Group #27304271 — subgroups

Subgroups — #27304271

Order 13568.

238 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 163 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	#20276311	C₂	1	1	yes	maximal, characteristic
3	#20272848	C₂	1	2	yes	maximal, complemented
4	C₃₂×Dic₅₃	C₂	1	2	yes	maximal, complemented
5	C₂²×C₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₄	1	1	yes	characteristic
7	#16810578	C₄	1	1	yes	characteristic
8	#16810457	C₄	1	4	yes	complemented
9	C₂×#13133123	C₂²	1	2	yes	—
10	#16810398	C₄	1	2	yes	—
11	C₁₆×Dic₅₃	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₃₂×C₅₃	C₄	1	4	yes	complemented
14	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#11719917	C₈	1	1	yes	characteristic
17	#13133176	C₈	1	1	yes	characteristic
18	#13133157	C₈	1	4	yes	—
19	C₂×#11719908	C₂×C₄	1	2	yes	—
20	#13133127	C₈	1	2	yes	—
21	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
22	#13133123	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
26	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
27	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
28	C₃₂×C₅₃	C₂×C₄	1	2	yes	—
29	C₃₂×C₅₃	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
31	C₂×C₈×C₅₃	C₄²	1	2	yes	—
32	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
34	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₂²⋊Dic₅₃	C₁₆	1	2	yes	—
36	C₄⋊Dic₅₃	C₁₆	1	2	yes	—
37	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
38	#11719917	C₁₆	1	4	yes	—
39	C₂×#88322	C₂×C₈	1	2	yes	—
40	C₁₆×C₅₃	C₄²	1	2	yes	—
41	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	#11719908	C₂×C₈	1	4	yes	—
44	C₂³×C₅₃	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
46	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
48	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
49	C₂×Dic₅₃	C₂×C₁₆	1	4	yes	—
50	C₈×C₅₃	C₄×C₈	1	2	yes	—
51	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
52	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
53	#88322	C₂×C₁₆	1	4	yes	—
54	#13549	—	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	C₂²×C₁₆	1	1	yes	characteristic
61	Dic₅₃	—	2	4	no	—
62	C₂×M₆₄	—	53	1	no	—
63	C₂²×C₃₂	D₅₃	1	1	yes	characteristic
64	#2108	—	53	1	no	—
65	#1397	—	53	2	no	—
66	C₄×C₃₂	—	53	2	no	—
67	C₂×C₅₃	—	2	2	no	—
68	C₂×C₅₃	C₂×C₄×C₁₆	1	1	yes	characteristic
69	C₂×C₅₃	#2258	1	2	yes	—
70	C₂×M₃₂	—	53	1	no	—
71	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
72	#405	—	53	1	no	—
73	M₆₄	—	53	4	no	—
74	C₂×C₃₂	—	53	2	no	—
75	C₂×C₃₂	Dic₅₃	1	4	yes	complemented
76	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
77	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
78	#346	—	53	2	no	—
79	C₄×C₁₆	—	53	2	no	—
80	C₅₃	#13549	1	1	yes	characteristic, complemented, Sylow
81	C₂×M₁₆	—	53	1	no	—
82	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
83	#117	—	53	1	no	—
84	M₃₂	—	53	4	no	—
85	C₂×C₁₆	—	53	2	no	—
86	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
87	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
88	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
89	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
90	#97	—	53	2	no	—
91	C₄×C₈	—	53	2	no	—
92	C₃₂	—	53	4	no	—
93	C₃₂	C₂×Dic₅₃	1	2	yes	—
94	C₃₂	C₂×Dic₅₃	1	2	yes	—
95	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
96	M₁₆	—	53	4	no	—
97	C₂×C₈	—	53	2	no	—
98	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
99	C₂×C₈	C₈×D₅₃	1	2	yes	—
100	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
101	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
102	C₄⋊C₄	—	53	2	no	—
103	C₂²⋊C₄	—	53	2	no	—
104	C₄²	—	53	2	no	—
105	C₁₆	—	53	4	no	—
106	C₁₆	C₄×Dic₅₃	1	2	yes	—
107	C₁₆	C₂×C₄×D₅₃	1	1	yes	characteristic
108	C₁₆	C₂²×Dic₅₃	1	1	yes	characteristic
109	C₂³	C₁₆×D₅₃	1	1	yes	characteristic
110	C₂×C₄	—	53	4	no	—
111	C₂×C₄	C₈×Dic₅₃	1	2	yes	—
112	C₂×C₄	C₁₆×D₅₃	1	2	yes	—
113	C₂×C₄	C₁₆×D₅₃	1	1	yes	characteristic
114	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
115	C₈	—	53	4	no	—
116	C₈	C₈×Dic₅₃	1	2	yes	—
117	C₈	C₂×C₄×Dic₅₃	1	1	yes	characteristic
118	C₈	C₂×C₈×D₅₃	1	1	yes	characteristic
119	C₂²	—	2	2	no	—
120	C₂²	C₁₆×Dic₅₃	1	2	yes	—
121	C₂²	C₂×C₁₆×D₅₃	1	1	yes	characteristic
122	C₄	—	106	4	no	—
123	C₄	C₁₆×Dic₅₃	1	2	yes	—
124	C₄	C₂×C₈×Dic₅₃	1	1	yes	characteristic
125	C₄	C₂×C₁₆×D₅₃	1	1	yes	characteristic
126	C₂	—	2	2	no	—
127	C₂	C₂×C₁₆×Dic₅₃	1	1	yes	characteristic
128	C₂	#20277008	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810457

