Group #20274454 — subgroups

Subgroups — #20274454

Order 6784.

388 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 189 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×#11719917	C₂	1	2	yes	maximal
2	#16810392	C₂	1	4	yes	maximal, complemented
3	C₂×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719917	—	2	4	no	—
5	C₂×#11719917	—	2	2	no	—
6	C₂×#11719917	C₄	1	8	yes	complemented
7	C₂×#11719917	C₂²	1	4	yes	—
8	C₂²×#88322	—	2	2	no	—
9	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
10	#117×C₅₃	—	2	2	no	—
11	#117×C₅₃	—	2	2	no	—
12	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
13	C₂×C₂²⋊C₄×C₅₃	C₄	1	2	yes	—
14	C₂×C₄²×C₅₃	C₄	1	2	yes	—
15	C₂⁴×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	C₂×C₄	1	4	yes	—
17	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₅₃	D₄	1	2	yes	—
20	C₂²×C₄×C₅₃	D₄	1	2	yes	—
21	C₂²×C₄×C₅₃	D₄	1	2	yes	—
22	C₂²×C₄×C₅₃	Q₈	1	2	yes	—
23	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₅₃	—	2	4	no	—
25	C₂²⋊C₄×C₅₃	—	2	4	no	—
26	C₄²×C₅₃	—	2	2	no	—
27	C₄²×C₅₃	—	2	2	no	—
28	#11719917	—	2	8	no	—
29	#11719917	—	2	4	no	—
30	#11719917	C₂×C₄	1	8	yes	—
31	C₂×#88322	—	2	8	no	—
32	C₂×#88322	—	2	4	no	—
33	C₂³×C₅₃	—	4	2	no	—
34	C₂³×C₅₃	C₄²	1	2	yes	—
35	C₂³×C₅₃	C₂²⋊C₄	1	2	yes	—
36	C₂³×C₅₃	C₄⋊C₄	1	2	yes	—
37	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	—	2	4	no	—
39	C₂×C₄×C₅₃	—	4	4	no	—
40	C₂×C₄×C₅₃	C₄²	1	2	yes	—
41	C₂×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
42	C₂×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
43	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
46	C₂×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
47	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
48	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
52	#88322	—	2	8	no	—
53	C₂²×C₅₃	—	2	2	no	—
54	C₂²×C₅₃	—	2	4	no	—
55	C₂²×C₅₃	—	4	2	no	—
56	C₂²×C₅₃	—	4	2	no	—
57	C₂²×C₅₃	#95	1	4	yes	—
58	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
59	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₄×C₅₃	—	4	4	no	—
62	C₄×C₅₃	#95	1	2	yes	—
63	C₄×C₅₃	#95	1	2	yes	—
64	C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
65	C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
66	#1742	—	53	1	no	maximal, Sylow
67	C₂×C₅₃	—	2	4	no	—
68	C₂×C₅₃	—	2	1	no	—
69	C₂×C₅₃	—	2	1	no	—
70	C₂×C₅₃	C₂×#95	1	1	yes	characteristic
71	C₂×C₅₃	#413	1	2	yes	—
72	C₂²×M₁₆	—	53	2	no	—
73	C₂×#117	D₅₃	1	1	yes	characteristic
74	#343	—	53	4	no	—
75	C₅₃	#1742	1	1	yes	characteristic, complemented, Sylow
76	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
77	C₂×M₁₆	—	53	8	no	—
78	C₂×M₁₆	—	53	4	no	—
79	C₂×M₁₆	—	106	4	no	—
80	C₂×M₁₆	—	106	2	no	—
81	C₂²×C₈	—	106	2	no	—
82	#117	—	2	2	no	—
83	#117	—	2	2	no	—
84	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
85	C₂×C₂²⋊C₄	Dic₅₃	1	2	yes	—
86	C₂×C₄²	Dic₅₃	1	2	yes	—
87	C₂⁴	C₂×Dic₅₃	1	1	yes	characteristic
88	C₂²×C₄	Dic₁₀₆	1	2	yes	—
89	C₂²×C₄	C₄×D₅₃	1	4	yes	—
90	C₂²×C₄	D₂₁₂	1	2	yes	—
91	C₂²×C₄	C₂×Dic₅₃	1	4	yes	—
92	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
93	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
94	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
95	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
96	M₁₆	—	53	8	no	—
97	M₁₆	—	106	4	no	—
98	M₁₆	—	106	8	no	—
99	C₂×C₈	—	106	4	no	—
100	C₂×C₈	—	106	8	no	—
101	C₄⋊C₄	—	2	4	no	—
102	C₂²⋊C₄	—	2	4	no	—
103	C₄²	—	2	2	no	—
104	C₄²	—	2	2	no	—
105	C₂³	—	4	2	no	—
106	C₂³	C₄×Dic₅₃	1	2	yes	—
107	C₂³	C₄⋊Dic₅₃	1	2	yes	—
108	C₂³	C₂²⋊Dic₅₃	1	2	yes	—
109	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
110	C₂×C₄	—	2	4	no	—
111	C₂×C₄	—	4	4	no	—
112	C₂×C₄	C₄×Dic₅₃	1	2	yes	—
113	C₂×C₄	#11719919	1	4	yes	—
114	C₂×C₄	#11719919	1	4	yes	—
115	C₂×C₄	C₄⋊Dic₅₃	1	2	yes	—
116	C₂×C₄	#11719921	1	4	yes	—
117	C₂×C₄	#11719921	1	4	yes	—
118	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
119	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
120	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
121	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
122	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
123	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
124	C₈	—	106	8	no	—
125	C₂²	—	2	2	no	—
126	C₂²	—	2	4	no	—
127	C₂²	—	4	2	no	—
128	C₂²	—	4	2	no	—
129	C₂²	#13133124	1	4	yes	—
130	C₂²	C₂×C₄×Dic₅₃	1	1	yes	characteristic
131	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
132	C₂²	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
133	C₄	—	4	4	no	—
134	C₄	#13133124	1	2	yes	—
135	C₄	#13133124	1	2	yes	—
136	C₄	C₂×#11719921	1	2	yes	—
137	C₄	C₂×#11719919	1	2	yes	—
138	C₂	—	2	4	no	—
139	C₂	—	2	1	no	—
140	C₂	—	2	1	no	—
141	C₂	C₂×#13133124	1	1	yes	characteristic
142	C₂	#16810607	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11719917

