Group #28345396 — subgroups

Subgroups — #28345396

Order 13696.

364 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 229 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	#20292758	C₂	1	4	yes	maximal, complemented
2	Q₈×C₁₀₇⋊C₈	C₂	1	4	yes	maximal, complemented
3	#20292654	C₂	1	2	yes	maximal
4	#20292651	C₂	1	1	yes	maximal, characteristic
5	#20292583	C₂	1	2	yes	maximal
6	C₂×C₄×C₁₀₇⋊C₈	C₂	1	1	yes	maximal, characteristic
7	#533×C₁₀₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#13133802	C₂²	1	2	yes	—
9	C₂²×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₄⋊(C₁₀₇⋊C₈)	C₂²	1	4	yes	—
12	C₄⋊(C₁₀₇⋊C₈)	C₂²	1	8	yes	—
13	#16812020	C₂²	1	4	yes	—
14	C₄×C₁₀₇⋊C₈	C₂²	1	2	yes	—
15	C₄×C₁₀₇⋊C₈	C₂²	1	4	yes	—
16	C₄×C₁₀₇⋊C₈	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₁₀₇	C₄	1	2	yes	—
18	#125×C₁₀₇	C₄	1	2	yes	—
19	C₂²⋊Q₈×C₁₀₇	C₄	1	4	yes	—
20	C₄×Q₈×C₁₀₇	C₂²	1	4	yes	—
21	#117×C₁₀₇	C₂²	1	2	yes	—
22	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
23	#13133802	—	2	4	no	—
24	C₂×C₁₀₇⋊C₈	—	2	2	no	—
25	C₂×C₁₀₇⋊C₈	Q₈	1	4	yes	complemented
26	C₂×C₁₀₇⋊C₈	C₂³	1	2	yes	—
27	C₂×C₁₀₇⋊C₈	C₂³	1	4	yes	—
28	C₂×C₁₀₇⋊C₈	C₂³	1	1	yes	characteristic
29	C₂×C₁₀₇⋊C₈	C₂³	1	1	yes	characteristic
30	C₂×Q₈×C₁₀₇	C₂×C₄	1	4	yes	—
31	C₂²×C₄×C₁₀₇	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₁₀₇	C₂×C₄	1	4	yes	—
34	C₄⋊C₄×C₁₀₇	C₂×C₄	1	8	yes	—
35	C₄⋊C₄×C₁₀₇	C₂×C₄	1	4	yes	—
36	C₂²⋊C₄×C₁₀₇	C₂×C₄	1	4	yes	—
37	C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₁₀₇	C₂³	1	2	yes	—
40	C₄²×C₁₀₇	C₂³	1	4	yes	—
41	C₂³×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
42	Q₈×C₁₀₇	—	2	8	no	—
43	C₂×C₄×C₁₀₇	—	2	4	no	—
44	C₂×C₄×C₁₀₇	C₂²×C₄	1	2	yes	—
45	C₂×C₄×C₁₀₇	C₂²×C₄	1	8	yes	—
46	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₇	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₁₀₇	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₁₀₇	#41	1	2	yes	—
50	C₂×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
51	C₁₀₇⋊C₈	—	2	2	no	—
52	C₁₀₇⋊C₈	—	2	4	no	—
53	C₁₀₇⋊C₈	C₂×Q₈	1	2	yes	—
54	C₁₀₇⋊C₈	C₂×Q₈	1	2	yes	—
55	C₂²×C₁₀₇	—	2	2	no	—
56	C₂²×C₁₀₇	C₄×Q₈	1	2	yes	—
57	C₂²×C₁₀₇	C₂³×C₄	1	1	yes	characteristic
58	C₄×C₁₀₇	—	2	2	no	—
59	C₄×C₁₀₇	—	2	8	no	—
60	C₄×C₁₀₇	C₄×Q₈	1	2	yes	—
61	C₄×C₁₀₇	#131	1	4	yes	—
