Subgroups — C₃₁×#19361 #21476120

Order 7936.

570 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3252	C₂	1	2	yes	maximal
2	C₃₁×#3160	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2966	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1692	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1691	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1676	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1675	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1674	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1673	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1567	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1559	C₂	1	2	yes	maximal, complemented
12	C₃₁×#533	C₂²	1	2	yes	—
13	C₃₁×#529	C₂²	1	1	yes	characteristic
14	C₃₁×#500	C₂²	1	1	yes	characteristic
15	C₃₁×#500	C₂²	1	1	yes	characteristic
16	C₃₁×#499	—	2	1	no	—
17	C₃₁×#499	—	2	1	no	—
18	C₃₁×#498	C₂²	1	2	yes	—
19	C₃₁×#479	—	2	2	no	—
20	C₃₁×#477	C₂²	1	2	yes	—
21	C₃₁×#476	C₂²	1	2	yes	—
22	C₃₁×#470	—	2	2	no	—
23	C₃₁×#468	—	2	2	no	—
24	C₃₁×#464	—	2	2	no	—
25	C₃₁×#462	C₂²	1	2	yes	—
26	C₃₁×#461	C₂²	1	2	yes	—
27	C₃₁×#446	C₂²	1	1	yes	characteristic
28	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#436	C₂²	1	1	yes	characteristic
30	C₃₁×#435	—	2	1	no	—
31	C₈×D₄×C₃₁	—	2	1	no	—
32	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#433	—	2	1	no	—
34	C₃₁×#432	C₂²	1	1	yes	characteristic
35	C₃₁×#427	—	2	1	no	—
36	C₃₁×#427	—	2	1	no	—
37	C₃₁×#423	C₂²	1	2	yes	—
38	C₃₁×#405	—	2	1	no	—
39	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
40	C₃₁×#335	C₂²	1	1	yes	characteristic
41	C₃₁×#335	C₂²	1	1	yes	characteristic
42	C₃₁×#335	C₂²	1	2	yes	—
43	C₃₁×#333	C₂²	1	2	yes	—
44	C₃₁×#333	C₂²	1	2	yes	—
45	C₃₁×#332	C₂²	1	2	yes	—
46	C₃₁×#332	C₂²	1	2	yes	—
47	C₂×Q₁₆×C₃₁	—	2	2	no	—
48	C₂×SD₁₆×C₃₁	—	2	2	no	—
49	C₂×M₁₆×C₃₁	—	2	2	no	—
50	C₂×M₁₆×C₃₁	—	2	1	no	—
51	C₂²×C₈×C₃₁	—	2	1	no	—
52	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
54	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
55	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
56	C₃₁×#126	—	2	1	no	—
57	C₃₁×#125	—	2	2	no	—
58	C₃₁×#125	D₄	1	1	yes	characteristic
59	C₃₁×#124	D₄	1	1	yes	characteristic
60	C₃₁×#123	—	2	1	no	—
61	C₃₁×C₂²⋊Q₈	—	2	2	no	—
62	C₃₁×C₂²⋊Q₈	—	2	2	no	—
63	C₃₁×C₂²⋊Q₈	D₄	1	1	yes	characteristic
64	C₃₁×#121	D₄	1	1	yes	characteristic
65	C₄×Q₈×C₃₁	—	2	2	no	—
66	C₄×Q₈×C₃₁	—	2	2	no	—
67	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
68	C₄×D₄×C₃₁	—	2	1	no	—
69	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₃₁×#117	—	2	1	no	—
71	C₃₁×#117	—	2	2	no	—
72	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
73	C₃₁×#107	—	2	1	no	—
74	C₃₁×#107	—	2	1	no	—
75	C₃₁×#107	—	2	1	no	—
76	C₃₁×#107	—	2	1	no	—
77	C₃₁×#107	—	2	2	no	—
78	C₃₁×#106	—	2	2	no	—
79	C₃₁×#106	—	2	2	no	—
80	C₃₁×#106	—	2	2	no	—
81	C₃₁×C₄⋊C₈	—	2	1	no	—
