Subgroups — C₃₁×#18702 #21475461

Order 7936.

868 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3550	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3255	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3254	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#3160	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1652	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1651	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1650	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1649	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1561	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1493	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1492	C₂	1	1	yes	maximal, characteristic
12	C₃₁×#563	—	2	2	no	—
13	C₃₁×#559	—	2	1	no	—
14	Q₈²×C₃₁	—	2	1	no	—
15	C₃₁×#556	—	2	1	no	—
16	C₃₁×#556	C₂²	1	2	yes	—
17	D₄×Q₈×C₃₁	—	2	1	no	—
18	D₄×Q₈×C₃₁	C₂²	1	2	yes	—
19	C₃₁×#542	—	2	2	no	—
20	C₃₁×#542	—	2	2	no	—
21	C₃₁×#536	—	2	2	no	—
22	C₃₁×#533	C₂²	1	1	yes	characteristic
23	C₃₁×#533	C₂²	1	1	yes	characteristic
24	C₃₁×#533	C₂²	1	1	yes	characteristic
25	C₃₁×#532	C₂²	1	1	yes	characteristic
26	C₃₁×#500	C₂²	1	1	yes	characteristic
27	C₃₁×#500	C₂²	1	1	yes	characteristic
28	C₃₁×#499	—	2	2	no	—
29	C₃₁×#498	C₂²	1	1	yes	characteristic
30	C₃₁×#498	C₂²	1	1	yes	characteristic
31	C₃₁×#477	—	2	1	no	—
32	C₃₁×#475	—	2	1	no	—
33	C₃₁×#475	C₂²	1	2	yes	—
34	C₃₁×#474	C₂²	1	2	yes	—
35	C₃₁×#470	—	2	2	no	—
36	C₃₁×#468	—	2	2	no	—
37	C₃₁×#462	—	2	1	no	—
38	C₃₁×#462	C₂²	1	2	yes	—
39	C₃₁×#461	C₂²	1	2	yes	—
40	C₃₁×#460	—	2	1	no	—
41	C₃₁×#427	—	2	2	no	—
42	C₃₁×#423	C₂²	1	1	yes	characteristic
43	C₃₁×#423	C₂²	1	1	yes	characteristic
44	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
45	C₃₁×#332	C₂²	1	2	yes	—
46	C₃₁×#332	C₂²	1	2	yes	—
47	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
48	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
49	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
50	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
51	C₂×#41×C₃₁	—	2	1	no	—
52	C₂²×Q₈×C₃₁	—	2	2	no	—
53	C₂×Q₁₆×C₃₁	—	2	2	no	—
54	C₂×SD₁₆×C₃₁	—	2	2	no	—
55	C₂×M₁₆×C₃₁	—	2	1	no	—
56	C₂×M₁₆×C₃₁	—	2	1	no	—
57	C₂²×C₈×C₃₁	—	2	1	no	—
58	C₃₁×C₄⋊Q₈	—	2	4	no	—
59	C₃₁×C₄⋊Q₈	—	2	2	no	—
60	C₃₁×C₄⋊Q₈	—	2	2	no	—
61	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
62	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
63	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
