Group #28345395 — subgroups

Subgroups — #28345395

Order 13696.

364 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 197 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	#20293184	C₂	1	1	yes	maximal, characteristic
2	#20292757	C₂	1	4	yes	maximal, complemented
3	C₈×Dic₂₁₄	C₂	1	4	yes	maximal, complemented
4	#20292653	C₂	1	2	yes	maximal
5	#20292651	C₂	1	1	yes	maximal, characteristic
6	#20292582	C₂	1	2	yes	maximal
7	C₂×C₄×C₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#13133802	C₂²	1	2	yes	—
9	#16812109	C₄	1	2	yes	—
10	#16812101	C₄	1	2	yes	—
11	#16812088	C₄	1	4	yes	—
12	C₄×Dic₂₁₄	C₂²	1	4	yes	—
13	#16812068	C₂²	1	2	yes	—
14	C₄⋊(C₁₀₇⋊C₈)	C₂²	1	4	yes	—
15	Dic₁₀₇⋊C₈	C₂²	1	8	yes	—
16	#16812019	C₂²	1	4	yes	—
17	C₈×Dic₁₀₇	C₂²	1	4	yes	—
18	C₂²×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₁₀₇	C₂²	1	2	yes	—
22	C₄×C₈×C₁₀₇	C₂²	1	2	yes	—
23	C₂×Dic₂₁₄	C₂×C₄	1	4	yes	—
24	#13133802	—	2	4	no	—
25	C₂×C₁₀₇⋊C₈	C₂³	1	4	yes	—
26	C₄⋊Dic₁₀₇	C₂×C₄	1	4	yes	—
27	C₄⋊Dic₁₀₇	C₂×C₄	1	4	yes	—
28	Dic₁₀₇⋊C₄	C₂×C₄	1	8	yes	—
29	C₂²⋊Dic₁₀₇	C₂×C₄	1	4	yes	—
30	C₄×Dic₁₀₇	C₂³	1	4	yes	—
31	C₂²×C₄×C₁₀₇	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₁₀₇	—	2	2	no	—
34	C₂×C₈×C₁₀₇	Q₈	1	4	yes	complemented
35	C₂×C₈×C₁₀₇	C₂³	1	2	yes	—
36	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
38	C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
40	C₄²×C₁₀₇	C₂³	1	2	yes	—
41	C₂³×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₀₇	—	2	4	no	—
43	C₂×C₄×C₁₀₇	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₇	C₂²×C₄	1	2	yes	—
46	C₂×C₄×C₁₀₇	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₁₀₇	#41	1	2	yes	—
48	C₂×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
49	C₂×Dic₁₀₇	C₂²×C₄	1	8	yes	—
50	Dic₂₁₄	—	2	8	no	—
51	C₈×C₁₀₇	—	2	2	no	—
52	C₈×C₁₀₇	C₂×Q₈	1	2	yes	—
53	C₈×C₁₀₇	C₂×Q₈	1	2	yes	—
54	C₁₀₇⋊C₈	—	2	4	no	—
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	C₄×Q₈	1	2	yes	—
60	C₄×C₁₀₇	#131	1	4	yes	—
61	C₄×C₁₀₇	C₂²×Q₈	1	1	yes	characteristic
62	C₄×C₁₀₇	C₂×#41	1	1	yes	characteristic
63	Dic₁₀₇	—	2	8	no	—
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	—	107	1	no	—
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	D₁₀₇	1	1	yes	characteristic
76	C₂×M₁₆	—	107	2	no	—
77	C₂²×C₈	C₂×D₁₀₇	1	1	yes	characteristic
78	C₂²×C₈	C₂×D₁₀₇	1	1	yes	characteristic
79	C₄⋊Q₈	—	107	2	no	—
80	#125	—	107	2	no	—
81	C₂²⋊Q₈	—	107	4	no	—
82	C₄×Q₈	—	107	4	no	—
83	#117	—	107	2	no	—
84	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
85	C₄⋊C₈	—	107	8	no	—
86	C₄⋊C₈	—	107	4	no	—
87	#97	—	107	4	no	—
88	C₄×C₈	—	107	4	no	—
89	C₄×C₈	C₂×D₁₀₇	1	2	yes	—
90	C₄×C₈	C₂×D₁₀₇	1	2	yes	—
91	C₂×Q₈	—	107	4	no	—
92	C₂²×C₄	C₄×D₁₀₇	1	2	yes	—
93	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
94	M₁₆	—	214	4	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	107	4	no	—
97	C₂×C₈	Dic₂₁₄	1	4	yes	complemented
98	C₂×C₈	C₂²×D₁₀₇	1	2	yes	—
99	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
100	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
101	C₄⋊C₄	—	107	4	no	—
102	C₄⋊C₄	—	107	8	no	—
103	C₄⋊C₄	—	107	4	no	—
104	C₂²⋊C₄	—	107	4	no	—
105	C₄²	—	107	4	no	—
106	C₄²	C₄×D₁₀₇	1	1	yes	characteristic
107	C₄²	C₄×D₁₀₇	1	1	yes	characteristic
108	C₄²	C₂²×D₁₀₇	1	2	yes	—
109	C₂³	C₂×C₄×D₁₀₇	1	1	yes	characteristic
110	Q₈	—	214	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	107	8	no	—
113	C₂×C₄	C₂×C₄×D₁₀₇	1	2	yes	—
114	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
115	C₂×C₄	C₂×C₄×D₁₀₇	1	2	yes	—
116	C₂×C₄	C₂×Dic₂₁₄	1	2	yes	—
117	C₂×C₄	#13133818	1	2	yes	—
118	C₂×C₄	C₂³×D₁₀₇	1	1	yes	characteristic
119	C₈	—	2	2	no	—
120	C₈	—	214	4	no	—
121	C₈	C₂×Dic₂₁₄	1	2	yes	—
122	C₈	C₂×Dic₂₁₄	1	2	yes	—
123	C₂²	—	2	2	no	—
124	C₂²	C₄×Dic₂₁₄	1	2	yes	—
125	C₂²	C₂²×C₄×D₁₀₇	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	214	8	no	—
128	C₄	C₄×Dic₂₁₄	1	2	yes	—
129	C₄	#16812115	1	4	yes	—
130	C₄	C₂²×Dic₂₁₄	1	1	yes	characteristic
131	C₄	C₂×#13133818	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×C₄×Dic₂₁₄	1	1	yes	characteristic
134	C₂	C₂×#16812115	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20293184

