Subgroups — C₃₁×#11592 #21468351

Order 7936.

402 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×#135	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#335	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1332	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1331	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#135	C₄	1	1	yes	characteristic
7	C₄×#41×C₃₁	—	2	1	no	—
8	C₂×C₃₁×M₃₂	—	2	1	no	—
9	C₄×Q₁₆×C₃₁	C₄	1	1	yes	characteristic
10	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₄×SD₁₆×C₃₁	—	2	1	no	—
12	C₄×SD₁₆×C₃₁	—	2	1	no	—
13	C₄×D₈×C₃₁	C₄	1	1	yes	characteristic
14	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#427	C₄	1	1	yes	characteristic
16	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
17	C₃₁×#416	—	2	1	no	—
18	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#346	—	2	1	no	—
20	C₄×C₁₆×C₃₁	C₂²	1	2	yes	—
21	C₃₁×#335	—	2	1	no	—
22	C₃₁×#335	C₂²	1	2	yes	—
23	C₂×#41×C₃₁	—	2	1	no	—
24	C₃₁×#135	—	2	2	no	—
25	C₃₁×#135	C₈	1	4	yes	complemented
26	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂×SD₁₆×C₃₁	—	2	1	no	—
28	C₂×D₈×C₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₃₁	—	4	1	no	—
30	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
32	C₄×Q₈×C₃₁	—	2	1	no	—
33	C₄×Q₈×C₃₁	—	2	1	no	—
34	C₄×D₄×C₃₁	—	2	1	no	—
35	C₄×D₄×C₃₁	—	2	1	no	—
36	C₄×D₄×C₃₁	—	4	1	no	—
37	C₄×D₄×C₃₁	—	4	1	no	—
38	C₃₁×#117	—	2	1	no	—
39	C₃₁×#117	—	4	1	no	—
40	C₂×C₄²×C₃₁	—	4	1	no	—
41	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
42	C₃₁×M₃₂	—	2	4	no	—
43	C₂×C₁₆×C₃₁	—	2	2	no	—
44	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
45	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
46	C₃₁×#106	—	2	1	no	—
47	C₃₁×C₄⋊C₈	—	2	2	no	—
48	C₃₁×C₄⋊C₈	—	4	1	no	—
49	C₃₁×#103	—	2	1	no	—
50	C₃₁×#103	—	2	1	no	—
51	C₃₁×#102	—	2	1	no	—
52	C₃₁×#102	—	2	1	no	—
53	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
55	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
56	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
57	#41×C₃₁	—	2	4	no	—
58	#41×C₃₁	—	4	2	no	—
59	C₂×Q₈×C₃₁	—	2	1	no	—
60	C₂×D₄×C₃₁	—	2	1	no	—
61	C₂×D₄×C₃₁	—	4	1	no	—
62	C₂²×C₄×C₃₁	—	4	1	no	—
63	C₂²×C₄×C₃₁	—	4	1	no	—
64	C₂²×C₄×C₃₁	—	4	1	no	—
65	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
68	Q₁₆×C₃₁	—	2	1	no	—
69	Q₁₆×C₃₁	C₂×C₈	1	2	yes	complemented
70	SD₁₆×C₃₁	—	2	4	no	—
71	D₈×C₃₁	—	2	1	no	—
72	D₈×C₃₁	C₂×C₈	1	2	yes	complemented
73	C₂×C₈×C₃₁	—	2	2	no	—
74	C₂×C₈×C₃₁	—	4	2	no	—
75	C₂×C₈×C₃₁	—	8	2	no	—
76	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₃₁	C₂×C₈	1	2	yes	—
79	C₂×C₈×C₃₁	M₁₆	1	2	yes	—
80	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
82	C₄⋊C₄×C₃₁	—	2	1	no	—
83	C₄⋊C₄×C₃₁	—	2	1	no	—
84	C₄⋊C₄×C₃₁	—	4	1	no	—
85	C₄⋊C₄×C₃₁	—	4	1	no	—
86	C₂²⋊C₄×C₃₁	—	4	1	no	—
87	C₂²⋊C₄×C₃₁	—	4	1	no	—
88	C₂²⋊C₄×C₃₁	—	4	1	no	—
89	C₄²×C₃₁	—	4	1	no	—
90	C₄²×C₃₁	—	4	1	no	—
91	C₄²×C₃₁	—	4	1	no	—
92	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
93	C₄²×C₃₁	D₈	1	1	yes	characteristic
94	C₄²×C₃₁	SD₁₆	1	1	yes	characteristic
95	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₁₆×C₃₁	—	2	4	no	—
97	#11592	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂³×C₃₁	—	4	1	no	—
99	C₂³×C₃₁	#102	1	1	yes	characteristic
100	Q₈×C₃₁	—	2	2	no	—
101	Q₈×C₃₁	—	4	1	no	—
102	D₄×C₃₁	—	2	2	no	—
103	D₄×C₃₁	—	4	4	no	—
104	D₄×C₃₁	—	4	1	no	—
105	C₂×C₄×C₃₁	—	2	1	no	—
106	C₂×C₄×C₃₁	—	2	1	no	—
107	C₂×C₄×C₃₁	—	4	1	no	—
108	C₂×C₄×C₃₁	—	4	1	no	—
109	C₂×C₄×C₃₁	—	4	4	no	—
