Group #16854652 — subgroups

Subgroups — #16854652

Order 3968.

348 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 221 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	8	yes	maximal, complemented
2	#14521076	C₂	1	2	yes	maximal
3	#14521073	C₂	1	1	yes	maximal, characteristic
4	#14521005	C₂	1	2	yes	maximal
5	C₂×#11758391	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#533	C₂	1	1	yes	maximal, characteristic
7	C₃₁×C₄⋊Q₈	C₄	1	2	yes	—
8	C₃₁×#125	C₄	1	2	yes	—
9	C₃₁×C₂²⋊Q₈	C₄	1	4	yes	—
10	C₄×Q₈×C₃₁	C₂²	1	4	yes	—
11	C₃₁×#117	C₂²	1	2	yes	—
12	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×#92474	C₂²	1	2	yes	—
14	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
15	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
16	C₄⋊(C₃₁⋊C₈)	C₂²	1	4	yes	—
17	C₄⋊(C₃₁⋊C₈)	C₂²	1	8	yes	—
18	#11758391	C₂²	1	4	yes	—
19	#11758391	C₂²	1	4	yes	—
20	C₄×C₃₁⋊C₈	C₂²	1	4	yes	—
21	C₂×Q₈×C₃₁	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
25	C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
26	C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
27	C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
28	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₃₁	C₂³	1	4	yes	—
31	C₄²×C₃₁	C₂³	1	2	yes	—
32	#92474	—	2	4	no	—
33	C₂×C₃₁⋊C₈	—	2	2	no	—
34	C₂×C₃₁⋊C₈	Q₈	1	4	yes	complemented
35	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
36	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
37	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
38	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
39	Q₈×C₃₁	—	4	4	no	—
40	C₂×C₄×C₃₁	—	4	2	no	—
41	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
42	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
43	C₂×C₄×C₃₁	C₂²×C₄	1	8	yes	—
44	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₃₁	#41	1	2	yes	—
47	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
48	C₃₁⋊C₈	—	2	2	no	—
49	C₃₁⋊C₈	—	2	4	no	—
50	C₃₁⋊C₈	C₂×Q₈	1	4	yes	—
51	#2961	—	31	1	no	maximal, Sylow
52	C₂²×C₃₁	—	2	2	no	—
53	C₂²×C₃₁	C₄×Q₈	1	2	yes	—
54	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
55	C₄×C₃₁	—	2	2	no	—
56	C₄×C₃₁	—	2	2	no	—
57	C₄×C₃₁	—	2	8	no	—
58	C₄×C₃₁	C₄×Q₈	1	2	yes	—
59	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
60	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
61	#533	D₃₁	1	1	yes	characteristic
62	#446	—	31	8	no	—
63	#424	—	31	2	no	—
64	#423	—	31	1	no	—
65	#405	—	31	2	no	—
66	C₂×#97	—	31	1	no	—
67	C₂×C₃₁	—	2	2	no	—
68	C₂×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
69	C₂×C₃₁	#568	1	2	yes	—
70	C₂×M₁₆	—	31	2	no	—
71	C₂²×C₈	—	31	1	no	—
72	C₂²×C₈	—	31	1	no	—
73	C₄⋊Q₈	Dic₃₁	1	2	yes	—
74	#125	Dic₃₁	1	2	yes	—
75	C₂²⋊Q₈	Dic₃₁	1	4	yes	—
76	C₄×Q₈	C₂×D₃₁	1	4	yes	—
77	#117	C₂×D₃₁	1	2	yes	—
78	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
79	C₄⋊C₈	—	31	4	no	—
80	C₄⋊C₈	—	31	8	no	—
81	#97	—	31	4	no	—
82	#97	—	31	4	no	—
83	C₄×C₈	—	31	4	no	—
84	C₃₁	#2961	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₈	C₂×Dic₃₁	1	4	yes	—
86	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
87	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
88	M₁₆	—	62	4	no	—
89	C₂×C₈	—	31	4	no	—
90	C₂×C₈	—	31	2	no	—
91	C₂×C₈	—	31	4	no	—
92	C₂×C₈	—	31	2	no	—
93	C₂×C₈	—	62	2	no	—
94	C₄⋊C₄	C₂×Dic₃₁	1	4	yes	—
95	C₄⋊C₄	C₂×Dic₃₁	1	8	yes	—
96	C₄⋊C₄	C₂×Dic₃₁	1	4	yes	—
97	C₂²⋊C₄	C₂×Dic₃₁	1	4	yes	—
98	C₄²	C₂×Dic₃₁	1	1	yes	characteristic
99	C₄²	C₂×Dic₃₁	1	1	yes	characteristic
100	C₄²	C₂²×D₃₁	1	4	yes	—
101	C₄²	C₂²×D₃₁	1	2	yes	—
102	C₂³	C₂²×Dic₃₁	1	1	yes	characteristic
103	Q₈	—	4	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	Q₈×D₃₁	1	2	yes	—
106	C₂×C₄	#92499	1	2	yes	—
107	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
108	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
109	C₂×C₄	C₂²×Dic₃₁	1	8	yes	—
110	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
111	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
112	C₈	—	31	4	no	—
113	C₈	—	62	2	no	—
114	C₈	—	62	4	no	—
115	C₂²	—	2	2	no	—
116	C₂²	Q₈×Dic₃₁	1	2	yes	—
117	C₂²	C₂³×Dic₃₁	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	2	8	no	—
121	C₄	Q₈×Dic₃₁	1	2	yes	—
122	C₄	C₂×Q₈×D₃₁	1	1	yes	characteristic
123	C₄	C₂×#92499	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	C₂×Q₈×Dic₃₁	1	1	yes	characteristic
126	C₂	#14521816	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14521180

