Subgroups — C₃₁×#61057 #21517816

Order 7936.

2696 conjugacy classes of subgroups.

The subgroup lattice has 926 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#578	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃₁×#135	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3489	C₂	1	2	yes	maximal, complemented
4	D₄×Q₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3285	C₂	1	2	yes	maximal, complemented
6	C₃₁×#3283	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3280	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3269	C₂	1	4	yes	maximal, complemented
9	C₃₁×#3266	C₂	1	4	yes	maximal, complemented
10	C₃₁×#3156	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#3061	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#467	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#3046	C₂	1	2	yes	maximal, complemented
14	C₃₁×#3013	C₂	1	2	yes	maximal, complemented
15	C₃₁×#3007	C₂	1	2	yes	maximal, complemented
16	C₃₁×#2951	C₂	1	1	yes	maximal, characteristic, complemented
17	C₃₁×#2929	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×C₃₁×2₋¹⁺⁴	C₂²	1	2	yes	complemented
19	C₂²×#41×C₃₁	C₂²	1	2	yes	—
20	C₃₁×#578	—	2	8	no	—
21	C₂×C₃₁×#137	C₂²	1	2	yes	complemented
22	C₂×C₃₁×#137	C₂²	1	4	yes	complemented
23	C₂×C₃₁×#137	C₂²	1	2	yes	complemented
24	C₂×C₃₁×#135	—	2	8	no	—
25	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#135	C₂²	1	2	yes	—
27	C₂×C₃₁×#135	C₂²	1	2	yes	—
28	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#135	C₂²	1	2	yes	complemented
32	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
33	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
34	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	complemented
35	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
36	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
37	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic, complemented
38	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
39	D₄×Q₈×C₃₁	C₂²	1	4	yes	—
40	C₃₁×#548	C₂²	1	4	yes	complemented
41	C₃₁×#547	—	2	4	no	—
42	C₃₁×#547	C₂²	1	4	yes	complemented
43	C₃₁×#546	—	2	4	no	—
44	C₃₁×#546	C₂²	1	4	yes	—
45	C₃₁×#537	—	2	2	no	—
46	C₃₁×#537	C₂²	1	2	yes	complemented
47	C₃₁×#536	—	2	2	no	—
48	C₃₁×#536	C₂²	1	2	yes	complemented
49	C₃₁×#525	—	2	2	no	—
50	C₃₁×#525	C₂²	1	1	yes	characteristic
51	C₂×C₃₁×C₂²⋊Q₈	—	2	2	no	—
52	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
53	C₃₁×#520	C₂²	1	2	yes	complemented
54	C₃₁×#497	C₂²	1	4	yes	complemented
55	C₃₁×#495	C₂²	1	2	yes	complemented
56	C₃₁×#494	C₂²	1	2	yes	complemented
57	C₃₁×#468	C₂²	1	4	yes	complemented
58	C₃₁×#467	C₂²	1	2	yes	—
59	C₃₁×#467	C₂²	1	2	yes	—
60	C₃₁×#467	C₂²	1	2	yes	complemented
61	C₃₁×#467	C₂²	1	2	yes	complemented
62	C₃₁×#467	C₂²	1	2	yes	complemented
63	C₃₁×#466	C₂²	1	2	yes	complemented
64	C₃₁×#464	C₂²	1	4	yes	complemented
65	C₃₁×#463	C₂²	1	4	yes	complemented
66	C₃₁×#462	C₂²	1	4	yes	complemented
67	C₃₁×#461	C₂²	1	4	yes	complemented
68	C₃₁×#452	C₂²	1	4	yes	complemented
69	C₃₁×#452	C₂²	1	4	yes	complemented
70	C₃₁×#452	C₂²	1	4	yes	complemented
71	C₃₁×#451	C₂²	1	4	yes	—
72	C₃₁×#451	C₂²	1	4	yes	complemented
73	C₃₁×#450	C₂²	1	4	yes	—
74	C₃₁×#449	C₂²	1	4	yes	complemented
75	C₃₁×#448	C₂²	1	4	yes	complemented
76	C₃₁×#440	C₂²	1	4	yes	complemented
77	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	complemented
78	C₄×D₈×C₃₁	C₂²	1	2	yes	complemented
79	C₃₁×#435	C₂²	1	2	yes	—
80	C₃₁×#435	C₂²	1	2	yes	complemented
81	C₈×D₄×C₃₁	C₂²	1	2	yes	—
82	C₈×D₄×C₃₁	C₂²	1	2	yes	complemented
83	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic, complemented
84	C₃₁×#424	C₂²	1	1	yes	characteristic, complemented
85	C₃₁×#417	C₂²	1	2	yes	complemented
86	C₃₁×#417	C₂²	1	2	yes	complemented
87	C₂×C₃₁×#103	C₂²	1	2	yes	complemented
88	C₃₁×#408	C₂²	1	1	yes	characteristic, complemented
89	C₃₁×#407	C₂²	1	1	yes	characteristic
90	C₃₁×#405	C₂²	1	1	yes	characteristic, complemented
91	C₃₁×2₋¹⁺⁴	—	2	8	no	—
92	C₃₁×2₋¹⁺⁴	—	4	4	no	—
93	C₂×#41×C₃₁	—	2	2	no	—
94	C₂×#41×C₃₁	—	2	4	no	—
95	C₂×#41×C₃₁	—	2	2	no	—
96	C₂×#41×C₃₁	—	2	2	no	—
97	C₂×#41×C₃₁	—	2	8	no	—
98	C₂×#41×C₃₁	—	4	4	no	—
99	C₂×#41×C₃₁	C₂³	1	2	yes	—
100	C₂×#41×C₃₁	C₂³	1	4	yes	—
101	C₂×#41×C₃₁	C₂³	1	2	yes	—
102	C₂×#41×C₃₁	C₂³	1	2	yes	—
103	C₂×#41×C₃₁	C₂³	1	2	yes	—
104	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
105	C₂²×Q₈×C₃₁	—	2	2	no	—
106	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
107	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
108	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
