Group #20272643 — subgroups

Subgroups — #20272643

Order 6784.

294 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 41 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	#16810347	C₂	1	2	yes	maximal, complemented
2	C₅₃×#379	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊#88322	—	2	2	no	—
4	C₂⁵×C₅₃	C₄	1	1	yes	characteristic
5	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
6	C₂×C₂²⋊C₄×C₅₃	C₄	1	1	yes	characteristic
7	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
8	C₂⁴×C₅₃	—	2	1	no	—
9	C₂⁴×C₅₃	—	2	1	no	—
10	C₂⁴×C₅₃	—	4	4	no	—
11	C₂⁴×C₅₃	—	4	1	no	—
12	C₂⁴×C₅₃	—	4	1	no	—
13	C₂⁴×C₅₃	C₈	1	2	yes	complemented
14	C₂⁴×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	—	2	1	no	—
16	C₂²×C₄×C₅₃	D₄	1	2	yes	—
17	C₂²⋊C₄×C₅₃	—	4	1	no	—
18	C₂²⋊C₄×C₅₃	—	4	4	no	—
19	C₂²⋊C₄×C₅₃	—	4	1	no	—
20	C₂×#88322	—	4	2	no	—
21	C₂³×C₅₃	—	2	4	no	—
22	C₂³×C₅₃	—	2	2	no	—
23	C₂³×C₅₃	—	2	1	no	—
24	C₂³×C₅₃	—	2	1	no	—
25	C₂³×C₅₃	—	4	4	no	—
26	C₂³×C₅₃	—	4	2	no	—
27	C₂³×C₅₃	—	4	2	no	—
28	C₂³×C₅₃	—	4	2	no	—
29	C₂³×C₅₃	—	4	4	no	—
30	C₂³×C₅₃	—	4	2	no	—
31	C₂³×C₅₃	—	4	4	no	—
32	C₂³×C₅₃	—	4	8	no	—
33	C₂³×C₅₃	—	4	4	no	—
34	C₂³×C₅₃	—	4	1	no	—
35	C₂³×C₅₃	—	4	1	no	—
36	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂³×C₅₃	C₂×C₈	1	1	yes	characteristic
38	C₂³×C₅₃	M₁₆	1	1	yes	characteristic
39	C₂×C₄×C₅₃	—	2	2	no	—
40	C₂×C₄×C₅₃	—	4	2	no	—
41	C₂×C₄×C₅₃	—	4	2	no	—
42	C₂×C₄×C₅₃	—	4	1	no	—
43	#88322	—	8	2	no	—
44	C₂²×C₅₃	—	2	2	no	—
45	C₂²×C₅₃	—	2	4	no	—
46	C₂²×C₅₃	—	2	1	no	—
47	C₂²×C₅₃	—	2	1	no	—
48	C₂²×C₅₃	—	4	1	no	—
49	C₂²×C₅₃	—	4	1	no	—
50	C₂²×C₅₃	—	4	4	no	—
51	C₂²×C₅₃	—	4	2	no	—
52	C₂²×C₅₃	—	4	2	no	—
53	C₂²×C₅₃	—	4	4	no	—
54	C₂²×C₅₃	—	4	4	no	—
55	C₂²×C₅₃	—	4	8	no	—
56	C₂²×C₅₃	—	4	2	no	—
57	C₂²×C₅₃	—	4	4	no	—
58	C₂²×C₅₃	—	4	2	no	—
59	C₂²×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
60	C₂²×C₅₃	#99	1	1	yes	characteristic
61	C₂²×C₅₃	#100	1	1	yes	characteristic
62	C₄×C₅₃	—	4	2	no	—
63	C₄×C₅₃	—	8	1	no	—
64	#1316	—	53	1	no	maximal, Sylow
65	C₂×C₅₃	—	2	1	no	—
66	C₂×C₅₃	—	2	1	no	—
67	C₂×C₅₃	—	4	4	no	—
68	C₂×C₅₃	—	4	1	no	—
69	C₂×C₅₃	—	4	1	no	—
70	C₂×C₅₃	#323	1	1	yes	characteristic
71	C₂×C₅₃	AΣL(1,𝔽₁₆)	1	2	yes	—
72	#379	D₅₃	1	1	yes	characteristic
73	#323	—	53	2	no	—
74	C₅₃	#1316	1	1	yes	characteristic, complemented, Sylow
75	C₂⁵	Dic₅₃	1	1	yes	characteristic
76	C₂×C₂²⋊C₄	—	2	2	no	—
77	C₂×C₂²⋊C₄	Dic₅₃	1	1	yes	characteristic
78	C₂×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic
79	C₂²⋊C₈	—	106	2	no	—
80	C₂⁴	—	2	1	no	—
81	C₂⁴	—	2	1	no	—
82	C₂⁴	—	4	4	no	—
83	C₂⁴	—	4	1	no	—
84	C₂⁴	—	4	1	no	—
85	C₂⁴	#88322	1	2	yes	complemented
86	C₂⁴	C₂×Dic₅₃	1	1	yes	characteristic
87	C₂²×C₄	—	2	1	no	—
88	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
89	C₂×C₈	—	212	2	no	—
90	C₂²⋊C₄	—	4	1	no	—
91	C₂²⋊C₄	—	4	4	no	—
92	C₂²⋊C₄	—	4	1	no	—
93	C₂³	—	2	4	no	—
94	C₂³	—	2	2	no	—
95	C₂³	—	2	1	no	—
96	C₂³	—	2	1	no	—
97	C₂³	—	4	4	no	—
98	C₂³	—	4	2	no	—
99	C₂³	—	4	2	no	—
100	C₂³	—	4	2	no	—
101	C₂³	—	4	4	no	—
102	C₂³	—	4	2	no	—
103	C₂³	—	4	4	no	—
104	C₂³	—	4	8	no	—
105	C₂³	—	4	4	no	—
106	C₂³	—	4	1	no	—
107	C₂³	—	4	1	no	—
108	C₂³	C₂×#88322	1	1	yes	characteristic
109	C₂³	#11719917	1	1	yes	characteristic
110	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	1	no	—
115	C₈	—	424	2	no	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	4	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	—	4	1	no	—
121	C₂²	—	4	1	no	—
122	C₂²	—	4	4	no	—
123	C₂²	—	4	2	no	—
124	C₂²	—	4	2	no	—
125	C₂²	—	4	4	no	—
126	C₂²	—	4	4	no	—
127	C₂²	—	4	8	no	—
128	C₂²	—	4	2	no	—
129	C₂²	—	4	4	no	—
130	C₂²	—	4	2	no	—
131	C₂²	C₂²⋊#88322	1	1	yes	characteristic
132	C₂²	#13133132	1	1	yes	characteristic
133	C₂²	#13133134	1	1	yes	characteristic
134	C₄	—	4	2	no	—
135	C₄	—	8	1	no	—
136	C₂	—	2	1	no	—
137	C₂	—	2	1	no	—
138	C₂	—	4	4	no	—
139	C₂	—	4	1	no	—
140	C₂	—	4	1	no	—
141	C₂	#16810347	1	1	yes	characteristic
142	C₂	#16810410	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16810347

