Group #20273595 — subgroups

Subgroups — #20273595

Order 6784.

166 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 63 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	C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#11719913	C₂	1	1	yes	maximal, characteristic
3	#16810702	C₂	1	1	yes	maximal, characteristic
4	#16810376	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810355	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810352	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719913	C₄	1	1	yes	characteristic
9	C₂×#11719912	—	2	1	no	—
10	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
11	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
12	#13133149	C₄	1	1	yes	characteristic
13	C₄⋊#88322	C₂²	1	1	yes	characteristic
14	C₄⋊#88322	C₂²	1	1	yes	characteristic
15	#13133146	—	2	1	no	—
16	#13133142	C₄	1	1	yes	characteristic
17	#13133139	C₄	1	1	yes	characteristic
18	#13133129	—	2	1	no	—
19	C₈×Dic₅₃	—	2	1	no	—
20	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₅₃	—	2	1	no	—
25	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×C₅₃	—	2	1	no	—
27	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	#11719921	—	2	1	no	—
33	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
34	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
35	#11719919	—	2	1	no	—
36	C₄×Dic₅₃	—	2	1	no	—
37	C₂×#88322	—	2	1	no	—
38	C₂×#88322	—	2	1	no	—
39	#11719913	—	2	2	no	—
40	#11719912	—	4	2	no	—
41	C₂²×D₅₃	—	2	1	no	—
42	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
45	C₂×Dic₅₃	—	2	1	no	—
46	C₂×Dic₅₃	—	2	1	no	—
47	C₂×Dic₅₃	—	2	1	no	—
48	D₂₁₂	—	2	1	no	—
49	D₂₁₂	—	2	1	no	—
50	C₄×D₅₃	—	4	1	no	—
51	C₄×D₅₃	—	4	1	no	—
52	Dic₁₀₆	—	2	1	no	—
53	Dic₁₀₆	M₁₆	1	2	yes	complemented
54	C₈×C₅₃	—	2	2	no	—
55	C₈×C₅₃	—	2	1	no	—
56	C₈×C₅₃	—	2	1	no	—
57	C₈×C₅₃	—	2	1	no	—
58	C₈×C₅₃	—	2	1	no	—
59	#88322	—	4	1	no	—
60	#88322	—	4	1	no	—
61	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
62	C₂×D₅₃	—	4	1	no	—
63	C₂×D₅₃	—	4	1	no	—
64	C₂×D₅₃	—	4	1	no	—
65	C₄×C₅₃	—	2	1	no	—
66	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
67	C₄×C₅₃	#131	1	1	yes	characteristic
68	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
69	C₄×C₅₃	#137	1	1	yes	characteristic
70	Dic₅₃	—	2	2	no	—
71	Dic₅₃	—	4	1	no	—
72	Dic₅₃	—	4	1	no	—
73	#1588	—	53	1	no	maximal, Sylow
74	C₂×C₅₃	#435	1	1	yes	characteristic
75	C₂×C₅₃	#440	1	1	yes	characteristic
76	C₂×C₅₃	#443	1	1	yes	characteristic
77	D₅₃	—	8	1	no	—
78	C₈×Q₈	—	53	1	no	—
79	C₄×SD₁₆	—	53	1	no	—
80	#436	—	53	1	no	—
81	#335	—	53	1	no	—
82	Q₈⋊C₈	—	53	1	no	—
83	D₄⋊C₈	—	53	1	no	—
84	#322	D₅₃	1	1	yes	characteristic, complemented
85	C₅₃	#1588	1	1	yes	characteristic, complemented, Sylow
86	C₂×SD₁₆	—	53	1	no	—
87	C₂×M₁₆	—	106	1	no	—
88	C₄×Q₈	—	53	1	no	—
89	C₄×D₄	—	53	1	no	—
90	#106	—	53	1	no	—
91	C₄⋊C₈	—	53	1	no	—
92	C₄⋊C₈	—	53	1	no	—
93	C₄⋊C₈	—	106	1	no	—
94	#103	—	53	1	no	—
95	#102	—	53	1	no	—
96	C₂²⋊C₈	—	106	1	no	—
97	C₄×C₈	—	106	1	no	—
98	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
99	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
100	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
101	C₂×Q₈	—	53	1	no	—
102	C₂×D₄	—	53	1	no	—
103	C₂²×C₄	—	106	1	no	—
104	SD₁₆	—	106	2	no	—
105	M₁₆	—	212	2	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	106	1	no	—
108	C₂×C₈	—	106	1	no	—
109	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
110	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
111	C₂×C₈	D₂₁₂	1	1	yes	characteristic
112	C₂×C₈	D₂₁₂	1	1	yes	characteristic
113	C₄⋊C₄	—	53	1	no	—
114	C₄⋊C₄	—	53	1	no	—
115	C₄⋊C₄	—	106	1	no	—
116	C₂²⋊C₄	—	106	1	no	—
117	C₄²	—	106	1	no	—
118	C₄²	C₂²×D₅₃	1	1	yes	characteristic
119	C₂³	—	106	1	no	—
120	Q₈	—	53	2	no	—
121	Q₈	—	106	1	no	—
122	D₄	—	106	1	no	—
123	D₄	—	106	1	no	—
124	C₂×C₄	—	106	1	no	—
125	C₂×C₄	—	106	1	no	—
126	C₂×C₄	—	106	1	no	—
127	C₂×C₄	—	212	1	no	—
128	C₂×C₄	—	212	1	no	—
129	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
130	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
131	C₂×C₄	#11719945	1	1	yes	characteristic
132	C₈	—	2	2	no	—
133	C₈	—	2	1	no	—
134	C₈	—	2	1	no	—
135	C₈	—	2	1	no	—
136	C₈	—	2	1	no	—
137	C₈	—	212	1	no	—
138	C₈	—	212	1	no	—
139	C₂²	—	212	1	no	—
140	C₂²	—	212	1	no	—
141	C₂²	—	212	1	no	—
142	C₂²	C₄×D₂₁₂	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	106	2	no	—
145	C₄	—	212	1	no	—
146	C₄	—	212	1	no	—
147	C₄	M₁₆×D₅₃	1	1	yes	characteristic
148	C₄	#13133225	1	1	yes	characteristic
149	C₄	#13133245	1	1	yes	characteristic
150	C₄	#13133250	1	1	yes	characteristic
151	C₂	—	424	1	no	—
152	C₂	#16810696	1	1	yes	characteristic
153	C₂	#16810725	1	1	yes	characteristic
154	C₂	#16810741	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈×Dic₁₀₆

