Group #28407484 — subgroups

Subgroups — #28407484

Order 14464.

330 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 148 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	#20299650	C₂	1	4	yes	maximal, complemented
2	#20299647	C₂	1	4	yes	maximal, complemented
3	#20299623	C₂	1	4	yes	maximal
4	#20299616	C₂	1	2	yes	maximal
5	#20299154	C₂	1	1	yes	maximal, characteristic
6	#16813689	C₂²	1	2	yes	—
7	#16813689	C₂²	1	2	yes	—
8	#16813689	C₂²	1	4	yes	—
9	C₄⋊#11744753	C₂²	1	4	yes	—
10	#16813679	C₂²	1	4	yes	—
11	#16813677	C₂²	1	4	yes	—
12	C₄×#11744753	C₂²	1	4	yes	—
13	C₄×#11744753	C₂²	1	4	yes	—
14	#16813621	C₄	1	2	yes	—
15	#16813614	C₄	1	1	yes	characteristic
16	#16813600	C₄	1	4	yes	—
17	#16813587	C₂²	1	4	yes	—
18	#16813580	C₂²	1	2	yes	—
19	C₄²×D₁₁₃	C₂²	1	1	yes	characteristic
20	#125×C₁₁₃	C₄	1	1	yes	characteristic
21	C₂×#11744753	C₂³	1	4	yes	—
22	C₂×#11744753	C₂³	1	4	yes	—
23	C₂×Dic₂₂₆	C₂×C₄	1	4	yes	—
24	C₂×C₄×D₁₁₃	C₂×C₄	1	2	yes	—
25	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
26	C₄⋊Dic₁₁₃	C₂×C₄	1	2	yes	—
27	#14268626	C₂×C₄	1	4	yes	—
28	#14268626	C₂×C₄	1	4	yes	—
29	#14268624	C₂×C₄	1	4	yes	—
30	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₁₁₃	C₂³	1	2	yes	—
32	C₄×Dic₁₁₃	C₂³	1	4	yes	—
33	C₄⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	—
34	C₄⋊C₄×C₁₁₃	C₂×C₄	1	4	yes	—
35	C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
39	C₂×C₄×C₁₁₃	C₂²×C₄	1	4	yes	—
40	C₂×Dic₁₁₃	C₂²×C₄	1	4	yes	—
41	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
42	C₂×Dic₁₁₃	C₂⁴	1	1	yes	characteristic
43	C₄×D₁₁₃	—	2	4	no	—
44	C₄×D₁₁₃	—	2	2	no	—
45	Dic₂₂₆	—	2	8	no	—
46	#11744753	—	2	4	no	—
47	#11744753	—	2	4	no	—
48	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
49	C₂×D₁₁₃	—	2	1	no	—
50	C₂×D₁₁₃	—	2	2	no	—
51	C₄×C₁₁₃	—	2	1	no	—