109	C₂²×D₄×C₃₁	—	2	2	no	—
110	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
111	C₂³×C₄×C₃₁	—	2	2	no	—
112	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
113	C₃₁×#137	—	2	8	no	—
114	C₃₁×#137	—	2	16	no	—
115	C₃₁×#137	—	4	4	no	—
116	C₃₁×#135	—	2	8	no	—
117	C₃₁×#135	—	2	4	no	—
118	C₃₁×#135	—	2	8	no	—
119	C₃₁×#135	—	2	4	no	—
120	C₃₁×#135	—	2	8	no	—
121	C₃₁×#135	—	2	4	no	—
122	C₃₁×#135	D₄	1	8	yes	complemented
123	C₂×Q₁₆×C₃₁	—	2	4	no	—
124	C₂×Q₁₆×C₃₁	—	2	8	no	—
125	C₂×Q₁₆×C₃₁	—	2	4	no	—
126	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
127	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
128	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
129	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
130	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
131	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
132	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
133	C₂×SD₁₆×C₃₁	—	2	8	no	—
134	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
135	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
136	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
137	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
138	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
139	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
140	C₂×D₈×C₃₁	—	2	4	no	—
141	C₂×D₈×C₃₁	C₂³	1	2	yes	—
142	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
143	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
144	C₃₁×#131	—	2	4	no	—
145	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
146	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
147	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
148	C₂²×C₈×C₃₁	—	2	4	no	—
149	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
150	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
151	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
152	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
153	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
154	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
155	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
156	C₃₁×C₄⋊Q₈	—	2	4	no	—
157	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
158	C₃₁×#124	—	2	4	no	—
159	C₃₁×#124	C₂³	1	4	yes	—
160	C₃₁×#123	—	2	4	no	—
161	C₃₁×#123	—	2	4	no	—
162	C₃₁×#123	—	2	4	no	—
163	C₃₁×#123	—	2	4	no	—
164	C₃₁×#123	D₄	1	2	yes	—
165	C₃₁×C₂²⋊Q₈	—	2	2	no	—
166	C₃₁×C₂²⋊Q₈	—	2	2	no	—
167	C₃₁×C₂²⋊Q₈	—	2	2	no	—
168	C₃₁×C₂²⋊Q₈	—	2	2	no	—
169	C₃₁×C₂²⋊Q₈	—	2	4	no	—
170	C₃₁×C₂²⋊Q₈	—	2	2	no	—
171	C₃₁×C₂²⋊Q₈	—	2	4	no	—
172	C₃₁×C₂²⋊Q₈	—	2	4	no	—
173	C₃₁×C₂²⋊Q₈	D₄	1	1	yes	characteristic
174	C₃₁×C₂²⋊Q₈	D₄	1	1	yes	characteristic
175	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
176	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
177	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
178	C₃₁×#121	—	2	2	no	—
179	C₃₁×#121	—	2	4	no	—
180	C₃₁×#121	—	2	4	no	—
181	C₃₁×#121	—	2	4	no	—
182	C₃₁×#121	—	2	4	no	—
183	C₃₁×#121	D₄	1	1	yes	characteristic
184	C₃₁×#121	D₄	1	1	yes	characteristic
185	C₃₁×#121	C₂³	1	4	yes	—
186	C₃₁×#120	—	2	4	no	—
187	C₃₁×#120	D₄	1	2	yes	—
188	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
189	C₄×D₄×C₃₁	—	2	4	no	—
190	C₄×D₄×C₃₁	—	2	4	no	—
191	C₄×D₄×C₃₁	—	2	4	no	—
192	C₄×D₄×C₃₁	C₂³	1	4	yes	—
193	C₄×D₄×C₃₁	C₂³	1	2	yes	—
194	C₄×D₄×C₃₁	C₂³	1	2	yes	—
195	C₃₁×#117	—	2	2	no	—
196	C₃₁×#117	C₂³	1	1	yes	characteristic
197	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
198	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
199	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
200	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
201	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
202	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
203	C₃₁×#107	C₂³	1	2	yes	—
204	C₃₁×#107	C₂³	1	2	yes	—
205	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
206	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
207	C₃₁×#103	C₂³	1	4	yes	—
208	C₃₁×#103	C₂³	1	4	yes	—
209	C₃₁×#103	C₂³	1	4	yes	—
210	C₃₁×#103	C₂³	1	4	yes	—
211	C₃₁×#102	C₂³	1	4	yes	—
212	C₃₁×#102	C₂³	1	4	yes	—
213	C₃₁×C₂²⋊C₈	C₂³	1	2	yes	—
214	C₃₁×C₂²⋊C₈	C₂³	1	2	yes	—
215	C₃₁×C₂²⋊C₈	C₂³	1	2	yes	—
216	C₃₁×C₂²⋊C₈	C₂³	1	2	yes	—
217	C₃₁×#97	C₂³	1	2	yes	—
218	C₄×C₈×C₃₁	C₂³	1	2	yes	—
219	C₂⁴×C₃₁	—	2	2	no	—
220	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
221	#41×C₃₁	—	2	2	no	—
222	#41×C₃₁	—	2	4	no	—
223	#41×C₃₁	—	2	2	no	—
224	#41×C₃₁	—	2	4	no	—
225	#41×C₃₁	—	2	4	no	—
226	#41×C₃₁	—	2	8	no	—
227	#41×C₃₁	—	2	4	no	—
