Subgroups — C₃₁×#12950 #21469709

Order 7936.

412 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#443	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#365	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#360	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#360	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1941	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1383	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1376	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#131	—	2	1	no	—
10	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
12	C₃₁×#443	—	2	2	no	—
13	C₃₁×#443	—	2	2	no	—
14	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#104	—	2	1	no	—
18	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#389	—	2	1	no	—
20	C₃₁×#365	—	2	2	no	—
21	C₃₁×#360	—	2	2	no	—
22	C₃₁×#360	C₄	1	4	yes	complemented
23	C₃₁×#340	—	2	1	no	—
24	C₂×#41×C₃₁	—	2	1	no	—
25	C₃₁×#135	—	2	1	no	—
26	C₃₁×#135	—	2	1	no	—
27	C₃₁×#135	C₂×C₄	1	2	yes	—
28	C₃₁×#135	D₄	1	2	yes	—
29	C₂×Q₁₆×C₃₁	Q₈	1	1	yes	characteristic
30	C₂×SD₁₆×C₃₁	—	2	1	no	—
31	C₂×D₈×C₃₁	Q₈	1	1	yes	characteristic
32	C₃₁×#131	—	2	4	no	—
33	C₃₁×#131	—	4	2	no	—
34	C₂×M₁₆×C₃₁	—	2	1	no	—
35	C₂×M₁₆×C₃₁	—	4	1	no	—
36	C₂²×C₈×C₃₁	—	4	1	no	—
37	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
40	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
41	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
42	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
43	C₃₁×M₃₂	—	2	1	no	—
44	C₃₁×M₃₂	—	2	1	no	—
45	C₃₁×M₃₂	—	2	1	no	—
46	C₃₁×M₃₂	C₂×C₄	1	2	yes	—
47	C₂×C₁₆×C₃₁	—	2	1	no	—
48	C₂×C₁₆×C₃₁	—	2	1	no	—
49	C₃₁×#108	—	2	1	no	—
50	C₃₁×#108	D₄	1	2	yes	—
51	C₃₁×#106	—	2	1	no	—
52	C₃₁×#106	—	2	1	no	—
53	C₃₁×#106	—	2	1	no	—
54	C₃₁×#106	C₂×C₄	1	2	yes	—
55	C₃₁×#104	—	2	4	no	—
56	C₄×C₈×C₃₁	—	2	1	no	—
57	C₄×C₈×C₃₁	—	2	1	no	—
58	C₃₁×#95	—	4	1	no	—
59	#41×C₃₁	—	2	2	no	—
60	#41×C₃₁	—	2	2	no	—
61	#41×C₃₁	—	4	1	no	—
62	#41×C₃₁	—	4	1	no	—
63	C₂×Q₈×C₃₁	—	2	1	no	—
64	C₂×D₄×C₃₁	—	2	1	no	—
65	C₂×D₄×C₃₁	—	4	1	no	—
66	C₂²×C₄×C₃₁	—	2	1	no	—
67	C₂²×C₄×C₃₁	—	4	1	no	—
68	C₂²×C₄×C₃₁	—	4	1	no	—
69	C₂²×C₄×C₃₁	—	4	1	no	—
70	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
71	Q₁₆×C₃₁	—	2	1	no	—
72	Q₁₆×C₃₁	C₄⋊C₄	1	2	yes	complemented
73	SD₁₆×C₃₁	—	2	2	no	—
74	SD₁₆×C₃₁	—	2	2	no	—
75	D₈×C₃₁	—	2	1	no	—
76	D₈×C₃₁	C₄⋊C₄	1	2	yes	complemented
77	M₁₆×C₃₁	—	2	2	no	—
78	M₁₆×C₃₁	—	4	4	no	—
79	M₁₆×C₃₁	—	4	1	no	—
80	C₂×C₈×C₃₁	—	2	1	no	—
81	C₂×C₈×C₃₁	—	2	1	no	—
82	C₂×C₈×C₃₁	—	4	4	no	—
83	C₂×C₈×C₃₁	—	4	1	no	—
84	C₂×C₈×C₃₁	SD₁₆	1	1	yes	characteristic
85	C₂×C₈×C₃₁	SD₁₆	1	1	yes	characteristic
86	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
90	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
91	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
92	C₄⋊C₄×C₃₁	—	2	2	no	—
93	C₄⋊C₄×C₃₁	—	4	1	no	—
94	C₄⋊C₄×C₃₁	—	4	1	no	—
95	C₄⋊C₄×C₃₁	—	4	1	no	—
96	C₄⋊C₄×C₃₁	—	8	2	no	—
97	C₄²×C₃₁	—	2	1	no	—
98	C₄²×C₃₁	—	2	1	no	—
99	C₁₆×C₃₁	—	2	2	no	—
100	C₁₆×C₃₁	—	4	1	no	—
101	#12950	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂³×C₃₁	—	4	1	no	—
103	C₂³×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
104	Q₈×C₃₁	—	2	2	no	—
105	Q₈×C₃₁	—	4	1	no	—
106	D₄×C₃₁	—	2	2	no	—
107	D₄×C₃₁	—	4	2	no	—
108	D₄×C₃₁	—	4	2	no	—
109	D₄×C₃₁	—	4	1	no	—
110	C₂×C₄×C₃₁	—	2	2	no	—
111	C₂×C₄×C₃₁	—	4	1	no	—
112	C₂×C₄×C₃₁	—	4	1	no	—
113	C₂×C₄×C₃₁	—	4	1	no	—
