Group #28407751 — subgroups

Subgroups — #28407751

Order 14464.

220 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 67 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	#20299717	C₂	1	2	yes	maximal, complemented
2	#20299696	C₂	1	4	yes	maximal, complemented
3	C₂×#16813709	C₂	1	1	yes	maximal, characteristic
4	#16813719	—	2	2	no	—
5	#16813718	C₂²	1	4	yes	—
6	#16813717	—	2	2	no	—
7	#16813709	—	2	2	no	—
8	#16813709	—	2	2	no	—
9	C₂×#14268657	C₄	1	1	yes	characteristic
10	C₂×#14268656	C₂²	1	2	yes	—
11	C₂×#14268655	C₂²	1	1	yes	characteristic
12	C₂²×#11744753	C₄	1	2	yes	—
13	C₂×#14268646	C₄	1	1	yes	characteristic
14	#14268659	—	2	4	no	—
15	#14268657	—	2	2	no	—
16	#14268656	—	2	4	no	—
17	#14268655	D₄	1	2	yes	—
18	#14268655	D₄	1	2	yes	—
19	C₂×#11744753	—	2	2	no	—
20	C₂×#11744753	—	2	2	no	—
21	C₂×#11744753	C₂×C₄	1	2	yes	—
22	C₂×#11744753	C₂×C₄	1	1	yes	characteristic
23	C₂×#11744753	C₂³	1	1	yes	characteristic
24	#14268646	—	2	2	no	—
25	#14268646	—	2	2	no	—
26	C₂×Dic₂₂₆	C₈	1	1	yes	characteristic
27	C₂×C₂²⋊D₁₁₃	C₈	1	2	yes	—
28	C₂²×Dic₁₁₃	C₂×C₄	1	2	yes	—
29	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂×D₄×C₁₁₃	C₈	1	1	yes	characteristic
31	C₂³×C₁₁₃	C₂×C₈	1	2	yes	—
32	C₂²×D₁₁₃	C₂×C₈	1	1	yes	characteristic
33	D₄×C₁₁₃	—	2	2	no	—
34	C₂×C₄×C₁₁₃	C₂×C₈	1	1	yes	characteristic
35	C₂²⋊D₁₁₃	—	2	4	no	—
36	C₂×Dic₁₁₃	—	2	2	no	—
37	C₂×Dic₁₁₃	—	2	2	no	—
38	C₂×Dic₁₁₃	C₂×C₈	1	2	yes	—
39	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₄×D₁₁₃	C₂²⋊C₄	1	2	yes	—
41	C₄×D₁₁₃	M₁₆	1	2	yes	—
42	Dic₂₂₆	—	2	2	no	—
43	#11744753	—	2	2	no	—
44	#11744753	C₂²⋊C₄	1	1	yes	characteristic
45	#11744753	C₂²⋊C₄	1	1	yes	characteristic
46	#11744753	C₂×D₄	1	2	yes	—
47	C₂²×C₁₁₃	—	2	2	no	—
48	C₂²×C₁₁₃	—	4	2	no	—
49	C₂²×C₁₁₃	C₂²×C₈	1	1	yes	characteristic
50	C₂×D₁₁₃	—	4	1	no	—
