Group #27270918 — subgroups

Subgroups — #27270918

Order 13568.

202 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 71 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	#20272797	C₂	1	2	yes	maximal
2	C₅₃×#1991	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133157	—	2	2	no	—
4	C₂×#125×C₅₃	C₄	1	1	yes	characteristic
5	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
6	C₅₃×#382	C₄	1	1	yes	characteristic
7	C₅₃×#340	—	2	2	no	—
8	#13133157	—	2	4	no	—
9	#13133157	—	4	2	no	—
10	C₂×#11719908	—	4	2	no	—
11	C₂²×C₈×C₅₃	D₄	1	2	yes	—
12	#125×C₅₃	—	2	1	no	—
13	#125×C₅₃	—	4	1	no	—
14	#125×C₅₃	C₈	1	2	yes	—
15	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
16	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
17	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
18	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₄×C₈×C₅₃	—	4	1	no	—
20	#95×C₅₃	—	4	1	no	—
21	C₂²×C₄×C₅₃	—	2	1	no	—
22	C₂²×C₄×C₅₃	—	4	1	no	—
23	C₂²×C₄×C₅₃	—	4	1	no	—
24	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₈×C₅₃	—	2	2	no	—
26	C₂×C₈×C₅₃	—	2	2	no	—
27	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
28	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
29	C₄⋊C₄×C₅₃	—	2	2	no	—
30	C₄⋊C₄×C₅₃	—	4	2	no	—
31	C₄⋊C₄×C₅₃	—	4	2	no	—
32	C₄⋊C₄×C₅₃	—	4	1	no	—
33	C₄⋊C₄×C₅₃	—	4	2	no	—
34	C₄²×C₅₃	—	2	1	no	—
35	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
36	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
37	#11719908	—	4	4	no	—
38	C₂³×C₅₃	#100	1	1	yes	characteristic
39	C₂×C₄×C₅₃	—	2	1	no	—
40	C₂×C₄×C₅₃	—	2	1	no	—
41	C₂×C₄×C₅₃	—	4	2	no	—
42	C₂×C₄×C₅₃	—	4	2	no	—
43	C₂×C₄×C₅₃	—	8	2	no	—
44	C₂×C₄×C₅₃	—	8	2	no	—
45	C₂×C₄×C₅₃	—	8	1	no	—
46	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
47	C₂×C₄×C₅₃	#99	1	1	yes	characteristic
48	C₂×C₄×C₅₃	#103	1	2	yes	—
49	C₂×C₄×C₅₃	#104	1	2	yes	—
50	C₈×C₅₃	—	2	2	no	—
51	C₈×C₅₃	—	2	2	no	—
52	#7135	—	53	1	no	maximal, Sylow
53	C₂²×C₅₃	—	2	2	no	—
54	C₂²×C₅₃	#323	1	1	yes	characteristic
55	C₂²×C₅₃	#329	1	1	yes	characteristic
56	C₂²×C₅₃	#333	1	1	yes	characteristic
57	C₄×C₅₃	—	4	1	no	—
58	C₄×C₅₃	—	4	1	no	—
59	C₄×C₅₃	—	8	2	no	—
60	C₄×C₅₃	—	16	1	no	—
61	C₄×C₅₃	Q₈⋊C₈	1	2	yes	—
62	C₄×C₅₃	#328	1	2	yes	—
63	#1991	D₅₃	1	1	yes	characteristic
64	#1376	—	53	2	no	—
65	C₂×C₅₃	—	2	1	no	—
66	C₂×C₅₃	—	2	1	no	—
67	C₂×C₅₃	#1272	1	1	yes	characteristic
68	C₂×C₅₃	#1366	1	2	yes	—
69	C₂×#125	Dic₅₃	1	1	yes	characteristic
70	C₂×M₃₂	—	106	2	no	—
71	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
72	#382	Dic₅₃	1	1	yes	characteristic
73	#340	—	2	2	no	—
74	C₅₃	#7135	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
76	#125	—	2	1	no	—
77	#125	—	4	1	no	—
78	#125	#88322	1	2	yes	—
79	C₂×C₄⋊C₄	—	2	1	no	—
80	C₂×C₄⋊C₄	—	2	1	no	—
81	C₂×C₄⋊C₄	—	4	1	no	—
82	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
83	M₃₂	—	106	4	no	—
84	M₃₂	—	212	2	no	—
85	C₂×C₁₆	—	212	2	no	—
86	C₄×C₈	—	4	1	no	—
87	#95	—	4	1	no	—
88	C₂²×C₄	—	2	1	no	—
89	C₂²×C₄	—	4	1	no	—
90	C₂²×C₄	—	4	1	no	—
91	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	#11719924	1	2	yes	—
95	C₂×C₈	#11719925	1	2	yes	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	4	2	no	—
99	C₄⋊C₄	—	4	1	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄²	—	2	1	no	—
102	C₄²	C₂×#88322	1	1	yes	characteristic
103	C₄²	#11719917	1	1	yes	characteristic
104	C₁₆	—	212	4	no	—
105	C₂³	#13133134	1	1	yes	characteristic
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	—	8	2	no	—
112	C₂×C₄	—	8	1	no	—
113	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
114	C₂×C₄	#13133132	1	1	yes	characteristic
115	C₂×C₄	#13133141	1	2	yes	—
116	C₂×C₄	#13133144	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	2	2	no	—
119	C₂²	—	2	2	no	—
120	C₂²	#16810347	1	1	yes	characteristic
121	C₂²	#16810363	1	1	yes	characteristic
122	C₂²	#16810372	1	1	yes	characteristic
123	C₄	—	4	1	no	—
124	C₄	—	4	1	no	—
125	C₄	—	8	2	no	—
126	C₄	—	16	1	no	—
127	C₄	Q₈⋊#88322	1	2	yes	—
128	C₄	#16810360	1	2	yes	—
129	C₂	—	2	1	no	—
130	C₂	—	2	1	no	—
131	C₂	#20272541	1	1	yes	characteristic
132	C₂	#20272776	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20272797

