Subgroups — C₃₁×#60591 #21517350

Order 7936.

2074 conjugacy classes of subgroups.

The subgroup lattice has 910 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3509	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₃₁×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3123	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3120	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3116	C₂	1	2	yes	maximal, complemented
6	C₃₁×#3110	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3107	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3101	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3097	C₂	1	2	yes	maximal, complemented
10	C₃₁×#3077	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#3076	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#3039	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₃₁×#461	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#3009	C₂	1	2	yes	maximal
15	C₃₁×#2949	C₂	1	2	yes	maximal, complemented
16	C₃₁×#2947	C₂	1	2	yes	maximal, complemented
17	C₃₁×#2925	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×C₃₁×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#137	C₂²	1	2	yes	—
20	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
21	C₃₁×#557	—	2	1	no	—
22	C₃₁×#557	—	2	1	no	—
23	C₃₁×#556	—	2	1	no	—
24	C₃₁×#556	—	2	1	no	—
25	C₃₁×#552	—	2	2	no	—
26	C₃₁×#552	—	2	2	no	—
27	C₃₁×#541	C₂²	1	4	yes	complemented
28	C₃₁×#536	C₂²	1	2	yes	—
29	C₃₁×#533	C₂²	1	2	yes	complemented
30	C₃₁×#533	C₂²	1	2	yes	complemented
31	C₃₁×#533	C₂²	1	1	yes	characteristic, complemented
32	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
33	C₃₁×#520	—	2	1	no	—
34	C₃₁×#520	C₂²	1	2	yes	—
35	C₃₁×#520	C₂²	1	1	yes	characteristic
36	C₃₁×#520	C₂²	1	1	yes	characteristic
37	C₃₁×#519	C₂²	1	2	yes	complemented
38	C₃₁×#518	—	2	2	no	—
39	C₃₁×#518	—	2	1	no	—
40	C₃₁×#518	C₂²	1	2	yes	—
41	C₃₁×#518	C₂²	1	1	yes	characteristic
42	C₄×#41×C₃₁	—	2	1	no	—
43	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
44	C₄×#41×C₃₁	C₂²	1	2	yes	—
45	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
46	C₂×C₄×D₄×C₃₁	—	2	2	no	—
47	C₂×C₄×D₄×C₃₁	—	2	1	no	—
48	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
49	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
50	C₃₁×#485	C₂²	1	4	yes	complemented
51	C₃₁×#485	C₂²	1	4	yes	complemented
52	C₃₁×#484	C₂²	1	4	yes	complemented
53	C₃₁×#483	C₂²	1	4	yes	complemented
54	C₃₁×#479	C₂²	1	2	yes	complemented
55	C₃₁×#478	C₂²	1	2	yes	complemented
56	C₃₁×#478	C₂²	1	4	yes	complemented
57	C₃₁×#477	C₂²	1	1	yes	characteristic, complemented
58	C₃₁×#476	C₂²	1	1	yes	characteristic, complemented
59	C₃₁×#476	C₂²	1	2	yes	complemented
60	C₃₁×#475	C₂²	1	1	yes	characteristic, complemented
61	C₃₁×#474	C₂²	1	1	yes	characteristic, complemented
62	C₃₁×#474	C₂²	1	2	yes	complemented
63	C₃₁×#462	C₂²	1	2	yes	complemented
64	C₃₁×#462	C₂²	1	2	yes	complemented
65	C₃₁×#461	C₂²	1	2	yes	complemented
66	C₃₁×#461	C₂²	1	2	yes	complemented
67	C₃₁×#461	C₂²	1	2	yes	complemented
68	C₃₁×#461	C₂²	1	4	yes	complemented
69	C₃₁×#461	C₂²	1	2	yes	complemented
70	C₃₁×#452	C₂²	1	4	yes	—
71	C₃₁×#452	C₂²	1	4	yes	—
72	C₃₁×#450	C₂²	1	4	yes	—
73	C₃₁×#450	C₂²	1	4	yes	—
74	C₃₁×#441	C₂²	1	4	yes	complemented
75	C₃₁×#440	C₂²	1	4	yes	complemented
76	C₃₁×#440	C₂²	1	4	yes	complemented
77	C₃₁×#440	C₂²	1	4	yes	complemented
78	C₄×Q₁₆×C₃₁	C₂²	1	4	yes	complemented
79	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	complemented
80	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	complemented
81	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	complemented
82	C₃₁×#433	C₂²	1	4	yes	complemented
83	C₃₁×#432	C₂²	1	2	yes	complemented
84	C₃₁×#431	C₂²	1	2	yes	complemented
85	C₃₁×#428	C₂²	1	2	yes	complemented
86	C₃₁×#427	C₂²	1	2	yes	complemented
87	C₃₁×#423	C₂²	1	1	yes	characteristic, complemented
88	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
89	C₃₁×#419	C₂²	1	2	yes	complemented
90	C₃₁×#418	C₂²	1	2	yes	—
91	C₃₁×#416	C₂²	1	2	yes	—
92	C₂×C₃₁×#103	C₂²	1	2	yes	complemented
93	C₃₁×#408	C₂²	1	2	yes	—
94	C₃₁×#405	C₂²	1	2	yes	complemented
95	C₃₁×H(5,𝔽₂)	—	2	4	no	—
96	C₃₁×H(5,𝔽₂)	—	2	4	no	—
97	C₂×#41×C₃₁	—	2	1	no	—
98	C₂×#41×C₃₁	C₂³	1	2	yes	—
99	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
100	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
101	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
102	C₂²×D₄×C₃₁	—	2	2	no	—
103	C₂²×D₄×C₃₁	—	2	1	no	—
