Group #16848314 — subgroups

Subgroups — #16848314

Order 3968.

512 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 229 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	#14520935	C₂	1	8	yes	maximal, complemented
2	#14520912	C₂	1	2	yes	maximal
3	#14520909	C₂	1	2	yes	maximal
4	C₄×C₂²⋊Dic₃₁	C₂	1	2	yes	maximal
5	C₂²×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
7	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
8	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₄×C₃₁	C₄	1	8	yes	complemented
10	C₂×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
11	C₂×C₂²⋊Dic₃₁	C₂²	1	4	yes	—
12	C₂×C₄⋊Dic₃₁	C₂²	1	4	yes	—
13	C₂×C₄×Dic₃₁	C₂²	1	4	yes	—
14	C₃₁⋊#95	C₂²	1	4	yes	—
15	C₃₁⋊#95	C₂²	1	4	yes	—
16	C₃₁⋊#95	C₂²	1	8	yes	—
17	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
18	C₂²×C₄×C₃₁	—	2	4	no	—
19	C₂²×C₄×C₃₁	—	4	2	no	—
20	C₂²×C₄×C₃₁	—	4	2	no	—
21	C₂²×C₄×C₃₁	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₃₁	C₂×C₄	1	4	yes	—
23	C₂²×C₄×C₃₁	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
25	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
26	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
27	C₂²×Dic₃₁	C₂³	1	8	yes	—
28	C₄⋊C₄×C₃₁	C₂×C₄	1	16	yes	—
29	C₂²⋊Dic₃₁	—	2	8	no	—
30	C₄⋊Dic₃₁	—	4	4	no	—
31	C₄×Dic₃₁	—	4	4	no	—
32	C₂³×C₃₁	—	2	2	no	—
33	C₂³×C₃₁	—	2	2	no	—
34	C₂³×C₃₁	C₂²×C₄	1	2	yes	—
35	C₂³×C₃₁	#41	1	4	yes	—
36	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
37	C₂×C₄×C₃₁	—	2	8	no	—
38	C₂×C₄×C₃₁	—	2	4	no	—
39	C₂×C₄×C₃₁	—	2	4	no	—
40	C₂×C₄×C₃₁	—	2	4	no	—
41	C₂×C₄×C₃₁	—	4	4	no	—
42	C₂×C₄×C₃₁	—	4	4	no	—
43	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
44	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
45	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
46	C₂×Dic₃₁	—	2	8	no	—
47	C₂×Dic₃₁	—	2	8	no	—
48	C₂×Dic₃₁	—	2	8	no	—
49	#2313	—	31	1	no	maximal, Sylow
50	C₂²×C₃₁	—	2	2	no	—
51	C₂²×C₃₁	—	2	2	no	—
52	C₂²×C₃₁	—	2	8	no	—
53	C₂²×C₃₁	#117	1	4	yes	—
54	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
55	C₂²×C₃₁	C₂×#41	1	2	yes	—
56	C₂²×C₃₁	H(5,𝔽₂)	1	2	yes	—
57	C₂²×C₃₁	2₋¹⁺⁴	1	2	yes	—
58	C₄×C₃₁	—	2	4	no	—
59	C₄×C₃₁	—	2	4	no	—
