Group #28345178 — subgroups

Subgroups — #28345178

Order 13696.

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

normal subgroups only

This group has 484 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	C₂×#16812117	C₂	1	2	yes	maximal, complemented
2	#20293177	C₂	1	1	yes	maximal, characteristic
3	#20292697	C₂	1	8	yes	maximal, complemented
4	#20292595	C₂	1	2	yes	maximal, complemented
5	C₂×#16812020	C₂	1	1	yes	maximal, characteristic, complemented
6	#423×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13133818	C₂²	1	2	yes	—
8	#16812117	—	2	8	no	—
9	C₂×#13133801	C₂²	1	4	yes	—
10	C₂²×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₈×D₁₀₇	C₂²	1	4	yes	—
13	#16812109	C₄	1	1	yes	characteristic
14	#16812106	C₄	1	1	yes	characteristic
15	#16812097	C₄	1	2	yes	—
16	#16812083	C₄	1	4	yes	—
17	C₄×D₄₂₈	C₂²	1	4	yes	—
18	(C₂×D₁₀₇)⋊C₈	C₂²	1	8	yes	—
19	#16812020	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	#13133818	—	2	4	no	—
24	#13133818	—	2	4	no	—
25	C₂×Dic₂₁₄	C₂×C₄	1	2	yes	—
26	C₂×D₄₂₈	C₂×C₄	1	2	yes	—
27	C₂×D₄₂₈	C₂×C₄	1	4	yes	—
28	C₂×C₂²⋊D₁₀₇	C₂×C₄	1	4	yes	—
29	C₂×C₄×D₁₀₇	C₂³	1	4	yes	—
30	#13133801	—	2	8	no	—
31	C₂×C₁₀₇⋊C₈	—	2	2	no	—
32	C₂×C₁₀₇⋊C₈	D₄	1	4	yes	complemented
33	C₂×C₁₀₇⋊C₈	C₂³	1	2	yes	—
34	C₂×C₁₀₇⋊C₈	C₂³	1	2	yes	—
35	C₈×D₁₀₇	—	2	8	no	—
36	C₄⋊Dic₁₀₇	C₂×C₄	1	4	yes	—
37	(C₂×D₁₀₇)⋊C₄	C₂×C₄	1	8	yes	—
38	C₂²×C₄×C₁₀₇	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
40	M₁₆×C₁₀₇	—	2	4	no	—
41	C₂×C₈×C₁₀₇	C₂³	1	4	yes	—
42	C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
44	C₄²×C₁₀₇	C₂³	1	2	yes	—
45	C₂³×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×D₁₀₇	C₂²×C₄	1	4	yes	—
47	C₂×C₄×C₁₀₇	—	4	2	no	—
48	C₂×C₄×C₁₀₇	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₁₀₇	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₀₇	#41	1	2	yes	—
53	C₂×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
54	C₂²⋊D₁₀₇	—	2	8	no	—
55	C₂×Dic₁₀₇	C₂²×C₄	1	4	yes	—
56	D₄₂₈	—	2	4	no	—
57	D₄₂₈	—	4	4	no	—
58	C₄×D₁₀₇	—	2	4	no	—
59	C₄×D₁₀₇	—	2	4	no	—
60	Dic₂₁₄	—	2	4	no	—
61	C₈×C₁₀₇	—	2	4	no	—
62	C₁₀₇⋊C₈	—	2	2	no	—
63	C₁₀₇⋊C₈	C₂×D₄	1	4	yes	—
64	C₂²×C₁₀₇	—	2	2	no	—
65	C₂²×C₁₀₇	C₄×D₄	1	2	yes	—
66	C₂²×C₁₀₇	C₂³×C₄	1	1	yes	characteristic
67	C₂×D₁₀₇	—	2	4	no	—
68	C₂×D₁₀₇	—	2	8	no	—
69	C₄×C₁₀₇	—	2	2	no	—
70	C₄×C₁₀₇	—	2	2	no	—
71	C₄×C₁₀₇	C₄×D₄	1	2	yes	—
72	C₄×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