104	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
105	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
106	C₂³×C₄×C₃₁	—	2	1	no	—
107	C₂³×C₄×C₃₁	—	2	2	no	—
108	C₃₁×#137	—	2	8	no	—
109	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
110	C₂×SD₁₆×C₃₁	—	2	8	no	—
111	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
112	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
113	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
114	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
115	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
116	C₃₁×C₄⋊Q₈	—	2	2	no	—
117	C₃₁×C₄⋊Q₈	—	2	1	no	—
118	C₃₁×C₄⋊Q₈	—	2	1	no	—
119	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
120	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
121	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
122	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
123	C₃₁×#126	—	2	4	no	—
124	C₃₁×#126	—	2	4	no	—
125	C₃₁×#125	—	2	4	no	—
126	C₃₁×#125	—	2	4	no	—
127	C₃₁×#125	—	2	2	no	—
128	C₃₁×#125	—	2	2	no	—
129	C₃₁×#125	C₂³	1	2	yes	—
130	C₃₁×#124	—	2	2	no	—
131	C₃₁×#123	—	2	4	no	—
132	C₃₁×C₂²⋊Q₈	—	2	1	no	—
133	C₃₁×C₂²⋊Q₈	—	2	1	no	—
134	C₃₁×C₂²⋊Q₈	—	2	2	no	—
135	C₃₁×C₂²⋊Q₈	—	2	2	no	—
136	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
137	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
138	C₄×Q₈×C₃₁	—	2	1	no	—
139	C₄×Q₈×C₃₁	—	2	2	no	—
140	C₄×Q₈×C₃₁	—	2	2	no	—
141	C₄×Q₈×C₃₁	—	2	2	no	—
142	C₄×Q₈×C₃₁	—	2	2	no	—
143	C₄×Q₈×C₃₁	—	2	1	no	—
144	C₄×Q₈×C₃₁	—	2	1	no	—
145	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
146	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
147	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
148	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
149	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
150	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
151	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
152	C₄×D₄×C₃₁	—	2	2	no	—
153	C₄×D₄×C₃₁	—	2	1	no	—
154	C₄×D₄×C₃₁	—	2	1	no	—
155	C₄×D₄×C₃₁	—	2	2	no	—
156	C₄×D₄×C₃₁	—	2	1	no	—
157	C₄×D₄×C₃₁	—	2	2	no	—
158	C₄×D₄×C₃₁	—	2	1	no	—
159	C₄×D₄×C₃₁	—	2	2	no	—
160	C₄×D₄×C₃₁	—	2	1	no	—
161	C₄×D₄×C₃₁	—	2	2	no	—
162	C₄×D₄×C₃₁	—	2	2	no	—
163	C₄×D₄×C₃₁	—	2	2	no	—
164	C₄×D₄×C₃₁	—	2	2	no	—
165	C₄×D₄×C₃₁	—	2	2	no	—
166	C₄×D₄×C₃₁	—	2	1	no	—
167	C₄×D₄×C₃₁	—	2	2	no	—
168	C₄×D₄×C₃₁	—	2	1	no	—
169	C₄×D₄×C₃₁	D₄	1	2	yes	—
170	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
171	C₄×D₄×C₃₁	D₄	1	2	yes	—
172	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
173	C₄×D₄×C₃₁	C₂³	1	2	yes	—
174	C₄×D₄×C₃₁	C₂³	1	2	yes	—
175	C₄×D₄×C₃₁	C₂³	1	2	yes	—
176	C₄×D₄×C₃₁	C₂³	1	2	yes	—
177	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
178	C₄×D₄×C₃₁	C₂³	1	2	yes	—
179	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
180	C₃₁×#117	—	2	2	no	—
181	C₃₁×#117	—	2	2	no	—
182	C₃₁×#117	—	2	1	no	—
183	C₃₁×#117	—	2	2	no	—
184	C₃₁×#117	—	2	4	no	—
185	C₃₁×#117	—	2	2	no	—
186	C₃₁×#117	—	2	1	no	—
187	C₃₁×#117	C₂³	1	2	yes	—
188	C₃₁×#117	C₂³	1	1	yes	characteristic
189	C₃₁×#117	C₂³	1	2	yes	—
190	C₃₁×#117	C₂³	1	1	yes	characteristic
191	C₃₁×#117	C₂³	1	2	yes	—
192	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
193	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
194	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
195	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
196	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
197	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
198	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
199	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
200	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
201	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
202	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
203	C₂×C₄²×C₃₁	—	2	2	no	—
204	C₂×C₄²×C₃₁	—	2	1	no	—
205	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
206	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
207	C₃₁×#107	C₂³	1	2	yes	—
208	C₃₁×#107	C₂³	1	2	yes	—
209	C₃₁×#107	C₂³	1	4	yes	—
210	C₃₁×#106	C₂³	1	2	yes	—
211	C₃₁×#106	C₂³	1	2	yes	—
212	C₃₁×#106	C₂³	1	4	yes	—
213	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
214	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
215	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
216	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
217	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
218	C₃₁×#103	C₂³	1	4	yes	—
