Group #16854752 — subgroups

Subgroups — #16854752

Order 3968.

452 conjugacy classes of subgroups; only the 217 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 217 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	#14521127	C₂	1	8	yes	maximal, complemented
2	#14521102	C₂	1	2	yes	maximal, complemented
3	#14521013	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×D₄×C₃₁	C₄	1	1	yes	characteristic
6	C₂³×C₄×C₃₁	C₂²	1	2	yes	—
7	C₄×D₄×C₃₁	C₄	1	4	yes	—
8	C₄×D₄×C₃₁	C₂²	1	4	yes	—
9	C₂×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
10	C₂×C₂²⋊C₄×C₃₁	C₄	1	2	yes	—
11	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×#92474	C₂²	1	8	yes	complemented
13	C₂²⋊(C₃₁⋊C₈)	C₂²	1	4	yes	—
14	C₂²⋊(C₃₁⋊C₈)	C₂²	1	8	yes	complemented
15	C₄⋊(C₃₁⋊C₈)	C₂²	1	4	yes	complemented
16	C₄×C₃₁⋊C₈	C₂²	1	4	yes	complemented
17	C₂⁴×C₃₁	C₂×C₄	1	2	yes	—
18	C₂×D₄×C₃₁	—	2	4	no	—
19	C₂×D₄×C₃₁	C₂×C₄	1	4	yes	—
20	C₂²×C₄×C₃₁	—	4	4	no	—
21	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₃₁	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
27	C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
28	C₂²⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
29	C₄²×C₃₁	C₂×C₄	1	2	yes	—
30	C₄²×C₃₁	C₂³	1	2	yes	—
31	#92474	—	4	8	no	—
32	C₂×C₃₁⋊C₈	C₂³	1	8	yes	—
33	C₂³×C₃₁	—	2	4	no	—
34	C₂³×C₃₁	—	4	2	no	—
35	C₂³×C₃₁	—	4	2	no	—
36	C₂³×C₃₁	C₂²×C₄	1	4	yes	—
37	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
38	D₄×C₃₁	—	2	8	no	—
39	C₂×C₄×C₃₁	—	2	8	no	—
40	C₂×C₄×C₃₁	—	2	2	no	—
41	C₂×C₄×C₃₁	—	4	4	no	—
42	C₂×C₄×C₃₁	—	4	2	no	—
43	C₂×C₄×C₃₁	—	4	1	no	—
44	C₂×C₄×C₃₁	M₁₆	1	4	yes	complemented
45	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
50	C₃₁⋊C₈	—	2	8	no	—
51	#2975	—	31	1	no	maximal, Sylow
52	C₂²×C₃₁	—	2	1	no	—
53	C₂²×C₃₁	—	2	1	no	—
54	C₂²×C₃₁	—	2	4	no	—
55	C₂²×C₃₁	—	2	4	no	—
56	C₂²×C₃₁	—	2	4	no	—
57	C₂²×C₃₁	—	4	8	no	—
58	C₂²×C₃₁	C₂×M₁₆	1	2	yes	—
59	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
60	C₄×C₃₁	—	2	1	no	—
61	C₄×C₃₁	—	2	4	no	—
62	C₄×C₃₁	—	2	2	no	—
63	C₄×C₃₁	C₂×M₁₆	1	4	yes	—
64	C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
65	C₂×C₄×D₄	D₃₁	1	1	yes	characteristic
66	#436	—	31	8	no	—
67	#431	—	31	2	no	—
68	#407	—	31	4	no	—
69	C₂×C₃₁	—	2	2	no	—
70	C₂×C₃₁	—	4	4	no	—
71	C₂×C₃₁	#518	1	1	yes	characteristic
72	C₂×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
73	C₂×C₃₁	#568	1	1	yes	characteristic
74	C₂²×D₄	Dic₃₁	1	1	yes	characteristic
75	C₂³×C₄	C₂×D₃₁	1	2	yes	—
76	C₂×M₁₆	—	31	8	no	—
77	C₄×D₄	Dic₃₁	1	4	yes	—
78	C₄×D₄	C₂×D₃₁	1	4	yes	—
79	C₂×C₄⋊C₄	Dic₃₁	1	1	yes	characteristic
80	C₂×C₂²⋊C₄	Dic₃₁	1	2	yes	—
81	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
82	C₄⋊C₈	—	31	4	no	—
83	C₂²⋊C₈	—	31	4	no	—
84	C₂²⋊C₈	—	31	8	no	—
85	C₄×C₈	—	31	4	no	—
86	C₃₁	#2975	1	1	yes	characteristic, complemented, Sylow
87	C₂⁴	C₂×Dic₃₁	1	2	yes	—
88	C₂×D₄	—	2	4	no	—
89	C₂×D₄	C₂×Dic₃₁	1	4	yes	—
90	C₂²×C₄	—	4	4	no	—
91	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
92	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
93	C₂²×C₄	C₂×Dic₃₁	1	4	yes	—
94	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
95	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
96	C₂²×C₄	C₂²×D₃₁	1	4	yes	—
97	M₁₆	—	124	8	no	—
98	C₂×C₈	—	31	8	no	—
99	C₄⋊C₄	C₂×Dic₃₁	1	4	yes	—
100	C₂²⋊C₄	C₂×Dic₃₁	1	8	yes	—
101	C₄²	C₂×Dic₃₁	1	2	yes	—
102	C₄²	C₂²×D₃₁	1	2	yes	—
103	C₂³	—	2	4	no	—
104	C₂³	—	4	2	no	—
105	C₂³	—	4	2	no	—
106	C₂³	C₂²×Dic₃₁	1	4	yes	—
107	C₂³	C₂²×Dic₃₁	1	1	yes	characteristic
108	D₄	—	2	8	no	—
109	C₂×C₄	—	2	8	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	4	4	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	#92474	1	4	yes	complemented
115	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
116	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
117	C₂×C₄	C₂²×Dic₃₁	1	4	yes	—
118	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
119	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
120	C₈	—	62	8	no	—
121	C₂²	—	2	1	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	4	no	—
124	C₂²	—	2	4	no	—
125	C₂²	—	2	4	no	—
126	C₂²	—	4	8	no	—
127	C₂²	C₂×#92474	1	2	yes	—
128	C₂²	C₂³×Dic₃₁	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	2	4	no	—
131	C₄	—	2	2	no	—
132	C₄	C₂×#92474	1	4	yes	—
133	C₄	#11758561	1	2	yes	—
134	C₂	—	2	2	no	—
135	C₂	—	4	4	no	—
136	C₂	#14521524	1	1	yes	characteristic
137	C₂	C₂²×#92474	1	1	yes	characteristic
138	C₂	#14521816	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521127

