Group #27271501 — subgroups

Subgroups — #27271501

Order 13568.

162 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 47 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	#20272661	C₂	1	2	yes	maximal
2	C₅₃×#1836	C₂	1	1	yes	maximal, characteristic
3	#16810349	—	2	2	no	—
4	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
5	C₅₃×#384	C₂²	1	1	yes	characteristic
6	C₅₃×#382	C₄	1	1	yes	characteristic
7	C₅₃×#340	—	2	2	no	—
8	C₂²⋊#88322	—	4	2	no	—
9	C₂²×C₈×C₅₃	—	2	1	no	—
10	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
11	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
12	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
13	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
14	#97×C₅₃	—	2	1	no	—
15	#97×C₅₃	C₈	1	2	yes	complemented
16	#95×C₅₃	—	4	1	no	—
17	C₂²×C₄×C₅₃	—	2	1	no	—
18	C₂²×C₄×C₅₃	—	2	2	no	—
19	C₂²×C₄×C₅₃	—	4	1	no	—
20	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
21	C₂×C₈×C₅₃	—	2	2	no	—
22	C₂×C₈×C₅₃	—	4	1	no	—
23	C₂×C₈×C₅₃	—	4	1	no	—
24	C₄⋊C₄×C₅₃	—	4	2	no	—
25	C₄⋊C₄×C₅₃	—	4	2	no	—
26	C₄⋊C₄×C₅₃	—	8	2	no	—
27	C₄²×C₅₃	—	2	1	no	—
28	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
29	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
30	C₂×#88322	—	8	2	no	—
31	C₂³×C₅₃	#99	1	1	yes	characteristic
32	C₂×C₄×C₅₃	—	2	1	no	—
33	C₂×C₄×C₅₃	—	2	1	no	—
34	C₂×C₄×C₅₃	—	2	1	no	—
35	C₂×C₄×C₅₃	—	2	1	no	—
36	C₂×C₄×C₅₃	—	4	1	no	—
37	C₂×C₄×C₅₃	—	4	1	no	—
38	C₂×C₄×C₅₃	—	4	2	no	—
39	C₂×C₄×C₅₃	—	8	2	no	—
40	C₂×C₄×C₅₃	—	8	1	no	—
41	C₂×C₄×C₅₃	—	8	1	no	—
42	C₂×C₄×C₅₃	—	8	1	no	—
43	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
44	C₂×C₄×C₅₃	#101	1	1	yes	characteristic
45	C₈×C₅₃	—	4	2	no	—
46	#88322	—	16	2	no	—
47	#7367	—	53	1	no	maximal, Sylow
48	C₂²×C₅₃	—	2	1	no	—
49	C₂²×C₅₃	—	2	1	no	—
50	C₂²×C₅₃	#324	1	1	yes	characteristic
51	C₂²×C₅₃	#354	1	1	yes	characteristic
52	C₂²×C₅₃	#355	1	1	yes	characteristic
53	C₄×C₅₃	—	2	1	no	—
54	C₄×C₅₃	—	2	1	no	—
55	C₄×C₅₃	—	4	1	no	—
56	C₄×C₅₃	—	4	1	no	—
57	C₄×C₅₃	—	8	2	no	—
58	C₄×C₅₃	—	16	1	no	—
59	#1836	D₅₃	1	1	yes	characteristic
60	#1325	—	53	2	no	—
61	C₂×C₅₃	—	2	1	no	—
62	C₂×C₅₃	—	2	1	no	—
63	C₂×C₅₃	#1324	1	1	yes	characteristic
64	C₂×C₅₃	#1412	1	1	yes	characteristic
65	C₂×C₅₃	#1413	1	1	yes	characteristic
66	C₂×#97	Dic₅₃	1	1	yes	characteristic
67	#384	C₂×D₅₃	1	1	yes	characteristic
68	#382	Dic₅₃	1	1	yes	characteristic
69	#340	—	2	2	no	—
70	#324	—	106	2	no	—
71	C₅₃	#7367	1	1	yes	characteristic, complemented, Sylow
72	C₂²×C₈	—	2	1	no	—
73	C₂×C₄⋊C₄	—	2	1	no	—
74	C₂×C₄⋊C₄	—	4	1	no	—
75	C₂×C₄⋊C₄	C₂²⋊D₅₃	1	2	yes	—
76	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
77	C₂²⋊C₈	—	212	2	no	—
78	#97	—	2	1	no	—
79	#97	#88322	1	2	yes	complemented
80	#95	—	4	1	no	—
81	C₂²×C₄	—	2	1	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	4	1	no	—
84	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	4	1	no	—
87	C₂×C₈	—	4	1	no	—
88	C₂×C₈	—	424	2	no	—
89	C₄⋊C₄	—	4	2	no	—
90	C₄⋊C₄	—	4	2	no	—
91	C₄⋊C₄	—	8	2	no	—
92	C₄²	—	2	1	no	—
93	C₄²	C₂×#88322	1	1	yes	characteristic
94	C₄²	#11719917	1	1	yes	characteristic
95	C₂³	#13133132	1	1	yes	characteristic
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	4	1	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	8	2	no	—
104	C₂×C₄	—	8	1	no	—
105	C₂×C₄	—	8	1	no	—
106	C₂×C₄	—	8	1	no	—
107	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
108	C₂×C₄	#13133136	1	1	yes	characteristic
109	C₈	—	4	2	no	—
110	C₈	—	848	2	no	—
111	C₂²	—	2	1	no	—
112	C₂²	—	2	1	no	—
113	C₂²	#16810349	1	1	yes	characteristic
114	C₂²	#16810418	1	1	yes	characteristic
115	C₂²	#16810420	1	1	yes	characteristic
116	C₄	—	2	1	no	—
117	C₄	—	2	1	no	—
118	C₄	—	4	1	no	—
119	C₄	—	4	1	no	—
120	C₄	—	8	2	no	—
121	C₄	—	16	1	no	—
122	C₂	—	2	1	no	—
123	C₂	—	2	1	no	—
124	C₂	#20272659	1	1	yes	characteristic
125	C₂	#20272886	1	1	yes	characteristic
126	C₂	#20272888	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20272661

