Group #20276521 — subgroups

Subgroups — #20276521

Order 6784.

138 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 67 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	#16810577	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810402	C₂	1	1	yes	maximal, characteristic
4	#16810402	C₂	1	1	yes	maximal, characteristic
5	#16810398	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₆×Dic₅₃	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	C₂×#11719908	C₂²	1	1	yes	characteristic
12	C₂×#11719908	C₂²	1	1	yes	characteristic
13	#13133128	C₄	1	1	yes	characteristic
14	#13133127	C₄	1	1	yes	characteristic
15	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
18	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
22	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
23	#11719921	C₈	1	1	yes	characteristic
24	#11719921	C₈	1	1	yes	characteristic
25	#11719919	C₈	1	1	yes	characteristic
26	#11719919	C₈	1	1	yes	characteristic
27	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×#88322	C₂×C₄	1	1	yes	characteristic
29	C₂×#88322	C₂×C₄	1	1	yes	characteristic
30	#11719912	—	2	2	no	—
31	C₈×D₅₃	—	2	1	no	—
32	C₈×D₅₃	—	2	1	no	—
33	C₁₆×C₅₃	—	2	1	no	—
34	C₁₆×C₅₃	—	2	1	no	—
35	#11719908	—	2	1	no	—
36	#11719908	—	2	1	no	—
37	C₂²×D₅₃	C₂×C₈	1	1	yes	characteristic
38	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
39	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
40	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×Dic₅₃	C₂×C₈	1	1	yes	characteristic
42	C₂×Dic₅₃	C₂×C₈	1	1	yes	characteristic
43	C₂×Dic₅₃	C₂×C₈	1	1	yes	characteristic
44	C₄×D₅₃	—	2	1	no	—
45	C₄×D₅₃	—	2	1	no	—
46	C₈×C₅₃	—	2	1	no	—
47	C₈×C₅₃	#41	1	1	yes	characteristic
48	C₈×C₅₃	#41	1	1	yes	characteristic
49	#88322	—	2	1	no	—
50	#88322	M₁₆	1	2	yes	—
51	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
52	C₂×D₅₃	—	2	1	no	—
53	C₂×D₅₃	—	2	1	no	—
54	C₂×D₅₃	—	2	1	no	—
55	C₄×C₅₃	—	2	1	no	—
56	C₄×C₅₃	—	2	1	no	—
57	C₄×C₅₃	#117	1	1	yes	characteristic
58	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
59	Dic₅₃	—	2	1	no	—
60	Dic₅₃	—	2	1	no	—
61	Dic₅₃	—	2	1	no	—
62	#2164	—	53	1	no	maximal, Sylow
63	C₂×C₅₃	#431	1	1	yes	characteristic
64	C₂×C₅₃	#504	1	1	yes	characteristic
65	C₂×C₅₃	#504	1	1	yes	characteristic
66	D₅₃	—	4	1	no	—
67	#405	—	53	1	no	—
68	C₄⋊C₁₆	—	53	1	no	—
69	C₄⋊C₁₆	D₅₃	1	1	yes	characteristic, complemented
70	C₂²⋊C₁₆	—	53	1	no	—
71	C₂²⋊C₁₆	—	53	1	no	—
72	#346	—	53	1	no	—
73	C₄×C₁₆	—	53	1	no	—
74	C₅₃	#2164	1	1	yes	characteristic, complemented, Sylow
75	C₂×M₁₆	—	53	1	no	—
76	C₂²×C₈	—	53	1	no	—
77	#117	—	53	1	no	—
78	C₂×C₁₆	—	53	1	no	—
79	C₂×C₁₆	—	53	1	no	—
80	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
81	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
82	#97	—	53	1	no	—
83	#97	—	53	1	no	—
84	C₄×C₈	—	53	1	no	—
85	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
86	C₂²×C₄	—	53	1	no	—
87	M₁₆	—	106	2	no	—
88	C₂×C₈	—	53	1	no	—
89	C₂×C₈	—	53	1	no	—
90	C₂×C₈	—	106	1	no	—
91	C₂×C₈	—	106	1	no	—
92	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
93	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
94	C₄⋊C₄	—	53	1	no	—
95	C₄⋊C₄	—	53	1	no	—
96	C₂²⋊C₄	—	53	1	no	—
97	C₂²⋊C₄	—	53	1	no	—
98	C₄²	—	53	1	no	—
99	C₄²	C₄×D₅₃	1	1	yes	characteristic
100	C₁₆	—	2	1	no	—
101	C₁₆	—	2	1	no	—
102	C₁₆	—	106	1	no	—
103	C₁₆	—	106	1	no	—
104	C₂³	—	53	1	no	—
105	C₂×C₄	—	53	1	no	—
106	C₂×C₄	—	53	1	no	—
107	C₂×C₄	—	53	1	no	—
108	C₂×C₄	—	106	1	no	—
109	C₂×C₄	—	106	1	no	—
110	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
111	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
112	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
113	C₈	—	2	1	no	—
114	C₈	—	53	2	no	—
115	C₈	—	106	1	no	—
116	C₈	#11719947	1	1	yes	characteristic
117	C₈	#11719949	1	1	yes	characteristic
118	C₂²	—	106	1	no	—
119	C₂²	—	106	1	no	—
120	C₂²	—	106	1	no	—
121	C₂²	C₂×C₈×D₅₃	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	—	106	1	no	—
125	C₄	—	106	1	no	—
126	C₄	—	106	1	no	—
127	C₄	#13133175	1	1	yes	characteristic
128	C₄	M₁₆×D₅₃	1	1	yes	characteristic
129	C₂	—	212	1	no	—
130	C₂	#16810675	1	1	yes	characteristic
131	C₂	#16811039	1	1	yes	characteristic
132	C₂	#16811041	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810577