114	C₂×C₄×C₃₁	—	4	2	no	—
115	C₂×C₄×C₃₁	—	4	2	no	—
116	C₂×C₄×C₃₁	—	4	2	no	—
117	C₂×C₄×C₃₁	—	8	1	no	—
118	C₂×C₄×C₃₁	—	8	1	no	—
119	C₂×C₄×C₃₁	—	8	1	no	—
120	C₂×C₄×C₃₁	—	8	1	no	—
121	C₂×C₄×C₃₁	—	8	1	no	—
122	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
124	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
127	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₈×C₃₁	—	2	1	no	—
129	C₈×C₃₁	—	2	1	no	—
130	C₈×C₃₁	—	4	2	no	—
131	C₈×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
133	C₈×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
134	C₈×C₃₁	#123	1	1	yes	characteristic
135	C₂×#443	C₂×C₃₁	1	1	yes	characteristic
136	C₂×#365	C₂×C₃₁	1	1	yes	characteristic, complemented
137	C₂×#360	C₂×C₃₁	1	1	yes	characteristic
138	C₂×#360	C₂×C₃₁	1	1	yes	characteristic
139	#1941	C₂×C₃₁	1	1	yes	characteristic, complemented
140	#1383	C₂×C₃₁	1	1	yes	characteristic, complemented
141	#1376	C₂×C₃₁	1	1	yes	characteristic, complemented
142	C₂²×C₃₁	—	2	1	no	—
143	C₂²×C₃₁	—	2	1	no	—
144	C₂²×C₃₁	—	4	2	no	—
145	C₂²×C₃₁	—	8	1	no	—
146	C₂²×C₃₁	—	8	1	no	—
147	C₂²×C₃₁	#442	1	1	yes	characteristic
148	C₂²×C₃₁	#449	1	1	yes	characteristic
149	C₂²×C₃₁	#476	1	1	yes	characteristic
150	C₄×C₃₁	—	4	2	no	—
151	C₄×C₃₁	—	4	2	no	—
152	C₄×C₃₁	—	8	2	no	—
153	C₄×C₃₁	—	16	1	no	—
154	C₄×C₃₁	#385	1	1	yes	characteristic
155	C₄×C₃₁	C₄×SD₁₆	1	1	yes	characteristic
156	C₄×C₃₁	#450	1	1	yes	characteristic
157	C₄×C₃₁	#478	1	1	yes	characteristic
158	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
159	C₂×#131	—	2	1	no	—
160	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
161	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
162	#443	—	2	2	no	—
163	#443	—	2	2	no	—
164	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
165	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
166	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
167	C₂×#104	—	2	1	no	—
168	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
169	#389	—	2	1	no	—
170	#365	—	2	2	no	—
171	#360	—	2	2	no	—
172	#360	C₄×C₃₁	1	4	yes	complemented
173	#340	—	2	1	no	—
174	C₂×C₃₁	—	2	1	no	—
175	C₂×C₃₁	—	2	1	no	—
176	C₂×C₃₁	—	8	2	no	—
177	C₂×C₃₁	#1864	1	1	yes	characteristic
178	C₂×C₃₁	#2230	1	1	yes	characteristic
179	C₂×C₃₁	#2231	1	1	yes	characteristic
180	C₂×#41	—	2	1	no	—
181	#135	—	2	1	no	—
182	#135	—	2	1	no	—
183	#135	C₂×C₄×C₃₁	1	2	yes	—
184	#135	D₄×C₃₁	1	2	yes	—
185	C₂×Q₁₆	Q₈×C₃₁	1	1	yes	characteristic
186	C₂×SD₁₆	—	2	1	no	—
187	C₂×D₈	Q₈×C₃₁	1	1	yes	characteristic
188	#131	—	2	4	no	—
189	#131	—	4	2	no	—
190	C₂×M₁₆	—	2	1	no	—
191	C₂×M₁₆	—	4	1	no	—
192	C₂²×C₈	—	4	1	no	—
193	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
194	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	4	1	no	—
198	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
199	M₃₂	—	2	1	no	—
200	M₃₂	—	2	1	no	—
201	M₃₂	—	2	1	no	—
202	M₃₂	C₂×C₄×C₃₁	1	2	yes	—
203	C₂×C₁₆	—	2	1	no	—
204	C₂×C₁₆	—	2	1	no	—
205	#108	—	2	1	no	—
206	#108	D₄×C₃₁	1	2	yes	—
207	#106	—	2	1	no	—
208	#106	—	2	1	no	—
209	#106	—	2	1	no	—
210	#106	C₂×C₄×C₃₁	1	2	yes	—
211	#104	—	2	4	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	—	2	1	no	—
214	#95	—	4	1	no	—
215	C₃₁	#12950	1	1	yes	characteristic, direct product factor, Sylow
216	#41	—	2	2	no	—
217	#41	—	2	2	no	—
218	#41	—	4	1	no	—
219	#41	—	4	1	no	—
220	C₂×Q₈	—	2	1	no	—
221	C₂×D₄	—	2	1	no	—
222	C₂×D₄	—	4	1	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	4	1	no	—