73	C₄×C₁₀₇	C₂×#41	1	1	yes	characteristic
74	Dic₁₀₇	—	2	4	no	—
75	C₂×C₁₀₇	—	2	2	no	—
76	C₂×C₁₀₇	C₂×C₄×D₄	1	1	yes	characteristic
77	C₂×C₁₀₇	#568	1	2	yes	—
78	D₁₀₇	—	4	4	no	—
79	#2932	—	107	1	no	maximal, Sylow
80	C₁₀₇	#2932	1	1	yes	characteristic, complemented, Sylow
81	C₂×#131	—	107	2	no	—
82	#532	—	107	1	no	—
83	#435	—	107	8	no	—
84	#423	D₁₀₇	1	1	yes	characteristic, complemented
85	#408	—	107	2	no	—
86	C₂×#97	—	107	1	no	—
87	C₂×#41	—	107	2	no	—
88	#131	—	214	8	no	—
89	C₂×M₁₆	—	107	4	no	—
90	C₂×M₁₆	C₂×D₁₀₇	1	2	yes	—
91	C₂²×C₈	—	107	1	no	—
92	C₂²×C₈	—	107	1	no	—
93	C₂²×C₈	—	107	4	no	—
94	C₄⋊Q₈	—	107	1	no	—
95	#127	—	107	1	no	—
96	#124	—	107	2	no	—
97	#121	—	107	4	no	—
98	C₄×D₄	—	107	4	no	—
99	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
100	C₄⋊C₈	C₂×D₁₀₇	1	4	yes	—
101	C₂²⋊C₈	—	107	8	no	—
102	#97	—	107	4	no	—
103	#41	—	214	4	no	—
104	#41	—	214	4	no	—
105	C₂×Q₈	—	107	2	no	—
106	C₂×D₄	—	107	4	no	—
107	C₂×D₄	—	107	4	no	—
108	C₂×D₄	—	107	2	no	—
109	C₂²×C₄	—	107	4	no	—
110	C₂²×C₄	C₄×D₁₀₇	1	2	yes	—
111	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
112	M₁₆	—	2	4	no	—
113	M₁₆	—	214	8	no	—
114	C₂×C₈	—	107	4	no	—
115	C₂×C₈	—	107	2	no	—
116	C₂×C₈	—	107	2	no	—
117	C₂×C₈	—	214	8	no	—
118	C₂×C₈	—	214	2	no	—
119	C₂×C₈	C₂²×D₁₀₇	1	4	yes	—
120	C₄⋊C₄	—	107	4	no	—
121	C₂²⋊C₄	—	107	8	no	—
122	C₄²	C₄×D₁₀₇	1	1	yes	characteristic
123	C₄²	C₄×D₁₀₇	1	1	yes	characteristic
124	C₄²	C₂²×D₁₀₇	1	2	yes	—
125	C₂³	—	107	4	no	—
126	C₂³	C₂×C₄×D₁₀₇	1	1	yes	characteristic
127	Q₈	—	214	4	no	—
128	D₄	—	428	4	no	—
129	D₄	—	214	8	no	—
130	D₄	—	214	4	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	107	4	no	—
133	C₂×C₄	—	214	4	no	—
134	C₂×C₄	—	214	4	no	—
135	C₂×C₄	C₂×C₄×D₁₀₇	1	2	yes	—
136	C₂×C₄	C₂×C₄×D₁₀₇	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
138	C₂×C₄	D₄×D₁₀₇	1	2	yes	—
139	C₂×C₄	#13133819	1	2	yes	—
140	C₂×C₄	C₂³×D₁₀₇	1	1	yes	characteristic
141	C₈	—	2	4	no	—
142	C₈	—	107	4	no	—
143	C₈	—	214	2	no	—
144	C₂²	—	2	2	no	—
145	C₂²	—	214	8	no	—
146	C₂²	—	214	4	no	—
147	C₂²	#16812071	1	2	yes	—
148	C₂²	C₂²×C₄×D₁₀₇	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	214	4	no	—
152	C₄	#16812071	1	2	yes	—
153	C₄	C₂×D₄×D₁₀₇	1	1	yes	characteristic
154	C₄	C₂×#13133819	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	428	4	no	—
157	C₂	C₂×#16812071	1	1	yes	characteristic
158	C₂	#20293393	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812117

