Group #28345328 — subgroups

Subgroups — #28345328

Order 13696.

456 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 181 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	C₂²×Dic₄₂₈	C₂	1	1	yes	maximal, characteristic
2	#20293104	C₂	1	2	yes	maximal
3	#20292733	C₂	1	4	yes	maximal, complemented
4	#20292721	C₂	1	4	yes	maximal, complemented
5	#20292635	C₂	1	2	yes	maximal
6	#20292582	C₂	1	1	yes	maximal, characteristic
7	#427×C₁₀₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₂₁₄	C₂²	1	2	yes	—
9	C₂×Dic₄₂₈	C₄	1	8	yes	complemented
10	C₂×Dic₄₂₈	C₂²	1	2	yes	—
11	C₂×Dic₄₂₈	C₂²	1	2	yes	—
12	C₂×#13133802	C₂²	1	1	yes	characteristic
13	C₄×Dic₂₁₄	—	2	4	no	—
14	#16812074	C₂²	1	8	yes	—
15	#16812068	C₂²	1	1	yes	characteristic
16	#16812065	—	2	4	no	—
17	#16812032	C₂²	1	8	yes	—
18	#16812019	C₂²	1	2	yes	—
19	C₈×Dic₁₀₇	C₂²	1	2	yes	—
20	C₂²×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
21	#117×C₁₀₇	C₂²	1	2	yes	—
22	#107×C₁₀₇	C₂²	1	2	yes	—
23	#106×C₁₀₇	C₂²	1	2	yes	—
24	C₂×Dic₂₁₄	—	2	2	no	—
25	C₂×Dic₂₁₄	—	2	4	no	—
26	C₂×Dic₂₁₄	C₂×C₄	1	8	yes	—
27	C₂×Dic₂₁₄	C₂³	1	4	yes	—
28	C₂²×Dic₁₀₇	—	2	2	no	—
29	Dic₄₂₈	C₂×C₄	1	8	yes	—
30	Dic₄₂₈	C₂×C₄	1	8	yes	—
31	#13133802	—	2	2	no	—
32	C₂×C₁₀₇⋊C₈	C₂³	1	2	yes	—
33	C₄⋊Dic₁₀₇	D₄	1	2	yes	—
34	Dic₁₀₇⋊C₄	—	2	8	no	—
35	C₂²⋊Dic₁₀₇	D₄	1	2	yes	—
36	C₄×Dic₁₀₇	—	2	8	no	—
37	C₄×Dic₁₀₇	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
39	C₂×C₈×C₁₀₇	C₂×C₄	1	4	yes	—
40	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
41	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
42	C₄⋊C₄×C₁₀₇	C₂³	1	4	yes	—
43	C₂²⋊C₄×C₁₀₇	—	2	2	no	—
44	C₄²×C₁₀₇	—	2	2	no	—
45	C₂³×C₁₀₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₀₇	—	2	4	no	—
47	C₂×C₄×C₁₀₇	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₀₇	#41	1	2	yes	—
50	C₂×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₁₀₇	—	2	8	no	—
52	C₂×Dic₁₀₇	—	2	4	no	—
53	C₂×Dic₁₀₇	C₂×D₄	1	4	yes	—
54	Dic₂₁₄	—	2	8	no	—
55	Dic₂₁₄	—	2	4	no	—
56	Dic₂₁₄	C₂²×C₄	1	8	yes	—
57	C₈×C₁₀₇	C₂²×C₄	1	2	yes	—
58	C₈×C₁₀₇	C₂²×C₄	1	2	yes	—
59	C₁₀₇⋊C₈	—	2	2	no	—
60	C₂²×C₁₀₇	—	2	2	no	—
61	C₂²×C₁₀₇	C₄×D₄	1	2	yes	—
62	C₂²×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₁₀₇	—	4	4	no	—
64	C₄×C₁₀₇	C₄×D₄	1	2	yes	—
