Group #27271507 — subgroups

Subgroups — #27271507

Order 13568.

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

normal subgroups only

This group has 190 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×#1845	C₂	1	1	yes	maximal, characteristic
3	#16810349	—	2	2	no	—
4	C₂×#107×C₅₃	—	2	2	no	—
5	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
6	C₅₃×#384	C₄	1	1	yes	characteristic
7	C₅₃×#384	C₂²	1	1	yes	characteristic
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	—	4	1	no	—
13	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
14	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
15	#107×C₅₃	—	4	4	no	—
16	C₄×C₈×C₅₃	—	2	1	no	—
17	C₄×C₈×C₅₃	C₈	1	2	yes	complemented
18	C₂²×C₄×C₅₃	—	2	1	no	—
19	C₂²×C₄×C₅₃	—	2	2	no	—
20	C₂²×C₄×C₅₃	—	4	1	no	—
21	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₈×C₅₃	—	2	2	no	—
23	C₂×C₈×C₅₃	—	2	2	no	—
24	C₂×C₈×C₅₃	—	2	2	no	—
25	C₄⋊C₄×C₅₃	—	4	2	no	—
26	C₄⋊C₄×C₅₃	—	4	2	no	—
27	C₄⋊C₄×C₅₃	—	8	2	no	—
28	C₄⋊C₄×C₅₃	—	8	2	no	—
29	C₄²×C₅₃	—	2	1	no	—
30	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
31	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
32	C₂×#88322	—	8	2	no	—
33	C₂³×C₅₃	#99	1	1	yes	characteristic
34	C₂×C₄×C₅₃	—	2	2	no	—
35	C₂×C₄×C₅₃	—	2	2	no	—
36	C₂×C₄×C₅₃	—	2	1	no	—
37	C₂×C₄×C₅₃	—	2	1	no	—
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	—	8	2	no	—
42	C₂×C₄×C₅₃	—	8	2	no	—
43	C₂×C₄×C₅₃	—	8	1	no	—
44	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
45	C₂×C₄×C₅₃	#101	1	1	yes	characteristic
46	C₈×C₅₃	—	2	4	no	—
47	#88322	—	16	2	no	—
48	#7370	—	53	1	no	maximal, Sylow
49	C₂²×C₅₃	—	2	1	no	—
50	C₂²×C₅₃	—	2	1	no	—
51	C₂²×C₅₃	#324	1	1	yes	characteristic
52	C₂²×C₅₃	#353	1	1	yes	characteristic
53	C₂²×C₅₃	#356	1	1	yes	characteristic
54	C₄×C₅₃	—	2	2	no	—
55	C₄×C₅₃	—	2	2	no	—
56	C₄×C₅₃	—	2	1	no	—
57	C₄×C₅₃	—	2	1	no	—
58	C₄×C₅₃	—	8	2	no	—
59	C₄×C₅₃	—	16	1	no	—
60	#1845	D₅₃	1	1	yes	characteristic
61	#1325	—	53	2	no	—
62	C₂×C₅₃	—	2	1	no	—
63	C₂×C₅₃	—	2	1	no	—
64	C₂×C₅₃	#1325	1	1	yes	characteristic
65	C₂×C₅₃	#1408	1	1	yes	characteristic
66	C₂×C₅₃	#1411	1	1	yes	characteristic
67	C₂×#107	—	2	2	no	—
68	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
69	#384	Dic₅₃	1	1	yes	characteristic
70	#384	C₂×D₅₃	1	1	yes	characteristic
71	#324	—	106	2	no	—
72	C₅₃	#7370	1	1	yes	characteristic, complemented, Sylow
73	C₂²×C₈	—	2	1	no	—
74	C₂×C₄⋊C₄	—	2	1	no	—
75	C₂×C₄⋊C₄	—	4	1	no	—
76	C₂×C₄⋊C₄	—	4	1	no	—
77	C₂×C₄⋊C₄	C₂²⋊D₅₃	1	2	yes	—
78	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
79	#107	—	4	4	no	—
80	C₂²⋊C₈	—	212	2	no	—
81	C₄×C₈	—	2	1	no	—
82	C₄×C₈	#88322	1	2	yes	complemented
83	C₂²×C₄	—	2	1	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	4	1	no	—
86	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	424	2	no	—
91	C₄⋊C₄	—	4	2	no	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	8	2	no	—
94	C₄⋊C₄	—	8	2	no	—
95	C₄²	—	2	1	no	—
96	C₄²	C₂×#88322	1	1	yes	characteristic
97	C₄²	#11719917	1	1	yes	characteristic
98	C₂³	#13133132	1	1	yes	characteristic
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	8	2	no	—
107	C₂×C₄	—	8	2	no	—
108	C₂×C₄	—	8	1	no	—
109	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
110	C₂×C₄	#13133136	1	1	yes	characteristic
111	C₈	—	2	4	no	—
112	C₈	—	848	2	no	—
113	C₂²	—	2	1	no	—
114	C₂²	—	2	1	no	—
115	C₂²	#16810349	1	1	yes	characteristic
116	C₂²	#16810416	1	1	yes	characteristic
117	C₂²	#16810422	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	2	1	no	—
121	C₄	—	2	1	no	—
122	C₄	—	8	2	no	—
123	C₄	—	16	1	no	—
124	C₂	—	2	1	no	—
125	C₂	—	2	1	no	—
126	C₂	#20272661	1	1	yes	characteristic
127	C₂	#20272878	1	1	yes	characteristic
128	C₂	#20272884	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20272661

