Subgroups — C₃₁×#9694 #21466453

Order 7936.

570 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2868	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1842	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#332	C₂	1	4	yes	maximal
5	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
6	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
7	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₃₁×C₄⋊C₈	—	2	2	no	—
9	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₃₁	—	2	1	no	—
12	C₃₁×#384	—	2	2	no	—
13	C₃₁×#384	C₄	1	4	yes	—
14	C₄³×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#336	—	2	1	no	—
16	C₃₁×#332	—	2	8	no	—
17	C₂²×Q₈×C₃₁	—	2	2	no	—
18	C₂²×C₈×C₃₁	—	2	2	no	—
19	C₂²×C₈×C₃₁	—	2	2	no	—
20	C₃₁×C₄⋊Q₈	—	2	4	no	—
21	C₃₁×C₄⋊Q₈	—	2	4	no	—
22	C₃₁×C₄⋊Q₈	—	4	4	no	—
23	C₃₁×C₄⋊Q₈	—	4	2	no	—
24	C₃₁×C₄⋊Q₈	—	4	2	no	—
25	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
26	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
27	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
28	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₃₁	D₄	1	2	yes	—
33	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
34	C₃₁×C₄⋊C₈	—	2	4	no	—
35	C₃₁×C₄⋊C₈	—	2	2	no	—
36	C₃₁×C₄⋊C₈	—	4	4	no	—
37	C₃₁×C₄⋊C₈	D₄	1	4	yes	—
38	C₄×C₈×C₃₁	—	4	2	no	—
39	C₂×Q₈×C₃₁	—	4	4	no	—
40	C₂×Q₈×C₃₁	—	4	2	no	—
41	C₂×Q₈×C₃₁	—	4	2	no	—
42	C₂×Q₈×C₃₁	—	4	4	no	—
43	C₂²×C₄×C₃₁	—	2	4	no	—
44	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₃₁	—	2	4	no	—
50	C₂×C₈×C₃₁	—	4	2	no	—
51	C₂×C₈×C₃₁	—	4	2	no	—
52	C₂×C₈×C₃₁	—	4	2	no	—
53	C₂×C₈×C₃₁	—	4	4	no	—
54	C₄⋊C₄×C₃₁	—	2	8	no	—
55	C₄⋊C₄×C₃₁	—	4	8	no	—
56	C₄⋊C₄×C₃₁	—	4	8	no	—
57	C₄⋊C₄×C₃₁	—	4	2	no	—
58	C₄⋊C₄×C₃₁	—	4	2	no	—
59	C₄²×C₃₁	—	2	4	no	—
60	C₄²×C₃₁	—	2	2	no	—
61	C₄²×C₃₁	D₈	1	2	yes	—
62	C₄²×C₃₁	SD₁₆	1	2	yes	—
63	C₄²×C₃₁	SD₁₆	1	4	yes	—
64	C₄²×C₃₁	Q₁₆	1	4	yes	—
65	C₄²×C₃₁	C₂×D₄	1	2	yes	—
66	C₄²×C₃₁	#41	1	2	yes	—
67	#9694	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	Q₈×C₃₁	—	8	4	no	—
70	Q₈×C₃₁	—	8	4	no	—
71	C₂×C₄×C₃₁	—	2	2	no	—
72	C₂×C₄×C₃₁	—	2	2	no	—
73	C₂×C₄×C₃₁	—	2	2	no	—
74	C₂×C₄×C₃₁	—	4	4	no	—
75	C₂×C₄×C₃₁	—	4	4	no	—
76	C₂×C₄×C₃₁	—	4	4	no	—
77	C₂×C₄×C₃₁	#101	1	4	yes	—
78	C₂×C₄×C₃₁	#102	1	4	yes	—
79	C₂×C₄×C₃₁	#103	1	4	yes	—
80	C₂×C₄×C₃₁	#103	1	4	yes	—
81	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
82	C₂×C₄×C₃₁	#121	1	2	yes	—
83	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
84	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
85	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
88	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
89	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
90	C₈×C₃₁	—	4	4	no	—
91	C₈×C₃₁	—	8	2	no	—
92	#2868	C₂×C₃₁	1	1	yes	characteristic
93	#1842	C₂×C₃₁	1	1	yes	characteristic
94	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic
95	C₂×#332	C₂×C₃₁	1	4	yes	—
96	C₂²×C₃₁	#390	1	1	yes	characteristic
97	C₂²×C₃₁	C₂×#101	1	2	yes	—
98	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
99	C₂²×C₃₁	C₂×#103	1	2	yes	—
100	C₂²×C₃₁	#418	1	1	yes	characteristic
101	C₄×C₃₁	—	2	4	no	—
102	C₄×C₃₁	—	2	2	no	—
103	C₄×C₃₁	—	8	4	no	—
104	C₄×C₃₁	#332	1	8	yes	complemented
105	C₄×C₃₁	#459	1	2	yes	—
106	C₄×C₃₁	#460	1	2	yes	—
107	C₄×C₃₁	#487	1	2	yes	—
108	C₄×C₃₁	#492	1	1	yes	characteristic
109	C₄×C₃₁	#494	1	1	yes	characteristic
110	C₂×C₄⋊Q₈	—	2	1	no	—
111	C₂×C₄⋊Q₈	—	2	1	no	—
112	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
113	C₂×C₄⋊C₈	—	2	2	no	—
114	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
115	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
116	C₂×C₄×C₈	—	2	1	no	—
117	#384	—	2	2	no	—
118	#384	C₄×C₃₁	1	4	yes	—
