Group #20274064 — subgroups

Subgroups — #20274064

Order 6784.

172 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16810996	C₂	1	1	yes	maximal, characteristic
2	#16810909	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810824	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810368	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810366	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#334	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719913	—	2	1	no	—
9	C₂×#11719913	—	2	1	no	—
10	C₂×#11719913	—	2	1	no	—
11	#13133217	C₂²	1	1	yes	characteristic
12	#13133214	C₂²	1	1	yes	characteristic
13	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
14	#13133149	—	2	1	no	—
15	C₄⋊#88322	C₂²	1	1	yes	characteristic
16	#13133146	—	2	1	no	—
17	#13133142	—	2	1	no	—
18	#13133139	—	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	—	2	1	no	—
24	C₂×D₂₁₂	—	2	1	no	—
25	C₂×D₂₁₂	—	2	1	no	—
26	C₂×Dic₁₀₆	—	2	1	no	—
27	C₂×Dic₁₀₆	D₄	1	1	yes	characteristic
28	C₂×C₈×C₅₃	—	2	1	no	—
29	C₂×C₈×C₅₃	—	2	1	no	—
30	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	C₄⋊Dic₅₃	—	2	1	no	—
34	C₄⋊Dic₅₃	—	2	1	no	—
35	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
36	#11719919	—	2	1	no	—
37	C₄×Dic₅₃	—	2	1	no	—
38	C₂×#88322	—	2	1	no	—
39	#11719913	—	4	2	no	—
40	#11719913	—	4	2	no	—
41	#11719913	—	4	2	no	—
42	C₂²×D₅₃	—	4	1	no	—
43	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
46	C₂×Dic₅₃	—	2	1	no	—
47	C₂×Dic₅₃	—	2	1	no	—
48	C₂×Dic₅₃	—	4	1	no	—
49	D₂₁₂	—	4	1	no	—
50	D₂₁₂	—	4	1	no	—
51	D₂₁₂	—	4	1	no	—
52	D₂₁₂	—	4	1	no	—
53	D₂₁₂	—	8	1	no	—
54	Dic₁₀₆	—	2	1	no	—
55	Dic₁₀₆	—	4	1	no	—
56	Dic₁₀₆	—	4	1	no	—
57	Dic₁₀₆	SD₁₆	1	2	yes	complemented
58	C₈×C₅₃	—	2	1	no	—
59	C₈×C₅₃	—	4	1	no	—
60	C₈×C₅₃	—	4	1	no	—
61	C₈×C₅₃	SD₁₆	1	2	yes	complemented
62	#88322	—	4	1	no	—
63	C₂²×C₅₃	#121	1	1	yes	characteristic
64	C₂×D₅₃	—	8	1	no	—
65	C₂×D₅₃	—	8	1	no	—
66	C₂×D₅₃	—	8	1	no	—
67	C₄×C₅₃	—	2	1	no	—
68	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
69	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
70	C₄×C₅₃	#135	1	1	yes	characteristic
71	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
72	Dic₅₃	—	2	2	no	—
73	Dic₅₃	—	4	1	no	—
74	Dic₅₃	—	8	1	no	—
75	#1666	—	53	1	no	maximal, Sylow
76	C₂×C₅₃	#460	1	1	yes	characteristic
77	C₂×C₅₃	#465	1	1	yes	characteristic
