Group #20272620 — subgroups

Subgroups — #20272620

Order 6784.

200 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 61 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	#16810587	C₂	1	1	yes	maximal, characteristic
2	#16810483	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#407	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719917	—	2	1	no	—
5	C₂×C₄⋊Dic₅₃	—	2	1	no	—
6	C₂×C₂²⋊Dic₅₃	C₄	1	2	yes	—
7	C₂²⋊#88322	—	2	1	no	—
8	C₂²⋊#88322	C₄	1	2	yes	complemented
9	#13133124	—	2	1	no	—
10	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×M₁₆×C₅₃	C₄	1	2	yes	complemented
12	C₂²⋊C₈×C₅₃	—	2	1	no	—
13	C₂²⋊C₈×C₅₃	—	2	1	no	—
14	C₂⁴×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	—	4	2	no	—
16	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	Q₈	1	1	yes	characteristic
22	C₂²×Dic₅₃	—	2	2	no	—
23	M₁₆×C₅₃	—	4	2	no	—
24	C₂×C₈×C₅₃	—	2	2	no	—
25	C₂²⋊Dic₅₃	—	4	2	no	—
26	C₄⋊Dic₅₃	—	4	2	no	—
27	#11719917	—	4	2	no	—
28	C₂×#88322	—	2	2	no	—
29	C₂³×C₅₃	—	2	1	no	—
30	C₂³×C₅₃	—	2	1	no	—
31	C₂³×C₅₃	—	4	1	no	—
32	C₂³×C₅₃	—	4	1	no	—
33	C₂³×C₅₃	C₄²	1	1	yes	characteristic
34	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₅₃	—	2	2	no	—
37	C₂×C₄×C₅₃	—	2	2	no	—
38	C₂×C₄×C₅₃	—	2	2	no	—
39	C₂×C₄×C₅₃	—	4	1	no	—
40	C₂×C₄×C₅₃	—	4	1	no	—
41	C₂×C₄×C₅₃	—	4	1	no	—
42	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×Dic₅₃	—	4	2	no	—
47	C₂×Dic₅₃	—	4	2	no	—
48	C₂×Dic₅₃	—	4	2	no	—
49	C₈×C₅₃	—	4	2	no	—
50	#88322	—	4	2	no	—
51	C₂²×C₅₃	—	2	1	no	—
52	C₂²×C₅₃	—	2	1	no	—
53	C₂²×C₅₃	—	2	1	no	—
54	C₂²×C₅₃	—	2	1	no	—
55	C₂²×C₅₃	—	2	1	no	—
56	C₂²×C₅₃	—	2	1	no	—
57	C₂²×C₅₃	—	2	1	no	—
58	C₂²×C₅₃	—	2	1	no	—
59	C₂²×C₅₃	—	4	2	no	—
60	C₂²×C₅₃	—	4	2	no	—
61	C₂²×C₅₃	#95	1	1	yes	characteristic
62	C₂²×C₅₃	#100	1	1	yes	characteristic
63	C₂²×C₅₃	#101	1	1	yes	characteristic
64	C₄×C₅₃	—	2	1	no	—
65	C₄×C₅₃	—	2	1	no	—
66	C₄×C₅₃	—	2	1	no	—
67	C₄×C₅₃	#99	1	2	yes	—
68	Dic₅₃	—	8	2	no	—
69	#1305	—	53	1	no	maximal, Sylow
70	C₂×C₅₃	—	2	1	no	—
71	C₂×C₅₃	—	2	1	no	—
72	C₂×C₅₃	—	4	1	no	—
73	C₂×C₅₃	—	4	1	no	—
74	C₂×C₅₃	#342	1	1	yes	characteristic
75	C₂×C₅₃	#343	1	1	yes	characteristic
76	C₂×C₅₃	#344	1	1	yes	characteristic
77	#407	—	53	1	no	—
78	#407	D₅₃	1	1	yes	characteristic
79	#380	—	53	1	no	—
80	C₅₃	#1305	1	1	yes	characteristic, complemented, Sylow
81	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
82	C₂×M₁₆	—	106	1	no	—
83	C₂×M₁₆	Dic₅₃	1	2	yes	complemented
84	C₂×C₄⋊C₄	—	106	1	no	—
85	C₂×C₂²⋊C₄	—	53	2	no	—
86	C₂²⋊C₈	—	2	1	no	—
87	C₂²⋊C₈	—	2	1	no	—
88	C₂²⋊C₈	—	53	2	no	—
89	C₂²⋊C₈	—	106	1	no	—
90	#95	—	106	1	no	—
91	C₂⁴	C₄×D₅₃	1	1	yes	characteristic
92	C₂²×C₄	—	4	2	no	—
93	C₂²×C₄	—	106	2	no	—
94	C₂²×C₄	Dic₁₀₆	1	1	yes	characteristic
95	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
96	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
97	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
98	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
100	M₁₆	—	4	2	no	—
101	M₁₆	—	212	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	106	2	no	—
104	C₄⋊C₄	—	212	2	no	—
105	C₂²⋊C₄	—	212	2	no	—
106	C₂³	—	2	1	no	—
107	C₂³	—	2	1	no	—
108	C₂³	—	4	1	no	—
109	C₂³	—	4	1	no	—
110	C₂³	C₄×Dic₅₃	1	1	yes	characteristic
111	C₂³	#11719919	1	1	yes	characteristic
112	C₂³	#11719921	1	1	yes	characteristic
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	212	2	no	—
120	C₂×C₄	—	212	2	no	—
121	C₂×C₄	—	212	2	no	—
122	C₂×C₄	#11719919	1	1	yes	characteristic
123	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
124	C₂×C₄	#11719921	1	1	yes	characteristic
125	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
126	C₈	—	4	2	no	—
127	C₈	—	212	2	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	—	4	2	no	—
137	C₂²	—	4	2	no	—
138	C₂²	#13133124	1	1	yes	characteristic
139	C₂²	#13133133	1	1	yes	characteristic
140	C₂²	#13133135	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	424	2	no	—
145	C₄	#13133132	1	2	yes	—
146	C₂	—	2	1	no	—
147	C₂	—	2	1	no	—
148	C₂	—	4	1	no	—
149	C₂	—	4	1	no	—
150	C₂	#16810390	1	1	yes	characteristic
151	C₂	#16810391	1	1	yes	characteristic
152	C₂	#16810395	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810587