219	C₃₁×#103	C₂³	1	4	yes	—
220	C₃₁×#103	C₂³	1	2	yes	—
221	C₃₁×#103	C₂³	1	2	yes	—
222	C₃₁×#103	C₂³	1	4	yes	—
223	C₃₁×#103	C₂³	1	4	yes	—
224	C₃₁×#102	C₂³	1	2	yes	—
225	C₃₁×#102	C₂³	1	2	yes	—
226	C₃₁×#102	C₂³	1	4	yes	—
227	C₃₁×#102	C₂³	1	4	yes	—
228	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
229	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
230	C₃₁×#97	C₂³	1	4	yes	—
231	C₄×C₈×C₃₁	C₂³	1	4	yes	—
232	C₂⁴×C₃₁	—	2	2	no	—
233	C₂⁴×C₃₁	—	2	1	no	—
234	#41×C₃₁	—	2	8	no	—
235	#41×C₃₁	—	2	4	no	—
236	#41×C₃₁	—	2	4	no	—
237	#41×C₃₁	—	2	2	no	—
238	#41×C₃₁	—	2	2	no	—
239	#41×C₃₁	—	2	2	no	—
240	#41×C₃₁	#41	1	4	yes	—
241	C₂×Q₈×C₃₁	—	2	1	no	—
242	C₂×Q₈×C₃₁	—	2	1	no	—
243	C₂×Q₈×C₃₁	—	2	2	no	—
244	C₂×Q₈×C₃₁	—	2	4	no	—
245	C₂×Q₈×C₃₁	—	2	4	no	—
246	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
247	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
248	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
249	C₂×D₄×C₃₁	—	2	16	no	—
250	C₂×D₄×C₃₁	—	2	8	no	—
251	C₂×D₄×C₃₁	—	2	2	no	—
252	C₂×D₄×C₃₁	—	2	1	no	—
253	C₂×D₄×C₃₁	—	2	4	no	—
254	C₂×D₄×C₃₁	—	2	4	no	—
255	C₂×D₄×C₃₁	—	2	2	no	—
256	C₂×D₄×C₃₁	—	2	1	no	—
257	C₂×D₄×C₃₁	—	2	2	no	—
258	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
259	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
260	C₂×D₄×C₃₁	#41	1	4	yes	—
261	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
262	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
263	C₂²×C₄×C₃₁	—	2	2	no	—
264	C₂²×C₄×C₃₁	—	2	1	no	—
265	C₂²×C₄×C₃₁	—	2	2	no	—
266	C₂²×C₄×C₃₁	—	2	2	no	—
267	C₂²×C₄×C₃₁	—	2	1	no	—
268	C₂²×C₄×C₃₁	—	2	2	no	—
269	C₂²×C₄×C₃₁	—	2	1	no	—
270	C₂²×C₄×C₃₁	—	2	2	no	—
271	C₂²×C₄×C₃₁	—	2	1	no	—
272	C₂²×C₄×C₃₁	—	2	2	no	—
273	C₂²×C₄×C₃₁	—	2	2	no	—
274	C₂²×C₄×C₃₁	—	2	2	no	—
275	C₂²×C₄×C₃₁	—	8	2	no	—
276	C₂²×C₄×C₃₁	—	8	4	no	—
277	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
278	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
279	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
280	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
281	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
282	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
283	Q₁₆×C₃₁	—	2	8	no	—
284	SD₁₆×C₃₁	—	2	8	no	—
285	SD₁₆×C₃₁	—	2	8	no	—
286	SD₁₆×C₃₁	—	2	8	no	—
287	M₁₆×C₃₁	—	2	4	no	—
288	C₂×C₈×C₃₁	—	2	4	no	—
289	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
290	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
291	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
292	C₄⋊C₄×C₃₁	—	2	2	no	—
293	C₄⋊C₄×C₃₁	—	2	1	no	—
294	C₄⋊C₄×C₃₁	—	2	2	no	—
295	C₄⋊C₄×C₃₁	—	2	1	no	—
296	C₄⋊C₄×C₃₁	—	2	2	no	—
297	C₄⋊C₄×C₃₁	—	2	2	no	—
298	C₄⋊C₄×C₃₁	—	2	2	no	—
299	C₄⋊C₄×C₃₁	—	2	2	no	—
300	C₄⋊C₄×C₃₁	—	2	1	no	—
301	C₄⋊C₄×C₃₁	—	2	1	no	—
302	C₄⋊C₄×C₃₁	—	2	1	no	—
303	C₄⋊C₄×C₃₁	—	2	4	no	—
304	C₄⋊C₄×C₃₁	—	2	4	no	—
305	C₄⋊C₄×C₃₁	—	2	2	no	—
306	C₄⋊C₄×C₃₁	—	2	2	no	—
307	C₄⋊C₄×C₃₁	—	2	4	no	—
308	C₄⋊C₄×C₃₁	—	2	4	no	—
309	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
310	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
311	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
312	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
313	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
314	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
315	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
316	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
317	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
318	C₂²⋊C₄×C₃₁	—	2	2	no	—
319	C₂²⋊C₄×C₃₁	—	2	1	no	—
320	C₂²⋊C₄×C₃₁	—	2	2	no	—
321	C₂²⋊C₄×C₃₁	—	2	4	no	—
322	C₂²⋊C₄×C₃₁	—	2	4	no	—
323	C₂²⋊C₄×C₃₁	—	2	2	no	—
324	C₂²⋊C₄×C₃₁	—	2	1	no	—
325	C₂²⋊C₄×C₃₁	—	2	2	no	—
326	C₂²⋊C₄×C₃₁	—	2	2	no	—
327	C₂²⋊C₄×C₃₁	—	2	2	no	—
328	C₂²⋊C₄×C₃₁	—	2	2	no	—
329	C₂²⋊C₄×C₃₁	—	2	1	no	—
330	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
331	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
332	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
333	C₄²×C₃₁	—	2	1	no	—
334	C₄²×C₃₁	—	2	2	no	—
335	C₄²×C₃₁	—	2	1	no	—
336	C₄²×C₃₁	—	2	2	no	—
337	C₄²×C₃₁	—	2	1	no	—
338	C₄²×C₃₁	—	2	4	no	—
339	C₄²×C₃₁	—	2	4	no	—
340	C₄²×C₃₁	—	2	2	no	—
341	C₄²×C₃₁	C₂×D₄	1	2	yes	—
342	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
343	C₄²×C₃₁	C₂×D₄	1	2	yes	—
344	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
345	C₄²×C₃₁	C₂⁴	1	2	yes	—
346	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