228	#41×C₃₁	—	2	4	no	—
229	#41×C₃₁	—	4	8	no	—
230	#41×C₃₁	—	4	2	no	—
231	#41×C₃₁	—	4	2	no	—
232	#41×C₃₁	—	4	4	no	—
233	#41×C₃₁	—	4	8	no	—
234	#41×C₃₁	—	4	8	no	—
235	#41×C₃₁	—	4	16	no	—
236	#41×C₃₁	C₂×D₄	1	8	yes	—
237	C₂×Q₈×C₃₁	—	2	2	no	—
238	C₂×Q₈×C₃₁	—	2	4	no	—
239	C₂×Q₈×C₃₁	—	2	2	no	—
240	C₂×Q₈×C₃₁	—	2	4	no	—
241	C₂×Q₈×C₃₁	—	2	8	no	—
242	C₂×Q₈×C₃₁	—	2	4	no	—
243	C₂×Q₈×C₃₁	—	2	4	no	—
244	C₂×Q₈×C₃₁	—	4	2	no	—
245	C₂×Q₈×C₃₁	—	4	4	no	—
246	C₂×Q₈×C₃₁	—	4	8	no	—
247	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
248	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
249	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
250	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
251	C₂×D₄×C₃₁	—	2	2	no	—
252	C₂×D₄×C₃₁	—	2	4	no	—
253	C₂×D₄×C₃₁	—	2	4	no	—
254	C₂×D₄×C₃₁	—	2	2	no	—
255	C₂×D₄×C₃₁	—	2	2	no	—
256	C₂×D₄×C₃₁	—	2	2	no	—
257	C₂×D₄×C₃₁	—	2	4	no	—
258	C₂×D₄×C₃₁	—	2	2	no	—
259	C₂×D₄×C₃₁	—	4	2	no	—
260	C₂×D₄×C₃₁	—	4	8	no	—
261	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
262	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
263	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
264	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
265	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
266	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
267	C₂²×C₄×C₃₁	—	2	2	no	—
268	C₂²×C₄×C₃₁	—	2	2	no	—
269	C₂²×C₄×C₃₁	—	2	2	no	—
270	C₂²×C₄×C₃₁	—	2	2	no	—
271	C₂²×C₄×C₃₁	—	2	4	no	—
272	C₂²×C₄×C₃₁	—	2	2	no	—
273	C₂²×C₄×C₃₁	—	2	2	no	—
274	C₂²×C₄×C₃₁	—	2	4	no	—
275	C₂²×C₄×C₃₁	—	2	2	no	—
276	C₂²×C₄×C₃₁	—	2	2	no	—
277	C₂²×C₄×C₃₁	—	2	2	no	—
278	C₂²×C₄×C₃₁	—	2	4	no	—
279	C₂²×C₄×C₃₁	—	4	8	no	—
280	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
281	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
282	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
283	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
284	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
285	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
286	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
287	Q₁₆×C₃₁	—	2	4	no	—
288	Q₁₆×C₃₁	—	2	8	no	—
289	Q₁₆×C₃₁	—	2	4	no	—
290	Q₁₆×C₃₁	—	2	2	no	—
291	Q₁₆×C₃₁	—	2	4	no	—
292	Q₁₆×C₃₁	—	4	4	no	—
293	Q₁₆×C₃₁	C₂×D₄	1	4	yes	—
294	SD₁₆×C₃₁	—	2	4	no	—
295	SD₁₆×C₃₁	—	2	4	no	—
296	SD₁₆×C₃₁	—	2	8	no	—
297	SD₁₆×C₃₁	—	2	4	no	—
298	SD₁₆×C₃₁	—	4	4	no	—
299	SD₁₆×C₃₁	C₂×D₄	1	8	yes	—
300	D₈×C₃₁	—	2	4	no	—
301	D₈×C₃₁	—	2	2	no	—
302	D₈×C₃₁	C₂×D₄	1	4	yes	—
303	M₁₆×C₃₁	—	2	2	no	—
304	M₁₆×C₃₁	—	2	2	no	—
305	M₁₆×C₃₁	—	2	4	no	—
306	C₂×C₈×C₃₁	—	2	2	no	—
307	C₂×C₈×C₃₁	—	2	2	no	—
308	C₂×C₈×C₃₁	—	2	2	no	—
309	C₂×C₈×C₃₁	—	2	2	no	—
310	C₂×C₈×C₃₁	—	2	2	no	—
311	C₂×C₈×C₃₁	—	2	4	no	—
312	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
313	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
314	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
315	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
316	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
317	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
318	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
319	C₄⋊C₄×C₃₁	—	2	4	no	—
320	C₄⋊C₄×C₃₁	—	2	2	no	—
321	C₄⋊C₄×C₃₁	—	2	4	no	—
322	C₄⋊C₄×C₃₁	—	2	4	no	—
323	C₄⋊C₄×C₃₁	—	2	4	no	—
324	C₄⋊C₄×C₃₁	—	2	4	no	—
325	C₄⋊C₄×C₃₁	—	2	4	no	—
326	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
327	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
328	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
329	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
330	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
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	C₂×D₄	1	2	yes	—
339	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
340	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
341	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
342	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
343	C₄²×C₃₁	—	2	4	no	—
344	C₄²×C₃₁	C₂⁴	1	2	yes	—
345	#61057	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
346	C₂³×C₃₁	—	2	2	no	—
347	C₂³×C₃₁	—	2	2	no	—
348	C₂³×C₃₁	—	2	2	no	—
349	C₂³×C₃₁	—	2	2	no	—
350	C₂³×C₃₁	—	4	1	no	—
351	C₂³×C₃₁	—	4	4	no	—
352	C₂³×C₃₁	—	4	4	no	—
353	C₂³×C₃₁	—	8	2	no	—
354	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
355	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
356	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
357	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
358	Q₈×C₃₁	—	2	2	no	—
359	Q₈×C₃₁	—	2	2	no	—
360	Q₈×C₃₁	—	2	2	no	—
361	Q₈×C₃₁	—	2	4	no	—