64	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
65	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
66	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
67	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
68	C₃₁×#127	D₄	1	1	yes	characteristic
69	C₃₁×#126	—	2	4	no	—
70	C₃₁×#126	—	2	4	no	—
71	C₃₁×#125	—	2	1	no	—
72	C₃₁×#125	—	2	1	no	—
73	C₃₁×#125	D₄	1	2	yes	—
74	C₃₁×#124	—	2	4	no	—
75	C₃₁×#124	—	2	2	no	—
76	C₃₁×#124	—	2	2	no	—
77	C₃₁×#124	—	2	1	no	—
78	C₃₁×#123	—	2	4	no	—
79	C₃₁×#123	—	2	4	no	—
80	C₃₁×C₂²⋊Q₈	—	2	4	no	—
81	C₃₁×C₂²⋊Q₈	—	2	2	no	—
82	C₃₁×C₂²⋊Q₈	—	2	1	no	—
83	C₃₁×C₂²⋊Q₈	—	2	1	no	—
84	C₃₁×C₂²⋊Q₈	—	2	1	no	—
85	C₃₁×C₂²⋊Q₈	—	2	2	no	—
86	C₃₁×C₂²⋊Q₈	—	2	4	no	—
87	C₃₁×C₂²⋊Q₈	—	2	1	no	—
88	C₃₁×C₂²⋊Q₈	—	2	1	no	—
89	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
90	C₃₁×#121	—	2	1	no	—
91	C₃₁×#121	D₄	1	2	yes	—
92	C₄×Q₈×C₃₁	—	2	2	no	—
93	C₄×Q₈×C₃₁	—	2	2	no	—
94	C₄×Q₈×C₃₁	—	2	1	no	—
95	C₄×Q₈×C₃₁	—	2	1	no	—
96	C₄×Q₈×C₃₁	—	2	1	no	—
97	C₄×Q₈×C₃₁	—	2	1	no	—
98	C₄×Q₈×C₃₁	—	2	1	no	—
99	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
100	C₄×D₄×C₃₁	—	2	2	no	—
101	C₄×D₄×C₃₁	—	2	2	no	—
102	C₄×D₄×C₃₁	—	2	1	no	—
103	C₄×D₄×C₃₁	C₂³	1	2	yes	—
104	C₃₁×#117	—	2	1	no	—
105	C₃₁×#117	—	2	2	no	—
106	C₃₁×#117	—	2	1	no	—
107	C₃₁×#117	D₄	1	2	yes	—
108	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
109	C₃₁×#107	—	2	2	no	—
110	C₃₁×#107	—	2	2	no	—
111	C₃₁×#107	—	2	2	no	—
112	C₃₁×#106	—	2	2	no	—
113	C₃₁×#106	—	2	2	no	—
114	C₃₁×#106	—	2	2	no	—
115	C₃₁×C₄⋊C₈	—	2	1	no	—
116	C₃₁×C₄⋊C₈	—	2	1	no	—
117	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
118	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
119	C₃₁×#103	—	2	2	no	—
120	C₃₁×#103	—	2	2	no	—
121	C₃₁×#103	—	2	2	no	—
122	C₃₁×#102	—	2	2	no	—
123	C₄×C₈×C₃₁	—	2	1	no	—
124	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
125	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
126	#41×C₃₁	—	8	1	no	—
127	#41×C₃₁	—	8	1	no	—
128	C₂×Q₈×C₃₁	—	2	4	no	—
129	C₂×Q₈×C₃₁	—	2	1	no	—
130	C₂×Q₈×C₃₁	—	2	1	no	—
131	C₂×Q₈×C₃₁	—	2	1	no	—
132	C₂×Q₈×C₃₁	—	2	1	no	—
133	C₂×Q₈×C₃₁	—	2	1	no	—
134	C₂×Q₈×C₃₁	—	2	1	no	—
135	C₂×Q₈×C₃₁	—	8	2	no	—
136	C₂×Q₈×C₃₁	—	8	2	no	—
137	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
138	C₂×D₄×C₃₁	—	2	2	no	—
139	C₂×D₄×C₃₁	—	2	1	no	—
140	C₂×D₄×C₃₁	—	2	1	no	—
141	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
142	C₂²×C₄×C₃₁	—	2	4	no	—
143	C₂²×C₄×C₃₁	—	2	2	no	—
