₂²×C₃₁	—	16	2	no	—
457	C₂²×C₃₁	—	16	4	no	—
458	C₂²×C₃₁	#535	1	2	yes	—
459	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
460	C₄×C₃₁	—	2	2	no	—
461	C₄×C₃₁	—	2	4	no	—
462	C₄×C₃₁	—	2	2	no	—
463	C₄×C₃₁	—	2	2	no	—
464	C₄×C₃₁	—	4	2	no	—
465	C₄×C₃₁	—	4	2	no	—
466	C₄×C₃₁	—	4	1	no	—
467	C₄×C₃₁	—	4	2	no	—
468	C₄×C₃₁	—	4	1	no	—
469	C₄×C₃₁	—	4	2	no	—
470	C₄×C₃₁	#535	1	2	yes	—
471	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
472	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
473	C₂×2₋¹⁺⁴	C₂²×C₃₁	1	1	yes	characteristic
474	C₂×H(5,𝔽₂)	C₂²×C₃₁	1	1	yes	characteristic
475	C₂×#137	C₂²×C₃₁	1	1	yes	characteristic
476	C₂×#137	C₂²×C₃₁	1	1	yes	characteristic, complemented
477	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	1	yes	characteristic
478	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	1	yes	characteristic, complemented
479	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
480	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
481	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
482	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
483	#559	—	2	2	no	—
484	#556	—	2	2	no	—
485	#551	—	2	4	no	—
486	#548	—	2	2	no	—
487	#546	—	2	4	no	—
488	#546	—	2	2	no	—
489	#542	C₂²×C₃₁	1	4	yes	complemented
490	#536	C₂²×C₃₁	1	1	yes	characteristic
491	#536	C₂²×C₃₁	1	1	yes	characteristic
492	#536	C₂²×C₃₁	1	2	yes	complemented
493	#535	C₂²×C₃₁	1	1	yes	characteristic
494	#535	C₂²×C₃₁	1	1	yes	characteristic
495	#535	C₂²×C₃₁	1	2	yes	complemented
496	#533	C₂²×C₃₁	1	1	yes	characteristic, complemented
497	#532	C₂²×C₃₁	1	1	yes	characteristic
498	#532	C₂²×C₃₁	1	2	yes	complemented
499	#529	C₂²×C₃₁	1	4	yes	complemented
500	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
501	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic, complemented
502	C₂×#124	C₂²×C₃₁	1	2	yes	complemented
503	#519	C₂²×C₃₁	1	1	yes	characteristic, complemented
504	#518	C₂²×C₃₁	1	1	yes	characteristic, complemented
505	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
506	#497	C₂²×C₃₁	1	2	yes	complemented
507	#496	C₂²×C₃₁	1	2	yes	complemented
508	#495	C₂²×C₃₁	1	4	yes	complemented
509	#495	C₂²×C₃₁	1	4	yes	complemented
510	#495	C₂²×C₃₁	1	2	yes	complemented
511	#495	C₂²×C₃₁	1	2	yes	complemented
512	#489	C₂²×C₃₁	1	4	yes	complemented
513	#488	C₂²×C₃₁	1	2	yes	complemented
514	#488	C₂²×C₃₁	1	2	yes	complemented
515	#488	C₂²×C₃₁	1	4	yes	complemented
516	#487	C₂²×C₃₁	1	2	yes	complemented
517	#486	C₂²×C₃₁	1	2	yes	complemented
518	#464	C₂²×C₃₁	1	4	yes	complemented
519	#464	C₂²×C₃₁	1	4	yes	complemented
520	#463	C₂²×C₃₁	1	4	yes	complemented
521	#463	C₂²×C₃₁	1	4	yes	complemented
522	#462	C₂²×C₃₁	1	4	yes	complemented
523	#461	C₂²×C₃₁	1	4	yes	complemented
524	#460	C₂²×C₃₁	1	4	yes	complemented
