Group #27295899 — subgroups

Subgroups — #27295899

Order 13568.

206 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 91 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	#20276623	C₂	1	2	yes	maximal
2	C₈×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
3	#20272785	C₂	1	1	yes	maximal, characteristic
4	#20272695	C₂	1	2	yes	maximal
5	C₈×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₁₀₆	C₂²	1	2	yes	—
7	C₄×Dic₂₁₂	C₄	1	1	yes	characteristic
8	C₄⋊#11719908	C₂²	1	2	yes	—
9	#16810441	—	2	2	no	—
10	#16810398	—	2	2	no	—
11	#16810376	C₄	1	1	yes	characteristic
12	#16810355	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×Dic₂₁₂	C₈	1	1	yes	characteristic
17	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
18	C₂×#11719908	—	2	2	no	—
19	#13133151	C₈	1	1	yes	characteristic
20	C₄⋊#88322	C₂×C₄	1	2	yes	—
21	#13133146	—	2	2	no	—
22	#13133142	C₈	1	2	yes	—
23	C₈×Dic₅₃	—	2	2	no	—
24	C₂×C₁₆×C₅₃	—	2	1	no	—
25	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₅₃	D₄	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	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
31	C₂×C₈×C₅₃	—	2	1	no	—
32	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
36	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
38	#11719919	—	2	2	no	—
39	C₄×Dic₅₃	—	2	2	no	—
40	C₂×#88322	—	2	2	no	—
41	Dic₂₁₂	—	2	2	no	—
42	C₁₆×C₅₃	—	2	2	no	—
43	C₁₆×C₅₃	—	2	1	no	—
44	C₁₆×C₅₃	Q₁₆	1	2	yes	complemented
45	#11719908	—	4	2	no	—
46	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
49	C₂×Dic₅₃	—	2	2	no	—
50	C₂×Dic₅₃	—	2	2	no	—
51	Dic₁₀₆	—	2	2	no	—
52	Dic₁₀₆	—	2	2	no	—
53	C₈×C₅₃	—	2	2	no	—
54	C₈×C₅₃	—	2	1	no	—
55	C₈×C₅₃	—	2	1	no	—
56	C₈×C₅₃	M₃₂	1	2	yes	—
57	C₈×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
58	C₈×C₅₃	#135	1	1	yes	characteristic
59	#88322	—	4	2	no	—
60	#12054	—	53	1	no	maximal, Sylow
61	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
62	C₄×C₅₃	—	2	1	no	—
63	C₄×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
64	C₄×C₅₃	#443	1	1	yes	characteristic
65	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
66	C₄×C₅₃	#504	1	1	yes	characteristic
67	Dic₅₃	—	4	2	no	—
68	Dic₅₃	—	4	2	no	—
69	#2183	—	53	2	no	—
70	C₈×Q₁₆	—	53	1	no	—
71	#1370	—	53	1	no	—
72	Q₈⋊C₁₆	—	53	2	no	—
73	C₈×C₁₆	D₅₃	1	1	yes	characteristic
74	C₂×C₅₃	C₈×Q₁₆	1	1	yes	characteristic
