Group #27295961 — subgroups

Subgroups — #27295961

Order 13568.

236 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 87 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	#20276548	C₂	1	2	yes	maximal, complemented
2	C₈×D₄₂₄	C₂	1	1	yes	maximal, characteristic
3	#20272779	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272671	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1312	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13133156	—	2	2	no	—
7	C₄×D₄₂₄	C₄	1	1	yes	characteristic
8	C₈×D₂₁₂	C₂²	1	2	yes	—
9	C₄⋊#11719908	C₂²	1	2	yes	—
10	#16810402	—	2	2	no	—
11	#16810376	C₄	1	1	yes	characteristic
12	#16810352	C₄	1	2	yes	—
13	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₈²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×D₄₂₄	C₈	1	1	yes	characteristic
17	C₂×C₈×D₅₃	—	2	2	no	—
18	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
19	#13133156	—	4	4	no	—
20	C₂×#11719908	—	2	2	no	—
21	#13133151	C₈	1	1	yes	characteristic
22	C₄⋊#88322	C₂×C₄	1	2	yes	—
23	#13133139	C₈	1	2	yes	—
24	#13133129	—	2	2	no	—
25	C₂×C₁₆×C₅₃	—	2	1	no	—
26	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
27	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
30	C₂×D₂₁₂	C₂×C₈	1	2	yes	—
31	C₂×C₄×D₅₃	—	2	2	no	—
32	C₂×C₈×C₅₃	—	2	1	no	—
33	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
37	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	#11719921	—	2	2	no	—
39	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
40	C₂×#88322	—	2	2	no	—
41	D₄₂₄	—	2	2	no	—
42	C₈×D₅₃	—	4	2	no	—
43	C₈×D₅₃	—	4	2	no	—
44	C₁₆×C₅₃	—	2	2	no	—
45	C₁₆×C₅₃	—	4	1	no	—
46	#11719908	—	4	2	no	—
47	C₂²×D₅₃	—	2	2	no	—
48	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
50	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
51	C₂×Dic₅₃	—	2	2	no	—
52	D₂₁₂	—	2	2	no	—
53	D₂₁₂	—	2	2	no	—
54	C₄×D₅₃	—	4	2	no	—
55	C₄×D₅₃	—	4	2	no	—
56	C₈×C₅₃	—	2	1	no	—
57	C₈×C₅₃	—	2	1	no	—
58	C₈×C₅₃	—	4	1	no	—
59	C₈×C₅₃	M₃₂	1	2	yes	complemented
60	C₈×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
61	#88322	—	4	2	no	—
62	#12064	—	53	1	no	maximal, Sylow
63	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
64	C₂×D₅₃	—	4	2	no	—
65	C₂×D₅₃	—	4	2	no	—
66	C₂×D₅₃	—	4	2	no	—
67	C₄×C₅₃	—	2	1	no	—
68	C₄×C₅₃	#442	1	1	yes	characteristic
69	C₄×C₅₃	#444	1	1	yes	characteristic
70	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
71	C₄×C₅₃	#504	1	1	yes	characteristic
72	Dic₅₃	—	4	2	no	—
73	#2169	—	53	2	no	—
74	C₈×D₈	—	53	1	no	—
75	#1367	—	53	1	no	—
76	D₄⋊C₁₆	—	53	2	no	—
77	#1312	D₅₃	1	1	yes	characteristic, complemented
78	C₂×C₅₃	#1580	1	1	yes	characteristic
79	C₂×C₅₃	#2169	1	1	yes	characteristic
80	C₂×C₅₃	#2171	1	1	yes	characteristic
81	D₅₃	—	8	2	no	—
82	C₂×M₃₂	—	106	2	no	—
83	C₄×D₈	—	53	1	no	—
84	C₈×D₄	—	53	2	no	—
85	C₄⋊C₁₆	—	53	2	no	—
86	C₂²⋊C₁₆	—	106	2	no	—
87	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
88	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
89	#335	—	53	1	no	—
90	D₄⋊C₈	—	53	2	no	—
91	C₈²	C₂×D₅₃	1	1	yes	characteristic
92	C₅₃	#12064	1	1	yes	characteristic, complemented, Sylow
93	C₂×D₈	—	53	1	no	—
94	C₂²×C₈	—	106	2	no	—
95	C₄×D₄	—	53	2	no	—
96	M₃₂	—	212	4	no	—
97	C₂×C₁₆	—	2	1	no	—
98	C₂×C₁₆	—	106	2	no	—
99	C₂×C₁₆	D₂₁₂	1	2	yes	—
100	#107	—	53	1	no	—
101	C₄⋊C₈	—	53	2	no	—
102	#102	—	53	2	no	—
103	C₂²⋊C₈	—	106	2	no	—
104	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
105	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
106	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
107	C₂×D₄	—	53	2	no	—
108	C₂²×C₄	—	106	2	no	—
109	D₈	—	106	2	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	106	2	no	—
112	C₂×C₈	—	212	2	no	—
113	C₂×C₈	—	212	2	no	—
114	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
115	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
116	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
117	C₂×C₈	#11719945	1	1	yes	characteristic
118	C₄⋊C₄	—	53	2	no	—
119	C₂²⋊C₄	—	106	2	no	—
120	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
121	C₁₆	—	2	2	no	—
122	C₁₆	—	4	1	no	—
123	C₁₆	—	212	2	no	—
124	C₂³	—	106	2	no	—
125	D₄	—	106	2	no	—
126	D₄	—	106	2	no	—
127	C₂×C₄	—	106	2	no	—
128	C₂×C₄	—	212	2	no	—
129	C₂×C₄	—	212	2	no	—
130	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
131	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
132	C₂×C₄	#13133223	1	1	yes	characteristic
133	C₈	—	2	1	no	—
134	C₈	—	2	1	no	—
135	C₈	—	4	1	no	—
136	C₈	—	212	2	no	—
137	C₈	#13133156	1	2	yes	complemented
138	C₈	#13133245	1	2	yes	—
139	C₂²	—	212	2	no	—
140	C₂²	—	212	2	no	—
141	C₂²	—	212	2	no	—
142	C₂²	C₈×D₂₁₂	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	212	2	no	—
145	C₄	#16810736	1	1	yes	characteristic
146	C₄	#16810746	1	1	yes	characteristic
147	C₄	C₂×#13133156	1	1	yes	characteristic
148	C₄	#16811039	1	1	yes	characteristic
149	C₂	—	424	2	no	—
150	C₂	#20273540	1	1	yes	characteristic
151	C₂	#20276548	1	1	yes	characteristic
152	C₂	#20276558	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276548

