Group #28307798 — subgroups

Subgroups — #28307798

Order 13568.

282 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 136 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	#20290916	C₂	1	1	yes	maximal, characteristic
2	#20289919	C₂	1	1	yes	maximal, characteristic
3	#20289632	C₂	1	2	yes	maximal, complemented
4	#20289631	C₂	1	2	yes	maximal, complemented
5	C₄×#13133156	C₂	1	1	yes	maximal, characteristic
6	C₂×D₅₃⋊C₁₆	C₄	1	2	yes	—
7	#16811641	C₄	1	2	yes	—
8	#16811512	C₂²	1	2	yes	—
9	#16811511	C₄	1	2	yes	—
10	#16811464	C₂²	1	2	yes	—
11	C₈×#88324	C₄	1	2	yes	—
12	C₂×#13133156	C₄	1	1	yes	characteristic
13	C₂×#13133156	C₄	1	1	yes	characteristic
14	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
15	#16810398	C₂²	1	1	yes	characteristic
16	#16810398	C₂²	1	1	yes	characteristic
17	C₄×#11719908	C₄	1	1	yes	characteristic
18	C₄×C₁₆×C₅₃	C₄	1	1	yes	characteristic
19	D₅₃⋊C₁₆	—	2	4	no	—
20	C₂×#11719910	C₂×C₄	1	4	yes	—
21	#13133284	—	2	2	no	—
22	#13133274	—	2	2	no	—
23	C₄×#88324	C₂×C₄	1	4	yes	—
24	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
27	#13133156	C₈	1	4	yes	complemented
28	#13133156	C₈	1	4	yes	complemented
29	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
30	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
31	C₄×#88322	C₂×C₄	1	1	yes	characteristic
32	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
33	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
34	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
38	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
39	C₂×#88324	—	2	4	no	—
40	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₄²×C₅₃	C₄²	1	1	yes	characteristic
43	C₄×Dic₅₃	M₁₆	1	2	yes	—
44	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×#88322	C₄²	1	1	yes	characteristic
46	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
47	C₈×D₅₃	C₂×C₈	1	2	yes	—
48	C₈×D₅₃	C₂×C₈	1	2	yes	—
49	C₈×D₅₃	M₁₆	1	4	yes	—
50	#11719910	—	2	4	no	—
51	C₁₆×C₅₃	C₂×C₈	1	2	yes	—
52	C₁₆×C₅₃	C₂×C₈	1	2	yes	—
53	#11719908	C₂×C₈	1	2	yes	—
54	#11719908	C₂×C₈	1	2	yes	—
55	C₂²×D₅₃	#97	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#97	1	2	yes	—
57	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
58	C₂×Dic₅₃	#97	1	1	yes	characteristic
59	C₂×Dic₅₃	C₂×M₁₆	1	2	yes	—
60	C₄×D₅₃	—	4	2	no	—
61	C₄×D₅₃	C₄×C₈	1	2	yes	—
62	C₄×D₅₃	#97	1	2	yes	—
63	#88324	—	4	4	no	—
64	C₈×C₅₃	#97	1	2	yes	—
65	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
66	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
67	#88322	C₄×C₈	1	2	yes	—
68	#88322	C₂×M₁₆	1	2	yes	—
69	#11406	—	53	1	no	maximal, Sylow
