Subgroups — C₃₁×#8901 #21465660

Order 7936.

372 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2991	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1837	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1574	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1489	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1476	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1299	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
9	C₈×Q₈×C₃₁	—	2	1	no	—
10	C₃₁×#436	—	2	2	no	—
11	C₃₁×#436	—	2	1	no	—
12	C₃₁×#431	—	2	1	no	—
13	C₃₁×#431	C₂²	1	1	yes	characteristic
14	C₃₁×#431	C₂²	1	1	yes	characteristic
15	C₃₁×#423	—	2	1	no	—
16	C₃₁×#423	C₂²	1	1	yes	characteristic
17	C₃₁×#423	C₂²	1	1	yes	characteristic
18	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×#341	—	2	2	no	—
21	C₃₁×#335	—	2	1	no	—
22	C₃₁×#334	—	2	1	no	—
23	C₃₁×Q₈⋊C₈	—	2	1	no	—
24	C₃₁×Q₈⋊C₈	C₄	1	1	yes	characteristic
25	C₃₁×Q₈⋊C₈	C₄	1	1	yes	characteristic
26	C₃₁×D₄⋊C₈	—	2	1	no	—
27	C₃₁×D₄⋊C₈	C₄	1	1	yes	characteristic
28	C₃₁×D₄⋊C₈	C₄	1	1	yes	characteristic
29	C₂×#41×C₃₁	Q₈	1	1	yes	characteristic
30	C₂×M₁₆×C₃₁	—	2	2	no	—
31	C₂×M₁₆×C₃₁	—	2	1	no	—
32	C₂×M₁₆×C₃₁	—	2	1	no	—
33	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
34	C₂²×C₈×C₃₁	—	2	1	no	—
35	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
36	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
37	C₄×D₄×C₃₁	—	2	1	no	—
38	C₄×D₄×C₃₁	—	2	1	no	—
39	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
41	C₃₁×#117	—	2	1	no	—
42	C₃₁×#117	Q₈	1	1	yes	characteristic
43	C₂×C₄²×C₃₁	—	2	1	no	—
44	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
45	C₃₁×C₄⋊C₈	—	2	2	no	—
46	C₃₁×C₄⋊C₈	—	2	1	no	—
47	C₃₁×C₄⋊C₈	—	2	1	no	—
48	C₃₁×C₄⋊C₈	—	2	1	no	—
49	C₃₁×C₄⋊C₈	—	2	1	no	—
50	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
51	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
52	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
53	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
54	C₃₁×C₂²⋊C₈	—	2	2	no	—
55	C₃₁×C₂²⋊C₈	—	2	1	no	—
56	C₃₁×C₂²⋊C₈	—	2	1	no	—
57	C₃₁×#97	—	2	1	no	—
58	C₄×C₈×C₃₁	—	2	2	no	—
59	C₄×C₈×C₃₁	—	2	1	no	—
60	C₄×C₈×C₃₁	—	2	1	no	—
61	C₄×C₈×C₃₁	—	2	1	no	—
62	C₄×C₈×C₃₁	—	2	1	no	—
63	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
65	#41×C₃₁	—	2	2	no	—
66	#41×C₃₁	—	4	1	no	—
67	C₂×Q₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
68	C₂×D₄×C₃₁	—	2	1	no	—
69	C₂×D₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₃₁	—	2	1	no	—
71	C₂²×C₄×C₃₁	—	2	1	no	—
72	C₂²×C₄×C₃₁	—	2	1	no	—
73	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
75	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
76	M₁₆×C₃₁	—	2	2	no	—
77	M₁₆×C₃₁	—	4	1	no	—
78	M₁₆×C₃₁	—	4	2	no	—
79	M₁₆×C₃₁	—	4	2	no	—
80	M₁₆×C₃₁	—	8	2	no	—
81	C₂×C₈×C₃₁	—	2	1	no	—
82	C₂×C₈×C₃₁	—	2	1	no	—
83	C₂×C₈×C₃₁	—	2	1	no	—
84	C₂×C₈×C₃₁	—	2	1	no	—
85	C₂×C₈×C₃₁	—	2	1	no	—
86	C₂×C₈×C₃₁	—	2	2	no	—
87	C₂×C₈×C₃₁	—	2	1	no	—
88	C₂×C₈×C₃₁	—	4	1	no	—
89	C₂×C₈×C₃₁	—	4	1	no	—
90	C₂×C₈×C₃₁	—	4	1	no	—
91	C₂×C₈×C₃₁	—	4	1	no	—
92	C₄⋊C₄×C₃₁	—	2	1	no	—
93	C₄⋊C₄×C₃₁	—	2	1	no	—
94	C₄⋊C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
95	C₄⋊C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
96	C₂²⋊C₄×C₃₁	—	2	1	no	—
97	C₂²⋊C₄×C₃₁	—	2	1	no	—
98	C₂²⋊C₄×C₃₁	—	2	1	no	—
99	C₄²×C₃₁	—	2	1	no	—
100	C₄²×C₃₁	—	2	1	no	—
101	C₄²×C₃₁	—	2	1	no	—
102	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₃₁	#41	1	1	yes	characteristic