52	C₄×C₁₁₃	—	2	4	no	—
53	C₄×C₁₁₃	—	2	2	no	—
54	Dic₁₁₃	—	2	4	no	—
55	Dic₁₁₃	#131	1	4	yes	—
56	Dic₁₁₃	2₋¹⁺⁴	1	2	yes	—
57	C₂×C₁₁₃	#519	1	1	yes	characteristic
58	C₂×C₁₁₃	C₂×#131	1	2	yes	—
59	D₁₁₃	—	4	1	no	—
60	#2986	—	113	1	no	maximal, Sylow
61	C₁₁₃	#2986	1	1	yes	characteristic, complemented, Sylow
62	#533	—	113	1	no	—
63	#446	—	113	4	no	—
64	C₈×Q₈	—	113	4	no	—
65	#433	—	113	4	no	—
66	#431	—	113	2	no	—
67	C₄⋊Q₈	—	113	2	no	—
68	#125	—	113	1	no	—
69	#125	#90673	1	1	yes	characteristic
70	C₂²⋊Q₈	—	113	4	no	—
71	C₄×Q₈	—	113	4	no	—
72	#117	—	113	2	no	—
73	C₂×C₄²	—	113	1	no	—
74	C₄⋊C₈	—	113	4	no	—
75	C₄⋊C₈	—	113	2	no	—
76	C₄⋊C₈	—	113	2	no	—
77	C₄⋊C₈	—	113	4	no	—
78	C₂²⋊C₈	—	113	4	no	—
79	#97	—	113	4	no	—
80	C₄×C₈	—	113	4	no	—
81	C₄×C₈	—	113	4	no	—
82	C₂×Q₈	—	113	4	no	—
83	C₂²×C₄	—	113	1	no	—
84	C₂²×C₄	—	113	2	no	—
85	C₂×C₈	—	113	4	no	—
86	C₂×C₈	—	113	4	no	—
87	C₄⋊C₄	—	113	2	no	—
88	C₄⋊C₄	—	113	4	no	—
89	C₄⋊C₄	—	113	4	no	—
90	C₄⋊C₄	C₂×#90673	1	2	yes	—
91	C₄⋊C₄	C₂×#90673	1	4	yes	—
92	C₂²⋊C₄	—	113	4	no	—
93	C₄²	—	113	2	no	—
94	C₄²	—	113	4	no	—
95	C₄²	—	113	1	no	—
96	C₄²	C₂×#90673	1	1	yes	characteristic
97	C₂³	—	113	1	no	—
98	Q₈	—	226	8	no	—
99	C₂×C₄	—	113	1	no	—
100	C₂×C₄	—	113	2	no	—
101	C₂×C₄	—	113	4	no	—
102	C₂×C₄	—	226	2	no	—
103	C₂×C₄	—	226	4	no	—
104	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
105	C₂×C₄	C₂²×#90673	1	2	yes	—
106	C₂×C₄	C₂²×#90673	1	4	yes	—
107	C₈	—	226	4	no	—
108	C₈	—	226	4	no	—
109	C₂²	—	226	1	no	—
110	C₂²	—	226	2	no	—
111	C₂²	C₂³×#90673	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	2	2	no	—
114	C₄	—	2	4	no	—
115	C₄	—	113	2	no	—
116	C₄	—	113	4	no	—
117	C₄	—	226	4	no	—
118	C₂	—	452	1	no	—
119	C₂	#20299666	1	1	yes	characteristic
120	C₂	D₁₁₃⋊#131	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20299650

