Subgroups — C₃₁×#8880 #21465639

Order 7936.

432 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×#2932	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1940	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1570	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1502	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1480	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1303	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1281	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#131	—	2	1	no	—
9	C₃₁×#532	C₂²	1	1	yes	characteristic
10	C₃₁×#435	—	2	2	no	—
11	C₃₁×#435	—	2	2	no	—
12	C₃₁×#431	C₂²	1	1	yes	characteristic
13	C₃₁×#431	C₂²	1	1	yes	characteristic
14	C₃₁×#423	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₃₁×#408	—	2	1	no	—
18	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×#336	—	2	1	no	—
21	C₃₁×#334	—	2	2	no	—
22	C₃₁×#327	C₄	1	2	yes	—
23	C₃₁×#327	C₄	1	2	yes	—
24	C₃₁×D₄⋊C₈	—	2	2	no	—
25	C₂×#41×C₃₁	—	2	1	no	—
26	C₃₁×#131	—	4	2	no	—
27	C₃₁×#131	—	4	2	no	—
28	C₂×M₁₆×C₃₁	—	2	2	no	—
29	C₂×M₁₆×C₃₁	—	2	1	no	—
30	C₂²×C₈×C₃₁	—	2	2	no	—
31	C₂²×C₈×C₃₁	—	2	1	no	—
32	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
34	C₃₁×C₄⋊Q₈	Q₈	1	1	yes	characteristic
35	C₃₁×#127	Q₈	1	1	yes	characteristic
36	C₃₁×#124	—	2	1	no	—
37	C₃₁×#121	—	2	1	no	—
38	C₃₁×#121	C₂×C₄	1	2	yes	—
39	C₄×D₄×C₃₁	—	2	1	no	—
40	C₄×D₄×C₃₁	D₄	1	2	yes	—
41	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
42	C₂×C₄²×C₃₁	—	2	1	no	—
43	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
44	C₃₁×C₄⋊C₈	—	2	1	no	—
45	C₃₁×C₄⋊C₈	—	2	1	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	—	4	2	no	—
50	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
51	C₃₁×C₂²⋊C₈	—	2	4	no	—
52	C₃₁×C₂²⋊C₈	—	2	1	no	—
53	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
54	C₃₁×#97	—	2	1	no	—
55	C₃₁×#97	—	2	1	no	—
56	C₄×C₈×C₃₁	—	2	1	no	—
57	C₄×C₈×C₃₁	—	2	1	no	—
58	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
59	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
60	#41×C₃₁	—	4	1	no	—
61	#41×C₃₁	—	4	1	no	—
62	#41×C₃₁	—	4	1	no	—
63	#41×C₃₁	—	4	1	no	—
64	C₂×Q₈×C₃₁	—	2	1	no	—
65	C₂×D₄×C₃₁	—	2	2	no	—
66	C₂×D₄×C₃₁	—	2	1	no	—
67	C₂×D₄×C₃₁	—	2	1	no	—
68	C₂×D₄×C₃₁	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₃₁	—	2	1	no	—
70	C₂²×C₄×C₃₁	—	2	2	no	—
71	C₂²×C₄×C₃₁	—	2	1	no	—
72	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
75	M₁₆×C₃₁	—	4	4	no	—
76	M₁₆×C₃₁	—	4	1	no	—
77	M₁₆×C₃₁	—	4	1	no	—
78	C₂×C₈×C₃₁	—	2	1	no	—
79	C₂×C₈×C₃₁	—	2	1	no	—
80	C₂×C₈×C₃₁	—	2	2	no	—
81	C₂×C₈×C₃₁	—	2	1	no	—
82	C₂×C₈×C₃₁	—	2	2	no	—
83	C₂×C₈×C₃₁	—	2	1	no	—
84	C₂×C₈×C₃₁	—	2	1	no	—
85	C₂×C₈×C₃₁	—	4	4	no	—
86	C₂×C₈×C₃₁	—	4	1	no	—
87	C₂×C₈×C₃₁	—	4	1	no	—
88	C₂×C₈×C₃₁	—	4	1	no	—
89	C₂×C₈×C₃₁	—	4	1	no	—
90	C₂×C₈×C₃₁	SD₁₆	1	2	yes	—
91	C₄⋊C₄×C₃₁	—	2	1	no	—
92	C₄⋊C₄×C₃₁	—	4	2	no	—
93	C₄⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
94	C₂²⋊C₄×C₃₁	—	2	2	no	—
95	C₂²⋊C₄×C₃₁	—	2	2	no	—
96	C₄²×C₃₁	—	4	1	no	—
97	C₄²×C₃₁	—	4	1	no	—
98	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
101	C₄²×C₃₁	#41	1	1	yes	characteristic
102	#8880	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂³×C₃₁	—	2	2	no	—
104	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
105	Q₈×C₃₁	—	4	1	no	—
106	Q₈×C₃₁	—	4	1	no	—
107	D₄×C₃₁	—	4	2	no	—
108	D₄×C₃₁	—	4	2	no	—
109	D₄×C₃₁	—	4	2	no	—
110	D₄×C₃₁	—	4	1	no	—
111	D₄×C₃₁	—	4	1	no	—