144	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
147	Q₁₆×C₃₁	—	8	2	no	—
148	SD₁₆×C₃₁	—	8	2	no	—
149	M₁₆×C₃₁	—	8	1	no	—
150	M₁₆×C₃₁	—	8	1	no	—
151	C₂×C₈×C₃₁	—	2	4	no	—
152	C₂×C₈×C₃₁	—	2	1	no	—
153	C₂×C₈×C₃₁	—	2	1	no	—
154	C₂×C₈×C₃₁	—	2	2	no	—
155	C₂×C₈×C₃₁	—	2	2	no	—
156	C₄⋊C₄×C₃₁	—	2	4	no	—
157	C₄⋊C₄×C₃₁	—	2	4	no	—
158	C₄⋊C₄×C₃₁	—	2	1	no	—
159	C₄⋊C₄×C₃₁	—	2	1	no	—
160	C₄⋊C₄×C₃₁	—	2	1	no	—
161	C₄⋊C₄×C₃₁	—	2	1	no	—
162	C₄⋊C₄×C₃₁	—	2	2	no	—
163	C₄⋊C₄×C₃₁	—	2	2	no	—
164	C₄⋊C₄×C₃₁	—	2	2	no	—
165	C₄⋊C₄×C₃₁	—	2	1	no	—
166	C₄⋊C₄×C₃₁	—	2	1	no	—
167	C₄⋊C₄×C₃₁	—	2	2	no	—
168	C₄⋊C₄×C₃₁	—	2	1	no	—
169	C₄⋊C₄×C₃₁	—	2	1	no	—
170	C₄⋊C₄×C₃₁	—	2	1	no	—
171	C₄⋊C₄×C₃₁	—	2	1	no	—
172	C₄⋊C₄×C₃₁	—	4	1	no	—
173	C₄⋊C₄×C₃₁	—	4	1	no	—
174	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
175	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
176	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
177	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
178	C₂²⋊C₄×C₃₁	—	2	4	no	—
179	C₂²⋊C₄×C₃₁	—	2	4	no	—
180	C₂²⋊C₄×C₃₁	—	2	4	no	—
181	C₂²⋊C₄×C₃₁	—	2	2	no	—
182	C₂²⋊C₄×C₃₁	—	2	2	no	—
183	C₂²⋊C₄×C₃₁	—	2	2	no	—
184	C₂²⋊C₄×C₃₁	—	2	2	no	—
185	C₂²⋊C₄×C₃₁	—	4	1	no	—
186	C₄²×C₃₁	—	2	4	no	—
187	C₄²×C₃₁	—	2	2	no	—
188	C₄²×C₃₁	—	2	2	no	—
189	C₄²×C₃₁	—	4	1	no	—
190	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
191	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
192	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
193	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
194	#18702	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
195	C₂³×C₃₁	—	2	2	no	—
196	C₂³×C₃₁	#120	1	1	yes	characteristic
197	Q₈×C₃₁	—	2	2	no	—
198	Q₈×C₃₁	—	2	2	no	—
199	Q₈×C₃₁	—	4	1	no	—
200	Q₈×C₃₁	—	4	1	no	—
201	Q₈×C₃₁	—	4	1	no	—
202	Q₈×C₃₁	—	4	1	no	—
203	Q₈×C₃₁	—	4	2	no	—
204	Q₈×C₃₁	—	4	1	no	—
205	Q₈×C₃₁	—	4	1	no	—
206	Q₈×C₃₁	—	4	1	no	—
207	Q₈×C₃₁	—	4	1	no	—
208	Q₈×C₃₁	—	8	1	no	—
209	Q₈×C₃₁	—	8	4	no	—
210	D₄×C₃₁	—	2	2	no	—
211	D₄×C₃₁	—	2	2	no	—
212	D₄×C₃₁	—	4	2	no	—
213	D₄×C₃₁	—	8	1	no	—
214	D₄×C₃₁	—	8	2	no	—
215	C₂×C₄×C₃₁	—	2	1	no	—
216	C₂×C₄×C₃₁	—	2	1	no	—
217	C₂×C₄×C₃₁	—	2	1	no	—
218	C₂×C₄×C₃₁	—	2	1	no	—
219	C₂×C₄×C₃₁	—	2	2	no	—
220	C₂×C₄×C₃₁	—	2	2	no	—
221	C₂×C₄×C₃₁	—	2	2	no	—
222	C₂×C₄×C₃₁	—	2	2	no	—
223	C₂×C₄×C₃₁	—	2	2	no	—
224	C₂×C₄×C₃₁	—	2	2	no	—
225	C₂×C₄×C₃₁	—	4	2	no	—
226	C₂×C₄×C₃₁	—	4	2	no	—
227	C₂×C₄×C₃₁	—	4	1	no	—
228	C₂×C₄×C₃₁	—	4	1	no	—
229	C₂×C₄×C₃₁	—	8	4	no	—