362	Q₈×C₃₁	—	2	4	no	—
363	Q₈×C₃₁	—	2	2	no	—
364	Q₈×C₃₁	—	4	8	no	—
365	Q₈×C₃₁	—	4	4	no	—
366	Q₈×C₃₁	—	4	4	no	—
367	Q₈×C₃₁	C₂²×D₄	1	4	yes	—
368	D₄×C₃₁	—	2	8	no	—
369	D₄×C₃₁	—	2	4	no	—
370	D₄×C₃₁	—	2	4	no	—
371	D₄×C₃₁	—	2	2	no	—
372	D₄×C₃₁	—	2	2	no	—
373	D₄×C₃₁	—	2	2	no	—
374	D₄×C₃₁	—	2	2	no	—
375	D₄×C₃₁	—	4	2	no	—
376	D₄×C₃₁	—	4	4	no	—
377	D₄×C₃₁	—	4	4	no	—
378	D₄×C₃₁	—	4	4	no	—
379	D₄×C₃₁	—	4	8	no	—
380	D₄×C₃₁	—	4	4	no	—
381	D₄×C₃₁	—	8	4	no	—
382	D₄×C₃₁	C₂²×D₄	1	4	yes	—
383	C₂×C₄×C₃₁	—	2	2	no	—
384	C₂×C₄×C₃₁	—	2	2	no	—
385	C₂×C₄×C₃₁	—	2	2	no	—
386	C₂×C₄×C₃₁	—	2	2	no	—
387	C₂×C₄×C₃₁	—	2	8	no	—
388	C₂×C₄×C₃₁	—	2	1	no	—
389	C₂×C₄×C₃₁	—	2	1	no	—
390	C₂×C₄×C₃₁	—	2	2	no	—
391	C₂×C₄×C₃₁	—	2	1	no	—
392	C₂×C₄×C₃₁	—	2	1	no	—
393	C₂×C₄×C₃₁	—	2	2	no	—
394	C₂×C₄×C₃₁	—	2	4	no	—
395	C₂×C₄×C₃₁	—	2	4	no	—
396	C₂×C₄×C₃₁	—	2	2	no	—
397	C₂×C₄×C₃₁	—	4	4	no	—
398	C₂×C₄×C₃₁	—	4	4	no	—
399	C₂×C₄×C₃₁	—	4	4	no	—
400	C₂×C₄×C₃₁	—	4	8	no	—
401	C₂×C₄×C₃₁	—	4	8	no	—
402	C₂×C₄×C₃₁	—	4	4	no	—
403	C₂×C₄×C₃₁	—	4	4	no	—
404	C₂×C₄×C₃₁	—	8	4	no	—
405	C₂×C₄×C₃₁	—	8	2	no	—
406	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
407	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
408	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
409	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
410	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
411	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
412	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
413	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
414	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
415	C₈×C₃₁	—	2	1	no	—
416	C₈×C₃₁	—	2	1	no	—
417	C₈×C₃₁	—	2	2	no	—
418	C₈×C₃₁	—	2	2	no	—
419	C₈×C₃₁	C₂²×D₄	1	2	yes	—
420	C₈×C₃₁	C₂²×D₄	1	2	yes	—
421	C₂×#578	C₂×C₃₁	1	1	yes	characteristic, complemented
422	C₂²×#135	C₂×C₃₁	1	1	yes	characteristic, complemented
423	#3489	C₂×C₃₁	1	2	yes	complemented
424	D₄×Q₁₆	C₂×C₃₁	1	2	yes	complemented
425	#3285	C₂×C₃₁	1	2	yes	complemented
426	#3283	C₂×C₃₁	1	2	yes	complemented
427	#3280	C₂×C₃₁	1	2	yes	complemented
428	#3269	C₂×C₃₁	1	4	yes	complemented
429	#3266	C₂×C₃₁	1	4	yes	complemented
430	#3156	C₂×C₃₁	1	1	yes	characteristic, complemented
431	#3061	C₂×C₃₁	1	1	yes	characteristic, complemented
432	C₂×#467	C₂×C₃₁	1	1	yes	characteristic, complemented
433	#3046	C₂×C₃₁	1	2	yes	complemented
434	#3013	C₂×C₃₁	1	2	yes	complemented
435	#3007	C₂×C₃₁	1	2	yes	complemented
436	#2951	C₂×C₃₁	1	1	yes	characteristic, complemented
437	#2929	C₂×C₃₁	1	1	yes	characteristic, complemented
438	C₂²×C₃₁	—	2	1	no	—
439	C₂²×C₃₁	—	2	1	no	—
440	C₂²×C₃₁	—	2	1	no	—
441	C₂²×C₃₁	—	2	1	no	—
442	C₂²×C₃₁	—	2	1	no	—
443	C₂²×C₃₁	—	2	2	no	—
444	C₂²×C₃₁	—	2	1	no	—
445	C₂²×C₃₁	—	2	2	no	—
446	C₂²×C₃₁	—	2	4	no	—
447	C₂²×C₃₁	—	2	2	no	—
448	C₂²×C₃₁	—	2	1	no	—
449	C₂²×C₃₁	—	4	4	no	—
450	C₂²×C₃₁	—	4	2	no	—
451	C₂²×C₃₁	—	4	2	no	—
452	C₂²×C₃₁	—	4	2	no	—
453	C₂²×C₃₁	—	4	2	no	—
454	C₂²×C₃₁	—	4	2	no	—
455	C₂²×C₃₁	—	8	8	no	—
456	C₂²×C₃₁	D₄²	1	2	yes	—
457	C₂²×C₃₁	#578	1	2	yes	complemented
458	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
459	C₄×C₃₁	—	2	2	no	—
460	C₄×C₃₁	—	2	1	no	—
461	C₄×C₃₁	—	2	1	no	—
462	C₄×C₃₁	—	2	4	no	—
463	C₄×C₃₁	—	2	2	no	—
464	C₄×C₃₁	—	2	2	no	—
465	C₄×C₃₁	—	4	4	no	—
466	C₄×C₃₁	—	4	2	no	—
467	C₄×C₃₁	—	4	4	no	—
468	C₄×C₃₁	D₄²	1	2	yes	—
469	C₄×C₃₁	C₂³×D₄	1	1	yes	characteristic
470	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
471	C₂×2₋¹⁺⁴	C₂²×C₃₁	1	2	yes	complemented
472	C₂²×#41	C₂²×C₃₁	1	2	yes	—
473	#578	—	2	8	no	—
474	C₂×#137	C₂²×C₃₁	1	2	yes	complemented
475	C₂×#137	C₂²×C₃₁	1	4	yes	complemented
476	C₂×#137	C₂²×C₃₁	1	2	yes	complemented
477	C₂×#135	—	2	8	no	—
478	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
479	C₂×#135	C₂²×C₃₁	1	2	yes	—
480	C₂×#135	C₂²×C₃₁	1	2	yes	—
481	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
482	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
483	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
484	C₂×#135	C₂²×C₃₁	1	2	yes	complemented
485	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
486	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
487	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	complemented
488	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
489	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
490	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic, complemented
491	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
492	D₄×Q₈	C₂²×C₃₁	1	4	yes	—
493	#548	C₂²×C₃₁	1	4	yes	complemented
494	#547	—	2	4	no	—