525	#459	C₂²×C₃₁	1	4	yes	complemented
526	#452	C₂²×C₃₁	1	4	yes	complemented
527	#452	C₂²×C₃₁	1	4	yes	complemented
528	#451	C₂²×C₃₁	1	4	yes	complemented
529	#450	C₂²×C₃₁	1	4	yes	complemented
530	#449	C₂²×C₃₁	1	4	yes	—
531	#449	C₂²×C₃₁	1	4	yes	complemented
532	#448	C₂²×C₃₁	1	4	yes	complemented
533	#447	C₂²×C₃₁	1	4	yes	—
534	#433	C₂²×C₃₁	1	4	yes	complemented
535	#431	C₂²×C₃₁	1	4	yes	complemented
536	#424	C₂²×C₃₁	1	1	yes	characteristic, complemented
537	#424	C₂²×C₃₁	1	1	yes	characteristic, complemented
538	#419	C₂²×C₃₁	1	1	yes	characteristic, complemented
539	#419	C₂²×C₃₁	1	1	yes	characteristic, complemented
540	#418	C₂²×C₃₁	1	1	yes	characteristic
541	#418	C₂²×C₃₁	1	1	yes	characteristic
542	#416	C₂²×C₃₁	1	1	yes	characteristic
543	#416	C₂²×C₃₁	1	1	yes	characteristic
544	#416	C₂²×C₃₁	1	1	yes	characteristic, complemented
545	#416	C₂²×C₃₁	1	1	yes	characteristic, complemented
546	#408	C₂²×C₃₁	1	1	yes	characteristic
547	#408	C₂²×C₃₁	1	1	yes	characteristic
548	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
549	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
550	C₂×C₃₁	—	2	2	no	—
551	C₂×C₃₁	—	4	2	no	—
552	C₂×C₃₁	—	8	1	no	—
553	C₂×C₃₁	—	8	2	no	—
554	C₂×C₃₁	—	8	2	no	—
555	C₂×C₃₁	—	8	1	no	—
556	C₂×C₃₁	C₂×#535	1	1	yes	characteristic
557	C₂×C₃₁	#3584	1	2	yes	—
558	2₋¹⁺⁴	—	4	2	no	—
559	2₋¹⁺⁴	—	4	2	no	—
560	H(5,𝔽₂)	—	4	2	no	—
561	H(5,𝔽₂)	—	4	2	no	—
562	C₂×#41	—	2	1	no	—
563	C₂×#41	—	2	2	no	—
564	C₂×#41	—	2	1	no	—
565	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
566	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
567	C₂×#41	C₂³×C₃₁	1	2	yes	—
568	C₂×#41	C₂³×C₃₁	1	2	yes	—
569	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
570	C₂²×Q₈	—	2	1	no	—
571	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
572	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
573	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
574	C₂²×D₄	—	2	2	no	—
575	C₂²×D₄	—	2	1	no	—
576	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
577	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
578	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
579	#137	—	4	2	no	—
580	#137	—	4	2	no	—
581	AGL(1,ℤ/8)	—	4	2	no	—
582	AGL(1,ℤ/8)	—	4	2	no	—
583	C₂×Q₁₆	—	4	2	no	—
584	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
585	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
586	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
587	C₂×SD₁₆	—	4	2	no	—
588	C₂×SD₁₆	—	4	2	no	—
589	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
590	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
591	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
592	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
593	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
