Group #28345391 — subgroups

Subgroups — #28345391

Order 13696.

364 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 189 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	#20292754	C₂	1	4	yes	maximal, complemented
3	#20292751	C₂	1	4	yes	maximal, complemented
4	#20292655	C₂	1	2	yes	maximal
5	#20292585	C₂	1	2	yes	maximal
6	C₂×C₄×C₁₀₇⋊C₈	C₂	1	1	yes	maximal, characteristic
7	#423×C₁₀₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
10	#16812109	C₄	1	2	yes	—
11	#16812101	C₄	1	2	yes	—
12	#16812088	C₄	1	4	yes	—
13	C₄×Dic₂₁₄	C₂²	1	4	yes	—
14	#16812068	C₂²	1	2	yes	—
15	Dic₁₀₇⋊C₈	C₂²	1	8	yes	—
16	#16812019	C₂²	1	4	yes	—
17	C₄×C₁₀₇⋊C₈	C₂²	1	2	yes	—
18	C₄×C₁₀₇⋊C₈	C₂²	1	2	yes	—
19	C₈×Dic₁₀₇	C₂²	1	4	yes	—
20	C₂×M₁₆×C₁₀₇	C₂²	1	2	yes	—
21	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₁₀₇	C₂²	1	4	yes	—
23	C₂×Dic₂₁₄	C₂×C₄	1	4	yes	—
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	1	yes	characteristic
28	C₂×C₁₀₇⋊C₈	C₂³	1	1	yes	characteristic
29	C₄⋊Dic₁₀₇	C₂×C₄	1	4	yes	—
30	C₄⋊Dic₁₀₇	C₂×C₄	1	4	yes	—
31	Dic₁₀₇⋊C₄	C₂×C₄	1	8	yes	—
32	C₂²⋊Dic₁₀₇	C₂×C₄	1	4	yes	—
33	C₄×Dic₁₀₇	C₂³	1	4	yes	—
34	C₂²×C₄×C₁₀₇	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
36	M₁₆×C₁₀₇	—	2	4	no	—
37	C₂×C₈×C₁₀₇	C₂³	1	4	yes	—
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	4	no	—
52	C₁₀₇⋊C₈	—	2	2	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	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	D₁₀₇	1	1	yes	characteristic
74	#405	—	107	2	no	—
75	C₂×C₄×C₈	—	107	1	no	—
76	C₂×M₁₆	C₂×D₁₀₇	1	2	yes	—
77	C₂²×C₈	—	107	1	no	—
78	C₂²×C₈	—	107	1	no	—
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	C₂×D₁₀₇	1	4	yes	—
87	#97	—	107	4	no	—
88	C₄×C₈	—	107	4	no	—
89	C₄×C₈	—	107	2	no	—
90	C₄×C₈	—	107	2	no	—
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	—	2	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	—	214	2	no	—
100	C₂×C₈	C₂²×D₁₀₇	1	4	yes	—
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	Q₈×D₁₀₇	1	2	yes	—
117	C₂×C₄	#13133817	1	2	yes	—
118	C₂×C₄	C₂³×D₁₀₇	1	1	yes	characteristic
119	C₈	—	2	4	no	—
120	C₈	—	107	2	no	—
121	C₈	—	107	2	no	—
122	C₈	—	214	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	#16812074	1	2	yes	—
125	C₂²	C₂²×C₄×D₁₀₇	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	214	8	no	—
128	C₄	#16812074	1	2	yes	—
129	C₄	#16812117	1	4	yes	—
130	C₄	C₂×Q₈×D₁₀₇	1	1	yes	characteristic
131	C₄	C₂×#13133817	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×#16812074	1	1	yes	characteristic
134	C₂	C₂×#16812117	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20293184

