Subgroups — C₃₁×#19712 #21476471

Order 7936.

704 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3228	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3210	C₂	1	2	yes	maximal, complemented
3	C₃₁×#3170	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3169	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1723	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1722	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1720	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1717	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1452	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#533	C₂²	1	2	yes	—
11	C₃₁×#532	C₂²	1	2	yes	—
12	C₃₁×#500	C₂²	1	1	yes	characteristic
13	C₃₁×#500	C₂²	1	1	yes	characteristic
14	C₃₁×#500	C₂²	1	2	yes	—
15	C₃₁×#498	C₂²	1	2	yes	—
16	C₃₁×#494	C₂²	1	1	yes	characteristic
17	C₃₁×#494	C₂²	1	2	yes	—
18	C₃₁×#493	C₂²	1	1	yes	characteristic
19	C₃₁×#492	C₂²	1	2	yes	—
20	C₃₁×#487	C₂²	1	2	yes	—
21	C₃₁×#487	C₂²	1	2	yes	—
22	C₃₁×#486	C₂²	1	2	yes	—
23	C₃₁×#486	C₂²	1	2	yes	—
24	C₃₁×#486	C₂²	1	2	yes	—
25	C₃₁×#486	C₂²	1	2	yes	—
26	C₃₁×#482	—	2	4	no	—
27	C₃₁×#478	—	2	4	no	—
28	C₃₁×#467	—	2	2	no	—
29	C₃₁×#466	—	2	2	no	—
30	C₃₁×#464	—	2	4	no	—
31	C₃₁×#452	—	2	4	no	—
32	C₄×Q₁₆×C₃₁	—	2	2	no	—
33	C₄×D₈×C₃₁	—	2	2	no	—
34	C₃₁×#431	C₂²	1	1	yes	characteristic
35	C₃₁×#431	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#322	C₂²	1	2	yes	—
38	C₃₁×#322	C₂²	1	2	yes	—
39	C₂×#41×C₃₁	—	2	2	no	—
40	C₂×Q₁₆×C₃₁	—	2	2	no	—
41	C₂×Q₁₆×C₃₁	—	2	4	no	—
42	C₂×SD₁₆×C₃₁	—	2	4	no	—
43	C₂×D₈×C₃₁	—	2	2	no	—
44	C₂²×C₈×C₃₁	D₄	1	2	yes	—
45	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
46	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
47	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
48	C₃₁×#127	C₂³	1	2	yes	—
49	C₃₁×#125	—	2	2	no	—
50	C₃₁×#124	—	2	2	no	—
51	C₃₁×C₂²⋊Q₈	—	2	2	no	—
52	C₃₁×C₂²⋊Q₈	—	2	2	no	—
53	C₃₁×#121	—	2	2	no	—
54	C₃₁×#121	—	2	2	no	—
55	C₄×Q₈×C₃₁	—	2	2	no	—
56	C₄×Q₈×C₃₁	—	2	2	no	—
57	C₄×D₄×C₃₁	—	2	2	no	—
58	C₄×D₄×C₃₁	—	2	2	no	—
59	C₃₁×#117	—	2	2	no	—
60	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
61	C₃₁×#107	—	2	2	no	—
62	C₃₁×#107	—	2	2	no	—
63	C₃₁×#107	—	2	4	no	—
64	C₃₁×#106	—	2	4	no	—
65	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
66	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
67	C₃₁×#103	—	2	4	no	—
68	C₃₁×#103	—	2	4	no	—
69	C₃₁×#102	—	2	4	no	—
70	C₃₁×#102	—	2	4	no	—
71	C₃₁×#102	—	2	4	no	—
72	C₃₁×#102	—	2	4	no	—
73	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
74	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
75	C₄×C₈×C₃₁	D₄	1	2	yes	—
76	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₃₁	C₂³	1	2	yes	—
79	C₄×C₈×C₃₁	C₂³	1	2	yes	—
80	#41×C₃₁	—	8	4	no	—
81	C₂×Q₈×C₃₁	—	2	2	no	—
82	C₂×Q₈×C₃₁	—	2	2	no	—
83	C₂×Q₈×C₃₁	—	2	2	no	—
84	C₂×D₄×C₃₁	—	2	2	no	—
85	C₂×D₄×C₃₁	—	2	2	no	—
86	C₂×D₄×C₃₁	—	2	2	no	—
87	C₂×D₄×C₃₁	—	4	2	no	—
88	C₂²×C₄×C₃₁	—	4	2	no	—
89	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
92	Q₁₆×C₃₁	—	4	4	no	—
93	Q₁₆×C₃₁	—	8	4	no	—
94	SD₁₆×C₃₁	—	8	4	no	—
95	D₈×C₃₁	—	4	4	no	—
96	C₂×C₈×C₃₁	—	4	2	no	—
97	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
100	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
101	C₄⋊C₄×C₃₁	—	2	2	no	—
102	C₄⋊C₄×C₃₁	—	2	2	no	—
103	C₄⋊C₄×C₃₁	—	2	2	no	—
104	C₄⋊C₄×C₃₁	—	2	2	no	—
105	C₄⋊C₄×C₃₁	—	2	2	no	—
106	C₄⋊C₄×C₃₁	—	2	2	no	—
107	C₄⋊C₄×C₃₁	—	4	2	no	—