230	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
231	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
232	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
233	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
234	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
235	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
236	C₈×C₃₁	—	2	2	no	—
237	C₈×C₃₁	—	2	2	no	—
238	C₈×C₃₁	—	4	2	no	—
239	C₈×C₃₁	—	4	2	no	—
240	#3550	C₂×C₃₁	1	1	yes	characteristic
241	#3255	C₂×C₃₁	1	1	yes	characteristic, complemented
242	#3254	C₂×C₃₁	1	1	yes	characteristic
243	#3160	C₂×C₃₁	1	1	yes	characteristic, complemented
244	#1652	C₂×C₃₁	1	2	yes	complemented
245	#1651	C₂×C₃₁	1	2	yes	complemented
246	#1650	C₂×C₃₁	1	2	yes	complemented
247	#1649	C₂×C₃₁	1	2	yes	complemented
248	#1561	C₂×C₃₁	1	1	yes	characteristic, complemented
249	#1493	C₂×C₃₁	1	1	yes	characteristic, complemented
250	#1492	C₂×C₃₁	1	1	yes	characteristic
251	C₂²×C₃₁	—	2	1	no	—
252	C₂²×C₃₁	—	2	1	no	—
253	C₂²×C₃₁	—	2	1	no	—
254	C₂²×C₃₁	—	4	2	no	—
255	C₂²×C₃₁	—	4	2	no	—
256	C₂²×C₃₁	—	4	2	no	—
257	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
258	C₄×C₃₁	—	2	1	no	—
259	C₄×C₃₁	—	2	1	no	—
260	C₄×C₃₁	—	2	1	no	—
261	C₄×C₃₁	—	4	2	no	—
262	C₄×C₃₁	—	4	2	no	—
263	C₄×C₃₁	—	4	2	no	—
264	C₄×C₃₁	—	4	2	no	—
265	C₄×C₃₁	—	4	2	no	—
266	C₄×C₃₁	—	8	1	no	—
267	C₄×C₃₁	—	8	1	no	—
268	C₄×C₃₁	#456	1	2	yes	—
269	C₄×C₃₁	#576	1	1	yes	characteristic
270	C₄×C₃₁	#577	1	1	yes	characteristic
271	C₄×C₃₁	#577	1	1	yes	characteristic
272	C₄×C₃₁	#578	1	1	yes	characteristic
273	#563	—	2	2	no	—
274	#559	—	2	1	no	—
275	Q₈²	—	2	1	no	—
276	#556	—	2	1	no	—
277	#556	C₂²×C₃₁	1	2	yes	—
278	D₄×Q₈	—	2	1	no	—
279	D₄×Q₈	C₂²×C₃₁	1	2	yes	—
280	#542	—	2	2	no	—
281	#542	—	2	2	no	—
282	#536	—	2	2	no	—
283	#533	C₂²×C₃₁	1	1	yes	characteristic
284	#533	C₂²×C₃₁	1	1	yes	characteristic
285	#533	C₂²×C₃₁	1	1	yes	characteristic
286	#532	C₂²×C₃₁	1	1	yes	characteristic
287	#500	C₂²×C₃₁	1	1	yes	characteristic
288	#500	C₂²×C₃₁	1	1	yes	characteristic
289	#499	—	2	2	no	—
290	#498	C₂²×C₃₁	1	1	yes	characteristic
291	#498	C₂²×C₃₁	1	1	yes	characteristic
292	#477	—	2	1	no	—
293	#475	—	2	1	no	—
294	#475	C₂²×C₃₁	1	2	yes	—
295	#474	C₂²×C₃₁	1	2	yes	—
296	#470	—	2	2	no	—
297	#468	—	2	2	no	—
298	#462	—	2	1	no	—
299	#462	C₂²×C₃₁	1	2	yes	—
300	#461	C₂²×C₃₁	1	2	yes	—
301	#460	—	2	1	no	—
302	#427	—	2	2	no	—
303	#423	C₂²×C₃₁	1	1	yes	characteristic
304	#423	C₂²×C₃₁	1	1	yes	characteristic