347	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
348	#60591	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
349	C₂³×C₃₁	—	2	8	no	—
350	C₂³×C₃₁	—	2	2	no	—
351	C₂³×C₃₁	—	2	1	no	—
352	C₂³×C₃₁	—	2	4	no	—
353	C₂³×C₃₁	—	8	1	no	—
354	C₂³×C₃₁	—	8	2	no	—
355	C₂³×C₃₁	—	8	1	no	—
356	C₂³×C₃₁	—	8	2	no	—
357	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
358	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
359	Q₈×C₃₁	—	2	2	no	—
360	Q₈×C₃₁	—	2	1	no	—
361	Q₈×C₃₁	—	2	4	no	—
362	Q₈×C₃₁	—	2	4	no	—
363	Q₈×C₃₁	—	4	2	no	—
364	Q₈×C₃₁	—	4	4	no	—
365	Q₈×C₃₁	—	4	2	no	—
366	Q₈×C₃₁	C₂×#41	1	2	yes	—
367	D₄×C₃₁	—	2	8	no	—
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	4	no	—
373	D₄×C₃₁	—	2	2	no	—
374	D₄×C₃₁	—	2	1	no	—
375	D₄×C₃₁	C₂×#41	1	2	yes	—
376	D₄×C₃₁	C₂×#41	1	4	yes	—
377	C₂×C₄×C₃₁	—	2	8	no	—
378	C₂×C₄×C₃₁	—	2	1	no	—
379	C₂×C₄×C₃₁	—	2	4	no	—
380	C₂×C₄×C₃₁	—	2	2	no	—
381	C₂×C₄×C₃₁	—	2	4	no	—
382	C₂×C₄×C₃₁	—	2	2	no	—
383	C₂×C₄×C₃₁	—	2	2	no	—
384	C₂×C₄×C₃₁	—	2	1	no	—
385	C₂×C₄×C₃₁	—	2	2	no	—
386	C₂×C₄×C₃₁	—	2	2	no	—
387	C₂×C₄×C₃₁	—	2	2	no	—
388	C₂×C₄×C₃₁	—	2	1	no	—
389	C₂×C₄×C₃₁	—	2	4	no	—
390	C₂×C₄×C₃₁	—	4	1	no	—
391	C₂×C₄×C₃₁	—	4	2	no	—
392	C₂×C₄×C₃₁	—	4	4	no	—
393	C₂×C₄×C₃₁	—	4	4	no	—
394	C₂×C₄×C₃₁	—	4	2	no	—
395	C₂×C₄×C₃₁	—	8	2	no	—
396	C₂×C₄×C₃₁	—	8	1	no	—
397	C₂×C₄×C₃₁	—	8	2	no	—
398	C₂×C₄×C₃₁	—	8	2	no	—
399	C₂×C₄×C₃₁	—	8	2	no	—
400	C₂×C₄×C₃₁	—	8	1	no	—
401	C₂×C₄×C₃₁	—	8	2	no	—
402	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
403	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
404	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
405	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
406	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
407	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
408	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
409	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
410	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
411	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
412	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
413	C₈×C₃₁	—	2	2	no	—
414	C₈×C₃₁	—	2	2	no	—
415	C₈×C₃₁	—	2	4	no	—
416	#3509	C₂×C₃₁	1	1	yes	characteristic, complemented
417	C₄×H(5,𝔽₂)	C₂×C₃₁	1	1	yes	characteristic
418	#3123	C₂×C₃₁	1	4	yes	complemented
419	#3120	C₂×C₃₁	1	4	yes	complemented
420	#3116	C₂×C₃₁	1	2	yes	complemented
421	#3110	C₂×C₃₁	1	2	yes	complemented
422	#3107	C₂×C₃₁	1	2	yes	complemented
423	#3101	C₂×C₃₁	1	2	yes	complemented
424	#3097	C₂×C₃₁	1	2	yes	complemented
425	#3077	C₂×C₃₁	1	1	yes	characteristic, complemented
426	#3076	C₂×C₃₁	1	1	yes	characteristic, complemented
427	#3039	C₂×C₃₁	1	1	yes	characteristic, complemented
428	C₂×#461	C₂×C₃₁	1	1	yes	characteristic, complemented
429	#3009	C₂×C₃₁	1	2	yes	—
430	#2949	C₂×C₃₁	1	2	yes	complemented
431	#2947	C₂×C₃₁	1	2	yes	complemented
432	#2925	C₂×C₃₁	1	1	yes	characteristic, complemented
433	C₂²×C₃₁	—	2	1	no	—
434	C₂²×C₃₁	—	2	1	no	—
435	C₂²×C₃₁	—	2	2	no	—
436	C₂²×C₃₁	—	2	2	no	—
437	C₂²×C₃₁	—	2	4	no	—
438	C₂²×C₃₁	—	2	2	no	—
439	C₂²×C₃₁	—	2	2	no	—
440	C₂²×C₃₁	—	4	2	no	—
441	C₂²×C₃₁	—	4	1	no	—
442	C₂²×C₃₁	—	4	2	no	—
443	C₂²×C₃₁	—	4	1	no	—
444	C₂²×C₃₁	—	8	4	no	—
445	C₂²×C₃₁	—	8	8	no	—
446	C₂²×C₃₁	#525	1	2	yes	—
447	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
448	C₄×C₃₁	—	2	1	no	—
449	C₄×C₃₁	—	2	2	no	—
450	C₄×C₃₁	—	2	1	no	—
451	C₄×C₃₁	—	2	2	no	—
452	C₄×C₃₁	—	2	2	no	—
453	C₄×C₃₁	—	2	2	no	—
454	C₄×C₃₁	—	2	1	no	—
455	C₄×C₃₁	—	4	1	no	—
456	C₄×C₃₁	—	4	2	no	—
457	C₄×C₃₁	—	4	4	no	—
458	C₄×C₃₁	—	4	2	no	—
459	C₄×C₃₁	—	4	1	no	—
460	C₄×C₃₁	—	4	2	no	—
461	C₄×C₃₁	#525	1	2	yes	—
462	C₄×C₃₁	#577	1	2	yes	—
463	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
464	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
465	C₂×H(5,𝔽₂)	C₂²×C₃₁	1	1	yes	characteristic
466	C₂×#137	C₂²×C₃₁	1	2	yes	—
467	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
468	#557	—	2	1	no	—
469	#557	—	2	1	no	—
470	#556	—	2	1	no	—
471	#556	—	2	1	no	—
472	#552	—	2	2	no	—
473	#552	—	2	2	no	—
474	#541	C₂²×C₃₁	1	4	yes	complemented
475	#536	C₂²×C₃₁	1	2	yes	—
476	#533	C₂²×C₃₁	1	2	yes	complemented
477	#533	C₂²×C₃₁	1	2	yes	complemented
478	#533	C₂²×C₃₁	1	1	yes	characteristic, complemented
479	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
480	#520	—	2	1	no	—
481	#520	C₂²×C₃₁	1	2	yes	—
482	#520	C₂²×C₃₁	1	1	yes	characteristic
483	#520	C₂²×C₃₁	1	1	yes	characteristic
484	#519	C₂²×C₃₁	1	2	yes	complemented
