Group #20277329 — subgroups

Subgroups — #20277329

Order 6784.

816 conjugacy classes of subgroups; only the 309 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 309 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	C₂²×C₂²⋊Dic₅₃	C₂	1	2	yes	maximal
2	#16810498	C₂	1	8	yes	maximal, complemented
3	#16810483	C₂	1	4	yes	maximal, complemented
4	C₂²×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₅₃	C₂²	1	4	yes	—
6	C₂×C₄⋊Dic₅₃	C₂²	1	8	yes	complemented
7	C₂×C₂²⋊Dic₅₃	—	2	8	no	—
8	C₂×C₂²⋊Dic₅₃	C₂²	1	4	yes	—
9	C₂×C₂²⋊Dic₅₃	C₂²	1	4	yes	—
10	#13133124	C₂²	1	8	yes	complemented
11	C₂⁵×C₅₃	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₅₃	C₂²	1	2	yes	—
13	C₂×C₂²⋊C₄×C₅₃	C₄	1	8	yes	complemented
14	C₂×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
15	C₂⁴×C₅₃	—	4	4	no	—
16	C₂⁴×C₅₃	C₂×C₄	1	4	yes	—
17	C₂⁴×C₅₃	D₄	1	4	yes	—
18	C₂⁴×C₅₃	Q₈	1	4	yes	—
19	C₂⁴×C₅₃	C₂³	1	2	yes	—
20	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	—	4	4	no	—
22	C₂²×C₄×C₅₃	C₂×C₄	1	8	yes	complemented
23	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
24	C₂²×Dic₅₃	—	2	8	no	—
25	C₂²×Dic₅₃	—	4	8	no	—
26	C₂²×Dic₅₃	C₂³	1	8	yes	—
27	C₂²⋊C₄×C₅₃	C₂×C₄	1	16	yes	complemented
28	C₂²⋊Dic₅₃	—	2	8	no	—
29	C₂²⋊Dic₅₃	—	2	8	no	—
30	C₄⋊Dic₅₃	—	4	8	no	—
31	C₂³×C₅₃	—	2	8	no	—
32	C₂³×C₅₃	—	2	4	no	—
33	C₂³×C₅₃	—	4	8	no	—
34	C₂³×C₅₃	—	4	8	no	—
35	C₂³×C₅₃	—	4	8	no	—
36	C₂³×C₅₃	C₄⋊C₄	1	16	yes	complemented
37	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
38	C₂³×C₅₃	C₂²×C₄	1	4	yes	—
39	C₂³×C₅₃	C₂×D₄	1	4	yes	—
40	C₂³×C₅₃	C₂×D₄	1	2	yes	—
41	C₂³×C₅₃	C₂×Q₈	1	4	yes	—
42	C₂³×C₅₃	C₂×Q₈	1	2	yes	—
43	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
44	C₂×C₄×C₅₃	—	2	8	no	—
45	C₂×C₄×C₅₃	—	4	8	no	—
46	C₂×C₄×C₅₃	C₂²×C₄	1	8	yes	—
47	C₂×Dic₅₃	—	2	8	no	—
48	C₂×Dic₅₃	—	2	16	no	—
49	C₂×Dic₅₃	—	4	16	no	—
50	C₂²×C₅₃	—	2	4	no	—
51	C₂²×C₅₃	—	2	8	no	—
52	C₂²×C₅₃	—	2	8	no	—
53	C₂²×C₅₃	—	2	4	no	—
54	C₂²×C₅₃	—	2	8	no	—
55	C₂²×C₅₃	—	2	4	no	—
56	C₂²×C₅₃	—	4	16	no	—
57	C₂²×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
58	C₂²×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
59	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
60	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
61	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
62	C₂²×C₅₃	H(5,𝔽₂)	1	4	yes	—
63	C₄×C₅₃	—	2	8	no	—
64	Dic₅₃	—	4	8	no	—
65	#2315	—	53	1	no	maximal, Sylow
66	C₂×C₅₃	—	2	4	no	—
67	C₂×C₅₃	—	2	8	no	—
68	C₂×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₅₃	#518	1	2	yes	—
70	C₂×C₅₃	#534	1	2	yes	—
71	C₂×C₅₃	#543	1	2	yes	—
72	C₂²×C₂²⋊C₄	—	53	2	no	—
73	C₂²×C₂²⋊C₄	D₅₃	1	1	yes	characteristic
74	#385	—	53	8	no	—
75	#380	—	53	4	no	—
76	C₅₃	#2315	1	1	yes	characteristic, complemented, Sylow
77	C₂⁵	C₂×D₅₃	1	1	yes	characteristic
78	C₂³×C₄	—	53	4	no	—
79	C₂³×C₄	C₂×D₅₃	1	2	yes	—
80	C₂×C₄⋊C₄	—	53	8	no	—
81	C₂×C₂²⋊C₄	—	53	4	no	—
82	C₂×C₂²⋊C₄	—	53	4	no	—
83	C₂×C₂²⋊C₄	—	106	8	no	—
84	C₂×C₂²⋊C₄	Dic₅₃	1	8	yes	complemented
85	C₂×C₂²⋊C₄	C₂×D₅₃	1	4	yes	—
86	#95	—	53	8	no	—
87	C₂⁴	—	4	4	no	—
88	C₂⁴	Dic₁₀₆	1	4	yes	—
89	C₂⁴	D₂₁₂	1	4	yes	—
90	C₂⁴	C₂×Dic₅₃	1	4	yes	—
91	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
92	C₂⁴	C₂²×D₅₃	1	2	yes	—
93	C₂²×C₄	—	4	4	no	—
94	C₂²×C₄	—	53	8	no	—
95	C₂²×C₄	—	106	8	no	—
96	C₂²×C₄	—	212	8	no	—
97	C₂²×C₄	C₂×Dic₅₃	1	8	yes	complemented
98	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
99	C₄⋊C₄	—	212	8	no	—
100	C₂²⋊C₄	—	106	8	no	—
101	C₂²⋊C₄	—	106	8	no	—
102	C₂²⋊C₄	C₂×Dic₅₃	1	16	yes	complemented
103	C₂³	—	2	4	no	—
104	C₂³	—	2	8	no	—
105	C₂³	—	4	8	no	—
106	C₂³	—	4	8	no	—
107	C₂³	—	4	8	no	—
108	C₂³	C₄⋊Dic₅₃	1	16	yes	complemented
109	C₂³	C₂×Dic₁₀₆	1	2	yes	—
110	C₂³	C₂×Dic₁₀₆	1	4	yes	—
111	C₂³	C₂×D₂₁₂	1	2	yes	—
112	C₂³	C₂×D₂₁₂	1	4	yes	—
113	C₂³	C₂²×Dic₅₃	1	4	yes	—
114	C₂³	C₂²×Dic₅₃	1	2	yes	—
115	C₂³	C₂³×D₅₃	1	1	yes	characteristic
116	C₂×C₄	—	2	8	no	—
117	C₂×C₄	—	4	8	no	—
118	C₂×C₄	—	106	8	no	—
119	C₂×C₄	—	106	16	no	—
120	C₂×C₄	—	212	16	no	—
121	C₂×C₄	C₂²×Dic₅₃	1	8	yes	—
122	C₂²	—	2	4	no	—
123	C₂²	—	2	8	no	—
124	C₂²	—	2	8	no	—
125	C₂²	—	2	4	no	—
126	C₂²	—	2	4	no	—
127	C₂²	—	2	8	no	—
128	C₂²	—	4	16	no	—
129	C₂²	C₂×C₄⋊Dic₅₃	1	4	yes	—
130	C₂²	C₂×C₄⋊Dic₅₃	1	8	yes	—
131	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
132	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
133	C₂²	C₂²×Dic₁₀₆	1	1	yes	characteristic
134	C₂²	#13133262	1	4	yes	—
135	C₄	—	2	8	no	—
136	C₄	—	212	8	no	—
137	C₂	—	2	4	no	—
138	C₂	—	2	8	no	—
139	C₂	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
140	C₂	#16811098	1	2	yes	—
141	C₂	#16811185	1	2	yes	—
142	C₂	#16811249	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₂²⋊Dic₅₃

