Group #28312339 — subgroups

Subgroups — #28312339

Order 13568.

330 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 116 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	#20289829	C₂	1	4	yes	maximal, complemented
3	C₂²×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#16811555	—	2	4	no	—
5	C₂×#13133267	C₂²	1	4	yes	complemented
6	C₂²×C₈×D₅₃	C₄	1	1	yes	characteristic
7	C₂²×#11719908	C₂²	1	1	yes	characteristic
8	C₂×C₁₆×D₅₃	—	2	4	no	—
9	C₂×C₁₆×D₅₃	C₄	1	2	yes	—
10	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
11	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
12	#13133267	—	2	4	no	—
13	C₂²×C₄×D₅₃	C₈	1	1	yes	characteristic
14	C₂²×#88322	C₈	1	1	yes	characteristic
15	C₂×C₈×D₅₃	—	2	2	no	—
16	C₂×C₈×D₅₃	—	2	2	no	—
17	C₂×C₈×D₅₃	C₈	1	2	yes	—
18	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
19	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
20	C₂×#11719908	D₄	1	4	yes	—
21	C₂×#11719908	C₂³	1	1	yes	characteristic
22	C₁₆×D₅₃	—	2	4	no	—
23	C₁₆×D₅₃	—	2	4	no	—
24	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₅₃	—	2	2	no	—
26	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
27	C₂³×D₅₃	C₁₆	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
29	C₂²×Dic₅₃	C₁₆	1	1	yes	characteristic
30	C₂×C₄×D₅₃	—	2	2	no	—
31	C₂×C₄×D₅₃	—	2	2	no	—
32	C₂×C₄×D₅₃	C₁₆	1	2	yes	—
33	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
34	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
35	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
37	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×#88322	—	2	2	no	—
39	C₂×#88322	C₂×C₈	1	2	yes	—
40	C₈×D₅₃	—	2	4	no	—
41	C₈×D₅₃	—	2	2	no	—
42	C₈×D₅₃	—	2	2	no	—
43	C₁₆×C₅₃	—	2	2	no	—
44	#11719908	C₂²⋊C₄	1	2	yes	—
45	#11719908	C₂×D₄	1	2	yes	—
46	C₂³×C₅₃	C₂×C₁₆	1	1	yes	characteristic
47	C₂²×D₅₃	—	2	2	no	—
48	C₂²×D₅₃	—	2	4	no	—
49	C₂²×D₅₃	C₂×C₁₆	1	2	yes	—
50	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
51	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
52	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
53	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
54	C₂×Dic₅₃	—	2	2	no	—
55	C₂×Dic₅₃	C₂×C₁₆	1	2	yes	—
56	C₄×D₅₃	—	2	4	no	—
57	C₄×D₅₃	—	2	2	no	—
58	C₄×D₅₃	—	2	2	no	—
59	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
60	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
62	#88322	—	2	2	no	—
63	#13553	—	53	1	no	maximal, Sylow
64	C₂²×C₅₃	—	2	2	no	—
65	C₂²×C₅₃	C₂²⋊C₁₆	1	2	yes	—
66	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
67	C₂×D₅₃	—	2	8	no	—
68	C₂×D₅₃	—	2	2	no	—
69	C₂×D₅₃	—	2	2	no	—