485	#518	—	2	2	no	—
486	#518	—	2	1	no	—
487	#518	C₂²×C₃₁	1	2	yes	—
488	#518	C₂²×C₃₁	1	1	yes	characteristic
489	C₄×#41	—	2	1	no	—
490	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
491	C₄×#41	C₂²×C₃₁	1	2	yes	—
492	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
493	C₂×C₄×D₄	—	2	2	no	—
494	C₂×C₄×D₄	—	2	1	no	—
495	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
496	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
497	#485	C₂²×C₃₁	1	4	yes	complemented
498	#485	C₂²×C₃₁	1	4	yes	complemented
499	#484	C₂²×C₃₁	1	4	yes	complemented
500	#483	C₂²×C₃₁	1	4	yes	complemented
501	#479	C₂²×C₃₁	1	2	yes	complemented
502	#478	C₂²×C₃₁	1	2	yes	complemented
503	#478	C₂²×C₃₁	1	4	yes	complemented
504	#477	C₂²×C₃₁	1	1	yes	characteristic, complemented
505	#476	C₂²×C₃₁	1	1	yes	characteristic, complemented
506	#476	C₂²×C₃₁	1	2	yes	complemented
507	#475	C₂²×C₃₁	1	1	yes	characteristic, complemented
508	#474	C₂²×C₃₁	1	1	yes	characteristic, complemented
509	#474	C₂²×C₃₁	1	2	yes	complemented
510	#462	C₂²×C₃₁	1	2	yes	complemented
511	#462	C₂²×C₃₁	1	2	yes	complemented
512	#461	C₂²×C₃₁	1	2	yes	complemented
513	#461	C₂²×C₃₁	1	2	yes	complemented
514	#461	C₂²×C₃₁	1	2	yes	complemented
515	#461	C₂²×C₃₁	1	4	yes	complemented
516	#461	C₂²×C₃₁	1	2	yes	complemented
517	#452	C₂²×C₃₁	1	4	yes	—
518	#452	C₂²×C₃₁	1	4	yes	—
519	#450	C₂²×C₃₁	1	4	yes	—
520	#450	C₂²×C₃₁	1	4	yes	—
521	#441	C₂²×C₃₁	1	4	yes	complemented
522	#440	C₂²×C₃₁	1	4	yes	complemented
523	#440	C₂²×C₃₁	1	4	yes	complemented
524	#440	C₂²×C₃₁	1	4	yes	complemented
525	C₄×Q₁₆	C₂²×C₃₁	1	4	yes	complemented
526	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	complemented
527	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	complemented
528	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	complemented
529	#433	C₂²×C₃₁	1	4	yes	complemented
530	#432	C₂²×C₃₁	1	2	yes	complemented
531	#431	C₂²×C₃₁	1	2	yes	complemented
532	#428	C₂²×C₃₁	1	2	yes	complemented
533	#427	C₂²×C₃₁	1	2	yes	complemented
534	#423	C₂²×C₃₁	1	1	yes	characteristic, complemented
535	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
536	#419	C₂²×C₃₁	1	2	yes	complemented
537	#418	C₂²×C₃₁	1	2	yes	—
538	#416	C₂²×C₃₁	1	2	yes	—
539	C₂×#103	C₂²×C₃₁	1	2	yes	complemented
540	#408	C₂²×C₃₁	1	2	yes	—
541	#405	C₂²×C₃₁	1	2	yes	complemented
542	C₂×C₃₁	—	2	2	no	—
543	C₂×C₃₁	—	4	2	no	—
544	C₂×C₃₁	—	4	4	no	—
545	C₂×C₃₁	—	4	2	no	—
546	C₂×C₃₁	C₂×#525	1	1	yes	characteristic
547	C₂×C₃₁	C₂×#577	1	1	yes	characteristic
548	C₂×C₃₁	#3586	1	1	yes	characteristic
549	H(5,𝔽₂)	—	2	4	no	—
550	H(5,𝔽₂)	—	2	4	no	—
551	C₂×#41	—	2	1	no	—
552	C₂×#41	C₂³×C₃₁	1	2	yes	—
553	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
554	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
555	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
556	C₂²×D₄	—	2	2	no	—
557	C₂²×D₄	—	2	1	no	—
558	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
559	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
560	C₂³×C₄	—	2	1	no	—
561	C₂³×C₄	—	2	2	no	—
562	#137	—	2	8	no	—
563	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
564	C₂×SD₁₆	—	2	8	no	—
565	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
566	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
567	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
568	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
569	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
570	C₄⋊Q₈	—	2	2	no	—
571	C₄⋊Q₈	—	2	1	no	—
572	C₄⋊Q₈	—	2	1	no	—
573	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
574	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
575	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
576	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
577	#126	—	2	4	no	—
578	#126	—	2	4	no	—
579	#125	—	2	4	no	—
580	#125	—	2	4	no	—
581	#125	—	2	2	no	—
582	#125	—	2	2	no	—
583	#125	C₂³×C₃₁	1	2	yes	—
584	#124	—	2	2	no	—
585	#123	—	2	4	no	—
586	C₂²⋊Q₈	—	2	1	no	—
587	C₂²⋊Q₈	—	2	1	no	—
588	C₂²⋊Q₈	—	2	2	no	—
589	C₂²⋊Q₈	—	2	2	no	—
590	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
591	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
592	C₄×Q₈	—	2	1	no	—
593	C₄×Q₈	—	2	2	no	—
594	C₄×Q₈	—	2	2	no	—
595	C₄×Q₈	—	2	2	no	—
596	C₄×Q₈	—	2	2	no	—
597	C₄×Q₈	—	2	1	no	—
598	C₄×Q₈	—	2	1	no	—
599	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
600	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
601	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
