Group #16853975 — subgroups

Subgroups — #16853975

Order 3968.

548 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14520985	C₂	1	4	yes	maximal, complemented
2	#14520972	C₂	1	8	yes	maximal, complemented
3	#14520905	C₂	1	2	yes	maximal
4	C₃₁×#379	C₂	1	1	yes	maximal, characteristic
5	C₂⁵×C₃₁	C₂²	1	1	yes	characteristic
6	C₂×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
7	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
8	C₂×C₂²⋊Dic₃₁	C₂²	1	4	yes	—
9	C₂×C₂²⋊Dic₃₁	C₂²	1	4	yes	—
10	C₂×C₂²⋊Dic₃₁	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₃₁	C₂²	1	4	yes	complemented
12	C₂×Dic₃₁⋊C₄	C₂²	1	8	yes	complemented
13	C₃₁⋊#95	C₂²	1	4	yes	complemented
14	C₂⁴×C₃₁	—	4	4	no	—
15	C₂⁴×C₃₁	—	4	2	no	—
16	C₂⁴×C₃₁	Q₈	1	4	yes	—
17	C₂⁴×C₃₁	C₂³	1	2	yes	—
18	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
19	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
20	C₂²×Dic₃₁	C₂³	1	8	yes	—
21	C₂²⋊C₄×C₃₁	—	4	4	no	—
22	C₂²⋊C₄×C₃₁	—	4	2	no	—
23	C₂²⋊Dic₃₁	—	4	4	no	—
24	C₂²⋊Dic₃₁	—	4	4	no	—
25	C₂²⋊Dic₃₁	—	4	4	no	—
26	C₄⋊Dic₃₁	—	4	4	no	—
27	Dic₃₁⋊C₄	—	4	8	no	—
28	C₂³×C₃₁	—	2	4	no	—
29	C₂³×C₃₁	—	2	4	no	—
30	C₂³×C₃₁	—	2	4	no	—
31	C₂³×C₃₁	—	2	4	no	—
32	C₂³×C₃₁	—	2	2	no	—
33	C₂³×C₃₁	—	4	8	no	—
34	C₂³×C₃₁	—	4	4	no	—
35	C₂³×C₃₁	—	4	16	no	—
36	C₂³×C₃₁	C₂×Q₈	1	4	yes	—
37	C₂³×C₃₁	C₂×Q₈	1	2	yes	—
38	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
39	C₂×C₄×C₃₁	—	2	8	no	—
40	C₂×C₄×C₃₁	—	2	4	no	—
41	C₂×Dic₃₁	—	2	8	no	—
42	C₂×Dic₃₁	—	2	8	no	—
43	C₂×Dic₃₁	—	2	8	no	—
44	#2842	—	31	1	no	maximal, Sylow
45	C₂²×C₃₁	—	2	4	no	—
46	C₂²×C₃₁	—	2	4	no	—
47	C₂²×C₃₁	—	2	4	no	—
48	C₂²×C₃₁	—	2	4	no	—
49	C₂²×C₃₁	—	2	2	no	—
50	C₂²×C₃₁	—	4	16	no	—
51	C₂²×C₃₁	—	4	8	no	—
52	C₂²×C₃₁	—	4	4	no	—
53	C₂²×C₃₁	C₂²×Q₈	1	1	yes	characteristic
54	C₂²×C₃₁	H(5,𝔽₂)	1	4	yes	—
