Group #20276231 — subgroups

Subgroups — #20276231

Order 6784.

222 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 65 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	#16810672	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810671	C₂	1	2	yes	maximal, complemented
3	#16810587	C₂	1	2	yes	maximal
4	#16810381	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#407	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₂²	1	2	yes	—
7	C₂×#11719917	C₂²	1	2	yes	—
8	#13133153	—	2	2	no	—
9	#13133152	—	2	4	no	—
10	C₂²⋊#88322	—	2	4	no	—
11	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²⋊C₈×C₅₃	—	2	2	no	—
15	C₂⁴×C₅₃	Q₈	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	—	2	1	no	—
17	C₂²×C₄×C₅₃	—	2	2	no	—
18	C₂²×C₄×C₅₃	—	2	1	no	—
19	C₂²×C₄×C₅₃	—	2	2	no	—
20	C₂²×C₄×C₅₃	Q₈	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	M₁₆×C₅₃	—	2	2	no	—
23	M₁₆×C₅₃	—	2	2	no	—
24	C₂×C₈×C₅₃	—	2	1	no	—
25	C₂×C₈×C₅₃	D₄	1	2	yes	—
26	#11719917	—	2	2	no	—
27	#11719917	—	2	2	no	—
28	#11719917	—	2	2	no	—
29	#11719917	—	2	2	no	—
30	C₂×#88322	—	2	2	no	—
31	C₂×#88322	D₄	1	4	yes	—
32	C₂³×C₅₃	—	2	2	no	—
33	C₂³×C₅₃	—	2	1	no	—
34	C₂³×C₅₃	—	4	2	no	—
35	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₅₃	—	2	2	no	—
37	C₂×C₄×C₅₃	—	2	1	no	—
38	C₂×C₄×C₅₃	—	4	1	no	—
39	C₂×C₄×C₅₃	—	4	2	no	—
40	C₂×C₄×C₅₃	—	4	1	no	—
41	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
42	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
44	C₂×C₄×C₅₃	#41	1	2	yes	—
45	C₈×C₅₃	—	2	2	no	—
46	C₈×C₅₃	—	4	1	no	—
47	#88322	—	2	4	no	—
48	#88322	—	4	2	no	—
49	C₂²×C₅₃	—	2	4	no	—
50	C₂²×C₅₃	—	2	2	no	—
51	C₂²×C₅₃	—	4	1	no	—
52	C₂²×C₅₃	—	4	2	no	—
53	C₂²×C₅₃	—	4	1	no	—
54	C₂²×C₅₃	—	4	1	no	—
55	C₂²×C₅₃	C₂²⋊Q₈	1	2	yes	—
56	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
57	C₄×C₅₃	—	4	1	no	—
58	C₄×C₅₃	#123	1	1	yes	characteristic
59	C₄×C₅₃	#123	1	2	yes	—
60	C₄×C₅₃	#127	1	1	yes	characteristic
61	#2091	—	53	1	no	maximal, Sylow
62	C₂×C₅₃	—	2	2	no	—
63	C₂×C₅₃	—	2	1	no	—
64	C₂×C₅₃	—	4	2	no	—
65	C₂×C₅₃	#399	1	1	yes	characteristic
66	#430	—	53	2	no	—
67	#430	—	53	1	no	—
68	#407	—	53	2	no	—
69	#407	D₅₃	1	1	yes	characteristic
70	#338	—	53	1	no	—
71	C₅₃	#2091	1	1	yes	characteristic, complemented, Sylow
72	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
73	C₂×M₁₆	—	53	2	no	—
74	C₂×M₁₆	—	53	2	no	—
75	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
76	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
77	#108	—	106	4	no	—
78	#108	—	106	2	no	—
79	C₂²⋊C₈	—	2	2	no	—
80	C₂²⋊C₈	—	106	4	no	—
81	C₂⁴	Dic₁₀₆	1	1	yes	characteristic
82	C₂²×C₄	—	2	1	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	2	1	no	—
86	C₂²×C₄	Dic₁₀₆	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
88	M₁₆	—	2	2	no	—
89	M₁₆	—	2	2	no	—
90	M₁₆	—	106	2	no	—
91	M₁₆	—	106	2	no	—
92	M₁₆	—	106	2	no	—
93	M₁₆	—	106	2	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	53	4	no	—
96	C₂×C₈	—	106	2	no	—
97	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
98	C₂³	—	2	2	no	—
99	C₂³	—	2	1	no	—
100	C₂³	—	4	2	no	—
101	C₂³	C₂×Dic₁₀₆	1	1	yes	characteristic
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	4	1	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	#11719945	1	1	yes	characteristic
108	C₂×C₄	D₄×D₅₃	1	2	yes	—
109	C₂×C₄	#11719947	1	2	yes	—
110	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	4	1	no	—
113	C₈	—	106	4	no	—
114	C₈	—	212	2	no	—
115	C₂²	—	2	4	no	—
116	C₂²	—	2	2	no	—
117	C₂²	—	4	1	no	—
118	C₂²	—	4	2	no	—
119	C₂²	—	4	1	no	—
120	C₂²	—	4	1	no	—
121	C₂²	#13133194	1	2	yes	—
122	C₂²	#13133196	1	1	yes	characteristic
123	C₄	—	4	1	no	—
124	C₄	#13133198	1	1	yes	characteristic
125	C₄	#13133199	1	2	yes	—
126	C₄	#13133213	1	1	yes	characteristic
127	C₂	—	2	2	no	—
128	C₂	—	2	1	no	—
129	C₂	—	4	2	no	—
130	C₂	#16810559	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810672

