Group #28407476 — subgroups

Subgroups — #28407476

Order 14464.

364 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 166 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	#20299648	C₂	1	6	yes	maximal, complemented
2	Q₈×#11744753	C₂	1	2	yes	maximal, complemented
3	#20299620	C₂	1	3	yes	maximal
4	#20299617	C₂	1	3	yes	maximal
5	C₄×Q₈×D₁₁₃	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#11744753	C₂²	1	6	yes	—
7	C₄⋊#11744753	C₂²	1	6	yes	—
8	#16813679	C₂²	1	1	yes	characteristic
9	#16813679	C₂²	1	3	yes	—
10	#16813677	C₂²	1	6	yes	—
11	C₄×#11744753	C₂²	1	6	yes	—
12	C₂×Q₈×D₁₁₃	C₄	1	1	yes	characteristic
13	Q₈×Dic₁₁₃	C₂²	1	1	yes	characteristic
14	C₄×Dic₂₂₆	C₄	1	3	yes	—
15	#16813587	C₂²	1	3	yes	—
16	C₄⋊C₄×D₁₁₃	C₄	1	3	yes	—
17	C₄²×D₁₁₃	C₂²	1	3	yes	—
18	C₄×Q₈×C₁₁₃	C₄	1	1	yes	characteristic
19	C₂×#11744753	C₂³	1	2	yes	—
20	C₂×#11744753	C₂³	1	6	yes	—
21	C₂×Dic₂₂₆	C₂×C₄	1	3	yes	—
22	Q₈×D₁₁₃	—	2	4	no	—
23	C₂×C₄×D₁₁₃	C₂×C₄	1	3	yes	—
24	C₂×C₄×D₁₁₃	C₂×C₄	1	3	yes	—
25	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
26	C₄⋊Dic₁₁₃	C₂×C₄	1	3	yes	—
27	#14268626	C₂×C₄	1	6	yes	—
28	C₄×Dic₁₁₃	C₂×C₄	1	3	yes	—
29	C₄×Dic₁₁₃	C₂³	1	3	yes	—
30	C₄×Dic₁₁₃	C₂³	1	3	yes	—
31	C₂×Q₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
32	C₄⋊C₄×C₁₁₃	C₂×C₄	1	3	yes	—
33	C₄²×C₁₁₃	C₂×C₄	1	3	yes	—
34	C₂²×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
35	Q₈×C₁₁₃	M₁₆	1	4	yes	—
36	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₁₃	C₂²×C₄	1	3	yes	—
38	C₂×C₄×C₁₁₃	C₂²×C₄	1	3	yes	—
39	C₂×Dic₁₁₃	C₂²×C₄	1	3	yes	—
40	C₂×Dic₁₁₃	C₂²×C₄	1	3	yes	—
41	C₂×Dic₁₁₃	C₂⁴	1	1	yes	characteristic
42	C₄×D₁₁₃	—	2	6	no	—
43	C₄×D₁₁₃	—	2	6	no	—
44	C₄×D₁₁₃	—	2	2	no	—
45	Dic₂₂₆	—	2	6	no	—
46	#11744753	—	2	2	no	—
47	#11744753	—	2	6	no	—
48	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
49	C₂×D₁₁₃	—	2	1	no	—
50	C₂×D₁₁₃	—	2	1	no	—
51	C₂×D₁₁₃	#131	1	2	yes	—
52	C₄×C₁₁₃	—	2	1	no	—
53	C₄×C₁₁₃	—	2	3	no	—