594	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
595	C₂×D₈	—	4	2	no	—
596	C₂×D₈	C₂³×C₃₁	1	4	yes	—
597	C₂×D₈	C₂³×C₃₁	1	2	yes	—
598	C₂×D₈	C₂³×C₃₁	1	2	yes	—
599	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
600	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
601	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
602	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
603	C₄⋊Q₈	—	2	1	no	—
604	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
605	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
606	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
607	#127	—	2	1	no	—
608	#127	C₂³×C₃₁	1	1	yes	characteristic
609	#127	C₂³×C₃₁	1	2	yes	—
610	#126	—	2	4	no	—
611	#126	—	2	4	no	—
612	#125	C₂³×C₃₁	1	2	yes	—
613	#124	—	2	1	no	—
614	#124	—	2	1	no	—
615	#124	—	2	4	no	—
616	#124	—	2	2	no	—
617	#124	—	2	4	no	—
618	#124	—	2	2	no	—
619	#124	C₂³×C₃₁	1	2	yes	—
620	#124	C₂³×C₃₁	1	2	yes	—
621	#124	C₂³×C₃₁	1	1	yes	characteristic
622	#124	C₂³×C₃₁	1	2	yes	—
623	#124	C₂³×C₃₁	1	2	yes	—
624	#124	C₂³×C₃₁	1	2	yes	—
625	#124	C₂³×C₃₁	1	2	yes	—
626	#124	C₂³×C₃₁	1	1	yes	characteristic
627	#123	—	2	2	no	—
628	#123	—	2	4	no	—
629	#123	—	2	2	no	—
630	#123	—	2	4	no	—
631	#123	—	2	4	no	—
632	C₂²⋊Q₈	—	2	2	no	—
633	C₂²⋊Q₈	—	2	2	no	—
634	C₂²⋊Q₈	—	2	4	no	—
635	C₂²⋊Q₈	—	2	4	no	—
636	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
637	#121	—	2	2	no	—
638	#121	—	2	4	no	—
639	#121	—	2	2	no	—
640	#121	—	2	4	no	—
641	#121	C₂³×C₃₁	1	4	yes	—
642	#120	—	2	2	no	—
643	#120	—	2	2	no	—
644	#120	—	2	4	no	—
645	C₄×Q₈	—	2	2	no	—
646	C₄×Q₈	—	2	2	no	—
647	C₄×Q₈	D₄×C₃₁	1	2	yes	—
648	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
649	C₄×D₄	—	2	4	no	—
650	C₄×D₄	—	2	2	no	—
651	C₄×D₄	—	2	2	no	—
652	C₄×D₄	—	2	4	no	—
653	C₄×D₄	D₄×C₃₁	1	4	yes	—
654	C₄×D₄	D₄×C₃₁	1	2	yes	—
655	C₄×D₄	C₂³×C₃₁	1	4	yes	—
656	#117	—	2	2	no	—
657	#117	—	2	4	no	—
658	#117	C₂³×C₃₁	1	1	yes	characteristic
659	#117	C₂³×C₃₁	1	1	yes	characteristic
660	#117	C₂³×C₃₁	1	1	yes	characteristic
661	#117	C₂³×C₃₁	1	2	yes	—
662	C₂×C₂²⋊C₄	—	2	2	no	—
663	C₂×C₂²⋊C₄	—	2	4	no	—
664	C₂×C₂²⋊C₄	—	2	2	no	—
665	C₂×C₂²⋊C₄	—	2	2	no	—
666	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
667	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
668	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
669	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
670	#103	C₂³×C₃₁	1	2	yes	—
671	#103	C₂³×C₃₁	1	2	yes	—
672	#103	C₂³×C₃₁	1	2	yes	—
673	#103	C₂³×C₃₁	1	2	yes	—
674	#103	C₂³×C₃₁	1	4	yes	—
675	#103	C₂³×C₃₁	1	4	yes	—
676	#102	C₂³×C₃₁	1	2	yes	—
677	#102	C₂³×C₃₁	1	2	yes	—
678	#102	C₂³×C₃₁	1	2	yes	—
679	#102	C₂³×C₃₁	1	2	yes	—
680	#102	C₂³×C₃₁	1	4	yes	—
681	#102	C₂³×C₃₁	1	4	yes	—
682	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
683	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
