Group #20272610 — subgroups

Subgroups — #20272610

Order 6784.

148 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 77 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	#16810676	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#423	C₂	1	1	yes	maximal, characteristic
4	C₂²×#88322	—	2	1	no	—
5	C₄⋊#88322	—	2	1	no	—
6	C₄⋊#88322	—	2	1	no	—
7	C₄⋊#88322	C₄	1	2	yes	—
8	C₂²⋊#88322	C₄	1	2	yes	—
9	C₄×#88322	—	2	1	no	—
10	C₂×M₁₆×C₅₃	—	2	1	no	—
11	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₅₃	—	2	1	no	—
13	C₄⋊C₈×C₅₃	C₄	1	2	yes	—
14	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	Q₈	1	1	yes	characteristic
17	M₁₆×C₅₃	—	4	1	no	—
18	M₁₆×C₅₃	—	4	1	no	—
19	C₂×C₈×C₅₃	—	2	2	no	—
20	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₅₃	D₄	1	1	yes	characteristic
23	C₄²×C₅₃	D₄	1	1	yes	characteristic
24	C₂×#88322	—	2	2	no	—
25	C₂×#88322	—	2	2	no	—
26	C₂×#88322	—	4	1	no	—
27	C₂×#88322	—	4	1	no	—
28	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₅₃	—	2	2	no	—
30	C₂×C₄×C₅₃	—	4	1	no	—
31	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
32	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
38	C₂×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
39	C₂×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
40	C₂×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
41	C₈×C₅₃	—	4	2	no	—
42	#88322	—	4	2	no	—
43	#88322	—	4	2	no	—
44	C₂²×C₅₃	—	2	1	no	—
45	C₂²×C₅₃	—	2	1	no	—
46	C₂²×C₅₃	#95	1	1	yes	characteristic
47	C₂²×C₅₃	#106	1	1	yes	characteristic
48	C₂²×C₅₃	#107	1	1	yes	characteristic
49	C₄×C₅₃	—	2	1	no	—
50	C₄×C₅₃	—	2	1	no	—
51	C₄×C₅₃	—	2	1	no	—
52	C₄×C₅₃	#102	1	1	yes	characteristic
53	C₄×C₅₃	#102	1	1	yes	characteristic
54	C₄×C₅₃	#103	1	1	yes	characteristic
55	C₄×C₅₃	#103	1	1	yes	characteristic
56	C₄×C₅₃	#108	1	2	yes	—
57	#1301	—	53	1	no	maximal, Sylow
58	C₂×C₅₃	—	2	1	no	—
59	C₂×C₅₃	—	2	1	no	—
60	C₂×C₅₃	#340	1	1	yes	characteristic
61	C₂×C₅₃	#341	1	1	yes	characteristic
62	C₂×C₅₃	#344	1	1	yes	characteristic
63	#431	—	53	1	no	—
64	#423	D₅₃	1	1	yes	characteristic
65	C₂×C₄⋊C₈	—	53	1	no	—
66	C₅₃	#1301	1	1	yes	characteristic, complemented, Sylow
67	C₂×M₁₆	—	2	1	no	—
68	C₂²×C₈	—	106	1	no	—
69	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
70	C₄⋊C₈	—	2	1	no	—
71	C₄⋊C₈	—	53	2	no	—
72	C₄⋊C₈	—	106	1	no	—
73	C₄⋊C₈	—	106	1	no	—
74	C₄⋊C₈	Dic₅₃	1	2	yes	—
75	C₂²⋊C₈	—	53	2	no	—
76	C₄×C₈	—	106	1	no	—
77	C₂²×C₄	Dic₁₀₆	1	1	yes	characteristic
78	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
79	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
80	M₁₆	—	4	1	no	—
81	M₁₆	—	4	1	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	106	2	no	—
84	C₂×C₈	—	106	2	no	—
85	C₂×C₈	—	212	1	no	—
86	C₂×C₈	—	212	1	no	—
87	C₄²	C₄×D₅₃	1	1	yes	characteristic
88	C₄²	D₂₁₂	1	1	yes	characteristic
89	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
90	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
91	C₂³	#11719919	1	1	yes	characteristic
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	4	1	no	—
94	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
95	C₂×C₄	#11719919	1	1	yes	characteristic
96	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
97	C₂×C₄	#11719921	1	1	yes	characteristic
98	C₂×C₄	#11719921	1	1	yes	characteristic
99	C₂×C₄	#11719922	1	1	yes	characteristic
100	C₂×C₄	#11719923	1	1	yes	characteristic
101	C₂×C₄	#11719924	1	1	yes	characteristic
102	C₂×C₄	#11719925	1	1	yes	characteristic
103	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
104	C₈	—	4	2	no	—
105	C₈	—	212	2	no	—
106	C₈	—	212	2	no	—
107	C₂²	—	2	1	no	—
108	C₂²	—	2	1	no	—
109	C₂²	#13133124	1	1	yes	characteristic
110	C₂²	#13133148	1	1	yes	characteristic
111	C₂²	#13133150	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	#13133137	1	1	yes	characteristic
116	C₄	#13133138	1	1	yes	characteristic
117	C₄	#13133140	1	1	yes	characteristic
118	C₄	#13133141	1	1	yes	characteristic
119	C₄	#13133152	1	2	yes	—
120	C₂	—	2	1	no	—
121	C₂	—	2	1	no	—
122	C₂	#16810385	1	1	yes	characteristic
123	C₂	#16810387	1	1	yes	characteristic
124	C₂	#16810394	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810676