C₂

yes

maximal, complemented

#20293177

C₂

yes

maximal, characteristic

#20292697

C₂

yes

maximal, complemented

#20292595

C₂

yes

maximal, complemented

C₂×#16812020

C₂

yes

maximal, characteristic, complemented

#423×C₁₀₇

C₂

yes

maximal, characteristic, complemented

C₂×#13133818

C₂²

yes

—

#16812117

—

C₂×#13133801

C₂²

yes

—

C₂²×C₁₀₇⋊C₈

C₂²

yes

characteristic

C₂²×C₁₀₇⋊C₈

C₂²

yes

characteristic

C₂×C₈×D₁₀₇

C₂²

yes

—

#16812109

C₄

yes

characteristic

#16812106

C₄

yes

characteristic

#16812097

C₄

yes

—

#16812083

C₄

yes

—

C₄×D₄₂₈

C₂²

yes

—

(C₂×D₁₀₇)⋊C₈

C₂²

yes

—

#16812020

C₂²

yes

—

C₂×M₁₆×C₁₀₇

C₂²

yes

—

C₂×C₄²×C₁₀₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₇

C₂²

yes

—

#13133818

—

#13133818

—

C₂×Dic₂₁₄

C₂×C₄

yes

—

C₂×D₄₂₈

C₂×C₄

yes

—

C₂×D₄₂₈

C₂×C₄

yes

—

C₂×C₂²⋊D₁₀₇

C₂×C₄

yes

—

C₂×C₄×D₁₀₇

C₂³

yes

—

#13133801

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

D₄

yes

complemented

C₂×C₁₀₇⋊C₈

C₂³

yes

—

C₂×C₁₀₇⋊C₈

C₂³

yes

—

C₈×D₁₀₇

—

C₄⋊Dic₁₀₇

C₂×C₄

yes

—

(C₂×D₁₀₇)⋊C₄

C₂×C₄

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

C₂²×D₁₀₇

C₂²×C₄

yes

—

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

C₂×D₄

yes

—

C₂×C₄×C₁₀₇

#41

yes

—

C₂×C₄×C₁₀₇

C₂⁴

yes

characteristic

C₂²⋊D₁₀₇

—

C₂×Dic₁₀₇

C₂²×C₄

yes

—

D₄₂₈

—

D₄₂₈

—

C₄×D₁₀₇

—

C₄×D₁₀₇

—

Dic₂₁₄

—

C₈×C₁₀₇

—

C₁₀₇⋊C₈

—

C₁₀₇⋊C₈

C₂×D₄

yes

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

C₄×D₄

yes

—

C₂²×C₁₀₇

C₂³×C₄

yes

characteristic

C₂×D₁₀₇

—

C₂×D₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₄×D₄

yes

—

C₄×C₁₀₇

C₂²×D₄

yes

characteristic

C₄×C₁₀₇

C₂×#41

yes

characteristic

Dic₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

C₂×C₄×D₄

yes

characteristic

C₂×C₁₀₇

#568

yes

—

D₁₀₇

—

#2932

—

107

maximal, Sylow

C₁₀₇

#2932

yes

characteristic, complemented, Sylow

C₂×#131

—

107

—

#532

—

107

—

#435

—

107

—

#423

D₁₀₇

yes

characteristic, complemented

#408

—

107

—

C₂×#97

—

107

—

C₂×#41

—

107

—

#131

—

214

—

C₂×M₁₆

—

107

—

C₂×M₁₆

C₂×D₁₀₇

yes

—

C₂²×C₈

—

107

—

C₂²×C₈

—

107

—

C₂²×C₈

—

107

—

C₄⋊Q₈

—

107

—

#127

—

107

—

#124

—

107

—

#121

—

107

—

C₄×D₄

—

107

—

C₂×C₄²

C₂×D₁₀₇

yes

characteristic

100

C₄⋊C₈

C₂×D₁₀₇

yes

—

101

C₂²⋊C₈

—

107

—

102

#97

—

107

—

103

#41

—

214

—

104

#41

—

214

—

105

C₂×Q₈

—

107

—

106

C₂×D₄

—

107

—

107