60	C₄×C₃₁	—	2	4	no	—
61	Dic₃₁	—	4	8	no	—
62	C₂²×C₄⋊C₄	D₃₁	1	1	yes	characteristic
63	#387	—	31	8	no	—
64	#381	—	31	2	no	—
65	#380	—	31	2	no	—
66	C₄×C₂²⋊C₄	—	31	2	no	—
67	C₂×C₃₁	—	2	4	no	—
68	C₂×C₃₁	C₂×#117	1	1	yes	characteristic
69	C₂×C₃₁	#518	1	1	yes	characteristic
70	C₂×C₃₁	#519	1	1	yes	characteristic
71	C₂×C₃₁	#539	1	4	yes	—
72	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
73	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
74	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
75	C₂×C₄⋊C₄	—	31	4	no	—
76	C₂×C₄⋊C₄	Dic₃₁	1	8	yes	complemented
77	C₂×C₄⋊C₄	C₂×D₃₁	1	4	yes	—
78	C₂×C₂²⋊C₄	—	31	4	no	—
79	C₂×C₄²	—	31	4	no	—
80	#95	—	31	4	no	—
81	#95	—	31	4	no	—
82	#95	—	31	8	no	—
83	C₃₁	#2313	1	1	yes	characteristic, complemented, Sylow
84	C₂⁴	C₂²×D₃₁	1	1	yes	characteristic
85	C₂²×C₄	—	2	4	no	—
86	C₂²×C₄	—	4	2	no	—
87	C₂²×C₄	—	4	2	no	—
88	C₂²×C₄	—	31	8	no	—
89	C₂²×C₄	C₂×Dic₃₁	1	4	yes	—
90	C₂²×C₄	C₂×Dic₃₁	1	4	yes	—
91	C₂²×C₄	C₂×Dic₃₁	1	4	yes	—
92	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
93	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
94	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
95	C₄⋊C₄	—	124	4	no	—
96	C₄⋊C₄	C₂×Dic₃₁	1	16	yes	—
97	C₂²⋊C₄	—	62	8	no	—
98	C₄²	—	124	4	no	—
99	C₂³	—	2	2	no	—
100	C₂³	—	2	2	no	—
101	C₂³	#92495	1	4	yes	—
102	C₂³	C₂²×Dic₃₁	1	2	yes	—
103	C₂³	C₂³×D₃₁	1	1	yes	characteristic
104	C₂×C₄	—	2	8	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	4	4	no	—
109	C₂×C₄	—	4	4	no	—
110	C₂×C₄	—	62	8	no	—
111	C₂×C₄	—	62	8	no	—
112	C₂×C₄	—	62	8	no	—
113	C₂×C₄	C₂²×Dic₃₁	1	4	yes	—
114	C₂×C₄	C₂²×Dic₃₁	1	4	yes	—
115	C₂×C₄	C₂²×Dic₃₁	1	4	yes	—
116	C₂²	—	2	8	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	2	no	—
119	C₂²	#11758503	1	4	yes	—
120	C₂²	C₂×#92495	1	2	yes	—
121	C₂²	#11758561	1	2	yes	—
122	C₂²	#11758564	1	2	yes	—
123	C₂²	C₂³×Dic₃₁	1	1	yes	characteristic
124	C₄	—	2	4	no	—
125	C₄	—	2	4	no	—
126	C₄	—	2	4	no	—
127	C₄	—	124	8	no	—
128	C₂	—	2	4	no	—
129	C₂	C₂×#11758503	1	1	yes	characteristic
130	C₂	#14521524	1	1	yes	characteristic
131	C₂	#14521527	1	1	yes	characteristic
132	C₂	#14521646	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14520935

