Group #28407487 — subgroups

Subgroups — #28407487

Order 14464.

314 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 144 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	#20299648	C₂	1	4	yes	maximal, complemented
3	#20299623	C₂	1	4	yes	maximal
4	#20299620	C₂	1	2	yes	maximal
5	#20299154	C₂	1	1	yes	maximal, characteristic
6	#16813689	C₂²	1	4	yes	—
7	C₄⋊#11744753	C₂²	1	2	yes	—
8	C₄⋊#11744753	C₂²	1	2	yes	—
9	C₄⋊#11744753	C₂²	1	4	yes	—
10	#16813679	C₂²	1	4	yes	—
11	#16813677	C₂²	1	4	yes	—
12	#16813677	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	2	yes	—
41	C₂×Dic₁₁₃	C₂²×C₄	1	4	yes	—
42	C₂×Dic₁₁₃	C₂⁴	1	1	yes	characteristic
43	C₄×D₁₁₃	—	2	1	no	—
44	C₄×D₁₁₃	—	2	1	no	—
45	C₄×D₁₁₃	—	4	2	no	—
46	Dic₂₂₆	—	4	4	no	—
47	#11744753	—	2	4	no	—
48	#11744753	—	2	4	no	—
49	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
50	C₂×D₁₁₃	—	2	1	no	—
51	C₂×D₁₁₃	—	2	2	no	—
52	C₄×C₁₁₃	—	2	1	no	—
53	C₄×C₁₁₃	—	2	4	no	—
54	C₄×C₁₁₃	—	2	2	no	—
55	Dic₁₁₃	—	2	2	no	—
56	Dic₁₁₃	—	2	4	no	—
57	Dic₁₁₃	2₋¹⁺⁴	1	1	yes	characteristic
58	Dic₁₁₃	2₋¹⁺⁴	1	1	yes	characteristic
59	C₂×C₁₁₃	#519	1	1	yes	characteristic
60	C₂×C₁₁₃	#568	1	2	yes	—
61	D₁₁₃	—	4	1	no	—
62	#2987	—	113	1	no	maximal, Sylow
63	C₁₁₃	#2987	1	1	yes	characteristic, complemented, Sylow
64	#533	—	113	1	no	—
65	#446	—	113	4	no	—
66	#446	—	113	4	no	—
67	#433	—	113	4	no	—
68	#432	—	113	2	no	—
69	C₄⋊Q₈	—	113	2	no	—
70	#125	—	113	1	no	—
71	#125	#90673	1	1	yes	characteristic
72	C₂²⋊Q₈	—	113	4	no	—
73	C₄×Q₈	—	113	4	no	—
74	#117	—	113	2	no	—
75	C₂×C₄²	—	113	1	no	—
76	C₄⋊C₈	—	113	2	no	—
77	C₄⋊C₈	—	113	2	no	—
78	C₄⋊C₈	—	113	4	no	—
79	C₄⋊C₈	—	113	4	no	—
80	C₂²⋊C₈	—	113	4	no	—
81	#97	—	113	4	no	—
82	#97	—	113	4	no	—
83	C₄×C₈	—	113	4	no	—
84	C₂×Q₈	—	113	4	no	—
85	C₂²×C₄	—	113	2	no	—
86	C₂²×C₄	—	113	1	no	—
87	C₂×C₈	—	113	4	no	—
88	C₂×C₈	—	113	4	no	—
89	C₄⋊C₄	—	113	2	no	—
90	C₄⋊C₄	—	113	4	no	—
91	C₄⋊C₄	—	113	4	no	—
92	C₄⋊C₄	C₂×#90673	1	2	yes	—
93	C₄⋊C₄	C₂×#90673	1	4	yes	—
94	C₂²⋊C₄	—	113	4	no	—
95	C₄²	—	113	4	no	—
96	C₄²	—	113	2	no	—
97	C₄²	—	113	1	no	—
98	C₄²	C₂×#90673	1	1	yes	characteristic
99	C₂³	—	113	1	no	—
100	Q₈	—	452	4	no	—
101	C₂×C₄	—	113	1	no	—
102	C₂×C₄	—	113	2	no	—
103	C₂×C₄	—	113	4	no	—
104	C₂×C₄	—	452	2	no	—
105	C₂×C₄	—	226	1	no	—
106	C₂×C₄	—	226	1	no	—
107	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
108	C₂×C₄	C₂²×#90673	1	2	yes	—
109	C₂×C₄	C₂²×#90673	1	4	yes	—
110	C₈	—	226	4	no	—
111	C₈	—	226	4	no	—
112	C₂²	—	226	1	no	—
113	C₂²	—	226	2	no	—
114	C₂²	C₂³×#90673	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	2	2	no	—
117	C₄	—	2	4	no	—
118	C₄	—	113	1	no	—
119	C₄	—	113	1	no	—
120	C₄	—	226	2	no	—
121	C₄	—	226	4	no	—
122	C₂	—	452	1	no	—
123	C₂	#20299666	1	1	yes	characteristic
124	C₂	#20299677	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20299650

