Group #16854646 — subgroups

Subgroups — #16854646

Order 3968.

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

normal subgroups only

This group has 364 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	#14521180	C₂	1	4	yes	maximal, complemented
2	Q₈×C₃₁⋊C₈	C₂	1	4	yes	maximal, complemented
3	#14521076	C₂	1	2	yes	maximal
4	#14521073	C₂	1	1	yes	maximal, characteristic
5	#14521005	C₂	1	2	yes	maximal
6	C₂×C₄×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#533	C₂	1	1	yes	maximal, characteristic
8	C₃₁×C₄⋊Q₈	C₄	1	2	yes	—
9	C₃₁×#125	C₄	1	2	yes	—
10	C₃₁×C₂²⋊Q₈	C₄	1	4	yes	—
11	C₄×Q₈×C₃₁	C₂²	1	4	yes	—
12	C₃₁×#117	C₂²	1	2	yes	—
13	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×#92474	C₂²	1	2	yes	—
15	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
16	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊(C₃₁⋊C₈)	C₂²	1	4	yes	—
18	C₄⋊(C₃₁⋊C₈)	C₂²	1	8	yes	—
19	#11758391	C₂²	1	4	yes	—
20	C₄×C₃₁⋊C₈	C₂²	1	2	yes	—
21	C₄×C₃₁⋊C₈	C₂²	1	4	yes	—
22	C₄×C₃₁⋊C₈	C₂²	1	2	yes	—
23	C₂×Q₈×C₃₁	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
27	C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
28	C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
29	C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
30	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
32	C₄²×C₃₁	C₂³	1	2	yes	—
33	C₄²×C₃₁	C₂³	1	4	yes	—
34	#92474	—	2	4	no	—
35	C₂×C₃₁⋊C₈	—	2	2	no	—
36	C₂×C₃₁⋊C₈	Q₈	1	4	yes	complemented
37	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
38	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
39	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
40	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
41	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
42	Q₈×C₃₁	—	2	8	no	—
43	C₂×C₄×C₃₁	—	2	4	no	—
44	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
45	C₂×C₄×C₃₁	C₂²×C₄	1	8	yes	—
46	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₃₁	#41	1	2	yes	—
50	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
51	C₃₁⋊C₈	—	2	2	no	—
52	C₃₁⋊C₈	—	2	4	no	—
53	C₃₁⋊C₈	C₂×Q₈	1	2	yes	—
54	C₃₁⋊C₈	C₂×Q₈	1	2	yes	—
55	#2960	—	31	1	no	maximal, Sylow
56	C₂²×C₃₁	—	2	2	no	—
57	C₂²×C₃₁	C₄×Q₈	1	2	yes	—
58	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
59	C₄×C₃₁	—	2	2	no	—
60	C₄×C₃₁	—	2	8	no	—
61	C₄×C₃₁	C₄×Q₈	1	2	yes	—