C₂

yes

maximal, characteristic, complemented

C₄×#11719913

C₂

yes

maximal, characteristic

#16810702

C₂

yes

maximal, characteristic

#16810376

C₂

yes

maximal, characteristic, complemented

#16810355

C₂

yes

maximal, characteristic, complemented

#16810352

C₂

yes

maximal, characteristic

C₅₃×#322

C₂

yes

maximal, characteristic, complemented

C₂×#11719913

C₄

yes

characteristic

C₂×#11719912

—

C₄×Dic₁₀₆

C₂²

yes

characteristic

C₄×D₂₁₂

C₂²

yes

characteristic

#13133149

C₄

yes

characteristic

C₄⋊#88322

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

characteristic

#13133146

—

#13133142

C₄

yes

characteristic

#13133139

C₄

yes

characteristic

#13133129

—

C₈×Dic₅₃

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×D₂₁₂

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

#11719921

—

C₄⋊Dic₅₃

C₂×C₄

yes

characteristic

C₄⋊Dic₅₃

C₂×C₄

yes

characteristic

#11719919

—

C₄×Dic₅₃

—

—

—

—

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

M₁₆

yes

complemented

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×M₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

C₈×Q₈

—

C₄×SD₁₆

—

#436

—

#335

—

Q₈⋊C₈

—

D₄⋊C₈

—

#322

D₅₃

yes

characteristic, complemented

C₅₃

#1588

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×M₁₆

—

106

—

C₄×Q₈

—

C₄×D₄

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

#103

—

#102

—

C₂²⋊C₈

—

106

—

C₄×C₈

—

106

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₄×C₈

C₂×D₅₃

yes

characteristic

100

C₄×C₈

C₂×D₅₃

yes

characteristic

101

C₂×Q₈

—

102

C₂×D₄

—

103

C₂²×C₄

—

106

—

104

SD₁₆

—

106

—

105

M₁₆

—

212

—

106

C₂×C₈

—

107

C₂×C₈

—

106

—

108

C₂×C₈

—

106

—

109

C₂×C₈

C₄×D₅₃

yes

characteristic

110

C₂×C₈

C₄×D₅₃

yes

characteristic

111

C₂×C₈

D₂₁₂

yes

characteristic

112

C₂×C₈

D₂₁₂

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

106

—

116

C₂²⋊C₄

—

106

—

117

C₄²

—

106

—

118

C₄²

C₂²×D₅₃

yes

characteristic

119

C₂³

—

106

—

120

Q₈

—

121

Q₈

—

106

—

122

D₄

—

106

—

123

D₄

—

106

—

124

C₂×C₄

—

106

—

125

C₂×C₄

—

106

—

126

C₂×C₄

—

106

—

127

C₂×C₄

—

212

—

128

C₂×C₄

—

212

—

129

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

130

C₂×C₄

C₂×D₂₁₂

yes

characteristic

131

C₂×C₄

#11719945

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

212

—

138

C₈

—

212

—

139

C₂²

—

212

—

140

C₂²

—

212

—

141

C₂²

—

212

—

142

C₂²

C₄×D₂₁₂

yes

characteristic

143

C₄

—

144

C₄

—

106

—

145

C₄

—

212

—

146

C₄

—

212

—

147

C₄

M₁₆×D₅₃

yes

characteristic

148

C₄

#13133225

yes

characteristic

149

C₄

#13133245

yes

characteristic

150

C₄

#13133250

yes

characteristic

151

C₂

—

424

—

152

C₂

#16810696

yes

characteristic

153

C₂

#16810725

yes

characteristic

154

C₂

#16810741

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#322	C₂
socle	C₂²×C₅₃	C₄×D₄
hypercenter	#322	D₅₃
hypocenter	C₅₃	#1588
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

subgroup

G/H

center

C₂×C₄

C₂×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#322

C₂

socle

C₂²×C₅₃

C₄×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

N	H ≅ G/N	type
1. C₈×Dic₁₀₆	C₂	semidirect
4. #16810376	C₂	semidirect
5. #16810355	C₂	semidirect
7. C₅₃×#322	C₂	semidirect
53. Dic₁₀₆	M₁₆	semidirect
84. #322	D₅₃	semidirect
85. C₅₃	#1588	semidirect

H ≅ G/N

type

1. C₈×Dic₁₀₆

C₂

semidirect

4. #16810376

C₂

semidirect

5. #16810355

C₂

semidirect

7. C₅₃×#322

C₂

semidirect

53. Dic₁₀₆

M₁₆

semidirect

84. #322

D₅₃

semidirect

85. C₅₃

#1588

semidirect

Subgroups — #20273595

Conjugacy classes of subgroups

Complements