Group #20273339 — subgroups

Subgroups — #20273339

Order 6784.

248 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 77 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	#16810680	C₂	1	2	yes	maximal, complemented
2	#16810363	C₂	1	4	yes	maximal, complemented
3	C₂×#124×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#88322	—	2	2	no	—
5	C₂²⋊#88322	—	2	2	no	—
6	#13133128	C₂²	1	4	yes	complemented
7	C₂²×Q₈×C₅₃	C₄	1	1	yes	characteristic
8	C₂²×D₄×C₅₃	C₄	1	1	yes	characteristic
9	#124×C₅₃	—	2	2	no	—
10	#124×C₅₃	C₄	1	2	yes	—
11	#124×C₅₃	C₂²	1	2	yes	—
12	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
13	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	C₂⁴×C₅₃	—	2	1	no	—
15	C₂×Q₈×C₅₃	—	2	1	no	—
16	C₂×Q₈×C₅₃	—	4	2	no	—
17	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
18	C₂×D₄×C₅₃	—	2	1	no	—
19	C₂×D₄×C₅₃	—	4	2	no	—
20	C₂×D₄×C₅₃	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₅₃	—	2	1	no	—
22	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	D₄	1	2	yes	—
24	C₂²⋊C₄×C₅₃	—	2	4	no	—
25	C₂²⋊C₄×C₅₃	—	2	4	no	—
26	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₅₃	D₄	1	2	yes	—
28	C₄²×C₅₃	C₂³	1	1	yes	characteristic
29	C₂×#88322	—	2	4	no	—
30	C₂³×C₅₃	—	2	2	no	—
31	C₂³×C₅₃	—	4	4	no	—
32	C₂³×C₅₃	—	4	2	no	—
33	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
34	Q₈×C₅₃	—	4	2	no	—
35	Q₈×C₅₃	—	4	2	no	—
36	D₄×C₅₃	—	4	2	no	—
37	D₄×C₅₃	—	4	2	no	—
38	C₂×C₄×C₅₃	—	2	2	no	—
39	C₂×C₄×C₅₃	—	2	2	no	—
40	C₂×C₄×C₅₃	—	4	2	no	—
41	C₂×C₄×C₅₃	—	4	2	no	—
42	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
46	#88322	—	4	4	no	—
47	C₂²×C₅₃	—	2	2	no	—
48	C₂²×C₅₃	—	4	2	no	—
49	C₂²×C₅₃	—	4	4	no	—
50	C₂²×C₅₃	—	8	4	no	—
51	C₂²×C₅₃	#100	1	2	yes	—
52	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₄×C₅₃	—	2	1	no	—
54	C₄×C₅₃	—	2	1	no	—
55	C₄×C₅₃	—	2	2	no	—
56	C₄×C₅₃	—	2	2	no	—
57	C₄×C₅₃	—	4	2	no	—
58	#1533	—	53	1	no	maximal, Sylow
59	C₂×C₅₃	—	2	2	no	—
60	C₂×C₅₃	—	8	2	no	—
61	C₂×C₅₃	C₂×#100	1	1	yes	characteristic
62	C₂×C₅₃	#421	1	2	yes	—
63	C₂×#124	D₅₃	1	1	yes	characteristic
64	#432	—	53	2	no	—
65	#329	—	53	4	no	—
66	C₅₃	#1533	1	1	yes	characteristic, complemented, Sylow
67	C₂²×Q₈	Dic₅₃	1	1	yes	characteristic
68	C₂²×D₄	Dic₅₃	1	1	yes	characteristic
69	#124	—	2	2	no	—
70	#124	Dic₅₃	1	2	yes	—
71	#124	C₂×D₅₃	1	2	yes	—
72	C₂×C₂²⋊C₄	—	2	2	no	—
73	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
74	C₄⋊C₈	—	106	2	no	—
75	C₂²⋊C₈	—	106	2	no	—
76	#97	—	53	4	no	—
77	C₂⁴	—	2	1	no	—
78	C₂×Q₈	—	2	1	no	—
79	C₂×Q₈	—	4	2	no	—
80	C₂×Q₈	C₂×Dic₅₃	1	2	yes	—
81	C₂×D₄	—	2	1	no	—
82	C₂×D₄	—	4	2	no	—
83	C₂×D₄	C₂×Dic₅₃	1	2	yes	—
84	C₂²×C₄	—	2	1	no	—
85	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
86	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
87	C₂×C₈	—	106	4	no	—
88	C₂²⋊C₄	—	2	4	no	—
89	C₂²⋊C₄	—	2	4	no	—
90	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
91	C₄²	C₂²⋊D₅₃	1	2	yes	—
92	C₄²	C₂²×D₅₃	1	1	yes	characteristic
93	C₂³	—	2	2	no	—
94	C₂³	—	4	4	no	—
95	C₂³	—	4	2	no	—
96	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
97	Q₈	—	4	2	no	—
98	Q₈	—	4	2	no	—
99	D₄	—	4	2	no	—
100	D₄	—	4	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
106	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
107	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
108	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
109	C₈	—	212	4	no	—
110	C₂²	—	2	2	no	—
111	C₂²	—	4	2	no	—
112	C₂²	—	4	4	no	—
113	C₂²	—	8	4	no	—
114	C₂²	#13133134	1	2	yes	—
115	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
116	C₄	—	2	1	no	—
117	C₄	—	2	1	no	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	4	2	no	—
121	C₂	—	2	2	no	—
122	C₂	—	8	2	no	—
123	C₂	C₂×#13133134	1	1	yes	characteristic
124	C₂	#16810640	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16810680

