Group #28345402 — subgroups

Subgroups — #28345402

Order 13696.

348 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 221 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	8	yes	maximal, complemented
2	#20292654	C₂	1	2	yes	maximal
3	#20292651	C₂	1	1	yes	maximal, characteristic
4	#20292583	C₂	1	2	yes	maximal
5	C₂×#16812020	C₂	1	1	yes	maximal, characteristic
6	#533×C₁₀₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133802	C₂²	1	2	yes	—
8	C₂²×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₄⋊(C₁₀₇⋊C₈)	C₂²	1	4	yes	—
11	C₄⋊(C₁₀₇⋊C₈)	C₂²	1	8	yes	—
12	#16812020	C₂²	1	4	yes	—
13	#16812020	C₂²	1	4	yes	—
14	C₄×C₁₀₇⋊C₈	C₂²	1	4	yes	—
15	C₄⋊Q₈×C₁₀₇	C₄	1	2	yes	—
16	#125×C₁₀₇	C₄	1	2	yes	—
17	C₂²⋊Q₈×C₁₀₇	C₄	1	4	yes	—
18	C₄×Q₈×C₁₀₇	C₂²	1	4	yes	—
19	#117×C₁₀₇	C₂²	1	2	yes	—
20	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
21	#13133802	—	2	4	no	—
22	C₂×C₁₀₇⋊C₈	—	2	2	no	—
23	C₂×C₁₀₇⋊C₈	Q₈	1	4	yes	complemented
24	C₂×C₁₀₇⋊C₈	C₂³	1	2	yes	—
25	C₂×C₁₀₇⋊C₈	C₂³	1	4	yes	—
26	C₂×C₁₀₇⋊C₈	C₂³	1	2	yes	—
27	C₂×Q₈×C₁₀₇	C₂×C₄	1	4	yes	—
28	C₂²×C₄×C₁₀₇	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₁₀₇	C₂×C₄	1	4	yes	—
31	C₄⋊C₄×C₁₀₇	C₂×C₄	1	8	yes	—
32	C₄⋊C₄×C₁₀₇	C₂×C₄	1	4	yes	—
33	C₂²⋊C₄×C₁₀₇	C₂×C₄	1	4	yes	—
34	C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₁₀₇	C₂³	1	2	yes	—
37	C₄²×C₁₀₇	C₂³	1	4	yes	—
38	C₂³×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
39	Q₈×C₁₀₇	—	4	4	no	—
40	C₂×C₄×C₁₀₇	—	4	2	no	—
41	C₂×C₄×C₁₀₇	C₂²×C₄	1	2	yes	—
42	C₂×C₄×C₁₀₇	C₂²×C₄	1	2	yes	—
43	C₂×C₄×C₁₀₇	C₂²×C₄	1	8	yes	—
44	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₇	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₁₀₇	#41	1	2	yes	—
47	C₂×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
48	C₁₀₇⋊C₈	—	2	2	no	—
49	C₁₀₇⋊C₈	—	2	4	no	—
50	C₁₀₇⋊C₈	C₂×Q₈	1	4	yes	—
51	C₂²×C₁₀₇	—	2	2	no	—
52	C₂²×C₁₀₇	C₄×Q₈	1	2	yes	—
53	C₂²×C₁₀₇	C₂³×C₄	1	1	yes	characteristic
54	C₄×C₁₀₇	—	2	2	no	—
55	C₄×C₁₀₇	—	2	2	no	—
56	C₄×C₁₀₇	—	2	8	no	—
57	C₄×C₁₀₇	C₄×Q₈	1	2	yes	—
58	C₄×C₁₀₇	C₂²×Q₈	1	1	yes	characteristic
59	C₄×C₁₀₇	C₂×#41	1	1	yes	characteristic
60	C₂×C₁₀₇	—	2	2	no	—
61	C₂×C₁₀₇	C₂×C₄×Q₈	1	1	yes	characteristic
62	C₂×C₁₀₇	#568	1	2	yes	—
63	#2961	—	107	1	no	maximal, Sylow
64	C₁₀₇	#2961	1	1	yes	characteristic, complemented, Sylow
65	#533	D₁₀₇	1	1	yes	characteristic
66	#446	—	107	8	no	—
67	#424	—	107	2	no	—
68	#423	—	107	1	no	—
69	#405	—	107	2	no	—
70	C₂×#97	—	107	1	no	—
71	C₂×M₁₆	—	107	2	no	—
72	C₂²×C₈	—	107	1	no	—
73	C₂²×C₈	—	107	1	no	—
74	C₄⋊Q₈	Dic₁₀₇	1	2	yes	—
75	#125	Dic₁₀₇	1	2	yes	—
76	C₂²⋊Q₈	Dic₁₀₇	1	4	yes	—
77	C₄×Q₈	C₂×D₁₀₇	1	4	yes	—
78	#117	C₂×D₁₀₇	1	2	yes	—
79	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
80	C₄⋊C₈	—	107	4	no	—
81	C₄⋊C₈	—	107	8	no	—
82	#97	—	107	4	no	—
83	#97	—	107	4	no	—
84	C₄×C₈	—	107	4	no	—
85	C₂×Q₈	C₂×Dic₁₀₇	1	4	yes	—
86	C₂²×C₄	C₂×Dic₁₀₇	1	2	yes	—
87	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
88	M₁₆	—	214	4	no	—
89	C₂×C₈	—	107	4	no	—
90	C₂×C₈	—	107	2	no	—
91	C₂×C₈	—	107	4	no	—
92	C₂×C₈	—	107	2	no	—
93	C₂×C₈	—	214	2	no	—
94	C₄⋊C₄	C₂×Dic₁₀₇	1	4	yes	—
95	C₄⋊C₄	C₂×Dic₁₀₇	1	8	yes	—
96	C₄⋊C₄	C₂×Dic₁₀₇	1	4	yes	—
97	C₂²⋊C₄	C₂×Dic₁₀₇	1	4	yes	—
98	C₄²	C₂×Dic₁₀₇	1	1	yes	characteristic
99	C₄²	C₂×Dic₁₀₇	1	1	yes	characteristic
100	C₄²	C₂²×D₁₀₇	1	4	yes	—
101	C₄²	C₂²×D₁₀₇	1	2	yes	—
102	C₂³	C₂²×Dic₁₀₇	1	1	yes	characteristic
103	Q₈	—	4	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	C₂²×Dic₁₀₇	1	2	yes	—
106	C₂×C₄	C₂²×Dic₁₀₇	1	2	yes	—
107	C₂×C₄	C₂²×Dic₁₀₇	1	8	yes	—
108	C₂×C₄	C₂²×Dic₁₀₇	1	1	yes	characteristic
109	C₂×C₄	Q₈×D₁₀₇	1	2	yes	—
110	C₂×C₄	#13133819	1	2	yes	—
111	C₂×C₄	C₂³×D₁₀₇	1	1	yes	characteristic
112	C₈	—	107	4	no	—
113	C₈	—	214	2	no	—
114	C₈	—	214	4	no	—
115	C₂²	—	2	2	no	—
116	C₂²	Q₈×Dic₁₀₇	1	2	yes	—
117	C₂²	C₂³×Dic₁₀₇	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	2	8	no	—
121	C₄	Q₈×Dic₁₀₇	1	2	yes	—
122	C₄	C₂×Q₈×D₁₀₇	1	1	yes	characteristic
123	C₄	C₂×#13133819	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	C₂×Q₈×Dic₁₀₇	1	1	yes	characteristic
126	C₂	#20293394	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20292758

