Group #20274196 — subgroups

Subgroups — #20274196

Order 6784.

184 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 55 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	#16810820	C₂	1	2	yes	maximal, complemented
2	#16810753	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810374	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810367	C₂	1	2	yes	maximal, complemented
5	C₅₃×#493	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719923	—	2	2	no	—
7	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
8	C₄⋊#88322	C₂²	1	1	yes	characteristic
9	C₄⋊#88322	C₂²	1	2	yes	—
10	#13133146	—	2	1	no	—
11	#13133138	—	2	2	no	—
12	#13133127	—	2	1	no	—
13	C₂×D₈×C₅₃	—	2	1	no	—
14	C₂×D₈×C₅₃	—	2	1	no	—
15	#127×C₅₃	C₂²	1	2	yes	—
16	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×D₄×C₅₃	—	2	2	no	—
18	C₂×D₄×C₅₃	—	2	2	no	—
19	C₂×D₄×C₅₃	—	2	2	no	—
20	C₂×Dic₁₀₆	D₄	1	1	yes	characteristic
21	D₈×C₅₃	—	4	2	no	—
22	D₈×C₅₃	—	4	2	no	—
23	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
25	C₄²×C₅₃	C₂³	1	1	yes	characteristic
26	#11719923	—	4	4	no	—
27	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
28	#11719919	—	2	1	no	—
29	C₄×Dic₅₃	—	2	1	no	—
30	C₂×#88322	—	2	1	no	—
31	C₂×#88322	—	2	2	no	—
32	C₂³×C₅₃	—	4	2	no	—
33	D₄×C₅₃	—	4	2	no	—
34	D₄×C₅₃	—	4	2	no	—
35	D₄×C₅₃	—	4	2	no	—
36	D₄×C₅₃	—	4	2	no	—
37	D₄×C₅₃	—	8	2	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	—	2	1	no	—
42	C₂×Dic₅₃	—	2	1	no	—
43	Dic₁₀₆	—	2	2	no	—
44	C₈×C₅₃	—	2	1	no	—
45	C₈×C₅₃	SD₁₆	1	2	yes	complemented
46	#88322	—	4	1	no	—
47	#88322	—	4	2	no	—
48	C₂²×C₅₃	—	8	2	no	—
49	C₂²×C₅₃	—	8	2	no	—
50	C₂²×C₅₃	—	8	2	no	—
51	C₂²×C₅₃	#121	1	1	yes	characteristic
52	C₄×C₅₃	—	2	1	no	—
53	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
54	C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
56	Dic₅₃	—	4	1	no	—
57	Dic₅₃	—	4	1	no	—
58	#1688	—	53	1	no	maximal, Sylow
59	C₂×C₅₃	—	16	2	no	—
60	C₂×C₅₃	#460	1	1	yes	characteristic
61	C₂×C₅₃	#469	1	1	yes	characteristic
62	C₂×C₅₃	#472	1	1	yes	characteristic
63	#493	D₅₃	1	1	yes	characteristic
64	#460	—	53	2	no	—
65	#446	—	53	1	no	—
66	#334	—	53	1	no	—
67	#331	—	53	2	no	—
68	C₅₃	#1688	1	1	yes	characteristic, complemented, Sylow
69	C₂×SD₁₆	—	106	2	no	—
70	C₂×D₈	—	2	1	no	—
71	C₂×D₈	—	2	1	no	—
72	#127	C₂×D₅₃	1	2	yes	—
73	C₄×Q₈	—	53	1	no	—
74	C₄⋊C₈	—	53	1	no	—
75	C₄⋊C₈	—	53	2	no	—
76	C₄⋊C₈	—	106	1	no	—
77	#102	—	106	2	no	—
78	#97	—	106	1	no	—
79	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
80	C₂×Q₈	—	53	1	no	—
81	C₂×D₄	—	2	2	no	—
82	C₂×D₄	—	2	2	no	—
83	C₂×D₄	—	2	2	no	—
84	SD₁₆	—	212	4	no	—
85	D₈	—	4	2	no	—
86	D₈	—	4	2	no	—
87	C₂×C₈	—	106	2	no	—
88	C₂×C₈	—	106	1	no	—
89	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
90	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
91	C₄⋊C₄	—	53	1	no	—
92	C₄⋊C₄	—	106	1	no	—
93	C₄²	—	106	1	no	—
94	C₄²	C₂²×D₅₃	1	1	yes	characteristic
95	C₂³	—	4	2	no	—
96	Q₈	—	106	2	no	—
97	D₄	—	4	2	no	—
98	D₄	—	4	2	no	—
99	D₄	—	4	2	no	—
100	D₄	—	4	2	no	—
101	D₄	—	8	2	no	—
102	C₂×C₄	—	106	1	no	—
103	C₂×C₄	—	106	1	no	—
104	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
105	C₂×C₄	#11719947	1	1	yes	characteristic
106	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
107	C₈	—	2	1	no	—
108	C₈	—	212	2	no	—
109	C₈	—	212	1	no	—
110	C₈	#11719923	1	2	yes	complemented
111	C₂²	—	8	2	no	—
112	C₂²	—	8	2	no	—
113	C₂²	—	8	2	no	—
114	C₂²	#13133188	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	212	1	no	—
117	C₄	—	212	1	no	—
118	C₄	C₂×#11719923	1	1	yes	characteristic
119	C₄	#13133241	1	2	yes	—
120	C₄	#13133243	1	1	yes	characteristic
121	C₂	—	16	2	no	—
122	C₂	#16810820	1	1	yes	characteristic
123	C₂	#16810873	1	1	yes	characteristic
124	C₂	#16810890	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810820

