Group #28407745 — subgroups

Subgroups — #28407745

Order 14464.

228 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 71 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	C₂×#16813719	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16813717	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊#14268659	C₂	1	4	yes	maximal, complemented
4	C₂×#16813709	C₂	1	1	yes	maximal, characteristic
5	#16813719	—	2	2	no	—
6	#16813717	—	2	2	no	—
7	C₂×#14268659	C₂²	1	2	yes	—
8	C₂×#14268659	C₂²	1	2	yes	—
9	#16813709	—	2	2	no	—
10	#16813709	—	2	2	no	—
11	C₂×#14268657	C₄	1	1	yes	characteristic
12	C₂×#14268656	C₄	1	2	yes	—
13	C₂×#14268655	C₂²	1	1	yes	characteristic
14	C₂²×#11744753	C₂²	1	2	yes	—
15	C₂×#14268646	C₄	1	1	yes	characteristic
16	#14268659	—	2	2	no	—
17	#14268659	—	2	2	no	—
18	#14268657	—	2	2	no	—
19	#14268656	—	2	4	no	—
20	#14268655	D₄	1	2	yes	—
21	#14268655	D₄	1	2	yes	—
22	C₂×#11744753	—	2	2	no	—
23	C₂×#11744753	—	2	2	no	—
24	C₂×#11744753	C₂×C₄	1	2	yes	—
25	C₂×#11744753	C₂×C₄	1	1	yes	characteristic
26	C₂×#11744753	C₂³	1	1	yes	characteristic
27	#14268646	—	2	2	no	—
28	#14268646	—	2	2	no	—
29	C₂×Dic₂₂₆	C₈	1	1	yes	characteristic
30	C₂×C₂²⋊D₁₁₃	C₈	1	2	yes	—
31	C₂²×Dic₁₁₃	C₂×C₄	1	2	yes	—
32	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂×D₄×C₁₁₃	C₈	1	1	yes	characteristic
34	C₂³×C₁₁₃	C₂×C₈	1	2	yes	—
35	C₂²×D₁₁₃	C₂×C₈	1	1	yes	characteristic
36	D₄×C₁₁₃	—	2	2	no	—
37	C₂×C₄×C₁₁₃	C₂×C₈	1	1	yes	characteristic
38	C₂²⋊D₁₁₃	—	2	4	no	—
39	C₂×Dic₁₁₃	—	2	2	no	—
40	C₂×Dic₁₁₃	—	2	2	no	—
41	C₂×Dic₁₁₃	C₂×C₈	1	2	yes	—
42	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
43	C₄×D₁₁₃	C₂²⋊C₄	1	2	yes	—
44	C₄×D₁₁₃	M₁₆	1	2	yes	—
45	Dic₂₂₆	—	2	2	no	—
46	#11744753	—	2	2	no	—
47	#11744753	C₂²⋊C₄	1	2	yes	—
48	#11744753	C₂×D₄	1	1	yes	characteristic
49	#11744753	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₁₁₃	—	2	2	no	—
51	C₂²×C₁₁₃	—	2	4	no	—
52	C₂²×C₁₁₃	C₂²×C₈	1	1	yes	characteristic
53	C₂×D₁₁₃	—	2	2	no	—
54	C₂×D₁₁₃	C₂²⋊C₈	1	2	yes	—
55	C₄×C₁₁₃	C₂²⋊C₈	1	2	yes	—
56	Dic₁₁₃	—	2	2	no	—
57	Dic₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	Dic₁₁₃	C₂×M₁₆	1	1	yes	characteristic
59	C₂×C₁₁₃	—	4	2	no	—
60	C₂×C₁₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
61	C₂×C₁₁₃	#504	1	2	yes	—
62	D₁₁₃	—	2	2	no	—
63	#2113	—	113	1	no	maximal, Sylow
64	C₁₁₃	#2113	1	1	yes	characteristic, complemented, Sylow
65	C₂×#131	—	113	1	no	—
66	C₂×M₃₂	—	113	1	no	—
67	C₂²×C₁₆	—	113	1	no	—
68	C₂²⋊C₁₆	—	113	4	no	—
69	C₂×#41	—	113	1	no	—
70	#131	—	226	2	no	—
71	#131	—	226	2	no	—
72	C₂×M₁₆	—	113	2	no	—
73	C₂×M₁₆	—	113	1	no	—
74	C₂²×C₈	—	113	1	no	—
75	C₂²×C₈	—	113	2	no	—
76	M₃₂	—	226	2	no	—
77	C₂×C₁₆	—	113	2	no	—
78	C₂×C₁₆	—	113	2	no	—
79	C₂×C₁₆	—	226	2	no	—
80	#41	—	226	2	no	—
81	#41	—	226	2	no	—
82	C₂×Q₈	—	113	1	no	—
83	C₂×D₄	—	113	2	no	—
84	C₂×D₄	#11744762	1	1	yes	characteristic
85	C₂²×C₄	—	113	1	no	—
86	C₂²×C₄	—	113	2	no	—
87	M₁₆	—	226	4	no	—
88	M₁₆	—	226	2	no	—
89	C₂×C₈	—	113	1	no	—
90	C₂×C₈	—	113	2	no	—
91	C₂×C₈	—	113	2	no	—
92	C₂×C₈	—	113	1	no	—
93	C₂×C₈	—	113	2	no	—
94	C₂×C₈	—	226	2	no	—
95	C₂×C₈	—	226	2	no	—
96	C₁₆	—	226	2	no	—
97	C₁₆	—	226	2	no	—
98	C₂³	—	113	1	no	—
99	C₂³	C₂×#11744762	1	2	yes	—
100	Q₈	—	226	2	no	—
101	D₄	—	2	2	no	—
102	D₄	—	226	4	no	—
103	C₂×C₄	—	113	1	no	—
104	C₂×C₄	—	113	2	no	—
105	C₂×C₄	—	113	2	no	—
106	C₂×C₄	—	113	2	no	—
107	C₂×C₄	—	226	2	no	—
108	C₂×C₄	—	226	2	no	—
109	C₂×C₄	C₂×#11744762	1	1	yes	characteristic
110	C₈	—	113	1	no	—
111	C₈	—	113	1	no	—
112	C₈	—	113	2	no	—
113	C₈	—	226	2	no	—
114	C₂²	—	2	2	no	—
115	C₂²	—	2	4	no	—
116	C₂²	—	113	2	no	—
117	C₂²	—	226	2	no	—
118	C₂²	C₂²×#11744762	1	1	yes	characteristic
119	C₄	—	113	1	no	—
120	C₄	—	113	1	no	—
121	C₄	—	226	2	no	—
122	C₄	C₂²⋊#11744762	1	2	yes	—
123	C₂	—	4	2	no	—
124	C₂	—	226	2	no	—
125	C₂	C₂×C₂²⋊#11744762	1	1	yes	characteristic
126	C₂	#20299719	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16813719