62	C₄×C₁₀₇	C₂²×Q₈	1	1	yes	characteristic
63	C₄×C₁₀₇	C₂×#41	1	1	yes	characteristic
64	C₂×C₁₀₇	—	2	2	no	—
65	C₂×C₁₀₇	C₂×C₄×Q₈	1	1	yes	characteristic
66	C₂×C₁₀₇	C₂×#131	1	2	yes	—
67	#2960	—	107	1	no	maximal, Sylow
68	C₁₀₇	#2960	1	1	yes	characteristic, complemented, Sylow
69	#533	D₁₀₇	1	1	yes	characteristic
70	#446	—	107	4	no	—
71	C₈×Q₈	—	107	4	no	—
72	#424	—	107	2	no	—
73	#423	—	107	1	no	—
74	#405	—	107	2	no	—
75	C₂×C₄×C₈	—	107	1	no	—
76	C₂×M₁₆	—	107	2	no	—
77	C₂²×C₈	—	107	1	no	—
78	C₂²×C₈	—	107	1	no	—
79	C₄⋊Q₈	Dic₁₀₇	1	2	yes	—
80	#125	Dic₁₀₇	1	2	yes	—
81	C₂²⋊Q₈	Dic₁₀₇	1	4	yes	—
82	C₄×Q₈	C₂×D₁₀₇	1	4	yes	—
83	#117	C₂×D₁₀₇	1	2	yes	—
84	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
85	C₄⋊C₈	—	107	4	no	—
86	C₄⋊C₈	—	107	8	no	—
87	#97	—	107	4	no	—
88	C₄×C₈	—	107	2	no	—
89	C₄×C₈	—	107	2	no	—
90	C₄×C₈	—	107	4	no	—
91	C₂×Q₈	C₂×Dic₁₀₇	1	4	yes	—
92	C₂²×C₄	C₂×Dic₁₀₇	1	2	yes	—
93	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
94	M₁₆	—	214	4	no	—
95	C₂×C₈	—	107	4	no	—
96	C₂×C₈	—	107	1	no	—
97	C₂×C₈	—	107	1	no	—
98	C₂×C₈	—	107	2	no	—
99	C₂×C₈	—	107	4	no	—
100	C₂×C₈	—	214	2	no	—
101	C₄⋊C₄	C₂×Dic₁₀₇	1	4	yes	—
102	C₄⋊C₄	C₂×Dic₁₀₇	1	8	yes	—
103	C₄⋊C₄	C₂×Dic₁₀₇	1	4	yes	—
104	C₂²⋊C₄	C₂×Dic₁₀₇	1	4	yes	—
105	C₄²	C₂×Dic₁₀₇	1	1	yes	characteristic
106	C₄²	C₂×Dic₁₀₇	1	1	yes	characteristic
107	C₄²	C₂²×D₁₀₇	1	2	yes	—
108	C₄²	C₂²×D₁₀₇	1	4	yes	—
109	C₂³	C₂²×Dic₁₀₇	1	1	yes	characteristic
110	Q₈	—	2	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	C₂²×Dic₁₀₇	1	2	yes	—
113	C₂×C₄	C₂²×Dic₁₀₇	1	8	yes	—
114	C₂×C₄	C₂²×Dic₁₀₇	1	1	yes	characteristic
115	C₂×C₄	C₂²×Dic₁₀₇	1	2	yes	—
116	C₂×C₄	Q₈×D₁₀₇	1	2	yes	—
117	C₂×C₄	#13133819	1	2	yes	—
118	C₂×C₄	C₂³×D₁₀₇	1	1	yes	characteristic
119	C₈	—	107	2	no	—
120	C₈	—	107	2	no	—
121	C₈	—	214	2	no	—
122	C₈	—	214	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	Q₈×Dic₁₀₇	1	2	yes	—
125	C₂²	C₂³×Dic₁₀₇	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	2	8	no	—
128	C₄	Q₈×Dic₁₀₇	1	2	yes	—
129	C₄	#16812116	1	4	yes	—
130	C₄	C₂×Q₈×D₁₀₇	1	1	yes	characteristic
131	C₄	C₂×#13133819	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×Q₈×Dic₁₀₇	1	1	yes	characteristic
134	C₂	C₂×#16812116	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20292758