C₂

yes

maximal, complemented

#14521102

C₂

yes

maximal, complemented

#14521013

C₂

yes

maximal, complemented

C₂×C₄×D₄×C₃₁

C₂

yes

maximal, characteristic

C₂²×D₄×C₃₁

C₄

yes

characteristic

C₂³×C₄×C₃₁

C₂²

yes

—

C₄×D₄×C₃₁

C₄

yes

—

C₄×D₄×C₃₁

C₂²

yes

—

C₂×C₄⋊C₄×C₃₁

C₄

yes

characteristic

C₂×C₂²⋊C₄×C₃₁

C₄

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×#92474

C₂²

yes

complemented

C₂²⋊(C₃₁⋊C₈)

C₂²

yes

—

C₂²⋊(C₃₁⋊C₈)

C₂²

yes

complemented

C₄⋊(C₃₁⋊C₈)

C₂²

yes

complemented

C₄×C₃₁⋊C₈

C₂²

yes

complemented

C₂⁴×C₃₁

C₂×C₄

yes

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

—

C₄⋊C₄×C₃₁

C₂×C₄

yes

—

C₂²⋊C₄×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

C₂³

yes

—

#92474

—

C₂×C₃₁⋊C₈

C₂³

yes

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

C₂²×C₄

yes

—

C₂³×C₃₁

C₂²×C₄

yes

characteristic

D₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

M₁₆

yes

complemented

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₃₁⋊C₈

—

#2975

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂×M₁₆

yes

—

C₂²×C₃₁

C₂³×C₄

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂×M₁₆

yes

—

C₄×C₃₁

H(5,𝔽₂)