110	C₂×C₄×C₃₁	—	4	1	no	—
111	C₂×C₄×C₃₁	—	4	1	no	—
112	C₂×C₄×C₃₁	—	8	2	no	—
113	C₂×C₄×C₃₁	—	8	1	no	—
114	C₂×C₄×C₃₁	—	8	1	no	—
115	C₂×C₄×C₃₁	C₂²⋊C₈	1	2	yes	—
116	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
117	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
118	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
119	C₂×C₄×C₃₁	#104	1	2	yes	—
120	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
123	C₈×C₃₁	—	2	1	no	—
124	C₈×C₃₁	—	2	1	no	—
125	C₈×C₃₁	—	8	2	no	—
126	C₈×C₃₁	C₂²⋊C₈	1	2	yes	—
127	C₈×C₃₁	C₂²×C₈	1	1	yes	characteristic
128	C₈×C₃₁	C₂×M₁₆	1	1	yes	characteristic
129	C₄×#135	C₂×C₃₁	1	1	yes	characteristic
130	C₄×M₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
131	C₂×#335	C₂×C₃₁	1	1	yes	characteristic, complemented
132	#1332	C₂×C₃₁	1	2	yes	complemented
133	#1331	C₂×C₃₁	1	2	yes	complemented
134	C₂²×C₃₁	—	2	1	no	—
135	C₂²×C₃₁	—	2	1	no	—
136	C₂²×C₃₁	—	4	2	no	—
137	C₂²×C₃₁	—	8	1	no	—
138	C₂²×C₃₁	—	8	1	no	—
139	C₂²×C₃₁	D₄⋊C₈	1	2	yes	—
140	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
141	C₄×C₃₁	—	2	1	no	—
142	C₄×C₃₁	—	2	1	no	—
143	C₄×C₃₁	—	2	1	no	—
144	C₄×C₃₁	—	4	2	no	—
145	C₄×C₃₁	—	8	1	no	—
146	C₄×C₃₁	—	8	1	no	—
147	C₄×C₃₁	D₄⋊C₈	1	2	yes	—
148	C₄×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
149	C₄×C₃₁	C₂×#104	1	1	yes	characteristic
150	C₄×C₃₁	#509	1	1	yes	characteristic
151	C₄×C₃₁	#510	1	1	yes	characteristic
152	C₂×#135	C₄×C₃₁	1	1	yes	characteristic
153	C₄×#41	—	2	1	no	—
154	C₂×M₃₂	—	2	1	no	—
155	C₄×Q₁₆	C₄×C₃₁	1	1	yes	characteristic
156	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
157	C₄×SD₁₆	—	2	1	no	—
158	C₄×SD₁₆	—	2	1	no	—
159	C₄×D₈	C₄×C₃₁	1	1	yes	characteristic
160	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
161	#427	C₄×C₃₁	1	1	yes	characteristic
162	C₂×C₄⋊C₈	—	2	1	no	—
163	#416	—	2	1	no	—
164	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
165	#346	—	2	1	no	—
166	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
167	#335	—	2	1	no	—
168	#335	C₂²×C₃₁	1	2	yes	—
169	C₂×C₃₁	—	2	2	no	—
170	C₂×C₃₁	—	8	2	no	—
171	C₂×C₃₁	C₂×D₄⋊C₈	1	1	yes	characteristic
172	C₂×C₃₁	#2139	1	1	yes	characteristic
173	C₂×C₃₁	#2143	1	1	yes	characteristic
174	C₂×#41	—	2	1	no	—
175	#135	—	2	2	no	—
176	#135	C₈×C₃₁	1	4	yes	complemented
177	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
178	C₂×SD₁₆	—	2	1	no	—
179	C₂×D₈	C₂×C₄×C₃₁	1	1	yes	characteristic
180	C₂²×C₈	—	4	1	no	—
181	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
182	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
183	C₄×Q₈	—	2	1	no	—
184	C₄×Q₈	—	2	1	no	—
185	C₄×D₄	—	2	1	no	—
186	C₄×D₄	—	2	1	no	—
187	C₄×D₄	—	4	1	no	—
188	C₄×D₄	—	4	1	no	—
189	#117	—	2	1	no	—
190	#117	—	4	1	no	—
191	C₂×C₄²	—	4	1	no	—
192	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
193	M₃₂	—	2	4	no	—
194	C₂×C₁₆	—	2	2	no	—
195	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
196	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
197	#106	—	2	1	no	—
198	C₄⋊C₈	—	2	2	no	—
199	C₄⋊C₈	—	4	1	no	—
200	#103	—	2	1	no	—
201	#103	—	2	1	no	—
202	#102	—	2	1	no	—
203	#102	—	2	1	no	—
204	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
205	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
206	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
207	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
208	C₃₁	#11592	1	1	yes	characteristic, direct product factor, Sylow
209	#41	—	2	4	no	—
210	#41	—	4	2	no	—
211	C₂×Q₈	—	2	1	no	—