C₂

yes

maximal

#16810498

C₂

yes

maximal, complemented

#16810483

C₂

yes

maximal, complemented

C₂²×C₂²⋊C₄×C₅₃

C₂

yes

maximal, characteristic

C₂³×Dic₅₃

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

complemented

C₂×C₂²⋊Dic₅₃

—

C₂×C₂²⋊Dic₅₃

C₂²

yes

—

C₂×C₂²⋊Dic₅₃

C₂²

yes

—

#13133124

C₂²

yes

complemented

C₂⁵×C₅₃

C₂²

yes

characteristic

C₂³×C₄×C₅₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₅₃

C₄

yes

complemented

C₂×C₂²⋊C₄×C₅₃

C₂²

yes

—

C₂⁴×C₅₃

—

C₂⁴×C₅₃

C₂×C₄

yes

—

C₂⁴×C₅₃

D₄

yes

—

C₂⁴×C₅₃

Q₈

yes

—

C₂⁴×C₅₃

C₂³

yes

—

C₂⁴×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

complemented

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

C₂³

yes

—

C₂²⋊C₄×C₅₃

C₂×C₄

yes

complemented

C₂²⋊Dic₅₃

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₄⋊C₄

yes

complemented

C₂³×C₅₃

C₂²×C₄

yes

—

C₂³×C₅₃

C₂²×C₄

yes

—

C₂³×C₅₃

C₂×D₄

yes

—

C₂³×C₅₃

C₂×D₄

yes

—

C₂³×C₅₃

C₂×Q₈

yes

—

C₂³×C₅₃

C₂×Q₈

yes

—

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂²×C₅₃

C₂²×Q₈

yes

characteristic

C₂²×C₅₃

H(5,𝔽₂)