C₂

yes

maximal, characteristic

#16810483

C₂

yes

maximal, characteristic

C₅₃×#407

C₂

yes

maximal, characteristic

C₂×#11719917

—

C₂×C₄⋊Dic₅₃

—

C₂×C₂²⋊Dic₅₃

C₄

yes

—

C₂²⋊#88322

—

C₂²⋊#88322

C₄

yes

complemented

#13133124

—

C₂³×C₄×C₅₃

C₂²

yes

characteristic

C₂×M₁₆×C₅₃

C₄

yes

complemented

C₂²⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₂⁴×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

Q₈

yes

characteristic

C₂²×Dic₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719917

—

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₄²

yes

characteristic

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

C₂³×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#99

yes

—

Dic₅₃

—

#1305

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

yes

characteristic

#380

—

C₅₃

#1305

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₅₃

yes

characteristic

C₂×M₁₆

—

106

—

C₂×M₁₆

Dic₅₃

yes

complemented

C₂×C₄⋊C₄

—

106

—

C₂×C₂²⋊C₄

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

106

—

#95

—

106

—

C₂⁴

C₄×D₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

Dic₁₀₆

yes

characteristic

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

D₂₁₂

yes

characteristic

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

100

M₁₆

—

101

M₁₆

—

212

—

102

C₂×C₈

—

103

C₂×C₈

—

106

—

104

C₄⋊C₄

—

212

—

105

C₂²⋊C₄

—

212

—

106

C₂³

—

107

C₂³

—

108

C₂³

—

109

C₂³

—

110

C₂³

C₄×Dic₅₃

yes

characteristic

111

C₂³

#11719919

yes

characteristic

112

C₂³

#11719921

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

212

—

120

C₂×C₄

—

212

—

121

C₂×C₄

—

212

—

122

C₂×C₄

#11719919

yes

characteristic

123

C₂×C₄

C₄⋊Dic₅₃

yes

characteristic

124

C₂×C₄

#11719921

yes

characteristic

125

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

126

C₈

—

127

C₈

—

212

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

#13133124

yes

characteristic

139

C₂²

#13133133

yes

characteristic

140

C₂²

#13133135

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

424

—

145

C₄

#13133132

yes

—

146

C₂

—

147

C₂

—

148

C₂

—

149

C₂

—

150

C₂

#16810390

yes

characteristic

151

C₂

#16810391

yes

characteristic

152

C₂

#16810395

yes

characteristic

	subgroup	G/H
center	C₂²	#13133124
commutator	C₂³×C₅₃	C₄²
Frattini	C₂³×C₄	C₂×D₅₃
Fitting	C₅₃×#407	C₂
socle	C₂²×C₅₃	#95
hypercenter	#407	D₅₃
hypocenter	C₅₃	#1305
LCS	G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂²

#13133124

commutator

C₂³×C₅₃

C₄²

Frattini

C₂³×C₄

C₂×D₅₃

Fitting

C₅₃×#407

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
8. C₂²⋊#88322	C₄	semidirect
11. C₂×M₁₆×C₅₃	C₄	semidirect
80. C₅₃	#1305	semidirect
83. C₂×M₁₆	Dic₅₃	semidirect

H ≅ G/N

type

8. C₂²⋊#88322

C₄

semidirect

11. C₂×M₁₆×C₅₃

C₄

semidirect

80. C₅₃

#1305

semidirect

83. C₂×M₁₆

Dic₅₃

semidirect

Subgroups — #20272620

Conjugacy classes of subgroups

Complements