225	C₂²×C₄	—	4	1	no	—
226	C₂²×C₄	—	4	1	no	—
227	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
228	Q₁₆	—	2	1	no	—
229	Q₁₆	C₄⋊C₄×C₃₁	1	2	yes	complemented
230	SD₁₆	—	2	2	no	—
231	SD₁₆	—	2	2	no	—
232	D₈	—	2	1	no	—
233	D₈	C₄⋊C₄×C₃₁	1	2	yes	complemented
234	M₁₆	—	2	2	no	—
235	M₁₆	—	4	4	no	—
236	M₁₆	—	4	1	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	4	4	no	—
240	C₂×C₈	—	4	1	no	—
241	C₂×C₈	SD₁₆×C₃₁	1	1	yes	characteristic
242	C₂×C₈	SD₁₆×C₃₁	1	1	yes	characteristic
243	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
244	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
245	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
246	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
247	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
248	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	4	1	no	—
251	C₄⋊C₄	—	4	1	no	—
252	C₄⋊C₄	—	4	1	no	—
253	C₄⋊C₄	—	8	2	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₁₆	—	2	2	no	—
257	C₁₆	—	4	1	no	—
258	C₂³	—	4	1	no	—
259	C₂³	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
260	Q₈	—	2	2	no	—
261	Q₈	—	4	1	no	—
262	D₄	—	2	2	no	—
263	D₄	—	4	2	no	—
264	D₄	—	4	2	no	—
265	D₄	—	4	1	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	—	8	1	no	—
275	C₂×C₄	—	8	1	no	—
276	C₂×C₄	—	8	1	no	—
277	C₂×C₄	—	8	1	no	—
278	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
279	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
280	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
281	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
282	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
283	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	4	2	no	—
287	C₈	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
288	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
289	C₈	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
290	C₈	C₃₁×#123	1	1	yes	characteristic
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	4	2	no	—
294	C₂²	—	8	1	no	—
295	C₂²	—	8	1	no	—
296	C₂²	C₃₁×#442	1	1	yes	characteristic
297	C₂²	C₃₁×#449	1	1	yes	characteristic
298	C₂²	C₃₁×#476	1	1	yes	characteristic
299	C₄	—	4	2	no	—
300	C₄	—	4	2	no	—
301	C₄	—	8	2	no	—
302	C₄	—	16	1	no	—
303	C₄	C₃₁×#385	1	1	yes	characteristic
304	C₄	C₄×SD₁₆×C₃₁	1	1	yes	characteristic
305	C₄	C₃₁×#450	1	1	yes	characteristic
306	C₄	C₃₁×#478	1	1	yes	characteristic
307	C₂	—	2	1	no	—
308	C₂	—	2	1	no	—
309	C₂	—	8	2	no	—
310	C₂	C₃₁×#1864	1	1	yes	characteristic
311	C₂	C₃₁×#2230	1	1	yes	characteristic
312	C₂	C₃₁×#2231	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	#476
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₂×C₃₁×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#365	C₂	semidirect
5. C₃₁×#1941	C₂	semidirect
6. C₃₁×#1383	C₂	semidirect
7. C₃₁×#1376	C₂	semidirect
22. C₃₁×#360	C₄	semidirect
72. Q₁₆×C₃₁	C₄⋊C₄	semidirect
76. D₈×C₃₁	C₄⋊C₄	semidirect
101. #12950	C₃₁	direct (N × H)
136. C₂×#365	C₂×C₃₁	semidirect
139. #1941	C₂×C₃₁	semidirect
140. #1383	C₂×C₃₁	semidirect
141. #1376	C₂×C₃₁	semidirect
172. #360	C₄×C₃₁	semidirect
215. C₃₁	#12950	direct (N × H)
229. Q₁₆	C₄⋊C₄×C₃₁	semidirect
233. D₈	C₄⋊C₄×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#443	2
C₂×C₃₁×#365	2
C₂×C₃₁×#360	2
C₃₁×#1941	2
C₃₁×#1383	2
C₃₁×#1376	2
#12950	31

Subgroups — C31×#12950 #21469709

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12950 #21469709