Subgroups — C₃₁×#8863 #21465622

Order 7936.

374 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2921	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1798	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1571	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1570	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1458	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1301	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1301	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
9	C₃₁×#433	—	2	2	no	—
10	C₃₁×#433	—	2	2	no	—
11	C₃₁×#431	C₂²	1	1	yes	characteristic
12	C₃₁×#431	C₂²	1	1	yes	characteristic
13	C₃₁×#423	C₂²	1	1	yes	characteristic
14	C₃₁×#423	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₃₁×#407	—	2	1	no	—
18	C₂×C₃₁×C₂²⋊C₈	—	2	1	no	—
19	C₄×M₁₆×C₃₁	—	2	1	no	—
20	C₃₁×#336	—	2	1	no	—
21	C₃₁×#335	—	2	2	no	—
22	C₃₁×#334	—	2	2	no	—
23	C₃₁×#324	C₄	1	2	yes	—
24	C₃₁×#323	C₄	1	2	yes	—
25	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
26	C₂×M₁₆×C₃₁	—	2	1	no	—
27	C₂×M₁₆×C₃₁	—	2	1	no	—
28	C₂²×C₈×C₃₁	—	2	1	no	—
29	C₂²×C₈×C₃₁	—	2	1	no	—
30	C₃₁×#117	—	2	1	no	—
31	C₃₁×#117	—	2	1	no	—
32	C₃₁×#117	—	2	1	no	—
33	C₃₁×#117	—	2	1	no	—
34	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₃₁	Q₈	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₃₁	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
40	C₃₁×C₄⋊C₈	—	2	1	no	—
41	C₃₁×C₄⋊C₈	—	2	1	no	—
42	C₃₁×C₄⋊C₈	—	2	1	no	—
43	C₃₁×C₄⋊C₈	—	2	1	no	—
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	D₄	1	2	yes	—
49	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
50	C₃₁×C₂²⋊C₈	—	2	4	no	—
51	C₃₁×C₂²⋊C₈	—	4	2	no	—
52	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
53	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
54	C₃₁×#97	—	2	2	no	—
55	C₄×C₈×C₃₁	—	2	2	no	—
56	C₄×C₈×C₃₁	—	2	1	no	—
57	C₄×C₈×C₃₁	—	2	1	no	—
58	C₂⁴×C₃₁	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	—	2	1	no	—
60	C₂²×C₄×C₃₁	—	2	1	no	—
61	C₂²×C₄×C₃₁	—	2	1	no	—
62	C₂²×C₄×C₃₁	—	2	1	no	—
63	C₂²×C₄×C₃₁	—	2	1	no	—
64	C₂²×C₄×C₃₁	—	2	1	no	—
65	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
71	M₁₆×C₃₁	—	4	1	no	—
72	M₁₆×C₃₁	—	4	1	no	—
73	M₁₆×C₃₁	—	4	1	no	—
74	M₁₆×C₃₁	—	4	1	no	—
75	C₂×C₈×C₃₁	—	2	2	no	—
76	C₂×C₈×C₃₁	—	2	2	no	—
77	C₂×C₈×C₃₁	—	2	2	no	—
78	C₂×C₈×C₃₁	—	2	2	no	—
79	C₂×C₈×C₃₁	—	4	1	no	—
80	C₂×C₈×C₃₁	—	4	1	no	—
81	C₂×C₈×C₃₁	—	4	1	no	—
82	C₂×C₈×C₃₁	—	4	1	no	—
83	C₂×C₈×C₃₁	—	4	1	no	—
84	C₂×C₈×C₃₁	—	4	1	no	—
85	C₄⋊C₄×C₃₁	—	2	1	no	—
86	C₄⋊C₄×C₃₁	—	2	1	no	—
87	C₄⋊C₄×C₃₁	—	2	1	no	—
88	C₄⋊C₄×C₃₁	—	2	1	no	—
89	C₂²⋊C₄×C₃₁	—	2	1	no	—
90	C₂²⋊C₄×C₃₁	—	2	1	no	—
91	C₂²⋊C₄×C₃₁	—	2	1	no	—
92	C₂²⋊C₄×C₃₁	—	2	1	no	—
93	C₄²×C₃₁	—	2	1	no	—
94	C₄²×C₃₁	—	2	1	no	—
95	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₃₁	#41	1	1	yes	characteristic
98	C₄²×C₃₁	#41	1	1	yes	characteristic
99	#8863	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂³×C₃₁	—	2	1	no	—
101	C₂³×C₃₁	—	2	1	no	—
102	C₂³×C₃₁	—	2	1	no	—
103	C₂³×C₃₁	—	4	1	no	—
104	C₂³×C₃₁	—	4	1	no	—
105	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₃₁	—	2	1	no	—
107	C₂×C₄×C₃₁	—	2	1	no	—
108	C₂×C₄×C₃₁	—	2	1	no	—