70	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
71	C₂×D₅₃	—	2	2	no	—
72	C₂×D₅₃	#322	1	2	yes	—
73	C₄×C₅₃	—	2	2	no	—
74	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
75	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
76	Dic₅₃	—	2	2	no	—
77	Dic₅₃	#322	1	2	yes	—
78	C₄×M₃₂	—	53	1	no	—
79	C₂×#346	—	53	1	no	—
80	#1450	—	53	1	no	—
81	#1313	—	53	2	no	—
82	#1313	—	53	2	no	—
83	C₂×C₅₃	C₂×#322	1	1	yes	characteristic
84	C₂×C₅₃	#2110	1	2	yes	—
85	D₅₃	—	2	2	no	—
86	C₂×M₃₂	—	53	1	no	—
87	C₂×M₃₂	—	53	1	no	—
88	C₂²×C₁₆	—	53	2	no	—
89	#431	—	53	2	no	—
90	C₂×C₄×C₈	—	53	1	no	—
91	#346	—	53	2	no	—
92	#346	—	53	2	no	—
93	#346	—	53	1	no	—
94	#346	—	53	1	no	—
95	C₄×C₁₆	—	53	1	no	—
96	C₄×C₁₆	#6157	1	1	yes	characteristic
97	#322	—	53	2	no	—
98	C₈²	—	53	2	no	—
99	C₅₃	#11406	1	1	yes	characteristic, complemented, Sylow
100	C₂²×C₈	—	53	1	no	—
101	C₂²×C₈	—	53	1	no	—
102	C₂×C₄²	—	53	1	no	—
103	M₃₂	—	53	4	no	—
104	M₃₂	—	53	4	no	—
105	C₂×C₁₆	—	53	1	no	—
106	C₂×C₁₆	—	53	1	no	—
107	C₂×C₁₆	—	53	4	no	—
108	C₂×C₁₆	—	106	4	no	—
109	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
110	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
111	C₄⋊C₈	—	106	2	no	—
112	C₂²⋊C₈	—	106	2	no	—
113	C₄×C₈	—	53	1	no	—
114	C₄×C₈	—	53	1	no	—
115	C₄×C₈	—	53	4	no	—
116	C₄×C₈	—	53	1	no	—
117	C₄×C₈	C₂×#6157	1	1	yes	characteristic
118	C₂²×C₄	—	53	1	no	—
119	C₂²×C₄	—	53	2	no	—
120	C₂×C₈	—	53	1	no	—
121	C₂×C₈	—	53	1	no	—
122	C₂×C₈	—	53	4	no	—
123	C₂×C₈	—	53	2	no	—
124	C₂×C₈	—	53	2	no	—
125	C₂×C₈	—	106	4	no	—
126	C₂×C₈	C₄×#6157	1	1	yes	characteristic
127	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
128	C₄²	—	53	2	no	—
129	C₄²	—	53	1	no	—
130	C₄²	C₄×#6157	1	1	yes	characteristic
131	C₁₆	—	53	2	no	—
132	C₁₆	—	53	2	no	—
133	C₁₆	—	106	4	no	—
134	C₁₆	D₅₃⋊C₈	1	2	yes	—
135	C₁₆	D₅₃⋊C₈	1	2	yes	—
136	C₂³	—	53	1	no	—
137	C₂×C₄	—	53	1	no	—
138	C₂×C₄	—	53	2	no	—
139	C₂×C₄	—	53	2	no	—
140	C₂×C₄	—	53	2	no	—
141	C₂×C₄	—	212	2	no	—
142	C₂×C₄	#13133271	1	2	yes	—
143	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
144	C₈	—	53	2	no	—
145	C₈	—	53	2	no	—
146	C₈	—	212	4	no	—
147	C₈	#13133271	1	2	yes	—
148	C₈	C₂×D₅₃⋊C₈	1	1	yes	characteristic
149	C₈	C₂×D₅₃⋊C₈	1	1	yes	characteristic
150	C₂²	—	53	2	no	—
151	C₂²	—	106	2	no	—
152	C₂²	C₂×#13133271	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	53	2	no	—
155	C₄	—	106	2	no	—
156	C₄	C₄×D₅₃⋊C₈	1	1	yes	characteristic
157	C₄	C₂×#13133271	1	1	yes	characteristic
158	C₂	—	106	2	no	—
159	C₂	#20289909	1	1	yes	characteristic
160	C₂	#20290934	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20290916