106	#8901	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂³×C₃₁	—	2	1	no	—
108	C₂³×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
109	Q₈×C₃₁	—	2	1	no	—
110	Q₈×C₃₁	#108	1	2	yes	—
111	D₄×C₃₁	—	2	1	no	—
112	D₄×C₃₁	—	2	1	no	—
113	D₄×C₃₁	—	4	2	no	—
114	C₂×C₄×C₃₁	—	2	1	no	—
115	C₂×C₄×C₃₁	—	2	1	no	—
116	C₂×C₄×C₃₁	—	2	1	no	—
117	C₂×C₄×C₃₁	—	2	1	no	—
118	C₂×C₄×C₃₁	—	2	1	no	—
119	C₂×C₄×C₃₁	—	2	1	no	—
120	C₂×C₄×C₃₁	—	2	1	no	—
121	C₂×C₄×C₃₁	—	2	1	no	—
122	C₂×C₄×C₃₁	—	2	1	no	—
123	C₂×C₄×C₃₁	—	4	2	no	—
124	C₂×C₄×C₃₁	—	4	1	no	—
125	C₂×C₄×C₃₁	—	4	1	no	—
126	C₂×C₄×C₃₁	—	4	1	no	—
127	C₂×C₄×C₃₁	—	4	1	no	—
128	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
132	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
133	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
134	C₈×C₃₁	—	4	1	no	—
135	C₈×C₃₁	—	4	1	no	—
136	C₈×C₃₁	—	4	2	no	—
137	C₈×C₃₁	—	4	1	no	—
138	C₈×C₃₁	—	4	1	no	—
139	C₈×C₃₁	—	4	1	no	—
140	C₈×C₃₁	—	4	2	no	—
141	C₈×C₃₁	—	4	1	no	—
142	C₈×C₃₁	—	8	1	no	—
143	#2991	C₂×C₃₁	1	1	yes	characteristic
144	#1837	C₂×C₃₁	1	1	yes	characteristic, complemented
145	#1574	C₂×C₃₁	1	1	yes	characteristic, complemented
146	#1489	C₂×C₃₁	1	1	yes	characteristic
147	#1476	C₂×C₃₁	1	1	yes	characteristic
148	#1299	C₂×C₃₁	1	1	yes	characteristic, complemented
149	#1278	C₂×C₃₁	1	1	yes	characteristic, complemented
150	C₂²×C₃₁	—	2	1	no	—
151	C₂²×C₃₁	—	2	1	no	—
152	C₂²×C₃₁	—	2	1	no	—
153	C₂²×C₃₁	—	4	1	no	—
154	C₂²×C₃₁	—	4	1	no	—
155	C₂²×C₃₁	—	4	1	no	—
156	C₂²×C₃₁	#385	1	1	yes	characteristic
157	C₄×C₃₁	—	2	1	no	—
158	C₄×C₃₁	—	2	2	no	—
159	C₄×C₃₁	—	2	1	no	—
160	C₄×C₃₁	—	2	1	no	—
161	C₄×C₃₁	—	4	1	no	—
162	C₄×C₃₁	—	4	1	no	—
163	C₄×C₃₁	C₂×#108	1	1	yes	characteristic
164	C₄×C₃₁	#430	1	1	yes	characteristic
165	C₄×C₃₁	#443	1	1	yes	characteristic
166	C₄×C₃₁	#444	1	1	yes	characteristic
167	C₄×C₃₁	#454	1	1	yes	characteristic
168	C₄×C₃₁	#455	1	1	yes	characteristic
169	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
170	C₈×Q₈	—	2	1	no	—
171	#436	—	2	2	no	—
172	#436	—	2	1	no	—
173	#431	—	2	1	no	—
174	#431	C₂²×C₃₁	1	1	yes	characteristic
175	#431	C₂²×C₃₁	1	1	yes	characteristic
176	#423	—	2	1	no	—
177	#423	C₂²×C₃₁	1	1	yes	characteristic
178	#423	C₂²×C₃₁	1	1	yes	characteristic
179	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
180	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
181	#341	—	2	2	no	—
182	#335	—	2	1	no	—
183	#334	—	2	1	no	—
184	Q₈⋊C₈	—	2	1	no	—
185	Q₈⋊C₈	C₄×C₃₁	1	1	yes	characteristic
186	Q₈⋊C₈	C₄×C₃₁	1	1	yes	characteristic
187	D₄⋊C₈	—	2	1	no	—
188	D₄⋊C₈	C₄×C₃₁	1	1	yes	characteristic
189	D₄⋊C₈	C₄×C₃₁	1	1	yes	characteristic
190	C₂×C₃₁	—	4	1	no	—
191	C₂×C₃₁	—	8	1	no	—
192	C₂×C₃₁	#1855	1	1	yes	characteristic
193	C₂×C₃₁	#1860	1	1	yes	characteristic
194	C₂×C₃₁	#1866	1	1	yes	characteristic
195	C₂×#41	Q₈×C₃₁	1	1	yes	characteristic
196	C₂×M₁₆	—	2	2	no	—
197	C₂×M₁₆	—	2	1	no	—
198	C₂×M₁₆	—	2	1	no	—
199	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
200	C₂²×C₈	—	2	1	no	—
201	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
202	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
203	C₄×D₄	—	2	1	no	—
204	C₄×D₄	—	2	1	no	—
205	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
206	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
207	#117	—	2	1	no	—
208	#117	Q₈×C₃₁	1	1	yes	characteristic
209	C₂×C₄²	—	2	1	no	—
210	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
211	C₄⋊C₈	—	2	2	no	—
212	C₄⋊C₈	—	2	1	no	—
213	C₄⋊C₈	—	2	1	no	—
