Group #20276860 — subgroups

Subgroups — #20276860

Order 6784.

142 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 51 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	#16811027	C₂	1	1	yes	maximal, characteristic, complemented
2	D₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#16810451	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810450	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810441	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810424	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#357	C₂	1	1	yes	maximal, characteristic
8	#13133216	—	2	1	no	—
9	D₄×Dic₅₃	—	2	1	no	—
10	C₂×#11719908	C₂²	1	1	yes	characteristic
11	#13133151	C₂²	1	1	yes	characteristic
12	#13133151	C₂²	1	1	yes	characteristic
13	#13133150	—	2	1	no	—
14	#13133138	—	2	1	no	—
15	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
16	C₂×D₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
18	#107×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×D₄×C₅₃	—	2	1	no	—
20	C₂²×Dic₅₃	—	4	1	no	—
21	C₂×Dic₁₀₆	—	4	1	no	—
22	D₈×C₅₃	—	2	1	no	—
23	D₈×C₅₃	Q₈	1	2	yes	complemented
24	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₅₃	—	2	1	no	—
26	C₂²⋊Dic₅₃	—	4	1	no	—
27	C₄⋊Dic₅₃	—	2	1	no	—
28	C₄⋊Dic₅₃	—	2	1	no	—
29	#11719919	—	4	1	no	—
30	C₄×Dic₅₃	D₄	1	1	yes	characteristic
31	C₂×#88322	D₄	1	1	yes	characteristic
32	C₁₆×C₅₃	—	2	1	no	—
33	#11719908	—	2	1	no	—
34	C₂³×C₅₃	—	4	1	no	—
35	D₄×C₅₃	—	2	2	no	—
36	D₄×C₅₃	—	4	1	no	—
37	C₂×C₄×C₅₃	—	4	1	no	—
38	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂×Dic₅₃	—	4	1	no	—
40	C₂×Dic₅₃	—	4	1	no	—
41	C₂×Dic₅₃	—	8	2	no	—
42	C₂×Dic₅₃	D₈	1	1	yes	characteristic
43	C₂×Dic₅₃	D₈	1	1	yes	characteristic
44	Dic₁₀₆	—	8	2	no	—
45	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
46	C₈×C₅₃	#41	1	1	yes	characteristic
47	#88322	—	2	1	no	—
48	C₂²×C₅₃	—	4	2	no	—
49	C₂²×C₅₃	—	8	1	no	—
50	C₂²×C₅₃	—	8	1	no	—
51	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
52	C₄×C₅₃	—	8	1	no	—
53	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
54	C₄×C₅₃	#137	1	1	yes	characteristic
55	Dic₅₃	—	2	1	no	—
56	Dic₅₃	—	8	1	no	—
57	Dic₅₃	—	8	1	no	—
58	Dic₅₃	SD₃₂	1	2	yes	—
59	#2226	—	53	1	no	maximal, Sylow
60	C₂×C₅₃	—	8	2	no	—
61	C₂×C₅₃	#474	1	1	yes	characteristic
62	C₂×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
63	C₂×C₅₃	#509	1	1	yes	characteristic
64	#500	—	53	1	no	—
65	C₄×D₈	—	53	1	no	—
66	#367	—	53	1	no	—
67	#367	—	53	1	no	—
68	C₄⋊C₁₆	—	53	1	no	—
69	#357	—	53	1	no	—
70	#357	D₅₃	1	1	yes	characteristic
71	C₅₃	#2226	1	1	yes	characteristic, complemented, Sylow
72	C₂×D₈	C₂×D₅₃	1	1	yes	characteristic
73	C₄⋊Q₈	—	106	1	no	—
74	C₄×D₄	—	106	1	no	—
75	C₂×C₁₆	—	53	1	no	—
76	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
77	#107	—	53	1	no	—
78	#107	—	53	1	no	—
79	#107	—	106	1	no	—
80	#107	C₂×D₅₃	1	1	yes	characteristic
81	#102	—	106	1	no	—
82	C₄×C₈	—	53	1	no	—
83	C₂×Q₈	—	212	1	no	—
84	C₂×D₄	—	2	1	no	—
85	C₂²×C₄	—	212	1	no	—
86	D₈	—	2	1	no	—
87	D₈	Dic₁₀₆	1	2	yes	complemented
88	C₂×C₈	—	53	1	no	—
89	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
90	C₄⋊C₄	—	2	1	no	—
91	C₄⋊C₄	—	106	1	no	—
92	C₄⋊C₄	—	106	1	no	—
93	C₄⋊C₄	—	212	1	no	—
94	C₂²⋊C₄	—	212	1	no	—
95	C₄²	—	53	1	no	—
96	C₁₆	—	2	1	no	—
97	C₁₆	—	106	1	no	—
98	C₂³	—	4	1	no	—
99	Q₈	—	424	2	no	—
100	D₄	—	2	2	no	—
101	D₄	—	4	1	no	—
102	C₂×C₄	—	4	1	no	—
103	C₂×C₄	—	53	1	no	—
104	C₂×C₄	—	53	1	no	—
105	C₂×C₄	—	424	2	no	—
106	C₂×C₄	—	212	1	no	—
107	C₂×C₄	—	212	1	no	—
108	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
109	C₈	—	106	1	no	—
110	C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
111	C₈	#11719947	1	1	yes	characteristic
112	C₂²	—	4	2	no	—
113	C₂²	—	8	1	no	—
114	C₂²	—	8	1	no	—
115	C₂²	D₈×D₅₃	1	1	yes	characteristic
116	C₄	—	8	1	no	—
117	C₄	—	53	2	no	—
118	C₄	—	106	1	no	—
119	C₄	—	424	1	no	—
120	C₄	—	424	1	no	—
121	C₄	#13133194	1	1	yes	characteristic
122	C₄	#13133246	1	1	yes	characteristic
123	C₂	—	8	2	no	—
124	C₂	#16810901	1	1	yes	characteristic
125	C₂	SD₃₂×D₅₃	1	1	yes	characteristic
126	C₂	#16811057	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811027

