Group #27293129 — subgroups

Subgroups — #27293129

Order 13568.

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

normal subgroups only

This group has 260 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	#20276514	C₂	1	2	yes	maximal
2	#20272633	C₂	1	2	yes	maximal, complemented
3	#20272632	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1450	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#11719908	C₄	1	4	yes	—
7	C₂²⋊#11719908	C₄	1	4	yes	—
8	C₄×#11719908	C₂²	1	2	yes	—
9	C₄×#11719908	C₂²	1	2	yes	—
10	C₅₃×#431	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#322	C₄	1	2	yes	—
13	C₈²×C₅₃	C₂²	1	1	yes	characteristic
14	C₈²×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#11719908	C₂×C₄	1	4	yes	—
16	C₂×#11719908	C₂×C₄	1	4	yes	—
17	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₄⋊C₈×C₅₃	C₈	1	4	yes	—
20	C₂²⋊C₈×C₅₃	C₈	1	4	yes	—
21	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
28	C₂×C₈×C₅₃	—	2	4	no	—
29	C₂×C₈×C₅₃	C₄²	1	2	yes	—
30	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
31	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
32	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
33	C₄²×C₅₃	C₄²	1	1	yes	characteristic
34	C₄²×C₅₃	C₂×C₈	1	2	yes	—
35	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	#11719908	—	2	4	no	—
37	#11719908	—	2	4	no	—
38	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
39	C₂×C₄×C₅₃	—	2	2	no	—
40	C₂×C₄×C₅₃	—	2	4	no	—
41	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
43	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
45	C₈×C₅₃	—	2	2	no	—
46	C₈×C₅₃	—	2	4	no	—
47	C₈×C₅₃	—	2	4	no	—
48	C₈×C₅₃	#131	1	4	yes	—
49	#11412	—	53	1	no	maximal, Sylow
50	C₂²×C₅₃	—	2	2	no	—
51	C₂²×C₅₃	—	2	1	no	—
52	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
53	C₄×C₅₃	—	2	2	no	—
54	C₄×C₅₃	—	2	1	no	—
55	C₄×C₅₃	#405	1	2	yes	—
56	C₄×C₅₃	#504	1	4	yes	—
57	#2162	—	53	2	no	—
58	#1450	D₅₃	1	1	yes	characteristic
59	#1312	—	53	2	no	—
60	#1311	—	53	1	no	—
61	C₈×C₁₆	—	53	1	no	—
62	C₂×C₅₃	—	4	1	no	—
63	C₂×C₅₃	#1450	1	1	yes	characteristic
64	C₂×C₅₃	#2108	1	2	yes	—
65	#431	Dic₅₃	1	2	yes	—
66	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
67	C₄⋊C₁₆	—	53	4	no	—
68	C₂²⋊C₁₆	—	53	4	no	—
69	C₄×C₁₆	—	53	2	no	—
70	C₄×C₁₆	—	53	2	no	—
71	#322	Dic₅₃	1	2	yes	—
72	C₈²	C₂×D₅₃	1	1	yes	characteristic
73	C₈²	C₂×D₅₃	1	1	yes	characteristic
74	C₅₃	#11412	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₈	C₄×D₅₃	1	2	yes	—
76	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
77	C₂×C₁₆	—	53	4	no	—
78	C₂×C₁₆	—	53	4	no	—
79	C₄⋊C₈	#88322	1	4	yes	—
80	C₂²⋊C₈	#88322	1	4	yes	—
81	C₄×C₈	C₄×D₅₃	1	2	yes	—
82	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
83	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
84	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
85	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
86	C₂²×C₄	C₂×#88322	1	2	yes	—
87	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
88	C₂×C₈	—	2	4	no	—
89	C₂×C₈	C₂×#88322	1	4	yes	—
90	C₂×C₈	C₂×#88322	1	4	yes	—
91	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
92	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
93	C₄²	C₂×#88322	1	2	yes	—
94	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
95	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
96	C₁₆	—	106	4	no	—
97	C₁₆	—	106	4	no	—
98	C₂³	C₄×#88322	1	1	yes	characteristic
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	C₄×#88322	1	1	yes	characteristic
102	C₂×C₄	C₄×#88322	1	2	yes	—
103	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
104	C₂×C₄	C₂²×#88322	1	2	yes	—
105	C₈	—	2	4	no	—
106	C₈	—	2	4	no	—
107	C₈	—	2	2	no	—
108	C₈	#13133223	1	4	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	—	2	1	no	—
111	C₂²	C₂×C₄×#88322	1	1	yes	characteristic
112	C₄	—	2	2	no	—
113	C₄	—	2	1	no	—
114	C₄	#16810578	1	2	yes	—
115	C₄	#16811040	1	4	yes	—
116	C₂	—	4	1	no	—
117	C₂	#20272998	1	1	yes	characteristic
118	C₂	#20276312	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20276514