82	C₃₁×C₄⋊C₈	—	2	1	no	—
83	C₃₁×C₄⋊C₈	—	2	2	no	—
84	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
85	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
86	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
87	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
88	C₃₁×#103	—	2	2	no	—
89	C₃₁×#103	—	2	2	no	—
90	C₃₁×#103	—	2	2	no	—
91	C₃₁×#102	—	2	2	no	—
92	C₃₁×C₂²⋊C₈	—	2	1	no	—
93	C₃₁×C₂²⋊C₈	—	2	1	no	—
94	C₃₁×C₂²⋊C₈	—	2	1	no	—
95	C₃₁×#97	—	2	1	no	—
96	C₃₁×#97	—	2	1	no	—
97	C₄×C₈×C₃₁	—	2	1	no	—
98	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
99	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
100	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
101	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
102	C₂×Q₈×C₃₁	—	2	2	no	—
103	C₂×Q₈×C₃₁	—	2	2	no	—
104	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂×D₄×C₃₁	—	2	1	no	—
106	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	—	2	1	no	—
108	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
111	Q₁₆×C₃₁	—	8	2	no	—
112	SD₁₆×C₃₁	—	8	2	no	—
113	M₁₆×C₃₁	—	4	2	no	—
114	M₁₆×C₃₁	—	8	2	no	—
115	C₂×C₈×C₃₁	—	2	2	no	—
116	C₂×C₈×C₃₁	—	2	2	no	—
117	C₂×C₈×C₃₁	—	2	1	no	—
118	C₂×C₈×C₃₁	—	2	1	no	—
119	C₂×C₈×C₃₁	—	2	2	no	—
120	C₂×C₈×C₃₁	—	2	2	no	—
121	C₂×C₈×C₃₁	—	4	2	no	—
122	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
126	C₄⋊C₄×C₃₁	—	2	1	no	—
127	C₄⋊C₄×C₃₁	—	2	1	no	—
128	C₄⋊C₄×C₃₁	—	2	1	no	—
129	C₄⋊C₄×C₃₁	—	2	1	no	—
130	C₄⋊C₄×C₃₁	—	2	2	no	—
131	C₄⋊C₄×C₃₁	—	2	2	no	—
132	C₄⋊C₄×C₃₁	—	2	2	no	—
133	C₄⋊C₄×C₃₁	—	2	2	no	—
134	C₄⋊C₄×C₃₁	—	4	2	no	—
135	C₄⋊C₄×C₃₁	—	4	2	no	—
136	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
138	C₂²⋊C₄×C₃₁	—	2	1	no	—
139	C₂²⋊C₄×C₃₁	—	2	1	no	—
140	C₂²⋊C₄×C₃₁	—	2	1	no	—
141	C₂²⋊C₄×C₃₁	—	2	1	no	—
142	C₂²⋊C₄×C₃₁	—	2	1	no	—
143	C₂²⋊C₄×C₃₁	—	4	2	no	—
144	C₄²×C₃₁	—	2	1	no	—
145	C₄²×C₃₁	—	4	2	no	—
146	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
147	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
148	C₄²×C₃₁	#41	1	1	yes	characteristic
149	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
150	#19361	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂³×C₃₁	—	2	1	no	—
152	C₂³×C₃₁	#121	1	1	yes	characteristic
153	Q₈×C₃₁	—	2	1	no	—
154	Q₈×C₃₁	—	2	1	no	—
155	Q₈×C₃₁	—	4	2	no	—
156	Q₈×C₃₁	—	4	2	no	—
157	Q₈×C₃₁	—	8	2	no	—
158	D₄×C₃₁	—	2	1	no	—
159	D₄×C₃₁	—	2	1	no	—
160	D₄×C₃₁	—	4	2	no	—
161	C₂×C₄×C₃₁	—	2	1	no	—
162	C₂×C₄×C₃₁	—	2	1	no	—
163	C₂×C₄×C₃₁	—	2	1	no	—
164	C₂×C₄×C₃₁	—	2	1	no	—
165	C₂×C₄×C₃₁	—	2	1	no	—
166	C₂×C₄×C₃₁	—	2	1	no	—
167	C₂×C₄×C₃₁	—	2	1	no	—
168	C₂×C₄×C₃₁	—	4	1	no	—
169	C₂×C₄×C₃₁	—	4	1	no	—
170	C₂×C₄×C₃₁	—	4	1	no	—
171	C₂×C₄×C₃₁	—	4	2	no	—
172	C₂×C₄×C₃₁	—	4	2	no	—
173	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
174	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
175	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