C₂

yes

maximal, complemented

#16810753

C₂

yes

maximal, characteristic, complemented

#16810374

C₂

yes

maximal, characteristic, complemented

#16810367

C₂

yes

maximal, complemented

C₅₃×#493

C₂

yes

maximal, characteristic

C₂×#11719923

—

C₄×Dic₁₀₆

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

—

#13133146

—

#13133138

—

#13133127

—

C₂×D₈×C₅₃

—

C₂×D₈×C₅₃

—

#127×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×Dic₁₀₆

D₄

yes

characteristic

D₈×C₅₃

—

D₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

#11719923

—

C₄⋊Dic₅₃

D₄

yes

characteristic

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

SD₁₆

yes

complemented

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#121

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₄×C₅₃

AGL(1,ℤ/8)

yes

characteristic

Dic₅₃

—

Dic₅₃

—

#1688

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

—

—

—

—

C₅₃

#1688

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

106

—

C₂×D₈

—

C₂×D₈

—

#127

C₂×D₅₃

yes

—

C₄×Q₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

#102

—

106

—

#97

—

106

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

212

—

D₈

—

D₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄²

—

106

—

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

—

Q₈

—

106

—

D₄

—

D₄

—

D₄

—

100

D₄

—

101

D₄

—

102

C₂×C₄

—

106

—

103

C₂×C₄

—

106

—

104

C₂×C₄

D₄×D₅₃

yes

characteristic

105

C₂×C₄

#11719947

yes

characteristic

106

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

107

C₈

—

108

C₈

—

212

—

109

C₈

—

212

—

110

C₈

#11719923

yes

complemented

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

#13133188

yes

characteristic

115

C₄

—

116

C₄

—

212

—

117

C₄

—

212

—

118

C₄

C₂×#11719923

yes

characteristic

119

C₄

#13133241

yes

—

120

C₄

#13133243

yes

characteristic

121

C₂

—

122

C₂

#16810820

yes

characteristic

123

C₂

#16810873

yes

characteristic

124

C₂

#16810890

yes

characteristic

	subgroup	G/H
center	C₂²	#13133188
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#493	C₂
socle	C₂²×C₅₃	#121
hypercenter	#493	D₅₃
hypocenter	C₅₃	#1688
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#13133188

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#493

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #16810820	C₂	semidirect
2. #16810753	C₂	semidirect
3. #16810374	C₂	semidirect
4. #16810367	C₂	semidirect
45. C₈×C₅₃	SD₁₆	semidirect
68. C₅₃	#1688	semidirect
110. C₈	#11719923	semidirect

H ≅ G/N

type

1. #16810820

C₂

semidirect

2. #16810753

C₂

semidirect

3. #16810374

C₂

semidirect

4. #16810367

C₂

semidirect

45. C₈×C₅₃

SD₁₆

semidirect

68. C₅₃

#1688

semidirect

110. C₈

#11719923

semidirect

Subgroups — #20274196

Conjugacy classes of subgroups

Complements