C₂

yes

maximal, characteristic

C₂×C₄⋊#88322

C₂

yes

maximal, characteristic

C₅₃×#423

C₂

yes

maximal, characteristic

C₂²×#88322

—

C₄⋊#88322

—

C₄⋊#88322

—

C₄⋊#88322

C₄

yes

—

C₂²⋊#88322

C₄

yes

—

C₄×#88322

—

C₂×M₁₆×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

Q₈

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

—

—

—

—

C₂³×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

SD₁₆

yes

characteristic

C₂×C₄×C₅₃

SD₁₆

yes

characteristic

C₂×C₄×C₅₃

Q₁₆

yes

characteristic

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

#1301

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

yes

characteristic

C₂×C₄⋊C₈

—

C₅₃

#1301

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

C₄⋊C₈

Dic₅₃

yes

—

C₂²⋊C₈

—

C₄×C₈

—

106

—

C₂²×C₄

Dic₁₀₆

yes

characteristic

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

212

—

C₂×C₈

—

212

—

C₄²

C₄×D₅₃

yes

characteristic

C₄²

D₂₁₂

yes

characteristic

C₄²

C₂×Dic₅₃

yes

characteristic

C₄²

C₂²⋊D₅₃

yes

characteristic

C₂³

#11719919

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₅₃

yes

characteristic

C₂×C₄

#11719919

yes

characteristic

C₂×C₄

C₄⋊Dic₅₃

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

100

C₂×C₄

#11719923

yes

characteristic

101

C₂×C₄

#11719924

yes

characteristic

102

C₂×C₄

#11719925

yes

characteristic

103

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

104

C₈

—

105

C₈

—

212

—

106

C₈

—

212

—

107

C₂²

—

108

C₂²

—

109

C₂²

#13133124

yes

characteristic

110

C₂²

#13133148

yes

characteristic

111

C₂²

#13133150

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

#13133137

yes

characteristic

116

C₄

#13133138

yes

characteristic

117

C₄

#13133140

yes

characteristic

118

C₄

#13133141

yes

characteristic

119

C₄

#13133152

yes

—

120

C₂

—

121

C₂

—

122

C₂

#16810385

yes

characteristic

123

C₂

#16810387

yes

characteristic

124

C₂

#16810394

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₄²
Frattini	C₂×C₄²	C₂×D₅₃
Fitting	C₅₃×#423	C₂
socle	C₂²×C₅₃	#95
hypercenter	#423	D₅₃
hypocenter	C₅₃	#1301
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂²⋊Dic₅₃

commutator

C₂×C₄×C₅₃

C₄²

Frattini

C₂×C₄²

C₂×D₅₃

Fitting

C₅₃×#423

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
66. C₅₃	#1301	semidirect

H ≅ G/N

type

66. C₅₃

#1301

semidirect

Subgroups — #20272610

Conjugacy classes of subgroups

Complements