C₂

yes

maximal, complemented

#14520912

C₂

yes

maximal

#14520909

C₂

yes

maximal

C₄×C₂²⋊Dic₃₁

C₂

yes

maximal

C₂²×C₄⋊C₄×C₃₁

C₂

yes

maximal, characteristic

C₂³×C₄×C₃₁

C₂²

yes

characteristic

C₂³×C₄×C₃₁

C₂²

yes

characteristic

C₂³×C₄×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₁

C₄

yes

complemented

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₂²⋊Dic₃₁

C₂²

yes

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×C₄×Dic₃₁

C₂²

yes

—

C₃₁⋊#95

C₂²

yes

—

C₃₁⋊#95

C₂²

yes

—

C₃₁⋊#95

C₂²

yes

—

C₂⁴×C₃₁

C₂³

yes

characteristic

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

—

C₂²×C₄×C₃₁

C₂³

yes

—

C₂²×C₄×C₃₁

C₂³

yes

—

C₂²×Dic₃₁

C₂³

yes

—

C₄⋊C₄×C₃₁

C₂×C₄

yes

—

C₂²⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄×Dic₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

C₂²×C₄

yes

—

C₂³×C₃₁

#41

yes

—

C₂³×C₃₁

C₂⁴

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

#2313

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

#117

yes

—

C₂²×C₃₁

C₂³×C₄

yes

characteristic

C₂²×C₃₁

C₂×#41

yes

—

C₂²×C₃₁

H(5,𝔽₂)

yes

—

C₂²×C₃₁

2₋¹⁺⁴

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

Dic₃₁

—

C₂²×C₄⋊C₄

D₃₁

yes

—

—

—

C₄×C₂²⋊C₄

—

C₂×C₃₁

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

—

C₂³×C₄

C₂×D₃₁

yes

characteristic

C₂³×C₄

C₂×D₃₁

yes

characteristic

C₂³×C₄

C₂×D₃₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₃₁

yes

complemented

C₂×C₄⋊C₄

C₂×D₃₁

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

#95

—

#95

—

#95

—

C₃₁

#2313

yes

characteristic, complemented, Sylow

C₂⁴

C₂²×D₃₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₃₁

yes

—

C₂²×C₄

C₂×Dic₃₁

yes

—

C₂²×C₄

C₂×Dic₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

—

C₄⋊C₄

—

124

—

C₄⋊C₄

C₂×Dic₃₁

yes

—

C₂²⋊C₄

—

C₄²

—

124

—

C₂³

—

100

C₂³

—

101

C₂³

#92495

yes

—

102

C₂³

C₂²×Dic₃₁

yes

—

103

C₂³

C₂³×D₃₁

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂²×Dic₃₁

yes

—

114

C₂×C₄

C₂²×Dic₃₁

yes

—

115

C₂×C₄

C₂²×Dic₃₁

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

#11758503

yes

—

120

C₂²

C₂×#92495

yes

—

121

C₂²

#11758561

yes

—

122

C₂²

#11758564

yes

—

123

C₂²

C₂³×Dic₃₁

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

124

—

128

C₂

—

129

C₂

C₂×#11758503

yes

characteristic

130

C₂

#14521524

yes

characteristic

131

C₂

#14521527

yes

characteristic

132

C₂

#14521646

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₁
commutator	C₂²×C₃₁	C₂³×C₄
Fitting	C₂²×C₄⋊C₄×C₃₁	C₂
socle, cosocle	C₂³×C₃₁	C₂⁴
prime radical	C₂⁴×C₃₁	C₂³
hypercenter	C₂²×C₄⋊C₄	D₃₁
hypocenter	C₃₁	#2313
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃₁

commutator

C₂²×C₃₁

C₂³×C₄

Fitting

C₂²×C₄⋊C₄×C₃₁

C₂

socle, cosocle

C₂³×C₃₁

C₂⁴

prime radical

C₂⁴×C₃₁

C₂³

hypercenter

C₂²×C₄⋊C₄

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

N	H ≅ G/N	type
1. #14520935	C₂	semidirect
9. C₂×C₄⋊C₄×C₃₁	C₄	semidirect
76. C₂×C₄⋊C₄	Dic₃₁	semidirect
83. C₃₁	#2313	semidirect

H ≅ G/N

type

1. #14520935

C₂

semidirect

9. C₂×C₄⋊C₄×C₃₁

C₄

semidirect

76. C₂×C₄⋊C₄

Dic₃₁

semidirect

83. C₃₁

#2313

semidirect

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

subgroup	index
#14520935	2
#14520912	2
#14520909	2
C₄×C₂²⋊Dic₃₁	2
C₂²×C₄⋊C₄×C₃₁	2
#2313	31

subgroup

index

#14520935

#14520912

#14520909

C₄×C₂²⋊Dic₃₁

C₂²×C₄⋊C₄×C₃₁

#2313

Subgroups — #16848314

Conjugacy classes of subgroups

Complements

Maximal subgroups