Group #20276472 — subgroups

Subgroups — #20276472

Order 6784.

160 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 79 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	C₂×#13133156	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#16810577	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810441	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810441	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719912	C₄	1	1	yes	characteristic
9	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
10	#13133174	C₄	1	1	yes	characteristic
11	#13133156	—	2	2	no	—
12	C₂×#11719908	C₂²	1	1	yes	characteristic
13	C₂×#11719908	C₂²	1	1	yes	characteristic
14	C₁₆×D₅₃	—	2	2	no	—
15	#13133128	C₄	1	1	yes	characteristic
16	#13133127	C₄	1	1	yes	characteristic
17	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
24	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
25	#11719921	C₈	1	1	yes	characteristic
26	#11719921	C₈	1	1	yes	characteristic
27	#11719919	C₈	1	1	yes	characteristic
28	#11719919	C₈	1	1	yes	characteristic
29	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×#88322	C₂×C₄	1	1	yes	characteristic
31	C₂×#88322	C₂×C₄	1	1	yes	characteristic
32	#11719912	—	2	2	no	—
33	C₈×D₅₃	D₄	1	2	yes	—
34	C₈×D₅₃	Q₈	1	2	yes	—
35	C₁₆×C₅₃	—	2	1	no	—
36	C₁₆×C₅₃	—	2	1	no	—
37	#11719908	—	2	1	no	—
38	#11719908	—	2	1	no	—
39	C₂²×D₅₃	C₂×C₈	1	1	yes	characteristic
40	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×Dic₅₃	C₂×C₈	1	1	yes	characteristic
44	C₂×Dic₅₃	C₂×C₈	1	1	yes	characteristic
45	C₂×Dic₅₃	C₂×C₈	1	1	yes	characteristic
46	C₄×D₅₃	C₄⋊C₄	1	2	yes	—
47	C₄×D₅₃	M₁₆	1	2	yes	—
48	C₈×C₅₃	—	2	1	no	—
49	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
51	#88322	—	2	1	no	—
52	#88322	C₄⋊C₄	1	2	yes	—
53	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
54	C₂×D₅₃	—	2	1	no	—
55	C₂×D₅₃	—	2	1	no	—
56	C₂×D₅₃	C₄⋊C₈	1	2	yes	—
57	C₄×C₅₃	—	2	1	no	—
58	C₄×C₅₃	—	2	1	no	—
59	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
61	Dic₅₃	—	2	1	no	—
62	Dic₅₃	—	2	1	no	—
63	Dic₅₃	C₄⋊C₈	1	2	yes	—
64	#2151	—	53	1	no	maximal, Sylow
65	C₂×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
66	C₂×C₅₃	#504	1	1	yes	characteristic
67	C₂×C₅₃	#504	1	1	yes	characteristic
68	D₅₃	—	2	2	no	—
69	C₂×M₃₂	—	53	1	no	—
70	C₂²×C₁₆	—	53	1	no	—
71	#405	—	53	1	no	—
72	C₄⋊C₁₆	—	53	1	no	—
73	C₄⋊C₁₆	—	53	1	no	—
74	C₄⋊C₁₆	—	53	1	no	—
75	C₄⋊C₁₆	D₅₃	1	1	yes	characteristic, complemented
76	C₅₃	#2151	1	1	yes	characteristic, complemented, Sylow
77	C₂×M₁₆	—	53	1	no	—
78	C₂²×C₈	—	53	1	no	—
79	#117	—	53	1	no	—
80	M₃₂	—	106	2	no	—
81	C₂×C₁₆	—	53	1	no	—
82	C₂×C₁₆	—	53	1	no	—
83	C₂×C₁₆	—	106	2	no	—
84	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
85	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
86	#97	—	53	1	no	—
87	#97	—	53	1	no	—
88	C₄×C₈	—	53	1	no	—
89	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
90	C₂²×C₄	—	53	1	no	—
91	M₁₆	—	106	2	no	—
92	C₂×C₈	—	53	1	no	—
93	C₂×C₈	—	53	2	no	—
94	C₂×C₈	—	53	2	no	—
95	C₂×C₈	—	53	1	no	—
96	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
97	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
98	C₄⋊C₄	—	53	1	no	—
99	C₄⋊C₄	—	53	1	no	—
100	C₂²⋊C₄	—	53	1	no	—
101	C₂²⋊C₄	—	53	1	no	—
102	C₄²	—	53	1	no	—
103	C₄²	C₄×D₅₃	1	1	yes	characteristic
104	C₁₆	—	2	1	no	—
105	C₁₆	—	2	1	no	—
106	C₁₆	—	106	1	no	—
107	C₁₆	—	106	1	no	—
108	C₂³	—	53	1	no	—
109	C₂×C₄	—	53	1	no	—
110	C₂×C₄	—	53	2	no	—
111	C₂×C₄	—	53	2	no	—
112	C₂×C₄	—	53	1	no	—
113	C₂×C₄	—	53	1	no	—
114	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
115	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
116	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
117	C₈	—	2	1	no	—
118	C₈	—	53	2	no	—
119	C₈	—	106	1	no	—
120	C₈	D₄×D₅₃	1	1	yes	characteristic
121	C₈	Q₈×D₅₃	1	1	yes	characteristic
122	C₂²	—	53	2	no	—
123	C₂²	—	106	1	no	—
124	C₂²	—	106	1	no	—
125	C₂²	C₂×C₈×D₅₃	1	1	yes	characteristic
126	C₄	—	2	1	no	—
127	C₄	—	2	1	no	—
128	C₄	—	53	2	no	—
129	C₄	—	106	1	no	—
130	C₄	—	106	1	no	—
131	C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
132	C₄	M₁₆×D₅₃	1	1	yes	characteristic
133	C₂	—	106	2	no	—
134	C₂	C₄⋊C₈×D₅₃	1	1	yes	characteristic
135	C₂	#16811039	1	1	yes	characteristic
136	C₂	#16811041	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133156

