Subgroups — C₃₁×#20013 #21476772

Order 7936.

572 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3365	C₂	1	2	yes	maximal
2	C₃₁×#3363	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1708	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1684	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1682	C₂	1	2	yes	maximal
6	C₃₁×#1653	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1630	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#1623	C₂	1	1	yes	maximal, characteristic
9	C₈×Q₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
10	C₃₁×#559	C₂²	1	1	yes	characteristic
11	Q₈²×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#557	C₂²	1	2	yes	—
13	C₃₁×#498	C₂²	1	1	yes	characteristic
14	C₃₁×#492	C₂²	1	1	yes	characteristic
15	C₃₁×#488	—	2	2	no	—
16	C₃₁×#487	—	2	2	no	—
17	C₃₁×#487	C₂²	1	2	yes	—
18	C₃₁×#486	C₂²	1	2	yes	—
19	C₃₁×#478	—	2	2	no	—
20	C₃₁×#477	—	2	2	no	—
21	C₃₁×#477	C₂²	1	2	yes	—
22	C₃₁×#475	C₂²	1	2	yes	—
23	C₃₁×#464	—	2	2	no	—
24	C₃₁×#462	—	2	2	no	—
25	C₃₁×#462	C₂²	1	2	yes	—
26	C₃₁×#460	C₂²	1	2	yes	—
27	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
28	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
29	C₄×Q₁₆×C₃₁	—	2	2	no	—
30	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
31	C₄×SD₁₆×C₃₁	—	2	2	no	—
32	C₃₁×#335	C₂²	1	2	yes	—
33	C₃₁×#333	C₂²	1	1	yes	characteristic
34	C₃₁×#332	C₂²	1	2	yes	—
35	C₃₁×#331	C₂²	1	1	yes	characteristic
36	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
37	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
38	C₈²×C₃₁	C₂²	1	1	yes	characteristic
39	C₂×Q₁₆×C₃₁	—	2	2	no	—
40	C₂×Q₁₆×C₃₁	D₄	1	2	yes	—
41	C₂×SD₁₆×C₃₁	—	2	2	no	—
42	C₂×SD₁₆×C₃₁	—	4	1	no	—
43	C₃₁×C₄⋊Q₈	—	2	2	no	—
44	C₃₁×C₄⋊Q₈	—	2	2	no	—
45	C₃₁×C₄⋊Q₈	—	4	1	no	—
46	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
47	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
48	C₃₁×#127	C₂³	1	1	yes	characteristic
49	C₃₁×#125	—	2	2	no	—
50	C₃₁×#125	—	4	2	no	—
51	C₃₁×#124	—	2	2	no	—
52	C₃₁×#123	—	4	1	no	—
53	C₄×Q₈×C₃₁	—	2	2	no	—
54	C₄×Q₈×C₃₁	—	2	2	no	—
55	C₄×Q₈×C₃₁	—	2	2	no	—
56	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
57	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
58	C₄×D₄×C₃₁	—	2	2	no	—
59	C₃₁×#107	—	2	2	no	—
60	C₃₁×#107	D₄	1	2	yes	—
61	C₃₁×#106	—	2	2	no	—
62	C₃₁×#106	—	4	1	no	—
63	C₃₁×C₄⋊C₈	—	2	2	no	—
64	C₃₁×C₄⋊C₈	—	2	2	no	—
65	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
66	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
67	C₃₁×#103	—	2	2	no	—
68	C₃₁×#103	—	2	2	no	—
69	C₃₁×#103	—	2	2	no	—
70	C₃₁×#103	—	2	2	no	—
71	C₃₁×#103	—	2	2	no	—
72	C₃₁×#103	D₄	1	2	yes	—
73	C₃₁×#103	D₄	1	2	yes	—
74	C₃₁×#102	—	2	2	no	—
75	C₃₁×#102	—	2	2	no	—
76	C₃₁×#102	—	2	2	no	—
77	C₄×C₈×C₃₁	—	2	2	no	—
78	C₄×C₈×C₃₁	—	2	2	no	—
79	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₃₁	C₂³	1	2	yes	—
81	C₂×Q₈×C₃₁	—	2	2	no	—
82	C₂×Q₈×C₃₁	—	2	1	no	—
83	C₂×Q₈×C₃₁	—	4	1	no	—
84	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
85	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
86	C₂×D₄×C₃₁	—	2	2	no	—
87	C₂×D₄×C₃₁	—	2	1	no	—
88	C₂²×C₄×C₃₁	—	4	1	no	—
89	Q₁₆×C₃₁	—	2	4	no	—
90	Q₁₆×C₃₁	—	4	4	no	—
91	SD₁₆×C₃₁	—	4	4	no	—
92	SD₁₆×C₃₁	—	8	2	no	—
93	C₂×C₈×C₃₁	—	2	1	no	—
