Group #27274547 — subgroups

Subgroups — #27274547

Order 13568.

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

normal subgroups only

This group has 294 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	#20274466	C₂	1	1	yes	maximal, characteristic
2	#20273007	C₂	1	1	yes	maximal, characteristic
3	#20272545	C₂	1	2	yes	maximal
4	#20272543	C₂	1	2	yes	maximal
5	C₅₃×#1450	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	2	yes	—
7	C₂×#13133128	C₂²	1	1	yes	characteristic
8	#16810469	C₂²	1	1	yes	characteristic
9	#16810377	C₄	1	4	yes	—
10	#16810345	C₄	1	4	yes	complemented
11	C₅₃×#431	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#322	C₄	1	2	yes	complemented
14	C₈²×C₅₃	C₄	1	2	yes	complemented
15	C₂²×#88322	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
17	C₄⋊#88322	—	2	2	no	—
18	C₂²⋊#88322	—	2	2	no	—
19	#13133128	—	2	2	no	—
20	C₄×#88322	C₂×C₄	1	4	yes	—
21	#13133124	—	2	2	no	—
22	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
24	C₄⋊C₈×C₅₃	—	2	2	no	—
25	C₂²⋊C₈×C₅₃	—	2	2	no	—
26	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
28	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	#41	1	2	yes	—
32	C₂²×Dic₅₃	—	2	2	no	—
33	C₂×C₈×C₅₃	—	2	4	no	—
34	C₂×C₈×C₅₃	—	2	4	no	—
35	C₂×C₈×C₅₃	C₄²	1	2	yes	—
36	C₂×C₈×C₅₃	C₄²	1	2	yes	—
37	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₄²×C₅₃	#41	1	2	yes	—
40	C₄×Dic₅₃	—	2	4	no	—
41	C₂×#88322	—	2	4	no	—
42	C₂×#88322	—	2	2	no	—
43	C₂×#88322	—	2	4	no	—
44	C₂³×C₅₃	#117	1	1	yes	characteristic
45	C₂×C₄×C₅₃	—	4	2	no	—
46	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
47	C₂×C₄×C₅₃	#117	1	2	yes	—
48	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
49	C₂×C₄×C₅₃	#117	1	2	yes	—
50	C₂×C₄×C₅₃	#131	1	4	yes	—
51	C₂×Dic₅₃	—	4	4	no	—
52	C₂×Dic₅₃	—	4	2	no	—
53	C₈×C₅₃	—	2	2	no	—
54	C₈×C₅₃	—	2	2	no	—
55	C₈×C₅₃	—	4	4	no	—
56	#88322	—	4	4	no	—
57	#88322	—	4	2	no	—
58	#8141	—	53	1	no	maximal, Sylow
59	C₂²×C₅₃	—	2	2	no	—
60	C₂²×C₅₃	#376	1	1	yes	characteristic
61	C₂²×C₅₃	#433	1	2	yes	—
62	C₄×C₅₃	—	2	2	no	—
63	C₄×C₅₃	—	2	2	no	—
64	C₄×C₅₃	#405	1	2	yes	—
65	C₄×C₅₃	#433	1	2	yes	—
66	Dic₅₃	—	8	2	no	—
67	#1746	—	53	1	no	—
68	#1452	—	53	1	no	—
69	#1450	D₅₃	1	1	yes	characteristic
70	#1275	—	53	2	no	—
71	#1274	—	53	2	no	—
72	C₂×C₅₃	—	2	2	no	—
73	C₂×C₅₃	#1746	1	1	yes	characteristic
74	C₂×C₅₃	#1747	1	2	yes	—
75	#431	—	53	2	no	—
76	#431	C₂×D₅₃	1	2	yes	—
77	C₂×#97	—	53	1	no	—
78	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
79	#376	—	53	1	no	—
80	#336	—	53	4	no	—
81	#322	—	53	4	no	—
82	#322	Dic₅₃	1	2	yes	complemented
83	C₈²	Dic₅₃	1	2	yes	complemented
84	C₅₃	#8141	1	1	yes	characteristic, complemented, Sylow
85	C₂²×C₈	—	53	2	no	—
86	C₂²×C₈	C₄×D₅₃	1	2	yes	—
87	C₂×C₄²	—	53	2	no	—
88	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
89	C₄⋊C₈	—	2	2	no	—
90	C₄⋊C₈	—	106	2	no	—
91	C₂²⋊C₈	—	2	2	no	—
92	C₂²⋊C₈	—	106	2	no	—
93	#97	—	106	2	no	—
94	C₄×C₈	—	53	4	no	—
95	C₄×C₈	C₄×D₅₃	1	2	yes	—
96	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
97	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
98	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
99	#95	—	106	2	no	—
100	C₂²×C₄	—	106	2	no	—
101	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
102	C₂²×C₄	#11719945	1	2	yes	—
103	C₂×C₈	—	2	4	no	—
104	C₂×C₈	—	2	4	no	—
105	C₂×C₈	—	106	4	no	—
106	C₂×C₈	—	106	2	no	—
107	C₂×C₈	—	106	4	no	—
108	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
109	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
110	C₄²	—	106	4	no	—
111	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
112	C₄²	#11719945	1	2	yes	—
113	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
114	C₂³	#13133174	1	1	yes	characteristic
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	212	4	no	—
117	C₂×C₄	—	212	2	no	—
118	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
119	C₂×C₄	#13133174	1	2	yes	—
120	C₂×C₄	#13133174	1	1	yes	characteristic
121	C₂×C₄	#13133176	1	2	yes	—
122	C₂×C₄	#13133223	1	4	yes	—
123	C₈	—	2	2	no	—
124	C₈	—	2	2	no	—
125	C₈	—	4	4	no	—
126	C₈	—	212	4	no	—
127	C₈	—	212	2	no	—
128	C₂²	—	2	2	no	—
129	C₂²	#16810469	1	1	yes	characteristic
130	C₂²	#16810682	1	2	yes	—
131	C₄	—	2	2	no	—
132	C₄	—	2	2	no	—
133	C₄	—	424	2	no	—
134	C₄	#16810578	1	2	yes	—
135	C₄	#16810682	1	2	yes	—
136	C₂	—	2	2	no	—
137	C₂	#20274466	1	1	yes	characteristic
138	C₂	#20274470	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20274466