119	C₄³	C₂²×C₃₁	1	1	yes	characteristic
120	#336	—	2	1	no	—
121	#332	—	2	8	no	—
122	C₂×C₃₁	C₂×#332	1	2	yes	—
123	C₂×C₃₁	#1549	1	2	yes	—
124	C₂×C₃₁	#1956	1	1	yes	characteristic
125	C₂×C₃₁	#1966	1	1	yes	characteristic
126	C₂×C₃₁	#1982	1	1	yes	characteristic
127	C₂²×Q₈	—	2	2	no	—
128	C₂²×C₈	—	2	2	no	—
129	C₂²×C₈	—	2	2	no	—
130	C₄⋊Q₈	—	2	4	no	—
131	C₄⋊Q₈	—	2	4	no	—
132	C₄⋊Q₈	—	4	4	no	—
133	C₄⋊Q₈	—	4	2	no	—
134	C₄⋊Q₈	—	4	2	no	—
135	C₂×C₄⋊C₄	—	2	4	no	—
136	C₂×C₄⋊C₄	—	2	4	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
139	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
140	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
141	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
142	C₂×C₄²	D₄×C₃₁	1	2	yes	—
143	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
144	C₄⋊C₈	—	2	4	no	—
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	—	4	4	no	—
147	C₄⋊C₈	D₄×C₃₁	1	4	yes	—
148	C₄×C₈	—	4	2	no	—
149	C₃₁	#9694	1	1	yes	characteristic, direct product factor, Sylow
150	C₂×Q₈	—	4	4	no	—
151	C₂×Q₈	—	4	2	no	—
152	C₂×Q₈	—	4	2	no	—
153	C₂×Q₈	—	4	4	no	—
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
156	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
157	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
158	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
159	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	—	4	2	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	—	4	2	no	—
164	C₂×C₈	—	4	4	no	—
165	C₄⋊C₄	—	2	8	no	—
166	C₄⋊C₄	—	4	8	no	—
167	C₄⋊C₄	—	4	8	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄²	—	2	4	no	—
171	C₄²	—	2	2	no	—
172	C₄²	D₈×C₃₁	1	2	yes	—
173	C₄²	SD₁₆×C₃₁	1	2	yes	—
174	C₄²	SD₁₆×C₃₁	1	4	yes	—
175	C₄²	Q₁₆×C₃₁	1	4	yes	—
176	C₄²	C₂×D₄×C₃₁	1	2	yes	—
177	C₄²	#41×C₃₁	1	2	yes	—
178	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
179	Q₈	—	8	4	no	—
180	Q₈	—	8	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	4	4	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	4	4	no	—
187	C₂×C₄	C₃₁×#101	1	4	yes	—
188	C₂×C₄	C₃₁×#102	1	4	yes	—
189	C₂×C₄	C₃₁×#103	1	4	yes	—
190	C₂×C₄	C₃₁×#103	1	4	yes	—
191	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
192	C₂×C₄	C₃₁×#121	1	2	yes	—
193	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
194	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
195	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
196	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
197	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
198	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
199	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
200	C₈	—	4	4	no	—
201	C₈	—	8	2	no	—
202	C₂²	C₃₁×#390	1	1	yes	characteristic
203	C₂²	C₂×C₃₁×#101	1	2	yes	—
204	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
205	C₂²	C₂×C₃₁×#103	1	2	yes	—
206	C₂²	C₃₁×#418	1	1	yes	characteristic
207	C₄	—	2	4	no	—
208	C₄	—	2	2	no	—
209	C₄	—	8	4	no	—
210	C₄	C₃₁×#332	1	8	yes	complemented
211	C₄	C₃₁×#459	1	2	yes	—
212	C₄	C₃₁×#460	1	2	yes	—
213	C₄	C₃₁×#487	1	2	yes	—
214	C₄	C₃₁×#492	1	1	yes	characteristic
215	C₄	C₃₁×#494	1	1	yes	characteristic
216	C₂	C₂×C₃₁×#332	1	2	yes	—
217	C₂	C₃₁×#1549	1	2	yes	—
218	C₂	C₃₁×#1956	1	1	yes	characteristic
219	C₂	C₃₁×#1966	1	1	yes	characteristic
220	C₂	C₃₁×#1982	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
67. #9694	C₃₁	direct (N × H)
104. C₄×C₃₁	#332	semidirect
149. C₃₁	#9694	direct (N × H)
210. C₄	C₃₁×#332	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#2868	2
C₃₁×#1842	2
C₄×C₃₁×C₄⋊C₈	2
C₂×C₃₁×#332	2
#9694	31

Subgroups — C31×#9694 #21466453

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9694 #21466453