C₂

yes

maximal

#16810392

C₂

yes

maximal, complemented

C₂×#117×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

—

C₂×#11719917

—

C₂×#11719917

C₄

yes

complemented

C₂×#11719917

C₂²

yes

—

C₂²×#88322

—

C₂³×C₄×C₅₃

C₂²

yes

characteristic

#117×C₅₃

—

#117×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₅₃

C₄

yes

—

C₂×C₄²×C₅₃

C₄

yes

—

C₂⁴×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

#11719917

—

#11719917

—

#11719917

C₂×C₄

yes

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

C₄²

yes

—

C₂³×C₅₃

C₂²⋊C₄

yes

—

C₂³×C₅₃

C₄⋊C₄

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄²

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#95

yes

—

C₂²×C₅₃

C₂×C₄²

yes

characteristic

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

#95

yes

—

C₄×C₅₃

#95

yes

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

—

#1742

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#95

yes

characteristic

C₂×C₅₃

#413

yes

—

C₂²×M₁₆

—

C₂×#117

D₅₃

yes

characteristic

#343

—

C₅₃

#1742

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₅₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

106

—

C₂×M₁₆

—

106

—

C₂²×C₈

—

106

—

#117

—

#117

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₂²⋊C₄

Dic₅₃

yes

—

C₂×C₄²

Dic₅₃

yes

—

C₂⁴

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

Dic₁₀₆

yes

—

C₂²×C₄

C₄×D₅₃

yes

—

C₂²×C₄

D₂₁₂

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

106

—

100

C₂×C₈

—

106

—

101

C₄⋊C₄

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

—

105

C₂³

—

106

C₂³

C₄×Dic₅₃

yes

—

107

C₂³

C₄⋊Dic₅₃

yes

—

108

C₂³

C₂²⋊Dic₅₃

yes

—

109

C₂³

C₂²×Dic₅₃

yes

characteristic

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₄×Dic₅₃

yes

—

113

C₂×C₄

#11719919

yes

—

114

C₂×C₄

#11719919

yes

—

115

C₂×C₄

C₄⋊Dic₅₃

yes

—

116

C₂×C₄

#11719921

yes

—

117

C₂×C₄

#11719921

yes

—

118

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

119

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

120

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

121

C₂×C₄

C₂×C₄×D₅₃

yes

—

122

C₂×C₄

C₂×D₂₁₂

yes

characteristic

123

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

124

C₈

—

106

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

#13133124

yes

—

130

C₂²

C₂×C₄×Dic₅₃

yes

characteristic

131

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

132

C₂²

C₂×C₄⋊Dic₅₃

yes

characteristic

133

C₄

—

134

C₄

#13133124

yes

—

135

C₄

#13133124

yes

—

136

C₄

C₂×#11719921

yes

—

137

C₄

C₂×#11719919

yes

—

138

C₂

—

139

C₂

—

140

C₂

—

141

C₂

C₂×#13133124

yes

characteristic

142

C₂

#16810607

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₅₃
commutator	C₂²×C₅₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#117×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	C₂×#117	D₅₃
hypocenter	C₅₃	#1742
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#117

subgroup

G/H

center

C₂×C₄

C₂²⋊Dic₅₃

commutator

C₂²×C₅₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×#117×C₅₃

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#117

N	H ≅ G/N	type
2. #16810392	C₂	semidirect
6. C₂×#11719917	C₄	semidirect
75. C₅₃	#1742	semidirect

H ≅ G/N

type

2. #16810392

C₂

semidirect

6. C₂×#11719917

C₄

semidirect

75. C₅₃

#1742

semidirect

Subgroups — #20274454

Conjugacy classes of subgroups

Complements