305	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
306	#332	C₂²×C₃₁	1	2	yes	—
307	#332	C₂²×C₃₁	1	2	yes	—
308	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
309	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
310	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
311	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
312	C₂×C₃₁	—	4	1	no	—
313	C₂×C₃₁	—	8	2	no	—
314	C₂×C₃₁	#3012	1	1	yes	characteristic
315	C₂×C₃₁	#3013	1	1	yes	characteristic
316	C₂×C₃₁	C₂×#456	1	1	yes	characteristic
317	C₂×#41	—	2	1	no	—
318	C₂²×Q₈	—	2	2	no	—
319	C₂×Q₁₆	—	2	2	no	—
320	C₂×SD₁₆	—	2	2	no	—
321	C₂×M₁₆	—	2	1	no	—
322	C₂×M₁₆	—	2	1	no	—
323	C₂²×C₈	—	2	1	no	—
324	C₄⋊Q₈	—	2	4	no	—
325	C₄⋊Q₈	—	2	2	no	—
326	C₄⋊Q₈	—	2	2	no	—
327	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
328	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
329	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
330	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
331	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
332	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
333	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
334	#127	D₄×C₃₁	1	1	yes	characteristic
335	#126	—	2	4	no	—
336	#126	—	2	4	no	—
337	#125	—	2	1	no	—
338	#125	—	2	1	no	—
339	#125	D₄×C₃₁	1	2	yes	—
340	#124	—	2	4	no	—
341	#124	—	2	2	no	—
342	#124	—	2	2	no	—
343	#124	—	2	1	no	—
344	#123	—	2	4	no	—
345	#123	—	2	4	no	—
346	C₂²⋊Q₈	—	2	4	no	—
347	C₂²⋊Q₈	—	2	2	no	—
348	C₂²⋊Q₈	—	2	1	no	—
349	C₂²⋊Q₈	—	2	1	no	—
350	C₂²⋊Q₈	—	2	1	no	—
351	C₂²⋊Q₈	—	2	2	no	—
352	C₂²⋊Q₈	—	2	4	no	—
353	C₂²⋊Q₈	—	2	1	no	—
354	C₂²⋊Q₈	—	2	1	no	—
355	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
356	#121	—	2	1	no	—
357	#121	D₄×C₃₁	1	2	yes	—
358	C₄×Q₈	—	2	2	no	—
359	C₄×Q₈	—	2	2	no	—
360	C₄×Q₈	—	2	1	no	—
361	C₄×Q₈	—	2	1	no	—
362	C₄×Q₈	—	2	1	no	—
363	C₄×Q₈	—	2	1	no	—
364	C₄×Q₈	—	2	1	no	—
365	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
366	C₄×D₄	—	2	2	no	—
367	C₄×D₄	—	2	2	no	—
368	C₄×D₄	—	2	1	no	—
369	C₄×D₄	C₂³×C₃₁	1	2	yes	—
370	#117	—	2	1	no	—
371	#117	—	2	2	no	—
372	#117	—	2	1	no	—
373	#117	D₄×C₃₁	1	2	yes	—
374	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
375	#107	—	2	2	no	—
376	#107	—	2	2	no	—
377	#107	—	2	2	no	—
378	#106	—	2	2	no	—
379	#106	—	2	2	no	—
380	#106	—	2	2	no	—
381	C₄⋊C₈	—	2	1	no	—
382	C₄⋊C₈	—	2	1	no	—
383	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