176	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
179	C₈×C₃₁	—	2	1	no	—
180	C₈×C₃₁	—	2	1	no	—
181	C₈×C₃₁	—	4	1	no	—
182	C₈×C₃₁	—	4	1	no	—
183	C₈×C₃₁	—	4	2	no	—
184	C₈×C₃₁	—	4	2	no	—
185	C₈×C₃₁	#137	1	2	yes	—
186	C₈×C₃₁	#137	1	2	yes	—
187	#3252	C₂×C₃₁	1	2	yes	—
188	#3160	C₂×C₃₁	1	1	yes	characteristic, complemented
189	#2966	C₂×C₃₁	1	1	yes	characteristic
190	#1692	C₂×C₃₁	1	2	yes	complemented
191	#1691	C₂×C₃₁	1	2	yes	complemented
192	#1676	C₂×C₃₁	1	1	yes	characteristic, complemented
193	#1675	C₂×C₃₁	1	1	yes	characteristic, complemented
194	#1674	C₂×C₃₁	1	1	yes	characteristic, complemented
195	#1673	C₂×C₃₁	1	1	yes	characteristic, complemented
196	#1567	C₂×C₃₁	1	1	yes	characteristic, complemented
197	#1559	C₂×C₃₁	1	2	yes	complemented
198	C₂²×C₃₁	—	2	1	no	—
199	C₂²×C₃₁	—	2	1	no	—
200	C₂²×C₃₁	—	2	1	no	—
201	C₂²×C₃₁	—	4	1	no	—
202	C₂²×C₃₁	—	4	1	no	—
203	C₂²×C₃₁	—	4	1	no	—
204	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
205	C₄×C₃₁	—	2	1	no	—
206	C₄×C₃₁	—	2	1	no	—
207	C₄×C₃₁	—	2	1	no	—
208	C₄×C₃₁	—	2	1	no	—
209	C₄×C₃₁	—	4	1	no	—
210	C₄×C₃₁	—	4	1	no	—
211	C₄×C₃₁	—	4	1	no	—
212	C₄×C₃₁	—	8	2	no	—
213	C₄×C₃₁	—	8	2	no	—
214	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
215	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
216	C₄×C₃₁	#576	1	1	yes	characteristic
217	C₄×C₃₁	#578	1	1	yes	characteristic
218	#533	C₂²×C₃₁	1	2	yes	—
219	#529	C₂²×C₃₁	1	1	yes	characteristic
220	#500	C₂²×C₃₁	1	1	yes	characteristic
221	#500	C₂²×C₃₁	1	1	yes	characteristic
222	#499	—	2	1	no	—
223	#499	—	2	1	no	—
224	#498	C₂²×C₃₁	1	2	yes	—
225	#479	—	2	2	no	—
226	#477	C₂²×C₃₁	1	2	yes	—
227	#476	C₂²×C₃₁	1	2	yes	—
228	#470	—	2	2	no	—
229	#468	—	2	2	no	—
230	#464	—	2	2	no	—
231	#462	C₂²×C₃₁	1	2	yes	—
232	#461	C₂²×C₃₁	1	2	yes	—
233	#446	C₂²×C₃₁	1	1	yes	characteristic
234	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
235	#436	C₂²×C₃₁	1	1	yes	characteristic
236	#435	—	2	1	no	—
237	C₈×D₄	—	2	1	no	—
238	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
239	#433	—	2	1	no	—
240	#432	C₂²×C₃₁	1	1	yes	characteristic
241	#427	—	2	1	no	—
242	#427	—	2	1	no	—
243	#423	C₂²×C₃₁	1	2	yes	—
244	#405	—	2	1	no	—
245	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
246	#335	C₂²×C₃₁	1	1	yes	characteristic
247	#335	C₂²×C₃₁	1	1	yes	characteristic
248	#335	C₂²×C₃₁	1	2	yes	—
249	#333	C₂²×C₃₁	1	2	yes	—
250	#333	C₂²×C₃₁	1	2	yes	—
251	#332	C₂²×C₃₁	1	2	yes	—
252	#332	C₂²×C₃₁	1	2	yes	—
253	C₂×C₃₁	—	4	1	no	—
254	C₂×C₃₁	—	8	1	no	—
255	C₂×C₃₁	#3039	1	1	yes	characteristic
256	C₂×C₃₁	#3061	1	1	yes	characteristic
257	C₂×C₃₁	#3067	1	1	yes	characteristic
258	C₂×Q₁₆	—	2	2	no	—
259	C₂×SD₁₆	—	2	2	no	—
260	C₂×M₁₆	—	2	2	no	—
261	C₂×M₁₆	—	2	1	no	—
262	C₂²×C₈	—	2	1	no	—
263	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
264	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
