Group #20299227 — subgroups

Subgroups — #20299227

Order 7232.

206 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 103 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	#16813621	C₂	1	2	yes	maximal, complemented
2	#16813620	C₂	1	2	yes	maximal, complemented
3	#16813612	C₂	1	2	yes	maximal, complemented
4	#16813612	C₂	1	2	yes	maximal, complemented
5	#16813601	C₂	1	1	yes	maximal, characteristic
6	#16813599	C₂	1	2	yes	maximal
7	#16813578	C₂	1	2	yes	maximal
8	C₂×#14268626	C₂	1	1	yes	maximal, characteristic
9	C₂²⋊Q₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic
10	C₂×Dic₂₂₆	C₂²	1	2	yes	—
11	C₂×Dic₂₂₆	C₂²	1	1	yes	characteristic
12	C₂²×Dic₁₁₃	C₂²	1	1	yes	characteristic
13	C₂²×Dic₁₁₃	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
15	C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
16	#14268626	C₂²	1	2	yes	—
17	#14268626	C₂²	1	2	yes	—
18	#14268626	C₂²	1	2	yes	—
19	#14268626	C₂²	1	2	yes	—
20	#14268626	C₂²	1	2	yes	—
21	#14268626	C₂²	1	2	yes	—
22	#14268626	C₂²	1	2	yes	—
23	C₂²⋊Dic₁₁₃	C₂²	1	2	yes	—
24	C₄×Dic₁₁₃	C₂²	1	2	yes	—
25	C₄×Dic₁₁₃	C₂²	1	2	yes	—
26	C₂×Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
27	C₂²×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
29	C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
30	C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
31	C₂³×C₁₁₃	C₂³	1	1	yes	characteristic
32	Q₈×C₁₁₃	—	4	1	no	—
33	C₂×C₄×C₁₁₃	—	4	1	no	—
34	C₂×C₄×C₁₁₃	C₂³	1	2	yes	—
35	C₂×C₄×C₁₁₃	C₂³	1	2	yes	—
36	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
38	C₂×Dic₁₁₃	—	4	1	no	—
39	C₂×Dic₁₁₃	Q₈	1	4	yes	complemented
40	C₂×Dic₁₁₃	C₂³	1	2	yes	—
41	C₂×Dic₁₁₃	C₂³	1	2	yes	—
42	C₂×Dic₁₁₃	C₂³	1	2	yes	—
43	C₂×Dic₁₁₃	C₂³	1	2	yes	—
44	Dic₂₂₆	—	4	1	no	—
45	Dic₂₂₆	—	4	2	no	—
46	C₂²×C₁₁₃	—	2	1	no	—
47	C₂²×C₁₁₃	—	2	1	no	—
48	C₂²×C₁₁₃	C₂×Q₈	1	2	yes	—
49	C₂²×C₁₁₃	C₂⁴	1	1	yes	characteristic
50	C₄×C₁₁₃	—	2	1	no	—
51	C₄×C₁₁₃	—	2	2	no	—
52	C₄×C₁₁₃	—	2	1	no	—
53	C₄×C₁₁₃	—	2	2	no	—
54	Dic₁₁₃	—	2	2	no	—
55	Dic₁₁₃	—	2	2	no	—
56	Dic₁₁₃	—	2	2	no	—
57	Dic₁₁₃	C₂×Q₈	1	4	yes	—
58	C₂×C₁₁₃	—	2	2	no	—
59	C₂×C₁₁₃	C₂²×Q₈	1	1	yes	characteristic
60	C₂×C₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
61	C₂×C₁₁₃	2₋¹⁺⁴	1	1	yes	characteristic
62	C₁₁₃	#544	1	1	yes	characteristic, complemented, Sylow
63	#544	—	113	1	no	maximal, Sylow
64	C₄⋊Q₈	—	113	2	no	—
65	C₄⋊Q₈	—	113	2	no	—
66	#125	—	113	2	no	—
67	#125	—	113	2	no	—
68	C₂²⋊Q₈	—	113	1	no	—
69	C₂²⋊Q₈	—	113	2	no	—
70	C₂²⋊Q₈	D₁₁₃	1	1	yes	characteristic
71	#117	—	113	2	no	—
72	C₂×C₄⋊C₄	—	113	1	no	—
73	C₂×Q₈	—	113	1	no	—
74	C₂×Q₈	—	113	2	no	—
75	C₂×Q₈	C₂×D₁₁₃	1	1	yes	characteristic
76	C₂²×C₄	—	113	1	no	—
77	C₂²×C₄	—	113	1	no	—
78	C₂²×C₄	C₂×D₁₁₃	1	1	yes	characteristic
79	C₄⋊C₄	—	113	2	no	—
80	C₄⋊C₄	—	113	2	no	—
81	C₄⋊C₄	—	113	2	no	—
82	C₄⋊C₄	—	113	2	no	—
83	C₄⋊C₄	—	113	2	no	—
84	C₄⋊C₄	—	113	1	no	—
85	C₄⋊C₄	—	113	2	no	—
86	C₄⋊C₄	—	113	2	no	—
87	C₄⋊C₄	—	113	2	no	—
88	C₄⋊C₄	C₂×D₁₁₃	1	1	yes	characteristic
89	C₄⋊C₄	C₂×D₁₁₃	1	2	yes	—
90	C₂²⋊C₄	—	113	2	no	—
91	C₂²⋊C₄	C₂×D₁₁₃	1	2	yes	—
92	C₄²	—	113	2	no	—
93	C₄²	—	113	2	no	—
94	C₂³	C₂²×D₁₁₃	1	1	yes	characteristic
95	Q₈	—	4	1	no	—
96	Q₈	—	452	1	no	—
97	Q₈	—	452	2	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	113	4	no	—
100	C₂×C₄	—	113	2	no	—
101	C₂×C₄	—	113	2	no	—
102	C₂×C₄	—	113	2	no	—
103	C₂×C₄	—	113	2	no	—
104	C₂×C₄	—	452	1	no	—
105	C₂×C₄	C₂²×D₁₁₃	1	2	yes	—
106	C₂×C₄	C₂²×D₁₁₃	1	2	yes	—
107	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
108	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
109	C₂²	—	2	1	no	—
110	C₂²	—	2	1	no	—
111	C₂²	Q₈×D₁₁₃	1	2	yes	—
112	C₂²	C₂³×D₁₁₃	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	—	2	2	no	—
116	C₄	—	2	2	no	—
117	C₄	—	113	4	no	—
118	C₄	—	226	2	no	—
119	C₄	—	226	2	no	—
120	C₄	—	226	2	no	—
121	C₂	—	2	2	no	—
122	C₂	C₂×Q₈×D₁₁₃	1	1	yes	characteristic
123	C₂	#16813667	1	1	yes	characteristic
124	C₂	#16813669	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16813621