C₂

yes

maximal, characteristic, complemented

#16810671

C₂

yes

maximal, complemented

#16810587

C₂

yes

maximal

#16810381

C₂

yes

maximal, characteristic, complemented

C₅₃×#407

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

C₂×#11719917

C₂²

yes

—

#13133153

—

#13133152

—

C₂²⋊#88322

—

C₂³×C₄×C₅₃

C₂²

yes

characteristic

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂²⋊C₈×C₅₃

—

C₂⁴×C₅₃

Q₈

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

Q₈

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

—

—

—

—

—

—

D₄

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²⋊Q₈

yes

—

C₂²×C₅₃

C₂²⋊Q₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

#123

yes

characteristic

C₄×C₅₃

#123

yes

—

C₄×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

—

—

—

D₅₃

yes

characteristic

#338

—

C₅₃

#2091

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₅₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

characteristic

C₂×M₁₆

C₂×D₅₃

yes

characteristic

#108

—

106

—

#108

—

106

—

C₂²⋊C₈

—

C₂²⋊C₈

—

106

—

C₂⁴

Dic₁₀₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₁₀₆

yes

characteristic

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

106

—

M₁₆

—

106

—

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

C₂²⋊D₅₃

yes

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

C₂×Dic₁₀₆

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

#11719945

yes

characteristic

108

C₂×C₄

D₄×D₅₃

yes

—

109

C₂×C₄

#11719947

yes

—

110

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

—

106

—

114

C₈

—

212

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

#13133194

yes

—

122

C₂²

#13133196

yes

characteristic

123

C₄

—

124

C₄

#13133198

yes

characteristic

125

C₄

#13133199

yes

—

126

C₄

#13133213

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#16810559

yes

characteristic

	subgroup	G/H
center	C₄	#13133213
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#407	C₂
socle	C₂×C₅₃	#399
hypercenter	#407	D₅₃
hypocenter	C₅₃	#2091
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #407

subgroup

G/H

center

C₄

#13133213

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#407

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #407

N	H ≅ G/N	type
1. #16810672	C₂	semidirect
2. #16810671	C₂	semidirect
4. #16810381	C₂	semidirect
71. C₅₃	#2091	semidirect

H ≅ G/N

type

1. #16810672

C₂

semidirect

2. #16810671

C₂

semidirect

4. #16810381

C₂

semidirect

71. C₅₃

#2091

semidirect

Subgroups — #20276231

Conjugacy classes of subgroups

Complements