C₂

yes

maximal, complemented

C₈×D₄₂₄

C₂

yes

maximal, characteristic

#20272779

C₂

yes

maximal, characteristic, complemented

#20272671

C₂

yes

maximal, complemented

C₅₃×#1312

C₂

yes

maximal, characteristic, complemented

C₂×#13133156

—

C₄×D₄₂₄

C₄

yes

characteristic

C₈×D₂₁₂

C₂²

yes

—

C₄⋊#11719908

C₂²

yes

—

#16810402

—

#16810376

C₄

yes

characteristic

#16810352

C₄

yes

—

C₄×C₁₆×C₅₃

C₂²

yes

characteristic

C₄×C₁₆×C₅₃

C₂²

yes

characteristic

C₈²×C₅₃

C₂²

yes

characteristic

C₂×D₄₂₄

C₈

yes

characteristic

C₂×C₈×D₅₃

—

C₄×D₂₁₂

C₂×C₄

yes

—

#13133156

—

C₂×#11719908

—

#13133151

C₈

yes

characteristic

C₄⋊#88322

C₂×C₄

yes

—

#13133139

C₈

yes

—

#13133129

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×D₂₁₂

C₂×C₈

yes

—

C₂×C₄×D₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₈

yes

characteristic

C₂×C₈×C₅₃

C₂×C₈

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

#41

yes

characteristic

C₄²×C₅₃

C₂²×C₄

yes

characteristic

#11719921

—

C₄⋊Dic₅₃

C₂×C₈

yes

—

C₂×#88322

—

D₄₂₄

—

C₈×D₅₃

—

C₈×D₅₃

—

C₁₆×C₅₃

—

C₁₆×C₅₃

—

#11719908

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

C₄×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

#131

yes

characteristic

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

M₃₂

yes

complemented

C₈×C₅₃

AGL(1,ℤ/8)