62	C₄×C₃₁	#131	1	4	yes	—
63	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
64	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
65	#533	D₃₁	1	1	yes	characteristic
66	#446	—	31	4	no	—
67	C₈×Q₈	—	31	4	no	—
68	#424	—	31	2	no	—
69	#423	—	31	1	no	—
70	#405	—	31	2	no	—
71	C₂×C₄×C₈	—	31	1	no	—
72	C₂×C₃₁	—	2	2	no	—
73	C₂×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
74	C₂×C₃₁	C₂×#131	1	2	yes	—
75	C₂×M₁₆	—	31	2	no	—
76	C₂²×C₈	—	31	1	no	—
77	C₂²×C₈	—	31	1	no	—
78	C₄⋊Q₈	Dic₃₁	1	2	yes	—
79	#125	Dic₃₁	1	2	yes	—
80	C₂²⋊Q₈	Dic₃₁	1	4	yes	—
81	C₄×Q₈	C₂×D₃₁	1	4	yes	—
82	#117	C₂×D₃₁	1	2	yes	—
83	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
84	C₄⋊C₈	—	31	4	no	—
85	C₄⋊C₈	—	31	8	no	—
86	#97	—	31	4	no	—
87	C₄×C₈	—	31	2	no	—
88	C₄×C₈	—	31	2	no	—
89	C₄×C₈	—	31	4	no	—
90	C₃₁	#2960	1	1	yes	characteristic, complemented, Sylow
91	C₂×Q₈	C₂×Dic₃₁	1	4	yes	—
92	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
93	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
94	M₁₆	—	62	4	no	—
95	C₂×C₈	—	31	4	no	—
96	C₂×C₈	—	31	1	no	—
97	C₂×C₈	—	31	1	no	—
98	C₂×C₈	—	31	2	no	—
99	C₂×C₈	—	31	4	no	—
100	C₂×C₈	—	62	2	no	—
101	C₄⋊C₄	C₂×Dic₃₁	1	4	yes	—
102	C₄⋊C₄	C₂×Dic₃₁	1	8	yes	—
103	C₄⋊C₄	C₂×Dic₃₁	1	4	yes	—
104	C₂²⋊C₄	C₂×Dic₃₁	1	4	yes	—
105	C₄²	C₂×Dic₃₁	1	1	yes	characteristic
106	C₄²	C₂×Dic₃₁	1	1	yes	characteristic
107	C₄²	C₂²×D₃₁	1	2	yes	—
108	C₄²	C₂²×D₃₁	1	4	yes	—
109	C₂³	C₂²×Dic₃₁	1	1	yes	characteristic
110	Q₈	—	2	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	Q₈×D₃₁	1	2	yes	—
113	C₂×C₄	#92499	1	2	yes	—
114	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
115	C₂×C₄	C₂²×Dic₃₁	1	8	yes	—
116	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
117	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
118	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
119	C₈	—	31	2	no	—
120	C₈	—	31	2	no	—
121	C₈	—	62	2	no	—
122	C₈	—	62	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	Q₈×Dic₃₁	1	2	yes	—
125	C₂²	C₂³×Dic₃₁	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	2	8	no	—
128	C₄	Q₈×Dic₃₁	1	2	yes	—
129	C₄	#11758525	1	4	yes	—
130	C₄	C₂×Q₈×D₃₁	1	1	yes	characteristic
131	C₄	C₂×#92499	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×Q₈×Dic₃₁	1	1	yes	characteristic
134	C₂	C₂×#11758525	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14521180

