Group #20276469 — subgroups

Subgroups — #20276469

Order 6784.

172 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 85 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	2	yes	maximal
2	C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810441	C₂	1	2	yes	maximal, complemented
5	C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₅₃	C₄	1	1	yes	characteristic
7	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
8	C₄²×D₅₃	C₄	1	1	yes	characteristic
9	#13133156	—	2	4	no	—
10	C₂×#11719908	C₂²	1	2	yes	—
11	C₄×#88322	C₄	1	1	yes	characteristic
12	C₈×Dic₅₃	C₄	1	1	yes	characteristic
13	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₅₃	C₂²	1	2	yes	—
15	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₅₃	C₈	1	1	yes	characteristic
17	C₂×C₄×D₅₃	C₈	1	1	yes	characteristic
18	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
21	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄×Dic₅₃	C₈	1	1	yes	characteristic
23	C₄×Dic₅₃	C₈	1	1	yes	characteristic
24	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×#88322	C₂×C₄	1	1	yes	characteristic
26	C₂×#88322	C₂×C₄	1	1	yes	characteristic
27	C₈×D₅₃	—	2	2	no	—
28	C₈×D₅₃	D₄	1	2	yes	—
29	C₈×D₅₃	Q₈	1	2	yes	—
30	C₁₆×C₅₃	—	2	2	no	—
31	#11719908	—	2	2	no	—
32	C₂²×D₅₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×Dic₅₃	C₂×C₈	1	1	yes	characteristic
37	C₂×Dic₅₃	C₂×C₈	1	1	yes	characteristic
38	C₂×Dic₅₃	C₂×C₈	1	1	yes	characteristic
39	C₄×D₅₃	—	2	2	no	—
40	C₄×D₅₃	—	2	2	no	—
41	C₄×D₅₃	C₄⋊C₄	1	2	yes	—
42	C₄×D₅₃	M₁₆	1	2	yes	—
43	C₈×C₅₃	—	2	1	no	—
44	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
46	#88322	—	2	1	no	—
47	#88322	C₄⋊C₄	1	2	yes	—
48	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
49	C₂×D₅₃	—	2	1	no	—
50	C₂×D₅₃	—	2	1	no	—
51	C₂×D₅₃	C₄⋊C₈	1	2	yes	—
52	C₄×C₅₃	—	2	1	no	—
53	C₄×C₅₃	M₃₂	1	2	yes	complemented
54	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
56	Dic₅₃	—	2	1	no	—
57	Dic₅₃	C₄⋊C₈	1	2	yes	—
58	Dic₅₃	M₃₂	1	2	yes	complemented
59	#2150	—	53	1	no	maximal, Sylow
60	C₂×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₅₃	C₂×M₃₂	1	1	yes	characteristic
62	C₂×C₅₃	C₂×M₃₂	1	1	yes	characteristic
63	D₅₃	—	2	2	no	—
64	C₂×M₃₂	—	53	2	no	—
65	C₂×C₄×C₈	—	53	1	no	—
66	C₄⋊C₁₆	—	53	2	no	—
67	C₄⋊C₁₆	—	53	1	no	—
68	C₄⋊C₁₆	D₅₃	1	1	yes	characteristic, complemented
69	C₅₃	#2150	1	1	yes	characteristic, complemented, Sylow
70	C₂²×C₈	—	53	1	no	—
71	C₂²×C₈	—	53	1	no	—
72	C₂×C₄²	—	53	1	no	—
73	M₃₂	—	106	4	no	—
74	C₂×C₁₆	—	53	2	no	—
75	C₂×C₁₆	C₂×D₅₃	1	2	yes	—
76	C₄×C₈	—	53	1	no	—
77	C₄×C₈	—	53	1	no	—
78	C₄×C₈	—	53	1	no	—
79	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
80	C₂²×C₄	—	53	1	no	—
81	C₂²×C₄	—	53	1	no	—
82	C₂²×C₄	—	53	1	no	—
83	C₂×C₈	—	53	1	no	—
84	C₂×C₈	—	53	2	no	—
85	C₂×C₈	—	53	2	no	—
86	C₂×C₈	—	53	1	no	—
87	C₂×C₈	—	106	2	no	—
88	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
89	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
90	C₄²	—	53	1	no	—
91	C₄²	—	53	1	no	—
92	C₄²	—	53	1	no	—
93	C₄²	C₄×D₅₃	1	1	yes	characteristic
94	C₁₆	—	2	2	no	—
95	C₁₆	—	106	2	no	—
96	C₂³	—	53	1	no	—
97	C₂×C₄	—	53	1	no	—
98	C₂×C₄	—	53	1	no	—
99	C₂×C₄	—	53	2	no	—
100	C₂×C₄	—	53	2	no	—
101	C₂×C₄	—	53	1	no	—
102	C₂×C₄	—	106	2	no	—
103	C₂×C₄	—	106	2	no	—
104	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
105	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
106	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
107	C₈	—	2	1	no	—
108	C₈	—	53	2	no	—
109	C₈	—	106	1	no	—
110	C₈	D₄×D₅₃	1	1	yes	characteristic
111	C₈	Q₈×D₅₃	1	1	yes	characteristic
112	C₂²	—	53	2	no	—
113	C₂²	—	106	1	no	—
114	C₂²	—	106	1	no	—
115	C₂²	C₂×C₈×D₅₃	1	1	yes	characteristic
116	C₄	—	2	1	no	—
117	C₄	—	53	2	no	—
118	C₄	—	53	2	no	—
119	C₄	—	106	1	no	—
120	C₄	#13133156	1	2	yes	complemented
121	C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
122	C₄	M₁₆×D₅₃	1	1	yes	characteristic
123	C₂	—	106	2	no	—
124	C₂	C₄⋊C₈×D₅₃	1	1	yes	characteristic
125	C₂	M₃₂×D₅₃	1	1	yes	characteristic
126	C₂	C₂×#13133156	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133156

