Group #28312348 — subgroups

Subgroups — #28312348

Order 13568.

290 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 108 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×#16811555	C₂	1	2	yes	maximal, complemented
2	#20289831	C₂	1	4	yes	maximal, complemented
3	C₂×#16811041	C₂	1	1	yes	maximal, characteristic
4	#16811555	—	2	2	no	—
5	#16811555	C₂²	1	4	yes	—
6	C₂×#13133267	—	2	2	no	—
7	C₂×#13133223	C₄	1	1	yes	characteristic
8	#16811041	—	2	2	no	—
9	#16811041	—	2	2	no	—
10	C₂×#13133156	C₂²	1	2	yes	—
11	C₂²×#11719908	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×D₅₃	C₄	1	2	yes	—
13	C₂×M₃₂×C₅₃	C₄	1	1	yes	characteristic
14	#13133267	—	2	4	no	—
15	C₂×#11719945	C₈	1	1	yes	characteristic
16	#13133223	—	2	2	no	—
17	#13133223	—	2	2	no	—
18	C₂×#11719917	C₈	1	1	yes	characteristic
19	C₂×#11719912	C₈	1	2	yes	—
20	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
21	#13133156	—	2	4	no	—
22	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
23	C₂×#11719908	D₄	1	2	yes	—
24	C₂×#11719908	D₄	1	2	yes	—
25	C₂×#11719908	C₂³	1	1	yes	characteristic
26	C₁₆×D₅₃	—	2	2	no	—
27	C₁₆×D₅₃	—	2	2	no	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	M₃₂×C₅₃	—	2	2	no	—
30	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₂²⋊D₅₃	C₁₆	1	2	yes	—
33	#11719945	—	2	2	no	—
34	#11719945	—	2	2	no	—
35	C₂×D₂₁₂	C₁₆	1	1	yes	characteristic
36	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
37	C₂×Dic₁₀₆	C₁₆	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
39	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
40	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
41	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	#11719917	—	2	2	no	—
43	C₂×#88322	C₂×C₈	1	2	yes	—
44	#11719912	—	2	4	no	—
45	C₈×D₅₃	—	2	2	no	—
46	C₈×D₅₃	—	2	2	no	—
47	C₁₆×C₅₃	—	2	2	no	—
48	#11719908	C₂²⋊C₄	1	1	yes	characteristic
49	#11719908	C₂²⋊C₄	1	1	yes	characteristic
50	#11719908	C₂×D₄	1	2	yes	—
51	C₂³×C₅₃	C₂×C₁₆	1	1	yes	characteristic
52	C₂²×D₅₃	C₂×C₁₆	1	2	yes	—
53	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
54	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
55	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
56	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
57	C₂²⋊D₅₃	—	2	4	no	—
58	C₂×Dic₅₃	C₂×C₁₆	1	2	yes	—
59	D₂₁₂	—	2	2	no	—
60	C₄×D₅₃	—	2	2	no	—
61	C₄×D₅₃	—	2	2	no	—
62	Dic₁₀₆	—	2	2	no	—
63	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
64	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
66	#88322	—	2	2	no	—
67	#13556	—	53	1	no	maximal, Sylow
68	C₂²×C₅₃	—	4	1	no	—
69	C₂²×C₅₃	C₂²⋊C₁₆	1	2	yes	—