C₂

yes

maximal, complemented

Q₈×C₃₁⋊C₈

C₂

yes

maximal, complemented

#14521076

C₂

yes

maximal

#14521073

C₂

yes

maximal, characteristic

#14521005

C₂

yes

maximal

C₂×C₄×C₃₁⋊C₈

C₂

yes

maximal, characteristic

C₃₁×#533

C₂

yes

maximal, characteristic

C₃₁×C₄⋊Q₈

C₄

yes

—

C₃₁×#125

C₄

yes

—

C₃₁×C₂²⋊Q₈

C₄

yes

—

C₄×Q₈×C₃₁

C₂²

yes

—

C₃₁×#117

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×#92474

C₂²

yes

—

C₂²×C₃₁⋊C₈

C₂²

yes

characteristic

C₂²×C₃₁⋊C₈

C₂²

yes

characteristic

C₄⋊(C₃₁⋊C₈)

C₂²

yes

—

C₄⋊(C₃₁⋊C₈)

C₂²

yes

—

#11758391

C₂²

yes

—

C₄×C₃₁⋊C₈

C₂²

yes

—

C₄×C₃₁⋊C₈

C₂²

yes

—

C₄×C₃₁⋊C₈

C₂²

yes

—

C₂×Q₈×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₄⋊C₄×C₃₁

C₂×C₄

yes

—

C₄⋊C₄×C₃₁

C₂×C₄

yes

—

C₄⋊C₄×C₃₁

C₂×C₄

yes

—

C₂²⋊C₄×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

C₂×C₄

yes

characteristic

C₄²×C₃₁

C₂×C₄

yes

characteristic

C₄²×C₃₁

C₂³

yes

—

C₄²×C₃₁

C₂³

yes

—

#92474

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

Q₈

yes

complemented

C₂×C₃₁⋊C₈

C₂³

yes

—

C₂×C₃₁⋊C₈

C₂³

yes

—

C₂×C₃₁⋊C₈

C₂³

yes

characteristic

C₂×C₃₁⋊C₈

C₂³

yes

characteristic

C₂³×C₃₁

C₂²×C₄

yes

characteristic

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

#41

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₃₁⋊C₈

—

C₃₁⋊C₈

—

C₃₁⋊C₈

C₂×Q₈

yes

—

C₃₁⋊C₈

C₂×Q₈

yes

—

#2960

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

C₄×Q₈

yes

—

C₂²×C₃₁

C₂³×C₄

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₄×Q₈

yes

—

C₄×C₃₁

#131

yes

—

C₄×C₃₁

C₂²×Q₈

yes

characteristic

C₄×C₃₁

C₂×#41

yes

characteristic

#533

D₃₁

yes

characteristic

#446

—

C₈×Q₈

—

#424

—

#423

—

#405

—

C₂×C₄×C₈

—

C₂×C₃₁

—

C₂×C₃₁

C₂×C₄×Q₈

yes

characteristic

C₂×C₃₁

C₂×#131

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

Dic₃₁

yes

—

#125

Dic₃₁

yes

—

C₂²⋊Q₈

Dic₃₁

yes

—

C₄×Q₈

C₂×D₃₁

yes

—

#117

C₂×D₃₁

yes

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₃₁

#2960

yes

characteristic, complemented, Sylow

C₂×Q₈

C₂×Dic₃₁

yes

—

C₂²×C₄

C₂×Dic₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₄⋊C₄

C₂×Dic₃₁

yes

—

102

C₄⋊C₄

C₂×Dic₃₁

yes

—

103

C₄⋊C₄

C₂×Dic₃₁

yes

—

104

C₂²⋊C₄

C₂×Dic₃₁

yes

—

105

C₄²

C₂×Dic₃₁

yes

characteristic

106

C₄²

C₂×Dic₃₁

yes

characteristic

107

C₄²

C₂²×D₃₁

yes

—

108

C₄²

C₂²×D₃₁

yes

—

109

C₂³

C₂²×Dic₃₁

yes

characteristic

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

Q₈×D₃₁

yes

—

113

C₂×C₄

#92499

yes

—

114

C₂×C₄

C₂²×Dic₃₁

yes

—

115

C₂×C₄

C₂²×Dic₃₁

yes

—

116

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

117

C₂×C₄

C₂²×Dic₃₁

yes

—

118

C₂×C₄

C₂³×D₃₁

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

—

123

C₂²

—

124

C₂²

Q₈×Dic₃₁

yes

—

125

C₂²

C₂³×Dic₃₁

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

Q₈×Dic₃₁

yes

—

129

C₄

#11758525

yes

—

130

C₄

C₂×Q₈×D₃₁

yes

characteristic

131

C₄

C₂×#92499

yes

characteristic

132

C₂

—

133

C₂

C₂×Q₈×Dic₃₁

yes

characteristic

134

C₂

C₂×#11758525

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Fitting	C₃₁×#533	C₂
prime radical	C₂³×C₃₁	C₂²×C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#533	D₃₁
hypocenter	C₃₁	#2960
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ #533

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₃₁

commutator, socle

C₂²×C₃₁

C₂³×C₄

Fitting

C₃₁×#533

C₂

prime radical

C₂³×C₃₁

C₂²×C₄

cosocle

C₂×C₄×C₃₁

C₂⁴

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂×C₄ ◁ #533

N	H ≅ G/N	type
1. #14521180	C₂	semidirect
2. Q₈×C₃₁⋊C₈	C₂	semidirect
36. C₂×C₃₁⋊C₈	Q₈	semidirect
90. C₃₁	#2960	semidirect

H ≅ G/N

type

1. #14521180

C₂

semidirect

2. Q₈×C₃₁⋊C₈

C₂

semidirect

36. C₂×C₃₁⋊C₈

Q₈

semidirect

90. C₃₁

#2960

semidirect

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

subgroup	index
#14521180	2
Q₈×C₃₁⋊C₈	2
#14521076	2
#14521073	2
#14521005	2
C₂×C₄×C₃₁⋊C₈	2
C₃₁×#533	2
#2960	31

subgroup

index

Q₈×C₃₁⋊C₈

C₂×C₄×C₃₁⋊C₈

C₃₁×#533

#2960

Subgroups — #16854646

Conjugacy classes of subgroups

Complements

Maximal subgroups