C₂

yes

maximal

C₅₃×#1836

C₂

yes

maximal, characteristic

#16810349

—

C₂×#97×C₅₃

C₄

yes

characteristic

C₅₃×#384

C₂²

yes

characteristic

C₅₃×#382

C₄

yes

characteristic

C₅₃×#340

—

C₂²⋊#88322

—

C₂²×C₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

#97×C₅₃

—

#97×C₅₃

C₈

yes

complemented

#95×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₈

yes

characteristic

C₄²×C₅₃

M₁₆

yes

characteristic

C₂×#88322

—

C₂³×C₅₃

#99

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₅₃

#101

yes

characteristic

C₈×C₅₃

—

#88322

—

#7367

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

#1836

D₅₃

yes

characteristic

#1325

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×#97

Dic₅₃

yes

characteristic

#384

C₂×D₅₃

yes

characteristic

#382

Dic₅₃

yes

characteristic

#340

—

#324

—

106

—

C₅₃

#7367

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₅₃

yes

—

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

C₂²⋊C₈

—

212

—

—

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊Dic₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

424

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

yes

C₄²

yes

C₂³

yes

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₂²⋊#88322

yes

characteristic

108

C₂×C₄

#13133136

yes

characteristic

109

C₈

—

110

C₈

—

848

—

111

C₂²

—

112

C₂²

—

113

C₂²

#16810349

yes

characteristic

114

C₂²

#16810418

yes

characteristic

115

C₂²

#16810420

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₂

—

123

C₂

—

124

C₂

#20272659

yes

characteristic

125

C₂

#20272886

yes

characteristic

126

C₂

#20272888

yes

characteristic

	subgroup	G/H
center	C₂²	#16810418
commutator	C₄²×C₅₃	C₂×C₈
Frattini	#384	C₂×D₅₃
Fitting	C₅₃×#1836	C₂
socle	C₂²×C₅₃	#354
prime radical	C₂³×C₅₃	#99
cosocle	C₅₃×#384	C₂²
hypercenter	#1836	D₅₃
hypocenter	C₅₃	#7367
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #384 ◁ #1836

subgroup

G/H

center

C₂²

#16810418

commutator

C₄²×C₅₃

C₂×C₈

Frattini

#384

C₂×D₅₃

Fitting

C₅₃×#1836

C₂

C₂²×C₅₃

C₂³×C₅₃

C₅₃×#384

C₂²

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #384 ◁ #1836

N	H ≅ G/N	type
15. #97×C₅₃	C₈	semidirect
71. C₅₃	#7367	semidirect
79. #97	#88322	semidirect

H ≅ G/N

type

15. #97×C₅₃

C₈

semidirect

71. C₅₃

#7367

semidirect

79. #97

#88322

semidirect

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

subgroup	index
#20272661	2
C₅₃×#1836	2
#7367	53

subgroup

index

#20272661

C₅₃×#1836

#7367

Subgroups — #27271501

Conjugacy classes of subgroups

Complements

Maximal subgroups