684	#97	C₂³×C₃₁	1	2	yes	—
685	C₄×C₈	C₂³×C₃₁	1	2	yes	—
686	C₄×C₈	C₂³×C₃₁	1	2	yes	—
687	C₄×C₈	C₂³×C₃₁	1	2	yes	—
688	C₃₁	#60819	1	1	yes	characteristic, direct product factor, Sylow
689	C₂⁴	—	2	1	no	—
690	C₂⁴	—	2	2	no	—
691	C₂⁴	—	2	1	no	—
692	#41	—	2	2	no	—
693	#41	—	2	2	no	—
694	#41	—	2	2	no	—
695	#41	—	2	2	no	—
696	#41	—	2	2	no	—
697	#41	—	2	2	no	—
698	#41	—	4	2	no	—
699	#41	—	4	2	no	—
700	#41	—	4	2	no	—
701	#41	—	4	2	no	—
702	#41	—	4	8	no	—
703	#41	—	4	4	no	—
704	#41	—	4	4	no	—
705	C₂×Q₈	—	2	1	no	—
706	C₂×Q₈	—	2	2	no	—
707	C₂×Q₈	—	2	1	no	—
708	C₂×Q₈	—	2	2	no	—
709	C₂×Q₈	—	2	2	no	—
710	C₂×Q₈	—	2	2	no	—
711	C₂×Q₈	—	2	2	no	—
712	C₂×Q₈	—	2	2	no	—
713	C₂×Q₈	—	4	2	no	—
714	C₂×Q₈	—	4	2	no	—
715	C₂×Q₈	—	4	4	no	—
716	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
717	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
718	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
719	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
720	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
721	C₂×D₄	—	2	2	no	—
722	C₂×D₄	—	2	1	no	—
723	C₂×D₄	—	2	2	no	—
724	C₂×D₄	—	2	1	no	—
725	C₂×D₄	—	2	4	no	—
726	C₂×D₄	—	2	1	no	—
727	C₂×D₄	—	2	2	no	—
728	C₂×D₄	—	2	1	no	—
729	C₂×D₄	—	2	4	no	—
730	C₂×D₄	—	2	2	no	—
731	C₂×D₄	—	2	2	no	—
732	C₂×D₄	—	2	2	no	—
733	C₂×D₄	—	2	2	no	—
734	C₂×D₄	—	2	2	no	—
735	C₂×D₄	—	4	2	no	—
736	C₂×D₄	—	4	2	no	—
737	C₂×D₄	—	4	8	no	—
738	C₂×D₄	—	4	4	no	—
739	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
740	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
741	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
742	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
743	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
744	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
745	C₂²×C₄	—	2	1	no	—
746	C₂²×C₄	—	2	2	no	—
747	C₂²×C₄	—	2	1	no	—
748	C₂²×C₄	—	2	2	no	—
749	C₂²×C₄	—	2	1	no	—
750	C₂²×C₄	—	2	2	no	—
751	C₂²×C₄	—	2	1	no	—
752	C₂²×C₄	—	2	4	no	—
753	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
754	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
755	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
756	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
757	Q₁₆	—	4	4	no	—
758	Q₁₆	—	4	2	no	—
759	Q₁₆	—	4	2	no	—
760	SD₁₆	—	4	4	no	—
761	SD₁₆	—	4	4	no	—
762	SD₁₆	—	4	2	no	—
763	SD₁₆	—	4	2	no	—
764	SD₁₆	—	4	2	no	—
765	SD₁₆	—	4	2	no	—
766	D₈	—	4	4	no	—
767	D₈	—	4	2	no	—
768	D₈	—	4	2	no	—
769	M₁₆	—	2	2	no	—
770	M₁₆	—	2	2	no	—
771	C₂×C₈	—	2	2	no	—
772	C₂×C₈	—	2	2	no	—
773	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
774	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
775	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
776	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
777	C₄⋊C₄	—	2	2	no	—
