Group #27293130 — subgroups

Subgroups — #27293130

Order 13568.

260 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 131 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	#20276512	C₂	1	2	yes	maximal
2	#20272997	C₂	1	1	yes	maximal, characteristic
3	#20272634	C₂	1	2	yes	maximal, complemented
4	C₈×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1311	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810674	C₄	1	2	yes	—
7	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
8	#16810441	C₄	1	4	yes	—
9	#16810402	C₄	1	4	yes	—
10	C₄×#11719908	C₂²	1	2	yes	—
11	#16810345	C₄	1	2	yes	—
12	C₈×#88322	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
14	C₈²×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
16	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×#11719908	C₂×C₄	1	4	yes	—
18	#13133146	C₈	1	4	yes	—
19	#13133129	C₈	1	4	yes	—
20	C₄×#88322	C₂×C₄	1	1	yes	characteristic
21	C₄×#88322	C₂×C₄	1	2	yes	—
22	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
24	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
26	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
27	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
28	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
29	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
30	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
31	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
32	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
33	C₂×#88322	C₄²	1	2	yes	—
34	C₂×#88322	C₂×C₈	1	4	yes	—
35	C₈×D₅₃	—	2	4	no	—
36	C₁₆×C₅₃	—	2	4	no	—
37	#11719908	—	2	4	no	—
38	C₂²×D₅₃	C₄×C₈	1	1	yes	characteristic
39	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
40	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
41	C₂×Dic₅₃	C₄×C₈	1	1	yes	characteristic
42	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
43	C₄×D₅₃	—	2	2	no	—
44	C₄×D₅₃	—	2	4	no	—
45	C₈×C₅₃	—	2	4	no	—
46	C₈×C₅₃	#131	1	4	yes	—
47	#88322	—	2	2	no	—
48	#88322	—	2	4	no	—
49	#11412	—	53	1	no	maximal, Sylow
50	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
51	C₂×D₅₃	—	2	2	no	—
52	C₂×D₅₃	—	2	1	no	—
53	C₄×C₅₃	#405	1	2	yes	—
54	C₄×C₅₃	#504	1	4	yes	—
55	Dic₅₃	—	2	2	no	—
56	Dic₅₃	—	2	1	no	—
57	#2162	—	53	2	no	—
58	#1450	—	53	1	no	—
59	#1312	—	53	2	no	—
60	#1311	D₅₃	1	1	yes	characteristic, complemented
61	C₈×C₁₆	—	53	1	no	—
62	C₂×C₅₃	#1450	1	1	yes	characteristic
63	C₂×C₅₃	#2108	1	2	yes	—
64	D₅₃	—	4	1	no	—
65	#431	—	53	2	no	—
66	C₂×C₄×C₈	—	53	1	no	—
67	C₄⋊C₁₆	—	53	4	no	—
68	C₂²⋊C₁₆	—	53	4	no	—
69	C₄×C₁₆	—	53	2	no	—
70	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
71	#322	—	53	2	no	—
72	C₈²	—	53	1	no	—
73	C₈²	C₂×D₅₃	1	1	yes	characteristic
74	C₅₃	#11412	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₈	—	53	2	no	—
76	C₂×C₄²	—	53	1	no	—
77	C₂×C₁₆	—	53	4	no	—
78	C₂×C₁₆	C₄×D₅₃	1	4	yes	—
79	C₄⋊C₈	—	53	4	no	—
80	C₂²⋊C₈	—	53	4	no	—
81	C₄×C₈	—	53	2	no	—
82	C₄×C₈	—	53	2	no	—
83	C₄×C₈	—	53	1	no	—
84	C₄×C₈	C₄×D₅₃	1	2	yes	—
85	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
86	C₂²×C₄	—	53	1	no	—
87	C₂²×C₄	—	53	2	no	—
88	C₂×C₈	—	53	2	no	—
89	C₂×C₈	—	53	4	no	—
90	C₂×C₈	—	106	4	no	—
91	C₂×C₈	C₈×D₅₃	1	4	yes	—
92	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
93	C₄²	—	53	1	no	—
94	C₄²	—	53	2	no	—
95	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
96	C₁₆	—	2	4	no	—
97	C₁₆	—	106	4	no	—
98	C₂³	—	53	1	no	—
99	C₂×C₄	—	53	2	no	—
100	C₂×C₄	—	53	1	no	—
101	C₂×C₄	—	106	4	no	—
102	C₂×C₄	—	106	2	no	—
103	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
104	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
105	C₈	—	2	4	no	—
106	C₈	—	106	4	no	—
107	C₈	—	106	2	no	—
108	C₈	#13133223	1	4	yes	—
109	C₂²	—	106	2	no	—
110	C₂²	—	106	1	no	—
111	C₂²	C₄×C₈×D₅₃	1	1	yes	characteristic
112	C₄	—	106	2	no	—
113	C₄	—	106	1	no	—
114	C₄	#16810577	1	2	yes	—
115	C₄	#16811041	1	4	yes	—
116	C₂	—	212	1	no	—
117	C₂	#20272997	1	1	yes	characteristic
118	C₂	#20276313	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20276512

