Group #20272564 — subgroups

Subgroups — #20272564

Order 6784.

160 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 81 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×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
2	C₄×#11719917	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#431	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719917	C₄	1	2	yes	—
5	C₂²×#88322	C₄	1	2	yes	—
6	C₄⋊#88322	—	2	1	no	—
7	C₄⋊#88322	—	2	1	no	—
8	#13133128	—	2	1	no	—
9	C₄×#88322	—	2	1	no	—
10	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₅₃	—	2	1	no	—
12	C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
13	C₄×C₈×C₅₃	—	2	1	no	—
14	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₅₃	—	2	2	no	—
18	C₄²×C₅₃	D₄	1	1	yes	characteristic
19	C₄²×C₅₃	D₄	1	1	yes	characteristic
20	C₄²×C₅₃	D₄	1	1	yes	characteristic
21	C₄²×C₅₃	Q₈	1	1	yes	characteristic
22	#11719917	—	2	2	no	—
23	#11719917	C₈	1	4	yes	complemented
24	C₂×#88322	—	2	2	no	—
25	C₂×#88322	—	2	2	no	—
26	C₂×#88322	—	4	2	no	—
27	C₂³×C₅₃	C₄²	1	1	yes	characteristic
28	C₂×C₄×C₅₃	—	2	1	no	—
29	C₂×C₄×C₅₃	—	2	1	no	—
30	C₂×C₄×C₅₃	—	4	1	no	—
31	C₂×C₄×C₅₃	C₂²⋊C₄	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	C₄⋊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	C₂×C₈	1	2	yes	—
38	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
39	C₈×C₅₃	—	4	2	no	—
40	#88322	—	2	4	no	—
41	#88322	—	4	2	no	—
42	C₂²×C₅₃	—	2	1	no	—
43	C₂²×C₅₃	—	2	1	no	—
44	C₂²×C₅₃	#95	1	1	yes	characteristic
45	C₂²×C₅₃	C₄×C₈	1	1	yes	characteristic
46	C₂²×C₅₃	#97	1	1	yes	characteristic
47	C₄×C₅₃	—	2	1	no	—
48	C₄×C₅₃	—	2	1	no	—
49	C₄×C₅₃	—	2	1	no	—
50	C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
51	C₄×C₅₃	#100	1	1	yes	characteristic
52	C₄×C₅₃	#101	1	1	yes	characteristic
53	C₄×C₅₃	C₄⋊C₈	1	2	yes	—
54	#1283	—	53	1	no	maximal, Sylow
55	C₂×C₅₃	—	2	1	no	—
56	C₂×C₅₃	—	2	1	no	—
57	C₂×C₅₃	#336	1	1	yes	characteristic
58	C₂×C₅₃	#343	1	1	yes	characteristic
59	C₂×C₅₃	#344	1	1	yes	characteristic
60	#431	D₅₃	1	1	yes	characteristic
61	C₂×C₄⋊C₈	—	53	1	no	—
62	C₄×M₁₆	—	53	1	no	—
63	C₅₃	#1283	1	1	yes	characteristic, complemented, Sylow
64	C₂×M₁₆	—	53	2	no	—
65	C₂²×C₈	—	53	2	no	—
66	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
67	C₄⋊C₈	—	2	1	no	—
68	C₄⋊C₈	—	106	1	no	—
69	C₄⋊C₈	—	106	1	no	—
70	C₂²⋊C₈	Dic₅₃	1	2	yes	—
71	#97	—	106	1	no	—
72	C₄×C₈	—	2	1	no	—
73	C₄×C₈	—	106	1	no	—
74	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
75	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
76	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
77	M₁₆	—	53	4	no	—
78	M₁₆	—	106	2	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	106	2	no	—
81	C₂×C₈	—	106	2	no	—
82	C₂×C₈	—	212	2	no	—
83	C₄²	Dic₁₀₆	1	1	yes	characteristic
84	C₄²	D₂₁₂	1	1	yes	characteristic
85	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
86	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
87	C₂³	C₄×Dic₅₃	1	1	yes	characteristic
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	4	1	no	—
91	C₂×C₄	C₈×D₅₃	1	2	yes	—
92	C₂×C₄	#11719912	1	2	yes	—
93	C₂×C₄	#11719919	1	1	yes	characteristic
94	C₂×C₄	#11719919	1	1	yes	characteristic
95	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
96	C₂×C₄	#11719921	1	1	yes	characteristic
97	C₂×C₄	#11719921	1	1	yes	characteristic
98	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
99	C₈	—	4	2	no	—
100	C₈	—	106	4	no	—
101	C₈	—	212	2	no	—
102	C₂²	—	2	1	no	—
103	C₂²	—	2	1	no	—
104	C₂²	#13133124	1	1	yes	characteristic
105	C₂²	C₈×Dic₅₃	1	1	yes	characteristic
106	C₂²	#13133127	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	#13133129	1	2	yes	—
111	C₄	#13133133	1	1	yes	characteristic
112	C₄	#13133135	1	1	yes	characteristic
113	C₄	#13133146	1	2	yes	—
114	C₂	—	2	1	no	—
115	C₂	—	2	1	no	—
116	C₂	#16810377	1	1	yes	characteristic
117	C₂	#16810391	1	1	yes	characteristic
118	C₂	#16810393	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#88322