94	C₂×C₈×C₃₁	—	2	2	no	—
95	C₂×C₈×C₃₁	—	2	2	no	—
96	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
98	C₄⋊C₄×C₃₁	—	2	2	no	—
99	C₄⋊C₄×C₃₁	—	2	2	no	—
100	C₄⋊C₄×C₃₁	—	2	2	no	—
101	C₄⋊C₄×C₃₁	—	2	2	no	—
102	C₄⋊C₄×C₃₁	—	2	2	no	—
103	C₄⋊C₄×C₃₁	—	4	2	no	—
104	C₄⋊C₄×C₃₁	—	4	2	no	—
105	C₄⋊C₄×C₃₁	—	4	2	no	—
106	C₄⋊C₄×C₃₁	—	4	2	no	—
107	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
108	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
109	C₂²⋊C₄×C₃₁	—	4	2	no	—
110	C₂²⋊C₄×C₃₁	—	4	1	no	—
111	C₄²×C₃₁	—	2	2	no	—
112	C₄²×C₃₁	—	2	2	no	—
113	C₄²×C₃₁	—	4	2	no	—
114	C₄²×C₃₁	—	4	1	no	—
115	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
116	#20013	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂³×C₃₁	—	4	1	no	—
118	Q₈×C₃₁	—	2	2	no	—
119	Q₈×C₃₁	—	2	2	no	—
120	Q₈×C₃₁	—	2	2	no	—
121	Q₈×C₃₁	—	4	2	no	—
122	Q₈×C₃₁	—	4	2	no	—
123	Q₈×C₃₁	—	8	1	no	—
124	Q₈×C₃₁	—	16	1	no	—
125	Q₈×C₃₁	#135	1	4	yes	—
126	D₄×C₃₁	—	4	2	no	—
127	D₄×C₃₁	—	4	2	no	—
128	D₄×C₃₁	—	8	1	no	—
129	C₂×C₄×C₃₁	—	2	2	no	—
130	C₂×C₄×C₃₁	—	2	2	no	—
131	C₂×C₄×C₃₁	—	2	2	no	—
132	C₂×C₄×C₃₁	—	2	2	no	—
133	C₂×C₄×C₃₁	—	4	2	no	—
134	C₂×C₄×C₃₁	—	4	1	no	—
135	C₂×C₄×C₃₁	—	16	1	no	—
136	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
137	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
138	C₈×C₃₁	—	2	2	no	—
139	C₈×C₃₁	—	2	2	no	—
140	C₈×C₃₁	—	2	2	no	—
141	C₈×C₃₁	—	4	2	no	—
142	C₈×C₃₁	—	4	2	no	—
143	#3365	C₂×C₃₁	1	2	yes	—
144	#3363	C₂×C₃₁	1	2	yes	complemented
145	#1708	C₂×C₃₁	1	1	yes	characteristic
146	#1684	C₂×C₃₁	1	2	yes	complemented
147	#1682	C₂×C₃₁	1	2	yes	—
148	#1653	C₂×C₃₁	1	2	yes	complemented
149	#1630	C₂×C₃₁	1	1	yes	characteristic
150	#1623	C₂×C₃₁	1	1	yes	characteristic
151	C₈×Q₁₆	C₂×C₃₁	1	2	yes	complemented
152	C₂²×C₃₁	—	8	2	no	—
153	C₂²×C₃₁	—	8	1	no	—
154	C₂²×C₃₁	D₄²	1	1	yes	characteristic
155	C₄×C₃₁	—	2	4	no	—
156	C₄×C₃₁	—	2	1	no	—
157	C₄×C₃₁	—	4	2	no	—
158	C₄×C₃₁	—	4	2	no	—
159	C₄×C₃₁	—	4	2	no	—
160	C₄×C₃₁	—	8	2	no	—
161	C₄×C₃₁	—	8	1	no	—
162	C₄×C₃₁	C₂×#135	1	2	yes	—
163	C₄×C₃₁	#578	1	2	yes	—
164	#559	C₂²×C₃₁	1	1	yes	characteristic
165	Q₈²	C₂²×C₃₁	1	1	yes	characteristic
166	#557	C₂²×C₃₁	1	2	yes	—
167	#498	C₂²×C₃₁	1	1	yes	characteristic
168	#492	C₂²×C₃₁	1	1	yes	characteristic
169	#488	—	2	2	no	—
170	#487	—	2	2	no	—
171	#487	C₂²×C₃₁	1	2	yes	—
172	#486	C₂²×C₃₁	1	2	yes	—
173	#478	—	2	2	no	—
174	#477	—	2	2	no	—
175	#477	C₂²×C₃₁	1	2	yes	—
176	#475	C₂²×C₃₁	1	2	yes	—
177	#464	—	2	2	no	—
178	#462	—	2	2	no	—
179	#462	C₂²×C₃₁	1	2	yes	—
180	#460	C₂²×C₃₁	1	2	yes	—
181	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
182	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
183	C₄×Q₁₆	—	2	2	no	—
184	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
185	C₄×SD₁₆	—	2	2	no	—
186	#335	C₂²×C₃₁	1	2	yes	—
187	#333	C₂²×C₃₁	1	1	yes	characteristic
188	#332	C₂²×C₃₁	1	2	yes	—
189	#331	C₂²×C₃₁	1	1	yes	characteristic
190	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
191	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
192	C₈²	C₂²×C₃₁	1	1	yes	characteristic
193	C₂×C₃₁	—	16	1	no	—
194	C₂×C₃₁	#3285	1	2	yes	—
195	C₂×C₃₁	#3291	1	1	yes	characteristic
196	C₂×Q₁₆	—	2	2	no	—
197	C₂×Q₁₆	D₄×C₃₁	1	2	yes	—
198	C₂×SD₁₆	—	2	2	no	—
199	C₂×SD₁₆	—	4	1	no	—
200	C₄⋊Q₈	—	2	2	no	—
