Group #27304574 — subgroups

Subgroups — #27304574

Order 13568.

218 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 127 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×#16810457	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#20272910	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×M₆₄	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	C₄	1	1	yes	characteristic
6	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
7	#16810457	—	2	2	no	—
8	C₂×#13133123	C₂²	1	2	yes	—
9	C₁₆×Dic₅₃	C₄	1	2	yes	—
10	C₁₆×Dic₅₃	C₂²	1	2	yes	—
11	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×M₆₄	—	2	2	no	—
13	C₂×C₃₂×C₅₃	C₂²	1	2	yes	—
14	C₂²×#88322	C₈	1	1	yes	characteristic
15	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
16	C₂×#11719908	—	2	2	no	—
17	C₂×#11719908	C₂×C₄	1	2	yes	—
18	C₈×Dic₅₃	C₈	1	2	yes	—
19	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
20	#13133123	—	2	2	no	—
21	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
24	C₂×C₁₆×C₅₃	Q₈	1	2	yes	—
25	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
26	C₃₂×C₅₃	—	2	2	no	—
27	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
28	C₂²×Dic₅₃	C₁₆	1	2	yes	—
29	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
30	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
31	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
32	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₄×Dic₅₃	C₁₆	1	1	yes	characteristic
34	C₄×Dic₅₃	C₁₆	1	1	yes	characteristic
35	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
36	C₂×#88322	—	2	2	no	—
37	C₂×#88322	C₂×C₈	1	2	yes	—
38	C₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
39	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₁₆×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	#11719908	—	2	2	no	—
42	C₂³×C₅₃	C₂×C₁₆	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
44	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
46	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
47	C₂×Dic₅₃	—	2	4	no	—
48	C₂×Dic₅₃	C₂×C₁₆	1	2	yes	—
49	C₂×Dic₅₃	C₂×C₁₆	1	2	yes	—
50	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
51	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
53	#88322	—	2	2	no	—
54	#13627	—	53	1	no	maximal, Sylow
55	C₂²×C₅₃	—	2	2	no	—
56	C₂²×C₅₃	C₄⋊C₁₆	1	2	yes	—
57	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
58	C₄×C₅₃	C₄⋊C₁₆	1	2	yes	—
59	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
60	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
61	Dic₅₃	—	2	2	no	—
62	Dic₅₃	M₆₄	1	4	yes	complemented
63	C₂×M₆₄	—	53	1	no	—
64	C₂×M₆₄	D₅₃	1	1	yes	characteristic
65	C₂×C₄×C₁₆	—	53	1	no	—
66	C₄⋊C₃₂	—	53	4	no	—
67	C₂×C₅₃	—	2	2	no	—
68	C₂×C₅₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
69	C₂×C₅₃	C₂×M₆₄	1	2	yes	—
70	C₂²×C₁₆	—	53	1	no	—
71	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
72	C₂×C₄×C₈	—	53	1	no	—
73	M₆₄	—	2	2	no	—
74	M₆₄	—	106	2	no	—
75	C₂×C₃₂	—	53	2	no	—
76	C₂×C₃₂	C₂×D₅₃	1	2	yes	—
77	C₄×C₁₆	—	53	2	no	—
78	C₄×C₁₆	—	53	2	no	—
79	C₅₃	#13627	1	1	yes	characteristic, complemented, Sylow
80	C₂²×C₈	—	53	1	no	—
81	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
82	C₂×C₄²	—	53	1	no	—
83	C₂×C₁₆	—	53	2	no	—
84	C₂×C₁₆	—	106	2	no	—
85	C₂×C₁₆	Dic₁₀₆	1	2	yes	—
86	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
87	C₂×C₁₆	C₂²⋊D₅₃	1	2	yes	—
88	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
89	C₄×C₈	—	53	2	no	—
90	C₄×C₈	—	53	2	no	—
91	C₃₂	—	2	2	no	—
92	C₃₂	—	106	2	no	—
93	C₂²×C₄	—	53	2	no	—
94	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
95	C₂×C₈	—	53	2	no	—
96	C₂×C₈	—	106	2	no	—
97	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
98	C₂×C₈	#11719912	1	2	yes	—
99	C₂×C₈	#11719919	1	2	yes	—
100	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
101	C₄²	—	53	1	no	—
102	C₄²	—	53	1	no	—
103	C₄²	—	53	2	no	—
104	C₁₆	—	106	2	no	—
105	C₁₆	#11719919	1	2	yes	—
106	C₁₆	C₂×Dic₁₀₆	1	1	yes	characteristic
107	C₁₆	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
108	C₂³	C₁₆×D₅₃	1	1	yes	characteristic
109	C₂×C₄	—	53	2	no	—
110	C₂×C₄	—	53	2	no	—
111	C₂×C₄	—	106	4	no	—
112	C₂×C₄	#13133146	1	2	yes	—
113	C₂×C₄	C₁₆×D₅₃	1	1	yes	characteristic
114	C₂×C₄	#13133156	1	2	yes	—
115	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
116	C₈	—	106	2	no	—
117	C₈	#13133146	1	2	yes	—
118	C₈	C₂×#11719919	1	1	yes	characteristic
119	C₈	C₂×#11719912	1	1	yes	characteristic
120	C₂²	—	2	2	no	—
121	C₂²	#16810441	1	2	yes	—
122	C₂²	C₂×C₁₆×D₅₃	1	1	yes	characteristic
123	C₄	—	53	4	no	—
124	C₄	—	106	2	no	—
125	C₄	#16810441	1	2	yes	—
126	C₄	C₂×#13133146	1	1	yes	characteristic
127	C₄	C₂×#13133156	1	1	yes	characteristic
128	C₂	—	2	2	no	—
129	C₂	C₂×#16810441	1	1	yes	characteristic
130	C₂	M₆₄×D₅₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810457