112	D₄×C₃₁	—	8	1	no	—
113	C₂×C₄×C₃₁	—	2	2	no	—
114	C₂×C₄×C₃₁	—	2	2	no	—
115	C₂×C₄×C₃₁	—	4	1	no	—
116	C₂×C₄×C₃₁	—	4	1	no	—
117	C₂×C₄×C₃₁	—	4	1	no	—
118	C₂×C₄×C₃₁	—	4	1	no	—
119	C₂×C₄×C₃₁	—	4	2	no	—
120	C₂×C₄×C₃₁	—	4	2	no	—
121	C₂×C₄×C₃₁	—	4	1	no	—
122	C₂×C₄×C₃₁	—	4	1	no	—
123	C₂×C₄×C₃₁	—	4	1	no	—
124	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
127	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
129	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
130	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
131	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
132	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₈×C₃₁	—	2	2	no	—
135	C₈×C₃₁	—	4	2	no	—
136	C₈×C₃₁	—	4	1	no	—
137	C₈×C₃₁	—	4	2	no	—
138	C₈×C₃₁	—	8	1	no	—
139	C₈×C₃₁	—	8	1	no	—
140	#2932	C₂×C₃₁	1	1	yes	characteristic
141	#1940	C₂×C₃₁	1	1	yes	characteristic, complemented
142	#1570	C₂×C₃₁	1	1	yes	characteristic, complemented
143	#1502	C₂×C₃₁	1	1	yes	characteristic
144	#1480	C₂×C₃₁	1	1	yes	characteristic
145	#1303	C₂×C₃₁	1	1	yes	characteristic, complemented
146	#1281	C₂×C₃₁	1	1	yes	characteristic, complemented
147	C₂²×C₃₁	—	2	1	no	—
148	C₂²×C₃₁	—	2	1	no	—
149	C₂²×C₃₁	—	4	2	no	—
150	C₂²×C₃₁	—	4	2	no	—
151	C₂²×C₃₁	—	4	2	no	—
152	C₂²×C₃₁	#385	1	1	yes	characteristic
153	C₂²×C₃₁	C₄×SD₁₆	1	1	yes	characteristic
154	C₂²×C₃₁	#442	1	1	yes	characteristic
155	C₄×C₃₁	—	2	1	no	—
156	C₄×C₃₁	—	2	1	no	—
157	C₄×C₃₁	—	2	1	no	—
158	C₄×C₃₁	—	2	1	no	—
159	C₄×C₃₁	—	4	2	no	—
160	C₄×C₃₁	—	4	2	no	—
161	C₄×C₃₁	—	8	1	no	—
162	C₄×C₃₁	#449	1	1	yes	characteristic
163	C₄×C₃₁	#450	1	1	yes	characteristic
164	C₄×C₃₁	#476	1	1	yes	characteristic
165	C₄×C₃₁	#478	1	1	yes	characteristic
166	C₂×#131	—	2	1	no	—
167	#532	C₂²×C₃₁	1	1	yes	characteristic
168	#435	—	2	2	no	—
169	#435	—	2	2	no	—
170	#431	C₂²×C₃₁	1	1	yes	characteristic
171	#431	C₂²×C₃₁	1	1	yes	characteristic
172	#423	C₂²×C₃₁	1	1	yes	characteristic
173	C₂×C₄⋊C₈	—	2	1	no	—
174	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
175	#408	—	2	1	no	—
176	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
177	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
178	#336	—	2	1	no	—
179	#334	—	2	2	no	—
180	#327	C₄×C₃₁	1	2	yes	—
181	#327	C₄×C₃₁	1	2	yes	—
182	D₄⋊C₈	—	2	2	no	—
183	C₂×C₃₁	—	2	1	no	—
184	C₂×C₃₁	—	2	1	no	—
185	C₂×C₃₁	—	8	2	no	—
186	C₂×C₃₁	#1861	1	1	yes	characteristic
187	C₂×C₃₁	#1864	1	1	yes	characteristic
188	C₂×C₃₁	#1868	1	1	yes	characteristic
189	C₂×#41	—	2	1	no	—
190	#131	—	4	2	no	—
191	#131	—	4	2	no	—
192	C₂×M₁₆	—	2	2	no	—
193	C₂×M₁₆	—	2	1	no	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	—	2	1	no	—
196	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
197	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
198	C₄⋊Q₈	Q₈×C₃₁	1	1	yes	characteristic
199	#127	Q₈×C₃₁	1	1	yes	characteristic
200	#124	—	2	1	no	—
201	#121	—	2	1	no	—
202	#121	C₂×C₄×C₃₁	1	2	yes	—
203	C₄×D₄	—	2	1	no	—
204	C₄×D₄	D₄×C₃₁	1	2	yes	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄²	—	2	1	no	—
207	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
208	C₄⋊C₈	—	2	1	no	—
209	C₄⋊C₈	—	2	1	no	—
210	C₄⋊C₈	—	2	1	no	—
211	C₄⋊C₈	—	2	1	no	—
212	C₄⋊C₈	—	2	1	no	—
213	C₄⋊C₈	—	4	2	no	—
214	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
215	C₂²⋊C₈	—	2	4	no	—
216	C₂²⋊C₈	—	2	1	no	—
217	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
218	#97	—	2	1	no	—
219	#97	—	2	1	no	—
220	C₄×C₈	—	2	1	no	—
