Group #28307830 — subgroups

Subgroups — #28307830

Order 13568.

306 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 156 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	#20291074	C₂	1	2	yes	maximal
2	#20289628	C₂	1	2	yes	maximal, complemented
3	C₈×#11719910	C₂	1	2	yes	maximal, complemented
4	C₈²×D₅₃	C₂	1	1	yes	maximal, characteristic
5	#16811544	C₄	1	4	yes	—
6	#16811518	C₄	1	4	yes	—
7	C₄×#11719910	C₂²	1	4	yes	—
8	C₄×C₈×D₅₃	C₄	1	1	yes	characteristic
9	C₄×C₈×D₅₃	C₄	1	1	yes	characteristic
10	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
11	C₈×#88322	C₄	1	1	yes	characteristic
12	C₈×#88322	C₂²	1	1	yes	characteristic
13	C₈×#88322	C₂²	1	1	yes	characteristic
14	C₈²×C₅₃	C₄	1	1	yes	characteristic
15	C₂×#11719910	C₂×C₄	1	8	yes	—
16	C₂×C₈×D₅₃	C₈	1	2	yes	—
17	C₂×C₈×D₅₃	C₈	1	2	yes	—
18	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
19	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
20	C₄×#88322	C₂×C₄	1	1	yes	characteristic
21	C₄×#88322	C₂×C₄	1	1	yes	characteristic
22	C₄×#88322	C₂³	1	1	yes	characteristic
23	C₈×Dic₅₃	C₈	1	2	yes	—
24	C₈×Dic₅₃	C₈	1	2	yes	—
25	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
30	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₅₃	C₄²	1	2	yes	—
33	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
34	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
35	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
37	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
38	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
39	C₂×#88322	C₂×C₈	1	2	yes	—
40	C₂×#88322	C₂×C₈	1	2	yes	—
41	C₂×#88322	C₂²×C₄	1	2	yes	—
42	C₈×D₅₃	—	2	4	no	—
43	C₈×D₅₃	—	2	4	no	—
44	C₈×D₅₃	C₁₆	1	8	yes	complemented
45	#11719910	—	2	8	no	—
46	C₂²×D₅₃	C₄×C₈	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
50	C₂×Dic₅₃	C₄×C₈	1	1	yes	characteristic
51	C₂×Dic₅₃	C₄×C₈	1	1	yes	characteristic
52	C₂×Dic₅₃	C₄×C₈	1	1	yes	characteristic
53	C₄×D₅₃	—	2	2	no	—
54	C₄×D₅₃	—	2	2	no	—
55	C₄×D₅₃	C₂×C₁₆	1	4	yes	—
56	C₈×C₅₃	—	2	2	no	—
57	C₈×C₅₃	C₂×C₁₆	1	4	yes	—
58	C₈×C₅₃	M₃₂	1	4	yes	—
59	#88322	—	2	2	no	—
60	#88322	C₂×C₁₆	1	4	yes	—
61	#88322	#131	1	4	yes	—
62	#11410	—	53	1	no	maximal, Sylow
63	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
64	C₂×D₅₃	—	2	1	no	—
65	C₂×D₅₃	—	2	1	no	—
66	C₂×D₅₃	C₄×C₁₆	1	2	yes	—
67	C₄×C₅₃	#405	1	2	yes	—
68	C₄×C₅₃	C₂²×C₁₆	1	2	yes	—