212	C₂×D₄	—	2	1	no	—
213	C₂×D₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	4	1	no	—
217	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
218	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
219	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
220	Q₁₆	—	2	1	no	—
221	Q₁₆	C₂×C₈×C₃₁	1	2	yes	complemented
222	SD₁₆	—	2	4	no	—
223	D₈	—	2	1	no	—
224	D₈	C₂×C₈×C₃₁	1	2	yes	complemented
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	8	2	no	—
228	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
229	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
230	C₂×C₈	C₂×C₈×C₃₁	1	2	yes	—
231	C₂×C₈	M₁₆×C₃₁	1	2	yes	—
232	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
233	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	4	1	no	—
237	C₄⋊C₄	—	4	1	no	—
238	C₂²⋊C₄	—	4	1	no	—
239	C₂²⋊C₄	—	4	1	no	—
240	C₂²⋊C₄	—	4	1	no	—
241	C₄²	—	4	1	no	—
242	C₄²	—	4	1	no	—
243	C₄²	—	4	1	no	—
244	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
245	C₄²	D₈×C₃₁	1	1	yes	characteristic
246	C₄²	SD₁₆×C₃₁	1	1	yes	characteristic
247	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
248	C₁₆	—	2	4	no	—
249	C₂³	—	4	1	no	—
250	C₂³	C₃₁×#102	1	1	yes	characteristic
251	Q₈	—	2	2	no	—
252	Q₈	—	4	1	no	—
253	D₄	—	2	2	no	—
254	D₄	—	4	4	no	—
255	D₄	—	4	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	8	2	no	—
264	C₂×C₄	—	8	1	no	—
265	C₂×C₄	—	8	1	no	—
266	C₂×C₄	C₃₁×C₂²⋊C₈	1	2	yes	—
267	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
268	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
269	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
270	C₂×C₄	C₃₁×#104	1	2	yes	—
271	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
272	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
273	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
274	C₈	—	2	1	no	—
275	C₈	—	2	1	no	—
276	C₈	—	8	2	no	—
277	C₈	C₃₁×C₂²⋊C₈	1	2	yes	—
278	C₈	C₂²×C₈×C₃₁	1	1	yes	characteristic
279	C₈	C₂×M₁₆×C₃₁	1	1	yes	characteristic
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	8	1	no	—
284	C₂²	—	8	1	no	—
285	C₂²	C₃₁×D₄⋊C₈	1	2	yes	—
286	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	4	2	no	—
291	C₄	—	8	1	no	—
292	C₄	—	8	1	no	—
293	C₄	C₃₁×D₄⋊C₈	1	2	yes	—
294	C₄	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
295	C₄	C₂×C₃₁×#104	1	1	yes	characteristic
296	C₄	C₃₁×#509	1	1	yes	characteristic
297	C₄	C₃₁×#510	1	1	yes	characteristic
298	C₂	—	2	2	no	—
299	C₂	—	8	2	no	—
300	C₂	C₂×C₃₁×D₄⋊C₈	1	1	yes	characteristic
301	C₂	C₃₁×#2139	1	1	yes	characteristic
302	C₂	C₃₁×#2143	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×D₈
commutator	C₈	C₂²×C₈×C₃₁
Frattini, cosocle	C₄×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×#102
prime radical	C₂×C₃₁×#135	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₃₁×M₃₂	C₂	semidirect
3. C₂×C₃₁×#335	C₂	semidirect
4. C₃₁×#1332	C₂	semidirect
5. C₃₁×#1331	C₂	semidirect
25. C₃₁×#135	C₈	semidirect
69. Q₁₆×C₃₁	C₂×C₈	semidirect
72. D₈×C₃₁	C₂×C₈	semidirect
97. #11592	C₃₁	direct (N × H)
130. C₄×M₃₂	C₂×C₃₁	semidirect
131. C₂×#335	C₂×C₃₁	semidirect
132. #1332	C₂×C₃₁	semidirect
133. #1331	C₂×C₃₁	semidirect
176. #135	C₈×C₃₁	semidirect
208. C₃₁	#11592	direct (N × H)
221. Q₁₆	C₂×C₈×C₃₁	semidirect
224. D₈	C₂×C₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₃₁×#135	2
C₄×C₃₁×M₃₂	2
C₂×C₃₁×#335	2
C₃₁×#1332	2
C₃₁×#1331	2
#11592	31

Subgroups — C31×#11592 #21468351

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11592 #21468351