Group #20279927 — subgroups

Subgroups — #20279927

Order 6784.

468 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 181 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	C₂×#13133217	C₂	1	1	yes	maximal, characteristic
2	#16810562	C₂	1	4	yes	maximal
3	#16810530	C₂	1	2	yes	maximal
4	#16810529	C₂	1	4	yes	maximal
5	#16810488	C₂	1	2	yes	maximal
6	C₄×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₀₆	C₂²	1	2	yes	—
9	#13133217	—	2	4	no	—
10	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
13	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
14	#13133124	C₂²	1	8	yes	—
15	#13133124	C₂²	1	8	yes	—
16	#13133124	C₂²	1	2	yes	—
17	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₅₃	—	2	4	no	—
19	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₅₃	—	2	2	no	—
24	C₂×Q₈×C₅₃	—	2	2	no	—
25	C₂×Q₈×C₅₃	Q₈	1	4	yes	complemented
26	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
27	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
29	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
31	C₂²×Dic₅₃	C₂³	1	4	yes	—
32	C₂²×Dic₅₃	C₂³	1	4	yes	—
33	C₂×Dic₁₀₆	—	2	4	no	—
34	C₂×Dic₁₀₆	—	2	8	no	—
35	C₄⋊C₄×C₅₃	—	2	4	no	—
36	C₄⋊C₄×C₅₃	—	2	2	no	—
37	C₄²×C₅₃	—	2	4	no	—
38	C₄²×C₅₃	D₄	1	4	yes	—
39	C₄⋊Dic₅₃	—	2	4	no	—
40	C₄⋊Dic₅₃	—	2	4	no	—
41	C₄⋊Dic₅₃	—	2	4	no	—
42	C₄×Dic₅₃	—	2	4	no	—
43	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
44	Q₈×C₅₃	—	2	8	no	—
45	C₂×C₄×C₅₃	—	2	2	no	—
46	C₂×C₄×C₅₃	—	2	2	no	—
47	C₂×C₄×C₅₃	—	2	2	no	—
48	C₂×C₄×C₅₃	—	2	2	no	—
49	C₂×C₄×C₅₃	—	2	4	no	—
50	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	—
54	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
55	C₂×C₄×C₅₃	#41	1	2	yes	—
56	C₂×C₄×C₅₃	#41	1	4	yes	—
57	C₂×Dic₅₃	—	2	4	no	—
58	C₂×Dic₅₃	—	2	8	no	—
59	C₂×Dic₅₃	—	2	4	no	—
60	C₂×Dic₅₃	—	2	4	no	—
61	C₂×Dic₅₃	—	2	4	no	—
62	Dic₁₀₆	—	4	8	no	—
63	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
64	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
65	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
66	C₂²×C₅₃	C₂×#41	1	2	yes	—
67	C₂²×C₅₃	2₋¹⁺⁴	1	2	yes	—
68	C₄×C₅₃	—	2	4	no	—
69	C₄×C₅₃	—	2	2	no	—
70	C₄×C₅₃	—	4	2	no	—
71	C₄×C₅₃	C₂²⋊Q₈	1	4	yes	—
72	C₄×C₅₃	#124	1	4	yes	—
73	Dic₅₃	—	4	4	no	—
74	Dic₅₃	—	4	4	no	—
75	#2568	—	53	1	no	maximal, Sylow
76	C₂×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
77	C₂×C₅₃	C₂×#124	1	1	yes	characteristic
78	C₂×C₅₃	#536	1	1	yes	characteristic
79	C₂×C₅₃	#556	1	2	yes	—
80	C₂×C₅₃	#557	1	2	yes	—
81	C₂×C₄⋊Q₈	—	53	1	no	—
82	C₂×C₄×Q₈	D₅₃	1	1	yes	characteristic
83	#400	—	53	4	no	—
84	#391	—	53	2	no	—
85	#391	—	53	4	no	—
86	#382	—	53	2	no	—
87	C₄×C₄⋊C₄	—	53	1	no	—
88	C₅₃	#2568	1	1	yes	characteristic, complemented, Sylow
89	C₂²×Q₈	—	53	2	no	—
90	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
91	C₄⋊Q₈	—	106	4	no	—
92	C₄×Q₈	—	2	4	no	—
93	C₂×C₄⋊C₄	—	53	2	no	—
94	C₂×C₄⋊C₄	—	53	2	no	—
95	C₂×C₄⋊C₄	—	53	2	no	—
96	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
97	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
98	C₂×C₄²	—	53	2	no	—
99	C₂×C₄²	C₂×D₅₃	1	2	yes	—
100	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
101	#95	—	53	2	no	—
102	#95	—	53	8	no	—
103	#95	—	53	8	no	—
104	C₂×Q₈	—	2	2	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	106	8	no	—
107	C₂×Q₈	—	106	4	no	—
108	C₂×Q₈	Dic₁₀₆	1	4	yes	complemented
109	C₂²×C₄	—	53	4	no	—
110	C₂²×C₄	—	53	4	no	—
111	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
112	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
113	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
114	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
115	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
116	C₄⋊C₄	—	2	4	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	106	4	no	—
119	C₄⋊C₄	—	106	4	no	—
120	C₄⋊C₄	—	106	4	no	—
121	C₄²	—	2	4	no	—
122	C₄²	—	106	4	no	—
123	C₄²	C₂²⋊D₅₃	1	4	yes	—
124	C₂³	C₂³×D₅₃	1	1	yes	characteristic
125	Q₈	—	2	8	no	—
126	Q₈	—	212	8	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	106	4	no	—
133	C₂×C₄	—	106	4	no	—
134	C₂×C₄	—	106	4	no	—
135	C₂×C₄	—	106	8	no	—
136	C₂×C₄	—	106	4	no	—
137	C₂×C₄	C₂×Dic₁₀₆	1	4	yes	—
138	C₂×C₄	C₂×Dic₁₀₆	1	2	yes	—
139	C₂×C₄	#11719945	1	2	yes	—
140	C₂×C₄	#11719949	1	4	yes	—
141	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
142	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
143	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
144	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
145	C₂²	C₂²×Dic₁₀₆	1	1	yes	characteristic
146	C₂²	C₂×#11719945	1	1	yes	characteristic
147	C₂²	C₂×#11719949	1	2	yes	—
148	C₂²	#13133264	1	2	yes	—
149	C₄	—	2	4	no	—
150	C₄	—	2	2	no	—
151	C₄	—	4	2	no	—
152	C₄	—	212	4	no	—
153	C₄	—	212	4	no	—
154	C₄	#13133196	1	4	yes	—
155	C₄	#13133206	1	4	yes	—
156	C₂	C₂×#13133196	1	1	yes	characteristic
157	C₂	C₂×#13133206	1	1	yes	characteristic
158	C₂	#16811194	1	1	yes	characteristic
159	C₂	#16811346	1	2	yes	—
160	C₂	#16811352	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133217

