Subgroups — C₃₁×#22725 #21479484

Order 7936.

750 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2355	C₂	1	2	yes	maximal
3	C₃₁×#1949	C₂	1	4	yes	maximal
4	C₃₁×#1947	C₂	1	2	yes	maximal
5	C₃₁×#1846	C₂	1	2	yes	maximal
6	C₃₁×#1844	C₂	1	1	yes	maximal, characteristic
7	C₄×C₃₁×#106	C₂	1	1	yes	maximal, characteristic
8	C₄×C₃₁×#106	C₂	1	2	yes	maximal
9	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₈×Q₈×C₃₁	C₄	1	8	yes	complemented
11	C₂×C₃₁×#107	C₂²	1	4	yes	—
12	C₂×C₃₁×#106	C₂²	1	2	yes	—
13	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#106	C₂²	1	2	yes	—
15	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
20	C₃₁×#391	—	2	2	no	—
21	C₃₁×#387	—	2	4	no	—
22	C₃₁×#384	C₂²	1	2	yes	—
23	C₃₁×#382	C₂²	1	4	yes	—
24	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
25	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
27	C₃₁×#340	C₂²	1	4	yes	—
28	C₃₁×#340	C₂²	1	2	yes	—
29	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
32	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
33	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	—
34	C₄×Q₈×C₃₁	Q₈	1	4	yes	—
35	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
37	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
41	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
42	C₂×C₄²×C₃₁	—	2	2	no	—
43	C₂×C₄²×C₃₁	—	2	4	no	—
44	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
45	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
46	C₃₁×#107	—	4	4	no	—
47	C₃₁×#106	—	2	4	no	—
48	C₃₁×#106	—	2	2	no	—
49	C₃₁×#106	—	4	2	no	—
50	C₃₁×#106	Q₈	1	4	yes	complemented
51	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
52	C₃₁×C₄⋊C₈	C₂×C₄	1	8	yes	—
53	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
54	C₄×C₈×C₃₁	C₂×C₄	1	8	yes	—
55	C₃₁×#95	—	2	4	no	—
56	C₃₁×#95	—	2	2	no	—
57	C₃₁×#95	—	2	4	no	—
58	C₃₁×#95	—	2	4	no	—
59	C₂×Q₈×C₃₁	—	2	4	no	—
60	C₂×Q₈×C₃₁	C₄⋊C₄	1	4	yes	—
61	C₂²×C₄×C₃₁	—	2	2	no	—
62	C₂²×C₄×C₃₁	—	2	2	no	—
63	C₂²×C₄×C₃₁	—	2	4	no	—
64	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
69	C₂×C₈×C₃₁	—	2	1	no	—
70	C₂×C₈×C₃₁	—	2	2	no	—
71	C₂×C₈×C₃₁	—	2	1	no	—
72	C₂×C₈×C₃₁	—	2	2	no	—
73	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
75	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
77	C₂×C₈×C₃₁	#41	1	2	yes	—
78	C₄⋊C₄×C₃₁	—	2	2	no	—
79	C₄⋊C₄×C₃₁	—	2	4	no	—
80	C₄⋊C₄×C₃₁	—	2	2	no	—
81	C₄⋊C₄×C₃₁	—	2	4	no	—
82	C₄⋊C₄×C₃₁	—	2	2	no	—
83	C₄⋊C₄×C₃₁	—	4	4	no	—
84	C₄⋊C₄×C₃₁	—	8	2	no	—
85	C₄⋊C₄×C₃₁	C₄⋊C₄	1	8	yes	—
86	C₄⋊C₄×C₃₁	C₄⋊C₄	1	4	yes	—
87	C₄⋊C₄×C₃₁	C₂×Q₈	1	4	yes	—
88	C₄²×C₃₁	—	4	4	no	—
89	C₄²×C₃₁	—	8	4	no	—
90	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
91	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
92	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
93	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
94	#22725	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
96	Q₈×C₃₁	—	2	8	no	—
97	C₂×C₄×C₃₁	—	2	4	no	—
98	C₂×C₄×C₃₁	—	2	4	no	—
99	C₂×C₄×C₃₁	—	2	2	no	—
100	C₂×C₄×C₃₁	—	2	4	no	—
101	C₂×C₄×C₃₁	—	4	2	no	—
102	C₂×C₄×C₃₁	—	4	2	no	—
103	C₂×C₄×C₃₁	—	4	4	no	—
104	C₂×C₄×C₃₁	—	4	2	no	—
105	C₂×C₄×C₃₁	—	4	2	no	—
106	C₂×C₄×C₃₁	—	4	4	no	—
107	C₂×C₄×C₃₁	—	4	4	no	—
108	C₂×C₄×C₃₁	—	4	2	no	—
109	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
110	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
112	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₃₁	#135	1	4	yes	—
114	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
117	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
118	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
119	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
120	C₈×C₃₁	—	2	2	no	—
121	C₈×C₃₁	—	2	4	no	—
122	C₈×C₃₁	C₄×Q₈	1	4	yes	—
123	C₂×C₈×Q₈	C₂×C₃₁	1	1	yes	characteristic
124	#2355	C₂×C₃₁	1	2	yes	—
125	#1949	C₂×C₃₁	1	4	yes	—
126	#1947	C₂×C₃₁	1	2	yes	—
127	#1846	C₂×C₃₁	1	2	yes	—
128	#1844	C₂×C₃₁	1	1	yes	characteristic
129	C₄×#106	C₂×C₃₁	1	2	yes	—
130	C₄×#106	C₂×C₃₁	1	1	yes	characteristic
131	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
132	C₂²×C₃₁	D₄×Q₈	1	1	yes	characteristic
133	C₂²×C₃₁	#550	1	1	yes	characteristic
134	C₂²×C₃₁	C₂×#135	1	2	yes	—
135	C₂²×C₃₁	#577	1	2	yes	—
136	C₄×C₃₁	—	2	4	no	—
137	C₄×C₃₁	—	2	4	no	—
138	C₄×C₃₁	—	2	2	no	—
139	C₄×C₃₁	—	4	4	no	—
140	C₄×C₃₁	—	8	2	no	—
141	C₄×C₃₁	—	8	4	no	—
142	C₄×C₃₁	#427	1	4	yes	—
143	C₄×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
144	C₄×C₃₁	#520	1	1	yes	characteristic
145	C₄×C₃₁	#557	1	1	yes	characteristic
146	C₄×C₃₁	Q₈²	1	1	yes	characteristic
147	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
148	C₈×Q₈	C₄×C₃₁	1	8	yes	complemented
149	C₂×#107	C₂²×C₃₁	1	4	yes	—
150	C₂×#106	C₂²×C₃₁	1	2	yes	—
151	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
152	C₂×#106	C₂²×C₃₁	1	2	yes	—
153	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