C₂

yes

maximal, characteristic

C₄⋊#11719908

C₂

yes

maximal, characteristic, complemented

#16810402

C₂

yes

maximal, characteristic

#16810402

C₂

yes

maximal, characteristic

#16810398

C₂

yes

maximal, characteristic, complemented

C₁₆×Dic₅₃

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

C₂×#11719908

C₂²

yes

characteristic

C₂×#11719908

C₂²

yes

characteristic

#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

—

C₈×D₅₃

—

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×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

yes

C₈×C₅₃

yes

—

M₁₆

yes

—

C₂²×C₅₃

C₂²×C₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#117

yes

characteristic

C₄×C₅₃

C₂×M₁₆

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#405

—

C₄⋊C₁₆

—

C₄⋊C₁₆

D₅₃

yes

characteristic, complemented

C₂²⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

C₄×C₁₆

—

C₅₃

#2164

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

#117

—

C₂×C₁₆

—

C₂×C₁₆

—

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

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₄×D₅₃

yes

characteristic

100

C₁₆

—

101

C₁₆

—

102

C₁₆

—

106

—

103

C₁₆

—

106

—

104

C₂³

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

106

—

109

C₂×C₄

—

106

—

110

C₂×C₄

C₈×D₅₃

yes

characteristic

111

C₂×C₄

C₈×D₅₃

yes

characteristic

112

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

—

106

—

116

C₈

#11719947

yes

characteristic

117

C₈

#11719949

yes

characteristic

118

C₂²

—

106

—

119

C₂²

—

106

—

120

C₂²

—

106

—

121

C₂²

C₂×C₈×D₅₃

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

106

—

125

C₄

—

106

—

126

C₄

—

106

—

127

C₄

#13133175

yes

characteristic

128

C₄

M₁₆×D₅₃

yes

characteristic

129

C₂

—

212

—

130

C₂

#16810675

yes

characteristic

131

C₂

#16811039

yes

characteristic

132

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	#2164
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
2. C₄⋊#11719908	C₂	semidirect
5. #16810398	C₂	semidirect
6. C₁₆×Dic₅₃	C₂	semidirect
7. C₅₃×C₄⋊C₁₆	C₂	semidirect
69. C₄⋊C₁₆	D₅₃	semidirect
74. C₅₃	#2164	semidirect

H ≅ G/N

type

2. C₄⋊#11719908

C₂

semidirect

5. #16810398

C₂

semidirect

6. C₁₆×Dic₅₃

C₂

semidirect

7. C₅₃×C₄⋊C₁₆

C₂

semidirect

69. C₄⋊C₁₆

D₅₃

semidirect

74. C₅₃

#2164

semidirect

Subgroups — #20276521

Conjugacy classes of subgroups

Complements