65	C₄×C₁₀₇	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₁₀₇	C₂×#41	1	1	yes	characteristic
67	Dic₁₀₇	—	2	8	no	—
68	Dic₁₀₇	—	2	4	no	—
69	C₂×C₁₀₇	—	2	2	no	—
70	C₂×C₁₀₇	C₂×C₄×D₄	1	1	yes	characteristic
71	C₂×C₁₀₇	#578	1	2	yes	—
72	#2950	—	107	1	no	maximal, Sylow
73	C₁₀₇	#2950	1	1	yes	characteristic, complemented, Sylow
74	C₂²×Q₁₆	—	107	1	no	—
75	#519	—	107	2	no	—
76	#441	—	107	4	no	—
77	C₄×Q₁₆	—	107	4	no	—
78	#427	D₁₀₇	1	1	yes	characteristic
79	#419	—	107	2	no	—
80	#405	—	107	1	no	—
81	C₂²×Q₈	—	107	2	no	—
82	C₂×Q₁₆	—	107	8	no	—
83	C₂×Q₁₆	—	107	2	no	—
84	C₂×Q₁₆	—	107	2	no	—
85	C₂×M₁₆	—	107	1	no	—
86	C₂²×C₈	C₂×D₁₀₇	1	1	yes	characteristic
87	C₄×Q₈	—	107	8	no	—
88	C₄×Q₈	—	214	4	no	—
89	#117	—	107	1	no	—
90	#117	—	214	4	no	—
91	#117	C₂×D₁₀₇	1	2	yes	—
92	#107	C₂×D₁₀₇	1	2	yes	—
93	#106	C₂×D₁₀₇	1	2	yes	—
94	#103	—	107	8	no	—
95	#97	—	107	2	no	—
96	C₄×C₈	—	107	2	no	—
97	C₂×Q₈	—	107	8	no	—
98	C₂×Q₈	—	107	4	no	—
99	C₂×Q₈	—	214	2	no	—
100	C₂×Q₈	—	214	4	no	—
101	C₂²×C₄	—	214	2	no	—
102	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
103	Q₁₆	—	107	8	no	—
104	Q₁₆	—	107	8	no	—
105	M₁₆	—	214	2	no	—
106	C₂×C₈	—	107	2	no	—
107	C₂×C₈	C₄×D₁₀₇	1	4	yes	—
108	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
109	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
110	C₄⋊C₄	—	107	2	no	—
111	C₄⋊C₄	—	214	8	no	—
112	C₄⋊C₄	C₂²×D₁₀₇	1	4	yes	—
113	C₂²⋊C₄	—	2	2	no	—
114	C₂²⋊C₄	—	107	2	no	—
115	C₄²	—	2	2	no	—
116	C₄²	—	107	2	no	—
117	C₄²	—	214	8	no	—
118	C₂³	D₄×D₁₀₇	1	1	yes	characteristic
119	Q₈	—	107	8	no	—
120	Q₈	—	214	8	no	—
121	Q₈	—	214	4	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	107	4	no	—
124	C₂×C₄	—	214	8	no	—
125	C₂×C₄	—	214	4	no	—
126	C₂×C₄	C₂×C₄×D₁₀₇	1	2	yes	—
127	C₂×C₄	D₄×D₁₀₇	1	1	yes	characteristic
128	C₂×C₄	#13133819	1	2	yes	—
129	C₂×C₄	C₂³×D₁₀₇	1	1	yes	characteristic
130	C₈	—	214	2	no	—
131	C₈	C₂×C₄×D₁₀₇	1	2	yes	—
132	C₈	C₂×C₄×D₁₀₇	1	2	yes	—
133	C₂²	—	2	2	no	—
134	C₂²	#16812071	1	2	yes	—
135	C₂²	C₂×D₄×D₁₀₇	1	1	yes	characteristic
136	C₄	—	4	4	no	—
137	C₄	—	214	4	no	—
138	C₄	—	214	8	no	—
139	C₄	#16812071	1	2	yes	—
140	C₄	C₂²×C₄×D₁₀₇	1	1	yes	characteristic
141	C₄	C₂×#13133819	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	C₂×#16812071	1	1	yes	characteristic
144	C₂	#20293441	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₄₂₈