265	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
266	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
267	#126	—	2	1	no	—
268	#125	—	2	2	no	—
269	#125	D₄×C₃₁	1	1	yes	characteristic
270	#124	D₄×C₃₁	1	1	yes	characteristic
271	#123	—	2	1	no	—
272	C₂²⋊Q₈	—	2	2	no	—
273	C₂²⋊Q₈	—	2	2	no	—
274	C₂²⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
275	#121	D₄×C₃₁	1	1	yes	characteristic
276	C₄×Q₈	—	2	2	no	—
277	C₄×Q₈	—	2	2	no	—
278	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
279	C₄×D₄	—	2	1	no	—
280	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
281	#117	—	2	1	no	—
282	#117	—	2	2	no	—
283	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
284	#107	—	2	1	no	—
285	#107	—	2	1	no	—
286	#107	—	2	1	no	—
287	#107	—	2	1	no	—
288	#107	—	2	2	no	—
289	#106	—	2	2	no	—
290	#106	—	2	2	no	—
291	#106	—	2	2	no	—
292	C₄⋊C₈	—	2	1	no	—
293	C₄⋊C₈	—	2	1	no	—
294	C₄⋊C₈	—	2	2	no	—
295	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
296	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
297	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
298	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
299	#103	—	2	2	no	—
300	#103	—	2	2	no	—
301	#103	—	2	2	no	—
302	#102	—	2	2	no	—
303	C₂²⋊C₈	—	2	1	no	—
304	C₂²⋊C₈	—	2	1	no	—
305	C₂²⋊C₈	—	2	1	no	—
306	#97	—	2	1	no	—
307	#97	—	2	1	no	—
308	C₄×C₈	—	2	1	no	—
309	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
310	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
311	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
312	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
313	C₃₁	#19361	1	1	yes	characteristic, direct product factor, Sylow
314	C₂×Q₈	—	2	2	no	—
315	C₂×Q₈	—	2	2	no	—
316	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
317	C₂×D₄	—	2	1	no	—
318	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
319	C₂²×C₄	—	2	1	no	—
320	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
321	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
322	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
323	Q₁₆	—	8	2	no	—
324	SD₁₆	—	8	2	no	—
325	M₁₆	—	4	2	no	—
326	M₁₆	—	8	2	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	—	2	2	no	—
333	C₂×C₈	—	4	2	no	—
334	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
335	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
336	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
337	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
338	C₄⋊C₄	—	2	1	no	—
339	C₄⋊C₄	—	2	1	no	—
340	C₄⋊C₄	—	2	1	no	—
341	C₄⋊C₄	—	2	1	no	—
342	C₄⋊C₄	—	2	2	no	—
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	2	2	no	—
345	C₄⋊C₄	—	2	2	no	—
346	C₄⋊C₄	—	4	2	no	—
347	C₄⋊C₄	—	4	2	no	—
348	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
349	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
350	C₂²⋊C₄	—	2	1	no	—
351	C₂²⋊C₄	—	2	1	no	—
352	C₂²⋊C₄	—	2	1	no	—
353	C₂²⋊C₄	—	2	1	no	—
