Subgroups — C₃₁×#7198 #21463957

Order 7936.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2162	C₂	1	2	yes	maximal
2	C₃₁×#1452	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#431	C₄	1	2	yes	—
4	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
5	C₃₁×C₄⋊C₁₆	—	2	2	no	—
6	C₃₁×C₂²⋊C₁₆	C₄	1	4	yes	—
7	C₄×C₁₆×C₃₁	—	2	2	no	—
8	C₃₁×#322	—	2	2	no	—
9	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
10	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
11	C₂×C₁₆×C₃₁	—	2	4	no	—
12	C₃₁×C₄⋊C₈	C₈	1	4	yes	—
13	C₃₁×C₂²⋊C₈	—	2	2	no	—
14	C₄×C₈×C₃₁	—	2	2	no	—
15	C₄×C₈×C₃₁	D₄	1	2	yes	—
16	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
17	C₄×C₈×C₃₁	Q₈	1	1	yes	characteristic
18	C₂²×C₄×C₃₁	C₄²	1	1	yes	characteristic
19	C₂²×C₄×C₃₁	M₁₆	1	2	yes	—
20	C₂×C₈×C₃₁	—	2	4	no	—
21	C₂×C₈×C₃₁	—	4	2	no	—
22	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
23	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
24	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
25	C₄²×C₃₁	C₄⋊C₄	1	1	yes	characteristic
26	C₄²×C₃₁	C₂×C₈	1	2	yes	—
27	C₁₆×C₃₁	—	4	4	no	—
28	#7198	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₂³×C₃₁	#97	1	1	yes	characteristic
30	C₂×C₄×C₃₁	—	2	2	no	—
31	C₂×C₄×C₃₁	—	2	4	no	—
32	C₂×C₄×C₃₁	#95	1	1	yes	characteristic
33	C₂×C₄×C₃₁	C₄×C₈	1	1	yes	characteristic
34	C₂×C₄×C₃₁	C₂²⋊C₈	1	2	yes	—
35	C₂×C₄×C₃₁	C₄⋊C₈	1	2	yes	—
36	C₈×C₃₁	—	2	2	no	—
37	C₈×C₃₁	—	2	2	no	—
38	C₈×C₃₁	—	4	4	no	—
39	#2162	C₂×C₃₁	1	2	yes	—
40	#1452	C₂×C₃₁	1	1	yes	characteristic
41	C₂²×C₃₁	—	2	1	no	—
42	C₂²×C₃₁	—	2	2	no	—
43	C₂²×C₃₁	#336	1	1	yes	characteristic
44	C₄×C₃₁	—	2	1	no	—
45	C₄×C₃₁	—	2	2	no	—
46	C₄×C₃₁	#344	1	2	yes	—
47	C₄×C₃₁	#350	1	4	yes	—
48	#431	C₄×C₃₁	1	2	yes	—
49	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
50	C₄⋊C₁₆	—	2	2	no	—
51	C₂²⋊C₁₆	C₄×C₃₁	1	4	yes	—
52	C₄×C₁₆	—	2	2	no	—
53	#322	—	2	2	no	—
54	C₂×C₃₁	—	4	1	no	—
55	C₂×C₃₁	#1281	1	1	yes	characteristic
56	C₂×C₃₁	#1379	1	2	yes	—
57	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
58	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
59	C₂×C₁₆	—	2	4	no	—
60	C₄⋊C₈	C₈×C₃₁	1	4	yes	—
61	C₂²⋊C₈	—	2	2	no	—
62	C₄×C₈	—	2	2	no	—
63	C₄×C₈	D₄×C₃₁	1	2	yes	—
64	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
65	C₄×C₈	Q₈×C₃₁	1	1	yes	characteristic
66	C₃₁	#7198	1	1	yes	characteristic, direct product factor, Sylow
67	C₂²×C₄	C₄²×C₃₁	1	1	yes	characteristic
68	C₂²×C₄	M₁₆×C₃₁	1	2	yes	—
69	C₂×C₈	—	2	4	no	—
70	C₂×C₈	—	4	2	no	—
71	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
72	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
73	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
74	C₄²	C₄⋊C₄×C₃₁	1	1	yes	characteristic
75	C₄²	C₂×C₈×C₃₁	1	2	yes	—
76	C₁₆	—	4	4	no	—
77	C₂³	C₃₁×#97	1	1	yes	characteristic
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	—	2	4	no	—
80	C₂×C₄	C₃₁×#95	1	1	yes	characteristic
81	C₂×C₄	C₄×C₈×C₃₁	1	1	yes	characteristic
82	C₂×C₄	C₃₁×C₂²⋊C₈	1	2	yes	—
83	C₂×C₄	C₃₁×C₄⋊C₈	1	2	yes	—
84	C₈	—	2	2	no	—
85	C₈	—	2	2	no	—
86	C₈	—	4	4	no	—
87	C₂²	—	2	2	no	—
88	C₂²	—	2	1	no	—
89	C₂²	C₃₁×#336	1	1	yes	characteristic
90	C₄	—	2	2	no	—
91	C₄	—	2	1	no	—
92	C₄	C₃₁×#344	1	2	yes	—
93	C₄	C₃₁×#350	1	4	yes	—
94	C₂	—	4	1	no	—
95	C₂	C₃₁×#1281	1	1	yes	characteristic
96	C₂	C₃₁×#1379	1	2	yes	—

	subgroup	G/H
center	C₄²×C₃₁	C₂²⋊C₄
commutator	C₂×C₄	C₄×C₈×C₃₁
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₃₁
socle	C₂²×C₃₁	#336
prime radical	C₂³×C₃₁	#97
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
28. #7198	C₃₁	direct (N × H)
66. C₃₁	#7198	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×#2162	2
C₃₁×#1452	2
#7198	31

Subgroups — C31×#7198 #21463957

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#7198 #21463957