108	C₄⋊C₄×C₃₁	—	4	2	no	—
109	C₂²⋊C₄×C₃₁	—	4	2	no	—
110	C₂²⋊C₄×C₃₁	—	4	2	no	—
111	C₂²⋊C₄×C₃₁	—	4	2	no	—
112	C₄²×C₃₁	—	4	2	no	—
113	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
117	#19712	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂³×C₃₁	—	4	2	no	—
119	C₂³×C₃₁	#127	1	1	yes	characteristic
120	Q₈×C₃₁	—	4	4	no	—
121	Q₈×C₃₁	—	4	4	no	—
122	Q₈×C₃₁	—	4	4	no	—
123	D₄×C₃₁	—	4	4	no	—
124	D₄×C₃₁	—	4	4	no	—
125	D₄×C₃₁	—	4	4	no	—
126	D₄×C₃₁	—	8	4	no	—
127	C₂×C₄×C₃₁	—	2	2	no	—
128	C₂×C₄×C₃₁	—	2	2	no	—
129	C₂×C₄×C₃₁	—	2	2	no	—
130	C₂×C₄×C₃₁	—	4	2	no	—
131	C₂×C₄×C₃₁	—	4	2	no	—
132	C₂×C₄×C₃₁	—	4	2	no	—
133	C₂×C₄×C₃₁	—	8	4	no	—
134	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
135	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
136	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
137	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
140	C₈×C₃₁	—	2	2	no	—
141	C₈×C₃₁	—	2	2	no	—
142	C₈×C₃₁	—	2	4	no	—
143	C₈×C₃₁	—	2	4	no	—
144	#3228	C₂×C₃₁	1	2	yes	complemented
145	#3210	C₂×C₃₁	1	2	yes	complemented
146	#3170	C₂×C₃₁	1	1	yes	characteristic, complemented
147	#3169	C₂×C₃₁	1	1	yes	characteristic
148	#1723	C₂×C₃₁	1	2	yes	complemented
149	#1722	C₂×C₃₁	1	2	yes	complemented
150	#1720	C₂×C₃₁	1	2	yes	complemented
151	#1717	C₂×C₃₁	1	2	yes	complemented
152	#1452	C₂×C₃₁	1	1	yes	characteristic, complemented
153	C₂²×C₃₁	—	2	1	no	—
154	C₂²×C₃₁	—	2	1	no	—
155	C₂²×C₃₁	—	2	1	no	—
156	C₂²×C₃₁	—	8	2	no	—
157	C₂²×C₃₁	—	8	2	no	—
158	C₂²×C₃₁	—	8	2	no	—
159	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
160	C₄×C₃₁	—	2	1	no	—
161	C₄×C₃₁	—	2	1	no	—
162	C₄×C₃₁	—	2	1	no	—
163	C₄×C₃₁	—	8	2	no	—
164	C₄×C₃₁	—	8	2	no	—
165	C₄×C₃₁	—	8	2	no	—
166	C₄×C₃₁	#575	1	2	yes	—
167	C₄×C₃₁	#576	1	2	yes	—
168	C₄×C₃₁	#578	1	2	yes	—
169	#533	C₂²×C₃₁	1	2	yes	—
170	#532	C₂²×C₃₁	1	2	yes	—
171	#500	C₂²×C₃₁	1	1	yes	characteristic
172	#500	C₂²×C₃₁	1	1	yes	characteristic
173	#500	C₂²×C₃₁	1	2	yes	—
174	#498	C₂²×C₃₁	1	2	yes	—
175	#494	C₂²×C₃₁	1	1	yes	characteristic
176	#494	C₂²×C₃₁	1	2	yes	—
177	#493	C₂²×C₃₁	1	1	yes	characteristic
178	#492	C₂²×C₃₁	1	2	yes	—
179	#487	C₂²×C₃₁	1	2	yes	—
180	#487	C₂²×C₃₁	1	2	yes	—
181	#486	C₂²×C₃₁	1	2	yes	—
182	#486	C₂²×C₃₁	1	2	yes	—
183	#486	C₂²×C₃₁	1	2	yes	—
184	#486	C₂²×C₃₁	1	2	yes	—
185	#482	—	2	4	no	—
186	#478	—	2	4	no	—
187	#467	—	2	2	no	—
188	#466	—	2	2	no	—
189	#464	—	2	4	no	—
190	#452	—	2	4	no	—
191	C₄×Q₁₆	—	2	2	no	—
192	C₄×D₈	—	2	2	no	—
193	#431	C₂²×C₃₁	1	1	yes	characteristic
194	#431	C₂²×C₃₁	1	1	yes	characteristic
195	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
196	#322	C₂²×C₃₁	1	2	yes	—
197	#322	C₂²×C₃₁	1	2	yes	—
198	C₂×C₃₁	—	4	1	no	—
199	C₂×C₃₁	—	16	2	no	—
200	C₂×C₃₁	#3150	1	1	yes	characteristic
201	C₂×C₃₁	#3155	1	1	yes	characteristic
202	C₂×C₃₁	#3156	1	1	yes	characteristic
203	C₂×#41	—	2	2	no	—
204	C₂×Q₁₆	—	2	2	no	—
205	C₂×Q₁₆	—	2	4	no	—
206	C₂×SD₁₆	—	2	4	no	—
207	C₂×D₈	—	2	2	no	—
208	C₂²×C₈	D₄×C₃₁	1	2	yes	—
209	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
210	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
211	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
212	#127	C₂³×C₃₁	1	2	yes	—
213	#125	—	2	2	no	—
214	#124	—	2	2	no	—
215	C₂²⋊Q₈	—	2	2	no	—
216	C₂²⋊Q₈	—	2	2	no	—
217	#121	—	2	2	no	—
218	#121	—	2	2	no	—
219	C₄×Q₈	—	2	2	no	—
220	C₄×Q₈	—	2	2	no	—
221	C₄×D₄	—	2	2	no	—
222	C₄×D₄	—	2	2	no	—
223	#117	—	2	2	no	—
