Group #20274080 — subgroups

Subgroups — #20274080

Order 6784.

172 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 61 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	#16811003	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810903	C₂	1	2	yes	maximal
3	#16810376	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810370	C₂	1	2	yes	maximal, complemented
5	C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₁₂	—	2	1	no	—
7	C₂×Dic₂₁₂	—	2	1	no	—
8	#13133217	C₂²	1	2	yes	—
9	#13133150	—	2	2	no	—
10	C₄⋊#88322	C₂²	1	2	yes	—
11	#13133138	—	2	2	no	—
12	C₂²×C₈×C₅₃	—	2	1	no	—
13	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²⋊C₈×C₅₃	—	2	1	no	—
16	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×D₄×C₅₃	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	—	2	1	no	—
19	C₂×Dic₁₀₆	—	2	2	no	—
20	C₂×C₈×C₅₃	—	2	1	no	—
21	C₂×C₈×C₅₃	—	4	2	no	—
22	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
23	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
25	C₂²⋊C₄×C₅₃	—	2	1	no	—
26	C₄²×C₅₃	C₂³	1	1	yes	characteristic
27	C₄⋊Dic₅₃	—	2	2	no	—
28	C₄⋊Dic₅₃	—	2	2	no	—
29	C₂×#88322	—	2	2	no	—
30	Dic₂₁₂	—	4	2	no	—
31	Dic₂₁₂	—	4	2	no	—
32	C₂³×C₅₃	—	2	1	no	—
33	D₄×C₅₃	—	2	1	no	—
34	D₄×C₅₃	Q₁₆	1	2	yes	complemented
35	C₂×C₄×C₅₃	—	2	1	no	—
36	C₂×C₄×C₅₃	—	4	1	no	—
37	C₂×C₄×C₅₃	—	4	1	no	—
38	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
41	C₂×Dic₅₃	—	4	2	no	—
42	Dic₁₀₆	—	4	2	no	—
43	Dic₁₀₆	—	4	2	no	—
44	C₈×C₅₃	—	2	1	no	—
45	C₈×C₅₃	—	4	1	no	—
46	C₈×C₅₃	D₈	1	2	yes	—
47	#88322	—	4	2	no	—
48	C₂²×C₅₃	—	2	2	no	—
49	C₂²×C₅₃	—	4	1	no	—
50	C₂²×C₅₃	—	4	1	no	—
51	C₂²×C₅₃	#121	1	1	yes	characteristic
52	C₄×C₅₃	—	2	1	no	—
53	C₄×C₅₃	—	4	1	no	—
54	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
55	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
56	C₄×C₅₃	#135	1	1	yes	characteristic
57	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
58	Dic₅₃	—	8	2	no	—
59	#1669	—	53	1	no	maximal, Sylow
60	C₂×C₅₃	—	4	2	no	—
61	C₂×C₅₃	#459	1	1	yes	characteristic
62	C₂×C₅₃	#467	1	1	yes	characteristic
63	C₂×C₅₃	#473	1	1	yes	characteristic
64	#494	—	53	1	no	—
65	#474	—	53	2	no	—
66	C₈×D₄	D₅₃	1	1	yes	characteristic
67	#335	—	53	1	no	—
68	#332	—	53	2	no	—
69	C₅₃	#1669	1	1	yes	characteristic, complemented, Sylow
70	C₂×Q₁₆	—	106	1	no	—
71	C₂×Q₁₆	—	106	1	no	—
72	C₂²×C₈	—	2	1	no	—
73	C₄⋊Q₈	—	53	2	no	—
74	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
75	#107	—	106	2	no	—
76	C₄⋊C₈	—	53	2	no	—
77	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
78	#102	—	106	2	no	—
79	C₂²⋊C₈	—	2	1	no	—
80	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
81	C₂×Q₈	—	106	2	no	—
82	C₂×D₄	D₂₁₂	1	1	yes	characteristic
83	C₂²×C₄	—	2	1	no	—
84	Q₁₆	—	212	2	no	—
85	Q₁₆	—	212	2	no	—
86	C₂×C₈	—	2	1	no	—
87	C₂×C₈	—	4	2	no	—
88	C₂×C₈	—	106	2	no	—
89	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
90	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
91	C₄⋊C₄	—	106	2	no	—
92	C₄⋊C₄	—	106	2	no	—
93	C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
94	C₂²⋊C₄	—	2	1	no	—
95	C₄²	C₂²×D₅₃	1	1	yes	characteristic
96	C₂³	—	2	1	no	—
97	Q₈	—	212	2	no	—
98	Q₈	—	212	2	no	—
99	D₄	—	2	1	no	—
100	D₄	Dic₂₁₂	1	2	yes	complemented
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	4	1	no	—
103	C₂×C₄	—	4	1	no	—
104	C₂×C₄	—	212	2	no	—
105	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
106	C₂×C₄	#11719945	1	1	yes	characteristic
107	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
108	C₈	—	2	1	no	—
109	C₈	—	4	1	no	—
110	C₈	—	212	2	no	—
111	C₈	#11719922	1	2	yes	—
112	C₂²	—	2	2	no	—
113	C₂²	—	4	1	no	—
114	C₂²	—	4	1	no	—
115	C₂²	#13133191	1	1	yes	characteristic
116	C₄	—	2	1	no	—
117	C₄	—	4	1	no	—
118	C₄	—	424	2	no	—
119	C₄	C₂×#11719922	1	1	yes	characteristic
120	C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
121	C₄	#13133239	1	1	yes	characteristic
122	C₄	#13133242	1	1	yes	characteristic
123	C₂	—	4	2	no	—
124	C₂	#16810818	1	1	yes	characteristic
125	C₂	#16810864	1	1	yes	characteristic
126	C₂	#16810898	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811003

