Group #20283060 — subgroups

Subgroups — #20283060

Order 6784.

924 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 237 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×#13133214	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13133191	C₂	1	6	yes	maximal, complemented
3	#16810525	C₂	1	6	yes	maximal, complemented
4	#16810493	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₄²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₁₂	C₂²	1	6	yes	—
7	C₂²×C₂²⋊D₅₃	C₂²	1	4	yes	—
8	#13133214	—	2	4	no	—
9	#13133191	—	2	24	no	—
10	C₂×C₄⋊Dic₅₃	C₂²	1	6	yes	—
11	C₂×#11719921	C₂²	1	12	yes	—
12	C₂³×C₄×C₅₃	C₂²	1	3	yes	—
13	C₂×C₄²×C₅₃	—	2	4	no	—
14	C₂×C₄²×C₅₃	C₂²	1	3	yes	—
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
17	C₂³×D₅₃	C₂³	1	4	yes	—
18	C₂²×C₄×C₅₃	—	2	6	no	—
19	C₂²×C₄×C₅₃	—	2	6	no	—
20	C₂²×C₄×C₅₃	D₄	1	12	yes	complemented
21	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
22	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
23	C₂×C₂²⋊D₅₃	—	2	24	no	—
24	C₂²×Dic₅₃	C₂³	1	4	yes	—
25	C₂×D₂₁₂	—	2	12	no	—
26	C₂×D₂₁₂	—	2	12	no	—
27	C₂×D₂₁₂	—	2	12	no	—
28	C₄²×C₅₃	—	2	6	no	—
29	C₄²×C₅₃	D₄	1	4	yes	complemented
30	#11719921	—	2	24	no	—
31	C₄⋊Dic₅₃	—	2	12	no	—
32	C₂³×C₅₃	—	2	1	no	—
33	C₂³×C₅₃	—	2	3	no	—
34	C₂³×C₅₃	C₂×D₄	1	6	yes	—
35	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
36	C₂²×D₅₃	—	2	4	no	—
37	C₂²×D₅₃	—	2	12	no	—
38	C₂²×D₅₃	—	2	12	no	—
39	C₂×C₄×C₅₃	—	2	3	no	—
40	C₂×C₄×C₅₃	—	2	3	no	—
41	C₂×C₄×C₅₃	—	2	24	no	—
42	C₂×C₄×C₅₃	C₂×D₄	1	6	yes	—
43	C₂×C₄×C₅₃	C₂×D₄	1	6	yes	—
44	C₂×C₄×C₅₃	C₂×D₄	1	6	yes	—
45	C₂×C₄×C₅₃	#41	1	6	yes	—
46	C₂²⋊D₅₃	—	4	16	no	—
47	C₂×Dic₅₃	—	2	12	no	—
48	D₂₁₂	—	4	24	no	—
49	C₂²×C₅₃	—	2	6	no	—
50	C₂²×C₅₃	—	2	6	no	—
51	C₂²×C₅₃	#127	1	4	yes	complemented
52	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
53	C₂²×C₅₃	C₂²×D₄	1	3	yes	—
54	C₂²×C₅₃	C₂×#41	1	3	yes	—
55	C₂×D₅₃	—	4	12	no	—
56	C₂×D₅₃	—	4	4	no	—
57	C₂×D₅₃	—	4	12	no	—
58	C₄×C₅₃	—	2	6	no	—
59	C₄×C₅₃	#121	1	12	yes	complemented
60	Dic₅₃	—	4	4	no	—
61	#2856	—	53	1	no	maximal, Sylow
62	C₂×C₅₃	—	2	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	—	8	4	no	—
67	C₂×#127	—	53	1	no	—
68	C₂×#121	—	53	6	no	—
69	C₂²×C₄²	D₅₃	1	1	yes	characteristic, complemented
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	—	53	6	no	—
75	C₂³×C₄	C₂×D₅₃	1	3	yes	—
76	#127	—	106	4	no	—
77	#121	—	106	24	no	—
78	C₂×C₄⋊C₄	—	53	6	no	—
79	C₂×C₂²⋊C₄	—	53	12	no	—
80	C₂×C₄²	—	2	4	no	—
81	C₂×C₄²	C₂×D₅₃	1	3	yes	—
82	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
83	C₂⁴	—	53	4	no	—
84	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
85	C₂×D₄	—	106	24	no	—
86	C₂×D₄	—	106	12	no	—
87	C₂×D₄	—	106	12	no	—
88	C₂×D₄	—	106	12	no	—
89	C₂²×C₄	—	2	6	no	—
90	C₂²×C₄	—	2	6	no	—
91	C₂²×C₄	—	53	4	no	—
92	C₂²×C₄	D₂₁₂	1	12	yes	complemented
93	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
94	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
95	C₄⋊C₄	—	106	12	no	—
96	C₂²⋊C₄	—	106	24	no	—
97	C₄²	—	2	6	no	—
98	C₄²	C₂²⋊D₅₃	1	4	yes	complemented
99	C₂³	—	2	1	no	—
100	C₂³	—	2	3	no	—
101	C₂³	—	106	4	no	—
102	C₂³	—	106	12	no	—
103	C₂³	—	106	12	no	—
104	C₂³	C₂×D₂₁₂	1	6	yes	—
105	C₂³	C₂³×D₅₃	1	1	yes	characteristic
106	D₄	—	212	16	no	—
107	D₄	—	212	24	no	—
108	C₂×C₄	—	2	3	no	—
109	C₂×C₄	—	2	3	no	—
110	C₂×C₄	—	2	24	no	—
111	C₂×C₄	—	106	12	no	—
112	C₂×C₄	C₂×D₂₁₂	1	6	yes	—
113	C₂×C₄	C₂×D₂₁₂	1	6	yes	—
114	C₂×C₄	#11719945	1	6	yes	—
115	C₂×C₄	C₂×C₂²⋊D₅₃	1	6	yes	—
116	C₂²	—	2	6	no	—
117	C₂²	—	2	6	no	—
118	C₂²	—	212	12	no	—
119	C₂²	—	212	4	no	—
120	C₂²	—	212	12	no	—
121	C₂²	#13133214	1	4	yes	complemented
122	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
123	C₂²	C₂²×D₂₁₂	1	3	yes	—
124	C₂²	C₂×#11719945	1	3	yes	—
125	C₄	—	2	6	no	—
126	C₄	—	212	4	no	—
127	C₄	#13133191	1	12	yes	complemented
128	C₂	—	2	4	no	—
129	C₂	—	424	4	no	—
130	C₂	C₂×#13133191	1	3	yes	—
131	C₂	C₂×#13133214	1	1	yes	characteristic
132	C₂	#16811173	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133214