yes

—

C₄×C₅₃

—

Dic₅₃

—

#2315

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₅₃

#518

yes

—

C₂×C₅₃

#534

yes

—

C₂×C₅₃

#543

yes

—

C₂²×C₂²⋊C₄

—

C₂²×C₂²⋊C₄

D₅₃

yes

characteristic

#385

—

#380

—

C₅₃

#2315

yes

characteristic, complemented, Sylow

C₂⁵

C₂×D₅₃

yes

characteristic

C₂³×C₄

—

C₂³×C₄

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

106

—

C₂×C₂²⋊C₄

Dic₅₃

yes

complemented

C₂×C₂²⋊C₄

C₂×D₅₃

yes

—

#95

—

C₂⁴

—

C₂⁴

Dic₁₀₆

yes

—

C₂⁴

D₂₁₂

yes

—

C₂⁴

C₂×Dic₅₃

yes

—

C₂⁴

C₂²×D₅₃

yes

characteristic

C₂⁴

C₂²×D₅₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

—

212

—

C₂²×C₄

C₂×Dic₅₃

yes

complemented

C₂²×C₄

C₂²×D₅₃

yes

—

C₄⋊C₄

—

212

—

100

C₂²⋊C₄

—

106

—

101

C₂²⋊C₄

—

106

—

102

C₂²⋊C₄

C₂×Dic₅₃

yes

complemented

103

C₂³

—

104

C₂³

—

105

C₂³

—

106

C₂³

—

107

C₂³

—

108

C₂³

C₄⋊Dic₅₃

yes

complemented

109

C₂³

C₂×Dic₁₀₆

yes

—

110

C₂³

C₂×Dic₁₀₆

yes

—

111

C₂³

C₂×D₂₁₂

yes

—

112

C₂³

C₂×D₂₁₂

yes

—

113

C₂³

C₂²×Dic₅₃

yes

—

114

C₂³

C₂²×Dic₅₃

yes

—

115

C₂³

C₂³×D₅₃

yes

characteristic

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

106

—

119

C₂×C₄

—

106

—

120

C₂×C₄

—

212

—

121

C₂×C₄

C₂²×Dic₅₃

yes

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₂×C₄⋊Dic₅₃

yes

—

130

C₂²

C₂×C₄⋊Dic₅₃

yes

—

131

C₂²

C₂³×Dic₅₃

yes

characteristic

132

C₂²

C₂²×D₂₁₂

yes

characteristic

133

C₂²

C₂²×Dic₁₀₆

yes

characteristic

134

C₂²

#13133262

yes

—

135

C₄

—

136

C₄

—

212

—

137

C₂

—

138

C₂

—

139

C₂

C₂²×C₄⋊Dic₅₃

yes

characteristic

140

C₂

#16811098

yes

—

141

C₂

#16811185

yes

—

142

C₂

#16811249

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂³×C₄
Fitting	C₂²×C₂²⋊C₄×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	C₂²×C₂²⋊C₄	D₅₃
hypocenter	C₅₃	#2315
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₂²⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator

C₂²×C₅₃

C₂³×C₄

Fitting

C₂²×C₂²⋊C₄×C₅₃

C₂

socle

C₂³×C₅₃

C₂⁴

hypercenter

C₂²×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ C₂²×C₂²⋊C₄

N	H ≅ G/N	type
2. #16810498	C₂	semidirect
3. #16810483	C₂	semidirect
6. C₂×C₄⋊Dic₅₃	C₂²	semidirect
10. #13133124	C₂²	semidirect
13. C₂×C₂²⋊C₄×C₅₃	C₄	semidirect
22. C₂²×C₄×C₅₃	C₂×C₄	semidirect
27. C₂²⋊C₄×C₅₃	C₂×C₄	semidirect
36. C₂³×C₅₃	C₄⋊C₄	semidirect
76. C₅₃	#2315	semidirect
84. C₂×C₂²⋊C₄	Dic₅₃	semidirect
97. C₂²×C₄	C₂×Dic₅₃	semidirect
102. C₂²⋊C₄	C₂×Dic₅₃	semidirect
108. C₂³	C₄⋊Dic₅₃	semidirect

H ≅ G/N

type

2. #16810498

C₂

semidirect

3. #16810483

C₂

semidirect

6. C₂×C₄⋊Dic₅₃

C₂²

semidirect

10. #13133124

C₂²

semidirect

13. C₂×C₂²⋊C₄×C₅₃

C₄

semidirect

22. C₂²×C₄×C₅₃

C₂×C₄

semidirect

27. C₂²⋊C₄×C₅₃

C₂×C₄

semidirect

36. C₂³×C₅₃

C₄⋊C₄

semidirect

76. C₅₃

#2315

semidirect

84. C₂×C₂²⋊C₄

Dic₅₃

semidirect

97. C₂²×C₄

C₂×Dic₅₃

semidirect

102. C₂²⋊C₄

C₂×Dic₅₃

semidirect

108. C₂³

C₄⋊Dic₅₃

semidirect

Subgroups — #20277329

Conjugacy classes of subgroups

Complements