51	C₂×D₁₁₃	C₂²⋊C₈	1	2	yes	—
52	C₄×C₁₁₃	C₂²⋊C₈	1	2	yes	—
53	Dic₁₁₃	—	2	2	no	—
54	Dic₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	Dic₁₁₃	C₂×M₁₆	1	1	yes	characteristic
56	C₂×C₁₁₃	—	2	1	no	—
57	C₂×C₁₁₃	—	4	2	no	—
58	C₂×C₁₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
59	D₁₁₃	—	2	2	no	—
60	#2115	—	113	1	no	maximal, Sylow
61	C₁₁₃	#2115	1	1	yes	characteristic, complemented, Sylow
62	C₂×#131	—	113	1	no	—
63	C₂×M₃₂	—	113	2	no	—
64	#349	—	113	4	no	—
65	C₂×#41	—	113	1	no	—
66	#131	—	226	2	no	—
67	#131	—	226	2	no	—
68	C₂×M₁₆	—	113	2	no	—
69	C₂×M₁₆	—	113	1	no	—
70	C₂²×C₈	—	113	1	no	—
71	C₂²×C₈	—	113	2	no	—
72	M₃₂	—	113	4	no	—
73	M₃₂	—	226	2	no	—
74	C₂×C₁₆	—	226	2	no	—
75	#41	—	226	2	no	—
76	#41	—	226	2	no	—
77	C₂×Q₈	—	113	1	no	—
78	C₂×D₄	—	113	2	no	—
79	C₂×D₄	#11744762	1	1	yes	characteristic
80	C₂²×C₄	—	113	1	no	—
81	C₂²×C₄	—	113	2	no	—
82	M₁₆	—	226	4	no	—
83	M₁₆	—	226	2	no	—
84	C₂×C₈	—	113	2	no	—
85	C₂×C₈	—	113	1	no	—
86	C₂×C₈	—	113	2	no	—
87	C₂×C₈	—	113	1	no	—
88	C₂×C₈	—	113	2	no	—
89	C₂×C₈	—	226	2	no	—
90	C₂×C₈	—	226	2	no	—
91	C₁₆	—	226	4	no	—
92	C₂³	—	113	1	no	—
93	C₂³	C₂×#11744762	1	2	yes	—
94	Q₈	—	226	2	no	—
95	D₄	—	2	2	no	—
96	D₄	—	226	4	no	—
97	C₂×C₄	—	113	1	no	—
98	C₂×C₄	—	113	2	no	—
99	C₂×C₄	—	113	2	no	—
100	C₂×C₄	—	113	2	no	—
101	C₂×C₄	—	226	2	no	—
102	C₂×C₄	—	226	2	no	—
103	C₂×C₄	C₂×#11744762	1	1	yes	characteristic
104	C₈	—	113	2	no	—
105	C₈	—	113	1	no	—
106	C₈	—	113	1	no	—
107	C₈	—	226	2	no	—
108	C₂²	—	2	2	no	—
109	C₂²	—	4	2	no	—
110	C₂²	—	113	2	no	—
111	C₂²	—	452	1	no	—
112	C₂²	C₂²×#11744762	1	1	yes	characteristic
113	C₄	—	113	1	no	—
114	C₄	—	113	1	no	—
115	C₄	—	226	2	no	—
116	C₄	C₂²⋊#11744762	1	2	yes	—
117	C₂	—	2	1	no	—
118	C₂	—	4	2	no	—
119	C₂	—	226	2	no	—
120	C₂	C₂×C₂²⋊#11744762	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20299717