C₂

yes

maximal

C₅₃×#1991

C₂

yes

maximal, characteristic

C₂×#13133157

—

C₂×#125×C₅₃

C₄

yes

characteristic

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#382

C₄

yes

characteristic

C₅₃×#340

—

#13133157

—

#13133157

—

C₂×#11719908

—

C₂²×C₈×C₅₃

D₄

yes

—

#125×C₅₃

—

#125×C₅₃

—

#125×C₅₃

C₈

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

—

#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

SD₁₆

yes

—

C₂×C₈×C₅₃

Q₁₆

yes

—

C₄⋊C₄×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

#11719908

—

C₂³×C₅₃

#100

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

C₂²⋊C₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

#104

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

#7135

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

Q₈⋊C₈

yes

—

C₄×C₅₃

#328

yes

—

#1991

D₅₃

yes

characteristic

#1376

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1272

yes

characteristic

C₂×C₅₃

#1366

yes

—

C₂×#125

Dic₅₃

yes

characteristic

C₂×M₃₂

—

106

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#382

Dic₅₃

yes

characteristic

#340

—

C₅₃

#7135

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂²⋊D₅₃

yes

—

—

—

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

M₃₂

—

106

—

M₃₂

—

212

—

C₂×C₁₆

—

212

—

C₄×C₈

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊Dic₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#11719924

yes

—

C₂×C₈

#11719925

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄²

—

102

C₄²

C₂×#88322

yes

characteristic

103

C₄²

#11719917

yes

characteristic

104

C₁₆

—

212

—

105

C₂³

#13133134

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂²⋊#88322

yes

characteristic

114

C₂×C₄

#13133132

yes

characteristic

115

C₂×C₄

#13133141

yes

—

116

C₂×C₄

#13133144

yes

—

117

C₈

—

118

C₈

—

119

C₂²

—

120

C₂²

#16810347

yes

characteristic

121

C₂²

#16810363

yes

characteristic

122

C₂²

#16810372

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

Q₈⋊#88322

yes

—

128

C₄

#16810360

yes

—

129

C₂

—

130

C₂

—

131

C₂

#20272541

yes

characteristic

132

C₂

#20272776

yes

—

	subgroup	G/H
center	C₂²	#16810372
commutator	C₄²×C₅₃	C₂×C₈
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₅₃×#1991	C₂
socle	C₂²×C₅₃	#333
prime radical	C₂³×C₅₃	#100
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	#1991	D₅₃
hypocenter	C₅₃	#7135
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1991

subgroup

G/H

center

C₂²

#16810372

commutator

C₄²×C₅₃

C₂×C₈

Frattini

C₂×C₄×C₈

C₂×D₅₃

Fitting

C₅₃×#1991

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄×C₈×C₅₃

C₂²

D₅₃

C₅₃

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1991

N	H ≅ G/N	type
74. C₅₃	#7135	semidirect

H ≅ G/N

type

74. C₅₃

#7135

semidirect

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

subgroup	index
#20272797	2
C₅₃×#1991	2
#7135	53

subgroup

index

#20272797

C₅₃×#1991

#7135

Subgroups — #27270918

Conjugacy classes of subgroups

Complements

Maximal subgroups