70	C₂×D₅₃	M₆₄	1	4	yes	complemented
71	C₄×C₅₃	C₂²⋊C₁₆	1	2	yes	—
72	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
73	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
74	Dic₅₃	—	2	2	no	—
75	C₂³×C₁₆	—	53	1	no	—
76	C₂×M₆₄	—	53	2	no	—
77	C₂²⋊C₃₂	—	53	4	no	—
78	C₂×C₅₃	—	2	2	no	—
79	C₂×C₅₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
80	C₂×C₅₃	C₂×M₆₄	1	2	yes	—
81	D₅₃	—	2	4	no	—
82	C₂³×C₈	—	53	1	no	—
83	C₂²×C₁₆	—	53	1	no	—
84	C₂²×C₁₆	—	53	2	no	—
85	C₂²×C₁₆	—	53	2	no	—
86	C₂²×C₁₆	—	106	4	no	—
87	C₂²×C₁₆	#6157	1	1	yes	characteristic
88	M₆₄	—	106	4	no	—
89	C₂×C₃₂	—	53	4	no	—
90	C₅₃	#13553	1	1	yes	characteristic, complemented, Sylow
91	C₂³×C₄	—	53	1	no	—
92	C₂²×C₈	—	53	1	no	—
93	C₂²×C₈	—	53	2	no	—
94	C₂²×C₈	—	53	2	no	—
95	C₂²×C₈	—	106	2	no	—
96	C₂²×C₈	—	106	2	no	—
97	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
98	C₂×C₁₆	—	2	2	no	—
99	C₂×C₁₆	—	53	4	no	—
100	C₂×C₁₆	—	53	1	no	—
101	C₂×C₁₆	—	53	1	no	—
102	C₂×C₁₆	—	106	4	no	—
103	C₂×C₁₆	—	106	4	no	—
104	C₂×C₁₆	C₂×#6157	1	2	yes	—
105	C₃₂	—	106	4	no	—
106	C₂⁴	—	53	1	no	—
107	C₂²×C₄	—	53	1	no	—
108	C₂²×C₄	—	53	2	no	—
109	C₂²×C₄	—	53	2	no	—
110	C₂²×C₄	—	106	2	no	—
111	C₂²×C₄	—	106	2	no	—
112	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
113	C₂×C₈	—	53	2	no	—
114	C₂×C₈	—	106	2	no	—
115	C₂×C₈	—	106	4	no	—
116	C₂×C₈	—	106	2	no	—
117	C₂×C₈	—	106	2	no	—
118	C₂×C₈	D₅₃⋊C₈	1	1	yes	characteristic
119	C₂×C₈	#11719936	1	2	yes	—
120	C₂×C₈	C₂²⋊#6157	1	2	yes	—
121	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
122	C₁₆	—	2	2	no	—
123	C₁₆	—	53	2	no	—
124	C₁₆	—	53	2	no	—
125	C₂³	—	53	2	no	—
126	C₂³	—	106	4	no	—
127	C₂³	—	106	2	no	—
128	C₂³	D₅₃⋊C₁₆	1	1	yes	characteristic
129	C₂×C₄	—	53	2	no	—
130	C₂×C₄	—	106	2	no	—
131	C₂×C₄	—	106	2	no	—
132	C₂×C₄	—	106	2	no	—
133	C₂×C₄	—	106	4	no	—
134	C₂×C₄	#13133274	1	2	yes	—
135	C₂×C₄	D₅₃⋊C₁₆	1	1	yes	characteristic
136	C₂×C₄	#13133292	1	2	yes	—
137	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
138	C₈	—	106	2	no	—
139	C₈	#13133274	1	2	yes	—
140	C₈	C₂×C₂²⋊#6157	1	1	yes	characteristic
141	C₈	C₂×#11719936	1	1	yes	characteristic
142	C₂²	—	2	2	no	—
143	C₂²	—	53	4	no	—
144	C₂²	—	106	8	no	—
145	C₂²	—	106	2	no	—
146	C₂²	—	106	2	no	—
147	C₂²	#16811518	1	2	yes	—
148	C₂²	C₂×D₅₃⋊C₁₆	1	1	yes	characteristic
149	C₄	—	106	2	no	—
150	C₄	#16811518	1	2	yes	—
151	C₄	C₂×#13133274	1	1	yes	characteristic
152	C₄	C₂×#13133292	1	1	yes	characteristic
153	C₂	—	2	2	no	—
154	C₂	—	106	4	no	—
155	C₂	C₂×#16811518	1	1	yes	characteristic
156	C₂	D₅₃⋊M₆₄	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16811555