154	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
155	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
156	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
157	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
158	#391	—	2	2	no	—
159	#387	—	2	4	no	—
160	#384	C₂²×C₃₁	1	2	yes	—
161	#382	C₂²×C₃₁	1	4	yes	—
162	C₄×C₄⋊C₄	—	2	2	no	—
163	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
164	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
165	#340	C₂²×C₃₁	1	4	yes	—
166	#340	C₂²×C₃₁	1	2	yes	—
167	C₂×C₃₁	Q₈×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₃₁	C₂×#427	1	1	yes	characteristic
169	C₂×C₃₁	#2912	1	1	yes	characteristic
170	C₂×C₃₁	#3381	1	2	yes	—
171	C₂×C₃₁	#3394	1	2	yes	—
172	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
173	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
174	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
175	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
176	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	—
177	C₄×Q₈	Q₈×C₃₁	1	4	yes	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
181	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
182	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
183	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
184	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
185	C₂×C₄²	—	2	2	no	—
186	C₂×C₄²	—	2	4	no	—
187	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
188	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
189	#107	—	4	4	no	—
190	#106	—	2	4	no	—
191	#106	—	2	2	no	—
192	#106	—	4	2	no	—
193	#106	Q₈×C₃₁	1	4	yes	complemented
194	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
195	C₄⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
196	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
197	C₄×C₈	C₂×C₄×C₃₁	1	8	yes	—
198	#95	—	2	4	no	—
199	#95	—	2	2	no	—
200	#95	—	2	4	no	—
201	#95	—	2	4	no	—
202	C₃₁	#22725	1	1	yes	characteristic, direct product factor, Sylow
203	C₂×Q₈	—	2	4	no	—
204	C₂×Q₈	C₄⋊C₄×C₃₁	1	4	yes	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
209	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
210	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
212	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
218	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
219	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
220	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
221	C₂×C₈	#41×C₃₁	1	2	yes	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	4	4	no	—
228	C₄⋊C₄	—	8	2	no	—
229	C₄⋊C₄	C₄⋊C₄×C₃₁	1	8	yes	—
230	C₄⋊C₄	C₄⋊C₄×C₃₁	1	4	yes	—
231	C₄⋊C₄	C₂×Q₈×C₃₁	1	4	yes	—
232	C₄²	—	4	4	no	—
233	C₄²	—	8	4	no	—
234	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
235	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
236	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
237	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
238	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
239	Q₈	—	2	8	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
253	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
254	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
255	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
256	C₂×C₄	C₃₁×#135	1	4	yes	—
257	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
258	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
259	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
260	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
261	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
262	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
263	C₈	—	2	2	no	—
264	C₈	—	2	4	no	—
265	C₈	C₄×Q₈×C₃₁	1	4	yes	—
266	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
267	C₂²	D₄×Q₈×C₃₁	1	1	yes	characteristic
268	C₂²	C₃₁×#550	1	1	yes	characteristic
269	C₂²	C₂×C₃₁×#135	1	2	yes	—
270	C₂²	C₃₁×#577	1	2	yes	—
271	C₄	—	2	4	no	—
272	C₄	—	2	4	no	—
273	C₄	—	2	2	no	—
274	C₄	—	4	4	no	—
275	C₄	—	8	2	no	—
276	C₄	—	8	4	no	—
277	C₄	C₃₁×#427	1	4	yes	—
278	C₄	C₂×C₄×Q₈×C₃₁	1	1	yes	characteristic
279	C₄	C₃₁×#520	1	1	yes	characteristic
280	C₄	C₃₁×#557	1	1	yes	characteristic
281	C₄	Q₈²×C₃₁	1	1	yes	characteristic
282	C₂	Q₈×C₄⋊C₄×C₃₁	1	1	yes	characteristic
283	C₂	C₂×C₃₁×#427	1	1	yes	characteristic
284	C₂	C₃₁×#2912	1	1	yes	characteristic
285	C₂	C₃₁×#3381	1	2	yes	—
286	C₂	C₃₁×#3394	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₈×Q₈×C₃₁	C₄	semidirect
50. C₃₁×#106	Q₈	semidirect
94. #22725	C₃₁	direct (N × H)
148. C₈×Q₈	C₄×C₃₁	semidirect
193. #106	Q₈×C₃₁	semidirect
202. C₃₁	#22725	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₈×Q₈×C₃₁	2
C₃₁×#2355	2
C₃₁×#1949	2
C₃₁×#1947	2
C₃₁×#1846	2
C₃₁×#1844	2
C₄×C₃₁×#106	2
#22725	31

Subgroups — C31×#22725 #21479484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22725 #21479484