201	C₄⋊Q₈	—	2	2	no	—
202	C₄⋊Q₈	—	4	1	no	—
203	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
204	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
205	#127	C₂³×C₃₁	1	1	yes	characteristic
206	#125	—	2	2	no	—
207	#125	—	4	2	no	—
208	#124	—	2	2	no	—
209	#123	—	4	1	no	—
210	C₄×Q₈	—	2	2	no	—
211	C₄×Q₈	—	2	2	no	—
212	C₄×Q₈	—	2	2	no	—
213	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
214	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
215	C₄×D₄	—	2	2	no	—
216	#107	—	2	2	no	—
217	#107	D₄×C₃₁	1	2	yes	—
218	#106	—	2	2	no	—
219	#106	—	4	1	no	—
220	C₄⋊C₈	—	2	2	no	—
221	C₄⋊C₈	—	2	2	no	—
222	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
223	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
224	#103	—	2	2	no	—
225	#103	—	2	2	no	—
226	#103	—	2	2	no	—
227	#103	—	2	2	no	—
228	#103	—	2	2	no	—
229	#103	D₄×C₃₁	1	2	yes	—
230	#103	D₄×C₃₁	1	2	yes	—
231	#102	—	2	2	no	—
232	#102	—	2	2	no	—
233	#102	—	2	2	no	—
234	C₄×C₈	—	2	2	no	—
235	C₄×C₈	—	2	2	no	—
236	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
237	C₄×C₈	C₂³×C₃₁	1	2	yes	—
238	C₃₁	#20013	1	1	yes	characteristic, direct product factor, Sylow
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	2	1	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
243	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	2	1	no	—
246	C₂²×C₄	—	4	1	no	—
247	Q₁₆	—	2	4	no	—
248	Q₁₆	—	4	4	no	—
249	SD₁₆	—	4	4	no	—
250	SD₁₆	—	8	2	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
255	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
266	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
267	C₂²⋊C₄	—	4	2	no	—
268	C₂²⋊C₄	—	4	1	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	4	2	no	—
272	C₄²	—	4	1	no	—
273	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
274	C₂³	—	4	1	no	—
275	Q₈	—	2	2	no	—
276	Q₈	—	2	2	no	—
277	Q₈	—	2	2	no	—
278	Q₈	—	4	2	no	—
279	Q₈	—	4	2	no	—
280	Q₈	—	8	1	no	—
281	Q₈	—	16	1	no	—
282	Q₈	C₃₁×#135	1	4	yes	—
283	D₄	—	4	2	no	—
284	D₄	—	4	2	no	—
285	D₄	—	8	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	16	1	no	—
293	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
294	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	2	2	no	—
297	C₈	—	2	2	no	—
298	C₈	—	4	2	no	—
299	C₈	—	4	2	no	—
300	C₂²	—	8	2	no	—
301	C₂²	—	8	1	no	—
302	C₂²	D₄²×C₃₁	1	1	yes	characteristic
303	C₄	—	2	4	no	—
304	C₄	—	2	1	no	—
305	C₄	—	4	2	no	—
306	C₄	—	4	2	no	—
307	C₄	—	4	2	no	—
308	C₄	—	8	2	no	—
309	C₄	—	8	1	no	—
310	C₄	C₂×C₃₁×#135	1	2	yes	—
311	C₄	C₃₁×#578	1	2	yes	—
312	C₂	—	16	1	no	—
313	C₂	C₃₁×#3285	1	2	yes	—
314	C₂	C₃₁×#3291	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#127	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#3363	C₂	semidirect
4. C₃₁×#1684	C₂	semidirect
6. C₃₁×#1653	C₂	semidirect
9. C₈×Q₁₆×C₃₁	C₂	semidirect
116. #20013	C₃₁	direct (N × H)
144. #3363	C₂×C₃₁	semidirect
146. #1684	C₂×C₃₁	semidirect
148. #1653	C₂×C₃₁	semidirect
151. C₈×Q₁₆	C₂×C₃₁	semidirect
238. C₃₁	#20013	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×#3365	2
C₃₁×#3363	2
C₃₁×#1708	2
C₃₁×#1684	2
C₃₁×#1682	2
C₃₁×#1653	2
C₃₁×#1630	2
C₃₁×#1623	2
C₈×Q₁₆×C₃₁	2
#20013	31

Subgroups — C31×#20013 #21476772

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20013 #21476772