Group #20276430 — subgroups

Subgroups — #20276430

Order 6784.

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

normal subgroups only

This group has 260 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	C₂²×D₄₂₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133157	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810423	C₂	1	4	yes	maximal, complemented
4	C₅₃×#427	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₂₁₂	—	2	1	no	—
6	C₂×D₄₂₄	—	2	2	no	—
7	C₂×D₄₂₄	—	2	1	no	—
8	C₂×D₄₂₄	C₄	1	4	yes	complemented
9	C₂×D₄₂₄	C₂²	1	2	yes	—
10	#13133157	—	2	2	no	—
11	C₂×#11719908	C₂²	1	2	yes	complemented
12	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	#117×C₅₃	—	2	1	no	—
14	#107×C₅₃	C₂²	1	2	yes	complemented
15	#106×C₅₃	—	2	1	no	—
16	C₂³×D₅₃	—	4	1	no	—
17	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
18	C₂×D₂₁₂	—	2	4	no	—
19	C₂×D₂₁₂	—	2	2	no	—
20	C₂×D₂₁₂	—	4	1	no	—
21	C₂×D₂₁₂	—	4	2	no	—
22	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂×C₈×C₅₃	D₄	1	2	yes	—
25	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₅₃	—	2	2	no	—
27	C₂²⋊C₄×C₅₃	—	4	1	no	—
28	C₄²×C₅₃	—	4	1	no	—
29	D₄₂₄	—	2	4	no	—
30	D₄₂₄	—	2	2	no	—
31	D₄₂₄	C₂×C₄	1	4	yes	—
32	#11719908	—	2	2	no	—
33	C₂³×C₅₃	D₈	1	1	yes	characteristic
34	C₂²×D₅₃	—	4	4	no	—
35	C₂²×D₅₃	—	4	2	no	—
36	C₂²×D₅₃	—	8	4	no	—
37	C₂×C₄×C₅₃	—	4	2	no	—
38	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
40	C₂×C₄×C₅₃	SD₁₆	1	2	yes	—
41	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	D₂₁₂	—	2	4	no	—
43	D₂₁₂	—	4	4	no	—
44	D₂₁₂	—	4	2	no	—
45	C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
46	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₅₃	—	2	2	no	—
49	C₂²×C₅₃	#102	1	2	yes	—
50	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
51	C₂×D₅₃	—	4	4	no	—
52	C₂×D₅₃	—	8	8	no	—
53	C₂×D₅₃	—	8	4	no	—
54	C₄×C₅₃	—	8	2	no	—
55	C₄×C₅₃	#102	1	2	yes	—
56	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
58	#2140	—	53	1	no	maximal, Sylow
59	C₂×C₅₃	—	2	2	no	—
60	C₂×C₅₃	C₂×#102	1	1	yes	characteristic
61	C₂×C₅₃	#509	1	2	yes	—
62	D₅₃	—	8	4	no	—
63	C₂²×D₈	—	53	1	no	—
64	C₂×M₃₂	—	53	1	no	—
65	#427	D₅₃	1	1	yes	characteristic, complemented
66	#357	—	53	4	no	—
67	C₅₃	#2140	1	1	yes	characteristic, complemented, Sylow
68	C₂²×D₄	—	106	1	no	—
69	C₂×D₈	—	53	4	no	—
70	C₂×D₈	—	53	2	no	—
71	C₂×D₈	—	106	2	no	—
72	C₂×D₈	—	106	1	no	—
73	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
74	#117	—	2	1	no	—
75	M₃₂	—	106	2	no	—
76	C₂×C₁₆	—	53	2	no	—
77	#107	C₂×D₅₃	1	2	yes	complemented
78	#106	—	2	1	no	—
79	C₂⁴	—	212	1	no	—
80	C₂×D₄	—	106	2	no	—
81	C₂×D₄	—	106	4	no	—
82	C₂×D₄	—	212	1	no	—
83	C₂×D₄	—	212	2	no	—
84	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
85	D₈	—	53	4	no	—
86	D₈	—	106	2	no	—
87	D₈	—	106	4	no	—
88	C₂×C₈	C₄×D₅₃	1	2	yes	—
89	C₂×C₈	D₂₁₂	1	2	yes	—
90	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
91	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
92	C₄⋊C₄	—	2	2	no	—
93	C₂²⋊C₄	—	4	1	no	—
94	C₄²	—	4	1	no	—
95	C₁₆	—	106	2	no	—
96	C₂³	—	424	4	no	—
97	C₂³	—	212	4	no	—
98	C₂³	—	212	2	no	—
99	C₂³	#11719922	1	1	yes	characteristic
100	D₄	—	106	4	no	—
101	D₄	—	212	4	no	—
102	D₄	—	212	2	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	#11719921	1	2	yes	—
105	C₂×C₄	#11719922	1	1	yes	characteristic
106	C₂×C₄	#11719924	1	2	yes	—
107	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
108	C₈	#11719921	1	2	yes	—
109	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
110	C₈	C₂×D₂₁₂	1	1	yes	characteristic
111	C₂²	—	2	2	no	—
112	C₂²	—	424	8	no	—
113	C₂²	—	424	4	no	—
114	C₂²	—	212	4	no	—
115	C₂²	#13133137	1	2	yes	—
116	C₂²	C₂×#11719922	1	1	yes	characteristic
117	C₄	—	8	2	no	—
118	C₄	#13133137	1	2	yes	—
119	C₄	C₂×#11719921	1	1	yes	characteristic
120	C₄	C₂×#11719924	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	424	4	no	—
123	C₂	C₂×#13133137	1	1	yes	characteristic
124	C₂	#16811058	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₄₂₄