495	#547	C₂²×C₃₁	1	4	yes	complemented
496	#546	—	2	4	no	—
497	#546	C₂²×C₃₁	1	4	yes	—
498	#537	—	2	2	no	—
499	#537	C₂²×C₃₁	1	2	yes	complemented
500	#536	—	2	2	no	—
501	#536	C₂²×C₃₁	1	2	yes	complemented
502	#525	—	2	2	no	—
503	#525	C₂²×C₃₁	1	1	yes	characteristic
504	C₂×C₂²⋊Q₈	—	2	2	no	—
505	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
506	#520	C₂²×C₃₁	1	2	yes	complemented
507	#497	C₂²×C₃₁	1	4	yes	complemented
508	#495	C₂²×C₃₁	1	2	yes	complemented
509	#494	C₂²×C₃₁	1	2	yes	complemented
510	#468	C₂²×C₃₁	1	4	yes	complemented
511	#467	C₂²×C₃₁	1	2	yes	—
512	#467	C₂²×C₃₁	1	2	yes	—
513	#467	C₂²×C₃₁	1	2	yes	complemented
514	#467	C₂²×C₃₁	1	2	yes	complemented
515	#467	C₂²×C₃₁	1	2	yes	complemented
516	#466	C₂²×C₃₁	1	2	yes	complemented
517	#464	C₂²×C₃₁	1	4	yes	complemented
518	#463	C₂²×C₃₁	1	4	yes	complemented
519	#462	C₂²×C₃₁	1	4	yes	complemented
520	#461	C₂²×C₃₁	1	4	yes	complemented
521	#452	C₂²×C₃₁	1	4	yes	complemented
522	#452	C₂²×C₃₁	1	4	yes	complemented
523	#452	C₂²×C₃₁	1	4	yes	complemented
524	#451	C₂²×C₃₁	1	4	yes	—
525	#451	C₂²×C₃₁	1	4	yes	complemented
526	#450	C₂²×C₃₁	1	4	yes	—
527	#449	C₂²×C₃₁	1	4	yes	complemented
528	#448	C₂²×C₃₁	1	4	yes	complemented
529	#440	C₂²×C₃₁	1	4	yes	complemented
530	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	complemented
531	C₄×D₈	C₂²×C₃₁	1	2	yes	complemented
532	#435	C₂²×C₃₁	1	2	yes	—
533	#435	C₂²×C₃₁	1	2	yes	complemented
534	C₈×D₄	C₂²×C₃₁	1	2	yes	—
535	C₈×D₄	C₂²×C₃₁	1	2	yes	complemented
536	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic, complemented
537	#424	C₂²×C₃₁	1	1	yes	characteristic, complemented
538	#417	C₂²×C₃₁	1	2	yes	complemented
539	#417	C₂²×C₃₁	1	2	yes	complemented
540	C₂×#103	C₂²×C₃₁	1	2	yes	complemented
541	#408	C₂²×C₃₁	1	1	yes	characteristic, complemented
542	#407	C₂²×C₃₁	1	1	yes	characteristic
543	#405	C₂²×C₃₁	1	1	yes	characteristic, complemented
544	C₂×C₃₁	—	2	2	no	—
545	C₂×C₃₁	—	2	2	no	—
546	C₂×C₃₁	—	4	2	no	—
547	C₂×C₃₁	—	4	1	no	—
548	C₂×C₃₁	—	4	4	no	—
549	C₂×C₃₁	—	8	2	no	—
550	C₂×C₃₁	C₂×D₄²	1	1	yes	characteristic
551	C₂×C₃₁	C₂×#578	1	1	yes	characteristic
552	C₂×C₃₁	#3584	1	1	yes	characteristic
553	2₋¹⁺⁴	—	2	8	no	—
554	2₋¹⁺⁴	—	4	4	no	—
555	C₂×#41	—	2	2	no	—
556	C₂×#41	—	2	4	no	—
557	C₂×#41	—	2	2	no	—
558	C₂×#41	—	2	2	no	—
559	C₂×#41	—	2	8	no	—
560	C₂×#41	—	4	4	no	—
561	C₂×#41	C₂³×C₃₁	1	2	yes	—
562	C₂×#41	C₂³×C₃₁	1	4	yes	—
563	C₂×#41	C₂³×C₃₁	1	2	yes	—
564	C₂×#41	C₂³×C₃₁	1	2	yes	—
565	C₂×#41	C₂³×C₃₁	1	2	yes	—
566	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
567	C₂²×Q₈	—	2	2	no	—
568	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
569	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
570	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
571	C₂²×D₄	—	2	2	no	—
572	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
573	C₂³×C₄	—	2	2	no	—
574	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
575	#137	—	2	8	no	—
576	#137	—	2	16	no	—
577	#137	—	4	4	no	—
578	#135	—	2	8	no	—
579	#135	—	2	4	no	—
580	#135	—	2	8	no	—
581	#135	—	2	4	no	—
582	#135	—	2	8	no	—
583	#135	—	2	4	no	—
584	#135	D₄×C₃₁	1	8	yes	complemented
585	C₂×Q₁₆	—	2	4	no	—
586	C₂×Q₁₆	—	2	8	no	—
587	C₂×Q₁₆	—	2	4	no	—
588	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
589	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
590	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
591	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
592	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
593	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
594	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
595	C₂×SD₁₆	—	2	8	no	—
596	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
597	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
598	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
599	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
600	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
601	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
602	C₂×D₈	—	2	4	no	—
603	C₂×D₈	C₂³×C₃₁	1	2	yes	—
604	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
605	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
606	#131	—	2	4	no	—
607	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
608	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
609	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
610	C₂²×C₈	—	2	4	no	—
611	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
612	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
613	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