384	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
385	#103	—	2	2	no	—
386	#103	—	2	2	no	—
387	#103	—	2	2	no	—
388	#102	—	2	2	no	—
389	C₄×C₈	—	2	1	no	—
390	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
391	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
392	C₃₁	#18702	1	1	yes	characteristic, direct product factor, Sylow
393	#41	—	8	1	no	—
394	#41	—	8	1	no	—
395	C₂×Q₈	—	2	4	no	—
396	C₂×Q₈	—	2	1	no	—
397	C₂×Q₈	—	2	1	no	—
398	C₂×Q₈	—	2	1	no	—
399	C₂×Q₈	—	2	1	no	—
400	C₂×Q₈	—	2	1	no	—
401	C₂×Q₈	—	2	1	no	—
402	C₂×Q₈	—	8	2	no	—
403	C₂×Q₈	—	8	2	no	—
404	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
405	C₂×D₄	—	2	2	no	—
406	C₂×D₄	—	2	1	no	—
407	C₂×D₄	—	2	1	no	—
408	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
409	C₂²×C₄	—	2	4	no	—
410	C₂²×C₄	—	2	2	no	—
411	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
412	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
413	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
414	Q₁₆	—	8	2	no	—
415	SD₁₆	—	8	2	no	—
416	M₁₆	—	8	1	no	—
417	M₁₆	—	8	1	no	—
418	C₂×C₈	—	2	4	no	—
419	C₂×C₈	—	2	1	no	—
420	C₂×C₈	—	2	1	no	—
421	C₂×C₈	—	2	2	no	—
422	C₂×C₈	—	2	2	no	—
423	C₄⋊C₄	—	2	4	no	—
424	C₄⋊C₄	—	2	4	no	—
425	C₄⋊C₄	—	2	1	no	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	—	2	1	no	—
429	C₄⋊C₄	—	2	2	no	—
430	C₄⋊C₄	—	2	2	no	—
431	C₄⋊C₄	—	2	2	no	—
432	C₄⋊C₄	—	2	1	no	—
433	C₄⋊C₄	—	2	1	no	—
434	C₄⋊C₄	—	2	2	no	—
435	C₄⋊C₄	—	2	1	no	—
436	C₄⋊C₄	—	2	1	no	—
437	C₄⋊C₄	—	2	1	no	—
438	C₄⋊C₄	—	2	1	no	—
439	C₄⋊C₄	—	4	1	no	—
440	C₄⋊C₄	—	4	1	no	—
441	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
442	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
443	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
444	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
445	C₂²⋊C₄	—	2	4	no	—
446	C₂²⋊C₄	—	2	4	no	—
447	C₂²⋊C₄	—	2	4	no	—
448	C₂²⋊C₄	—	2	2	no	—
449	C₂²⋊C₄	—	2	2	no	—
450	C₂²⋊C₄	—	2	2	no	—
451	C₂²⋊C₄	—	2	2	no	—
452	C₂²⋊C₄	—	4	1	no	—
453	C₄²	—	2	4	no	—
454	C₄²	—	2	2	no	—
455	C₄²	—	2	2	no	—
456	C₄²	—	4	1	no	—
457	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
458	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
459	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
460	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
461	C₂³	—	2	2	no	—
462	C₂³	C₃₁×#120	1	1	yes	characteristic
463	Q₈	—	2	2	no	—
464	Q₈	—	2	2	no	—
465	Q₈	—	4	2	no	—
466	Q₈	—	4	1	no	—
467	Q₈	—	4	1	no	—
468	Q₈	—	4	1	no	—
469	Q₈	—	4	1	no	—