C₂

yes

maximal, characteristic

#20293104

C₂

yes

maximal

#20292733

C₂

yes

maximal, complemented

#20292721

C₂

yes

maximal, complemented

#20292635

C₂

yes

maximal

#20292582

C₂

yes

maximal, characteristic

#427×C₁₀₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₄

C₂²

yes

—

C₂×Dic₄₂₈

C₄

yes

complemented

C₂×Dic₄₂₈

C₂²

yes

—

C₂×Dic₄₂₈

C₂²

yes

—

C₂×#13133802

C₂²

yes

characteristic

C₄×Dic₂₁₄

—

#16812074

C₂²

yes

—

#16812068

C₂²

yes

characteristic

#16812065

—

#16812032

C₂²

yes

—

#16812019

C₂²

yes

—

C₈×Dic₁₀₇

C₂²

yes

—

C₂²×C₈×C₁₀₇

C₂²

yes

characteristic

#117×C₁₀₇

C₂²

yes

—

#107×C₁₀₇

C₂²

yes

—

#106×C₁₀₇

C₂²

yes

—

C₂×Dic₂₁₄

—

C₂×Dic₂₁₄

—

C₂×Dic₂₁₄

C₂×C₄

yes

—

C₂×Dic₂₁₄

C₂³

yes

—

C₂²×Dic₁₀₇

—

Dic₄₂₈

C₂×C₄

yes

—

Dic₄₂₈

C₂×C₄

yes

—

#13133802

—

C₂×C₁₀₇⋊C₈

C₂³

yes

—

C₄⋊Dic₁₀₇

D₄

yes

—

Dic₁₀₇⋊C₄

—

C₂²⋊Dic₁₀₇

D₄

yes

—

C₄×Dic₁₀₇

—

C₄×Dic₁₀₇

C₂³

yes

—

C₂²×C₄×C₁₀₇

C₂³

yes

characteristic

C₂×C₈×C₁₀₇

C₂×C₄

yes

—

C₂×C₈×C₁₀₇

C₂³

yes

characteristic

C₂×C₈×C₁₀₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₇

C₂³

yes

—

C₂²⋊C₄×C₁₀₇

—

C₄²×C₁₀₇

—

C₂³×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂²×C₄

yes

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

#41

yes

—

C₂×C₄×C₁₀₇

C₂⁴

yes

characteristic

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

C₂×D₄

yes

—

Dic₂₁₄

—

Dic₂₁₄

—

Dic₂₁₄

C₂²×C₄

yes

—

C₈×C₁₀₇

C₂²×C₄

yes

—

C₈×C₁₀₇

C₂²×C₄

yes

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

C₄×D₄

yes

—

C₂²×C₁₀₇

C₂²×D₄

yes

characteristic

C₄×C₁₀₇

—

C₄×C₁₀₇

C₄×D₄

yes

—

C₄×C₁₀₇

C₂³×C₄

yes

characteristic

C₄×C₁₀₇

C₂×#41

yes

characteristic

Dic₁₀₇

—

Dic₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

C₂×C₄×D₄

yes

characteristic

C₂×C₁₀₇

#578

yes

—

#2950

—

107

maximal, Sylow

C₁₀₇

#2950

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

107

—

#519

—

107

—

#441

—

107

—

C₄×Q₁₆

—

107

—

#427

D₁₀₇

yes

characteristic

#419

—

107

—

#405

—

107

—

C₂²×Q₈

—

107

—

C₂×Q₁₆

—

107

—

C₂×Q₁₆

—

107

—

C₂×Q₁₆

—

107

—

C₂×M₁₆

—

107

—

C₂²×C₈

C₂×D₁₀₇

yes

characteristic

C₄×Q₈

—

107

—

C₄×Q₈

—

214

—

#117

—

107

—

#117

—

214

—

#117

C₂×D₁₀₇

yes

—

#107

C₂×D₁₀₇

yes

—

#106

C₂×D₁₀₇

yes

—

#103

—

107

—

#97

—

107

—

C₄×C₈

—

107

—

C₂×Q₈