221	C₄×C₈	—	2	1	no	—
222	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
223	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
224	C₃₁	#8880	1	1	yes	characteristic, direct product factor, Sylow
225	#41	—	4	1	no	—
226	#41	—	4	1	no	—
227	#41	—	4	1	no	—
228	#41	—	4	1	no	—
229	C₂×Q₈	—	2	1	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	2	1	no	—
232	C₂×D₄	—	2	1	no	—
233	C₂×D₄	C₄⋊C₄×C₃₁	1	2	yes	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
239	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
240	M₁₆	—	4	4	no	—
241	M₁₆	—	4	1	no	—
242	M₁₆	—	4	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	4	4	no	—
251	C₂×C₈	—	4	1	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	SD₁₆×C₃₁	1	2	yes	—
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
259	C₂²⋊C₄	—	2	2	no	—
260	C₂²⋊C₄	—	2	2	no	—
261	C₄²	—	4	1	no	—
262	C₄²	—	4	1	no	—
263	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
264	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
265	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
266	C₄²	#41×C₃₁	1	1	yes	characteristic
267	C₂³	—	2	2	no	—
268	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
269	Q₈	—	4	1	no	—
270	Q₈	—	4	1	no	—
271	D₄	—	4	2	no	—
272	D₄	—	4	2	no	—
273	D₄	—	4	2	no	—
274	D₄	—	4	1	no	—
275	D₄	—	4	1	no	—
276	D₄	—	8	1	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
289	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
290	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
291	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
292	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
293	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
294	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
295	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
296	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
297	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
298	C₈	—	2	2	no	—
299	C₈	—	4	2	no	—
300	C₈	—	4	1	no	—
301	C₈	—	4	2	no	—
302	C₈	—	8	1	no	—
303	C₈	—	8	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	4	2	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	4	2	no	—
309	C₂²	C₃₁×#385	1	1	yes	characteristic
310	C₂²	C₄×SD₁₆×C₃₁	1	1	yes	characteristic
311	C₂²	C₃₁×#442	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	4	2	no	—
317	C₄	—	4	2	no	—
318	C₄	—	8	1	no	—
319	C₄	C₃₁×#449	1	1	yes	characteristic
320	C₄	C₃₁×#450	1	1	yes	characteristic
321	C₄	C₃₁×#476	1	1	yes	characteristic
322	C₄	C₃₁×#478	1	1	yes	characteristic
323	C₂	—	2	1	no	—
324	C₂	—	2	1	no	—
325	C₂	—	8	2	no	—
326	C₂	C₃₁×#1861	1	1	yes	characteristic
327	C₂	C₃₁×#1864	1	1	yes	characteristic
328	C₂	C₃₁×#1868	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#120
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	#385
prime radical	C₃₁×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ 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×#1940	C₂	semidirect
3. C₃₁×#1570	C₂	semidirect
6. C₃₁×#1303	C₂	semidirect
7. C₃₁×#1281	C₂	semidirect
102. #8880	C₃₁	direct (N × H)
141. #1940	C₂×C₃₁	semidirect
142. #1570	C₂×C₃₁	semidirect
145. #1303	C₂×C₃₁	semidirect
146. #1281	C₂×C₃₁	semidirect
224. C₃₁	#8880	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×#2932	2
C₃₁×#1940	2
C₃₁×#1570	2
C₃₁×#1502	2
C₃₁×#1480	2
C₃₁×#1303	2
C₃₁×#1281	2
#8880	31

Subgroups — C31×#8880 #21465639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8880 #21465639