C₂

yes

maximal, complemented

#16813620

C₂

yes

maximal, complemented

#16813612

C₂

yes

maximal, complemented

#16813612

C₂

yes

maximal, complemented

#16813601

C₂

yes

maximal, characteristic

#16813599

C₂

yes

maximal

#16813578

C₂

yes

maximal

C₂×#14268626

C₂

yes

maximal, characteristic

C₂²⋊Q₈×C₁₁₃

C₂

yes

maximal, characteristic

C₂×Dic₂₂₆

C₂²

yes

—

C₂×Dic₂₂₆

C₂²

yes

characteristic

C₂²×Dic₁₁₃

C₂²

yes

characteristic

C₂²×Dic₁₁₃

C₂²

yes

characteristic

C₄⋊Dic₁₁₃

C₂²

yes

—

C₄⋊Dic₁₁₃

C₂²

yes

characteristic

#14268626

C₂²

yes

—

#14268626

C₂²

yes

—

#14268626

C₂²

yes

—

#14268626

C₂²

yes

—

#14268626

C₂²

yes

—

#14268626

C₂²

yes

—

#14268626

C₂²

yes

—

C₂²⋊Dic₁₁₃

C₂²

yes

—

C₄×Dic₁₁₃

C₂²

yes

—

C₄×Dic₁₁₃

C₂²

yes

—

C₂×Q₈×C₁₁₃

C₂²

yes

characteristic

C₂²×C₄×C₁₁₃

C₂²

yes

characteristic

C₄⋊C₄×C₁₁₃

C₂²

yes

characteristic

C₄⋊C₄×C₁₁₃

C₂²

yes

—

C₂²⋊C₄×C₁₁₃

C₂²

yes

—

C₂³×C₁₁₃

C₂³

yes

characteristic

Q₈×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

C₂³

yes

—

C₂×C₄×C₁₁₃

C₂³

yes

—

C₂×C₄×C₁₁₃

C₂³

yes

characteristic

C₂×C₄×C₁₁₃

C₂³

yes

characteristic

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

Q₈

yes

complemented

C₂×Dic₁₁₃

C₂³

yes

—

C₂×Dic₁₁₃

C₂³

yes

—

C₂×Dic₁₁₃

C₂³

yes

—

C₂×Dic₁₁₃

C₂³

yes

—

Dic₂₂₆

—

Dic₂₂₆

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

C₂×Q₈

yes

—

C₂²×C₁₁₃

C₂⁴

yes

characteristic

C₄×C₁₁₃

—

C₄×C₁₁₃

—

C₄×C₁₁₃

—

C₄×C₁₁₃

—

Dic₁₁₃

—

Dic₁₁₃

—

Dic₁₁₃

—

Dic₁₁₃

C₂×Q₈

yes

—

C₂×C₁₁₃

—

C₂×C₁₁₃

C₂²×Q₈

yes

characteristic

C₂×C₁₁₃

H(5,𝔽₂)

