Group #20283057 — subgroups

Subgroups — #20283057

Order 6784.

924 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 209 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	C₂×#13133188	C₂	1	6	yes	maximal, complemented
2	C₂×C₄²×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810526	C₂	1	6	yes	maximal, complemented
4	#16810493	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#127×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₂²⋊D₅₃	C₂²	1	4	yes	—
7	C₂²×C₄×D₅₃	C₂²	1	3	yes	—
8	#13133188	—	2	24	no	—
9	C₂×C₄⋊Dic₅₃	C₂²	1	6	yes	—
10	C₂×C₂²⋊Dic₅₃	C₂²	1	12	yes	—
11	C₂×C₄×Dic₅₃	C₂²	1	3	yes	—
12	C₄²×D₅₃	—	2	4	no	—
13	C₂²×D₄×C₅₃	C₂²	1	6	yes	—
14	#127×C₅₃	—	2	4	no	—
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂⁴×C₅₃	C₂³	1	4	yes	—
17	C₂³×D₅₃	C₂³	1	1	yes	characteristic
18	C₂×D₄×C₅₃	—	2	12	no	—
19	C₂×D₄×C₅₃	—	2	12	no	—
20	C₂×D₄×C₅₃	—	2	12	no	—
21	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
22	C₂×C₂²⋊D₅₃	—	2	24	no	—
23	C₂²×Dic₅₃	C₂³	1	4	yes	—
24	C₂²×Dic₅₃	C₂³	1	3	yes	—
25	C₂×C₄×D₅₃	—	2	6	no	—
26	C₂×C₄×D₅₃	—	2	6	no	—
27	C₂×C₄×D₅₃	D₄	1	12	yes	complemented
28	C₄²×C₅₃	D₄	1	4	yes	complemented
29	C₂²⋊Dic₅₃	—	2	24	no	—
30	C₄⋊Dic₅₃	—	2	12	no	—
31	C₄×Dic₅₃	—	2	6	no	—
32	C₂³×C₅₃	—	2	12	no	—
33	C₂³×C₅₃	—	2	4	no	—
34	C₂³×C₅₃	—	2	12	no	—
35	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
36	C₂²×D₅₃	—	2	1	no	—
37	C₂²×D₅₃	—	2	3	no	—
38	C₂²×D₅₃	C₂×D₄	1	6	yes	—
39	D₄×C₅₃	—	4	24	no	—
40	C₂×C₄×C₅₃	C₂×D₄	1	6	yes	—
41	C₂×C₄×C₅₃	C₂×D₄	1	6	yes	—
42	C₂×C₄×C₅₃	#41	1	6	yes	—
43	C₂²⋊D₅₃	—	4	16	no	—
44	C₂×Dic₅₃	—	2	3	no	—
45	C₂×Dic₅₃	—	2	3	no	—
46	C₂×Dic₅₃	—	2	12	no	—
47	C₂×Dic₅₃	C₂×D₄	1	6	yes	—
48	C₄×D₅₃	—	2	24	no	—
49	C₂²×C₅₃	—	4	12	no	—
50	C₂²×C₅₃	—	4	4	no	—
51	C₂²×C₅₃	—	4	12	no	—
52	C₂²×C₅₃	C₂²×D₄	1	3	yes	—
53	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
54	C₂²×C₅₃	C₂×#41	1	3	yes	—
55	C₂×D₅₃	—	2	6	no	—
56	C₂×D₅₃	—	2	6	no	—
57	C₂×D₅₃	#127	1	4	yes	complemented
58	C₄×C₅₃	#121	1	12	yes	complemented
59	Dic₅₃	—	2	6	no	—