C₂

yes

maximal, complemented

Q₈×C₁₀₇⋊C₈

C₂

yes

maximal, complemented

#20292654

C₂

yes

maximal

#20292651

C₂

yes

maximal, characteristic

#20292583

C₂

yes

maximal

C₂×C₄×C₁₀₇⋊C₈

C₂

yes

maximal, characteristic

#533×C₁₀₇

C₂

yes

maximal, characteristic

C₂×#13133802

C₂²

yes

—

C₂²×C₁₀₇⋊C₈

C₂²

yes

characteristic

C₂²×C₁₀₇⋊C₈

C₂²

yes

characteristic

C₄⋊(C₁₀₇⋊C₈)

C₂²

yes

—

C₄⋊(C₁₀₇⋊C₈)

C₂²

yes

—

#16812020

C₂²

yes

—

C₄×C₁₀₇⋊C₈

C₂²

yes

—

C₄×C₁₀₇⋊C₈

C₂²

yes

—

C₄×C₁₀₇⋊C₈

C₂²

yes

—

C₄⋊Q₈×C₁₀₇

C₄

yes

—

#125×C₁₀₇

C₄

yes

—

C₂²⋊Q₈×C₁₀₇

C₄

yes

—

C₄×Q₈×C₁₀₇

C₂²

yes

—

#117×C₁₀₇

C₂²

yes

—

C₂×C₄²×C₁₀₇

C₂²

yes

characteristic

#13133802

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

Q₈

yes

complemented

C₂×C₁₀₇⋊C₈

C₂³

yes

—

C₂×C₁₀₇⋊C₈

C₂³

yes

—

C₂×C₁₀₇⋊C₈

C₂³

yes

characteristic

C₂×C₁₀₇⋊C₈

C₂³

yes

characteristic

C₂×Q₈×C₁₀₇

C₂×C₄

yes

—

C₂²×C₄×C₁₀₇

C₂×C₄

yes

—

C₂²×C₄×C₁₀₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₇

C₂×C₄

yes

—

C₄⋊C₄×C₁₀₇

C₂×C₄

yes

—

C₄⋊C₄×C₁₀₇

C₂×C₄

yes

—

C₂²⋊C₄×C₁₀₇

C₂×C₄

yes

—

C₄²×C₁₀₇

C₂×C₄

yes

characteristic

C₄²×C₁₀₇

C₂×C₄

yes

characteristic

C₄²×C₁₀₇

C₂³

yes

—

C₄²×C₁₀₇

C₂³

yes

—

C₂³×C₁₀₇

C₂²×C₄

yes

characteristic

Q₈×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂²×C₄

yes

—

C₂×C₄×C₁₀₇

C₂²×C₄

yes

—

C₂×C₄×C₁₀₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂²×C₄

yes

—

C₂×C₄×C₁₀₇

C₂×Q₈

yes

—

C₂×C₄×C₁₀₇

#41

yes

—

C₂×C₄×C₁₀₇

C₂⁴

yes

characteristic

C₁₀₇⋊C₈

—

C₁₀₇⋊C₈

—

C₁₀₇⋊C₈

C₂×Q₈

yes

—

C₁₀₇⋊C₈

C₂×Q₈

yes

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

C₄×Q₈

yes

—

C₂²×C₁₀₇

C₂³×C₄

yes

characteristic

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₄×Q₈

yes

—

C₄×C₁₀₇

#131

yes

—

C₄×C₁₀₇

C₂²×Q₈

yes

characteristic

C₄×C₁₀₇

C₂×#41

yes

characteristic

C₂×C₁₀₇

—

C₂×C₁₀₇

C₂×C₄×Q₈

yes

characteristic

C₂×C₁₀₇

C₂×#131

yes

—

#2960

—

107

maximal, Sylow

C₁₀₇

#2960

yes

characteristic, complemented, Sylow

#533

D₁₀₇

yes

characteristic

#446

—

107

—

C₈×Q₈

—

107

—

#424

—

107

—

#423

—

107

—

#405

—

107

—

C₂×C₄×C₈

—

107

—

C₂×M₁₆

—

107

—

C₂²×C₈

—

107

—

C₂²×C₈

—

107

—

C₄⋊Q₈

Dic₁₀₇

yes

—

#125

Dic₁₀₇

yes

—

C₂²⋊Q₈

Dic₁₀₇

yes

—

C₄×Q₈

C₂×D₁₀₇

yes

—

#117

C₂×D₁₀₇

yes

—

C₂×C₄²

C₂×D₁₀₇

yes

characteristic

C₄⋊C₈

—

107