C₂

yes

maximal, characteristic

#20276311

C₂

yes

maximal, characteristic

#20272848

C₂

yes

maximal, complemented

C₃₂×Dic₅₃

C₂

yes

maximal, complemented

C₂²×C₃₂×C₅₃

C₂

yes

maximal, characteristic

C₂×#13133157

C₄

yes

characteristic

#16810578

C₄

yes

characteristic

#16810457

C₄

yes

complemented

C₂×#13133123

C₂²

yes

—

#16810398

C₄

yes

—

C₁₆×Dic₅₃

C₂²

yes

—

C₂²×C₁₆×C₅₃

C₂²

yes

characteristic

C₂×C₃₂×C₅₃

C₄

yes

complemented

C₂×C₃₂×C₅₃

C₂²

yes

characteristic

C₂×C₃₂×C₅₃

C₂²

yes

characteristic

C₂×#11719917

C₈

yes

characteristic

#13133176

C₈

yes

characteristic

#13133157

C₈

yes

—

C₂×#11719908

C₂×C₄

yes

—

#13133127

C₈

yes

—

C₈×Dic₅₃

C₂×C₄

yes

—

#13133123

C₂×C₄

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₁₆×C₅₃

C₂³

yes

characteristic

C₃₂×C₅₃

C₂×C₄

yes

—

C₃₂×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₈×C₅₃

C₄²

yes

—

C₂×C₈×C₅₃

C₂×C₈

yes

characteristic

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂²⋊Dic₅₃

C₁₆

yes

—

C₄⋊Dic₅₃

C₁₆

yes

—

C₄×Dic₅₃

C₂×C₈

yes

—

#11719917

C₁₆

yes

—

C₂×#88322

C₂×C₈

yes

—

C₁₆×C₅₃

C₄²

yes

—

C₁₆×C₅₃

C₂²×C₄

yes

characteristic

C₁₆×C₅₃

C₂²×C₄

yes

characteristic

#11719908

C₂×C₈

yes

—

C₂³×C₅₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅₃

C₄×C₈

yes

—

C₂×C₄×C₅₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×C₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×Dic₅₃