C₂

yes

maximal, characteristic

C₄×#11719917

C₂

yes

maximal, characteristic

C₅₃×#431

C₂

yes

maximal, characteristic

C₂×#11719917

C₄

yes

—

C₂²×#88322

C₄

yes

—

—

—

—

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

C₄

yes

—

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

—

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

Q₈

yes

characteristic

#11719917

—

#11719917

C₈

yes

—

—

—

C₂³×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

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

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

M₁₆

yes

—

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#95

yes

characteristic

C₂²×C₅₃

C₄×C₈

yes

characteristic

C₂²×C₅₃

#97

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊C₈

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₄⋊C₈

yes

—

#1283

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

characteristic

C₂×C₄⋊C₈

—

C₄×M₁₆

—

C₅₃

#1283

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

Dic₅₃

yes

—

#97

—

106

—

C₄×C₈

—

C₄×C₈

—

106

—

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

M₁₆

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

212

—

C₄²

Dic₁₀₆

yes

characteristic

C₄²

D₂₁₂

yes

characteristic

C₄²

C₂²⋊D₅₃

yes

characteristic

C₄²

C₂²⋊D₅₃

yes

characteristic

C₂³

C₄×Dic₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₈×D₅₃

yes

—

C₂×C₄

#11719912

yes

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₄⋊Dic₅₃

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

C₈

—

100

C₈

—

106

—

101

C₈

—

212

—

102

C₂²

—

103

C₂²

—

104

C₂²

#13133124

yes

characteristic

105

C₂²

C₈×Dic₅₃

yes

characteristic

106

C₂²

#13133127

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

#13133129

yes

—

111

C₄

#13133133

yes

characteristic

112

C₄

#13133135

yes

characteristic

113

C₄

#13133146

yes

—

114

C₂

—

115

C₂

—

116

C₂

#16810377

yes

characteristic

117

C₂

#16810391

yes

characteristic

118

C₂

#16810393

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#11719921
commutator	C₂²×C₅₃	C₄×C₈
Frattini	C₂×C₄²	C₂×D₅₃
Fitting	C₅₃×#431	C₂
socle	C₂²×C₅₃	#95
hypercenter	#431	D₅₃
hypocenter	C₅₃	#1283
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

#11719921

commutator

C₂²×C₅₃

C₄×C₈

Frattini

C₂×C₄²

C₂×D₅₃

Fitting

C₅₃×#431

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
23. #11719917	C₈	semidirect
63. C₅₃	#1283	semidirect

H ≅ G/N

type

23. #11719917

C₈

semidirect

63. C₅₃

#1283

semidirect

Subgroups — #20272564

Conjugacy classes of subgroups

Complements