Group #27277799 — subgroups

Subgroups — #27277799

Order 13568.

228 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 107 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	#20273478	C₂	1	2	yes	maximal
2	#20272577	C₂	1	4	yes	maximal
3	C₅₃×#1450	C₂	1	1	yes	maximal, characteristic
4	#16810680	C₂²	1	4	yes	—
5	#16810376	—	2	2	no	—
6	#16810374	—	2	2	no	—
7	C₅₃×#431	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#322	C₄	1	2	yes	—
10	C₈²×C₅₃	C₄	1	2	yes	—
11	C₄⋊#88322	—	2	4	no	—
12	C₂²⋊#88322	—	2	4	no	—
13	#13133128	—	2	4	no	—
14	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
15	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
16	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
17	C₄⋊C₈×C₅₃	—	2	2	no	—
18	C₂²⋊C₈×C₅₃	—	2	2	no	—
19	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
20	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
23	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	#41	1	2	yes	—
26	C₂×C₈×C₅₃	—	2	2	no	—
27	C₂×C₈×C₅₃	—	2	2	no	—
28	C₂×C₈×C₅₃	—	2	4	no	—
29	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
33	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₄²×C₅₃	#41	1	2	yes	—
36	C₂×#88322	—	4	4	no	—
37	C₂³×C₅₃	#125	1	1	yes	characteristic
38	C₂×C₄×C₅₃	—	2	1	no	—
39	C₂×C₄×C₅₃	—	2	1	no	—
40	C₂×C₄×C₅₃	—	4	2	no	—
41	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	#117	1	2	yes	—
43	C₂×C₄×C₅₃	#124	1	2	yes	—
44	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
45	C₈×C₅₃	—	2	1	no	—
46	C₈×C₅₃	—	2	1	no	—
47	C₈×C₅₃	—	4	4	no	—
48	C₈×C₅₃	#108	1	2	yes	—
49	C₈×C₅₃	#108	1	2	yes	—
50	#88322	—	8	4	no	—
51	#8796	—	53	1	no	maximal, Sylow
52	C₂²×C₅₃	—	2	2	no	—
53	C₂²×C₅₃	—	2	1	no	—
54	C₂²×C₅₃	#382	1	1	yes	characteristic
55	C₄×C₅₃	—	2	2	no	—
56	C₄×C₅₃	—	2	2	no	—
57	C₄×C₅₃	—	2	1	no	—
58	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
59	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
60	#1568	—	53	2	no	—
61	#1450	D₅₃	1	1	yes	characteristic
62	#1287	—	53	4	no	—
63	C₂×C₅₃	—	4	1	no	—
64	C₂×C₅₃	#1833	1	2	yes	—
65	C₂×C₅₃	#1837	1	1	yes	characteristic
66	#432	—	53	4	no	—
67	#431	C₂×D₅₃	1	2	yes	—
68	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
69	#335	—	106	2	no	—
70	#334	—	106	2	no	—
71	#322	Dic₅₃	1	2	yes	—
72	C₈²	Dic₅₃	1	2	yes	—
73	C₅₃	#8796	1	1	yes	characteristic, complemented, Sylow
74	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
75	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
76	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
77	C₄⋊C₈	—	2	2	no	—
78	C₄⋊C₈	—	106	4	no	—
79	C₂²⋊C₈	—	2	2	no	—
80	C₂²⋊C₈	—	106	4	no	—
81	#97	—	106	4	no	—
82	C₄×C₈	D₂₁₂	1	1	yes	characteristic
83	C₄×C₈	D₂₁₂	1	1	yes	characteristic
84	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
85	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
86	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
87	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
88	C₂²×C₄	#11719945	1	2	yes	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	4	no	—
92	C₂×C₈	—	212	4	no	—
93	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
94	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
95	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
96	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
97	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
98	C₄²	#11719945	1	2	yes	—
99	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
100	C₂³	#13133209	1	1	yes	characteristic
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
105	C₂×C₄	#13133176	1	2	yes	—
106	C₂×C₄	#13133205	1	2	yes	—
107	C₂×C₄	#13133209	1	1	yes	characteristic
108	C₈	—	2	1	no	—
109	C₈	—	2	1	no	—
110	C₈	—	4	4	no	—
111	C₈	—	424	4	no	—
112	C₈	#13133153	1	2	yes	—
113	C₈	#13133153	1	2	yes	—
114	C₂²	—	2	2	no	—
115	C₂²	—	2	1	no	—
116	C₂²	#16810489	1	1	yes	characteristic
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	2	1	no	—
120	C₄	C₂×#13133153	1	1	yes	characteristic
121	C₄	C₂×#13133153	1	1	yes	characteristic
122	C₂	—	4	1	no	—
123	C₂	#20274839	1	2	yes	—
124	C₂	#20274853	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20273478