109	C₂×C₄×C₃₁	—	2	1	no	—
110	C₂×C₄×C₃₁	—	2	1	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	1	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	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
122	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
125	C₂×C₄×C₃₁	#135	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	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
129	C₈×C₃₁	—	4	2	no	—
130	C₈×C₃₁	—	4	2	no	—
131	C₈×C₃₁	—	4	2	no	—
132	C₈×C₃₁	—	4	2	no	—
133	C₈×C₃₁	—	8	1	no	—
134	C₈×C₃₁	—	8	1	no	—
135	#2921	C₂×C₃₁	1	1	yes	characteristic
136	#1798	C₂×C₃₁	1	1	yes	characteristic
137	#1571	C₂×C₃₁	1	1	yes	characteristic, complemented
138	#1570	C₂×C₃₁	1	1	yes	characteristic, complemented
139	#1458	C₂×C₃₁	1	1	yes	characteristic
140	#1301	C₂×C₃₁	1	1	yes	characteristic, complemented
141	#1301	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	—	4	1	no	—
146	C₂²×C₃₁	—	4	1	no	—
147	C₂²×C₃₁	—	4	1	no	—
148	C₂²×C₃₁	—	8	1	no	—
149	C₂²×C₃₁	#385	1	1	yes	characteristic
150	C₂²×C₃₁	#427	1	1	yes	characteristic
151	C₂²×C₃₁	#428	1	1	yes	characteristic
152	C₄×C₃₁	—	2	1	no	—
153	C₄×C₃₁	—	2	1	no	—
154	C₄×C₃₁	—	2	1	no	—
155	C₄×C₃₁	—	2	1	no	—
156	C₄×C₃₁	—	4	1	no	—
157	C₄×C₃₁	—	4	1	no	—
158	C₄×C₃₁	—	4	1	no	—
159	C₄×C₃₁	#449	1	1	yes	characteristic
160	C₄×C₃₁	#452	1	1	yes	characteristic
161	C₄×C₃₁	#482	1	1	yes	characteristic
162	C₄×C₃₁	#485	1	1	yes	characteristic
163	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
164	#433	—	2	2	no	—
165	#433	—	2	2	no	—
166	#431	C₂²×C₃₁	1	1	yes	characteristic
167	#431	C₂²×C₃₁	1	1	yes	characteristic
168	#423	C₂²×C₃₁	1	1	yes	characteristic
169	#423	C₂²×C₃₁	1	1	yes	characteristic
170	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
171	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
172	#407	—	2	1	no	—
173	C₂×C₂²⋊C₈	—	2	1	no	—
174	C₄×M₁₆	—	2	1	no	—
175	#336	—	2	1	no	—
176	#335	—	2	2	no	—
177	#334	—	2	2	no	—
178	#324	C₄×C₃₁	1	2	yes	—
179	#323	C₄×C₃₁	1	2	yes	—
180	C₂×C₃₁	—	2	1	no	—
181	C₂×C₃₁	—	2	1	no	—
182	C₂×C₃₁	—	8	1	no	—
183	C₂×C₃₁	#1854	1	1	yes	characteristic
184	C₂×C₃₁	#1858	1	1	yes	characteristic
185	C₂×C₃₁	#1868	1	1	yes	characteristic
186	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
187	C₂×M₁₆	—	2	1	no	—
188	C₂×M₁₆	—	2	1	no	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	—	2	1	no	—
191	#117	—	2	1	no	—
192	#117	—	2	1	no	—
193	#117	—	2	1	no	—
194	#117	—	2	1	no	—
195	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
196	C₂×C₄⋊C₄	Q₈×C₃₁	1	1	yes	characteristic
197	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
198	C₂×C₂²⋊C₄	Q₈×C₃₁	1	1	yes	characteristic
199	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
200	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
201	C₄⋊C₈	—	2	1	no	—
202	C₄⋊C₈	—	2	1	no	—
203	C₄⋊C₈	—	2	1	no	—
204	C₄⋊C₈	—	2	1	no	—
205	C₄⋊C₈	—	2	1	no	—
206	C₄⋊C₈	—	2	1	no	—
207	C₄⋊C₈	—	2	1	no	—
208	C₄⋊C₈	—	2	1	no	—
209	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
210	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
211	C₂²⋊C₈	—	2	4	no	—
212	C₂²⋊C₈	—	4	2	no	—
213	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