C₂

yes

maximal, complemented

#20292654

C₂

yes

maximal

#20292651

C₂

yes

maximal, characteristic

#20292583

C₂

yes

maximal

C₂×#16812020

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

—

#16812020

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

—

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

—

C₂×C₄×C₁₀₇

C₂²×C₄

yes

characteristic

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

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

C₄×Q₈

yes

—

C₂²×C₁₀₇

C₂³×C₄

yes

characteristic

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₄×Q₈

yes

—

C₄×C₁₀₇

C₂²×Q₈

yes

characteristic

C₄×C₁₀₇

C₂×#41

yes

characteristic

C₂×C₁₀₇

—

C₂×C₁₀₇

C₂×C₄×Q₈

yes

characteristic

C₂×C₁₀₇

#568

yes

—

#2961

—

107

maximal, Sylow

C₁₀₇

#2961

yes

characteristic, complemented, Sylow

#533

D₁₀₇

yes

characteristic

#446

—

107

—

#424

—

107

—

#423

—

107

—

#405

—

107

—

C₂×#97

—

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

—

#97

—

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

—

214

—

C₄⋊C₄

C₂×Dic₁₀₇

yes

—

C₄⋊C₄

C₂×Dic₁₀₇

yes

—

C₄⋊C₄

C₂×Dic₁₀₇

yes

—

C₂²⋊C₄

C₂×Dic₁₀₇

yes

—

C₄²

C₂×Dic₁₀₇

yes

characteristic

C₄²

C₂×Dic₁₀₇

yes

characteristic

100

C₄²

C₂²×D₁₀₇

yes

—

101

C₄²

C₂²×D₁₀₇

yes

—

102

C₂³

C₂²×Dic₁₀₇

yes

characteristic

103

Q₈

—

104

C₂×C₄

—

105

C₂×C₄

C₂²×Dic₁₀₇

yes

—

106

C₂×C₄

C₂²×Dic₁₀₇

yes

—

107

C₂×C₄

C₂²×Dic₁₀₇

yes

—

108

C₂×C₄

C₂²×Dic₁₀₇

yes

characteristic

109

C₂×C₄

Q₈×D₁₀₇

yes

—

110

C₂×C₄

#13133819

yes

—

111

C₂×C₄

C₂³×D₁₀₇

yes

characteristic

112

C₈

—

107

—

113

C₈

—

214

—

114

C₈

—

214

—

115

C₂²

—

116

C₂²

Q₈×Dic₁₀₇

yes

—

117

C₂²

C₂³×Dic₁₀₇

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

Q₈×Dic₁₀₇

yes

—

122

C₄

C₂×Q₈×D₁₀₇

yes

characteristic

123

C₄

C₂×#13133819

yes

characteristic

124

C₂

—

125

C₂

C₂×Q₈×Dic₁₀₇

yes

characteristic

126

C₂

#20293394

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	#2961
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
23. C₂×C₁₀₇⋊C₈	Q₈	semidirect
64. C₁₀₇	#2961	semidirect

H ≅ G/N

type

1. #20292758

C₂

semidirect

23. C₂×C₁₀₇⋊C₈

Q₈

semidirect

64. C₁₀₇

#2961

semidirect

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

subgroup	index
#20292758	2
#20292654	2
#20292651	2
#20292583	2
C₂×#16812020	2
#533×C₁₀₇	2
#2961	107

subgroup

index

#533×C₁₀₇

107

Subgroups — #28345402

Conjugacy classes of subgroups

Complements

Maximal subgroups