60	Dic₅₃	—	4	4	no	—
61	#2856	—	53	1	no	maximal, Sylow
62	C₂×C₅₃	—	8	4	no	—
63	C₂×C₅₃	C₂×#121	1	3	yes	—
64	C₂×C₅₃	C₂×#127	1	1	yes	characteristic
65	C₂×C₅₃	#532	1	3	yes	—
66	D₅₃	—	2	4	no	—
67	C₂×#127	D₅₃	1	1	yes	characteristic, complemented
68	C₂×#121	—	53	6	no	—
69	C₂²×C₄²	—	53	1	no	—
70	#390	—	53	6	no	—
71	#384	—	53	1	no	—
72	C₅₃	#2856	1	1	yes	characteristic, complemented, Sylow
73	C₂²×D₄	—	53	4	no	—
74	C₂²×D₄	C₂×D₅₃	1	6	yes	—
75	C₂³×C₄	—	53	3	no	—
76	#127	—	2	4	no	—
77	#121	—	106	24	no	—
78	C₂×C₄⋊C₄	—	53	6	no	—
79	C₂×C₂²⋊C₄	—	53	12	no	—
80	C₂×C₄²	—	53	3	no	—
81	C₂×C₄²	—	106	4	no	—
82	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
83	C₂⁴	—	53	1	no	—
84	C₂⁴	C₂²×D₅₃	1	4	yes	—
85	C₂×D₄	—	2	12	no	—
86	C₂×D₄	—	2	12	no	—
87	C₂×D₄	—	2	12	no	—
88	C₂×D₄	—	106	24	no	—
89	C₂²×C₄	—	53	3	no	—
90	C₂²×C₄	—	53	12	no	—
91	C₂²×C₄	—	53	4	no	—
92	C₂²×C₄	—	106	6	no	—
93	C₂²×C₄	—	106	6	no	—
94	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
95	C₄⋊C₄	—	106	12	no	—
96	C₂²⋊C₄	—	106	24	no	—
97	C₄²	—	106	6	no	—
98	C₄²	C₂²⋊D₅₃	1	4	yes	complemented
99	C₂³	—	2	12	no	—
100	C₂³	—	2	4	no	—
101	C₂³	—	2	12	no	—
102	C₂³	—	53	6	no	—
103	C₂³	—	106	1	no	—
104	C₂³	—	106	3	no	—
105	C₂³	C₂³×D₅₃	1	1	yes	characteristic
106	D₄	—	4	24	no	—
107	D₄	—	212	16	no	—
108	C₂×C₄	—	53	6	no	—
109	C₂×C₄	—	106	3	no	—
110	C₂×C₄	—	106	3	no	—
111	C₂×C₄	—	106	24	no	—
112	C₂×C₄	—	106	12	no	—
113	C₂×C₄	D₄×D₅₃	1	6	yes	—
114	C₂×C₄	#11719947	1	6	yes	—
115	C₂×C₄	C₂×C₂²⋊D₅₃	1	6	yes	—
116	C₂²	—	4	12	no	—
117	C₂²	—	4	4	no	—
118	C₂²	—	4	12	no	—
119	C₂²	—	53	4	no	—
120	C₂²	—	106	6	no	—
121	C₂²	—	106	6	no	—
122	C₂²	C₂×D₄×D₅₃	1	3	yes	—
123	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
124	C₂²	C₂×#11719947	1	3	yes	—
125	C₄	—	106	6	no	—
126	C₄	—	212	4	no	—
127	C₄	#13133188	1	12	yes	complemented
128	C₂	—	8	4	no	—
129	C₂	—	106	4	no	—
130	C₂	C₂×#13133188	1	3	yes	—
131	C₂	#127×D₅₃	1	1	yes	characteristic
132	C₂	#16811171	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133188