214	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
215	#97	—	2	2	no	—
216	C₄×C₈	—	2	2	no	—
217	C₄×C₈	—	2	1	no	—
218	C₄×C₈	—	2	1	no	—
219	C₃₁	#8863	1	1	yes	characteristic, direct product factor, Sylow
220	C₂⁴	C₄⋊C₄×C₃₁	1	1	yes	characteristic
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
228	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
229	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
230	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
232	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
233	M₁₆	—	4	1	no	—
234	M₁₆	—	4	1	no	—
235	M₁₆	—	4	1	no	—
236	M₁₆	—	4	1	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	4	1	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	—	4	1	no	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	2	1	no	—
252	C₂²⋊C₄	—	2	1	no	—
253	C₂²⋊C₄	—	2	1	no	—
254	C₂²⋊C₄	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
258	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
259	C₄²	#41×C₃₁	1	1	yes	characteristic
260	C₄²	#41×C₃₁	1	1	yes	characteristic
261	C₂³	—	2	1	no	—
262	C₂³	—	2	1	no	—
263	C₂³	—	2	1	no	—
264	C₂³	—	4	1	no	—
265	C₂³	—	4	1	no	—
266	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
281	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
282	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
283	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
284	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
285	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
286	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
287	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
288	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
289	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
290	C₈	—	4	2	no	—
291	C₈	—	4	2	no	—
292	C₈	—	4	2	no	—
293	C₈	—	4	2	no	—
294	C₈	—	8	1	no	—
295	C₈	—	8	1	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	4	2	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	1	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	8	1	no	—
303	C₂²	C₃₁×#385	1	1	yes	characteristic
304	C₂²	C₃₁×#427	1	1	yes	characteristic
305	C₂²	C₃₁×#428	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	4	1	no	—
312	C₄	—	4	1	no	—
313	C₄	C₃₁×#449	1	1	yes	characteristic
314	C₄	C₃₁×#452	1	1	yes	characteristic
315	C₄	C₃₁×#482	1	1	yes	characteristic
316	C₄	C₃₁×#485	1	1	yes	characteristic
317	C₂	—	2	1	no	—
318	C₂	—	2	1	no	—
319	C₂	—	8	1	no	—
320	C₂	C₃₁×#1854	1	1	yes	characteristic
321	C₂	C₃₁×#1858	1	1	yes	characteristic
322	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	C₂⁴×C₃₁	C₄⋊C₄
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
3. C₃₁×#1571	C₂	semidirect
4. C₃₁×#1570	C₂	semidirect
6. C₃₁×#1301	C₂	semidirect
7. C₃₁×#1301	C₂	semidirect
99. #8863	C₃₁	direct (N × H)
137. #1571	C₂×C₃₁	semidirect
138. #1570	C₂×C₃₁	semidirect
140. #1301	C₂×C₃₁	semidirect
141. #1301	C₂×C₃₁	semidirect
219. C₃₁	#8863	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×#2921	2
C₃₁×#1798	2
C₃₁×#1571	2
C₃₁×#1570	2
C₃₁×#1458	2
C₃₁×#1301	2
#8863	31

Subgroups — C31×#8863 #21465622

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8863 #21465622