54	C₄×C₁₁₃	C₂×M₁₆	1	6	yes	—
55	Dic₁₁₃	—	2	3	no	—
56	Dic₁₁₃	—	2	3	no	—
57	Dic₁₁₃	2₋¹⁺⁴	1	2	yes	—
58	C₂×C₁₁₃	#519	1	1	yes	characteristic
59	C₂×C₁₁₃	C₂²×M₁₆	1	1	yes	characteristic
60	C₂×C₁₁₃	C₂×#131	1	1	yes	characteristic
61	D₁₁₃	—	2	2	no	—
62	#2984	—	113	1	no	maximal, Sylow
63	C₁₁₃	#2984	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄×Q₈	—	113	1	no	—
65	#446	—	113	6	no	—
66	C₈×Q₈	—	113	2	no	—
67	#432	—	113	3	no	—
68	#431	—	113	3	no	—
69	C₂²×Q₈	—	113	1	no	—
70	C₄×Q₈	—	113	1	no	—
71	C₄×Q₈	—	113	3	no	—
72	C₄×Q₈	—	113	3	no	—
73	C₄×Q₈	#90673	1	1	yes	characteristic
74	C₂×C₄⋊C₄	—	113	3	no	—
75	C₂×C₄²	—	113	3	no	—
76	C₄⋊C₈	—	113	6	no	—
77	C₄⋊C₈	—	113	6	no	—
78	C₂²⋊C₈	—	113	1	no	—
79	C₂²⋊C₈	—	113	3	no	—
80	#97	—	113	6	no	—
81	C₄×C₈	—	113	6	no	—
82	C₂×Q₈	—	113	3	no	—
83	C₂×Q₈	—	226	4	no	—
84	C₂×Q₈	C₂×#90673	1	1	yes	characteristic
85	C₂²×C₄	—	113	1	no	—
86	C₂²×C₄	—	113	3	no	—
87	C₂²×C₄	—	113	3	no	—
88	C₂×C₈	—	113	2	no	—
89	C₂×C₈	—	113	6	no	—
90	C₄⋊C₄	—	113	6	no	—
91	C₄⋊C₄	—	113	3	no	—
92	C₄⋊C₄	C₂×#90673	1	3	yes	—
93	C₄²	—	113	3	no	—
94	C₄²	—	113	3	no	—
95	C₄²	—	113	3	no	—
96	C₄²	C₂×#90673	1	3	yes	—
97	C₂³	—	113	1	no	—
98	Q₈	—	226	6	no	—
99	Q₈	#14268657	1	4	yes	—
100	C₂×C₄	—	113	1	no	—
101	C₂×C₄	—	113	3	no	—
102	C₂×C₄	—	113	3	no	—
103	C₂×C₄	—	226	2	no	—
104	C₂×C₄	—	226	6	no	—
105	C₂×C₄	—	226	6	no	—
106	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
107	C₂×C₄	C₂²×#90673	1	3	yes	—
108	C₂×C₄	C₂²×#90673	1	3	yes	—
109	C₈	—	226	2	no	—
110	C₈	—	226	6	no	—
111	C₂²	—	113	2	no	—
112	C₂²	—	226	1	no	—
113	C₂²	—	226	1	no	—
114	C₂²	C₂³×#90673	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	2	3	no	—
117	C₄	—	113	2	no	—
118	C₄	—	226	3	no	—
119	C₄	—	226	3	no	—
120	C₄	C₂×#14268657	1	6	yes	—
121	C₂	—	226	2	no	—
122	C₂	#20299666	1	1	yes	characteristic
123	C₂	C₂²×#14268657	1	1	yes	characteristic
124	C₂	D₁₁₃⋊#131	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20299648