C₂

yes

maximal, characteristic

#20292754

C₂

yes

maximal, complemented

#20292751

C₂

yes

maximal, complemented

#20292655

C₂

yes

maximal

#20292585

C₂

yes

maximal

C₂×C₄×C₁₀₇⋊C₈

C₂

yes

maximal, characteristic

#423×C₁₀₇

C₂

yes

maximal, characteristic

C₂²×C₁₀₇⋊C₈

C₂²

yes

characteristic

C₂²×C₁₀₇⋊C₈

C₂²

yes

characteristic

#16812109

C₄

yes

—

#16812101

C₄

yes

—

#16812088

C₄

yes

—

C₄×Dic₂₁₄

C₂²

yes

—

#16812068

C₂²

yes

—

Dic₁₀₇⋊C₈

C₂²

yes

—

#16812019

C₂²

yes

—

C₄×C₁₀₇⋊C₈

C₂²

yes

—

C₄×C₁₀₇⋊C₈

C₂²

yes

—

C₈×Dic₁₀₇

C₂²

yes

—

C₂×M₁₆×C₁₀₇

C₂²

yes

—

C₂×C₄²×C₁₀₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₇

C₂²

yes

—

C₂×Dic₂₁₄

C₂×C₄

yes

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

Q₈

yes

complemented

C₂×C₁₀₇⋊C₈

C₂³

yes

—

C₂×C₁₀₇⋊C₈

C₂³

yes

characteristic

C₂×C₁₀₇⋊C₈

C₂³

yes

characteristic

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

M₁₆×C₁₀₇

—

C₂×C₈×C₁₀₇

C₂³

yes

—

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

—

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×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

D₁₀₇

yes

characteristic

#405

—

107

—

C₂×C₄×C₈

—

107

—

C₂×M₁₆

C₂×D₁₀₇

yes

—

C₂²×C₈

—

107

—

C₂²×C₈

—

107

—

C₄⋊Q₈

—

107

—

#125

—

107

—

C₂²⋊Q₈

—

107

—

C₄×Q₈

—

107

—

#117

—

107

—

C₂×C₄²

C₂×D₁₀₇

yes

characteristic

C₄⋊C₈

—

107

—

C₄⋊C₈

C₂×D₁₀₇

yes

—

#97

—

107

—

C₄×C₈

—

107

—

C₄×C₈

—

107

—

C₄×C₈

—

107

—

C₂×Q₈

—

107

—

C₂²×C₄

C₄×D₁₀₇

yes

—

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

M₁₆

—

C₂×C₈

—

107

—

C₂×C₈

—

107

—

C₂×C₈

—

107

—

C₂×C₈

—

107

—

C₂×C₈

—

214

—

100

C₂×C₈

C₂²×D₁₀₇

yes

—

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

Q₈×D₁₀₇

yes

—

117

C₂×C₄

#13133817

yes

—

118

C₂×C₄

C₂³×D₁₀₇

yes

characteristic

119

C₈

—

120

C₈

—

107

—

121

C₈

—

107

—

122

C₈

—

214

—

123

C₂²

—

124

C₂²

#16812074

yes

—

125

C₂²

C₂²×C₄×D₁₀₇

yes

characteristic

126

C₄

—

127

C₄

—

214

—

128

C₄

#16812074

yes

—

129

C₄

#16812117

yes

—

130

C₄

C₂×Q₈×D₁₀₇

yes

characteristic

131

C₄

C₂×#13133817

yes

characteristic

132

C₂

—

133

C₂

C₂×#16812074

yes

characteristic

134

C₂

C₂×#16812117

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₇
commutator, socle	C₂²×C₁₀₇	C₂³×C₄
Fitting	#423×C₁₀₇	C₂
prime radical	C₂³×C₁₀₇	C₂²×C₄
cosocle	C₂×C₄×C₁₀₇	C₂⁴
hypercenter	#423	D₁₀₇
hypocenter	C₁₀₇	#2960
LCS	G ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₀₇

commutator, socle

C₂²×C₁₀₇

C₂³×C₄

Fitting

#423×C₁₀₇

C₂

prime radical

C₂³×C₁₀₇

C₂²×C₄

cosocle

C₂×C₄×C₁₀₇

C₂⁴

D₁₀₇

C₁₀₇

G ▷ C₂²×C₁₀₇ ▷ C₁₀₇

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
2. #20292754	C₂	semidirect
3. #20292751	C₂	semidirect
25. C₂×C₁₀₇⋊C₈	Q₈	semidirect
68. C₁₀₇	#2960	semidirect

H ≅ G/N

type

2. #20292754

C₂

semidirect

3. #20292751

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
#20293184	2
#20292754	2
#20292751	2
#20292655	2
#20292585	2
C₂×C₄×C₁₀₇⋊C₈	2
#423×C₁₀₇	2
#2960	107

subgroup

index

C₂×C₄×C₁₀₇⋊C₈

#423×C₁₀₇

#2960

107

Subgroups — #28345391

Conjugacy classes of subgroups

Complements

Maximal subgroups