C₂

yes

maximal, characteristic, complemented

C₂×#13133191

C₂

yes

maximal, complemented

#16810525

C₂

yes

maximal, complemented

#16810493

C₂

yes

maximal, characteristic, complemented

C₂²×C₄²×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₁₂

C₂²

yes

—

C₂²×C₂²⋊D₅₃

C₂²

yes

—

#13133214

—

#13133191

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719921

C₂²

yes

—

C₂³×C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂⁴×C₅₃

C₂³

yes

characteristic

C₂³×D₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

complemented

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂×C₂²⋊D₅₃

—

C₂²×Dic₅₃

C₂³

yes

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₄²×C₅₃

—

C₄²×C₅₃

D₄

yes

complemented

#11719921

—

C₄⋊Dic₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×D₄

yes

—

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#127

yes

complemented

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂²×C₅₃

C₂²×D₄

yes

—

C₂²×C₅₃

C₂×#41

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#121

yes

complemented

Dic₅₃

—

#2856

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×#121

yes

—

C₂×C₅₃

C₂×#127

yes

characteristic

C₂×C₅₃

#532

yes

—

D₅₃

—

C₂×#127

—

C₂×#121

—

C₂²×C₄²

D₅₃

yes

characteristic, complemented

#390

—

#384

—

C₅₃

#2856

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

—

C₂³×C₄

C₂×D₅₃

yes

—

#127

—

106

—

#121

—

106

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₂⁴

—

C₂⁴

C₂²×D₅₃

yes

characteristic

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₂₁₂

yes

complemented

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

106

—

C₄²

—

C₄²

C₂²⋊D₅₃

yes

complemented

C₂³

—

100

C₂³

—

101

C₂³

—

106

—

102

C₂³

—

106

—

103

C₂³

—

106

—

104

C₂³

C₂×D₂₁₂

yes

—

105

C₂³

C₂³×D₅₃

yes

characteristic

106

D₄

—

212

—

107

D₄

—

212

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

106

—

112

C₂×C₄

C₂×D₂₁₂

yes

—

113

C₂×C₄

C₂×D₂₁₂

yes

—

114

C₂×C₄

#11719945

yes

—

115

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

212

—

119

C₂²

—

212

—

120

C₂²

—

212

—

121

C₂²

#13133214

yes

complemented

122

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

123

C₂²

C₂²×D₂₁₂

yes

—

124

C₂²

C₂×#11719945

yes

—

125

C₄

—

126

C₄

—

212

—

127

C₄

#13133191

yes

complemented

128

C₂

—

129

C₂

—

424

—

130

C₂

C₂×#13133191

yes

—

131

C₂

C₂×#13133214

yes

characteristic

132

C₂

#16811173

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂⁴
Fitting	C₂²×C₄²×C₅₃	C₂
hypercenter	C₂²×C₄²	D₅₃
hypocenter	C₅₃	#2856
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄²

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator, socle

C₂³×C₅₃

C₂⁴

Fitting

C₂²×C₄²×C₅₃

C₂

hypercenter

C₂²×C₄²

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ C₂²×C₄²

N	H ≅ G/N	type
1. C₂×#13133214	C₂	semidirect
2. C₂×#13133191	C₂	semidirect
3. #16810525	C₂	semidirect
4. #16810493	C₂	semidirect
5. C₂²×C₄²×C₅₃	C₂	semidirect
20. C₂²×C₄×C₅₃	D₄	semidirect
29. C₄²×C₅₃	D₄	semidirect
51. C₂²×C₅₃	#127	semidirect
59. C₄×C₅₃	#121	semidirect
69. C₂²×C₄²	D₅₃	semidirect
72. C₅₃	#2856	semidirect
92. C₂²×C₄	D₂₁₂	semidirect
98. C₄²	C₂²⋊D₅₃	semidirect
121. C₂²	#13133214	semidirect
127. C₄	#13133191	semidirect

H ≅ G/N

type

1. C₂×#13133214

C₂

semidirect

2. C₂×#13133191

C₂

semidirect

3. #16810525

C₂

semidirect

4. #16810493

C₂

semidirect

5. C₂²×C₄²×C₅₃

C₂

semidirect

20. C₂²×C₄×C₅₃

D₄

semidirect

29. C₄²×C₅₃

D₄

semidirect

51. C₂²×C₅₃

#127

semidirect

59. C₄×C₅₃

#121

semidirect

69. C₂²×C₄²

D₅₃

semidirect

72. C₅₃

#2856

semidirect

92. C₂²×C₄

D₂₁₂

semidirect

98. C₄²

C₂²⋊D₅₃

semidirect

121. C₂²

#13133214

semidirect

127. C₄

#13133191

semidirect

Subgroups — #20283060

Conjugacy classes of subgroups

Complements