C₂

yes

maximal, characteristic

C₂×C₁₆×D₅₃

C₂

yes

maximal, characteristic

#16810577

C₂

yes

maximal, characteristic

C₄⋊#11719908

C₂

yes

maximal, characteristic, complemented

#16810441

C₂

yes

maximal, characteristic, complemented

#16810441

C₂

yes

maximal, characteristic, complemented

C₅₃×C₄⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₂×#11719912

C₄

yes

characteristic

C₂×C₈×D₅₃

C₂²

yes

characteristic

#13133174

C₄

yes

characteristic

#13133156

—

C₂×#11719908

C₂²

yes

characteristic

C₂×#11719908

C₂²

yes

characteristic

C₁₆×D₅₃

—

#13133128

C₄

yes

characteristic

#13133127

C₄

yes

characteristic

C₈×Dic₅₃

C₂²

yes

characteristic

C₂×C₁₆×C₅₃

C₂²

yes

characteristic

C₂×C₁₆×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

#11719921

C₈

yes

characteristic

#11719921

C₈

yes

characteristic

#11719919

C₈

yes

characteristic

#11719919

C₈

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

#11719912

—

C₈×D₅₃

D₄

yes

—

C₈×D₅₃

Q₈

yes

—

C₁₆×C₅₃

—

C₁₆×C₅₃

—

#11719908

—

#11719908

—

C₂²×D₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₂×C₈

yes

characteristic

C₂×Dic₅₃

C₂×C₈

yes

characteristic

C₂×Dic₅₃

C₂×C₈

yes

characteristic

C₄×D₅₃

C₄⋊C₄

yes

—

C₄×D₅₃

M₁₆

yes

—

C₈×C₅₃

—

C₈×C₅₃

C₂×D₄

yes

characteristic

C₈×C₅₃

C₂×Q₈

yes

characteristic

#88322

—

#88322

C₄⋊C₄

yes

—

C₂²×C₅₃

C₂²×C₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

C₄⋊C₈

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₂×M₁₆

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₄⋊C₈

yes

—

#2151

—

maximal, Sylow

C₂×C₅₃

C₂×C₄⋊C₈

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

D₅₃

yes

characteristic, complemented

C₅₃

#2151

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

106

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₂×C₁₆

C₂×D₅₃

yes

characteristic

C₂×C₁₆

C₂×D₅₃

yes

characteristic

#97

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂²×C₄

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₄²

—

103

C₄²

C₄×D₅₃

yes

characteristic

104

C₁₆

—

105

C₁₆

—

106

C₁₆

—

106

—

107

C₁₆

—

106

—

108

C₂³

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₈×D₅₃

yes

characteristic

115

C₂×C₄

C₈×D₅₃

yes

characteristic

116

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

117

C₈

—

118

C₈

—

119

C₈

—

106

—

120

C₈

D₄×D₅₃

yes

characteristic

121

C₈

Q₈×D₅₃

yes

characteristic

122

C₂²

—

123

C₂²

—

106

—

124

C₂²

—

106

—

125

C₂²

C₂×C₈×D₅₃

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

106

—

130

C₄

—

106

—

131

C₄

C₄⋊C₄×D₅₃

yes

characteristic

132

C₄

M₁₆×D₅₃

yes

characteristic

133

C₂

—

106

—

134

C₂

C₄⋊C₈×D₅₃

yes

characteristic

135

C₂

#16811039

yes

characteristic

136

C₂

#16811041

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₈
Fitting	C₅₃×C₄⋊C₁₆	C₂
hypercenter	C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	#2151
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₅₃

commutator, socle

C₂²×C₅₃

C₂²×C₈

Fitting

C₅₃×C₄⋊C₁₆

C₂

hypercenter

C₄⋊C₁₆

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
4. C₄⋊#11719908	C₂	semidirect
5. #16810441	C₂	semidirect
6. #16810441	C₂	semidirect
7. C₅₃×C₄⋊C₁₆	C₂	semidirect
75. C₄⋊C₁₆	D₅₃	semidirect
76. C₅₃	#2151	semidirect

H ≅ G/N

type

4. C₄⋊#11719908

C₂

semidirect

5. #16810441

C₂

semidirect

6. #16810441

C₂

semidirect

7. C₅₃×C₄⋊C₁₆

C₂

semidirect

75. C₄⋊C₁₆

D₅₃

semidirect

76. C₅₃

#2151

semidirect

Subgroups — #20276472

Conjugacy classes of subgroups

Complements