778	C₄⋊C₄	—	2	4	no	—
779	C₄⋊C₄	—	2	4	no	—
780	C₄⋊C₄	—	2	2	no	—
781	C₄⋊C₄	—	2	4	no	—
782	C₄⋊C₄	—	2	2	no	—
783	C₄⋊C₄	—	2	2	no	—
784	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
785	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
786	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
787	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
788	C₂²⋊C₄	—	2	1	no	—
789	C₂²⋊C₄	—	2	1	no	—
790	C₂²⋊C₄	—	2	4	no	—
791	C₂²⋊C₄	—	2	4	no	—
792	C₂²⋊C₄	—	2	2	no	—
793	C₂²⋊C₄	—	2	4	no	—
794	C₂²⋊C₄	—	2	4	no	—
795	C₂²⋊C₄	—	2	4	no	—
796	C₂²⋊C₄	—	2	4	no	—
797	C₂²⋊C₄	—	2	4	no	—
798	C₂²⋊C₄	—	2	4	no	—
799	C₂²⋊C₄	—	2	4	no	—
800	C₂²⋊C₄	—	2	2	no	—
801	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
802	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
803	C₄²	—	2	4	no	—
804	C₄²	—	2	1	no	—
805	C₄²	—	2	1	no	—
806	C₄²	C₂×D₄×C₃₁	1	4	yes	—
807	C₄²	C₂×D₄×C₃₁	1	2	yes	—
808	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
809	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
810	C₄²	C₂⁴×C₃₁	1	2	yes	—
811	C₂³	—	2	2	no	—
812	C₂³	—	2	2	no	—
813	C₂³	—	2	1	no	—
814	C₂³	—	2	1	no	—
815	C₂³	—	4	4	no	—
816	C₂³	—	4	2	no	—
817	C₂³	—	4	2	no	—
818	C₂³	—	8	2	no	—
819	C₂³	—	8	2	no	—
820	C₂³	—	8	4	no	—
821	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
822	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
823	Q₈	—	2	2	no	—
824	Q₈	—	2	2	no	—
825	Q₈	—	2	2	no	—
826	Q₈	—	2	1	no	—
827	Q₈	—	2	1	no	—
828	Q₈	—	2	2	no	—
829	Q₈	—	2	2	no	—
830	Q₈	—	2	1	no	—
831	Q₈	—	2	1	no	—
832	Q₈	—	4	4	no	—
833	Q₈	—	4	2	no	—
834	Q₈	—	4	2	no	—
835	Q₈	—	8	2	no	—
836	D₄	—	2	4	no	—
837	D₄	—	2	2	no	—
838	D₄	—	2	2	no	—
839	D₄	—	2	2	no	—
840	D₄	—	2	1	no	—
841	D₄	—	2	1	no	—
842	D₄	—	2	2	no	—
843	D₄	—	2	2	no	—
844	D₄	—	2	1	no	—
845	D₄	—	2	1	no	—
846	D₄	—	4	4	no	—
847	D₄	—	4	4	no	—
848	D₄	—	4	4	no	—
849	D₄	—	4	2	no	—
850	D₄	—	4	2	no	—
851	D₄	—	4	2	no	—
852	D₄	—	4	2	no	—
853	D₄	—	8	4	no	—
854	D₄	—	8	2	no	—
855	D₄	—	8	4	no	—
856	C₂×C₄	—	2	2	no	—
857	C₂×C₄	—	2	2	no	—
858	C₂×C₄	—	2	2	no	—
859	C₂×C₄	—	2	2	no	—
860	C₂×C₄	—	2	2	no	—
861	C₂×C₄	—	2	2	no	—
862	C₂×C₄	—	2	1	no	—
863	C₂×C₄	—	2	1	no	—
864	C₂×C₄	—	4	4	no	—
865	C₂×C₄	—	4	2	no	—
866	C₂×C₄	—	4	4	no	—
867	C₂×C₄	—	4	4	no	—
868	C₂×C₄	—	4	4	no	—
869	C₂×C₄	—	4	2	no	—
870	C₂×C₄	—	4	2	no	—
871	C₂×C₄	—	4	2	no	—
872	C₂×C₄	—	4	2	no	—
873	C₂×C₄	—	8	4	no	—
874	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
875	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
876	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
877	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
878	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
879	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
880	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