C₂

yes

maximal, characteristic

C₂×#13133157

C₂

yes

maximal, characteristic, complemented

#16810423

C₂

yes

maximal, complemented

C₅₃×#427

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₁₂

—

C₂×D₄₂₄

—

C₂×D₄₂₄

—

C₂×D₄₂₄

C₄

yes

complemented

C₂×D₄₂₄

C₂²

yes

—

#13133157

—

C₂×#11719908

C₂²

yes

complemented

C₂²×C₈×C₅₃

C₂²

yes

characteristic

#117×C₅₃

—

#107×C₅₃

C₂²

yes

complemented

#106×C₅₃

—

C₂³×D₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

D₄₂₄

—

D₄₂₄

—

D₄₂₄

C₂×C₄

yes

—

#11719908

—

C₂³×C₅₃

D₈

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

SD₁₆

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₈×C₅₃

C₂²⋊C₄

yes

—

C₈×C₅₃

C₂²×C₄

yes

characteristic

C₈×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₅₃

—

C₂²×C₅₃

#102

yes

—

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#102

yes

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

C₂×SD₁₆

yes

characteristic

#2140

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×#102

yes

characteristic

C₂×C₅₃

#509

yes

—

D₅₃

—

C₂²×D₈

—

C₂×M₃₂

—

#427

D₅₃

yes

characteristic, complemented

#357

—

C₅₃

#2140

yes

characteristic, complemented, Sylow

C₂²×D₄

—

106

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

106

—

C₂×D₈

—

106

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

#117

—

M₃₂

—

106

—

C₂×C₁₆

—

#107

C₂×D₅₃

yes

complemented

#106

—

C₂⁴

—

212

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

212

—

C₂×D₄

—

212

—

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

D₈

—

D₈

—

106

—

D₈

—

106

—

C₂×C₈

C₄×D₅₃

yes

—

C₂×C₈

D₂₁₂

yes

—

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

106

—

C₂³

—

424

—

C₂³

—

212

—

C₂³

—

212

—

C₂³

#11719922

yes

characteristic

100

D₄

—

106

—

101

D₄

—

212

—

102

D₄

—

212

—

103

C₂×C₄

—

104

C₂×C₄

#11719921

yes

—

105

C₂×C₄

#11719922

yes

characteristic

106

C₂×C₄

#11719924

yes

—

107

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

108

C₈

#11719921

yes

—

109

C₈

C₂×C₄×D₅₃

yes

characteristic

110

C₈

C₂×D₂₁₂

yes

characteristic

111

C₂²

—

112

C₂²

—

424

—

113

C₂²

—

424

—

114

C₂²

—

212

—

115

C₂²

#13133137

yes

—

116

C₂²

C₂×#11719922

yes

characteristic

117

C₄

—

118

C₄

#13133137

yes

—

119

C₄

C₂×#11719921

yes

characteristic

120

C₄

C₂×#11719924

yes

characteristic

121

C₂

—

122

C₂

—

424

—

123

C₂

C₂×#13133137

yes

characteristic

124

C₂

#16811058

yes

—

	subgroup	G/H
center	C₂²	C₂×#11719922
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#427	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	#427	D₅₃
hypocenter	C₅₃	#2140
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

subgroup

G/H

center

C₂²

C₂×#11719922

commutator

C₈×C₅₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#427

C₂

socle

C₂²×C₅₃

C₂×D₈

D₅₃

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

N	H ≅ G/N	type
2. C₂×#13133157	C₂	semidirect
3. #16810423	C₂	semidirect
4. C₅₃×#427	C₂	semidirect
8. C₂×D₄₂₄	C₄	semidirect
11. C₂×#11719908	C₂²	semidirect
14. #107×C₅₃	C₂²	semidirect
65. #427	D₅₃	semidirect
67. C₅₃	#2140	semidirect
77. #107	C₂×D₅₃	semidirect

H ≅ G/N

type

2. C₂×#13133157

C₂

semidirect

3. #16810423

C₂

semidirect

4. C₅₃×#427

C₂

semidirect

8. C₂×D₄₂₄

C₄

semidirect

11. C₂×#11719908

C₂²

semidirect

14. #107×C₅₃

C₂²

semidirect

65. #427

D₅₃

semidirect

67. C₅₃

#2140

semidirect

77. #107

C₂×D₅₃

semidirect

Subgroups — #20276430

Conjugacy classes of subgroups

Complements