C₂

yes

maximal, characteristic, complemented

C₂×#16813717

C₂

yes

maximal, characteristic, complemented

C₂²⋊#14268659

C₂

yes

maximal, complemented

C₂×#16813709

C₂

yes

maximal, characteristic

#16813719

—

#16813717

—

C₂×#14268659

C₂²

yes

—

C₂×#14268659

C₂²

yes

—

#16813709

—

#16813709

—

C₂×#14268657

C₄

yes

characteristic

C₂×#14268656

C₄

yes

—

C₂×#14268655

C₂²

yes

characteristic

C₂²×#11744753

C₂²

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

—

#11744753

C₂×D₄

yes

characteristic

#11744753

C₂×D₄

yes

characteristic

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×C₂²⋊C₈

yes

characteristic

C₂×C₁₁₃

#504

yes

—

D₁₁₃

—

#2113

—

113

maximal, Sylow

C₁₁₃

#2113

yes

characteristic, complemented, Sylow

C₂×#131

—

113

—

C₂×M₃₂

—

113

—

C₂²×C₁₆

—

113

—

C₂²⋊C₁₆

—

113

—

C₂×#41

—

113

—

#131

—

226

—

#131

—

226

—

C₂×M₁₆

—

113

—

C₂×M₁₆

—

113

—

C₂²×C₈

—

113

—

C₂²×C₈

—

113

—

M₃₂

—

226

—

C₂×C₁₆

—

113

—

C₂×C₁₆

—

113

—

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

—

C₁₆

—

226

—

C₂³

—

113

—

C₂³

C₂×#11744762

yes

—

100

Q₈

—

226

—

101

D₄

—

102

D₄

—

226

—

103

C₂×C₄

—

113

—

104

C₂×C₄

—

113

—

105

C₂×C₄

—

113

—

106

C₂×C₄

—

113

—

107

C₂×C₄

—

226

—

108

C₂×C₄

—

226

—

109

C₂×C₄

C₂×#11744762

yes

characteristic

110

C₈

—

113

—

111

C₈

—

113

—

112

C₈

—

113

—

113

C₈

—

226

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

113

—

117

C₂²

—

226

—

118

C₂²

C₂²×#11744762

yes

characteristic

119

C₄

—

113

—

120

C₄

—

113

—

121

C₄

—

226

—

122

C₄

C₂²⋊#11744762

yes

—

123

C₂

—

124

C₂

—

226

—

125

C₂

C₂×C₂²⋊#11744762

yes

characteristic

126

C₂

#20299719

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11744762
commutator, socle	C₂²×C₁₁₃	C₂²×C₈
Fitting	C₂×D₄×C₁₁₃	C₈
prime radical	C₂×#14268646	C₄
cosocle	C₂×#11744753	C₂³
hypercenter	C₂×D₄	#11744762
hypocenter	C₁₁₃	#2113
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂×D₄

subgroup

G/H

center, Frattini

C₂²

C₂²×#11744762

commutator, socle

C₂²×C₁₁₃

C₂²×C₈

Fitting

C₂×D₄×C₁₁₃

C₈

C₄

C₂³

C₂×D₄

C₁₁₃

G ▷ C₂²×C₁₁₃ ▷ C₁₁₃

C₁ ◁ C₂² ◁ C₂×D₄

N	H ≅ G/N	type
1. C₂×#16813719	C₂	semidirect
2. C₂×#16813717	C₂	semidirect
3. C₂²⋊#14268659	C₂	semidirect
64. C₁₁₃	#2113	semidirect

H ≅ G/N

type

1. C₂×#16813719

C₂

semidirect

2. C₂×#16813717

C₂

semidirect

3. C₂²⋊#14268659

C₂

semidirect

64. C₁₁₃

#2113

semidirect

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

subgroup	index
C₂×#16813719	2
C₂×#16813717	2
C₂²⋊#14268659	2
C₂×#16813709	2
#2113	113

subgroup

index

113

Subgroups — #28407745

Conjugacy classes of subgroups

Complements

Maximal subgroups