881	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
882	C₈	—	2	2	no	—
883	C₈	—	2	2	no	—
884	C₈	—	2	2	no	—
885	C₈	—	2	2	no	—
886	C₂²	—	2	2	no	—
887	C₂²	—	2	2	no	—
888	C₂²	—	2	4	no	—
889	C₂²	—	4	1	no	—
890	C₂²	—	4	2	no	—
891	C₂²	—	4	1	no	—
892	C₂²	—	4	2	no	—
893	C₂²	—	4	4	no	—
894	C₂²	—	4	2	no	—
895	C₂²	—	4	4	no	—
896	C₂²	—	4	2	no	—
897	C₂²	—	16	2	no	—
898	C₂²	—	16	2	no	—
899	C₂²	—	16	4	no	—
900	C₂²	C₃₁×#535	1	2	yes	—
901	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
902	C₄	—	2	2	no	—
903	C₄	—	2	2	no	—
904	C₄	—	2	4	no	—
905	C₄	—	2	2	no	—
906	C₄	—	4	1	no	—
907	C₄	—	4	2	no	—
908	C₄	—	4	1	no	—
909	C₄	—	4	2	no	—
910	C₄	—	4	2	no	—
911	C₄	—	4	2	no	—
912	C₄	C₃₁×#535	1	2	yes	—
913	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
914	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
915	C₂	—	2	2	no	—
916	C₂	—	4	2	no	—
917	C₂	—	8	2	no	—
918	C₂	—	8	1	no	—
919	C₂	—	8	2	no	—
920	C₂	—	8	1	no	—
921	C₂	C₂×C₃₁×#535	1	1	yes	characteristic
922	C₂	C₃₁×#3584	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃₁×#3510	C₂	semidirect
2. C₃₁×#3500	C₂	semidirect
3. C₃₁×#3225	C₂	semidirect
4. C₃₁×#3224	C₂	semidirect
5. C₃₁×#3212	C₂	semidirect
6. C₃₁×#3211	C₂	semidirect
7. C₃₁×#3153	C₂	semidirect
8. C₂×C₃₁×#495	C₂	semidirect
9. C₃₁×#3140	C₂	semidirect
10. C₃₁×#3131	C₂	semidirect
11. C₃₁×#3044	C₂	semidirect
12. C₃₁×#3043	C₂	semidirect
13. C₃₁×#3042	C₂	semidirect
14. C₃₁×#3041	C₂	semidirect
15. C₃₁×#3011	C₂	semidirect
16. C₃₁×#3010	C₂	semidirect
17. C₃₁×#3009	C₂	semidirect
18. C₃₁×#3008	C₂	semidirect
19. C₃₁×#2922	C₂	semidirect
23. C₂×C₃₁×#137	C₂²	semidirect
25. C₂×C₃₁×AGL(1,ℤ/8)	C₂²	semidirect
26. C₂²×Q₁₆×C₃₁	C₂²	semidirect
28. C₂²×SD₁₆×C₃₁	C₂²	semidirect
36. C₃₁×#542	C₂²	semidirect
39. C₃₁×#536	C₂²	semidirect
42. C₃₁×#535	C₂²	semidirect
43. C₃₁×#533	C₂²	semidirect
45. C₃₁×#532	C₂²	semidirect
46. C₃₁×#529	C₂²	semidirect
48. C₂×C₃₁×#124	C₂²	semidirect
49. C₂×C₃₁×#124	C₂²	semidirect
50. C₃₁×#519	C₂²	semidirect
51. C₃₁×#518	C₂²	semidirect
53. C₃₁×#497	C₂²	semidirect
54. C₃₁×#496	C₂²	semidirect
55. C₃₁×#495	C₂²	semidirect
56. C₃₁×#495	C₂²	semidirect
57. C₃₁×#495	C₂²	semidirect
58. C₃₁×#495	C₂²	semidirect
59. C₃₁×#489	C₂²	semidirect
60. C₃₁×#488	C₂²	semidirect
61. C₃₁×#488	C₂²	semidirect
62. C₃₁×#488	C₂²	semidirect
63. C₃₁×#487	C₂²	semidirect
64. C₃₁×#486	C₂²	semidirect
65. C₃₁×#464	C₂²	semidirect
66. C₃₁×#464	C₂²	semidirect
67. C₃₁×#463	C₂²	semidirect
68. C₃₁×#463	C₂²	semidirect
69. C₃₁×#462	C₂²	semidirect
70. C₃₁×#461	C₂²	semidirect
71. C₃₁×#460	C₂²	semidirect
72. C₃₁×#459	C₂²	semidirect
73. C₃₁×#452	C₂²	semidirect
74. C₃₁×#452	C₂²	semidirect
75. C₃₁×#451	C₂²	semidirect
76. C₃₁×#450	C₂²	semidirect
78. C₃₁×#449	C₂²	semidirect
79. C₃₁×#448	C₂²	semidirect
81. C₃₁×#433	C₂²	semidirect