C₂

yes

maximal

C₄×C₈×D₅₃

C₂

yes

maximal, characteristic

C₄⋊#11719908

C₂

yes

maximal, characteristic, complemented

#16810441

C₂

yes

maximal, complemented

C₅₃×C₄⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₅₃

C₄

yes

characteristic

C₂×C₈×D₅₃

C₂²

yes

characteristic

C₄²×D₅₃

C₄

yes

characteristic

#13133156

—

C₂×#11719908

C₂²

yes

—

C₄×#88322

C₄

yes

characteristic

C₈×Dic₅₃

C₄

yes

characteristic

C₈×Dic₅₃

C₂²

yes

characteristic

C₂×C₁₆×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄×D₅₃

C₈

yes

characteristic

C₂×C₄×D₅₃

C₈

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

C₄×Dic₅₃

C₈

yes

characteristic

C₄×Dic₅₃

C₈

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₈×D₅₃

—

C₈×D₅₃

D₄

yes

—

C₈×D₅₃

Q₈

yes

—

C₁₆×C₅₃

—

#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

—

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

M₃₂

yes

complemented

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₂×M₁₆

yes

characteristic

Dic₅₃

—

Dic₅₃

C₄⋊C₈

yes

—

Dic₅₃

M₃₂

yes

complemented

#2150

—

maximal, Sylow

C₂×C₅₃

C₂×C₄⋊C₈

yes

characteristic

C₂×C₅₃

C₂×M₃₂

yes

characteristic

C₂×C₅₃

C₂×M₃₂

yes

characteristic

D₅₃

—

C₂×M₃₂

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

D₅₃

yes

characteristic, complemented

C₅₃

#2150

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

106

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₃

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₄²

C₄×D₅₃

yes

characteristic

C₁₆

—

C₁₆

—

106

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

106

—

103

C₂×C₄

—

106

—

104

C₂×C₄

C₈×D₅₃

yes

characteristic

105

C₂×C₄

C₈×D₅₃

yes

characteristic

106

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

—

106

—

110

C₈

D₄×D₅₃

yes

characteristic

111

C₈

Q₈×D₅₃

yes

characteristic

112

C₂²

—

113

C₂²

—

106

—

114

C₂²

—

106

—

115

C₂²

C₂×C₈×D₅₃

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

106

—

120

C₄

#13133156

yes

complemented

121

C₄

C₄⋊C₄×D₅₃

yes

characteristic

122

C₄

M₁₆×D₅₃

yes

characteristic

123

C₂

—

106

—

124

C₂

C₄⋊C₈×D₅₃

yes

characteristic

125

C₂

M₃₂×D₅₃

yes

characteristic

126

C₂

C₂×#13133156

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	#2150
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
3. C₄⋊#11719908	C₂	semidirect
4. #16810441	C₂	semidirect
5. C₅₃×C₄⋊C₁₆	C₂	semidirect
53. C₄×C₅₃	M₃₂	semidirect
58. Dic₅₃	M₃₂	semidirect
68. C₄⋊C₁₆	D₅₃	semidirect
69. C₅₃	#2150	semidirect
120. C₄	#13133156	semidirect

H ≅ G/N

type

3. C₄⋊#11719908

C₂

semidirect

4. #16810441

C₂

semidirect

5. C₅₃×C₄⋊C₁₆

C₂

semidirect

53. C₄×C₅₃

M₃₂

semidirect

58. Dic₅₃

M₃₂

semidirect

68. C₄⋊C₁₆

D₅₃

semidirect

69. C₅₃

#2150

semidirect

120. C₄

#13133156

semidirect

Subgroups — #20276469

Conjugacy classes of subgroups

Complements