614	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
615	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
616	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
617	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
618	C₄⋊Q₈	—	2	4	no	—
619	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
620	#124	—	2	4	no	—
621	#124	C₂³×C₃₁	1	4	yes	—
622	#123	—	2	4	no	—
623	#123	—	2	4	no	—
624	#123	—	2	4	no	—
625	#123	—	2	4	no	—
626	#123	D₄×C₃₁	1	2	yes	—
627	C₂²⋊Q₈	—	2	2	no	—
628	C₂²⋊Q₈	—	2	2	no	—
629	C₂²⋊Q₈	—	2	2	no	—
630	C₂²⋊Q₈	—	2	2	no	—
631	C₂²⋊Q₈	—	2	4	no	—
632	C₂²⋊Q₈	—	2	2	no	—
633	C₂²⋊Q₈	—	2	4	no	—
634	C₂²⋊Q₈	—	2	4	no	—
635	C₂²⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
636	C₂²⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
637	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
638	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
639	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
640	#121	—	2	2	no	—
641	#121	—	2	4	no	—
642	#121	—	2	4	no	—
643	#121	—	2	4	no	—
644	#121	—	2	4	no	—
645	#121	D₄×C₃₁	1	1	yes	characteristic
646	#121	D₄×C₃₁	1	1	yes	characteristic
647	#121	C₂³×C₃₁	1	4	yes	—
648	#120	—	2	4	no	—
649	#120	D₄×C₃₁	1	2	yes	—
650	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
651	C₄×D₄	—	2	4	no	—
652	C₄×D₄	—	2	4	no	—
653	C₄×D₄	—	2	4	no	—
654	C₄×D₄	C₂³×C₃₁	1	4	yes	—
655	C₄×D₄	C₂³×C₃₁	1	2	yes	—
656	C₄×D₄	C₂³×C₃₁	1	2	yes	—
657	#117	—	2	2	no	—
658	#117	C₂³×C₃₁	1	1	yes	characteristic
659	C₂×C₄⋊C₄	—	2	2	no	—
660	C₂×C₄⋊C₄	—	2	2	no	—
661	C₂×C₄⋊C₄	—	2	4	no	—
662	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
663	C₂×C₂²⋊C₄	—	2	2	no	—
664	C₂×C₂²⋊C₄	—	2	4	no	—
665	#107	C₂³×C₃₁	1	2	yes	—
666	#107	C₂³×C₃₁	1	2	yes	—
667	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
668	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
669	#103	C₂³×C₃₁	1	4	yes	—
670	#103	C₂³×C₃₁	1	4	yes	—
671	#103	C₂³×C₃₁	1	4	yes	—
672	#103	C₂³×C₃₁	1	4	yes	—
673	#102	C₂³×C₃₁	1	4	yes	—
674	#102	C₂³×C₃₁	1	4	yes	—
675	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
676	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
677	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
678	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
679	#97	C₂³×C₃₁	1	2	yes	—
680	C₄×C₈	C₂³×C₃₁	1	2	yes	—
681	C₃₁	#61057	1	1	yes	characteristic, direct product factor, Sylow
682	C₂⁴	—	2	2	no	—
683	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
684	#41	—	2	2	no	—
685	#41	—	2	4	no	—
686	#41	—	2	2	no	—
687	#41	—	2	4	no	—
688	#41	—	2	4	no	—
689	#41	—	2	8	no	—
690	#41	—	2	4	no	—
691	#41	—	2	4	no	—
692	#41	—	4	8	no	—
693	#41	—	4	2	no	—
694	#41	—	4	2	no	—
695	#41	—	4	4	no	—
696	#41	—	4	8	no	—
697	#41	—	4	8	no	—
698	#41	—	4	16	no	—
699	#41	C₂×D₄×C₃₁	1	8	yes	—
700	C₂×Q₈	—	2	2	no	—
701	C₂×Q₈	—	2	4	no	—
702	C₂×Q₈	—	2	2	no	—
703	C₂×Q₈	—	2	4	no	—
704	C₂×Q₈	—	2	8	no	—
705	C₂×Q₈	—	2	4	no	—
706	C₂×Q₈	—	2	4	no	—
707	C₂×Q₈	—	4	2	no	—
708	C₂×Q₈	—	4	4	no	—
709	C₂×Q₈	—	4	8	no	—
710	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
711	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
712	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
713	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
714	C₂×D₄	—	2	2	no	—
715	C₂×D₄	—	2	4	no	—
716	C₂×D₄	—	2	4	no	—
717	C₂×D₄	—	2	2	no	—
718	C₂×D₄	—	2	2	no	—
719	C₂×D₄	—	2	2	no	—
720	C₂×D₄	—	2	4	no	—
721	C₂×D₄	—	2	2	no	—
722	C₂×D₄	—	4	2	no	—
723	C₂×D₄	—	4	8	no	—
724	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
725	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
726	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
727	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
728	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
729	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
730	C₂²×C₄	—	2	2	no	—
731	C₂²×C₄	—	2	2	no	—
732	C₂²×C₄	—	2	2	no	—
733	C₂²×C₄	—	2	2	no	—
734	C₂²×C₄	—	2	4	no	—
735	C₂²×C₄	—	2	2	no	—
736	C₂²×C₄	—	2	2	no	—
737	C₂²×C₄	—	2	4	no	—
738	C₂²×C₄	—	2	2	no	—
739	C₂²×C₄	—	2	2	no	—
740	C₂²×C₄	—	2	2	no	—
741	C₂²×C₄	—	2	4	no	—
742	C₂²×C₄	—	4	8	no	—
743	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
744	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
745	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
746	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
747	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
748	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
749	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