C₂

yes

maximal, characteristic

#20292757

C₂

yes

maximal, complemented

C₈×Dic₂₁₄

C₂

yes

maximal, complemented

#20292653

C₂

yes

maximal

#20292651

C₂

yes

maximal, characteristic

#20292582

C₂

yes

maximal

C₂×C₄×C₈×C₁₀₇

C₂

yes

maximal, characteristic

C₂×#13133802

C₂²

yes

—

#16812109

C₄

yes

—

#16812101

C₄

yes

—

#16812088

C₄

yes

—

C₄×Dic₂₁₄

C₂²

yes

—

#16812068

C₂²

yes

—

C₄⋊(C₁₀₇⋊C₈)

C₂²

yes

—

Dic₁₀₇⋊C₈

C₂²

yes

—

#16812019

C₂²

yes

—

C₈×Dic₁₀₇

C₂²

yes

—

C₂²×C₈×C₁₀₇

C₂²

yes

characteristic

C₂²×C₈×C₁₀₇

C₂²

yes

characteristic

C₂×C₄²×C₁₀₇

C₂²

yes

characteristic

C₄×C₈×C₁₀₇

C₂²

yes

—

C₄×C₈×C₁₀₇

C₂²

yes

—

C₂×Dic₂₁₄

C₂×C₄

yes

—

#13133802

—

C₂×C₁₀₇⋊C₈

C₂³

yes

—

C₄⋊Dic₁₀₇

C₂×C₄

yes

—

C₄⋊Dic₁₀₇

C₂×C₄

yes

—

Dic₁₀₇⋊C₄

C₂×C₄

yes

—

C₂²⋊Dic₁₀₇

C₂×C₄

yes

—

C₄×Dic₁₀₇

C₂³

yes

—

C₂²×C₄×C₁₀₇

C₂×C₄

yes

—

C₂²×C₄×C₁₀₇

C₂³

yes

characteristic

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

Q₈

yes

complemented

C₂×C₈×C₁₀₇

C₂³

yes

—

C₂×C₈×C₁₀₇

C₂³

yes

characteristic

C₂×C₈×C₁₀₇

C₂³

yes

characteristic

C₄²×C₁₀₇

C₂×C₄

yes

characteristic

C₄²×C₁₀₇

C₂×C₄

yes

characteristic

C₄²×C₁₀₇

C₂³

yes

—

C₂³×C₁₀₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₇

—

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

C₂×Dic₁₀₇

C₂²×C₄

yes

—

Dic₂₁₄

—

C₈×C₁₀₇

—

C₈×C₁₀₇

C₂×Q₈

yes

—

C₈×C₁₀₇

C₂×Q₈

yes

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

C₄×Q₈

yes

—

C₂²×C₁₀₇

C₂³×C₄

yes

characteristic

C₄×C₁₀₇

—

C₄×C₁₀₇

C₄×Q₈

yes

—

C₄×C₁₀₇

#131

yes

—

C₄×C₁₀₇

C₂²×Q₈

yes

characteristic

C₄×C₁₀₇

C₂×#41

yes

characteristic

Dic₁₀₇

—

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

—

107

—

#446

—

107

—

C₈×Q₈

—

107

—

#424

—

107

—

#423

—

107

—

#405

—

107

—

C₂×C₄×C₈

D₁₀₇

yes

characteristic

C₂×M₁₆

—

107

—

C₂²×C₈

C₂×D₁₀₇

yes

characteristic

C₂²×C₈

C₂×D₁₀₇

yes

characteristic

C₄⋊Q₈

—

107

—

#125

—

107

—

C₂²⋊Q₈

—

107

—

C₄×Q₈

—

107

—

#117

—

107

—

C₂×C₄²

C₂×D₁₀₇

yes

characteristic

C₄⋊C₈

—

107

—

C₄⋊C₈

—

107

—

#97

—

107

—

C₄×C₈

—

107

—

C₄×C₈

C₂×D₁₀₇

yes

—

C₄×C₈

C₂×D₁₀₇

yes

—

C₂×Q₈

—

107

—

C₂²×C₄

C₄×D₁₀₇

yes

—

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

M₁₆

—

214

—