C₂

yes

maximal

#20272633

C₂

yes

maximal, complemented

#20272632

C₂

yes

maximal, characteristic, complemented

C₈×#11719908

C₂

yes

maximal, characteristic, complemented

C₅₃×#1450

C₂

yes

maximal, characteristic

C₄⋊#11719908

C₄

yes

—

C₂²⋊#11719908

C₄

yes

—

C₄×#11719908

C₂²

yes

—

C₄×#11719908

C₂²

yes

—

C₅₃×#431

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#322

C₄

yes

—

C₈²×C₅₃

C₂²

yes

characteristic

C₈²×C₅₃

C₂²

yes

characteristic

C₂×#11719908

C₂×C₄

yes

—

C₂×#11719908

C₂×C₄

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₅₃

C₈

yes

—

C₂²⋊C₈×C₅₃

C₈

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₄²

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

—

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

—

C₄²×C₅₃

C₄²

yes

characteristic

C₄²×C₅₃

C₂×C₈

yes

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

#11719908

—

#11719908

—

C₂³×C₅₃

C₄×C₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄×C₈

yes

characteristic

C₂×C₄×C₅₃

C₄×C₈

yes

—

C₂×C₄×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₈

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#131

yes

—

#11412

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄×C₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#405

yes

—

C₄×C₅₃

#504

yes

—

#2162

—

#1450

D₅₃

yes

characteristic

#1312

—

#1311

—

C₈×C₁₆

—

C₂×C₅₃

—

C₂×C₅₃

#1450

yes

characteristic

C₂×C₅₃

#2108

yes

—

#431

Dic₅₃

yes

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

#322

Dic₅₃

yes

—

C₈²

C₂×D₅₃

yes

characteristic

C₈²

C₂×D₅₃

yes

characteristic

C₅₃

#11412

yes

characteristic, complemented, Sylow

C₂²×C₈

C₄×D₅₃

yes

—

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

#88322

yes

—

C₂²⋊C₈

#88322

yes

—

C₄×C₈

C₄×D₅₃

yes

—

C₄×C₈

C₂×Dic₅₃

yes

—

C₄×C₈

C₂×Dic₅₃

yes

—

C₄×C₈

C₂×Dic₅₃

yes

characteristic

C₄×C₈

C₂²×D₅₃

yes

characteristic

C₂²×C₄

C₂×#88322

yes

—

C₂²×C₄

C₄×Dic₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

C₂×#88322

yes

—

C₂×C₈

C₂×#88322

yes

—

C₂×C₈

C₄×Dic₅₃

yes

—

C₂×C₈

C₂×C₄×D₅₃

yes

—

C₄²

C₂×#88322

yes

—

C₄²

C₄×Dic₅₃

yes

characteristic

C₄²

C₂²×Dic₅₃

yes

characteristic

C₁₆

—

106

—

C₁₆

—

106

—

C₂³

C₄×#88322

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₄×#88322

yes

characteristic

102

C₂×C₄

C₄×#88322

yes

—

103

C₂×C₄

C₂×C₄×Dic₅₃

yes

characteristic

104

C₂×C₄

C₂²×#88322

yes

—

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

#13133223

yes

—

109

C₂²

—

110

C₂²

—

111

C₂²

C₂×C₄×#88322

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

#16810578

yes

—

115

C₄

#16811040

yes

—

116

C₂

—

117

C₂

#20272998

yes

characteristic

118

C₂

#20276312

yes

—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₅₃×#1450	C₂
prime radical	C₂³×C₅₃	C₄×C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#1450	D₅₃
hypocenter	C₅₃	#11412
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₈ ◁ #1450

subgroup

G/H

center, Frattini

C₄×C₈

C₂²×D₅₃

commutator, socle

C₂²×C₅₃

C₂×C₄×C₈

Fitting

C₅₃×#1450

C₂

prime radical

C₂³×C₅₃

C₄×C₈

cosocle

C₄×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₄×C₈ ◁ #1450

N	H ≅ G/N	type
2. #20272633	C₂	semidirect
3. #20272632	C₂	semidirect
4. C₈×#11719908	C₂	semidirect
74. C₅₃	#11412	semidirect

H ≅ G/N

type

2. #20272633

C₂

semidirect

3. #20272632

C₂

semidirect

4. C₈×#11719908

C₂

semidirect

74. C₅₃

#11412

semidirect

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

subgroup	index
#20276514	2
#20272633	2
#20272632	2
C₈×#11719908	2
C₅₃×#1450	2
#11412	53

subgroup

index

C₅₃×#1450

Subgroups — #27293129

Conjugacy classes of subgroups

Complements

Maximal subgroups