C₂

yes

maximal, characteristic

#20289919

C₂

yes

maximal, characteristic

#20289632

C₂

yes

maximal, complemented

#20289631

C₂

yes

maximal, complemented

C₄×#13133156

C₂

yes

maximal, characteristic

C₂×D₅₃⋊C₁₆

C₄

yes

—

#16811641

C₄

yes

—

#16811512

C₂²

yes

—

#16811511

C₄

yes

—

#16811464

C₂²

yes

—

C₈×#88324

C₄

yes

—

C₂×#13133156

C₄

yes

characteristic

C₂×#13133156

C₄

yes

characteristic

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810398

C₂²

yes

characteristic

#16810398

C₂²

yes

characteristic

C₄×#11719908

C₄

yes

characteristic

C₄×C₁₆×C₅₃

C₄

yes

characteristic

D₅₃⋊C₁₆

—

C₂×#11719910

C₂×C₄

yes

—

#13133284

—

#13133274

—

C₄×#88324

C₂×C₄

yes

—

C₂×C₈×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×D₅₃

C₂×C₄

yes

characteristic

C₄²×D₅₃

C₂×C₄

yes

characteristic

#13133156

C₈

yes

complemented

#13133156

C₈

yes

complemented

C₂×#11719908

C₂×C₄

yes

characteristic

C₂×#11719908

C₂×C₄

yes

characteristic

C₄×#88322

C₂×C₄

yes

characteristic

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₈×Dic₅₃

C₂³

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

C₄²

yes

characteristic

C₂×C₄×D₅₃

M₁₆

yes

—

C₂×#88324

—

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₄²

yes

characteristic

C₄×Dic₅₃

M₁₆

yes

—

C₄×Dic₅₃

C₂²×C₄

yes

characteristic

C₂×#88322

C₄²

yes

characteristic

C₂×#88322

C₂²×C₄

yes

characteristic

C₈×D₅₃

C₂×C₈

yes

—

C₈×D₅₃

C₂×C₈

yes

—

C₈×D₅₃

M₁₆

yes

—

#11719910

—

C₁₆×C₅₃

C₂×C₈

yes

—

C₁₆×C₅₃

C₂×C₈

yes

—

#11719908

C₂×C₈

yes

—

#11719908

C₂×C₈

yes

—

C₂²×D₅₃

#97

yes

characteristic

C₂×C₄×C₅₃

#97

yes

—

C₂×C₄×C₅₃

C₂×C₄²

yes

characteristic

C₂×Dic₅₃

#97

yes

characteristic

C₂×Dic₅₃

C₂×M₁₆

yes

—

C₄×D₅₃

—

C₄×D₅₃

C₄×C₈

yes

—

C₄×D₅₃

#97

yes

—

#88324

—

C₈×C₅₃

#97

yes

—

C₈×C₅₃

C₂²×C₈

yes

characteristic

C₈×C₅₃

C₂²×C₈

yes

characteristic

#88322

C₄×C₈

yes

—

#88322

C₂×M₁₆

yes

—

#11406

—

maximal, Sylow

C₂²×C₅₃

C₂×#97

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

#322

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄×C₈

yes

characteristic

C₄×C₅₃

C₂×#97

yes

characteristic

Dic₅₃

—

Dic₅₃

#322

yes

—

C₄×M₃₂

—

—

—

—

—

C₂×C₅₃

C₂×#322

yes

characteristic

C₂×C₅₃

#2110

yes

—

D₅₃

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

—

#431

—

C₂×C₄×C₈

—

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

#6157

yes

characteristic

#322

—

C₈²

—

C₅₃

#11406

yes

characteristic, complemented, Sylow

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂×C₄²

—

103

M₃₂

—

104

M₃₂

—

105

C₂×C₁₆

—

106

C₂×C₁₆

—

107

C₂×C₁₆

—

108

C₂×C₁₆

—

106

—

109

C₂×C₁₆

C₂×#6157

yes

characteristic

110

C₂×C₁₆

C₂×#6157

yes

characteristic

111

C₄⋊C₈

—

106

—

112

C₂²⋊C₈

—

106

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

C₂×#6157

yes

characteristic

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

106

—

126

C₂×C₈

C₄×#6157

yes

characteristic

127

C₂×C₈

C₂²×#6157

yes

characteristic

128

C₄²

—

129

C₄²

—

130

C₄²

C₄×#6157

yes

characteristic

131

C₁₆

—

132

C₁₆

—

133

C₁₆

—

106

—

134

C₁₆

D₅₃⋊C₈

yes

—

135

C₁₆

D₅₃⋊C₈

yes

—

136

C₂³

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

212

—

142

C₂×C₄

#13133271

yes

—

143

C₂×C₄

C₂×C₄×#6157

yes

characteristic

144

C₈

—

145

C₈

—

146

C₈

—

212

—

147

C₈

#13133271

yes

—

148

C₈

C₂×D₅₃⋊C₈

yes

characteristic

149

C₈

C₂×D₅₃⋊C₈

yes

characteristic

150

C₂²

—

151

C₂²

—

106

—

152

C₂²

C₂×#13133271

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

106

—

156

C₄

C₄×D₅₃⋊C₈

yes

characteristic

157

C₄

C₂×#13133271

yes

characteristic

158

C₂

—

106

—

159

C₂

#20289909

yes

characteristic

160

C₂

#20290934

yes

—

	subgroup	G/H
center	C₂×C₈	C₄×#6157
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₄×C₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₂×#97
prime radical	C₂²×D₅₃	#97
cosocle	C₈×Dic₅₃	C₂³
hypercenter	C₄×C₁₆	#6157
hypocenter	C₅₃	#11406
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆

subgroup

G/H

center

C₂×C₈

C₄×#6157

commutator

C₄×C₅₃

C₂×C₄×C₈

Frattini

C₂×C₈

C₂²×#6157

Fitting

C₄×C₁₆×C₅₃

C₄

socle

C₂²×C₅₃

C₂×#97

prime radical

C₂²×D₅₃

#97

cosocle

C₈×Dic₅₃

C₂³

C₄×C₁₆

C₅₃

G ▷ C₄×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₈ ◁ C₄×C₁₆

N	H ≅ G/N	type
3. #20289632	C₂	semidirect
4. #20289631	C₂	semidirect
27. #13133156	C₈	semidirect
28. #13133156	C₈	semidirect
99. C₅₃	#11406	semidirect

H ≅ G/N

type

3. #20289632

C₂

semidirect

4. #20289631

C₂

semidirect

27. #13133156

C₈

semidirect

28. #13133156

C₈

semidirect

99. C₅₃

#11406

semidirect

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

subgroup	index
#20290916	2
#20289919	2
#20289632	2
#20289631	2
C₄×#13133156	2
#11406	53

subgroup

index

Subgroups — #28307798

Conjugacy classes of subgroups

Complements

Maximal subgroups