C₂

yes

maximal

C₅₃×#1845

C₂

yes

maximal, characteristic

#16810349

—

C₂×#107×C₅₃

—

C₂×C₄×C₈×C₅₃

C₄

yes

characteristic

C₅₃×#384

C₄

yes

characteristic

C₅₃×#384

C₂²

yes

characteristic

C₂²⋊#88322

—

C₂²×C₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

#107×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

C₈

yes

complemented

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

—

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

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₅₃

#101

yes

characteristic

C₈×C₅₃

—

#88322

—

#7370

—

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

—

#1845

D₅₃

yes

characteristic

#1325

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₄×C₈

Dic₅₃

yes

characteristic

#384

Dic₅₃

yes

characteristic

#384

C₂×D₅₃

yes

characteristic

#324

—

106

—

C₅₃

#7370

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₅₃

yes

—

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

#107

—

C₂²⋊C₈

—

212

—

C₄×C₈

—

C₄×C₈

#88322

yes

complemented

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

—

C₄⋊C₄

—

C₄²

—

C₄²

yes

C₄²

yes

C₂³

yes

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

—

108

C₂×C₄

—

109

C₂×C₄

C₂²⋊#88322

yes

characteristic

110

C₂×C₄

#13133136

yes

characteristic

111

C₈

—

112

C₈

—

848

—

113

C₂²

—

114

C₂²

—

115

C₂²

#16810349

yes

characteristic

116

C₂²

#16810416

yes

characteristic

117

C₂²

#16810422

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₂

—

125

C₂

—

126

C₂

#20272661

yes

characteristic

127

C₂

#20272878

yes

characteristic

128

C₂

#20272884

yes

characteristic

	subgroup	G/H
center	C₂²	#16810416
commutator	C₄²×C₅₃	C₂×C₈
Frattini	#384	C₂×D₅₃
Fitting	C₅₃×#1845	C₂
socle	C₂²×C₅₃	#353
prime radical	C₂³×C₅₃	#99
cosocle	C₅₃×#384	C₂²
hypercenter	#1845	D₅₃
hypocenter	C₅₃	#7370
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #384 ◁ #1845

subgroup

G/H

center

C₂²

#16810416

commutator

C₄²×C₅₃

C₂×C₈

Frattini

#384

C₂×D₅₃

Fitting

C₅₃×#1845

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 ◁ #1845

N	H ≅ G/N	type
17. C₄×C₈×C₅₃	C₈	semidirect
72. C₅₃	#7370	semidirect
82. C₄×C₈	#88322	semidirect

H ≅ G/N

type

17. C₄×C₈×C₅₃

C₈

semidirect

72. C₅₃

#7370

semidirect

82. C₄×C₈

#88322

semidirect

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

subgroup	index
#20272661	2
C₅₃×#1845	2
#7370	53

subgroup

index

#20272661

C₅₃×#1845

#7370

Subgroups — #27271507

Conjugacy classes of subgroups

Complements

Maximal subgroups