69	C₄×C₅₃	C₂×M₃₂	1	2	yes	—
70	Dic₅₃	—	2	1	no	—
71	Dic₅₃	—	2	1	no	—
72	Dic₅₃	C₄×C₁₆	1	2	yes	—
73	#2162	—	53	2	no	—
74	C₂×C₈²	—	53	1	no	—
75	#1312	—	53	2	no	—
76	C₈×C₁₆	—	53	2	no	—
77	C₂×C₅₃	#1450	1	1	yes	characteristic
78	C₂×C₅₃	C₂×C₄×C₁₆	1	1	yes	characteristic
79	C₂×C₅₃	C₄×M₃₂	1	1	yes	characteristic
80	D₅₃	—	2	2	no	—
81	C₂×C₄×C₈	—	53	1	no	—
82	C₂×C₄×C₈	—	53	1	no	—
83	C₂×C₄×C₈	—	53	1	no	—
84	C₄⋊C₁₆	—	53	4	no	—
85	C₂²⋊C₁₆	—	53	4	no	—
86	C₄×C₁₆	—	53	4	no	—
87	C₈²	—	53	1	no	—
88	C₈²	—	53	1	no	—
89	C₈²	—	53	1	no	—
90	C₈²	#6157	1	1	yes	characteristic
91	C₅₃	#11410	1	1	yes	characteristic, complemented, Sylow
92	C₂²×C₈	—	53	2	no	—
93	C₂²×C₈	—	53	2	no	—
94	C₂²×C₈	—	53	2	no	—
95	C₂×C₄²	—	53	1	no	—
96	C₂×C₁₆	—	53	8	no	—
97	C₄×C₈	—	53	2	no	—
98	C₄×C₈	—	53	2	no	—
99	C₄×C₈	—	53	2	no	—
100	C₄×C₈	—	53	1	no	—
101	C₄×C₈	—	53	1	no	—
102	C₄×C₈	—	53	1	no	—
103	C₄×C₈	C₂×#6157	1	1	yes	characteristic
104	C₄×C₈	C₂×#6157	1	1	yes	characteristic
105	C₄×C₈	C₂×#6157	1	1	yes	characteristic
106	C₂²×C₄	—	53	1	no	—
107	C₂²×C₄	—	53	1	no	—
108	C₂²×C₄	—	53	1	no	—
109	C₂×C₈	—	53	2	no	—
110	C₂×C₈	—	53	2	no	—
111	C₂×C₈	—	53	8	no	—
112	C₂×C₈	—	53	2	no	—
113	C₂×C₈	—	106	4	no	—
114	C₂×C₈	—	106	4	no	—
115	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
116	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
117	C₂×C₈	C₄×#6157	1	2	yes	—
118	C₄²	—	53	1	no	—
119	C₄²	—	53	1	no	—
120	C₄²	—	53	1	no	—
121	C₄²	C₂²×#6157	1	1	yes	characteristic
122	C₁₆	—	106	8	no	—
123	C₂³	—	53	1	no	—
124	C₂×C₄	—	53	1	no	—
125	C₂×C₄	—	53	4	no	—
126	C₂×C₄	—	53	1	no	—
127	C₂×C₄	—	53	1	no	—
128	C₂×C₄	—	106	2	no	—
129	C₂×C₄	—	106	2	no	—
130	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
131	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
132	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	53	4	no	—
135	C₈	—	53	4	no	—
136	C₈	—	106	2	no	—
137	C₈	D₅₃⋊C₁₆	1	4	yes	—
138	C₈	#13133292	1	4	yes	—
139	C₂²	—	53	2	no	—
140	C₂²	—	106	1	no	—
141	C₂²	—	106	1	no	—
142	C₂²	C₄×D₅₃⋊C₈	1	1	yes	characteristic
143	C₄	—	53	2	no	—
144	C₄	—	106	1	no	—
145	C₄	—	106	1	no	—
146	C₄	#16811586	1	2	yes	—
147	C₄	C₂×D₅₃⋊C₁₆	1	2	yes	—
148	C₄	C₂×#13133292	1	2	yes	—
149	C₂	—	106	2	no	—
150	C₂	#20289919	1	1	yes	characteristic
151	C₂	C₄×D₅₃⋊C₁₆	1	1	yes	characteristic
152	C₂	C₄×#13133292	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20291074