C₂×D₄

—

107

—

108

C₂×D₄

—

107

—

109

C₂²×C₄

—

107

—

110

C₂²×C₄

C₄×D₁₀₇

yes

—

111

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

112

M₁₆

—

113

M₁₆

—

214

—

114

C₂×C₈

—

107

—

115

C₂×C₈

—

107

—

116

C₂×C₈

—

107

—

117

C₂×C₈

—

214

—

118

C₂×C₈

—

214

—

119

C₂×C₈

C₂²×D₁₀₇

yes

—

120

C₄⋊C₄

—

107

—

121

C₂²⋊C₄

—

107

—

122

C₄²

C₄×D₁₀₇

yes

characteristic

123

C₄²

C₄×D₁₀₇

yes

characteristic

124

C₄²

C₂²×D₁₀₇

yes

—

125

C₂³

—

107

—

126

C₂³

C₂×C₄×D₁₀₇

yes

characteristic

127

Q₈

—

214

—

128

D₄

—

428

—

129

D₄

—

214

—

130

D₄

—

214

—

131

C₂×C₄

—

132

C₂×C₄

—

107

—

133

C₂×C₄

—

214

—

134

C₂×C₄

—

214

—

135

C₂×C₄

C₂×C₄×D₁₀₇

yes

—

136

C₂×C₄

C₂×C₄×D₁₀₇

yes

—

137

C₂×C₄

C₂×C₄×D₁₀₇

yes

characteristic

138

C₂×C₄

D₄×D₁₀₇

yes

—

139

C₂×C₄

#13133819

yes

—

140

C₂×C₄

C₂³×D₁₀₇

yes

characteristic

141

C₈

—

142

C₈

—

107

—

143

C₈

—

214

—

144

C₂²

—

145

C₂²

—

214

—

146

C₂²

—

214

—

147

C₂²

#16812071

yes

—

148

C₂²

C₂²×C₄×D₁₀₇

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

214

—

152

C₄

#16812071

yes

—

153

C₄

C₂×D₄×D₁₀₇

yes

characteristic

154

C₄

C₂×#13133819

yes

characteristic

155

C₂

—

156

C₂

—

428

—

157

C₂

C₂×#16812071

yes

characteristic

158

C₂

#20293393

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₇
commutator, socle	C₂²×C₁₀₇	C₂³×C₄
Fitting	#423×C₁₀₇	C₂
prime radical	#20293177	C₂
cosocle	C₂×C₄×C₁₀₇	C₂⁴
hypercenter	#423	D₁₀₇
hypocenter	C₁₀₇	#2932
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

#20293177

C₂

cosocle

C₂×C₄×C₁₀₇

C₂⁴

D₁₀₇

C₁₀₇

G ▷ C₂²×C₁₀₇ ▷ C₁₀₇

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
1. C₂×#16812117	C₂	semidirect
3. #20292697	C₂	semidirect
4. #20292595	C₂	semidirect
5. C₂×#16812020	C₂	semidirect
6. #423×C₁₀₇	C₂	semidirect
32. C₂×C₁₀₇⋊C₈	D₄	semidirect
80. C₁₀₇	#2932	semidirect
84. #423	D₁₀₇	semidirect

H ≅ G/N

type

1. C₂×#16812117

C₂

semidirect

3. #20292697

C₂

semidirect

4. #20292595

C₂

semidirect

5. C₂×#16812020

C₂

semidirect

6. #423×C₁₀₇

C₂

semidirect

32. C₂×C₁₀₇⋊C₈

D₄

semidirect

80. C₁₀₇

#2932

semidirect

84. #423

D₁₀₇

semidirect

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

subgroup	index
C₂×#16812117	2
#20293177	2
#20292697	2
#20292595	2
C₂×#16812020	2
#423×C₁₀₇	2
#2932	107

subgroup

index

#423×C₁₀₇

107

Subgroups — #28345178

Conjugacy classes of subgroups

Complements

Maximal subgroups