C₂

yes

maximal, complemented

#20289829

C₂

yes

maximal, complemented

C₂²×C₁₆×D₅₃

C₂

yes

maximal, characteristic

#16811555

—

C₂×#13133267

C₂²

yes

complemented

C₂²×C₈×D₅₃

C₄

yes

characteristic

C₂²×#11719908

C₂²

yes

characteristic

C₂×C₁₆×D₅₃

—

C₂×C₁₆×D₅₃

C₄

yes

—

C₂×C₁₆×D₅₃

C₂²

yes

—

C₂²×C₁₆×C₅₃

C₄

yes

characteristic

#13133267

—

C₂²×C₄×D₅₃

C₈

yes

characteristic

C₂²×#88322

C₈

yes

characteristic

C₂×C₈×D₅₃

—

C₂×C₈×D₅₃

—

C₂×C₈×D₅₃

C₈

yes

—

C₂×C₈×D₅₃

C₂×C₄

yes

—

C₂×#11719908

C₂×C₄

yes

characteristic

C₂×#11719908

D₄

yes

—

C₂×#11719908

C₂³

yes

characteristic

C₁₆×D₅₃

—

C₁₆×D₅₃

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₂³×D₅₃

C₁₆

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂²×Dic₅₃

C₁₆

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₁₆

yes

—

C₂×C₄×D₅₃

C₂×C₈

yes

—

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

C₂×#88322

—

C₂×#88322

C₂×C₈

yes

—

C₈×D₅₃

—

C₈×D₅₃

—

C₈×D₅₃

—

C₁₆×C₅₃

—

#11719908

C₂²⋊C₄

yes

—

#11719908

C₂×D₄

yes

—

C₂³×C₅₃

C₂×C₁₆

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

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

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×C₁₆

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

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

—

C₂×D₅₃

—

C₂×D₅₃

M₆₄

yes

complemented

C₄×C₅₃

C₂²⋊C₁₆

yes

—

C₄×C₅₃

C₂×C₂²⋊C₈

yes

characteristic

C₄×C₅₃

C₂×M₃₂

yes

characteristic

Dic₅₃

—

C₂³×C₁₆

—

C₂×M₆₄

—

C₂²⋊C₃₂

—

C₂×C₅₃

—

C₂×C₅₃

C₂×C₂²⋊C₁₆

yes

characteristic

C₂×C₅₃

C₂×M₆₄

yes

—

D₅₃

—

C₂³×C₈

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂²×C₁₆

—

106

—

C₂²×C₁₆

#6157

yes

characteristic

M₆₄

—

106

—

C₂×C₃₂

—

C₅₃

#13553

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

106

—

C₂²×C₈

—

106

—

C₂²×C₈

C₂×#6157

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

106

—

103

C₂×C₁₆

—

106

—

104

C₂×C₁₆

C₂×#6157

yes

—

105

C₃₂

—

106

—

106

C₂⁴

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

106

—

111

C₂²×C₄

—

106

—

112

C₂²×C₄

D₅₃⋊C₈

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

106

—

115

C₂×C₈

—

106

—

116

C₂×C₈

—

106

—

117

C₂×C₈

—

106

—

118

C₂×C₈

D₅₃⋊C₈

yes

characteristic

119

C₂×C₈

#11719936

yes

—

120

C₂×C₈

C₂²⋊#6157

yes

—

121

C₂×C₈

C₂²×#6157

yes

characteristic

122

C₁₆

—

123

C₁₆

—

124

C₁₆

—

125

C₂³

—

126

C₂³

—

106

—

127

C₂³

—

106

—

128

C₂³

D₅₃⋊C₁₆

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

106

—

131

C₂×C₄

—

106

—

132

C₂×C₄

—

106

—

133

C₂×C₄

—

106

—

134

C₂×C₄

#13133274

yes

—

135

C₂×C₄

D₅₃⋊C₁₆

yes

characteristic

136

C₂×C₄

#13133292

yes

—

137

C₂×C₄

C₂×D₅₃⋊C₈

yes

characteristic

138

C₈

—

106

—

139

C₈

#13133274

yes

—

140

C₈

C₂×C₂²⋊#6157

yes

characteristic

141

C₈

C₂×#11719936

yes

characteristic

142

C₂²

—

143

C₂²

—

144

C₂²

—

106

—

145

C₂²

—

106

—

146

C₂²

—

106

—

147

C₂²

#16811518

yes

—

148

C₂²

C₂×D₅₃⋊C₁₆

yes

characteristic

149

C₄

—

106

—

150

C₄

#16811518

yes

—

151

C₄

C₂×#13133274

yes

characteristic

152

C₄

C₂×#13133292

yes

characteristic

153

C₂

—

154

C₂

—

106

—

155

C₂

C₂×#16811518

yes

characteristic

156

C₂

D₅₃⋊M₆₄

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#6157
commutator, socle	C₂²×C₅₃	C₂²×C₁₆
Fitting	C₂²×C₁₆×C₅₃	C₄
prime radical	C₂³×D₅₃	C₁₆
cosocle	C₂×#11719908	C₂³
hypercenter	C₂²×C₁₆	#6157
hypocenter	C₅₃	#13553
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×#6157

commutator, socle

C₂²×C₅₃

C₂²×C₁₆

Fitting

C₂²×C₁₆×C₅₃

C₄

prime radical

C₂³×D₅₃

C₁₆

C₂³

C₂²×C₁₆

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
1. C₂×#16811555	C₂	semidirect
2. #20289829	C₂	semidirect
5. C₂×#13133267	C₂²	semidirect
70. C₂×D₅₃	M₆₄	semidirect
90. C₅₃	#13553	semidirect

H ≅ G/N

type

1. C₂×#16811555

C₂

semidirect

2. #20289829

C₂

semidirect

5. C₂×#13133267

C₂²

semidirect

70. C₂×D₅₃

M₆₄

semidirect

90. C₅₃

#13553

semidirect

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

subgroup	index
C₂×#16811555	2
#20289829	2
C₂²×C₁₆×D₅₃	2
#13553	53

subgroup

index

C₂×#16811555

#20289829

C₂²×C₁₆×D₅₃

#13553

Subgroups — #28312339

Conjugacy classes of subgroups

Complements

Maximal subgroups