yes

characteristic

C₂×C₁₁₃

2₋¹⁺⁴

yes

characteristic

C₁₁₃

#544

yes

characteristic, complemented, Sylow

#544

—

113

maximal, Sylow

C₄⋊Q₈

—

113

—

C₄⋊Q₈

—

113

—

#125

—

113

—

#125

—

113

—

C₂²⋊Q₈

—

113

—

C₂²⋊Q₈

—

113

—

C₂²⋊Q₈

D₁₁₃

yes

characteristic

#117

—

113

—

C₂×C₄⋊C₄

—

113

—

C₂×Q₈

—

113

—

C₂×Q₈

—

113

—

C₂×Q₈

C₂×D₁₁₃

yes

characteristic

C₂²×C₄

—

113

—

C₂²×C₄

—

113

—

C₂²×C₄

C₂×D₁₁₃

yes

characteristic

C₄⋊C₄

—

113

—

C₄⋊C₄

—

113

—

C₄⋊C₄

—

113

—

C₄⋊C₄

—

113

—

C₄⋊C₄

—

113

—

C₄⋊C₄

—

113

—

C₄⋊C₄

—

113

—

C₄⋊C₄

—

113

—

C₄⋊C₄

—

113

—

C₄⋊C₄

C₂×D₁₁₃

yes

characteristic

C₄⋊C₄

C₂×D₁₁₃

yes

—

C₂²⋊C₄

—

113

—

C₂²⋊C₄

C₂×D₁₁₃

yes

—

C₄²

—

113

—

C₄²

—

113

—

C₂³

C₂²×D₁₁₃

yes

characteristic

Q₈

—

Q₈

—

452

—

Q₈

—

452

—

C₂×C₄

—

C₂×C₄

—

113

—

100

C₂×C₄

—

113

—

101

C₂×C₄

—

113

—

102

C₂×C₄

—

113

—

103

C₂×C₄

—

113

—

104

C₂×C₄

—

452

—

105

C₂×C₄

C₂²×D₁₁₃

yes

—

106

C₂×C₄

C₂²×D₁₁₃

yes

—

107

C₂×C₄

C₂²×D₁₁₃

yes

characteristic

108

C₂×C₄

C₂²×D₁₁₃

yes

characteristic

109

C₂²

—

110

C₂²

—

111

C₂²

Q₈×D₁₁₃

yes

—

112

C₂²

C₂³×D₁₁₃

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

113

—

118

C₄

—

226

—

119

C₄

—

226

—

120

C₄

—

226

—

121

C₂

—

122

C₂

C₂×Q₈×D₁₁₃

yes

characteristic

123

C₂

#16813667

yes

characteristic

124

C₂

#16813669

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₁₃
commutator, socle	C₂²×C₁₁₃	C₂⁴
Fitting	C₂²⋊Q₈×C₁₁₃	C₂
hypercenter	C₂²⋊Q₈	D₁₁₃
hypocenter	C₁₁₃	#544
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₁₃

commutator, socle

C₂²×C₁₁₃

C₂⁴

Fitting

C₂²⋊Q₈×C₁₁₃

C₂

hypercenter

C₂²⋊Q₈

D₁₁₃

C₁₁₃

G ▷ C₂²×C₁₁₃ ▷ C₁₁₃

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #16813621	C₂	semidirect
2. #16813620	C₂	semidirect
3. #16813612	C₂	semidirect
4. #16813612	C₂	semidirect
39. C₂×Dic₁₁₃	Q₈	semidirect
62. C₁₁₃	#544	semidirect

H ≅ G/N

type

1. #16813621

C₂

semidirect

2. #16813620

C₂

semidirect

3. #16813612

C₂

semidirect

4. #16813612

C₂

semidirect

39. C₂×Dic₁₁₃

Q₈

semidirect

62. C₁₁₃

#544

semidirect

Subgroups — #20299227

Conjugacy classes of subgroups

Complements