C₂

yes

maximal, characteristic

C₂×C₁₆×Dic₅₃

C₂

yes

maximal, characteristic

#20272910

C₂

yes

maximal, complemented

C₂×C₅₃×M₆₄

C₂

yes

maximal, characteristic

C₂²×#11719908

C₄

yes

characteristic

C₂×C₈×Dic₅₃

C₄

yes

characteristic

#16810457

—

C₂×#13133123

C₂²

yes

—

C₁₆×Dic₅₃

C₄

yes

—

C₁₆×Dic₅₃

C₂²

yes

—

C₂²×C₁₆×C₅₃

C₂²

yes

characteristic

C₅₃×M₆₄

—

C₂×C₃₂×C₅₃

C₂²

yes

—

C₂²×#88322

C₈

yes

characteristic

C₂×C₄×Dic₅₃

C₈

yes

characteristic

C₂×#11719908

—

C₂×#11719908

C₂×C₄

yes

—

C₈×Dic₅₃

C₈

yes

—

C₈×Dic₅₃

C₂×C₄

yes

—

#13133123

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

D₄

yes

—

C₂×C₁₆×C₅₃

Q₈

yes

—

C₂×C₁₆×C₅₃

C₂³

yes

characteristic

C₃₂×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂²×Dic₅₃

C₁₆

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₂×C₈

yes

characteristic

C₂×C₈×C₅₃

M₁₆

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₄×Dic₅₃

C₁₆

yes

characteristic

C₄×Dic₅₃

C₁₆

yes

characteristic

C₄×Dic₅₃

C₂×C₈

yes

—

C₂×#88322

—

C₂×#88322

C₂×C₈

yes

—

C₁₆×C₅₃

C₄⋊C₄

yes

—

C₁₆×C₅₃

C₂×D₄

yes

characteristic

C₁₆×C₅₃

C₂×Q₈

yes

characteristic

#11719908

—

C₂³×C₅₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅₃

C₄⋊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

—

C₂×Dic₅₃

C₂×C₁₆

yes

—

C₈×C₅₃

C₄⋊C₈

yes

—

C₈×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₈×C₅₃

C₂×M₁₆

yes

—

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄⋊C₁₆

yes

—

C₂²×C₅₃

C₂²×C₁₆

yes

characteristic

C₄×C₅₃

C₄⋊C₁₆

yes

—

C₄×C₅₃

C₂×C₄⋊C₈

yes

characteristic

C₄×C₅₃

C₂×M₃₂

yes

characteristic

Dic₅₃

—

Dic₅₃

M₆₄

yes

complemented

C₂×M₆₄

—

C₂×M₆₄

D₅₃

yes

characteristic

C₂×C₄×C₁₆

—

C₄⋊C₃₂

—

C₂×C₅₃

—

C₂×C₅₃

C₂×C₄⋊C₁₆

yes

characteristic

C₂×C₅₃

C₂×M₆₄

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₅₃

yes

characteristic

C₂×C₄×C₈

—

M₆₄

—

M₆₄

—

106

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₅₃

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

C₅₃

#13627

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₄×D₅₃

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₂×C₁₆