C₂

yes

maximal, complemented

#20299696

C₂

yes

maximal, complemented

C₂×#16813709

C₂

yes

maximal, characteristic

#16813719

—

#16813718

C₂²

yes

—

—

—

—

C₄

yes

characteristic

C₂×#14268656

C₂²

yes

—

C₂×#14268655

C₂²

yes

characteristic

C₂²×#11744753

C₄

yes

—

C₂×#14268646

C₄

yes

—

—

—

D₄

yes

—

#14268655

D₄

yes

—

C₂×#11744753

—

C₂×#11744753

—

C₂×#11744753

C₂×C₄

yes

—

C₂×#11744753

C₂×C₄

yes

characteristic

C₂×#11744753

C₂³

yes

characteristic

#14268646

—

#14268646

—

C₂×Dic₂₂₆

C₈

yes

characteristic

C₂×C₂²⋊D₁₁₃

C₈

yes

—

C₂²×Dic₁₁₃

C₂×C₄

yes

—

C₂×C₄×D₁₁₃

C₂×C₄

yes

characteristic

C₂×D₄×C₁₁₃

C₈

yes

characteristic

C₂³×C₁₁₃

C₂×C₈

yes

—

C₂²×D₁₁₃

C₂×C₈

yes

characteristic

D₄×C₁₁₃

—

C₂×C₄×C₁₁₃

C₂×C₈

yes

characteristic

C₂²⋊D₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

C₂×C₈

yes

—

C₂×Dic₁₁₃

C₂²×C₄

yes

characteristic

C₄×D₁₁₃

C₂²⋊C₄

yes

—

C₄×D₁₁₃

M₁₆

yes

—

Dic₂₂₆

—

#11744753

—

#11744753

C₂²⋊C₄

yes

characteristic

#11744753

C₂²⋊C₄

yes

characteristic

#11744753

C₂×D₄

yes

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

C₂²×C₈

yes

characteristic

C₂×D₁₁₃

—

C₂×D₁₁₃

C₂²⋊C₈

yes

—

C₄×C₁₁₃

C₂²⋊C₈

yes

—

Dic₁₁₃

—

Dic₁₁₃

C₂×C₂²⋊C₄

yes

characteristic

Dic₁₁₃

C₂×M₁₆

yes

characteristic

C₂×C₁₁₃

—

C₂×C₁₁₃

—

C₂×C₁₁₃

C₂×C₂²⋊C₈

yes

characteristic

D₁₁₃

—

#2115

—

113

maximal, Sylow

C₁₁₃

#2115

yes

characteristic, complemented, Sylow

C₂×#131

—

113

—

C₂×M₃₂

—

113

—

#349

—

113

—

C₂×#41

—

113

—

#131

—

226

—

#131

—

226

—

C₂×M₁₆

—

113

—

C₂×M₁₆

—

113

—

C₂²×C₈

—

113

—

C₂²×C₈

—

113

—

M₃₂

—

113

—

M₃₂

—

226

—

C₂×C₁₆

—

226

—

#41

—

226

—

#41

—

226

—

C₂×Q₈

—

113

—

C₂×D₄

—

113

—

C₂×D₄

#11744762

yes

characteristic

C₂²×C₄

—

113

—

C₂²×C₄

—

113

—

M₁₆

—

226

—

M₁₆

—

226

—

C₂×C₈

—

113

—

C₂×C₈

—

113

—

C₂×C₈

—

113

—

C₂×C₈

—

113

—

C₂×C₈

—

113

—

C₂×C₈

—

226

—

C₂×C₈

—

226

—

C₁₆

—

226

—

C₂³

—

113

—

C₂³

C₂×#11744762

yes

—

Q₈

—

226

—

D₄

—

D₄

—

226

—

C₂×C₄

—

113

—

C₂×C₄

—

113

—

C₂×C₄

—

113

—

100

C₂×C₄

—

113

—

101

C₂×C₄

—

226

—

102

C₂×C₄

—

226

—

103

C₂×C₄

C₂×#11744762

yes

characteristic

104

C₈

—

113

—

105

C₈

—

113

—

106

C₈

—

113

—

107

C₈

—

226

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

113

—

111

C₂²

—

452

—

112

C₂²

C₂²×#11744762

yes

characteristic

113

C₄

—

113

—

114

C₄

—

113

—

115

C₄

—

226

—

116

C₄

C₂²⋊#11744762

yes

—

117

C₂

—

118

C₂

—

119

C₂

—

226

—

120

C₂

C₂×C₂²⋊#11744762

yes

characteristic

	subgroup	G/H
center	C₂	C₂×C₂²⋊#11744762
commutator	C₂²×C₁₁₃	C₂²×C₈
Frattini	C₂²	C₂²×#11744762
Fitting	C₂×D₄×C₁₁₃	C₈
socle	C₂×C₁₁₃	C₂×C₂²⋊C₈
prime radical	C₂×#14268646	C₄
cosocle	C₂×#11744753	C₂³
hypercenter	C₂×D₄	#11744762
hypocenter	C₁₁₃	#2115
LCS	G ▷ C₂²×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₄

subgroup

G/H

center

C₂

C₂×C₂²⋊#11744762

commutator

C₂²×C₁₁₃

C₂²×C₈

Frattini

C₂²

C₂²×#11744762

Fitting

C₂×D₄×C₁₁₃

C₈

socle

C₂×C₁₁₃

C₂×C₂²⋊C₈

C₄

C₂³

C₂×D₄

C₁₁₃

G ▷ C₂²×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₄

N	H ≅ G/N	type
1. #20299717	C₂	semidirect
2. #20299696	C₂	semidirect
61. C₁₁₃	#2115	semidirect

H ≅ G/N

type

1. #20299717

C₂

semidirect

2. #20299696

C₂

semidirect

61. C₁₁₃

#2115

semidirect

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

subgroup	index
#20299717	2
#20299696	2
C₂×#16813709	2
#2115	113

subgroup

index

113

Subgroups — #28407751

Conjugacy classes of subgroups

Complements

Maximal subgroups