C₂

yes

maximal, complemented

Q₈×#11744753

C₂

yes

maximal, complemented

#20299620

C₂

yes

maximal

#20299617

C₂

yes

maximal

C₄×Q₈×D₁₁₃

C₂

yes

maximal, characteristic

C₄⋊#11744753

C₂²

yes

—

C₄⋊#11744753

C₂²

yes

—

#16813679

C₂²

yes

characteristic

#16813679

C₂²

yes

—

#16813677

C₂²

yes

—

C₄×#11744753

C₂²

yes

—

C₂×Q₈×D₁₁₃

C₄

yes

characteristic

Q₈×Dic₁₁₃

C₂²

yes

characteristic

C₄×Dic₂₂₆

C₄

yes

—

#16813587

C₂²

yes

—

C₄⋊C₄×D₁₁₃

C₄

yes

—

C₄²×D₁₁₃

C₂²

yes

—

C₄×Q₈×C₁₁₃

C₄

yes

characteristic

C₂×#11744753

C₂³

yes

—

C₂×#11744753

C₂³

yes

—

C₂×Dic₂₂₆

C₂×C₄

yes

—

Q₈×D₁₁₃

—

C₂×C₄×D₁₁₃

C₂×C₄

yes

—

C₂×C₄×D₁₁₃

C₂×C₄

yes

—

C₂×C₄×D₁₁₃

C₂³

yes

characteristic

C₄⋊Dic₁₁₃

C₂×C₄

yes

—

#14268626

C₂×C₄

yes

—

C₄×Dic₁₁₃

C₂×C₄

yes

—

C₄×Dic₁₁₃

C₂³

yes

—

C₄×Dic₁₁₃

C₂³

yes

—

C₂×Q₈×C₁₁₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₁₃

C₂×C₄

yes

—

C₄²×C₁₁₃

C₂×C₄

yes

—

C₂²×D₁₁₃

C₂²×C₄

yes

characteristic

Q₈×C₁₁₃

M₁₆

yes

—

C₂×C₄×C₁₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₁₃

C₂²×C₄

yes

—

C₂×Dic₁₁₃

C₂²×C₄

yes

—

C₂×Dic₁₁₃

C₂²×C₄

yes

—

C₂×Dic₁₁₃

C₂⁴

yes

characteristic

C₄×D₁₁₃

—

C₄×D₁₁₃

—

C₄×D₁₁₃

—

Dic₂₂₆

—

#11744753

—

#11744753

—

C₂²×C₁₁₃

C₂³×C₄

yes

characteristic

C₂×D₁₁₃

—

C₂×D₁₁₃

—

C₂×D₁₁₃

#131

yes

—

C₄×C₁₁₃

—

C₄×C₁₁₃

—

C₄×C₁₁₃

C₂×M₁₆

yes

—

Dic₁₁₃

—

Dic₁₁₃

—

Dic₁₁₃

2₋¹⁺⁴

yes

—

C₂×C₁₁₃

#519

yes

characteristic

C₂×C₁₁₃

C₂²×M₁₆

yes

characteristic

C₂×C₁₁₃

C₂×#131

yes

characteristic

D₁₁₃

—

#2984

—

113

maximal, Sylow

C₁₁₃

#2984

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

113

—

#446

—

113

—

C₈×Q₈

—

113

—

#432

—

113

—

#431

—

113

—

C₂²×Q₈

—

113

—

C₄×Q₈

—

113

—

C₄×Q₈

—

113

—

C₄×Q₈

—

113

—

C₄×Q₈

#90673

yes

characteristic

C₂×C₄⋊C₄

—

113

—

C₂×C₄²

—

113

—

C₄⋊C₈

—

113

—

C₄⋊C₈

—

113

—

C₂²⋊C₈

—

113

—

C₂²⋊C₈

—

113

—

#97

—

113

—

C₄×C₈

—

113

—

C₂×Q₈

—

113

—

C₂×Q₈

—

226

—

C₂×Q₈

C₂×#90673

yes

characteristic

C₂²×C₄

—

113

—

C₂²×C₄

—

113

—

C₂²×C₄

—

113

—

C₂×C₈

—

113

—

C₂×C₈

—

113

—

C₄⋊C₄

—

113

—

C₄⋊C₄

—

113

—

C₄⋊C₄

C₂×#90673

yes

—

C₄²

—

113

—

C₄²

—

113

—

C₄²

—

113

—

C₄²

C₂×#90673

yes

—

C₂³

—

113

—

Q₈

—

226

—

Q₈

#14268657

yes

—

100

C₂×C₄

—

113

—

101

C₂×C₄

—

113

—

102

C₂×C₄

—

113

—

103

C₂×C₄

—

226

—

104

C₂×C₄

—

226

—

105

C₂×C₄

—

226

—

106

C₂×C₄

C₂²×#90673

yes

characteristic

107

C₂×C₄

C₂²×#90673

yes

—

108

C₂×C₄

C₂²×#90673

yes

—

109

C₈

—

226

—

110

C₈

—

226

—

111

C₂²

—

113

—

112

C₂²

—

226

—

113

C₂²

—

226

—

114

C₂²

C₂³×#90673

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

113

—

118

C₄

—

226

—

119

C₄

—

226

—

120

C₄

C₂×#14268657

yes

—

121

C₂

—

226

—

122

C₂

#20299666

yes

characteristic

123

C₂

C₂²×#14268657

yes

characteristic

124

C₂

D₁₁₃⋊#131

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#90673
commutator, socle	C₂²×C₁₁₃	C₂³×C₄
Fitting	C₄×Q₈×C₁₁₃	C₄
prime radical	C₂²×D₁₁₃	C₂²×C₄
cosocle	C₂×Dic₁₁₃	C₂⁴
hypercenter	C₄×Q₈	#90673
hypocenter	C₁₁₃	#2984
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×#90673

commutator, socle

C₂²×C₁₁₃

C₂³×C₄

Fitting

C₄×Q₈×C₁₁₃

C₄

prime radical

C₂²×D₁₁₃

C₂²×C₄

cosocle

C₂×Dic₁₁₃

C₂⁴

C₄×Q₈

C₁₁₃

G ▷ C₂²×C₁₁₃ ▷ C₁₁₃

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
1. #20299648	C₂	semidirect
2. Q₈×#11744753	C₂	semidirect
63. C₁₁₃	#2984	semidirect

H ≅ G/N

type

1. #20299648

C₂

semidirect

2. Q₈×#11744753

C₂

semidirect

63. C₁₁₃

#2984

semidirect

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

subgroup	index
#20299648	2
Q₈×#11744753	2
#20299620	2
#20299617	2
C₄×Q₈×D₁₁₃	2
#2984	113

subgroup

index

C₄×Q₈×D₁₁₃

113

Subgroups — #28407476

Conjugacy classes of subgroups

Complements

Maximal subgroups