224	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
225	#107	—	2	2	no	—
226	#107	—	2	2	no	—
227	#107	—	2	4	no	—
228	#106	—	2	4	no	—
229	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
230	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
231	#103	—	2	4	no	—
232	#103	—	2	4	no	—
233	#102	—	2	4	no	—
234	#102	—	2	4	no	—
235	#102	—	2	4	no	—
236	#102	—	2	4	no	—
237	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
238	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
239	C₄×C₈	D₄×C₃₁	1	2	yes	—
240	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
241	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
242	C₄×C₈	C₂³×C₃₁	1	2	yes	—
243	C₄×C₈	C₂³×C₃₁	1	2	yes	—
244	C₃₁	#19712	1	1	yes	characteristic, direct product factor, Sylow
245	#41	—	8	4	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	4	2	no	—
253	C₂²×C₄	—	4	2	no	—
254	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
257	Q₁₆	—	4	4	no	—
258	Q₁₆	—	8	4	no	—
259	SD₁₆	—	8	4	no	—
260	D₈	—	4	4	no	—
261	C₂×C₈	—	4	2	no	—
262	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
263	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
264	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
265	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₄²	—	4	2	no	—
278	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
279	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
280	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
281	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
282	C₂³	—	4	2	no	—
283	C₂³	C₃₁×#127	1	1	yes	characteristic
284	Q₈	—	4	4	no	—
285	Q₈	—	4	4	no	—
286	Q₈	—	4	4	no	—
287	D₄	—	4	4	no	—
288	D₄	—	4	4	no	—
289	D₄	—	4	4	no	—
290	D₄	—	8	4	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	8	4	no	—
298	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
299	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
300	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
301	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
302	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
303	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
304	C₈	—	2	2	no	—
305	C₈	—	2	2	no	—
306	C₈	—	2	4	no	—
307	C₈	—	2	4	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	8	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	2	no	—
321	C₄	C₃₁×#575	1	2	yes	—
322	C₄	C₃₁×#576	1	2	yes	—
323	C₄	C₃₁×#578	1	2	yes	—
324	C₂	—	4	1	no	—
325	C₂	—	16	2	no	—
326	C₂	C₃₁×#3150	1	1	yes	characteristic
327	C₂	C₃₁×#3155	1	1	yes	characteristic
328	C₂	C₃₁×#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3169	C₂
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
1. C₃₁×#3228	C₂	semidirect
2. C₃₁×#3210	C₂	semidirect
3. C₃₁×#3170	C₂	semidirect
5. C₃₁×#1723	C₂	semidirect
6. C₃₁×#1722	C₂	semidirect
7. C₃₁×#1720	C₂	semidirect
8. C₃₁×#1717	C₂	semidirect
9. C₃₁×#1452	C₂	semidirect
117. #19712	C₃₁	direct (N × H)
144. #3228	C₂×C₃₁	semidirect
145. #3210	C₂×C₃₁	semidirect
146. #3170	C₂×C₃₁	semidirect
148. #1723	C₂×C₃₁	semidirect
149. #1722	C₂×C₃₁	semidirect
150. #1720	C₂×C₃₁	semidirect
151. #1717	C₂×C₃₁	semidirect
152. #1452	C₂×C₃₁	semidirect
244. C₃₁	#19712	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×#3228	2
C₃₁×#3210	2
C₃₁×#3170	2
C₃₁×#3169	2
C₃₁×#1723	2
C₃₁×#1722	2
C₃₁×#1720	2
C₃₁×#1717	2
C₃₁×#1452	2
#19712	31

Subgroups — C31×#19712 #21476471

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19712 #21476471