70	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
71	C₂×D₅₃	—	2	2	no	—
72	C₂×D₅₃	—	4	2	no	—
73	C₄×C₅₃	C₂²⋊C₁₆	1	2	yes	—
74	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
75	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
76	Dic₅₃	—	2	2	no	—
77	C₂×#504	—	53	1	no	—
78	C₂×M₆₄	—	53	2	no	—
79	#1400	—	53	4	no	—
80	C₂×C₅₃	—	2	1	no	—
81	C₂×C₅₃	—	2	2	no	—
82	C₂×C₅₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
83	D₅₃	—	4	2	no	—
84	C₂×#131	—	53	1	no	—
85	#504	—	106	2	no	—
86	#504	—	106	2	no	—
87	C₂×M₃₂	—	53	2	no	—
88	C₂×M₃₂	#6157	1	1	yes	characteristic
89	C₂²×C₁₆	—	53	1	no	—
90	C₂²×C₁₆	—	53	2	no	—
91	M₆₄	—	53	4	no	—
92	M₆₄	—	106	2	no	—
93	C₂×C₃₂	—	106	2	no	—
94	C₅₃	#13556	1	1	yes	characteristic, complemented, Sylow
95	C₂×#41	—	53	1	no	—
96	#131	—	106	2	no	—
97	#131	—	106	2	no	—
98	C₂×M₁₆	—	53	1	no	—
99	C₂×M₁₆	—	53	2	no	—
100	C₂²×C₈	—	53	2	no	—
101	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
102	M₃₂	—	2	2	no	—
103	M₃₂	—	106	4	no	—
104	C₂×C₁₆	—	53	2	no	—
105	C₂×C₁₆	—	53	2	no	—
106	C₂×C₁₆	—	53	1	no	—
107	C₂×C₁₆	—	53	1	no	—
108	C₂×C₁₆	—	106	2	no	—
109	C₂×C₁₆	—	106	2	no	—
110	C₂×C₁₆	C₂×#6157	1	2	yes	—
111	C₃₂	—	106	4	no	—
112	#41	—	106	2	no	—
113	#41	—	106	2	no	—
114	C₂×Q₈	—	53	1	no	—
115	C₂×D₄	—	53	2	no	—
116	C₂×D₄	—	53	1	no	—
117	C₂²×C₄	—	53	2	no	—
118	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
119	M₁₆	—	106	2	no	—
120	M₁₆	—	106	4	no	—
121	C₂×C₈	—	53	2	no	—
122	C₂×C₈	—	106	2	no	—
123	C₂×C₈	—	106	2	no	—
124	C₂×C₈	D₅₃⋊C₈	1	1	yes	characteristic
125	C₂×C₈	#11719936	1	2	yes	—
126	C₂×C₈	C₂²⋊#6157	1	2	yes	—
127	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
128	C₁₆	—	2	2	no	—
129	C₁₆	—	53	2	no	—
130	C₁₆	—	53	1	no	—
131	C₁₆	—	53	1	no	—
132	C₂³	—	53	2	no	—
133	C₂³	D₅₃⋊C₁₆	1	1	yes	characteristic
134	Q₈	—	106	2	no	—
135	D₄	—	106	4	no	—
136	D₄	—	106	2	no	—
137	C₂×C₄	—	53	2	no	—
138	C₂×C₄	—	106	2	no	—
139	C₂×C₄	—	106	2	no	—
140	C₂×C₄	#13133274	1	2	yes	—
141	C₂×C₄	D₅₃⋊C₁₆	1	1	yes	characteristic
142	C₂×C₄	#13133292	1	2	yes	—
143	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
144	C₈	—	106	2	no	—
145	C₈	#13133274	1	2	yes	—
146	C₈	C₂×C₂²⋊#6157	1	1	yes	characteristic
147	C₈	C₂×#11719936	1	1	yes	characteristic
148	C₂²	—	4	1	no	—
149	C₂²	—	106	2	no	—
150	C₂²	—	212	2	no	—
151	C₂²	#16811518	1	2	yes	—
152	C₂²	C₂×D₅₃⋊C₁₆	1	1	yes	characteristic
153	C₄	—	106	2	no	—
154	C₄	#16811518	1	2	yes	—
155	C₄	C₂×#13133274	1	1	yes	characteristic
156	C₄	C₂×#13133292	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	2	no	—
159	C₂	—	212	2	no	—
160	C₂	C₂×#16811518	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16811555