C₂

yes

maximal

#20272577

C₂

yes

maximal

C₅₃×#1450

C₂

yes

maximal, characteristic

#16810680

C₂²

yes

—

#16810376

—

#16810374

—

C₅₃×#431

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#322

C₄

yes

—

C₈²×C₅₃

C₄

yes

—

C₄⋊#88322

—

C₂²⋊#88322

—

#13133128

—

C₂²×C₈×C₅₃

Q₈

yes

characteristic

C₂²×C₈×C₅₃

Q₈

yes

characteristic

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

#41

yes

—

C₂×#88322

—

C₂³×C₅₃

#125

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

#124

yes

—

C₂×C₄×C₅₃

#125

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#108

yes

—

C₈×C₅₃

#108

yes

—

#88322

—

#8796

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#382

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

—

D₅₃

yes

characteristic

#1287

—

C₂×C₅₃

—

C₂×C₅₃

#1833

yes

—

C₂×C₅₃

yes

—

C₂×D₅₃

yes

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#335

—

106

—

#334

—

106

—

#322

Dic₅₃

yes

—

C₈²

Dic₅₃

yes

—

C₅₃

#8796

yes

characteristic, complemented, Sylow

C₂²×C₈

Dic₁₀₆

yes

characteristic

C₂²×C₈

Dic₁₀₆

yes

characteristic

C₂×C₄²

C₂²×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

C₂²⋊C₈

—

106

—

#97

—

106

—

C₄×C₈

D₂₁₂

yes

characteristic

C₄×C₈

D₂₁₂

yes

characteristic

C₄×C₈

C₂×Dic₅₃

yes

—

C₄×C₈

C₂×Dic₅₃

yes

characteristic

C₄×C₈

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂×Dic₁₀₆

yes

characteristic

C₂²×C₄

#11719945

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

212

—

C₂×C₈

C₄⋊Dic₅₃

yes

characteristic

C₂×C₈

C₄⋊Dic₅₃

yes

characteristic

C₂×C₈

C₄⋊Dic₅₃

yes

characteristic

C₂×C₈

C₄⋊Dic₅₃

yes

characteristic

C₄²

C₂×D₂₁₂

yes

characteristic

C₄²

#11719945

yes

—

C₄²

C₂²×Dic₅₃

yes

characteristic

100

C₂³

#13133209

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

105

C₂×C₄

#13133176

yes

—

106

C₂×C₄

#13133205

yes

—

107

C₂×C₄

#13133209

yes

characteristic

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

424

—

112

C₈

#13133153

yes

—

113

C₈

#13133153

yes

—

114

C₂²

—

115

C₂²

—

116

C₂²

#16810489

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

C₂×#13133153

yes

characteristic

121

C₄

C₂×#13133153

yes

characteristic

122

C₂

—

123

C₂

#20274839

yes

—

124

C₂

#20274853

yes

characteristic

	subgroup	G/H
center	C₂²	#16810489
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1450	C₂
socle	C₂²×C₅₃	#382
prime radical	C₂³×C₅₃	#125
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1450	D₅₃
hypocenter	C₅₃	#8796
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

subgroup

G/H

center

C₂²

#16810489

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1450

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

N	H ≅ G/N	type
73. C₅₃	#8796	semidirect

H ≅ G/N

type

73. C₅₃

#8796

semidirect

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

subgroup	index
#20273478	2
#20272577	2
C₅₃×#1450	2
#8796	53

subgroup

index

#20273478

#20272577

C₅₃×#1450

#8796

Subgroups — #27277799

Conjugacy classes of subgroups

Complements

Maximal subgroups