Group #20282967 — subgroups

Subgroups — #20282967

Order 6784.

580 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 125 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	#16810536	C₂	1	6	yes	maximal, complemented
2	#16810535	C₂	1	3	yes	maximal, complemented
3	#16810534	C₂	1	2	yes	maximal, complemented
4	#16810478	C₂	1	3	yes	maximal
5	C₅₃×#393	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁴×D₅₃	C₂²	1	1	yes	characteristic
7	C₂²×Dic₁₀₆	C₂²	1	3	yes	complemented
8	C₂×C₂²⋊Dic₅₃	C₂²	1	3	yes	—
9	C₂×#11719921	C₂²	1	6	yes	—
10	C₂×#11719921	C₂²	1	6	yes	—
11	#13133124	C₂²	1	6	yes	complemented
12	#13133124	C₂²	1	3	yes	complemented
13	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₂²⋊C₄×C₅₃	C₂²	1	3	yes	—
15	#95×C₅₃	C₂²	1	3	yes	complemented
16	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
17	C₂³×D₅₃	—	4	3	no	—
18	C₂³×D₅₃	—	4	3	no	—
19	C₂³×D₅₃	Q₈	1	4	yes	complemented
20	C₂³×D₅₃	C₂³	1	2	yes	—
21	C₂×Q₈×C₅₃	—	4	3	no	—
22	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
23	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
24	C₂²×Dic₅₃	C₂³	1	6	yes	—
25	C₂×Dic₁₀₆	—	4	3	no	—
26	C₂×Dic₁₀₆	—	4	6	no	—
27	C₂²⋊C₄×C₅₃	—	4	3	no	—
28	C₂²⋊Dic₅₃	—	4	3	no	—
29	#11719921	—	4	6	no	—
30	#11719921	—	4	6	no	—
31	C₂³×C₅₃	—	2	3	no	—
32	C₂³×C₅₃	—	2	3	no	—
33	C₂³×C₅₃	C₂×Q₈	1	2	yes	—
34	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
35	C₂²×D₅₃	—	2	6	no	—
36	C₂²×D₅₃	—	2	6	no	—
37	C₂²×D₅₃	—	4	12	no	—
38	C₂²×D₅₃	—	4	12	no	—
39	C₂²×D₅₃	—	4	4	no	—
40	C₂²×D₅₃	C₂×Q₈	1	4	yes	—
41	Q₈×C₅₃	—	4	4	no	—
42	C₂×C₄×C₅₃	—	2	6	no	—
43	C₂×C₄×C₅₃	—	2	3	no	—
44	C₂×C₄×C₅₃	—	2	6	no	—
45	C₂×C₄×C₅₃	—	2	3	no	—
46	C₂×Dic₅₃	—	2	6	no	—
47	C₂×Dic₅₃	—	2	6	no	—
48	C₂×Dic₅₃	—	2	6	no	—
49	Dic₁₀₆	—	4	12	no	—
50	C₂²×C₅₃	—	2	6	no	—
51	C₂²×C₅₃	—	4	3	no	—
52	C₂²×C₅₃	—	4	1	no	—
53	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
54	C₂²×C₅₃	H(5,𝔽₂)	1	3	yes	—
55	C₂²×C₅₃	H(5,𝔽₂)	1	3	yes	—
56	C₂×D₅₃	—	2	12	no	—
57	C₂×D₅₃	—	4	16	no	—
58	C₂×D₅₃	—	4	6	no	—
59	C₂×D₅₃	—	4	2	no	—
60	C₄×C₅₃	—	4	3	no	—
61	C₄×C₅₃	—	4	3	no	—
62	Dic₅₃	—	4	6	no	—
63	#2840	—	53	1	no	maximal, Sylow
64	C₂×C₅₃	—	4	2	no	—
65	C₂×C₅₃	#543	1	3	yes	—
66	C₂×C₅₃	H(3,𝔽₄)	1	3	yes	—
67	C₂×C₅₃	H(3,𝔽₄)	1	1	yes	characteristic
68	D₅₃	—	4	4	no	—
69	#393	—	53	2	no	—
70	#393	—	53	3	no	—
71	#393	—	53	6	no	—
72	#393	D₅₃	1	1	yes	characteristic, complemented
73	#379	—	53	3	no	—
74	C₅₃	#2840	1	1	yes	characteristic, complemented, Sylow
75	C₂⁵	—	53	1	no	—
76	C₂²×Q₈	—	53	3	no	—
77	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
78	C₂×C₂²⋊C₄	—	53	3	no	—
79	C₂×C₂²⋊C₄	—	53	6	no	—
80	C₂×C₂²⋊C₄	—	53	6	no	—
81	C₂×C₂²⋊C₄	C₂×D₅₃	1	3	yes	—
82	#95	—	53	6	no	—
83	#95	—	53	3	no	—
84	#95	C₂×D₅₃	1	3	yes	complemented
85	C₂⁴	—	53	2	no	—
86	C₂⁴	—	53	4	no	—
87	C₂⁴	—	212	3	no	—
88	C₂⁴	—	212	3	no	—
89	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
90	C₂×Q₈	—	4	3	no	—
91	C₂×Q₈	—	212	3	no	—
92	C₂×Q₈	—	212	6	no	—
93	C₂²×C₄	—	53	6	no	—
94	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
95	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
96	C₂²⋊C₄	—	4	3	no	—
97	C₂²⋊C₄	—	212	6	no	—
98	C₂²⋊C₄	—	212	6	no	—
99	C₂²⋊C₄	—	212	3	no	—
100	C₂³	—	2	3	no	—
101	C₂³	—	2	3	no	—
102	C₂³	—	53	4	no	—
103	C₂³	—	106	6	no	—
104	C₂³	—	106	6	no	—
105	C₂³	—	212	12	no	—
106	C₂³	—	212	12	no	—
107	C₂³	—	212	4	no	—
108	C₂³	Q₈×D₅₃	1	2	yes	—
109	C₂³	C₂³×D₅₃	1	1	yes	characteristic
110	Q₈	—	4	4	no	—
111	Q₈	—	212	12	no	—
112	C₂×C₄	—	2	3	no	—
113	C₂×C₄	—	2	3	no	—
114	C₂×C₄	—	2	6	no	—
115	C₂×C₄	—	2	6	no	—
116	C₂×C₄	—	106	6	no	—
117	C₂×C₄	—	106	6	no	—
118	C₂×C₄	—	106	6	no	—
119	C₂²	—	2	6	no	—
120	C₂²	—	4	3	no	—
121	C₂²	—	4	1	no	—
122	C₂²	—	106	12	no	—
123	C₂²	—	212	16	no	—
124	C₂²	—	212	6	no	—
125	C₂²	—	212	2	no	—
126	C₂²	C₂×Q₈×D₅₃	1	1	yes	characteristic
127	C₂²	#13133262	1	3	yes	—
128	C₂²	#13133263	1	3	yes	—
129	C₄	—	4	3	no	—
130	C₄	—	4	3	no	—
131	C₄	—	212	6	no	—
132	C₂	—	4	2	no	—
133	C₂	—	212	4	no	—
134	C₂	#16811248	1	3	yes	—
135	C₂	#16811365	1	3	yes	—
136	C₂	#16811366	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810536