—

C₄⋊C₈

—

107

—

#97

—

107

—

C₄×C₈

—

107

—

C₄×C₈

—

107

—

C₄×C₈

—

107

—

C₂×Q₈

C₂×Dic₁₀₇

yes

—

C₂²×C₄

C₂×Dic₁₀₇

yes

—

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

M₁₆

—

214

—

C₂×C₈

—

107

—

C₂×C₈

—

107

—

C₂×C₈

—

107

—

C₂×C₈

—

107

—

C₂×C₈

—

107

—

100

C₂×C₈

—

214

—

101

C₄⋊C₄

C₂×Dic₁₀₇

yes

—

102

C₄⋊C₄

C₂×Dic₁₀₇

yes

—

103

C₄⋊C₄

C₂×Dic₁₀₇

yes

—

104

C₂²⋊C₄

C₂×Dic₁₀₇

yes

—

105

C₄²

C₂×Dic₁₀₇

yes

characteristic

106

C₄²

C₂×Dic₁₀₇

yes

characteristic

107

C₄²

C₂²×D₁₀₇

yes

—

108

C₄²

C₂²×D₁₀₇

yes

—

109

C₂³

C₂²×Dic₁₀₇

yes

characteristic

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

C₂²×Dic₁₀₇

yes

—

113

C₂×C₄

C₂²×Dic₁₀₇

yes

—

114

C₂×C₄

C₂²×Dic₁₀₇

yes

characteristic

115

C₂×C₄

C₂²×Dic₁₀₇

yes

—

116

C₂×C₄

Q₈×D₁₀₇

yes

—

117

C₂×C₄

#13133819

yes

—

118

C₂×C₄

C₂³×D₁₀₇

yes

characteristic

119

C₈

—

107

—

120

C₈

—

107

—

121

C₈

—

214

—

122

C₈

—

214

—

123

C₂²

—

124

C₂²

Q₈×Dic₁₀₇

yes

—

125

C₂²

C₂³×Dic₁₀₇

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

Q₈×Dic₁₀₇

yes

—

129

C₄

#16812116

yes

—

130

C₄

C₂×Q₈×D₁₀₇

yes

characteristic

131

C₄

C₂×#13133819

yes

characteristic

132

C₂

—

133

C₂

C₂×Q₈×Dic₁₀₇

yes

characteristic

134

C₂

C₂×#16812116

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₇
commutator, socle	C₂²×C₁₀₇	C₂³×C₄
Fitting	#533×C₁₀₇	C₂
prime radical	C₂³×C₁₀₇	C₂²×C₄
cosocle	C₂×C₄×C₁₀₇	C₂⁴
hypercenter	#533	D₁₀₇
hypocenter	C₁₀₇	#2960
LCS	G ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂×C₄ ◁ #533

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₀₇

commutator, socle

C₂²×C₁₀₇

C₂³×C₄

Fitting

#533×C₁₀₇

C₂

prime radical

C₂³×C₁₀₇

C₂²×C₄

cosocle

C₂×C₄×C₁₀₇

C₂⁴

D₁₀₇

C₁₀₇

G ▷ C₂²×C₁₀₇ ▷ C₁₀₇

C₁ ◁ C₂×C₄ ◁ #533

N	H ≅ G/N	type
1. #20292758	C₂	semidirect
2. Q₈×C₁₀₇⋊C₈	C₂	semidirect
25. C₂×C₁₀₇⋊C₈	Q₈	semidirect
68. C₁₀₇	#2960	semidirect

H ≅ G/N

type

1. #20292758

C₂

semidirect

2. Q₈×C₁₀₇⋊C₈

C₂

semidirect

25. C₂×C₁₀₇⋊C₈

Q₈

semidirect

68. C₁₀₇

#2960

semidirect

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

subgroup	index
#20292758	2
Q₈×C₁₀₇⋊C₈	2
#20292654	2
#20292651	2
#20292583	2
C₂×C₄×C₁₀₇⋊C₈	2
#533×C₁₀₇	2
#2960	107

subgroup

index

Q₈×C₁₀₇⋊C₈

C₂×C₄×C₁₀₇⋊C₈

#533×C₁₀₇

#2960

107

Subgroups — #28345396

Conjugacy classes of subgroups

Complements

Maximal subgroups