C₂

yes

maximal, complemented

C₅₃×#379

C₂

yes

maximal, characteristic

C₂²⋊#88322

—

C₂⁵×C₅₃

C₄

yes

characteristic

C₂×C₂²⋊C₄×C₅₃

—

C₂×C₂²⋊C₄×C₅₃

C₄

yes

characteristic

C₂×C₂²⋊C₄×C₅₃

C₂²

yes

characteristic

C₂⁴×C₅₃

—

C₂⁴×C₅₃

—

C₂⁴×C₅₃

—

C₂⁴×C₅₃

—

C₂⁴×C₅₃

—

C₂⁴×C₅₃

C₈

yes

complemented

C₂⁴×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

C₂³×C₅₃

C₂×C₈

yes

characteristic

C₂³×C₅₃

M₁₆

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²⋊C₈

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

#1316

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#323

yes

characteristic

C₂×C₅₃

AΣL(1,𝔽₁₆)

yes

—

#379

D₅₃

yes

characteristic

#323

—

C₅₃

#1316

yes

characteristic, complemented, Sylow

C₂⁵

Dic₅₃

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

Dic₅₃

yes

characteristic

C₂×C₂²⋊C₄

C₂×D₅₃

yes

characteristic

C₂²⋊C₈

—

106

—

C₂⁴

—

C₂⁴

—

C₂⁴

—

C₂⁴

—

C₂⁴

—

C₂⁴

#88322

yes

complemented

C₂⁴

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂×C₈

—

212

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

—

102

C₂³

—

103

C₂³

—

104

C₂³

—

105

C₂³

—

106

C₂³

—

107

C₂³

—

108

C₂³

C₂×#88322

yes

characteristic

109

C₂³

#11719917

yes

characteristic

110

C₂³

C₂²⋊Dic₅₃

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₈

—

424

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

C₂²⋊#88322

yes

characteristic

132

C₂²

#13133132

yes

characteristic

133

C₂²

#13133134

yes

characteristic

134

C₄

—

135

C₄

—

136

C₂

—

137

C₂

—

138

C₂

—

139

C₂

—

140

C₂

—

141

C₂

#16810347

yes

characteristic

142

C₂

#16810410

yes

—

	subgroup	G/H
center	C₂²	#13133132
commutator	C₂³×C₅₃	C₂×C₈
Frattini	C₂×C₂²⋊C₄	C₂×D₅₃
Fitting	C₅₃×#379	C₂
socle	C₂²×C₅₃	#99
hypercenter	#379	D₅₃
hypocenter	C₅₃	#1316
LCS	G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₂²⋊C₄ ◁ #379

subgroup

G/H

center

C₂²

#13133132

commutator

C₂³×C₅₃

C₂×C₈

Frattini

C₂×C₂²⋊C₄

C₂×D₅₃

Fitting

C₅₃×#379

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₂²⋊C₄ ◁ #379

N	H ≅ G/N	type
1. #16810347	C₂	semidirect
13. C₂⁴×C₅₃	C₈	semidirect
74. C₅₃	#1316	semidirect
85. C₂⁴	#88322	semidirect

H ≅ G/N

type

1. #16810347

C₂

semidirect

13. C₂⁴×C₅₃

C₈

semidirect

74. C₅₃

#1316

semidirect

85. C₂⁴

#88322

semidirect

Subgroups — #20272643

Conjugacy classes of subgroups

Complements