55	C₂²×C₃₁	H(5,𝔽₂)	1	2	yes	—
56	C₄×C₃₁	—	4	4	no	—
57	Dic₃₁	—	4	8	no	—
58	#399	—	31	4	no	—
59	#396	—	31	8	no	—
60	#379	—	31	2	no	—
61	#379	D₃₁	1	1	yes	characteristic
62	C₂×C₃₁	—	4	4	no	—
63	C₂×C₃₁	—	4	2	no	—
64	C₂×C₃₁	#543	1	2	yes	—
65	C₂×C₃₁	#543	1	1	yes	characteristic
66	C₂×C₃₁	#560	1	4	yes	—
67	C₂⁵	C₂×D₃₁	1	1	yes	characteristic
68	C₂×C₄⋊C₄	—	31	8	no	—
69	C₂×C₄⋊C₄	—	31	4	no	—
70	C₂×C₂²⋊C₄	—	31	4	no	—
71	C₂×C₂²⋊C₄	—	31	4	no	—
72	C₂×C₂²⋊C₄	—	31	4	no	—
73	C₂×C₂²⋊C₄	C₂×D₃₁	1	4	yes	—
74	C₂×C₂²⋊C₄	C₂×D₃₁	1	2	yes	—
75	#95	—	31	4	no	—
76	C₃₁	#2842	1	1	yes	characteristic, complemented, Sylow
77	C₂⁴	—	4	4	no	—
78	C₂⁴	—	4	2	no	—
79	C₂⁴	Dic₆₂	1	4	yes	—
80	C₂⁴	C₂²×D₃₁	1	2	yes	—
81	C₂⁴	C₂²×D₃₁	1	1	yes	characteristic
82	C₂²×C₄	—	31	8	no	—
83	C₂²×C₄	C₂²×D₃₁	1	4	yes	—
84	C₄⋊C₄	—	124	8	no	—
85	C₄⋊C₄	—	124	4	no	—
86	C₂²⋊C₄	—	4	4	no	—
87	C₂²⋊C₄	—	4	2	no	—
88	C₂²⋊C₄	—	124	4	no	—
89	C₂²⋊C₄	—	124	4	no	—
90	C₂²⋊C₄	—	124	4	no	—
91	C₂³	—	2	4	no	—
92	C₂³	—	2	2	no	—
93	C₂³	—	2	4	no	—
94	C₂³	—	2	4	no	—
95	C₂³	—	2	4	no	—
96	C₂³	—	4	8	no	—
97	C₂³	—	4	4	no	—
98	C₂³	—	4	16	no	—
99	C₂³	C₂×Dic₆₂	1	4	yes	—
100	C₂³	C₂×Dic₆₂	1	2	yes	—
101	C₂³	C₂³×D₃₁	1	1	yes	characteristic
102	C₂×C₄	—	2	8	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	62	8	no	—
105	C₂×C₄	—	62	8	no	—
106	C₂×C₄	—	62	8	no	—
107	C₂²	—	2	4	no	—
108	C₂²	—	2	2	no	—
109	C₂²	—	2	4	no	—
110	C₂²	—	2	4	no	—
111	C₂²	—	2	4	no	—
112	C₂²	—	4	16	no	—
113	C₂²	—	4	8	no	—
114	C₂²	—	4	4	no	—
115	C₂²	C₂²×Dic₆₂	1	1	yes	characteristic
116	C₂²	#11758561	1	4	yes	—
117	C₂²	#11758561	1	2	yes	—
118	C₄	—	4	4	no	—
119	C₄	—	124	8	no	—
120	C₂	—	4	4	no	—
121	C₂	—	4	2	no	—
122	C₂	#14521675	1	2	yes	—
123	C₂	#14521675	1	1	yes	characteristic
124	C₂	#14521788	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14520985