C₂

yes

maximal, complemented

C₂×C₄²×D₅₃

C₂

yes

maximal, characteristic, complemented

#16810526

C₂

yes

maximal, complemented

#16810493

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂²×C₂²⋊D₅₃

C₂²

yes

—

C₂²×C₄×D₅₃

C₂²

yes

—

#13133188

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₂²⋊Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₄²×D₅₃

—

C₂²×D₄×C₅₃

C₂²

yes

—

#127×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂⁴×C₅₃

C₂³

yes

—

C₂³×D₅₃

C₂³

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂×C₂²⋊D₅₃

—

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

D₄

yes

complemented

C₄²×C₅₃

D₄

yes

complemented

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

C₂×D₄

yes

—

D₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

—

C₄×D₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²×D₄

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂²×C₅₃

C₂×#41

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

#2856

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×#121

yes

—

C₂×C₅₃

C₂×#127

yes

characteristic

C₂×C₅₃

#532

yes

—

D₅₃

—

C₂×#127

D₅₃

yes

characteristic, complemented

C₂×#121

—

C₂²×C₄²

—

#390

—

#384

—

C₅₃

#2856

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

C₂×D₅₃

yes

—

C₂³×C₄

—

#127

—

#121

—

106

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₂⁴

—

C₂⁴

C₂²×D₅₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

106

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

106

—

C₄²

—

106

—

C₄²

C₂²⋊D₅₃

yes

complemented

C₂³

—

100

C₂³

—

101

C₂³

—

102

C₂³

—

103

C₂³

—

106

—

104

C₂³

—

106

—

105

C₂³

C₂³×D₅₃

yes

characteristic

106

D₄

—

107

D₄

—

212

—

108

C₂×C₄

—

109

C₂×C₄

—

106

—

110

C₂×C₄

—

106

—

111

C₂×C₄

—

106

—

112

C₂×C₄

—

106

—

113

C₂×C₄

D₄×D₅₃

yes

—

114

C₂×C₄

#11719947

yes

—

115

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

106

—

121

C₂²

—

106

—

122

C₂²

C₂×D₄×D₅₃

yes

—

123

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

124

C₂²

C₂×#11719947

yes

—

125

C₄

—

106

—

126

C₄

—

212

—

127

C₄

#13133188

yes

complemented

128

C₂

—

129

C₂

—

106

—

130

C₂

C₂×#13133188

yes

—

131

C₂

#127×D₅₃

yes

characteristic

132

C₂

#16811171

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂⁴
Fitting	C₂×#127×C₅₃	C₂
hypercenter	C₂×#127	D₅₃
hypocenter	C₅₃	#2856
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×#127

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator, socle

C₂³×C₅₃

C₂⁴

Fitting

C₂×#127×C₅₃

C₂

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ C₂×#127

N	H ≅ G/N	type
1. C₂×#13133188	C₂	semidirect
2. C₂×C₄²×D₅₃	C₂	semidirect
3. #16810526	C₂	semidirect
4. #16810493	C₂	semidirect
5. C₂×#127×C₅₃	C₂	semidirect
27. C₂×C₄×D₅₃	D₄	semidirect
28. C₄²×C₅₃	D₄	semidirect
57. C₂×D₅₃	#127	semidirect
58. C₄×C₅₃	#121	semidirect
67. C₂×#127	D₅₃	semidirect
72. C₅₃	#2856	semidirect
98. C₄²	C₂²⋊D₅₃	semidirect
127. C₄	#13133188	semidirect

H ≅ G/N

type

1. C₂×#13133188

C₂

semidirect

2. C₂×C₄²×D₅₃

C₂

semidirect

3. #16810526

C₂

semidirect

4. #16810493

C₂

semidirect

5. C₂×#127×C₅₃

C₂

semidirect

27. C₂×C₄×D₅₃

D₄

semidirect

28. C₄²×C₅₃

D₄

semidirect

57. C₂×D₅₃

#127

semidirect

58. C₄×C₅₃

#121

semidirect

67. C₂×#127

D₅₃

semidirect

72. C₅₃

#2856

semidirect

98. C₄²

C₂²⋊D₅₃

semidirect

127. C₄

#13133188

semidirect

Subgroups — #20283057

Conjugacy classes of subgroups

Complements