750	Q₁₆	—	2	4	no	—
751	Q₁₆	—	2	8	no	—
752	Q₁₆	—	2	4	no	—
753	Q₁₆	—	2	2	no	—
754	Q₁₆	—	2	4	no	—
755	Q₁₆	—	4	4	no	—
756	Q₁₆	C₂×D₄×C₃₁	1	4	yes	—
757	SD₁₆	—	2	4	no	—
758	SD₁₆	—	2	4	no	—
759	SD₁₆	—	2	8	no	—
760	SD₁₆	—	2	4	no	—
761	SD₁₆	—	4	4	no	—
762	SD₁₆	C₂×D₄×C₃₁	1	8	yes	—
763	D₈	—	2	4	no	—
764	D₈	—	2	2	no	—
765	D₈	C₂×D₄×C₃₁	1	4	yes	—
766	M₁₆	—	2	2	no	—
767	M₁₆	—	2	2	no	—
768	M₁₆	—	2	4	no	—
769	C₂×C₈	—	2	2	no	—
770	C₂×C₈	—	2	2	no	—
771	C₂×C₈	—	2	2	no	—
772	C₂×C₈	—	2	2	no	—
773	C₂×C₈	—	2	2	no	—
774	C₂×C₈	—	2	4	no	—
775	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
776	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
777	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
778	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
779	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
780	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
781	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
782	C₄⋊C₄	—	2	4	no	—
783	C₄⋊C₄	—	2	2	no	—
784	C₄⋊C₄	—	2	4	no	—
785	C₄⋊C₄	—	2	4	no	—
786	C₄⋊C₄	—	2	4	no	—
787	C₄⋊C₄	—	2	4	no	—
788	C₄⋊C₄	—	2	4	no	—
789	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
790	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
791	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
792	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
793	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
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	4	no	—
801	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
802	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
803	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
804	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
805	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
806	C₄²	—	2	4	no	—
807	C₄²	C₂⁴×C₃₁	1	2	yes	—
808	C₂³	—	2	2	no	—
809	C₂³	—	2	2	no	—
810	C₂³	—	2	2	no	—
811	C₂³	—	2	2	no	—
812	C₂³	—	4	1	no	—
813	C₂³	—	4	4	no	—
814	C₂³	—	4	4	no	—
815	C₂³	—	8	2	no	—
816	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
817	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
818	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
819	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
820	Q₈	—	2	2	no	—
821	Q₈	—	2	2	no	—
822	Q₈	—	2	2	no	—
823	Q₈	—	2	4	no	—
824	Q₈	—	2	4	no	—
825	Q₈	—	2	2	no	—
826	Q₈	—	4	8	no	—
827	Q₈	—	4	4	no	—
828	Q₈	—	4	4	no	—
829	Q₈	C₂²×D₄×C₃₁	1	4	yes	—
830	D₄	—	2	8	no	—
831	D₄	—	2	4	no	—
832	D₄	—	2	4	no	—
833	D₄	—	2	2	no	—
834	D₄	—	2	2	no	—
835	D₄	—	2	2	no	—
836	D₄	—	2	2	no	—
837	D₄	—	4	2	no	—
838	D₄	—	4	4	no	—
839	D₄	—	4	4	no	—
840	D₄	—	4	4	no	—
841	D₄	—	4	8	no	—
842	D₄	—	4	4	no	—
843	D₄	—	8	4	no	—
844	D₄	C₂²×D₄×C₃₁	1	4	yes	—
845	C₂×C₄	—	2	2	no	—
846	C₂×C₄	—	2	2	no	—
847	C₂×C₄	—	2	2	no	—
848	C₂×C₄	—	2	2	no	—
849	C₂×C₄	—	2	8	no	—
850	C₂×C₄	—	2	1	no	—
851	C₂×C₄	—	2	1	no	—
852	C₂×C₄	—	2	2	no	—
853	C₂×C₄	—	2	1	no	—
854	C₂×C₄	—	2	1	no	—
855	C₂×C₄	—	2	2	no	—
856	C₂×C₄	—	2	4	no	—
857	C₂×C₄	—	2	4	no	—
858	C₂×C₄	—	2	2	no	—
859	C₂×C₄	—	4	4	no	—
860	C₂×C₄	—	4	4	no	—
861	C₂×C₄	—	4	4	no	—
862	C₂×C₄	—	4	8	no	—
863	C₂×C₄	—	4	8	no	—
864	C₂×C₄	—	4	4	no	—
865	C₂×C₄	—	4	4	no	—
866	C₂×C₄	—	8	4	no	—
867	C₂×C₄	—	8	2	no	—
868	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
869	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
870	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
871	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
872	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
873	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
874	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
875	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
876	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
877	C₈	—	2	1	no	—
878	C₈	—	2	1	no	—
879	C₈	—	2	2	no	—
880	C₈	—	2	2	no	—
881	C₈	C₂²×D₄×C₃₁	1	2	yes	—
882	C₈	C₂²×D₄×C₃₁	1	2	yes	—
883	C₂²	—	2	1	no	—
884	C₂²	—	2	1	no	—
885	C₂²	—	2	1	no	—
886	C₂²	—	2	1	no	—
887	C₂²	—	2	1	no	—
888	C₂²	—	2	2	no	—
889	C₂²	—	2	4	no	—
890	C₂²	—	2	2	no	—
891	C₂²	—	2	1	no	—
892	C₂²	—	2	1	no	—
893	C₂²	—	2	2	no	—
894	C₂²	—	4	4	no	—
895	C₂²	—	4	2	no	—
896	C₂²	—	4	2	no	—
897	C₂²	—	4	2	no	—
898	C₂²	—	4	2	no	—
899	C₂²	—	4	2	no	—
900	C₂²	—	8	8	no	—
901	C₂²	D₄²×C₃₁	1	2	yes	—
902	C₂²	C₃₁×#578	1	2	yes	complemented
903	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
904	C₄	—	2	1	no	—
905	C₄	—	2	1	no	—
906	C₄	—	2	4	no	—
907	C₄	—	2	2	no	—
908	C₄	—	2	2	no	—
909	C₄	—	2	2	no	—
910	C₄	—	4	2	no	—
911	C₄	—	4	4	no	—
912	C₄	—	4	4	no	—
913	C₄	D₄²×C₃₁	1	2	yes	—
914	C₄	C₂³×D₄×C₃₁	1	1	yes	characteristic