C₂

yes

maximal, complemented

#16810363

C₂

yes

maximal, complemented

C₂×#124×C₅₃

C₂

yes

maximal, characteristic

C₄⋊#88322

—

C₂²⋊#88322

—

#13133128

C₂²

yes

complemented

C₂²×Q₈×C₅₃

C₄

yes

characteristic

C₂²×D₄×C₅₃

C₄

yes

characteristic

#124×C₅₃

—

#124×C₅₃

C₄

yes

—

#124×C₅₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂⁴×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#100

yes

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

#1533

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#100

yes

characteristic

C₂×C₅₃

#421

yes

—

C₂×#124

D₅₃

yes

characteristic

#432

—

#329

—

C₅₃

#1533

yes

characteristic, complemented, Sylow

C₂²×Q₈

Dic₅₃

yes

characteristic

C₂²×D₄

Dic₅₃

yes

characteristic

#124

—

#124

Dic₅₃

yes

—

#124

C₂×D₅₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

106

—

#97

—

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₅₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₅₃

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂×C₈

—

106

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×Dic₅₃

yes

characteristic

C₄²

C₂²⋊D₅₃

yes

—

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊Dic₅₃

yes

characteristic

Q₈

—

Q₈

—

D₄

—

100

D₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂²⋊Dic₅₃

yes

—

106

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

107

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

108

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

109

C₈

—

212

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

#13133134

yes

—

115

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₂

—

122

C₂

—

123

C₂

C₂×#13133134

yes

characteristic

124

C₂

#16810640

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₂×#124×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	C₂×#124	D₅₃
hypocenter	C₅₃	#1533
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#124

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₂×#124×C₅₃

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#124

N	H ≅ G/N	type
1. #16810680	C₂	semidirect
2. #16810363	C₂	semidirect
6. #13133128	C₂²	semidirect
66. C₅₃	#1533	semidirect

H ≅ G/N

type

1. #16810680

C₂

semidirect

2. #16810363

C₂

semidirect

6. #13133128

C₂²

semidirect

66. C₅₃

#1533

semidirect

Subgroups — #20273339

Conjugacy classes of subgroups

Complements