Group #27304577 — subgroups

Subgroups — #27304577

Order 13568.

206 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 123 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	#20272910	C₂	1	4	yes	maximal, complemented
4	C₂²×C₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	C₄	1	1	yes	characteristic
6	#16810578	C₄	1	1	yes	characteristic
7	#16810457	—	2	2	no	—
8	C₂×#13133123	C₂²	1	2	yes	—
9	#16810398	C₄	1	2	yes	—
10	C₁₆×Dic₅₃	C₂²	1	2	yes	—
11	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₃₂×C₅₃	—	2	2	no	—
13	C₂×C₃₂×C₅₃	C₂²	1	2	yes	—
14	C₂×#11719917	C₈	1	1	yes	characteristic
15	#13133176	C₈	1	1	yes	characteristic
16	#13133157	—	2	2	no	—
17	C₂×#11719908	C₂×C₄	1	2	yes	—
18	#13133127	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	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
29	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
30	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
31	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₂²⋊Dic₅₃	C₁₆	1	2	yes	—
33	C₄⋊Dic₅₃	C₁₆	1	2	yes	—
34	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
35	#11719917	—	2	2	no	—
36	C₂×#88322	C₂×C₈	1	2	yes	—
37	C₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
38	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₁₆×C₅₃	C₂×Q₈	1	1	yes	characteristic
40	#11719908	—	2	2	no	—
41	C₂³×C₅₃	C₂×C₁₆	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
43	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
44	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
45	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
46	C₂×Dic₅₃	C₂×C₁₆	1	4	yes	—
47	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
48	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
50	#88322	—	2	2	no	—
51	#13628	—	53	1	no	maximal, Sylow
52	C₂²×C₅₃	—	2	2	no	—
53	C₂²×C₅₃	C₄⋊C₁₆	1	2	yes	—
54	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
55	C₄×C₅₃	C₄⋊C₁₆	1	2	yes	—
56	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
57	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
58	Dic₅₃	—	2	4	no	—
59	C₂×M₆₄	—	53	1	no	—
60	C₂²×C₃₂	D₅₃	1	1	yes	characteristic
61	#2108	—	53	1	no	—
62	C₄⋊C₃₂	—	53	4	no	—
63	C₂×C₅₃	—	2	2	no	—
64	C₂×C₅₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
65	C₂×C₅₃	#2258	1	2	yes	—
66	C₂×M₃₂	—	53	1	no	—
67	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
68	#405	—	53	1	no	—
69	M₆₄	—	106	2	no	—
70	C₂×C₃₂	—	2	2	no	—
71	C₂×C₃₂	—	53	2	no	—
72	C₂×C₃₂	C₂×D₅₃	1	2	yes	—
73	#346	—	53	2	no	—
74	C₄×C₁₆	—	53	2	no	—
75	C₅₃	#13628	1	1	yes	characteristic, complemented, Sylow
76	C₂×M₁₆	—	53	1	no	—
77	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
78	#117	—	53	1	no	—
79	M₃₂	—	106	2	no	—
80	C₂×C₁₆	—	53	2	no	—
81	C₂×C₁₆	Dic₁₀₆	1	2	yes	—
82	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
83	C₂×C₁₆	C₂²⋊D₅₃	1	2	yes	—
84	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
85	#97	—	53	2	no	—
86	C₄×C₈	—	53	2	no	—
87	C₃₂	—	2	2	no	—
88	C₃₂	—	106	2	no	—
89	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
90	M₁₆	—	106	2	no	—
91	C₂×C₈	—	53	2	no	—
92	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
93	C₂×C₈	#11719912	1	2	yes	—
94	C₂×C₈	#11719919	1	2	yes	—
95	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
96	C₄⋊C₄	—	53	2	no	—
97	C₂²⋊C₄	—	53	2	no	—
98	C₄²	—	53	2	no	—
99	C₁₆	—	106	2	no	—
100	C₁₆	#11719919	1	2	yes	—
101	C₁₆	C₂×Dic₁₀₆	1	1	yes	characteristic
102	C₁₆	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
103	C₂³	C₁₆×D₅₃	1	1	yes	characteristic
104	C₂×C₄	—	53	4	no	—
105	C₂×C₄	#13133146	1	2	yes	—
106	C₂×C₄	C₁₆×D₅₃	1	1	yes	characteristic
107	C₂×C₄	#13133156	1	2	yes	—
108	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
109	C₈	—	106	2	no	—
110	C₈	#13133146	1	2	yes	—
111	C₈	C₂×#11719919	1	1	yes	characteristic
112	C₈	C₂×#11719912	1	1	yes	characteristic
113	C₂²	—	2	2	no	—
114	C₂²	#16810441	1	2	yes	—
115	C₂²	C₂×C₁₆×D₅₃	1	1	yes	characteristic
116	C₄	—	106	4	no	—
117	C₄	#16810441	1	2	yes	—
118	C₄	C₂×#13133146	1	1	yes	characteristic
119	C₄	C₂×#13133156	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	C₂×#16810441	1	1	yes	characteristic
122	C₂	#20277008	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810457