C₂×C₈

—

C₂×C₈

—

107

—

C₂×C₈

Dic₂₁₄

yes

complemented

C₂×C₈

C₂²×D₁₀₇

yes

—

C₂×C₈

C₂²×D₁₀₇

yes

characteristic

100

C₂×C₈

C₂²×D₁₀₇

yes

characteristic

101

C₄⋊C₄

—

107

—

102

C₄⋊C₄

—

107

—

103

C₄⋊C₄

—

107

—

104

C₂²⋊C₄

—

107

—

105

C₄²

—

107

—

106

C₄²

C₄×D₁₀₇

yes

characteristic

107

C₄²

C₄×D₁₀₇

yes

characteristic

108

C₄²

C₂²×D₁₀₇

yes

—

109

C₂³

C₂×C₄×D₁₀₇

yes

characteristic

110

Q₈

—

214

—

111

C₂×C₄

—

112

C₂×C₄

—

107

—

113

C₂×C₄

C₂×C₄×D₁₀₇

yes

—

114

C₂×C₄

C₂×C₄×D₁₀₇

yes

characteristic

115

C₂×C₄

C₂×C₄×D₁₀₇

yes

—

116

C₂×C₄

C₂×Dic₂₁₄

yes

—

117

C₂×C₄

#13133818

yes

—

118

C₂×C₄

C₂³×D₁₀₇

yes

characteristic

119

C₈

—

120

C₈

—

214

—

121

C₈

C₂×Dic₂₁₄

yes

—

122

C₈

C₂×Dic₂₁₄

yes

—

123

C₂²

—

124

C₂²

C₄×Dic₂₁₄

yes

—

125

C₂²

C₂²×C₄×D₁₀₇

yes

characteristic

126

C₄

—

127

C₄

—

214

—

128

C₄

C₄×Dic₂₁₄

yes

—

129

C₄

#16812115

yes

—

130

C₄

C₂²×Dic₂₁₄

yes

characteristic

131

C₄

C₂×#13133818

yes

characteristic

132

C₂

—

133

C₂

C₂×C₄×Dic₂₁₄

yes

characteristic

134

C₂

C₂×#16812115

yes

—

	subgroup	G/H
center	C₂×C₈	C₂²×D₁₀₇
commutator, socle	C₂²×C₁₀₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₇
Fitting	C₂×C₄×C₈×C₁₀₇	C₂
prime radical	C₂³×C₁₀₇	C₂²×C₄
cosocle	C₂×C₄×C₁₀₇	C₂⁴
hypercenter	C₂×C₄×C₈	D₁₀₇
hypocenter	C₁₀₇	#2960
LCS	G ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₈

C₂²×D₁₀₇

commutator, socle

C₂²×C₁₀₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₁₀₇

Fitting

C₂×C₄×C₈×C₁₀₇

C₂

prime radical

C₂³×C₁₀₇

C₂²×C₄

cosocle

C₂×C₄×C₁₀₇

C₂⁴

hypercenter

C₂×C₄×C₈

D₁₀₇

C₁₀₇

G ▷ C₂²×C₁₀₇ ▷ C₁₀₇

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #20292757	C₂	semidirect
3. C₈×Dic₂₁₄	C₂	semidirect
34. C₂×C₈×C₁₀₇	Q₈	semidirect
68. C₁₀₇	#2960	semidirect
97. C₂×C₈	Dic₂₁₄	semidirect

H ≅ G/N

type

2. #20292757

C₂

semidirect

3. C₈×Dic₂₁₄

C₂

semidirect

34. C₂×C₈×C₁₀₇

Q₈

semidirect

68. C₁₀₇

#2960

semidirect

97. C₂×C₈

Dic₂₁₄

semidirect

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

subgroup	index
#20293184	2
#20292757	2
C₈×Dic₂₁₄	2
#20292653	2
#20292651	2
#20292582	2
C₂×C₄×C₈×C₁₀₇	2
#2960	107

subgroup

index

C₈×Dic₂₁₄

C₂×C₄×C₈×C₁₀₇

107

Subgroups — #28345395

Conjugacy classes of subgroups

Complements

Maximal subgroups