Group #27271483 — subgroups

Subgroups — #27271483

Order 13568.

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

normal subgroups only

This group has 180 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×#1992	C₂	1	1	yes	maximal, characteristic
3	#16810349	—	2	2	no	—
4	C₂×#125×C₅₃	C₄	1	1	yes	characteristic
5	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
6	C₅₃×#384	C₂²	1	1	yes	characteristic
7	C₅₃×#340	—	2	2	no	—
8	C₂²⋊#88322	—	4	2	no	—
9	C₂²×C₈×C₅₃	—	2	1	no	—
10	#125×C₅₃	—	2	1	no	—
11	#125×C₅₃	—	4	1	no	—
12	#125×C₅₃	C₈	1	2	yes	complemented
13	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
14	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
15	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
16	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
17	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
18	#97×C₅₃	—	4	1	no	—
19	C₂²×C₄×C₅₃	—	2	1	no	—
20	C₂²×C₄×C₅₃	—	2	2	no	—
21	C₂²×C₄×C₅₃	—	4	1	no	—
22	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₈×C₅₃	—	2	2	no	—
24	C₂×C₈×C₅₃	—	4	1	no	—
25	C₂×C₈×C₅₃	—	4	1	no	—
26	C₄⋊C₄×C₅₃	—	2	2	no	—
27	C₄⋊C₄×C₅₃	—	4	2	no	—
28	C₄⋊C₄×C₅₃	—	4	2	no	—
29	C₄⋊C₄×C₅₃	—	4	2	no	—
30	C₄⋊C₄×C₅₃	—	4	1	no	—
31	C₄⋊C₄×C₅₃	—	8	2	no	—
32	C₄²×C₅₃	—	2	1	no	—
33	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
34	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
35	C₂×#88322	—	8	2	no	—
36	C₂³×C₅₃	#99	1	1	yes	characteristic
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	—	2	1	no	—
41	C₂×C₄×C₅₃	—	4	1	no	—
42	C₂×C₄×C₅₃	—	4	1	no	—
43	C₂×C₄×C₅₃	—	4	2	no	—
44	C₂×C₄×C₅₃	—	4	2	no	—
45	C₂×C₄×C₅₃	—	8	1	no	—
46	C₂×C₄×C₅₃	—	8	1	no	—
47	C₂×C₄×C₅₃	—	8	2	no	—
48	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
49	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
50	C₈×C₅₃	—	4	2	no	—
51	#88322	—	16	2	no	—
52	#7358	—	53	1	no	maximal, Sylow
53	C₂²×C₅₃	—	2	1	no	—
54	C₂²×C₅₃	—	2	1	no	—
55	C₂²×C₅₃	#323	1	1	yes	characteristic
56	C₂²×C₅₃	#354	1	1	yes	characteristic
57	C₂²×C₅₃	#356	1	1	yes	characteristic
58	C₄×C₅₃	—	2	1	no	—
59	C₄×C₅₃	—	2	1	no	—
60	C₄×C₅₃	—	4	1	no	—
61	C₄×C₅₃	—	4	1	no	—
62	C₄×C₅₃	—	8	2	no	—
63	C₄×C₅₃	—	8	2	no	—
64	#1992	D₅₃	1	1	yes	characteristic
65	#1325	—	53	2	no	—
66	C₂×C₅₃	—	2	1	no	—
67	C₂×C₅₃	—	2	1	no	—
68	C₂×C₅₃	#1319	1	1	yes	characteristic
69	C₂×C₅₃	#1412	1	1	yes	characteristic
70	C₂×C₅₃	#1413	1	1	yes	characteristic
71	C₂×#125	Dic₅₃	1	1	yes	characteristic
72	C₂×#97	Dic₅₃	1	1	yes	characteristic
73	#384	C₂×D₅₃	1	1	yes	characteristic
74	#340	—	2	2	no	—
75	#324	—	106	2	no	—
76	C₅₃	#7358	1	1	yes	characteristic, complemented, Sylow
77	C₂²×C₈	—	2	1	no	—
78	#125	—	2	1	no	—
79	#125	—	4	1	no	—
80	#125	#88322	1	2	yes	complemented
81	C₂×C₄⋊C₄	—	2	1	no	—
82	C₂×C₄⋊C₄	—	4	1	no	—
83	C₂×C₄⋊C₄	—	4	1	no	—
84	C₂×C₄⋊C₄	C₂²⋊D₅₃	1	2	yes	—
85	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
86	C₂²⋊C₈	—	212	2	no	—
87	#97	—	4	1	no	—
88	C₂²×C₄	—	2	1	no	—
89	C₂²×C₄	—	2	2	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	—	4	1	no	—
94	C₂×C₈	—	4	1	no	—
95	C₂×C₈	—	424	2	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	4	2	no	—
99	C₄⋊C₄	—	4	2	no	—
100	C₄⋊C₄	—	4	1	no	—
101	C₄⋊C₄	—	8	2	no	—
102	C₄²	—	2	1	no	—
103	C₄²	C₂×#88322	1	1	yes	characteristic
104	C₄²	#11719917	1	1	yes	characteristic
105	C₂³	#13133132	1	1	yes	characteristic
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	8	1	no	—
115	C₂×C₄	—	8	1	no	—
116	C₂×C₄	—	8	2	no	—
117	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
118	C₂×C₄	#13133134	1	1	yes	characteristic
119	C₈	—	4	2	no	—
120	C₈	—	848	2	no	—
121	C₂²	—	2	1	no	—
122	C₂²	—	2	1	no	—
123	C₂²	#16810347	1	1	yes	characteristic
124	C₂²	#16810418	1	1	yes	characteristic
125	C₂²	#16810422	1	1	yes	characteristic
126	C₄	—	2	1	no	—
127	C₄	—	2	1	no	—
128	C₄	—	4	1	no	—
129	C₄	—	4	1	no	—
130	C₄	—	8	2	no	—
131	C₄	—	8	2	no	—
132	C₂	—	2	1	no	—
133	C₂	—	2	1	no	—
134	C₂	#20272649	1	1	yes	characteristic
135	C₂	#20272886	1	1	yes	characteristic
136	C₂	#20272888	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20272661