C₂

yes

maximal

#20289628

C₂

yes

maximal, complemented

C₈×#11719910

C₂

yes

maximal, complemented

C₈²×D₅₃

C₂

yes

maximal, characteristic

#16811544

C₄

yes

—

#16811518

C₄

yes

—

C₄×#11719910

C₂²

yes

—

C₄×C₈×D₅₃

C₄

yes

characteristic

C₄×C₈×D₅₃

C₄

yes

characteristic

C₄×C₈×D₅₃

C₂²

yes

characteristic

C₈×#88322

C₄

yes

characteristic

C₈×#88322

C₂²

yes

characteristic

C₈×#88322

C₂²

yes

characteristic

C₈²×C₅₃

C₄

yes

characteristic

C₂×#11719910

C₂×C₄

yes

—

C₂×C₈×D₅₃

C₈

yes

—

C₂×C₈×D₅₃

C₈

yes

—

C₂×C₈×D₅₃

C₂×C₄

yes

—

C₄²×D₅₃

C₂×C₄

yes

characteristic

C₄×#88322

C₂×C₄

yes

characteristic

C₄×#88322

C₂×C₄

yes

characteristic

C₄×#88322

C₂³

yes

characteristic

C₈×Dic₅₃

C₈

yes

—

C₈×Dic₅₃

C₈

yes

—

C₈×Dic₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

C₄²

yes

characteristic

C₂×C₄×D₅₃

C₂×C₈

yes

characteristic

C₂×C₄×D₅₃

C₂×C₈

yes

characteristic

C₂×C₈×C₅₃

C₄²

yes

—

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄×Dic₅₃

C₄²

yes

characteristic

C₄×Dic₅₃

C₂×C₈

yes

characteristic

C₄×Dic₅₃

C₂×C₈

yes

characteristic

C₂×#88322

C₂×C₈

yes

—

C₂×#88322

C₂×C₈

yes

—

C₂×#88322

C₂²×C₄

yes

—

C₈×D₅₃

—

C₈×D₅₃

—

C₈×D₅₃

C₁₆

yes

complemented

#11719910

—

C₂²×D₅₃

C₄×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×Dic₅₃

C₄×C₈

yes

characteristic

C₂×Dic₅₃

C₄×C₈

yes

characteristic

C₂×Dic₅₃

C₄×C₈

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

C₂×C₁₆

yes

—

C₈×C₅₃

—

C₈×C₅₃

C₂×C₁₆

yes

—

C₈×C₅₃

M₃₂

yes

—

#88322

—

#88322

C₂×C₁₆

yes

—

#88322

#131

yes

—

#11410

—

maximal, Sylow

C₂²×C₅₃

C₂×C₄×C₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

C₄×C₁₆

yes

—

C₄×C₅₃

#405

yes

—

C₄×C₅₃

C₂²×C₁₆

yes

—

C₄×C₅₃

C₂×M₃₂

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₄×C₁₆

yes

—

#2162

—

C₂×C₈²

—

#1312

—

C₈×C₁₆

—

C₂×C₅₃

#1450

yes

characteristic

C₂×C₅₃

C₂×C₄×C₁₆

yes

characteristic

C₂×C₅₃

C₄×M₃₂

yes

characteristic

D₅₃

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₈²

—

C₈²

—

C₈²

—

C₈²

#6157

yes

characteristic

C₅₃

#11410

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

C₂×#6157

yes

characteristic

104

C₄×C₈

C₂×#6157

yes

characteristic

105

C₄×C₈

C₂×#6157

yes

characteristic

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

106

—

114

C₂×C₈

—

106

—

115

C₂×C₈

D₅₃⋊C₈

yes

—

116

C₂×C₈

D₅₃⋊C₈

yes

—

117

C₂×C₈

C₄×#6157

yes

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

121

C₄²

C₂²×#6157

yes

characteristic

122

C₁₆

—

106

—

123

C₂³

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

106

—

129

C₂×C₄

—

106

—

130

C₂×C₄

C₂×C₄×#6157

yes

characteristic

131

C₂×C₄

C₂×D₅₃⋊C₈

yes

characteristic

132

C₂×C₄

C₂×D₅₃⋊C₈

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

106

—

137

C₈

D₅₃⋊C₁₆

yes

—

138

C₈

#13133292

yes

—

139

C₂²

—

140

C₂²

—

106

—

141

C₂²

—

106

—

142

C₂²

C₄×D₅₃⋊C₈

yes

characteristic

143

C₄

—

144

C₄

—

106

—

145

C₄

—

106

—

146

C₄

#16811586

yes

—

147

C₄

C₂×D₅₃⋊C₁₆

yes

—

148

C₄

C₂×#13133292

yes

—

149

C₂

—

106

—

150

C₂

#20289919

yes

characteristic

151

C₂

C₄×D₅₃⋊C₁₆

yes

characteristic

152

C₂

C₄×#13133292

yes

characteristic

	subgroup	G/H
center	C₄×C₈	C₂×#6157
commutator	C₂×C₅₃	C₂×C₄×C₁₆
Frattini	C₄²	C₂²×#6157
Fitting	C₈²×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄×C₈
prime radical	C₂²×D₅₃	C₄×C₈
cosocle	C₄×#88322	C₂³
hypercenter	C₈²	#6157
hypocenter	C₅₃	#11410
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₈ ◁ C₈²

subgroup

G/H

center

C₄×C₈

C₂×#6157

commutator

C₂×C₅₃

C₂×C₄×C₁₆

Frattini

C₄²

C₂²×#6157

Fitting

C₈²×C₅₃

C₄

socle

C₂²×C₅₃

C₂×C₄×C₈

prime radical

C₂²×D₅₃

C₄×C₈

C₂³

C₈²

C₅₃

G ▷ C₂×C₅₃ ▷ C₅₃

C₁ ◁ C₄×C₈ ◁ C₈²

N	H ≅ G/N	type
2. #20289628	C₂	semidirect
3. C₈×#11719910	C₂	semidirect
44. C₈×D₅₃	C₁₆	semidirect
91. C₅₃	#11410	semidirect

H ≅ G/N

type

2. #20289628

C₂

semidirect

3. C₈×#11719910

C₂

semidirect

44. C₈×D₅₃

C₁₆

semidirect

91. C₅₃

#11410

semidirect

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

subgroup	index
#20291074	2
#20289628	2
C₈×#11719910	2
C₈²×D₅₃	2
#11410	53

subgroup

index

C₈²×D₅₃

Subgroups — #28307830

Conjugacy classes of subgroups

Complements

Maximal subgroups