214	C₄⋊C₈	—	2	1	no	—
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
217	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
218	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
219	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
220	C₂²⋊C₈	—	2	2	no	—
221	C₂²⋊C₈	—	2	1	no	—
222	C₂²⋊C₈	—	2	1	no	—
223	#97	—	2	1	no	—
224	C₄×C₈	—	2	2	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	—	2	1	no	—
227	C₄×C₈	—	2	1	no	—
228	C₄×C₈	—	2	1	no	—
229	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
230	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
231	C₃₁	#8901	1	1	yes	characteristic, direct product factor, Sylow
232	#41	—	2	2	no	—
233	#41	—	4	1	no	—
234	C₂×Q₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
235	C₂×D₄	—	2	1	no	—
236	C₂×D₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
241	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
242	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
243	M₁₆	—	2	2	no	—
244	M₁₆	—	4	1	no	—
245	M₁₆	—	4	2	no	—
246	M₁₆	—	4	2	no	—
247	M₁₆	—	8	2	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	4	1	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
262	C₄⋊C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	2	1	no	—
266	C₄²	—	2	1	no	—
267	C₄²	—	2	1	no	—
268	C₄²	—	2	1	no	—
269	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
270	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
271	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
272	C₄²	#41×C₃₁	1	1	yes	characteristic
273	C₂³	—	2	1	no	—
274	C₂³	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
275	Q₈	—	2	1	no	—
276	Q₈	C₃₁×#108	1	2	yes	—
277	D₄	—	2	1	no	—
278	D₄	—	2	1	no	—
279	D₄	—	4	2	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
295	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
296	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
297	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
298	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
299	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
300	C₈	—	4	1	no	—
301	C₈	—	4	1	no	—
302	C₈	—	4	2	no	—
303	C₈	—	4	1	no	—
304	C₈	—	4	1	no	—
305	C₈	—	4	1	no	—
306	C₈	—	4	2	no	—
307	C₈	—	4	1	no	—
308	C₈	—	8	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	C₃₁×#385	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	2	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	C₂×C₃₁×#108	1	1	yes	characteristic
323	C₄	C₃₁×#430	1	1	yes	characteristic
324	C₄	C₃₁×#443	1	1	yes	characteristic
325	C₄	C₃₁×#444	1	1	yes	characteristic
326	C₄	C₃₁×#454	1	1	yes	characteristic
327	C₄	C₃₁×#455	1	1	yes	characteristic
328	C₂	—	4	1	no	—
329	C₂	—	8	1	no	—
330	C₂	C₃₁×#1855	1	1	yes	characteristic
331	C₂	C₃₁×#1860	1	1	yes	characteristic
332	C₂	C₃₁×#1866	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#120
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	#385
prime radical	C₂×#41×C₃₁	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1837	C₂	semidirect
3. C₃₁×#1574	C₂	semidirect
6. C₃₁×#1299	C₂	semidirect
7. C₃₁×#1278	C₂	semidirect
106. #8901	C₃₁	direct (N × H)
144. #1837	C₂×C₃₁	semidirect
145. #1574	C₂×C₃₁	semidirect
148. #1299	C₂×C₃₁	semidirect
149. #1278	C₂×C₃₁	semidirect
231. C₃₁	#8901	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×#2991	2
C₃₁×#1837	2
C₃₁×#1574	2
C₃₁×#1489	2
C₃₁×#1476	2
C₃₁×#1299	2
C₃₁×#1278	2
#8901	31

Subgroups — C31×#8901 #21465660

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8901 #21465660