C₂

yes

maximal

C₅₃×#1992

C₂

yes

maximal, characteristic

#16810349

—

C₂×#125×C₅₃

C₄

yes

characteristic

C₂×#97×C₅₃

C₄

yes

characteristic

C₅₃×#384

C₂²

yes

characteristic

C₅₃×#340

—

C₂²⋊#88322

—

C₂²×C₈×C₅₃

—

#125×C₅₃

—

#125×C₅₃

—

#125×C₅₃

C₈

yes

complemented

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

#97×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

—

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

#100

yes

characteristic

C₈×C₅₃

—

#88322

—

#7358

—

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

—

#1992

D₅₃

yes

characteristic

#1325

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

Dic₅₃

yes

characteristic

C₂×#97

Dic₅₃

yes

characteristic

#384

C₂×D₅₃

yes

characteristic

#340

—

#324

—

106

—

C₅₃

#7358

yes

characteristic, complemented, Sylow

C₂²×C₈

—

—

—

yes

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₅₃

yes

—

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

C₂²⋊C₈

—

212

—

#97

—

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

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄²

—

103

C₄²

C₂×#88322

yes

characteristic

104

C₄²

#11719917

yes

characteristic

105

C₂³

#13133132

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

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂²⋊#88322

yes

characteristic

118

C₂×C₄

#13133134

yes

characteristic

119

C₈

—

120

C₈

—

848

—

121

C₂²

—

122

C₂²

—

123

C₂²

#16810347

yes

characteristic

124

C₂²

#16810418

yes

characteristic

125

C₂²

#16810422

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₂

—

133

C₂

—

134

C₂

#20272649

yes

characteristic

135

C₂

#20272886

yes

characteristic

136

C₂

#20272888

yes

characteristic

	subgroup	G/H
center	C₂²	#16810418
commutator	C₄²×C₅₃	C₂×C₈
Frattini	#384	C₂×D₅₃
Fitting	C₅₃×#1992	C₂
socle	C₂²×C₅₃	#354
prime radical	C₂³×C₅₃	#99
cosocle	C₅₃×#384	C₂²
hypercenter	#1992	D₅₃
hypocenter	C₅₃	#7358
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #384 ◁ #1992

subgroup

G/H

center

C₂²

#16810418

commutator

C₄²×C₅₃

C₂×C₈

Frattini

#384

C₂×D₅₃

Fitting

C₅₃×#1992

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 ◁ #1992

N	H ≅ G/N	type
12. #125×C₅₃	C₈	semidirect
76. C₅₃	#7358	semidirect
80. #125	#88322	semidirect

H ≅ G/N

type

12. #125×C₅₃

C₈

semidirect

76. C₅₃

#7358

semidirect

80. #125

#88322

semidirect

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

subgroup	index
#20272661	2
C₅₃×#1992	2
#7358	53

subgroup

index

#20272661

C₅₃×#1992

#7358

Subgroups — #27271483

Conjugacy classes of subgroups

Complements

Maximal subgroups