C₂

yes

maximal, complemented

#16810535

C₂

yes

maximal, complemented

#16810534

C₂

yes

maximal, complemented

#16810478

C₂

yes

maximal

C₅₃×#393

C₂

yes

maximal, characteristic, complemented

C₂⁴×D₅₃

C₂²

yes

characteristic

C₂²×Dic₁₀₆

C₂²

yes

complemented

C₂×C₂²⋊Dic₅₃

C₂²

yes

—

C₂×#11719921

C₂²

yes

—

C₂×#11719921

C₂²

yes

—

#13133124

C₂²

yes

complemented

#13133124

C₂²

yes

complemented

C₂²×Q₈×C₅₃

C₂²

yes

characteristic, complemented

C₂×C₂²⋊C₄×C₅₃

C₂²

yes

—

#95×C₅₃

C₂²

yes

complemented

C₂⁴×C₅₃

C₂³

yes

characteristic

C₂³×D₅₃

—

C₂³×D₅₃

—

C₂³×D₅₃

Q₈

yes

complemented

C₂³×D₅₃

C₂³

yes

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂²⋊C₄×C₅₃

—

C₂²⋊Dic₅₃

—

#11719921

—

#11719921

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×Q₈

yes

—

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

C₂×Q₈

yes

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²×Q₈

yes

characteristic

C₂²×C₅₃

H(5,𝔽₂)

yes

—

C₂²×C₅₃

H(5,𝔽₂)