C₂

yes

maximal, characteristic, complemented

#16810903

C₂

yes

maximal

#16810376

C₂

yes

maximal, characteristic, complemented

#16810370

C₂

yes

maximal, complemented

C₈×D₄×C₅₃

C₂

yes

maximal, characteristic

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

#13133217

C₂²

yes

—

#13133150

—

C₄⋊#88322

C₂²

yes

—

#13133138

—

C₂²×C₈×C₅₃

—

C₄×D₄×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×D₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

Dic₂₁₂

—

Dic₂₁₂

—

C₂³×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

Q₁₆

yes

complemented

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

D₈

yes

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#121

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×D₈

yes

characteristic

C₄×C₅₃

C₂×Q₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#1669

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

C₈×D₄

D₅₃

yes

characteristic

#335

—

#332

—

C₅₃

#1669

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

106

—

C₂×Q₁₆

—

106

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄×D₄

C₂×D₅₃

yes

characteristic

#107

—

106

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₃

yes

characteristic

#102

—

106

—

C₂²⋊C₈

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

106

—

C₂×D₄

D₂₁₂

yes

characteristic

C₂²×C₄

—

Q₁₆

—

212

—

Q₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

D₂₁₂

yes

characteristic

C₂²⋊C₄

—

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

—

Q₈

—

212

—

Q₈

—

212

—

D₄

—

100

D₄

Dic₂₁₂

yes

complemented

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

212

—

105

C₂×C₄

C₂×D₂₁₂

yes

characteristic

106

C₂×C₄

#11719945

yes

characteristic

107

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

108

C₈

—

109

C₈

—

110

C₈

—

212

—

111

C₈

#11719922

yes

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

#13133191

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

424

—

119

C₄

C₂×#11719922

yes

characteristic

120

C₄

C₂×Dic₂₁₂

yes

characteristic

121

C₄

#13133239

yes

characteristic

122

C₄

#13133242

yes

characteristic

123

C₂

—

124

C₂

#16810818

yes

characteristic

125

C₂

#16810864

yes

characteristic

126

C₂

#16810898

yes

characteristic

	subgroup	G/H
center	C₂²	#13133191
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₈×D₄×C₅₃	C₂
socle	C₂²×C₅₃	#121
hypercenter	C₈×D₄	D₅₃
hypocenter	C₅₃	#1669
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄

subgroup

G/H

center

C₂²

#13133191

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₈×D₄×C₅₃

C₂

socle

C₂²×C₅₃

#121

hypercenter

C₈×D₄

D₅₃

hypocenter

C₅₃

#1669

LCS

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄

N	H ≅ G/N	type
1. #16811003	C₂	semidirect
3. #16810376	C₂	semidirect
4. #16810370	C₂	semidirect
34. D₄×C₅₃	Q₁₆	semidirect
69. C₅₃	#1669	semidirect
100. D₄	Dic₂₁₂	semidirect

H ≅ G/N

type

1. #16811003

C₂

semidirect

3. #16810376

C₂

semidirect

4. #16810370

C₂

semidirect

34. D₄×C₅₃

Q₁₆

semidirect

69. C₅₃

#1669

semidirect

100. D₄

Dic₂₁₂

semidirect

Subgroups — #20274080

Conjugacy classes of subgroups

Complements