82. C₃₁×#431	C₂²	semidirect
83. C₃₁×#424	C₂²	semidirect
84. C₃₁×#424	C₂²	semidirect
85. C₃₁×#419	C₂²	semidirect
86. C₃₁×#419	C₂²	semidirect
91. C₃₁×#416	C₂²	semidirect
92. C₃₁×#416	C₂²	semidirect
95. C₄×M₁₆×C₃₁	C₂²	semidirect
96. C₂×C₄×C₈×C₃₁	C₂²	semidirect
349. #60819	C₃₁	direct (N × H)
425. #3510	C₂×C₃₁	semidirect
426. #3500	C₂×C₃₁	semidirect
427. #3225	C₂×C₃₁	semidirect
428. #3224	C₂×C₃₁	semidirect
429. #3212	C₂×C₃₁	semidirect
430. #3211	C₂×C₃₁	semidirect
431. #3153	C₂×C₃₁	semidirect
432. C₂×#495	C₂×C₃₁	semidirect
433. #3140	C₂×C₃₁	semidirect
434. #3131	C₂×C₃₁	semidirect
435. #3044	C₂×C₃₁	semidirect
436. #3043	C₂×C₃₁	semidirect
437. #3042	C₂×C₃₁	semidirect
438. #3041	C₂×C₃₁	semidirect
439. #3011	C₂×C₃₁	semidirect
440. #3010	C₂×C₃₁	semidirect
441. #3009	C₂×C₃₁	semidirect
442. #3008	C₂×C₃₁	semidirect
443. #2922	C₂×C₃₁	semidirect
476. C₂×#137	C₂²×C₃₁	semidirect
478. C₂×AGL(1,ℤ/8)	C₂²×C₃₁	semidirect
479. C₂²×Q₁₆	C₂²×C₃₁	semidirect
481. C₂²×SD₁₆	C₂²×C₃₁	semidirect
489. #542	C₂²×C₃₁	semidirect
492. #536	C₂²×C₃₁	semidirect
495. #535	C₂²×C₃₁	semidirect
496. #533	C₂²×C₃₁	semidirect
498. #532	C₂²×C₃₁	semidirect
499. #529	C₂²×C₃₁	semidirect
501. C₂×#124	C₂²×C₃₁	semidirect
502. C₂×#124	C₂²×C₃₁	semidirect
503. #519	C₂²×C₃₁	semidirect
504. #518	C₂²×C₃₁	semidirect
506. #497	C₂²×C₃₁	semidirect
507. #496	C₂²×C₃₁	semidirect
508. #495	C₂²×C₃₁	semidirect
509. #495	C₂²×C₃₁	semidirect
510. #495	C₂²×C₃₁	semidirect
511. #495	C₂²×C₃₁	semidirect
512. #489	C₂²×C₃₁	semidirect
513. #488	C₂²×C₃₁	semidirect
514. #488	C₂²×C₃₁	semidirect
515. #488	C₂²×C₃₁	semidirect
516. #487	C₂²×C₃₁	semidirect
517. #486	C₂²×C₃₁	semidirect
518. #464	C₂²×C₃₁	semidirect
519. #464	C₂²×C₃₁	semidirect
520. #463	C₂²×C₃₁	semidirect
521. #463	C₂²×C₃₁	semidirect
522. #462	C₂²×C₃₁	semidirect
523. #461	C₂²×C₃₁	semidirect
524. #460	C₂²×C₃₁	semidirect
525. #459	C₂²×C₃₁	semidirect
526. #452	C₂²×C₃₁	semidirect
527. #452	C₂²×C₃₁	semidirect
528. #451	C₂²×C₃₁	semidirect
529. #450	C₂²×C₃₁	semidirect
531. #449	C₂²×C₃₁	semidirect
532. #448	C₂²×C₃₁	semidirect
534. #433	C₂²×C₃₁	semidirect
535. #431	C₂²×C₃₁	semidirect
536. #424	C₂²×C₃₁	semidirect
537. #424	C₂²×C₃₁	semidirect
538. #419	C₂²×C₃₁	semidirect
539. #419	C₂²×C₃₁	semidirect
544. #416	C₂²×C₃₁	semidirect
545. #416	C₂²×C₃₁	semidirect
548. C₄×M₁₆	C₂²×C₃₁	semidirect
549. C₂×C₄×C₈	C₂²×C₃₁	semidirect
688. C₃₁	#60819	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#3510	2
C₃₁×#3500	2
C₃₁×#3225	2
C₃₁×#3224	2
C₃₁×#3212	2
C₃₁×#3211	2
C₃₁×#3153	2
C₂×C₃₁×#495	2
C₃₁×#3140	2
C₃₁×#3131	2
C₃₁×#3044	2
C₃₁×#3043	2
C₃₁×#3042	2
C₃₁×#3041	2
C₃₁×#3011	2
C₃₁×#3010	2
C₃₁×#3009	2
C₃₁×#3008	2
C₃₁×#2922	2
#60819	31

Subgroups — C31×#60819 #21517578

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60819 #21517578