602	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
603	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
604	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
605	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
606	C₄×D₄	—	2	2	no	—
607	C₄×D₄	—	2	1	no	—
608	C₄×D₄	—	2	1	no	—
609	C₄×D₄	—	2	2	no	—
610	C₄×D₄	—	2	1	no	—
611	C₄×D₄	—	2	2	no	—
612	C₄×D₄	—	2	1	no	—
613	C₄×D₄	—	2	2	no	—
614	C₄×D₄	—	2	1	no	—
615	C₄×D₄	—	2	2	no	—
616	C₄×D₄	—	2	2	no	—
617	C₄×D₄	—	2	2	no	—
618	C₄×D₄	—	2	2	no	—
619	C₄×D₄	—	2	2	no	—
620	C₄×D₄	—	2	1	no	—
621	C₄×D₄	—	2	2	no	—
622	C₄×D₄	—	2	1	no	—
623	C₄×D₄	D₄×C₃₁	1	2	yes	—
624	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
625	C₄×D₄	D₄×C₃₁	1	2	yes	—
626	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
627	C₄×D₄	C₂³×C₃₁	1	2	yes	—
628	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
629	C₄×D₄	C₂³×C₃₁	1	2	yes	—
630	C₄×D₄	C₂³×C₃₁	1	2	yes	—
631	C₄×D₄	C₂³×C₃₁	1	2	yes	—
632	C₄×D₄	C₂³×C₃₁	1	2	yes	—
633	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
634	#117	—	2	2	no	—
635	#117	—	2	2	no	—
636	#117	—	2	1	no	—
637	#117	—	2	2	no	—
638	#117	—	2	4	no	—
639	#117	—	2	2	no	—
640	#117	—	2	1	no	—
641	#117	C₂³×C₃₁	1	1	yes	characteristic
642	#117	C₂³×C₃₁	1	2	yes	—
643	#117	C₂³×C₃₁	1	2	yes	—
644	#117	C₂³×C₃₁	1	1	yes	characteristic
645	#117	C₂³×C₃₁	1	2	yes	—
646	C₂×C₄⋊C₄	—	2	1	no	—
647	C₂×C₄⋊C₄	—	2	1	no	—
648	C₂×C₄⋊C₄	—	2	1	no	—
649	C₂×C₄⋊C₄	—	2	2	no	—
650	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
651	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
652	C₂×C₂²⋊C₄	—	2	1	no	—
653	C₂×C₂²⋊C₄	—	2	2	no	—
654	C₂×C₂²⋊C₄	—	2	2	no	—
655	C₂×C₂²⋊C₄	—	2	2	no	—
656	C₂×C₂²⋊C₄	—	2	2	no	—
657	C₂×C₄²	—	2	2	no	—
658	C₂×C₄²	—	2	1	no	—
659	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
660	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
661	#107	C₂³×C₃₁	1	2	yes	—
662	#107	C₂³×C₃₁	1	2	yes	—
663	#107	C₂³×C₃₁	1	4	yes	—
664	#106	C₂³×C₃₁	1	2	yes	—
665	#106	C₂³×C₃₁	1	2	yes	—
666	#106	C₂³×C₃₁	1	4	yes	—
667	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
668	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
669	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
670	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
671	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
672	#103	C₂³×C₃₁	1	4	yes	—
673	#103	C₂³×C₃₁	1	4	yes	—
674	#103	C₂³×C₃₁	1	2	yes	—
675	#103	C₂³×C₃₁	1	2	yes	—
676	#103	C₂³×C₃₁	1	4	yes	—
677	#103	C₂³×C₃₁	1	4	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	4	yes	—
685	C₄×C₈	C₂³×C₃₁	1	4	yes	—
686	C₃₁	#60591	1	1	yes	characteristic, direct product factor, Sylow
687	C₂⁴	—	2	2	no	—
688	C₂⁴	—	2	1	no	—
689	#41	—	2	8	no	—
690	#41	—	2	4	no	—
691	#41	—	2	4	no	—
692	#41	—	2	2	no	—
693	#41	—	2	2	no	—
694	#41	—	2	2	no	—
695	#41	#41×C₃₁	1	4	yes	—
696	C₂×Q₈	—	2	1	no	—
697	C₂×Q₈	—	2	1	no	—
698	C₂×Q₈	—	2	2	no	—
699	C₂×Q₈	—	2	4	no	—
700	C₂×Q₈	—	2	4	no	—
701	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
702	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
703	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
704	C₂×D₄	—	2	16	no	—
705	C₂×D₄	—	2	8	no	—
706	C₂×D₄	—	2	2	no	—
707	C₂×D₄	—	2	1	no	—
708	C₂×D₄	—	2	4	no	—
709	C₂×D₄	—	2	4	no	—
710	C₂×D₄	—	2	2	no	—
711	C₂×D₄	—	2	1	no	—
712	C₂×D₄	—	2	2	no	—
713	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
714	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
715	C₂×D₄	#41×C₃₁	1	4	yes	—
716	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
717	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
718	C₂²×C₄	—	2	2	no	—
719	C₂²×C₄	—	2	1	no	—
720	C₂²×C₄	—	2	2	no	—
721	C₂²×C₄	—	2	2	no	—
722	C₂²×C₄	—	2	1	no	—
723	C₂²×C₄	—	2	2	no	—
724	C₂²×C₄	—	2	1	no	—
725	C₂²×C₄	—	2	2	no	—
726	C₂²×C₄	—	2	1	no	—
727	C₂²×C₄	—	2	2	no	—
728	C₂²×C₄	—	2	2	no	—
729	C₂²×C₄	—	2	2	no	—
730	C₂²×C₄	—	8	2	no	—
731	C₂²×C₄	—	8	4	no	—
732	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
733	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
734	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
735	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
736	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
737	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
738	Q₁₆	—	2	8	no	—
739	SD₁₆	—	2	8	no	—
740	SD₁₆	—	2	8	no	—
741	SD₁₆	—	2	8	no	—
742	M₁₆	—	2	4	no	—
743	C₂×C₈	—	2	4	no	—
744	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