C₂

yes

maximal, complemented

#20299648

C₂

yes

maximal, complemented

#20299623

C₂

yes

maximal

#20299620

C₂

yes

maximal

#20299154

C₂

yes

maximal, characteristic

#16813689

C₂²

yes

—

C₄⋊#11744753

C₂²

yes

—

C₄⋊#11744753

C₂²

yes

—

C₄⋊#11744753

C₂²

yes

—

#16813679

C₂²

yes

—

#16813677

C₂²

yes

—

#16813677

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

—

C₄×D₁₁₃

—

Dic₂₂₆

—

#11744753

—

#11744753

—

C₂²×C₁₁₃

C₂³×C₄

yes

characteristic

C₂×D₁₁₃

—

C₂×D₁₁₃

—

C₄×C₁₁₃

—

C₄×C₁₁₃

—

C₄×C₁₁₃

—

Dic₁₁₃

—

Dic₁₁₃

—

Dic₁₁₃

2₋¹⁺⁴

yes

characteristic

Dic₁₁₃

2₋¹⁺⁴

yes

C₂×C₁₁₃

yes

C₂×C₁₁₃

yes

—

D₁₁₃

—

#2987

—

113

maximal, Sylow

C₁₁₃

#2987

yes

characteristic, complemented, Sylow

#533

—

113

—

#446

—

113

—

#446

—

113

—

#433

—

113

—

#432

—

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

—

#97

—

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

—

100

Q₈

—

452

—

101

C₂×C₄

—

113

—

102

C₂×C₄

—

113

—

103

C₂×C₄

—

113

—

104

C₂×C₄

—

452

—

105

C₂×C₄

—

226

—

106

C₂×C₄

—

226

—

107

C₂×C₄

C₂²×#90673

yes

characteristic

108

C₂×C₄

C₂²×#90673

yes

—

109

C₂×C₄

C₂²×#90673

yes

—

110

C₈

—

226

—

111

C₈

—

226

—

112

C₂²

—

226

—

113

C₂²

—

226

—

114

C₂²

C₂³×#90673

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

113

—

119

C₄

—

113

—

120

C₄

—

226

—

121

C₄

—

226

—

122

C₂

—

452

—

123

C₂

#20299666

yes

characteristic

124

C₂

#20299677

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	#2987
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. #20299648	C₂	semidirect
63. C₁₁₃	#2987	semidirect

H ≅ G/N

type

1. #20299650

C₂

semidirect

2. #20299648

C₂

semidirect

63. C₁₁₃

#2987

semidirect

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

subgroup	index
#20299650	2
#20299648	2
#20299623	2
#20299620	2
#20299154	2
#2987	113

subgroup

index

113

Subgroups — #28407487

Conjugacy classes of subgroups

Complements

Maximal subgroups