C₂

yes

maximal, characteristic

#20273007

C₂

yes

maximal, characteristic

#20272545

C₂

yes

maximal

#20272543

C₂

yes

maximal

C₅₃×#1450

C₂

yes

maximal, characteristic

#16810676

C₂²

yes

—

C₂×#13133128

C₂²

yes

characteristic

#16810469

C₂²

yes

characteristic

#16810377

C₄

yes

—

#16810345

C₄

yes

complemented

C₅₃×#431

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#322

C₄

yes

complemented

C₈²×C₅₃

C₄

yes

complemented

C₂²×#88322

C₂×C₄

yes

—

C₂×C₄×Dic₅₃

C₂×C₄

yes

—

—

—

—

C₂×C₄

yes

—

#13133124

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

—

C₂×C₈×C₅₃

C₄²

yes

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

#41

yes

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

#117

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

#117

yes

characteristic

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

#376

yes

characteristic

C₂²×C₅₃

#433

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#405

yes

—

C₄×C₅₃

#433

yes

—

Dic₅₃

—

#1746

—

#1452

—

#1450

D₅₃

yes

characteristic

#1275

—

#1274

—

C₂×C₅₃

—

C₂×C₅₃

#1746

yes

characteristic

C₂×C₅₃

#1747

yes

—

#431

—

#431

C₂×D₅₃

yes

—

C₂×#97

—

C₂×C₄×C₈

C₂×D₅₃

yes

—

—

—

Dic₅₃

yes

complemented

C₈²

Dic₅₃

yes

complemented

C₅₃

#8141

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₄×D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

C₂²⋊C₈

—

106

—

#97

—

106

—

C₄×C₈

—

C₄×C₈

C₄×D₅₃

yes

—

C₄×C₈

C₂×Dic₅₃

yes

—

C₄×C₈

C₂×Dic₅₃

yes

characteristic

C₄×C₈

C₂×Dic₅₃

yes

characteristic

#95

—

106

—

100

C₂²×C₄

—

106

—

101

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

102

C₂²×C₄

#11719945

yes

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

—

106

C₂×C₈

—

106

—

107

C₂×C₈

—

106

—

108

C₂×C₈

C₄×Dic₅₃

yes

—

109

C₂×C₈

C₄×Dic₅₃

yes

—

110

C₄²

—

106

—

111

C₄²

C₂×C₄×D₅₃

yes

characteristic

112

C₄²

#11719945

yes

—

113

C₄²

C₂²×Dic₅₃

yes

characteristic

114

C₂³

#13133174

yes

characteristic

115

C₂×C₄

—

116

C₂×C₄

—

212

—

117

C₂×C₄

—

212

—

118

C₂×C₄

C₂×C₄×Dic₅₃

yes

characteristic

119

C₂×C₄

#13133174

yes

—

120

C₂×C₄

#13133174

yes

characteristic

121

C₂×C₄

#13133176

yes

—

122

C₂×C₄

#13133223

yes

—

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

212

—

127

C₈

—

212

—

128

C₂²

—

129

C₂²

#16810469

yes

characteristic

130

C₂²

#16810682

yes

—

131

C₄

—

132

C₄

—

133

C₄

—

424

—

134

C₄

#16810578

yes

—

135

C₄

#16810682

yes

—

136

C₂

—

137

C₂

#20274466

yes

characteristic

138

C₂

#20274470

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1450	C₂
socle	C₂²×C₅₃	#376
prime radical	C₂³×C₅₃	#117
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1450	D₅₃
hypocenter	C₅₃	#8141
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #1450

subgroup

G/H

center

C₂×C₄

C₂×C₄×Dic₅₃

commutator

C₂×C₄×C₅₃

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1450

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ #1450

N	H ≅ G/N	type
10. #16810345	C₄	semidirect
13. C₅₃×#322	C₄	semidirect
14. C₈²×C₅₃	C₄	semidirect
82. #322	Dic₅₃	semidirect
83. C₈²	Dic₅₃	semidirect
84. C₅₃	#8141	semidirect

H ≅ G/N

type

10. #16810345

C₄

semidirect

13. C₅₃×#322

C₄

semidirect

14. C₈²×C₅₃

C₄

semidirect

82. #322

Dic₅₃

semidirect

83. C₈²

Dic₅₃

semidirect

84. C₅₃

#8141

semidirect

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

subgroup	index
#20274466	2
#20273007	2
#20272545	2
#20272543	2
C₅₃×#1450	2
#8141	53

subgroup

index

C₅₃×#1450

Subgroups — #27274547

Conjugacy classes of subgroups

Complements

Maximal subgroups