Group #28312342 — subgroups

Subgroups — #28312342

Order 13568.

298 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 112 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	1	yes	maximal, characteristic, complemented
2	C₂²×#13133267	C₂	1	1	yes	maximal, characteristic, complemented
3	#20289829	C₂	1	4	yes	maximal, complemented
4	C₂×#16811041	C₂	1	1	yes	maximal, characteristic
5	#16811555	—	2	2	no	—
6	C₂×#13133267	—	2	2	no	—
7	C₂×#13133267	C₂²	1	2	yes	—
8	C₂×#13133267	C₂²	1	2	yes	—
9	C₂×#13133223	C₄	1	1	yes	characteristic
10	#16811041	—	2	2	no	—
11	#16811041	—	2	2	no	—
12	C₂×#13133156	C₄	1	2	yes	—
13	C₂²×#11719908	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
15	C₂×M₃₂×C₅₃	C₄	1	1	yes	characteristic
16	#13133267	—	2	2	no	—
17	#13133267	—	2	2	no	—
18	C₂×#11719945	C₈	1	1	yes	characteristic
19	#13133223	—	2	2	no	—
20	#13133223	—	2	2	no	—
21	C₂×#11719917	C₈	1	1	yes	characteristic
22	C₂×#11719912	C₈	1	2	yes	—
23	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
24	#13133156	—	2	4	no	—
25	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
26	C₂×#11719908	D₄	1	2	yes	—
27	C₂×#11719908	D₄	1	2	yes	—
28	C₂×#11719908	C₂³	1	1	yes	characteristic
29	C₁₆×D₅₃	—	2	2	no	—
30	C₁₆×D₅₃	—	2	2	no	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	M₃₂×C₅₃	—	2	2	no	—
33	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₂²⋊D₅₃	C₁₆	1	2	yes	—
36	#11719945	—	2	2	no	—
37	#11719945	—	2	2	no	—
38	C₂×D₂₁₂	C₁₆	1	1	yes	characteristic
39	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
40	C₂×Dic₁₀₆	C₁₆	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
42	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
43	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
44	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	#11719917	—	2	2	no	—
46	C₂×#88322	C₂×C₈	1	2	yes	—
47	#11719912	—	2	4	no	—
48	C₈×D₅₃	—	2	2	no	—
49	C₈×D₅₃	—	2	2	no	—
50	C₁₆×C₅₃	—	2	2	no	—
51	#11719908	C₂²⋊C₄	1	2	yes	—
52	#11719908	C₂×D₄	1	1	yes	characteristic
53	#11719908	C₂×D₄	1	1	yes	characteristic
54	C₂³×C₅₃	C₂×C₁₆	1	1	yes	characteristic
55	C₂²×D₅₃	C₂×C₁₆	1	2	yes	—
56	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
57	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
58	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
59	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
60	C₂²⋊D₅₃	—	2	4	no	—
61	C₂×Dic₅₃	C₂×C₁₆	1	2	yes	—
62	D₂₁₂	—	2	2	no	—
63	C₄×D₅₃	—	2	2	no	—
64	C₄×D₅₃	—	2	2	no	—
65	Dic₁₀₆	—	2	2	no	—
66	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
67	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
69	#88322	—	2	2	no	—
70	#13554	—	53	1	no	maximal, Sylow
71	C₂²×C₅₃	—	2	2	no	—
72	C₂²×C₅₃	C₂²⋊C₁₆	1	2	yes	—