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

#2840

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#543

yes

—

C₂×C₅₃

H(3,𝔽₄)

yes

—

C₂×C₅₃

H(3,𝔽₄)

yes

characteristic

D₅₃

—

—

—

—

D₅₃

yes

characteristic, complemented

#379

—

C₅₃

#2840

yes

characteristic, complemented, Sylow

C₂⁵

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₅₃

yes

characteristic, complemented

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₅₃

yes

—

#95

—

#95

—

#95

C₂×D₅₃

yes

complemented

C₂⁴

—

C₂⁴

—

C₂⁴

—

212

—

C₂⁴

—

212

—

C₂⁴

C₂²×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

212

—

C₂×Q₈

—

212

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

212

—

C₂²⋊C₄

—

212

—

C₂²⋊C₄

—

212

—

100

C₂³

—

101

C₂³

—

102

C₂³

—

103

C₂³

—

106

—

104

C₂³

—

106

—

105

C₂³

—

212

—

106

C₂³

—

212

—

107

C₂³

—

212

—

108

C₂³

Q₈×D₅₃

yes

—

109

C₂³

C₂³×D₅₃

yes

characteristic

110

Q₈

—

111

Q₈

—

212

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

106

—

117

C₂×C₄

—

106

—

118

C₂×C₄

—

106

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

106

—

123

C₂²

—

212

—

124

C₂²

—

212

—

125

C₂²

—

212

—

126

C₂²

C₂×Q₈×D₅₃

yes

characteristic

127

C₂²

#13133262

yes

—

128

C₂²

#13133263

yes

—

129

C₄

—

130

C₄

—

131

C₄

—

212

—

132

C₂

—

133

C₂

—

212

—

134

C₂

#16811248

yes

—

135

C₂

#16811365

yes

—

136

C₂

#16811366

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂⁴
Fitting	C₅₃×#393	C₂
hypercenter	#393	D₅₃
hypocenter	C₅₃	#2840
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #393

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator, socle

C₂³×C₅₃

C₂⁴

Fitting

C₅₃×#393

C₂

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ #393

N	H ≅ G/N	type
1. #16810536	C₂	semidirect
2. #16810535	C₂	semidirect
3. #16810534	C₂	semidirect
5. C₅₃×#393	C₂	semidirect
7. C₂²×Dic₁₀₆	C₂²	semidirect
11. #13133124	C₂²	semidirect
12. #13133124	C₂²	semidirect
13. C₂²×Q₈×C₅₃	C₂²	semidirect
15. #95×C₅₃	C₂²	semidirect
19. C₂³×D₅₃	Q₈	semidirect
72. #393	D₅₃	semidirect
74. C₅₃	#2840	semidirect
77. C₂²×Q₈	C₂×D₅₃	semidirect
84. #95	C₂×D₅₃	semidirect

H ≅ G/N

type

1. #16810536

C₂

semidirect

2. #16810535

C₂

semidirect

3. #16810534

C₂

semidirect

5. C₅₃×#393

C₂

semidirect

7. C₂²×Dic₁₀₆

C₂²

semidirect

11. #13133124

C₂²

semidirect

12. #13133124

C₂²

semidirect

13. C₂²×Q₈×C₅₃

C₂²

semidirect

15. #95×C₅₃

C₂²

semidirect

19. C₂³×D₅₃

Q₈

semidirect

72. #393

D₅₃

semidirect

74. C₅₃

#2840

semidirect

77. C₂²×Q₈

C₂×D₅₃

semidirect

84. #95

C₂×D₅₃

semidirect

Subgroups — #20282967

Conjugacy classes of subgroups

Complements