C₂

yes

maximal

#20272997

C₂

yes

maximal, characteristic

#20272634

C₂

yes

maximal, complemented

C₈×#11719908

C₂

yes

maximal, characteristic, complemented

C₅₃×#1311

C₂

yes

maximal, characteristic, complemented

#16810674

C₄

yes

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810441

C₄

yes

—

#16810402

C₄

yes

—

C₄×#11719908

C₂²

yes

—

#16810345

C₄

yes

—

C₈×#88322

C₂²

yes

characteristic

C₄×C₁₆×C₅₃

C₂²

yes

—

C₈²×C₅₃

C₂²

yes

characteristic

C₂×C₈×D₅₃

C₂×C₄

yes

—

C₄²×D₅₃

C₂×C₄

yes

characteristic

C₂×#11719908

C₂×C₄

yes

—

#13133146

C₈

yes

—

#13133129

C₈

yes

—

C₄×#88322

C₂×C₄

yes

characteristic

C₄×#88322

C₂×C₄

yes

—

C₈×Dic₅₃

C₂×C₄

yes

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄×D₅₃

C₄²

yes

characteristic

C₂×C₄×D₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄×Dic₅₃

C₄²

yes

characteristic

C₄×Dic₅₃

C₂×C₈

yes

—

C₂×#88322

C₄²

yes

—

C₂×#88322

C₂×C₈

yes

—

C₈×D₅₃

—

C₁₆×C₅₃

—

#11719908

—

C₂²×D₅₃

C₄×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₈

yes

—

C₂×Dic₅₃

C₄×C₈

yes

characteristic

C₂×Dic₅₃

C₄×C₈

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#131

yes

—

—

—

—

C₂²×C₅₃

C₂×C₄×C₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

#405

yes

—

C₄×C₅₃

#504

yes

—

Dic₅₃

—

Dic₅₃

—

—

—

—

D₅₃

yes

characteristic, complemented

C₈×C₁₆

—

C₂×C₅₃

#1450

yes

characteristic

C₂×C₅₃

#2108

yes

—

D₅₃

—

#431

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₅₃

yes

—

#322

—

C₈²

—

C₈²

C₂×D₅₃

yes

characteristic

C₅₃

#11412

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₅₃

yes

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₅₃

yes

—

C₄×C₈

C₂²×D₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

C₈×D₅₃

yes

—

C₂×C₈

C₂×C₄×D₅₃

yes

—

C₄²

—

C₄²

—

C₄²

C₂×C₄×D₅₃

yes

characteristic

C₁₆

—

C₁₆

—

106

—

C₂³

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

106

—

102

C₂×C₄

—

106

—

103

C₂×C₄

C₄²×D₅₃

yes

characteristic

104

C₂×C₄

C₂×C₈×D₅₃

yes

—

105

C₈

—

106

C₈

—

106

—

107

C₈

—

106

—

108

C₈

#13133223

yes

—

109

C₂²

—

106

—

110

C₂²

—

106

—

111

C₂²

C₄×C₈×D₅₃

yes

characteristic

112

C₄

—

106

—

113

C₄

—

106

—

114

C₄

#16810577

yes

—

115

C₄

#16811041

yes

—

116

C₂

—

212

—

117

C₂

#20272997

yes

characteristic

118

C₂

#20276313

yes

—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₅₃×#1311	C₂
prime radical	C₂²×D₅₃	C₄×C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#1311	D₅₃
hypocenter	C₅₃	#11412
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₈ ◁ #1311

subgroup

G/H

center, Frattini

C₄×C₈

C₂²×D₅₃

commutator, socle

C₂²×C₅₃

C₂×C₄×C₈

Fitting

C₅₃×#1311

C₂

prime radical

C₂²×D₅₃

C₄×C₈

cosocle

C₄×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₄×C₈ ◁ #1311

N	H ≅ G/N	type
3. #20272634	C₂	semidirect
4. C₈×#11719908	C₂	semidirect
5. C₅₃×#1311	C₂	semidirect
60. #1311	D₅₃	semidirect
74. C₅₃	#11412	semidirect

H ≅ G/N

type

3. #20272634

C₂

semidirect

4. C₈×#11719908

C₂

semidirect

5. C₅₃×#1311

C₂

semidirect

60. #1311

D₅₃

semidirect

74. C₅₃

#11412

semidirect

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

subgroup	index
#20276512	2
#20272997	2
#20272634	2
C₈×#11719908	2
C₅₃×#1311	2
#11412	53

subgroup

index

C₅₃×#1311

Subgroups — #27293130

Conjugacy classes of subgroups

Complements

Maximal subgroups