354	C₂²⋊C₄	—	2	1	no	—
355	C₂²⋊C₄	—	4	2	no	—
356	C₄²	—	2	1	no	—
357	C₄²	—	4	2	no	—
358	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
359	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
360	C₄²	#41×C₃₁	1	1	yes	characteristic
361	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
362	C₂³	—	2	1	no	—
363	C₂³	C₃₁×#121	1	1	yes	characteristic
364	Q₈	—	2	1	no	—
365	Q₈	—	2	1	no	—
366	Q₈	—	4	2	no	—
367	Q₈	—	4	2	no	—
368	Q₈	—	8	2	no	—
369	D₄	—	2	1	no	—
370	D₄	—	2	1	no	—
371	D₄	—	4	2	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	2	1	no	—
374	C₂×C₄	—	2	1	no	—
375	C₂×C₄	—	2	1	no	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	4	1	no	—
380	C₂×C₄	—	4	1	no	—
381	C₂×C₄	—	4	1	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
385	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
386	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
387	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
388	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
389	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
390	C₈	—	2	1	no	—
391	C₈	—	2	1	no	—
392	C₈	—	4	1	no	—
393	C₈	—	4	1	no	—
394	C₈	—	4	2	no	—
395	C₈	—	4	2	no	—
396	C₈	C₃₁×#137	1	2	yes	—
397	C₈	C₃₁×#137	1	2	yes	—
398	C₂²	—	2	1	no	—
399	C₂²	—	2	1	no	—
400	C₂²	—	2	1	no	—
401	C₂²	—	4	1	no	—
402	C₂²	—	4	1	no	—
403	C₂²	—	4	1	no	—
404	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
405	C₄	—	2	1	no	—
406	C₄	—	2	1	no	—
407	C₄	—	2	1	no	—
408	C₄	—	2	1	no	—
409	C₄	—	4	1	no	—
410	C₄	—	4	1	no	—
411	C₄	—	4	1	no	—
412	C₄	—	8	2	no	—
413	C₄	—	8	2	no	—
414	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
415	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
416	C₄	C₃₁×#576	1	1	yes	characteristic
417	C₄	C₃₁×#578	1	1	yes	characteristic
418	C₂	—	4	1	no	—
419	C₂	—	8	1	no	—
420	C₂	C₃₁×#3039	1	1	yes	characteristic
421	C₂	C₃₁×#3061	1	1	yes	characteristic
422	C₂	C₃₁×#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#121	D₄
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×#3160	C₂	semidirect
4. C₃₁×#1692	C₂	semidirect
5. C₃₁×#1691	C₂	semidirect
6. C₃₁×#1676	C₂	semidirect
7. C₃₁×#1675	C₂	semidirect
8. C₃₁×#1674	C₂	semidirect
9. C₃₁×#1673	C₂	semidirect
10. C₃₁×#1567	C₂	semidirect
11. C₃₁×#1559	C₂	semidirect
150. #19361	C₃₁	direct (N × H)
188. #3160	C₂×C₃₁	semidirect
190. #1692	C₂×C₃₁	semidirect
191. #1691	C₂×C₃₁	semidirect
192. #1676	C₂×C₃₁	semidirect
193. #1675	C₂×C₃₁	semidirect
194. #1674	C₂×C₃₁	semidirect
195. #1673	C₂×C₃₁	semidirect
196. #1567	C₂×C₃₁	semidirect
197. #1559	C₂×C₃₁	semidirect
313. C₃₁	#19361	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×#3252	2
C₃₁×#3160	2
C₃₁×#2966	2
C₃₁×#1692	2
C₃₁×#1691	2
C₃₁×#1676	2
C₃₁×#1675	2
C₃₁×#1674	2
C₃₁×#1673	2
C₃₁×#1567	2
C₃₁×#1559	2
#19361	31

Subgroups — C31×#19361 #21476120

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19361 #21476120