915	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
916	C₂	—	2	2	no	—
917	C₂	—	2	2	no	—
918	C₂	—	4	1	no	—
919	C₂	—	4	2	no	—
920	C₂	—	4	4	no	—
921	C₂	—	8	2	no	—
922	C₂	C₂×D₄²×C₃₁	1	1	yes	characteristic
923	C₂	C₂×C₃₁×#578	1	1	yes	characteristic
924	C₂	C₃₁×#3584	1	1	yes	characteristic

	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 ◁ C₃₁×#525 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#578	C₂	semidirect
2. C₂²×C₃₁×#135	C₂	semidirect
3. C₃₁×#3489	C₂	semidirect
4. D₄×Q₁₆×C₃₁	C₂	semidirect
5. C₃₁×#3285	C₂	semidirect
6. C₃₁×#3283	C₂	semidirect
7. C₃₁×#3280	C₂	semidirect
8. C₃₁×#3269	C₂	semidirect
9. C₃₁×#3266	C₂	semidirect
10. C₃₁×#3156	C₂	semidirect
11. C₃₁×#3061	C₂	semidirect
12. C₂×C₃₁×#467	C₂	semidirect
13. C₃₁×#3046	C₂	semidirect
14. C₃₁×#3013	C₂	semidirect
15. C₃₁×#3007	C₂	semidirect
16. C₃₁×#2951	C₂	semidirect
17. C₃₁×#2929	C₂	semidirect
18. C₂×C₃₁×2₋¹⁺⁴	C₂²	semidirect
21. C₂×C₃₁×#137	C₂²	semidirect
22. C₂×C₃₁×#137	C₂²	semidirect
23. C₂×C₃₁×#137	C₂²	semidirect
31. C₂×C₃₁×#135	C₂²	semidirect
33. C₂²×Q₁₆×C₃₁	C₂²	semidirect
34. C₂²×Q₁₆×C₃₁	C₂²	semidirect
37. C₂×C₃₁×#131	C₂²	semidirect
40. C₃₁×#548	C₂²	semidirect
42. C₃₁×#547	C₂²	semidirect
46. C₃₁×#537	C₂²	semidirect
48. C₃₁×#536	C₂²	semidirect
53. C₃₁×#520	C₂²	semidirect
54. C₃₁×#497	C₂²	semidirect
55. C₃₁×#495	C₂²	semidirect
56. C₃₁×#494	C₂²	semidirect
57. C₃₁×#468	C₂²	semidirect
60. C₃₁×#467	C₂²	semidirect
61. C₃₁×#467	C₂²	semidirect
62. C₃₁×#467	C₂²	semidirect
63. C₃₁×#466	C₂²	semidirect
64. C₃₁×#464	C₂²	semidirect
65. C₃₁×#463	C₂²	semidirect
66. C₃₁×#462	C₂²	semidirect
67. C₃₁×#461	C₂²	semidirect
68. C₃₁×#452	C₂²	semidirect
69. C₃₁×#452	C₂²	semidirect
70. C₃₁×#452	C₂²	semidirect
72. C₃₁×#451	C₂²	semidirect
74. C₃₁×#449	C₂²	semidirect
75. C₃₁×#448	C₂²	semidirect
76. C₃₁×#440	C₂²	semidirect
77. C₄×Q₁₆×C₃₁	C₂²	semidirect
78. C₄×D₈×C₃₁	C₂²	semidirect
80. C₃₁×#435	C₂²	semidirect
82. C₈×D₄×C₃₁	C₂²	semidirect
83. C₂×C₃₁×#107	C₂²	semidirect
84. C₃₁×#424	C₂²	semidirect
85. C₃₁×#417	C₂²	semidirect
86. C₃₁×#417	C₂²	semidirect
87. C₂×C₃₁×#103	C₂²	semidirect
88. C₃₁×#408	C₂²	semidirect
90. C₃₁×#405	C₂²	semidirect
122. C₃₁×#135	D₄	semidirect
345. #61057	C₃₁	direct (N × H)
421. C₂×#578	C₂×C₃₁	semidirect
422. C₂²×#135	C₂×C₃₁	semidirect
423. #3489	C₂×C₃₁	semidirect
424. D₄×Q₁₆	C₂×C₃₁	semidirect
425. #3285	C₂×C₃₁	semidirect
426. #3283	C₂×C₃₁	semidirect
427. #3280	C₂×C₃₁	semidirect
428. #3269	C₂×C₃₁	semidirect
429. #3266	C₂×C₃₁	semidirect
430. #3156	C₂×C₃₁	semidirect
431. #3061	C₂×C₃₁	semidirect
432. C₂×#467	C₂×C₃₁	semidirect
433. #3046	C₂×C₃₁	semidirect
434. #3013	C₂×C₃₁	semidirect
435. #3007	C₂×C₃₁	semidirect
436. #2951	C₂×C₃₁	semidirect
437. #2929	C₂×C₃₁	semidirect
457. C₂²×C₃₁	#578	semidirect
471. C₂×2₋¹⁺⁴	C₂²×C₃₁	semidirect
474. C₂×#137	C₂²×C₃₁	semidirect
475. C₂×#137	C₂²×C₃₁	semidirect
476. C₂×#137	C₂²×C₃₁	semidirect
484. C₂×#135	C₂²×C₃₁	semidirect
486. C₂²×Q₁₆	C₂²×C₃₁	semidirect
487. C₂²×Q₁₆	C₂²×C₃₁	semidirect
490. C₂×#131	C₂²×C₃₁	semidirect
493. #548	C₂²×C₃₁	semidirect
495. #547	C₂²×C₃₁	semidirect
499. #537	C₂²×C₃₁	semidirect
501. #536	C₂²×C₃₁	semidirect
506. #520	C₂²×C₃₁	semidirect
507. #497	C₂²×C₃₁	semidirect
508. #495	C₂²×C₃₁	semidirect
509. #494	C₂²×C₃₁	semidirect
510. #468	C₂²×C₃₁	semidirect
513. #467	C₂²×C₃₁	semidirect
514. #467	C₂²×C₃₁	semidirect
515. #467	C₂²×C₃₁	semidirect
516. #466	C₂²×C₃₁	semidirect
517. #464	C₂²×C₃₁	semidirect
518. #463	C₂²×C₃₁	semidirect
519. #462	C₂²×C₃₁	semidirect
520. #461	C₂²×C₃₁	semidirect
521. #452	C₂²×C₃₁	semidirect
522. #452	C₂²×C₃₁	semidirect
523. #452	C₂²×C₃₁	semidirect
525. #451	C₂²×C₃₁	semidirect
527. #449	C₂²×C₃₁	semidirect
528. #448	C₂²×C₃₁	semidirect
529. #440	C₂²×C₃₁	semidirect
530. C₄×Q₁₆	C₂²×C₃₁	semidirect
531. C₄×D₈	C₂²×C₃₁	semidirect
533. #435	C₂²×C₃₁	semidirect
535. C₈×D₄	C₂²×C₃₁	semidirect
536. C₂×#107	C₂²×C₃₁	semidirect
537. #424	C₂²×C₃₁	semidirect
538. #417	C₂²×C₃₁	semidirect
539. #417	C₂²×C₃₁	semidirect
540. C₂×#103	C₂²×C₃₁	semidirect
541. #408	C₂²×C₃₁	semidirect
543. #405	C₂²×C₃₁	semidirect
584. #135	D₄×C₃₁	semidirect
681. C₃₁	#61057	direct (N × H)
902. C₂²	C₃₁×#578	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#578	2
C₂²×C₃₁×#135	2
C₃₁×#3489	2
D₄×Q₁₆×C₃₁	2
C₃₁×#3285	2
C₃₁×#3283	2
C₃₁×#3280	2
C₃₁×#3269	2
C₃₁×#3266	2
C₃₁×#3156	2
C₃₁×#3061	2
C₂×C₃₁×#467	2
C₃₁×#3046	2
C₃₁×#3013	2
C₃₁×#3007	2
C₃₁×#2951	2
C₃₁×#2929	2
#61057	31

Subgroups — C31×#61057 #21517816

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61057 #21517816