C₂

yes

maximal, complemented

#14520972

C₂

yes

maximal, complemented

#14520905

C₂

yes

maximal

C₃₁×#379

C₂

yes

maximal, characteristic

C₂⁵×C₃₁

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₃₁

C₂²

yes

—

C₂×C₂²⋊C₄×C₃₁

C₂²

yes

—

C₂×C₂²⋊Dic₃₁

C₂²

yes

—

C₂×C₂²⋊Dic₃₁

C₂²

yes

—

C₂×C₂²⋊Dic₃₁

C₂²

yes

—

C₂×C₄⋊Dic₃₁

C₂²

yes

complemented

C₂×Dic₃₁⋊C₄

C₂²

yes

complemented

C₃₁⋊#95

C₂²

yes

complemented

C₂⁴×C₃₁

—

C₂⁴×C₃₁

—

C₂⁴×C₃₁

Q₈

yes

—

C₂⁴×C₃₁

C₂³

yes

—

C₂⁴×C₃₁

C₂³

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

—

C₂²×Dic₃₁

C₂³

yes

—

C₂²⋊C₄×C₃₁

—

C₂²⋊C₄×C₃₁

—

C₂²⋊Dic₃₁

—

C₂²⋊Dic₃₁

—

C₂²⋊Dic₃₁

—

C₄⋊Dic₃₁

—

Dic₃₁⋊C₄

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

C₂×Q₈

yes

—

C₂³×C₃₁

C₂×Q₈

yes

—

C₂³×C₃₁

C₂⁴

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

#2842

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂²×Q₈

yes

characteristic

C₂²×C₃₁

H(5,𝔽₂)

yes

—

C₂²×C₃₁

H(5,𝔽₂)

yes

—

C₄×C₃₁

—

Dic₃₁

—

—

—

—

D₃₁

yes

characteristic

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

#543

yes

—

C₂×C₃₁

#543

yes

characteristic

C₂×C₃₁

#560

yes

—

C₂⁵

C₂×D₃₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₃₁

yes

—

C₂×C₂²⋊C₄

C₂×D₃₁

yes

—

#95

—

C₃₁

#2842

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂⁴

—

C₂⁴

Dic₆₂

yes

—

C₂⁴

C₂²×D₃₁

yes

—

C₂⁴

C₂²×D₃₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₁

yes

—

C₄⋊C₄

—

124

—

C₄⋊C₄

—

124

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

124

—

C₂²⋊C₄

—

124

—

C₂²⋊C₄

—

124

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂×Dic₆₂

yes

—

100

C₂³

C₂×Dic₆₂

yes

—

101

C₂³

C₂³×D₃₁

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂²×Dic₆₂

yes

characteristic

116

C₂²

#11758561

yes

—

117

C₂²

#11758561

yes

—

118

C₄

—

119

C₄

—

124

—

120

C₂

—

121

C₂

—

122

C₂

#14521675

yes

—

123

C₂

#14521675

yes

characteristic

124

C₂

#14521788

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₁
commutator, socle, cosocle	C₂³×C₃₁	C₂⁴
Fitting	C₃₁×#379	C₂
prime radical	C₂⁵×C₃₁	C₂²
hypercenter	#379	D₃₁
hypocenter	C₃₁	#2842
LCS	G ▷ C₂³×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ #379

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃₁

commutator, socle, cosocle

C₂³×C₃₁

C₂⁴

Fitting

C₃₁×#379

C₂

prime radical

C₂⁵×C₃₁

C₂²

D₃₁

C₃₁

G ▷ C₂³×C₃₁ ▷ C₃₁

C₁ ◁ C₂³ ◁ #379

N	H ≅ G/N	type
1. #14520985	C₂	semidirect
2. #14520972	C₂	semidirect
11. C₂×C₄⋊Dic₃₁	C₂²	semidirect
12. C₂×Dic₃₁⋊C₄	C₂²	semidirect
13. C₃₁⋊#95	C₂²	semidirect
76. C₃₁	#2842	semidirect

H ≅ G/N

type

1. #14520985

C₂

semidirect

2. #14520972

C₂

semidirect

11. C₂×C₄⋊Dic₃₁

C₂²

semidirect

12. C₂×Dic₃₁⋊C₄

C₂²

semidirect

13. C₃₁⋊#95

C₂²

semidirect

76. C₃₁

#2842

semidirect

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

subgroup	index
#14520985	2
#14520972	2
#14520905	2
C₃₁×#379	2
#2842	31

subgroup

index

C₃₁×#379

Subgroups — #16853975

Conjugacy classes of subgroups

Complements

Maximal subgroups