C₂

yes

maximal, complemented

#20289831

C₂

yes

maximal, complemented

C₂×#16811041

C₂

yes

maximal, characteristic

#16811555

—

#16811555

C₂²

yes

—

C₂×#13133267

—

C₂×#13133223

C₄

yes

—

—

C₂²

yes

—

C₂²×#11719908

C₂²

yes

characteristic

C₂×C₁₆×D₅₃

C₄

yes

—

C₂×M₃₂×C₅₃

C₄

yes

characteristic

#13133267

—

C₂×#11719945

C₈

yes

—

—

C₈

yes

characteristic

C₂×#11719912

C₈

yes

—

C₂×C₈×D₅₃

C₂×C₄

yes

—

#13133156

—

C₂×#11719908

C₂×C₄

yes

characteristic

C₂×#11719908

D₄

yes

—

C₂×#11719908

D₄

yes

—

C₂×#11719908

C₂³

yes

characteristic

C₁₆×D₅₃

—

C₁₆×D₅₃

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₂²⋊D₅₃

C₁₆

yes

—

#11719945

—

#11719945

—

C₂×D₂₁₂

C₁₆

yes

characteristic

C₂×C₄×D₅₃

C₂×C₈

yes

—

C₂×Dic₁₀₆

C₁₆

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂×C₈

yes

characteristic

C₂×C₈×C₅₃

M₁₆

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

#11719917

—

C₂×#88322

C₂×C₈

yes

—

#11719912

—

C₈×D₅₃

—

C₈×D₅₃

—

C₁₆×C₅₃

—

#11719908

C₂²⋊C₄

yes

characteristic

#11719908

C₂²⋊C₄

yes

characteristic

#11719908

C₂×D₄

yes

—

C₂³×C₅₃

C₂×C₁₆

yes

characteristic

C₂²×D₅₃

C₂×C₁₆

yes

—

C₂×C₄×C₅₃

C₂²⋊C₈

yes

—

C₂×C₄×C₅₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅₃

M₃₂

yes

—

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂²⋊D₅₃

—

C₂×Dic₅₃

C₂×C₁₆

yes

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

C₂²⋊C₈

yes

—

C₈×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₅₃

C₂×M₁₆

yes

—

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²⋊C₁₆

yes

—

C₂²×C₅₃

C₂²×C₁₆

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

C₂²⋊C₁₆

yes

—

C₄×C₅₃

C₂×C₂²⋊C₈

yes

characteristic

C₄×C₅₃

C₂×M₃₂

yes

characteristic

Dic₅₃

—

C₂×#504

—

C₂×M₆₄

—

#1400

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂×C₂²⋊C₁₆

yes

characteristic

D₅₃

—

C₂×#131

—

#504

—

106

—

#504

—

106

—

C₂×M₃₂

—

C₂×M₃₂

#6157

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

—

M₆₄

—

M₆₄

—

106

—

C₂×C₃₂

—

106

—

C₅₃

#13556

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

106

—

#131

—

106

—

C₂×M₁₆

—

C₂×M₁₆

—

100

C₂²×C₈

—

101

C₂²×C₈

C₂×#6157

yes

characteristic

102

M₃₂

—

103

M₃₂

—

106

—

104

C₂×C₁₆

—

105

C₂×C₁₆

—

106

C₂×C₁₆

—

107

C₂×C₁₆

—

108

C₂×C₁₆

—

106

—

109

C₂×C₁₆

—

106

—

110

C₂×C₁₆

C₂×#6157

yes

—

111

C₃₂

—

106

—

112

#41

—

106

—

113

#41

—

106

—

114

C₂×Q₈

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂²×C₄

—

118

C₂²×C₄

D₅₃⋊C₈

yes

characteristic

119

M₁₆

—

106

—

120

M₁₆

—

106

—

121

C₂×C₈

—

122

C₂×C₈

—

106

—

123

C₂×C₈

—

106

—

124

C₂×C₈

D₅₃⋊C₈

yes

characteristic

125

C₂×C₈

#11719936

yes

—

126

C₂×C₈

C₂²⋊#6157

yes

—

127

C₂×C₈

C₂²×#6157

yes

characteristic

128

C₁₆

—

129

C₁₆

—

130

C₁₆

—

131

C₁₆

—

132

C₂³

—

133

C₂³

D₅₃⋊C₁₆

yes

characteristic

134

Q₈

—

106

—

135

D₄

—

106

—

136

D₄

—

106

—

137

C₂×C₄

—

138

C₂×C₄

—

106

—

139

C₂×C₄

—

106

—

140

C₂×C₄

#13133274

yes

—

141

C₂×C₄

D₅₃⋊C₁₆

yes

characteristic

142

C₂×C₄

#13133292

yes

—

143

C₂×C₄

C₂×D₅₃⋊C₈

yes

characteristic

144

C₈

—

106

—

145

C₈

#13133274

yes

—

146

C₈

C₂×C₂²⋊#6157

yes

characteristic

147

C₈

C₂×#11719936

yes

characteristic

148

C₂²

—

149

C₂²

—

106

—

150

C₂²

—

212

—

151

C₂²

#16811518

yes

—

152

C₂²

C₂×D₅₃⋊C₁₆

yes

characteristic

153

C₄

—

106

—

154

C₄

#16811518

yes

—

155

C₄

C₂×#13133274

yes

characteristic

156

C₄

C₂×#13133292

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

—

212

—

160

C₂

C₂×#16811518

yes

characteristic

	subgroup	G/H
center	C₈	C₂×C₂²⋊#6157
commutator	C₂²×C₅₃	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₂×M₃₂×C₅₃	C₄
socle	C₂×C₅₃	C₂×C₂²⋊C₁₆
prime radical	C₂×#11719945	C₈
cosocle	C₂×#11719908	C₂³
hypercenter	C₂×M₃₂	#6157
hypocenter	C₅₃	#13556
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₈

C₂×C₂²⋊#6157

commutator

C₂²×C₅₃

C₂²×C₁₆

Frattini

C₂×C₈

C₂²×#6157

Fitting

C₂×M₃₂×C₅₃

C₄

socle

C₂×C₅₃

C₂×C₂²⋊C₁₆

C₈

C₂³

C₂×M₃₂

C₅₃

G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
1. C₂×#16811555	C₂	semidirect
2. #20289831	C₂	semidirect
94. C₅₃	#13556	semidirect

H ≅ G/N

type

1. C₂×#16811555

C₂

semidirect

2. #20289831

C₂

semidirect

94. C₅₃

#13556

semidirect

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

subgroup	index
C₂×#16811555	2
#20289831	2
C₂×#16811041	2
#13556	53

subgroup

index

Subgroups — #28312348

Conjugacy classes of subgroups

Complements

Maximal subgroups