Group #20290643 — subgroups

Subgroups — #20290643

Order 6784.

240 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 94 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	#16811489	C₂	1	2	yes	maximal
2	#16811489	C₂	1	2	yes	maximal
3	#16811489	C₂	1	2	yes	maximal
4	C₄²×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#88324	C₂²	1	4	yes	—
6	C₂×C₄×Dic₅₃	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₅₃	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₅₃	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₅₃	C₄	1	1	yes	characteristic
13	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
20	C₂×#88324	—	2	4	no	—
21	C₂×#88324	—	2	4	no	—
22	C₂×#88324	—	2	4	no	—
23	C₄²×C₅₃	—	2	2	no	—
24	C₄×Dic₅₃	—	2	2	no	—
25	C₄×Dic₅₃	—	2	2	no	—
26	C₄×Dic₅₃	—	2	2	no	—
27	C₄×Dic₅₃	—	2	2	no	—
28	C₄×Dic₅₃	—	2	2	no	—
29	C₄×Dic₅₃	C₈	1	4	yes	complemented
30	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₅₃	—	2	1	no	—
32	C₂×C₄×C₅₃	—	2	1	no	—
33	C₂×C₄×C₅₃	—	2	1	no	—
34	C₂×C₄×C₅₃	—	2	1	no	—
35	C₂×C₄×C₅₃	—	2	1	no	—
36	C₂×C₄×C₅₃	—	2	1	no	—
37	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
38	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
39	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
40	C₂×Dic₅₃	—	2	1	no	—
41	C₂×Dic₅₃	—	2	1	no	—
42	C₂×Dic₅₃	—	2	1	no	—
43	C₂×Dic₅₃	—	2	1	no	—
44	C₂×Dic₅₃	—	2	1	no	—
45	C₂×Dic₅₃	—	2	1	no	—
46	C₂×Dic₅₃	C₂×C₈	1	2	yes	—
47	C₂×Dic₅₃	C₂×C₈	1	2	yes	—
48	C₂×Dic₅₃	M₁₆	1	2	yes	—
49	C₂×Dic₅₃	#41	1	2	yes	—
50	C₂×Dic₅₃	#41	1	2	yes	—
51	C₂×Dic₅₃	#41	1	2	yes	—
52	#88324	—	4	4	no	—
53	C₂²×C₅₃	#117	1	1	yes	characteristic
54	C₂²×C₅₃	#117	1	1	yes	characteristic
55	C₂²×C₅₃	#117	1	1	yes	characteristic
56	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
57	C₂²×C₅₃	C₂×M₁₆	1	1	yes	characteristic
58	C₂²×C₅₃	C₂×M₁₆	1	1	yes	characteristic
59	C₂²×C₅₃	C₂×M₁₆	1	1	yes	characteristic
60	C₄×C₅₃	—	2	2	no	—
61	C₄×C₅₃	—	2	2	no	—
62	C₄×C₅₃	—	2	2	no	—
63	Dic₅₃	—	2	2	no	—
64	Dic₅₃	—	2	2	no	—
65	Dic₅₃	—	2	2	no	—
66	Dic₅₃	#126	1	4	yes	—
67	#1839	—	53	1	no	maximal, Sylow
68	C₂×C₅₃	#383	1	1	yes	characteristic
69	C₂×C₅₃	#431	1	1	yes	characteristic
70	C₂×C₅₃	#431	1	1	yes	characteristic
71	C₂×C₅₃	#431	1	1	yes	characteristic
72	C₂×C₅₃	#432	1	1	yes	characteristic
73	C₂×C₅₃	#432	1	1	yes	characteristic
74	C₂×C₅₃	#432	1	1	yes	characteristic
75	C₄³	—	53	1	no	—
76	#336	—	53	2	no	—
77	#336	—	53	2	no	—
78	#336	—	53	2	no	—
79	C₅₃	#1839	1	1	yes	characteristic, complemented, Sylow
80	C₂²×C₈	—	53	4	no	—
81	C₂×C₄²	—	53	1	no	—
82	C₂×C₄²	—	53	1	no	—
83	C₂×C₄²	—	53	1	no	—
84	C₂×C₄²	—	53	1	no	—
85	C₂×C₄²	—	53	1	no	—
86	C₂×C₄²	—	53	1	no	—
87	C₂×C₄²	#6157	1	1	yes	characteristic
88	C₂²×C₄	—	53	1	no	—
89	C₂²×C₄	—	53	1	no	—
90	C₂²×C₄	—	53	1	no	—
91	C₂²×C₄	—	53	1	no	—
92	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
93	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
94	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
95	C₂×C₈	—	106	4	no	—
96	C₂×C₈	—	106	4	no	—
97	C₂×C₈	—	106	4	no	—
98	C₄²	—	2	2	no	—
99	C₄²	—	53	4	no	—
100	C₄²	—	106	2	no	—
101	C₄²	—	106	2	no	—
102	C₄²	—	106	2	no	—
103	C₄²	—	106	2	no	—
104	C₄²	—	106	2	no	—
105	C₂³	C₂²×#6157	1	1	yes	characteristic
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	53	2	no	—
113	C₂×C₄	—	53	2	no	—
114	C₂×C₄	—	53	2	no	—
115	C₂×C₄	—	53	2	no	—
116	C₂×C₄	—	53	2	no	—
117	C₂×C₄	—	53	2	no	—
118	C₂×C₄	—	106	1	no	—
119	C₂×C₄	—	106	1	no	—
120	C₂×C₄	—	106	1	no	—
121	C₂×C₄	—	106	1	no	—
122	C₂×C₄	—	106	1	no	—
123	C₂×C₄	—	106	1	no	—
124	C₂×C₄	D₅₃⋊C₈	1	2	yes	—
125	C₂×C₄	#11719936	1	2	yes	—
126	C₂×C₄	#11719940	1	2	yes	—
127	C₈	—	212	4	no	—
128	C₂²	#13133296	1	1	yes	characteristic
129	C₂²	#13133296	1	1	yes	characteristic
130	C₂²	#13133296	1	1	yes	characteristic
131	C₂²	C₂×D₅₃⋊C₈	1	1	yes	characteristic
132	C₂²	C₂×#11719940	1	1	yes	characteristic
133	C₂²	D₅₃⋊M₁₆	1	1	yes	characteristic
134	C₂²	C₂×#11719936	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	53	4	no	—
139	C₄	—	106	2	no	—
140	C₄	—	106	2	no	—
141	C₄	—	106	2	no	—
142	C₂	#16811568	1	1	yes	characteristic
143	C₂	#16811640	1	1	yes	characteristic
144	C₂	#16811641	1	1	yes	characteristic
145	C₂	#16811642	1	1	yes	characteristic
146	C₂	#16811643	1	1	yes	characteristic
147	C₂	#16811644	1	1	yes	characteristic
148	C₂	#16811645	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811489