73	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
74	C₂×D₅₃	—	2	2	no	—
75	C₂×D₅₃	—	2	4	no	—
76	C₄×C₅₃	C₂²⋊C₁₆	1	2	yes	—
77	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
78	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
79	Dic₅₃	—	2	2	no	—
80	C₂×#504	—	53	1	no	—
81	C₂×M₆₄	—	53	1	no	—
82	C₂²×C₃₂	—	53	1	no	—
83	C₂²⋊C₃₂	—	53	4	no	—
84	C₂×C₅₃	—	2	2	no	—
85	C₂×C₅₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
86	C₂×C₅₃	#2258	1	2	yes	—
87	D₅₃	—	4	2	no	—
88	C₂×#131	—	53	1	no	—
89	#504	—	106	2	no	—
90	#504	—	106	2	no	—
91	C₂×M₃₂	—	53	2	no	—
92	C₂×M₃₂	#6157	1	1	yes	characteristic
93	C₂²×C₁₆	—	53	1	no	—
94	C₂²×C₁₆	—	53	2	no	—
95	M₆₄	—	106	2	no	—
96	C₂×C₃₂	—	53	2	no	—
97	C₂×C₃₂	—	53	2	no	—
98	C₂×C₃₂	—	106	2	no	—
99	C₅₃	#13554	1	1	yes	characteristic, complemented, Sylow
100	C₂×#41	—	53	1	no	—
101	#131	—	106	2	no	—
102	#131	—	106	2	no	—
103	C₂×M₁₆	—	53	1	no	—
104	C₂×M₁₆	—	53	2	no	—
105	C₂²×C₈	—	53	2	no	—
106	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
107	M₃₂	—	2	2	no	—
108	M₃₂	—	106	4	no	—
109	C₂×C₁₆	—	53	2	no	—
110	C₂×C₁₆	—	53	2	no	—
111	C₂×C₁₆	—	53	1	no	—
112	C₂×C₁₆	—	53	1	no	—
113	C₂×C₁₆	—	106	2	no	—
114	C₂×C₁₆	—	106	2	no	—
115	C₂×C₁₆	C₂×#6157	1	2	yes	—
116	C₃₂	—	106	2	no	—
117	C₃₂	—	106	2	no	—
118	#41	—	106	2	no	—
119	#41	—	106	2	no	—
120	C₂×Q₈	—	53	1	no	—
121	C₂×D₄	—	53	2	no	—
122	C₂×D₄	—	53	1	no	—
123	C₂²×C₄	—	53	2	no	—
124	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
125	M₁₆	—	106	2	no	—
126	M₁₆	—	106	4	no	—
127	C₂×C₈	—	53	2	no	—
128	C₂×C₈	—	106	2	no	—
129	C₂×C₈	—	106	2	no	—
130	C₂×C₈	D₅₃⋊C₈	1	1	yes	characteristic
131	C₂×C₈	#11719936	1	2	yes	—
132	C₂×C₈	C₂²⋊#6157	1	2	yes	—
133	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
134	C₁₆	—	2	2	no	—
135	C₁₆	—	53	1	no	—
136	C₁₆	—	53	1	no	—
137	C₁₆	—	53	2	no	—
138	C₂³	—	53	2	no	—
139	C₂³	D₅₃⋊C₁₆	1	1	yes	characteristic
140	Q₈	—	106	2	no	—
141	D₄	—	106	4	no	—
142	D₄	—	106	2	no	—
143	C₂×C₄	—	53	2	no	—
144	C₂×C₄	—	106	2	no	—
145	C₂×C₄	—	106	2	no	—
146	C₂×C₄	#13133274	1	2	yes	—
147	C₂×C₄	D₅₃⋊C₁₆	1	1	yes	characteristic
148	C₂×C₄	#13133292	1	2	yes	—
149	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
150	C₈	—	106	2	no	—
151	C₈	#13133274	1	2	yes	—
152	C₈	C₂×C₂²⋊#6157	1	1	yes	characteristic
153	C₈	C₂×#11719936	1	1	yes	characteristic
154	C₂²	—	2	2	no	—
155	C₂²	—	106	4	no	—
156	C₂²	—	106	2	no	—
157	C₂²	#16811518	1	2	yes	—
158	C₂²	C₂×D₅₃⋊C₁₆	1	1	yes	characteristic
159	C₄	—	106	2	no	—
160	C₄	#16811518	1	2	yes	—
161	C₄	C₂×#13133274	1	1	yes	characteristic
162	C₄	C₂×#13133292	1	1	yes	characteristic
163	C₂	—	2	2	no	—
164	C₂	—	212	2	no	—
165	C₂	C₂×#16811518	1	1	yes	characteristic
166	C₂	#20291139	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16811555