78	C₂×C₅₃	#471	1	1	yes	characteristic
79	D₅₃	—	16	1	no	—
80	#492	—	53	1	no	—
81	#475	—	53	1	no	—
82	#460	—	53	1	no	—
83	C₈×Q₈	—	53	1	no	—
84	#334	D₅₃	1	1	yes	characteristic, complemented
85	#332	—	53	1	no	—
86	#331	—	53	1	no	—
87	C₅₃	#1666	1	1	yes	characteristic, complemented, Sylow
88	C₂×SD₁₆	—	106	1	no	—
89	C₂×SD₁₆	—	106	1	no	—
90	C₂×SD₁₆	—	106	1	no	—
91	C₄⋊Q₈	—	53	1	no	—
92	#127	—	53	1	no	—
93	C₄×Q₈	—	53	1	no	—
94	#106	—	106	1	no	—
95	C₄⋊C₈	—	53	1	no	—
96	C₄⋊C₈	—	106	1	no	—
97	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
98	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
99	#103	—	106	1	no	—
100	#102	—	106	1	no	—
101	C₄×C₈	—	106	1	no	—
102	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
103	C₂×Q₈	—	53	1	no	—
104	C₂×Q₈	—	106	1	no	—
105	C₂×D₄	—	106	1	no	—
106	C₂×D₄	—	106	1	no	—
107	C₂×D₄	—	106	1	no	—
108	SD₁₆	—	212	2	no	—
109	SD₁₆	—	212	2	no	—
110	SD₁₆	—	212	2	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	106	1	no	—
114	C₂×C₈	D₂₁₂	1	1	yes	characteristic
115	C₂×C₈	D₂₁₂	1	1	yes	characteristic
116	C₄⋊C₄	—	53	1	no	—
117	C₄⋊C₄	—	106	1	no	—
118	C₄⋊C₄	—	106	1	no	—
119	C₄⋊C₄	—	106	1	no	—
120	C₄²	—	106	1	no	—
121	C₄²	C₂²×D₅₃	1	1	yes	characteristic
122	C₂³	—	212	1	no	—
123	Q₈	—	53	2	no	—
124	Q₈	—	106	1	no	—
125	Q₈	—	212	1	no	—
126	Q₈	—	212	1	no	—
127	D₄	—	424	1	no	—
128	D₄	—	212	1	no	—
129	D₄	—	212	1	no	—
130	D₄	—	212	1	no	—
131	D₄	—	212	1	no	—
132	C₂×C₄	—	106	1	no	—
133	C₂×C₄	—	106	1	no	—
134	C₂×C₄	—	212	1	no	—
135	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
136	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
137	C₂×C₄	#11719949	1	1	yes	characteristic
138	C₈	—	2	1	no	—
139	C₈	—	4	1	no	—
140	C₈	—	4	1	no	—
141	C₈	—	212	1	no	—
142	C₈	#11719913	1	2	yes	complemented
143	C₂²	—	424	1	no	—
144	C₂²	—	424	1	no	—
145	C₂²	—	424	1	no	—
146	C₂²	#13133192	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	106	2	no	—
149	C₄	—	424	1	no	—
150	C₄	—	212	1	no	—
151	C₄	SD₁₆×D₅₃	1	1	yes	characteristic
152	C₄	C₂×#11719913	1	1	yes	characteristic
153	C₄	#13133238	1	1	yes	characteristic
154	C₄	#13133245	1	1	yes	characteristic
155	C₂	—	848	1	no	—
156	C₂	#16810824	1	1	yes	characteristic
157	C₂	#16810853	1	1	yes	characteristic
158	C₂	#16810887	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810996