—

107

—

C₂×Q₈

—

107

—

C₂×Q₈

—

214

—

100

C₂×Q₈

—

214

—

101

C₂²×C₄

—

214

—

102

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

103

Q₁₆

—

107

—

104

Q₁₆

—

107

—

105

M₁₆

—

214

—

106

C₂×C₈

—

107

—

107

C₂×C₈

C₄×D₁₀₇

yes

—

108

C₂×C₈

C₂²×D₁₀₇

yes

characteristic

109

C₂×C₈

C₂²×D₁₀₇

yes

characteristic

110

C₄⋊C₄

—

107

—

111

C₄⋊C₄

—

214

—

112

C₄⋊C₄

C₂²×D₁₀₇

yes

—

113

C₂²⋊C₄

—

114

C₂²⋊C₄

—

107

—

115

C₄²

—

116

C₄²

—

107

—

117

C₄²

—

214

—

118

C₂³

D₄×D₁₀₇

yes

characteristic

119

Q₈

—

107

—

120

Q₈

—

214

—

121

Q₈

—

214

—

122

C₂×C₄

—

123

C₂×C₄

—

107

—

124

C₂×C₄

—

214

—

125

C₂×C₄

—

214

—

126

C₂×C₄

C₂×C₄×D₁₀₇

yes

—

127

C₂×C₄

D₄×D₁₀₇

yes

characteristic

128

C₂×C₄

#13133819

yes

—

129

C₂×C₄

C₂³×D₁₀₇

yes

characteristic

130

C₈

—

214

—

131

C₈

C₂×C₄×D₁₀₇

yes

—

132

C₈

C₂×C₄×D₁₀₇

yes

—

133

C₂²

—

134

C₂²

#16812071

yes

—

135

C₂²

C₂×D₄×D₁₀₇

yes

characteristic

136

C₄

—

137

C₄

—

214

—

138

C₄

—

214

—

139

C₄

#16812071

yes

—

140

C₄

C₂²×C₄×D₁₀₇

yes

characteristic

141

C₄

C₂×#13133819

yes

characteristic

142

C₂

—

143

C₂

C₂×#16812071

yes

characteristic

144

C₂

#20293441

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₇
commutator	C₄×C₁₀₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₇
Fitting	#427×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₂²×D₄
prime radical	C₂³×C₁₀₇	C₂×D₄
cosocle	C₂×C₄×C₁₀₇	C₂⁴
hypercenter	#427	D₁₀₇
hypocenter	C₁₀₇	#2950
LCS	G ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂²

C₂×D₄×D₁₀₇

commutator

C₄×C₁₀₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₁₀₇

Fitting

#427×C₁₀₇

C₂

socle

C₂²×C₁₀₇

C₂²×D₄

prime radical

C₂³×C₁₀₇

C₂×D₄

cosocle

C₂×C₄×C₁₀₇

C₂⁴

D₁₀₇

C₁₀₇

G ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
3. #20292733	C₂	semidirect
4. #20292721	C₂	semidirect
9. C₂×Dic₄₂₈	C₄	semidirect
73. C₁₀₇	#2950	semidirect

H ≅ G/N

type

3. #20292733

C₂

semidirect

4. #20292721

C₂

semidirect

9. C₂×Dic₄₂₈

C₄

semidirect

73. C₁₀₇

#2950

semidirect

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

subgroup	index
C₂²×Dic₄₂₈	2
#20293104	2
#20292733	2
#20292721	2
#20292635	2
#20292582	2
#427×C₁₀₇	2
#2950	107

subgroup

index

C₂²×Dic₄₂₈

#427×C₁₀₇

107

Subgroups — #28345328

Conjugacy classes of subgroups

Complements

Maximal subgroups