C₂

yes

maximal, characteristic, complemented

C₂²×#13133267

C₂

yes

maximal, characteristic, complemented

#20289829

C₂

yes

maximal, complemented

C₂×#16811041

C₂

yes

maximal, characteristic

#16811555

—

C₂×#13133267

—

C₂×#13133267

C₂²

yes

—

C₂×#13133267

C₂²

yes

—

C₂×#13133223

C₄

yes

—

—

C₄

yes

—

C₂²×#11719908

C₂²

yes

characteristic

C₂×C₁₆×D₅₃

C₂²

yes

—

C₂×M₃₂×C₅₃

C₄

yes

—

—

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

—

#11719908

C₂×D₄

yes

characteristic

#11719908

C₂×D₄

yes

characteristic

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

—

C₂²×C₃₂

—

C₂²⋊C₃₂

—

C₂×C₅₃

—

C₂×C₅₃

C₂×C₂²⋊C₁₆

yes

characteristic

C₂×C₅₃

#2258

yes

—

D₅₃

—

C₂×#131

—

#504

—

106

—

#504

—

106

—

C₂×M₃₂

—

C₂×M₃₂

#6157

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

—

M₆₄

—

106

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

106

—

C₅₃

#13554

yes

characteristic, complemented, Sylow

100

C₂×#41

—

101

#131

—

106

—

102

#131

—

106

—

103

C₂×M₁₆

—

104

C₂×M₁₆

—

105

C₂²×C₈

—

106

C₂²×C₈

C₂×#6157

yes

characteristic

107

M₃₂

—

108

M₃₂

—

106

—

109

C₂×C₁₆

—

110

C₂×C₁₆

—

111

C₂×C₁₆

—

112

C₂×C₁₆

—

113

C₂×C₁₆

—

106

—

114

C₂×C₁₆

—

106

—

115

C₂×C₁₆

C₂×#6157

yes

—

116

C₃₂

—

106

—

117

C₃₂

—

106

—

118

#41

—

106

—

119

#41

—

106

—

120

C₂×Q₈

—

121

C₂×D₄

—

122

C₂×D₄

—

123

C₂²×C₄

—

124

C₂²×C₄

D₅₃⋊C₈

yes

characteristic

125

M₁₆

—

106

—

126

M₁₆

—

106

—

127

C₂×C₈

—

128

C₂×C₈

—

106

—

129

C₂×C₈

—

106

—

130

C₂×C₈

D₅₃⋊C₈

yes

characteristic

131

C₂×C₈

#11719936

yes

—

132

C₂×C₈

C₂²⋊#6157

yes

—

133

C₂×C₈

C₂²×#6157

yes

characteristic

134

C₁₆

—

135

C₁₆

—

136

C₁₆

—

137

C₁₆

—

138

C₂³

—

139

C₂³

D₅₃⋊C₁₆

yes

characteristic

140

Q₈

—

106

—

141

D₄

—

106

—

142

D₄

—

106

—

143

C₂×C₄

—

144

C₂×C₄

—

106

—

145

C₂×C₄

—

106

—

146

C₂×C₄

#13133274

yes

—

147

C₂×C₄

D₅₃⋊C₁₆

yes

characteristic

148

C₂×C₄

#13133292

yes

—

149

C₂×C₄

C₂×D₅₃⋊C₈

yes

characteristic

150

C₈

—

106

—

151

C₈

#13133274

yes

—

152

C₈

C₂×C₂²⋊#6157

yes

characteristic

153

C₈

C₂×#11719936

yes

characteristic

154

C₂²

—

155

C₂²

—

106

—

156

C₂²

—

106

—

157

C₂²

#16811518

yes

—

158

C₂²

C₂×D₅₃⋊C₁₆

yes

characteristic

159

C₄

—

106

—

160

C₄

#16811518

yes

—

161

C₄

C₂×#13133274

yes

characteristic

162

C₄

C₂×#13133292

yes

characteristic

163

C₂

—

164

C₂

—

212

—

165

C₂

C₂×#16811518

yes

characteristic

166

C₂

#20291139

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#6157
commutator, socle	C₂²×C₅₃	C₂²×C₁₆
Fitting	C₂×M₃₂×C₅₃	C₄
prime radical	C₂×#11719945	C₈
cosocle	C₂×#11719908	C₂³
hypercenter	C₂×M₃₂	#6157
hypocenter	C₅₃	#13554
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×#6157

commutator, socle

C₂²×C₅₃

C₂²×C₁₆

Fitting

C₂×M₃₂×C₅₃

C₄

C₈

C₂³

C₂×M₃₂

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
1. C₂×#16811555	C₂	semidirect
2. C₂²×#13133267	C₂	semidirect
3. #20289829	C₂	semidirect
99. C₅₃	#13554	semidirect

H ≅ G/N

type

1. C₂×#16811555

C₂

semidirect

2. C₂²×#13133267

C₂

semidirect

3. #20289829

C₂

semidirect

99. C₅₃

#13554

semidirect

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

subgroup	index
C₂×#16811555	2
C₂²×#13133267	2
#20289829	2
C₂×#16811041	2
#13554	53

subgroup

index

Subgroups — #28312342

Conjugacy classes of subgroups

Complements

Maximal subgroups