C₂

yes

maximal, characteristic

#20276311

C₂

yes

maximal, characteristic

#20272910

C₂

yes

maximal, complemented

C₂²×C₃₂×C₅₃

C₂

yes

maximal, characteristic

C₂×#13133157

C₄

yes

characteristic

#16810578

C₄

yes

characteristic

#16810457

—

C₂×#13133123

C₂²

yes

—

#16810398

C₄

yes

—

C₁₆×Dic₅₃

C₂²

yes

—

C₂²×C₁₆×C₅₃

C₂²

yes

characteristic

C₂×C₃₂×C₅₃

—

C₂×C₃₂×C₅₃

C₂²

yes

—

C₂×#11719917

C₈

yes

characteristic

#13133176

C₈

yes

characteristic

#13133157

—

C₂×#11719908

C₂×C₄

yes

—

#13133127

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

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

C₂²⋊Dic₅₃

C₁₆

yes

—

C₄⋊Dic₅₃

C₁₆

yes

—

C₄×Dic₅₃

C₂×C₈

yes

—

#11719917

—

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×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

—

C₂×M₆₄

—

C₂²×C₃₂

D₅₃

yes

characteristic

#2108

—

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

—

106

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₅₃

yes

—

#346

—

C₄×C₁₆

—

C₅₃

#13628

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

C₄×D₅₃

yes

characteristic

#117

—

M₃₂

—

106

—

C₂×C₁₆

—

C₂×C₁₆

Dic₁₀₆

yes

—

C₂×C₁₆

C₄×D₅₃

yes

characteristic

C₂×C₁₆

C₂²⋊D₅₃

yes

—

C₂×C₁₆

C₂²×D₅₃

yes

characteristic

#97

—

C₄×C₈

—

C₃₂

—

C₃₂

—

106

—

C₂²×C₄

C₈×D₅₃

yes

characteristic

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

C₈×D₅₃

yes

characteristic

C₂×C₈

#11719912

yes

—

C₂×C₈

#11719919

yes

—

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

106

—

100

C₁₆

#11719919

yes

—

101

C₁₆

C₂×Dic₁₀₆

yes

characteristic

102

C₁₆

C₂×C₂²⋊D₅₃

yes

characteristic

103

C₂³

C₁₆×D₅₃

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

#13133146

yes

—

106

C₂×C₄

C₁₆×D₅₃

yes

characteristic

107

C₂×C₄

#13133156

yes

—

108

C₂×C₄

C₂×C₈×D₅₃

yes

characteristic

109

C₈

—

106

—

110

C₈

#13133146

yes

—

111

C₈

C₂×#11719919

yes

characteristic

112

C₈

C₂×#11719912

yes

characteristic

113

C₂²

—

114

C₂²

#16810441

yes

—

115

C₂²

C₂×C₁₆×D₅₃

yes

characteristic

116

C₄

—

106

—

117

C₄

#16810441

yes

—

118

C₄

C₂×#13133146

yes

characteristic

119

C₄

C₂×#13133156

yes

characteristic

120

C₂

—

121

C₂

C₂×#16810441

yes

characteristic

122

C₂

#20277008

yes

—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₁₆
Fitting	C₂²×C₃₂×C₅₃	C₂
prime radical	C₂³×C₅₃	C₂×C₁₆
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	C₂²×C₃₂	D₅₃
hypocenter	C₅₃	#13628
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₃₂

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₅₃

commutator, socle

C₂²×C₅₃

C₂²×C₁₆

Fitting

C₂²×C₃₂×C₅₃

C₂

prime radical

C₂³×C₅₃

C₂×C₁₆

cosocle

C₂×C₁₆×C₅₃

C₂³

hypercenter

C₂²×C₃₂

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₁₆ ◁ C₂²×C₃₂

N	H ≅ G/N	type
3. #20272910	C₂	semidirect
75. C₅₃	#13628	semidirect

H ≅ G/N

type

3. #20272910

C₂

semidirect

75. C₅₃

#13628

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
#20272910	2
C₂²×C₃₂×C₅₃	2
#13628	53

subgroup

index

C₂²×C₃₂×C₅₃

Subgroups — #27304577

Conjugacy classes of subgroups

Complements

Maximal subgroups