yes

—

#88322

—

#12064

—

maximal, Sylow

C₂²×C₅₃

C₈×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₂×M₃₂

yes

characteristic

C₄×C₅₃

#504

yes

characteristic

Dic₅₃

—

#2169

—

C₈×D₈

—

#1367

—

D₄⋊C₁₆

—

#1312

D₅₃

yes

characteristic, complemented

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

C₂×M₃₂

—

106

—

C₄×D₈

—

C₈×D₄

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

106

—

C₄×C₁₆

C₂×D₅₃

yes

characteristic

C₄×C₁₆

C₂×D₅₃

yes

characteristic

#335

—

D₄⋊C₈

—

C₈²

C₂×D₅₃

yes

characteristic

C₅₃

#12064

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂²×C₈

—

106

—

C₄×D₄

—

M₃₂

—

212

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₂×C₁₆

D₂₁₂

yes

—

100

#107

—

101

C₄⋊C₈

—

102

#102

—

103

C₂²⋊C₈

—

106

—

104

C₄×C₈

C₄×D₅₃

yes

characteristic

105

C₄×C₈

C₄×D₅₃

yes

characteristic

106

C₄×C₈

C₂²×D₅₃

yes

characteristic

107

C₂×D₄

—

108

C₂²×C₄

—

106

—

109

D₈

—

106

—

110

C₂×C₈

—

111

C₂×C₈

—

106

—

112

C₂×C₈

—

212

—

113

C₂×C₈

—

212

—

114

C₂×C₈

C₈×D₅₃

yes

characteristic

115

C₂×C₈

C₈×D₅₃

yes

characteristic

116

C₂×C₈

C₂×D₂₁₂

yes

characteristic

117

C₂×C₈

#11719945

yes

characteristic

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

106

—

120

C₄²

C₂×C₄×D₅₃

yes

characteristic

121

C₁₆

—

122

C₁₆

—

123

C₁₆

—

212

—

124

C₂³

—

106

—

125

D₄

—

106

—

126

D₄

—

106

—

127

C₂×C₄

—

106

—

128

C₂×C₄

—

212

—

129

C₂×C₄

—

212

—

130

C₂×C₄

C₄×D₂₁₂

yes

characteristic

131

C₂×C₄

C₂×C₈×D₅₃

yes

characteristic

132

C₂×C₄

#13133223

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

212

—

137

C₈

#13133156

yes

complemented

138

C₈

#13133245

yes

—

139

C₂²

—

212

—

140

C₂²

—

212

—

141

C₂²

—

212

—

142

C₂²

C₈×D₂₁₂

yes

characteristic

143

C₄

—

144

C₄

—

212

—

145

C₄

#16810736

yes

characteristic

146

C₄

#16810746

yes

characteristic

147

C₄

C₂×#13133156

yes

characteristic

148

C₄

#16811039

yes

characteristic

149

C₂

—

424

—

150

C₂

#20273540

yes

characteristic

151

C₂

#20276548

yes

characteristic

152

C₂

#20276558

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1312	C₂
socle	C₂²×C₅₃	C₈×D₄
prime radical	C₂×D₄₂₄	C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#1312	D₅₃
hypocenter	C₅₃	#12064
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ #1312

subgroup

G/H

center

C₂×C₈

C₂×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₅₃

Fitting

C₅₃×#1312

C₂

socle

C₂²×C₅₃

C₈×D₄

prime radical

C₂×D₄₂₄

C₈

cosocle

C₄×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ #1312

N	H ≅ G/N	type
1. #20276548	C₂	semidirect
3. #20272779	C₂	semidirect
4. #20272671	C₂	semidirect
5. C₅₃×#1312	C₂	semidirect
59. C₈×C₅₃	M₃₂	semidirect
77. #1312	D₅₃	semidirect
92. C₅₃	#12064	semidirect
137. C₈	#13133156	semidirect

H ≅ G/N

type

1. #20276548

C₂

semidirect

3. #20272779

C₂

semidirect

4. #20272671

C₂

semidirect

5. C₅₃×#1312

C₂

semidirect

59. C₈×C₅₃

M₃₂

semidirect

77. #1312

D₅₃

semidirect

92. C₅₃

#12064

semidirect

137. C₈

#13133156

semidirect

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

subgroup	index
#20276548	2
C₈×D₄₂₄	2
#20272779	2
#20272671	2
C₅₃×#1312	2
#12064	53

subgroup

index

C₈×D₄₂₄

C₅₃×#1312

Subgroups — #27295961

Conjugacy classes of subgroups

Complements

Maximal subgroups