C₂×C₁₆

yes

—

C₈×C₅₃

C₄×C₈

yes

—

C₈×C₅₃

C₂×C₄²

yes

characteristic

C₈×C₅₃

C₂²×C₈

yes

characteristic

#88322

C₂×C₁₆

yes

—

#13549

—

maximal, Sylow

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

C₂²×C₁₆

yes

characteristic

Dic₅₃

—

C₂×M₆₄

—

C₂²×C₃₂

D₅₃

yes

characteristic

#2108

—

#1397

—

C₄×C₃₂

—

C₂×C₅₃

—

C₂×C₅₃

C₂×C₄×C₁₆

yes

characteristic

C₂×C₅₃

#2258

yes

—

C₂×M₃₂

—

C₂²×C₁₆

C₂×D₅₃

yes

characteristic

#405

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

Dic₅₃

yes

complemented

C₂×C₃₂

C₂×D₅₃

yes

characteristic

C₂×C₃₂

C₂×D₅₃

yes

characteristic

#346

—

C₄×C₁₆

—

C₅₃

#13549

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

C₄×D₅₃

yes

characteristic

#117

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₅₃

yes

—

C₂×C₁₆

C₄×D₅₃

yes

characteristic

C₂×C₁₆

C₂×Dic₅₃

yes

—

C₂×C₁₆

C₂²×D₅₃

yes

characteristic

#97

—

C₄×C₈

—

C₃₂

—

C₃₂

C₂×Dic₅₃

yes

—

C₃₂

C₂×Dic₅₃

yes

—

C₂²×C₄

C₈×D₅₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

C₈×D₅₃

yes

characteristic

C₂×C₈

C₈×D₅₃

yes

—

100

C₂×C₈

C₄×Dic₅₃

yes

—

101

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

102

C₄⋊C₄

—

103

C₂²⋊C₄

—

104

C₄²

—

105

C₁₆

—

106

C₁₆

C₄×Dic₅₃

yes

—

107

C₁₆

C₂×C₄×D₅₃

yes

characteristic

108

C₁₆

C₂²×Dic₅₃

yes

characteristic

109

C₂³

C₁₆×D₅₃

yes

characteristic

110

C₂×C₄

—

111

C₂×C₄

C₈×Dic₅₃

yes

—

112

C₂×C₄

C₁₆×D₅₃

yes

—

113

C₂×C₄

C₁₆×D₅₃

yes

characteristic

114

C₂×C₄

C₂×C₈×D₅₃

yes

characteristic

115

C₈

—

116

C₈

C₈×Dic₅₃

yes

—

117

C₈

C₂×C₄×Dic₅₃

yes

characteristic

118

C₈

C₂×C₈×D₅₃

yes

characteristic

119

C₂²

—

120

C₂²

C₁₆×Dic₅₃

yes

—

121

C₂²

C₂×C₁₆×D₅₃

yes

characteristic

122

C₄

—

106

—

123

C₄

C₁₆×Dic₅₃

yes

—

124

C₄

C₂×C₈×Dic₅₃

yes

characteristic

125

C₄

C₂×C₁₆×D₅₃

yes

characteristic

126

C₂

—

127

C₂

C₂×C₁₆×Dic₅₃

yes

characteristic

128

C₂

#20277008

yes

—

	subgroup	G/H
center	C₂×C₃₂	C₂×D₅₃
commutator	C₂×C₅₃	C₂×C₄×C₁₆
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₂²×C₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×C₁₆
prime radical	C₂³×C₅₃	C₂×C₁₆
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	C₂²×C₃₂	D₅₃
hypocenter	C₅₃	#13549
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₃₂ ◁ C₂²×C₃₂

subgroup

G/H

center

C₂×C₃₂

C₂×D₅₃

commutator

C₂×C₅₃

C₂×C₄×C₁₆

Frattini

C₂×C₁₆

C₂²×D₅₃

Fitting

C₂²×C₃₂×C₅₃

C₂

socle

C₂²×C₅₃

C₂²×C₁₆

prime radical

C₂³×C₅₃

C₂×C₁₆

cosocle

C₂×C₁₆×C₅₃

C₂³

hypercenter

C₂²×C₃₂

D₅₃

C₅₃

G ▷ C₂×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₃₂ ◁ C₂²×C₃₂

N	H ≅ G/N	type
3. #20272848	C₂	semidirect
4. C₃₂×Dic₅₃	C₂	semidirect
8. #16810457	C₄	semidirect
13. C₂×C₃₂×C₅₃	C₄	semidirect
75. C₂×C₃₂	Dic₅₃	semidirect
80. C₅₃	#13549	semidirect

H ≅ G/N

type

3. #20272848

C₂

semidirect

4. C₃₂×Dic₅₃

C₂

semidirect

8. #16810457

C₄

semidirect

13. C₂×C₃₂×C₅₃

C₄

semidirect

75. C₂×C₃₂

Dic₅₃

semidirect

80. C₅₃

#13549

semidirect

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

subgroup	index
C₂×#16810457	2
#20276311	2
#20272848	2
C₃₂×Dic₅₃	2
C₂²×C₃₂×C₅₃	2
#13549	53

subgroup

index

C₂×#16810457

#20276311

#20272848

C₃₂×Dic₅₃

C₂²×C₃₂×C₅₃

#13549

Subgroups — #27304271

Conjugacy classes of subgroups

Complements

Maximal subgroups