75	C₂×C₅₃	#2169	1	1	yes	characteristic
76	C₂×C₅₃	#2170	1	1	yes	characteristic
77	C₈×Q₈	—	53	2	no	—
78	C₄×Q₁₆	—	53	1	no	—
79	C₄⋊C₁₆	—	53	2	no	—
80	C₄⋊C₁₆	—	106	2	no	—
81	#346	—	106	2	no	—
82	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
83	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
84	#335	—	53	1	no	—
85	Q₈⋊C₈	—	53	2	no	—
86	C₈²	C₂×D₅₃	1	1	yes	characteristic
87	C₅₃	#12054	1	1	yes	characteristic, complemented, Sylow
88	C₂×Q₁₆	—	53	1	no	—
89	C₄×Q₈	—	53	2	no	—
90	C₂×C₁₆	—	2	1	no	—
91	C₂×C₁₆	—	106	2	no	—
92	C₂×C₁₆	D₂₁₂	1	1	yes	characteristic
93	C₂×C₁₆	D₂₁₂	1	1	yes	characteristic
94	#107	—	53	1	no	—
95	C₄⋊C₈	—	53	2	no	—
96	C₄⋊C₈	—	106	2	no	—
97	#103	—	53	2	no	—
98	C₄×C₈	—	106	2	no	—
99	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
100	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
101	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
102	C₂×Q₈	—	53	2	no	—
103	Q₁₆	—	106	2	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	106	2	no	—
106	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
107	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
108	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
109	C₂×C₈	#11719945	1	1	yes	characteristic
110	C₄⋊C₄	—	53	2	no	—
111	C₄⋊C₄	—	106	2	no	—
112	C₄²	—	106	2	no	—
113	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
114	C₁₆	—	2	2	no	—
115	C₁₆	—	2	1	no	—
116	C₁₆	—	212	2	no	—
117	C₁₆	Dic₂₁₂	1	2	yes	complemented
118	Q₈	—	106	2	no	—
119	Q₈	—	106	2	no	—
120	C₂×C₄	—	106	2	no	—
121	C₂×C₄	—	106	2	no	—
122	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
123	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
124	C₂×C₄	#13133223	1	1	yes	characteristic
125	C₈	—	2	2	no	—
126	C₈	—	2	1	no	—
127	C₈	—	2	1	no	—
128	C₈	—	212	2	no	—
129	C₈	#13133156	1	2	yes	—
130	C₈	C₂×Dic₂₁₂	1	1	yes	characteristic
131	C₈	#13133239	1	1	yes	characteristic
132	C₂²	C₈×D₂₁₂	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	212	2	no	—
135	C₄	—	212	2	no	—
136	C₄	C₄×Dic₂₁₂	1	1	yes	characteristic
137	C₄	#16810741	1	1	yes	characteristic
138	C₄	C₂×#13133156	1	1	yes	characteristic
139	C₄	#16811039	1	1	yes	characteristic
140	C₂	C₈×Dic₂₁₂	1	1	yes	characteristic
141	C₂	#20276548	1	1	yes	characteristic
142	C₂	#20276553	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276623