470	Q₈	—	4	1	no	—
471	Q₈	—	4	1	no	—
472	Q₈	—	4	1	no	—
473	Q₈	—	4	1	no	—
474	Q₈	—	8	1	no	—
475	Q₈	—	8	4	no	—
476	D₄	—	2	2	no	—
477	D₄	—	2	2	no	—
478	D₄	—	4	2	no	—
479	D₄	—	8	1	no	—
480	D₄	—	8	2	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	1	no	—
483	C₂×C₄	—	2	1	no	—
484	C₂×C₄	—	2	1	no	—
485	C₂×C₄	—	2	1	no	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	4	2	no	—
492	C₂×C₄	—	4	2	no	—
493	C₂×C₄	—	4	1	no	—
494	C₂×C₄	—	4	1	no	—
495	C₂×C₄	—	8	4	no	—
496	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
497	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
498	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
499	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
500	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
501	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
502	C₈	—	2	2	no	—
503	C₈	—	2	2	no	—
504	C₈	—	4	2	no	—
505	C₈	—	4	2	no	—
506	C₂²	—	2	1	no	—
507	C₂²	—	2	1	no	—
508	C₂²	—	2	1	no	—
509	C₂²	—	4	2	no	—
510	C₂²	—	4	2	no	—
511	C₂²	—	4	2	no	—
512	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
513	C₄	—	2	1	no	—
514	C₄	—	2	1	no	—
515	C₄	—	2	1	no	—
516	C₄	—	4	2	no	—
517	C₄	—	4	2	no	—
518	C₄	—	4	2	no	—
519	C₄	—	4	2	no	—
520	C₄	—	4	2	no	—
521	C₄	—	8	1	no	—
522	C₄	—	8	1	no	—
523	C₄	C₃₁×#456	1	2	yes	—
524	C₄	C₃₁×#576	1	1	yes	characteristic
525	C₄	C₃₁×#577	1	1	yes	characteristic
526	C₄	C₃₁×#577	1	1	yes	characteristic
527	C₄	C₃₁×#578	1	1	yes	characteristic
528	C₂	—	4	1	no	—
529	C₂	—	8	2	no	—
530	C₂	C₃₁×#3012	1	1	yes	characteristic
531	C₂	C₃₁×#3013	1	1	yes	characteristic
532	C₂	C₂×C₃₁×#456	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#3255	C₂	semidirect
4. C₃₁×#3160	C₂	semidirect
5. C₃₁×#1652	C₂	semidirect
6. C₃₁×#1651	C₂	semidirect
7. C₃₁×#1650	C₂	semidirect
8. C₃₁×#1649	C₂	semidirect
9. C₃₁×#1561	C₂	semidirect
10. C₃₁×#1493	C₂	semidirect
194. #18702	C₃₁	direct (N × H)
241. #3255	C₂×C₃₁	semidirect
243. #3160	C₂×C₃₁	semidirect
244. #1652	C₂×C₃₁	semidirect
245. #1651	C₂×C₃₁	semidirect
246. #1650	C₂×C₃₁	semidirect
247. #1649	C₂×C₃₁	semidirect
248. #1561	C₂×C₃₁	semidirect
249. #1493	C₂×C₃₁	semidirect
392. C₃₁	#18702	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×#3550	2
C₃₁×#3255	2
C₃₁×#3254	2
C₃₁×#3160	2
C₃₁×#1652	2
C₃₁×#1651	2
C₃₁×#1650	2
C₃₁×#1649	2
C₃₁×#1561	2
C₃₁×#1493	2
C₃₁×#1492	2
#18702	31

Subgroups — C31×#18702 #21475461

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#18702 #21475461