745	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
746	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
747	C₄⋊C₄	—	2	2	no	—
748	C₄⋊C₄	—	2	1	no	—
749	C₄⋊C₄	—	2	2	no	—
750	C₄⋊C₄	—	2	1	no	—
751	C₄⋊C₄	—	2	2	no	—
752	C₄⋊C₄	—	2	2	no	—
753	C₄⋊C₄	—	2	2	no	—
754	C₄⋊C₄	—	2	2	no	—
755	C₄⋊C₄	—	2	1	no	—
756	C₄⋊C₄	—	2	1	no	—
757	C₄⋊C₄	—	2	1	no	—
758	C₄⋊C₄	—	2	4	no	—
759	C₄⋊C₄	—	2	4	no	—
760	C₄⋊C₄	—	2	2	no	—
761	C₄⋊C₄	—	2	2	no	—
762	C₄⋊C₄	—	2	4	no	—
763	C₄⋊C₄	—	2	4	no	—
764	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
765	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
766	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
767	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
768	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
769	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
770	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
771	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
772	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
773	C₂²⋊C₄	—	2	2	no	—
774	C₂²⋊C₄	—	2	1	no	—
775	C₂²⋊C₄	—	2	2	no	—
776	C₂²⋊C₄	—	2	4	no	—
777	C₂²⋊C₄	—	2	4	no	—
778	C₂²⋊C₄	—	2	2	no	—
779	C₂²⋊C₄	—	2	1	no	—
780	C₂²⋊C₄	—	2	2	no	—
781	C₂²⋊C₄	—	2	2	no	—
782	C₂²⋊C₄	—	2	2	no	—
783	C₂²⋊C₄	—	2	2	no	—
784	C₂²⋊C₄	—	2	1	no	—
785	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
786	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
787	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
788	C₄²	—	2	1	no	—
789	C₄²	—	2	2	no	—
790	C₄²	—	2	1	no	—
791	C₄²	—	2	2	no	—
792	C₄²	—	2	1	no	—
793	C₄²	—	2	4	no	—
794	C₄²	—	2	4	no	—
795	C₄²	—	2	2	no	—
796	C₄²	C₂×D₄×C₃₁	1	2	yes	—
797	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
798	C₄²	C₂×D₄×C₃₁	1	2	yes	—
799	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
800	C₄²	C₂⁴×C₃₁	1	2	yes	—
801	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
802	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
803	C₂³	—	2	8	no	—
804	C₂³	—	2	2	no	—
805	C₂³	—	2	1	no	—
806	C₂³	—	2	4	no	—
807	C₂³	—	8	1	no	—
808	C₂³	—	8	2	no	—
809	C₂³	—	8	1	no	—
810	C₂³	—	8	2	no	—
811	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
812	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
813	Q₈	—	2	2	no	—
814	Q₈	—	2	1	no	—
815	Q₈	—	2	4	no	—
816	Q₈	—	2	4	no	—
817	Q₈	—	4	2	no	—
818	Q₈	—	4	4	no	—
819	Q₈	—	4	2	no	—
820	Q₈	C₂×#41×C₃₁	1	2	yes	—
821	D₄	—	2	8	no	—
822	D₄	—	2	8	no	—
823	D₄	—	2	4	no	—
824	D₄	—	2	4	no	—
825	D₄	—	2	2	no	—
826	D₄	—	2	4	no	—
827	D₄	—	2	2	no	—
828	D₄	—	2	1	no	—
829	D₄	C₂×#41×C₃₁	1	2	yes	—
830	D₄	C₂×#41×C₃₁	1	4	yes	—
831	C₂×C₄	—	2	8	no	—
832	C₂×C₄	—	2	1	no	—
833	C₂×C₄	—	2	4	no	—
834	C₂×C₄	—	2	2	no	—
835	C₂×C₄	—	2	4	no	—
836	C₂×C₄	—	2	2	no	—
837	C₂×C₄	—	2	2	no	—
838	C₂×C₄	—	2	1	no	—
839	C₂×C₄	—	2	2	no	—
840	C₂×C₄	—	2	2	no	—
841	C₂×C₄	—	2	2	no	—
842	C₂×C₄	—	2	1	no	—
843	C₂×C₄	—	2	4	no	—
844	C₂×C₄	—	4	1	no	—
845	C₂×C₄	—	4	2	no	—
846	C₂×C₄	—	4	4	no	—
847	C₂×C₄	—	4	4	no	—
848	C₂×C₄	—	4	2	no	—
849	C₂×C₄	—	8	2	no	—
850	C₂×C₄	—	8	1	no	—
851	C₂×C₄	—	8	2	no	—
852	C₂×C₄	—	8	2	no	—
853	C₂×C₄	—	8	2	no	—
854	C₂×C₄	—	8	1	no	—
855	C₂×C₄	—	8	2	no	—
856	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
857	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
858	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
859	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
860	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
861	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
862	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
863	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
864	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
865	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
866	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
867	C₈	—	2	2	no	—
868	C₈	—	2	2	no	—
869	C₈	—	2	4	no	—
870	C₂²	—	2	1	no	—
871	C₂²	—	2	1	no	—
872	C₂²	—	2	2	no	—
873	C₂²	—	2	4	no	—
874	C₂²	—	2	2	no	—
875	C₂²	—	2	2	no	—
876	C₂²	—	2	2	no	—
877	C₂²	—	4	2	no	—
878	C₂²	—	4	1	no	—
879	C₂²	—	4	2	no	—
880	C₂²	—	4	1	no	—
881	C₂²	—	8	4	no	—
882	C₂²	—	8	8	no	—
883	C₂²	C₃₁×#525	1	2	yes	—
884	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
885	C₄	—	2	2	no	—
886	C₄	—	2	2	no	—
887	C₄	—	2	2	no	—
888	C₄	—	2	1	no	—
889	C₄	—	2	1	no	—
890	C₄	—	2	2	no	—
891	C₄	—	2	1	no	—
892	C₄	—	4	4	no	—
893	C₄	—	4	2	no	—
894	C₄	—	4	1	no	—
895	C₄	—	4	2	no	—
896	C₄	—	4	1	no	—
897	C₄	—	4	2	no	—
898	C₄	C₃₁×#525	1	2	yes	—