C₂

yes

maximal, complemented

#20299647

C₂

yes

maximal, complemented

#20299623

C₂

yes

maximal

#20299616

C₂

yes

maximal

#20299154

C₂

yes

maximal, characteristic

#16813689

C₂²

yes

—

#16813689

C₂²

yes

—

#16813689

C₂²

yes

—

C₄⋊#11744753

C₂²

yes

—

#16813679

C₂²

yes

—

#16813677

C₂²

yes

—

C₄×#11744753

C₂²

yes

—

C₄×#11744753

C₂²

yes

—

#16813621

C₄

yes

—

#16813614

C₄

yes

characteristic

#16813600

C₄

yes

—

#16813587

C₂²

yes

—

#16813580

C₂²

yes

—

C₄²×D₁₁₃

C₂²

yes

characteristic

#125×C₁₁₃

C₄

yes

characteristic

C₂×#11744753

C₂³

yes

—

C₂×#11744753

C₂³

yes

—

C₂×Dic₂₂₆

C₂×C₄

yes

—

C₂×C₄×D₁₁₃

C₂×C₄

yes

—

C₂×C₄×D₁₁₃

C₂³

yes

characteristic

C₄⋊Dic₁₁₃

C₂×C₄

yes

—

#14268626

C₂×C₄

yes

—

#14268626

C₂×C₄

yes

—

#14268624

C₂×C₄

yes

—

C₄×Dic₁₁₃

C₂×C₄

yes

characteristic

C₄×Dic₁₁₃

C₂³

yes

—

C₄×Dic₁₁₃

C₂³

yes

—

C₄⋊C₄×C₁₁₃

C₂×C₄

yes

—

C₄⋊C₄×C₁₁₃

C₂×C₄

yes

—

C₄²×C₁₁₃

C₂×C₄

yes

characteristic

C₂²×D₁₁₃

C₂²×C₄

yes

characteristic

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

—

Dic₂₂₆

—

#11744753

—

#11744753

—

C₂²×C₁₁₃

C₂³×C₄

yes

characteristic

C₂×D₁₁₃

—

C₂×D₁₁₃

—

C₄×C₁₁₃

—

C₄×C₁₁₃

—

C₄×C₁₁₃

—

Dic₁₁₃

—

Dic₁₁₃

#131

yes

—

Dic₁₁₃

2₋¹⁺⁴

yes

—

C₂×C₁₁₃

#519

yes

characteristic

C₂×C₁₁₃

C₂×#131

yes

—

D₁₁₃

—

#2986

—

113

maximal, Sylow

C₁₁₃

#2986

yes

characteristic, complemented, Sylow

#533

—

113

—

#446

—

113

—

C₈×Q₈

—

113

—

#433

—

113

—

#431

—

113

—

C₄⋊Q₈

—

113

—

#125

—

113

—

#125

#90673

yes

characteristic

C₂²⋊Q₈

—

113

—

C₄×Q₈

—

113

—

#117

—

113

—

C₂×C₄²

—

113

—

C₄⋊C₈

—

113

—

C₄⋊C₈

—

113

—

C₄⋊C₈

—

113

—

C₄⋊C₈

—

113

—

C₂²⋊C₈

—

113

—

#97

—

113

—

C₄×C₈

—

113

—

C₄×C₈

—

113

—

C₂×Q₈

—

113

—

C₂²×C₄

—

113

—

C₂²×C₄

—

113

—

C₂×C₈

—

113

—

C₂×C₈

—

113

—

C₄⋊C₄

—

113

—

C₄⋊C₄

—

113

—

C₄⋊C₄

—

113

—

C₄⋊C₄

C₂×#90673

yes

—

C₄⋊C₄

C₂×#90673

yes

—

C₂²⋊C₄

—

113

—

C₄²

—

113

—

C₄²

—

113

—

C₄²

—

113

—

C₄²

C₂×#90673

yes

characteristic

C₂³

—

113

—

Q₈

—

226

—

C₂×C₄

—

113

—

100

C₂×C₄

—

113

—

101

C₂×C₄

—

113

—

102

C₂×C₄

—

226

—

103

C₂×C₄

—

226

—

104

C₂×C₄

C₂²×#90673

yes

characteristic

105

C₂×C₄

C₂²×#90673

yes

—

106

C₂×C₄

C₂²×#90673

yes

—

107

C₈

—

226

—

108

C₈

—

226

—

109

C₂²

—

226

—

110

C₂²

—

226

—

111

C₂²

C₂³×#90673

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

113

—

116

C₄

—

113

—

117

C₄

—

226

—

118

C₂

—

452

—

119

C₂

#20299666

yes

characteristic

120

C₂

D₁₁₃⋊#131

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×#90673
commutator, socle	C₂²×C₁₁₃	C₂³×C₄
Fitting	#125×C₁₁₃	C₄
prime radical	C₂²×D₁₁₃	C₂²×C₄
cosocle	C₂×Dic₁₁₃	C₂⁴
hypercenter	#125	#90673
hypocenter	C₁₁₃	#2986
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×#90673

commutator, socle

C₂²×C₁₁₃

C₂³×C₄

Fitting

#125×C₁₁₃

C₄

prime radical

C₂²×D₁₁₃

C₂²×C₄

cosocle

C₂×Dic₁₁₃

C₂⁴

C₁₁₃

G ▷ C₂²×C₁₁₃ ▷ C₁₁₃

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. #20299650	C₂	semidirect
2. #20299647	C₂	semidirect
61. C₁₁₃	#2986	semidirect

H ≅ G/N

type

1. #20299650

C₂

semidirect

2. #20299647

C₂

semidirect

61. C₁₁₃

#2986

semidirect

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

subgroup	index
#20299650	2
#20299647	2
#20299623	2
#20299616	2
#20299154	2
#2986	113

subgroup

index

113

Subgroups — #28407484

Conjugacy classes of subgroups

Complements

Maximal subgroups