C₂

yes

maximal, complemented

#14521076

C₂

yes

maximal

#14521073

C₂

yes

maximal, characteristic

#14521005

C₂

yes

maximal

C₂×#11758391

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

—

#11758391

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

—

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

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

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

—

#2961

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

C₄×Q₈

yes

—

C₂²×C₃₁

C₂³×C₄

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₄×Q₈

yes

—

C₄×C₃₁

C₂²×Q₈

yes

characteristic

C₄×C₃₁

C₂×#41

yes

characteristic

#533

D₃₁

yes

—

—

—

—

C₂×#97

—

C₂×C₃₁

—

C₂×C₃₁

C₂×C₄×Q₈

yes

characteristic

C₂×C₃₁

#568

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

—

#97

—

C₄×C₈

—

C₃₁

#2961

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

—

C₄⋊C₄

C₂×Dic₃₁

yes

—

C₄⋊C₄

C₂×Dic₃₁

yes

—

C₄⋊C₄

C₂×Dic₃₁

yes

—

C₂²⋊C₄

C₂×Dic₃₁

yes

—

C₄²

C₂×Dic₃₁

yes

characteristic

C₄²

C₂×Dic₃₁

yes

characteristic

100

C₄²

C₂²×D₃₁

yes

—

101

C₄²

C₂²×D₃₁

yes

—

102

C₂³

C₂²×Dic₃₁

yes

characteristic

103

Q₈

—

104

C₂×C₄

—

105

C₂×C₄

Q₈×D₃₁

yes

—

106

C₂×C₄

#92499

yes

—

107

C₂×C₄

C₂²×Dic₃₁

yes

—

108

C₂×C₄

C₂²×Dic₃₁

yes

—

109

C₂×C₄

C₂²×Dic₃₁

yes

—

110

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

111

C₂×C₄

C₂³×D₃₁

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

—

115

C₂²

—

116

C₂²

Q₈×Dic₃₁

yes

—

117

C₂²

C₂³×Dic₃₁

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

Q₈×Dic₃₁

yes

—

122

C₄

C₂×Q₈×D₃₁

yes

characteristic

123

C₄

C₂×#92499

yes

characteristic

124

C₂

—

125

C₂

C₂×Q₈×Dic₃₁

yes

characteristic

126

C₂

#14521816

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	#2961
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
34. C₂×C₃₁⋊C₈	Q₈	semidirect
84. C₃₁	#2961	semidirect

H ≅ G/N

type

1. #14521180

C₂

semidirect

34. C₂×C₃₁⋊C₈

Q₈

semidirect

84. C₃₁

#2961

semidirect

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

subgroup	index
#14521180	2
#14521076	2
#14521073	2
#14521005	2
C₂×#11758391	2
C₃₁×#533	2
#2961	31

subgroup

index

C₃₁×#533

Subgroups — #16854652

Conjugacy classes of subgroups

Complements

Maximal subgroups