899	C₄	C₃₁×#577	1	2	yes	—
900	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
901	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
902	C₂	—	2	2	no	—
903	C₂	—	4	2	no	—
904	C₂	—	4	4	no	—
905	C₂	—	4	2	no	—
906	C₂	C₂×C₃₁×#525	1	1	yes	characteristic
907	C₂	C₂×C₃₁×#577	1	1	yes	characteristic
908	C₂	C₃₁×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂×C₃₁×H(5,𝔽₂)	C₂²
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×#3509	C₂	semidirect
3. C₃₁×#3123	C₂	semidirect
4. C₃₁×#3120	C₂	semidirect
5. C₃₁×#3116	C₂	semidirect
6. C₃₁×#3110	C₂	semidirect
7. C₃₁×#3107	C₂	semidirect
8. C₃₁×#3101	C₂	semidirect
9. C₃₁×#3097	C₂	semidirect
10. C₃₁×#3077	C₂	semidirect
11. C₃₁×#3076	C₂	semidirect
12. C₃₁×#3039	C₂	semidirect
13. C₂×C₃₁×#461	C₂	semidirect
15. C₃₁×#2949	C₂	semidirect
16. C₃₁×#2947	C₂	semidirect
17. C₃₁×#2925	C₂	semidirect
27. C₃₁×#541	C₂²	semidirect
29. C₃₁×#533	C₂²	semidirect
30. C₃₁×#533	C₂²	semidirect
31. C₃₁×#533	C₂²	semidirect
32. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
37. C₃₁×#519	C₂²	semidirect
50. C₃₁×#485	C₂²	semidirect
51. C₃₁×#485	C₂²	semidirect
52. C₃₁×#484	C₂²	semidirect
53. C₃₁×#483	C₂²	semidirect
54. C₃₁×#479	C₂²	semidirect
55. C₃₁×#478	C₂²	semidirect
56. C₃₁×#478	C₂²	semidirect
57. C₃₁×#477	C₂²	semidirect
58. C₃₁×#476	C₂²	semidirect
59. C₃₁×#476	C₂²	semidirect
60. C₃₁×#475	C₂²	semidirect
61. C₃₁×#474	C₂²	semidirect
62. C₃₁×#474	C₂²	semidirect
63. C₃₁×#462	C₂²	semidirect
64. C₃₁×#462	C₂²	semidirect
65. C₃₁×#461	C₂²	semidirect
66. C₃₁×#461	C₂²	semidirect
67. C₃₁×#461	C₂²	semidirect
68. C₃₁×#461	C₂²	semidirect
69. C₃₁×#461	C₂²	semidirect
74. C₃₁×#441	C₂²	semidirect
75. C₃₁×#440	C₂²	semidirect
76. C₃₁×#440	C₂²	semidirect
77. C₃₁×#440	C₂²	semidirect
78. C₄×Q₁₆×C₃₁	C₂²	semidirect
79. C₄×SD₁₆×C₃₁	C₂²	semidirect
80. C₄×SD₁₆×C₃₁	C₂²	semidirect
81. C₄×SD₁₆×C₃₁	C₂²	semidirect
82. C₃₁×#433	C₂²	semidirect
83. C₃₁×#432	C₂²	semidirect
84. C₃₁×#431	C₂²	semidirect
85. C₃₁×#428	C₂²	semidirect
86. C₃₁×#427	C₂²	semidirect
87. C₃₁×#423	C₂²	semidirect
88. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
89. C₃₁×#419	C₂²	semidirect
92. C₂×C₃₁×#103	C₂²	semidirect
94. C₃₁×#405	C₂²	semidirect
348. #60591	C₃₁	direct (N × H)
416. #3509	C₂×C₃₁	semidirect
418. #3123	C₂×C₃₁	semidirect
419. #3120	C₂×C₃₁	semidirect
420. #3116	C₂×C₃₁	semidirect
421. #3110	C₂×C₃₁	semidirect
422. #3107	C₂×C₃₁	semidirect
423. #3101	C₂×C₃₁	semidirect
424. #3097	C₂×C₃₁	semidirect
425. #3077	C₂×C₃₁	semidirect
426. #3076	C₂×C₃₁	semidirect
427. #3039	C₂×C₃₁	semidirect
428. C₂×#461	C₂×C₃₁	semidirect
430. #2949	C₂×C₃₁	semidirect
431. #2947	C₂×C₃₁	semidirect
432. #2925	C₂×C₃₁	semidirect
474. #541	C₂²×C₃₁	semidirect
476. #533	C₂²×C₃₁	semidirect
477. #533	C₂²×C₃₁	semidirect
478. #533	C₂²×C₃₁	semidirect
479. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
484. #519	C₂²×C₃₁	semidirect
497. #485	C₂²×C₃₁	semidirect
498. #485	C₂²×C₃₁	semidirect
499. #484	C₂²×C₃₁	semidirect
500. #483	C₂²×C₃₁	semidirect
501. #479	C₂²×C₃₁	semidirect
502. #478	C₂²×C₃₁	semidirect
503. #478	C₂²×C₃₁	semidirect
504. #477	C₂²×C₃₁	semidirect
505. #476	C₂²×C₃₁	semidirect
506. #476	C₂²×C₃₁	semidirect
507. #475	C₂²×C₃₁	semidirect
508. #474	C₂²×C₃₁	semidirect
509. #474	C₂²×C₃₁	semidirect
510. #462	C₂²×C₃₁	semidirect
511. #462	C₂²×C₃₁	semidirect
512. #461	C₂²×C₃₁	semidirect
513. #461	C₂²×C₃₁	semidirect
514. #461	C₂²×C₃₁	semidirect
515. #461	C₂²×C₃₁	semidirect
516. #461	C₂²×C₃₁	semidirect
521. #441	C₂²×C₃₁	semidirect
522. #440	C₂²×C₃₁	semidirect
523. #440	C₂²×C₃₁	semidirect
524. #440	C₂²×C₃₁	semidirect
525. C₄×Q₁₆	C₂²×C₃₁	semidirect
526. C₄×SD₁₆	C₂²×C₃₁	semidirect
527. C₄×SD₁₆	C₂²×C₃₁	semidirect
528. C₄×SD₁₆	C₂²×C₃₁	semidirect
529. #433	C₂²×C₃₁	semidirect
530. #432	C₂²×C₃₁	semidirect
531. #431	C₂²×C₃₁	semidirect
532. #428	C₂²×C₃₁	semidirect
533. #427	C₂²×C₃₁	semidirect
534. #423	C₂²×C₃₁	semidirect
535. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
536. #419	C₂²×C₃₁	semidirect
539. C₂×#103	C₂²×C₃₁	semidirect
541. #405	C₂²×C₃₁	semidirect
686. C₃₁	#60591	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×#3509	2
C₄×C₃₁×H(5,𝔽₂)	2
C₃₁×#3123	2
C₃₁×#3120	2
C₃₁×#3116	2
C₃₁×#3110	2
C₃₁×#3107	2
C₃₁×#3101	2
C₃₁×#3097	2
C₃₁×#3077	2
C₃₁×#3076	2
C₃₁×#3039	2
C₂×C₃₁×#461	2
C₃₁×#3009	2
C₃₁×#2949	2
C₃₁×#2947	2
C₃₁×#2925	2
#60591	31

Subgroups — C31×#60591 #21517350

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60591 #21517350