yes

—

C₂×C₄×D₄

D₃₁

yes

—

—

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

#518

yes

characteristic

C₂×C₃₁

C₂²×M₁₆

yes

characteristic

C₂×C₃₁

#568

yes

characteristic

C₂²×D₄

Dic₃₁

yes

characteristic

C₂³×C₄

C₂×D₃₁

yes

—

C₂×M₁₆

—

C₄×D₄

Dic₃₁

yes

—

C₄×D₄

C₂×D₃₁

yes

—

C₂×C₄⋊C₄

Dic₃₁

yes

characteristic

C₂×C₂²⋊C₄

Dic₃₁

yes

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₃₁

#2975

yes

characteristic, complemented, Sylow

C₂⁴

C₂×Dic₃₁

yes

—

C₂×D₄

—

C₂×D₄

C₂×Dic₃₁

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₃₁

yes

characteristic

C₂²×C₄

C₂×Dic₃₁

yes

—

C₂²×C₄

C₂×Dic₃₁

yes

—

C₂²×C₄

C₂×Dic₃₁

yes

characteristic

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂²×C₄

C₂²×D₃₁

yes

—

M₁₆

—

124

—

C₂×C₈

—

C₄⋊C₄

C₂×Dic₃₁

yes

—

100

C₂²⋊C₄

C₂×Dic₃₁

yes

—

101

C₄²

C₂×Dic₃₁

yes

—

102

C₄²

C₂²×D₃₁

yes

—

103

C₂³

—

104

C₂³

—

105

C₂³

—

106

C₂³

C₂²×Dic₃₁

yes

—

107

C₂³

C₂²×Dic₃₁

yes

characteristic

108

D₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

#92474

yes

complemented

115

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

116

C₂×C₄

C₂²×Dic₃₁

yes

—

117

C₂×C₄

C₂²×Dic₃₁

yes

—

118

C₂×C₄

C₂²×Dic₃₁

yes

—

119

C₂×C₄

C₂³×D₃₁

yes

characteristic

120

C₈

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

C₂×#92474

yes

—

128

C₂²

C₂³×Dic₃₁

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

C₂×#92474

yes

—

133

C₄

#11758561

yes

—

134

C₂

—

135

C₂

—

136

C₂

#14521524

yes

characteristic

137

C₂

C₂²×#92474

yes

characteristic

138

C₂

#14521816

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Fitting	C₂×C₄×D₄×C₃₁	C₂
prime radical	C₂²×D₄×C₃₁	C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₂×C₄×D₄	D₃₁
hypocenter	C₃₁	#2975
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₃₁

commutator, socle

C₂²×C₃₁

C₂³×C₄

Fitting

C₂×C₄×D₄×C₃₁

C₂

prime radical

C₂²×D₄×C₃₁

C₄

cosocle

C₂×C₄×C₃₁

C₂⁴

hypercenter

C₂×C₄×D₄

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄

N	H ≅ G/N	type
1. #14521127	C₂	semidirect
2. #14521102	C₂	semidirect
3. #14521013	C₂	semidirect
12. C₂×#92474	C₂²	semidirect
14. C₂²⋊(C₃₁⋊C₈)	C₂²	semidirect
15. C₄⋊(C₃₁⋊C₈)	C₂²	semidirect
16. C₄×C₃₁⋊C₈	C₂²	semidirect
44. C₂×C₄×C₃₁	M₁₆	semidirect
86. C₃₁	#2975	semidirect
114. C₂×C₄	#92474	semidirect

H ≅ G/N

type

1. #14521127

C₂

semidirect

2. #14521102

C₂

semidirect

3. #14521013

C₂

semidirect

12. C₂×#92474

C₂²

semidirect

14. C₂²⋊(C₃₁⋊C₈)

C₂²

semidirect

15. C₄⋊(C₃₁⋊C₈)

C₂²

semidirect

16. C₄×C₃₁⋊C₈

C₂²

semidirect

44. C₂×C₄×C₃₁

M₁₆

semidirect

86. C₃₁

#2975

semidirect

114. C₂×C₄

#92474

semidirect

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

subgroup	index
#14521127	2
#14521102	2
#14521013	2
C₂×C₄×D₄×C₃₁	2
#2975	31

subgroup

index

C₂×C₄×D₄×C₃₁

Subgroups — #16854752

Conjugacy classes of subgroups

Complements

Maximal subgroups