C₂

yes

maximal

C₈×Dic₂₁₂

C₂

yes

maximal, characteristic

#20272785

C₂

yes

maximal, characteristic

#20272695

C₂

yes

maximal

C₈×C₁₆×C₅₃

C₂

yes

maximal, characteristic

C₈×Dic₁₀₆

C₂²

yes

—

C₄×Dic₂₁₂

C₄

yes

characteristic

C₄⋊#11719908

C₂²

yes

—

#16810441

—

#16810398

—

#16810376

C₄

yes

characteristic

#16810355

C₄

yes

—

C₄×C₁₆×C₅₃

C₂²

yes

characteristic

C₄×C₁₆×C₅₃

C₂²

yes

characteristic

C₈²×C₅₃

C₂²

yes

characteristic

C₂×Dic₂₁₂

C₈

yes

characteristic

C₄×Dic₁₀₆

C₂×C₄

yes

—

C₂×#11719908

—

#13133151

C₈

yes

characteristic

C₄⋊#88322

C₂×C₄

yes

—

#13133146

—

#13133142

C₈

yes

—

C₈×Dic₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

D₄

yes

characteristic

C₂×C₁₆×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

C₂×C₈

yes

—

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

C₄⋊Dic₅₃

C₂×C₈

yes

—

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

—

Dic₂₁₂

—

C₁₆×C₅₃

—

C₁₆×C₅₃

—

C₁₆×C₅₃

Q₁₆

yes

complemented

#11719908

—

C₂×C₄×C₅₃

C₄×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

#131

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

M₃₂

yes

—

C₈×C₅₃

C₂×Q₁₆

yes

C₈×C₅₃

yes

—

—

C₂²×C₅₃

C₈×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₄×Q₁₆

yes

characteristic

C₄×C₅₃

#443

yes

characteristic

C₄×C₅₃

C₂×M₃₂

yes

characteristic

C₄×C₅₃

#504

yes

characteristic

Dic₅₃

—

Dic₅₃

—

#2183

—

C₈×Q₁₆

—

#1370

—

Q₈⋊C₁₆

—

C₈×C₁₆

D₅₃

yes

characteristic

C₂×C₅₃

C₈×Q₁₆

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₈×Q₈

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

106

—

#346

—

106

—

C₄×C₁₆

C₂×D₅₃

yes

characteristic

C₄×C₁₆

C₂×D₅₃

yes

characteristic

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₅₃

yes

characteristic

C₅₃

#12054

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₂×C₁₆

D₂₁₂

yes

characteristic

C₂×C₁₆

D₂₁₂

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

#103

—

C₄×C₈

—

106

—

C₄×C₈

C₄×D₅₃

yes

characteristic

100

C₄×C₈

C₄×D₅₃

yes

characteristic

101

C₄×C₈

C₂²×D₅₃

yes

characteristic

102

C₂×Q₈

—

103

Q₁₆

—

106

—

104

C₂×C₈

—

105

C₂×C₈

—

106

—

106

C₂×C₈

C₈×D₅₃

yes

characteristic

107

C₂×C₈

C₈×D₅₃

yes

characteristic

108

C₂×C₈

C₂×D₂₁₂

yes

characteristic

109

C₂×C₈

#11719945

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

106

—

112

C₄²

—

106

—

113

C₄²

C₂×C₄×D₅₃

yes

characteristic

114

C₁₆

—

115

C₁₆

—

116

C₁₆

—

212

—

117

C₁₆

Dic₂₁₂

yes

complemented

118

Q₈

—

106

—

119

Q₈

—

106

—

120

C₂×C₄

—

106

—

121

C₂×C₄

—

106

—

122

C₂×C₄

C₄×D₂₁₂

yes

characteristic

123

C₂×C₄

C₂×C₈×D₅₃

yes

characteristic

124

C₂×C₄

#13133223

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

212

—

129

C₈

#13133156

yes

—

130

C₈

C₂×Dic₂₁₂

yes

characteristic

131

C₈

#13133239

yes

characteristic

132

C₂²

C₈×D₂₁₂

yes

characteristic

133

C₄

—

134

C₄

—

212

—

135

C₄

—

212

—

136

C₄

C₄×Dic₂₁₂

yes

characteristic

137

C₄

#16810741

yes

characteristic

138

C₄

C₂×#13133156

yes

characteristic

139

C₄

#16811039

yes

characteristic

140

C₂

C₈×Dic₂₁₂

yes

characteristic

141

C₂

#20276548

yes

characteristic

142

C₂

#20276553

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₈×C₁₆×C₅₃	C₂
socle, prime radical	C₂²×C₅₃	C₈×D₄
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₈×C₁₆	D₅₃
hypocenter	C₅₃	#12054
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ C₈×C₁₆

subgroup

G/H

center

C₂×C₈

C₂×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₅₃

Fitting

C₈×C₁₆×C₅₃

C₂

socle, prime radical

C₂²×C₅₃

C₈×D₄

cosocle

C₄×C₈×C₅₃

C₂³

hypercenter

C₈×C₁₆

D₅₃

hypocenter

C₅₃

#12054

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ C₈×C₁₆

N	H ≅ G/N	type
44. C₁₆×C₅₃	Q₁₆	semidirect
87. C₅₃	#12054	semidirect
117. C₁₆	Dic₂₁₂	semidirect

H ≅ G/N

type

44. C₁₆×C₅₃

Q₁₆

semidirect

87. C₅₃

#12054

semidirect

117. C₁₆

Dic₂₁₂

semidirect

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

subgroup	index
#20276623	2
C₈×Dic₂₁₂	2
#20272785	2
#20272695	2
C₈×C₁₆×C₅₃	2
#12054	53

subgroup

index

C₈×Dic₂₁₂

C₈×C₁₆×C₅₃

Subgroups — #27295899

Conjugacy classes of subgroups

Complements

Maximal subgroups