Dic₁₀₆

yes

—

C₂×C₁₆

C₄×D₅₃

yes

characteristic

C₂×C₁₆

C₂²⋊D₅₃

yes

—

C₂×C₁₆

C₂²×D₅₃

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₃₂

—

C₃₂

—

106

—

C₂²×C₄

—

C₂²×C₄

C₈×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

C₈×D₅₃

yes

characteristic

C₂×C₈

#11719912

yes

—

C₂×C₈

#11719919

yes

—

100

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

101

C₄²

—

102

C₄²

—

103

C₄²

—

104

C₁₆

—

106

—

105

C₁₆

#11719919

yes

—

106

C₁₆

C₂×Dic₁₀₆

yes

characteristic

107

C₁₆

C₂×C₂²⋊D₅₃

yes

characteristic

108

C₂³

C₁₆×D₅₃

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

106

—

112

C₂×C₄

#13133146

yes

—

113

C₂×C₄

C₁₆×D₅₃

yes

characteristic

114

C₂×C₄

#13133156

yes

—

115

C₂×C₄

C₂×C₈×D₅₃

yes

characteristic

116

C₈

—

106

—

117

C₈

#13133146

yes

—

118

C₈

C₂×#11719919

yes

characteristic

119

C₈

C₂×#11719912

yes

characteristic

120

C₂²

—

121

C₂²

#16810441

yes

—

122

C₂²

C₂×C₁₆×D₅₃

yes

characteristic

123

C₄

—

124

C₄

—

106

—

125

C₄

#16810441

yes

—

126

C₄

C₂×#13133146

yes

characteristic

127

C₄

C₂×#13133156

yes

characteristic

128

C₂

—

129

C₂

C₂×#16810441

yes

characteristic

130

C₂

M₆₄×D₅₃

yes

—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₁₆
Fitting	C₂×C₅₃×M₆₄	C₂
prime radical	C₂³×C₅₃	C₂×C₁₆
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	C₂×M₆₄	D₅₃
hypocenter	C₅₃	#13627
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₅₃

commutator, socle

C₂²×C₅₃

C₂²×C₁₆

Fitting

C₂×C₅₃×M₆₄

C₂

prime radical

C₂³×C₅₃

C₂×C₁₆

cosocle

C₂×C₁₆×C₅₃

C₂³

hypercenter

C₂×M₆₄

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

N	H ≅ G/N	type
3. #20272910	C₂	semidirect
62. Dic₅₃	M₆₄	semidirect
79. C₅₃	#13627	semidirect

H ≅ G/N

type

3. #20272910

C₂

semidirect

62. Dic₅₃

M₆₄

semidirect

79. C₅₃

#13627

semidirect

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

subgroup	index
C₂×#16810457	2
C₂×C₁₆×Dic₅₃	2
#20272910	2
C₂×C₅₃×M₆₄	2
#13627	53

subgroup

index

C₂×#16810457

C₂×C₁₆×Dic₅₃

#20272910

C₂×C₅₃×M₆₄

#13627

Subgroups — #27304574

Conjugacy classes of subgroups

Complements

Maximal subgroups