C₂

yes

maximal, characteristic, complemented

D₈×Dic₅₃

C₂

yes

maximal, characteristic

#16810451

C₂

yes

maximal, characteristic, complemented

#16810450

C₂

yes

maximal, characteristic, complemented

#16810441

C₂

yes

maximal, characteristic, complemented

#16810424

C₂

yes

maximal, characteristic

C₅₃×#357

C₂

yes

maximal, characteristic

#13133216

—

D₄×Dic₅₃

—

C₂×#11719908

C₂²

yes

characteristic

#13133151

C₂²

yes

characteristic

#13133151

C₂²

yes

characteristic

#13133150

—

#13133138

—

C₈×Dic₅₃

C₂²

yes

characteristic

C₂×D₈×C₅₃

C₂²

yes

characteristic

C₂×C₁₆×C₅₃

C₂²

yes

characteristic

#107×C₅₃

C₂²

yes

characteristic

C₂×D₄×C₅₃

—

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

D₈×C₅₃

—

D₈×C₅₃

Q₈

yes

complemented

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

D₄

yes

characteristic

C₂×#88322

D₄

yes

characteristic

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

D₈

yes

characteristic

C₂×Dic₅₃

D₈

yes

characteristic

Dic₁₀₆

—

C₈×C₅₃

C₂×Q₈

yes

C₈×C₅₃

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₄×C₅₃

#137

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

SD₃₂

yes

—

#2226

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#474

yes

characteristic

C₂×C₅₃

C₂×SD₃₂

yes

C₂×C₅₃

yes

—

C₄×D₈

—

#367

—

#367

—

C₄⋊C₁₆

—

#357

—

#357

D₅₃

yes

characteristic

C₅₃

#2226

yes

characteristic, complemented, Sylow

C₂×D₈

C₂×D₅₃

yes

characteristic

C₄⋊Q₈

—

106

—

C₄×D₄

—

106

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₃

yes

—

—

—

106

—

#107

C₂×D₅₃

yes

characteristic

#102

—

106

—

C₄×C₈

—

C₂×Q₈

—

212

—

C₂×D₄

—

C₂²×C₄

—

212

—

D₈

—

D₈

Dic₁₀₆

yes

complemented

C₂×C₈

—

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₂²⋊C₄

—

212

—

C₄²

—

C₁₆

—

C₁₆

—

106

—

C₂³

—

Q₈

—

424

—

100

D₄

—

101

D₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

424

—

106

C₂×C₄

—

212

—

107

C₂×C₄

—

212

—

108

C₂×C₄

D₄×D₅₃

yes

characteristic

109

C₈

—

106

—

110

C₈

C₂×Dic₁₀₆

yes

characteristic

111

C₈

#11719947

yes

characteristic

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

D₈×D₅₃

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

106

—

119

C₄

—

424

—

120

C₄

—

424

—

121

C₄

#13133194

yes

characteristic

122

C₄

#13133246

yes

characteristic

123

C₂

—

124

C₂

#16810901

yes

characteristic

125

C₂

SD₃₂×D₅₃

yes

characteristic

126

C₂

#16811057

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₅₃
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#357	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	#357	D₅₃
hypocenter	C₅₃	#2226
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

subgroup

G/H

center

C₂²

D₈×D₅₃

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#357

C₂

socle

C₂²×C₅₃

C₂×D₈

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

N	H ≅ G/N	type
1. #16811027	C₂	semidirect
3. #16810451	C₂	semidirect
4. #16810450	C₂	semidirect
5. #16810441	C₂	semidirect
23. D₈×C₅₃	Q₈	semidirect
71. C₅₃	#2226	semidirect
87. D₈	Dic₁₀₆	semidirect

H ≅ G/N

type

1. #16811027

C₂

semidirect

3. #16810451

C₂

semidirect

4. #16810450

C₂

semidirect

5. #16810441

C₂

semidirect

23. D₈×C₅₃

Q₈

semidirect

71. C₅₃

#2226

semidirect

87. D₈

Dic₁₀₆

semidirect

Subgroups — #20276860

Conjugacy classes of subgroups

Complements