C₂

yes

maximal, characteristic

#16810909

C₂

yes

maximal, characteristic, complemented

#16810824

C₂

yes

maximal, characteristic

C₈×Dic₁₀₆

C₂

yes

maximal, characteristic, complemented

#16810368

C₂

yes

maximal, characteristic, complemented

#16810366

C₂

yes

maximal, characteristic

C₅₃×#334

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂²

yes

characteristic

#13133214

C₂²

yes

characteristic

C₄×Dic₁₀₆

C₂²

yes

characteristic

#13133149

—

C₄⋊#88322

C₂²

yes

—

—

—

C₈×Dic₅₃

—

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

D₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

characteristic

#11719919

—

C₄×Dic₅₃

—

—

—

—

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

SD₁₆

yes

complemented

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

SD₁₆

yes

complemented

#88322

—

C₂²×C₅₃

#121

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₄×C₅₃

C₂×SD₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#492

—

#475

—

#460

—

C₈×Q₈

—

#334

D₅₃

yes

characteristic, complemented

#332

—

#331

—

C₅₃

#1666

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

106

—

C₂×SD₁₆

—

106

—

C₂×SD₁₆

—

106

—

C₄⋊Q₈

—

#127

—

C₄×Q₈

—

#106

—

106

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₄⋊C₈

C₂×D₅₃

yes

characteristic

C₄⋊C₈

C₂×D₅₃

yes

characteristic

#103

—

106

—

100

#102

—

106

—

101

C₄×C₈

—

106

—

102

C₄×C₈

C₂×D₅₃

yes

characteristic

103

C₂×Q₈

—

104

C₂×Q₈

—

106

—

105

C₂×D₄

—

106

—

106

C₂×D₄

—

106

—

107

C₂×D₄

—

106

—

108

SD₁₆

—

212

—

109

SD₁₆

—

212

—

110

SD₁₆

—

212

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

106

—

114

C₂×C₈

D₂₁₂

yes

characteristic

115

C₂×C₈

D₂₁₂

yes

characteristic

116

C₄⋊C₄

—

117

C₄⋊C₄

—

106

—

118

C₄⋊C₄

—

106

—

119

C₄⋊C₄

—

106

—

120

C₄²

—

106

—

121

C₄²

C₂²×D₅₃

yes

characteristic

122

C₂³

—

212

—

123

Q₈

—

124

Q₈

—

106

—

125

Q₈

—

212

—

126

Q₈

—

212

—

127

D₄

—

424

—

128

D₄

—

212

—

129

D₄

—

212

—

130

D₄

—

212

—

131

D₄

—

212

—

132

C₂×C₄

—

106

—

133

C₂×C₄

—

106

—

134

C₂×C₄

—

212

—

135

C₂×C₄

C₂×D₂₁₂

yes

characteristic

136

C₂×C₄

D₄×D₅₃

yes

characteristic

137

C₂×C₄

#11719949

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

212

—

142

C₈

#11719913

yes

complemented

143

C₂²

—

424

—

144

C₂²

—

424

—

145

C₂²

—

424

—

146

C₂²

#13133192

yes

characteristic

147

C₄

—

148

C₄

—

106

—

149

C₄

—

424

—

150

C₄

—

212

—

151

C₄

SD₁₆×D₅₃

yes

characteristic

152

C₄

C₂×#11719913

yes

characteristic

153

C₄

#13133238

yes

characteristic

154

C₄

#13133245

yes

characteristic

155

C₂

—

848

—

156

C₂

#16810824

yes

characteristic

157

C₂

#16810853

yes

characteristic

158

C₂

#16810887

yes

characteristic

	subgroup	G/H
center	C₂²	#13133192
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#334	C₂
socle	C₂²×C₅₃	#121
hypercenter	#334	D₅₃
hypocenter	C₅₃	#1666
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

subgroup

G/H

center

C₂²

#13133192

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#334

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #334

N	H ≅ G/N	type
2. #16810909	C₂	semidirect
4. C₈×Dic₁₀₆	C₂	semidirect
5. #16810368	C₂	semidirect
7. C₅₃×#334	C₂	semidirect
57. Dic₁₀₆	SD₁₆	semidirect
61. C₈×C₅₃	SD₁₆	semidirect
84. #334	D₅₃	semidirect
87. C₅₃	#1666	semidirect
142. C₈	#11719913	semidirect

H ≅ G/N

type

2. #16810909

C₂

semidirect

4. C₈×Dic₁₀₆

C₂

semidirect

5. #16810368

C₂

semidirect

7. C₅₃×#334

C₂

semidirect

57. Dic₁₀₆

SD₁₆

semidirect

61. C₈×C₅₃

SD₁₆

semidirect

84. #334

D₅₃

semidirect

87. C₅₃

#1666

semidirect

142. C₈

#11719913

semidirect

Subgroups — #20274064

Conjugacy classes of subgroups

Complements