C₂

yes

maximal

#16811489

C₂

yes

maximal

#16811489

C₂

yes

maximal

C₄²×Dic₅₃

C₂

yes

maximal, characteristic

C₂²×#88324

C₂²

yes

—

C₂×C₄×Dic₅₃

C₄

yes

characteristic

C₂×C₄×Dic₅₃

C₄

yes

characteristic

C₂×C₄×Dic₅₃

C₄

yes

characteristic

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₂³

yes

—

—

—

C₄²×C₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₈

yes

complemented

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

M₁₆

yes

—

C₂×C₄×C₅₃

M₁₆

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×C₈

yes

—

C₂×Dic₅₃

C₂×C₈

yes

—

C₂×Dic₅₃

M₁₆

yes

—

C₂×Dic₅₃

#41

yes

—

C₂×Dic₅₃

#41

yes

—

C₂×Dic₅₃

#41

yes

—

#88324

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

C₂²×C₈

yes

characteristic

C₂²×C₅₃

C₂×M₁₆

yes

characteristic

C₂²×C₅₃

C₂×M₁₆

yes

characteristic

C₂²×C₅₃

C₂×M₁₆

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#126

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₄³

—

#336

—

#336

—

#336

—

C₅₃

#1839

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

#6157

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄²

—

C₄²

—

100

C₄²

—

106

—

101

C₄²

—

106

—

102

C₄²

—

106

—

103

C₄²

—

106

—

104

C₄²

—

106

—

105

C₂³

C₂²×#6157

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

106

—

119

C₂×C₄

—

106

—

120

C₂×C₄

—

106

—

121

C₂×C₄

—

106

—

122

C₂×C₄

—

106

—

123

C₂×C₄

—

106

—

124

C₂×C₄

D₅₃⋊C₈

yes

—

125

C₂×C₄

#11719936

yes

—

126

C₂×C₄

#11719940

yes

—

127

C₈

—

212

—

128

C₂²

#13133296

yes

characteristic

129

C₂²

#13133296

yes

characteristic

130

C₂²

#13133296

yes

characteristic

131

C₂²

C₂×D₅₃⋊C₈

yes

characteristic

132

C₂²

C₂×#11719940

yes

characteristic

133

C₂²

D₅₃⋊M₁₆

yes

characteristic

134

C₂²

C₂×#11719936

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

106

—

140

C₄

—

106

—

141

C₄

—

106

—

142

C₂

#16811568

yes

characteristic

143

C₂

#16811640

yes

characteristic

144

C₂

#16811641

yes

characteristic

145

C₂

#16811642

yes

characteristic

146

C₂

#16811643

yes

characteristic

147

C₂

#16811644

yes

characteristic

148

C₂

#16811645

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×#6157
commutator	C₂²×C₅₃	C₂²×C₈
Fitting	C₂×C₄²×C₅₃	C₄
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	C₂×C₄²	#6157
hypocenter	C₅₃	#1839
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

subgroup

G/H

center, Frattini

C₂³

C₂²×#6157

commutator

C₂²×C₅₃

C₂²×C₈

Fitting

C₂×C₄²×C₅₃

C₄

socle

C₂³×C₅₃

C₂²×C₄

C₂×C₄²

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ C₂×C₄²

N	H ≅ G/N	type
29. C₄×Dic₅₃	C₈	semidirect
79. C₅₃	#1839	semidirect

H ≅ G/N

type

29. C₄×Dic₅₃

C₈

semidirect

79. C₅₃

#1839

semidirect

Subgroups — #20290643

Conjugacy classes of subgroups

Complements