C₂

yes

maximal, characteristic

#16810562

C₂

yes

maximal

#16810530

C₂

yes

maximal

#16810529

C₂

yes

maximal

#16810488

C₂

yes

maximal

C₄×C₄⋊Dic₅₃

C₂

yes

maximal, characteristic

C₂×C₄×Q₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

—

#13133217

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

#13133124

C₂²

yes

—

#13133124

C₂²

yes

—

#13133124

C₂²

yes

—

C₂²×Q₈×C₅₃

C₂²

yes

characteristic

C₄×Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

Q₈

yes

complemented

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

D₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₂³×C₅₃

C₂⁴

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂²×C₅₃

C₂²×Q₈

yes

characteristic

C₂²×C₅₃

C₂×#41

yes

characteristic

C₂²×C₅₃

C₂×#41

yes

—

C₂²×C₅₃

2₋¹⁺⁴

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

#124

yes

—

Dic₅₃

—

Dic₅₃

—

#2568

—

maximal, Sylow

C₂×C₅₃

C₂×C₂²⋊Q₈

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#557

yes

—

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

D₅₃

yes

—

—

—

—

C₄×C₄⋊C₄

—

C₅₃

#2568

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

C₂×D₅₃

yes

characteristic

C₄⋊Q₈

—

106

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

—

100

C₂×C₄²

C₂×D₅₃

yes

characteristic

101

#95

—

102

#95

—

103

#95

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

106

—

107

C₂×Q₈

—

106

—

108

C₂×Q₈

Dic₁₀₆

yes

complemented

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂²×D₅₃

yes

—

112

C₂²×C₄

C₂²×D₅₃

yes

—

113

C₂²×C₄

C₂²×D₅₃

yes

characteristic

114

C₂²×C₄

C₂²×D₅₃

yes

characteristic

115

C₂²×C₄

C₂²×D₅₃

yes

characteristic

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

106

—

119

C₄⋊C₄

—

106

—

120

C₄⋊C₄

—

106

—

121

C₄²

—

122

C₄²

—

106

—

123

C₄²

C₂²⋊D₅₃

yes

—

124

C₂³

C₂³×D₅₃

yes

characteristic

125

Q₈

—

126

Q₈

—

212

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

106

—

133

C₂×C₄

—

106

—

134

C₂×C₄

—

106

—

135

C₂×C₄

—

106

—

136

C₂×C₄

—

106

—

137

C₂×C₄

C₂×Dic₁₀₆

yes

—

138

C₂×C₄

C₂×Dic₁₀₆

yes

—

139

C₂×C₄

#11719945

yes

—

140

C₂×C₄

#11719949

yes

—

141

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

142

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

143

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

144

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

145

C₂²

C₂²×Dic₁₀₆

yes

characteristic

146

C₂²

C₂×#11719945

yes

characteristic

147

C₂²

C₂×#11719949

yes

—

148

C₂²

#13133264

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

212

—

153

C₄

—

212

—

154

C₄

#13133196

yes

—

155

C₄

#13133206

yes

—

156

C₂

C₂×#13133196

yes

characteristic

157

C₂

C₂×#13133206

yes

characteristic

158

C₂

#16811194

yes

characteristic

159

C₂

#16811346

yes

—

160

C₂

#16811352

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂⁴
Fitting	C₂×C₄×Q₈×C₅₃	C₂
hypercenter	C₂×C₄×Q₈	D₅₃
hypocenter	C₅₃	#2568
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator, socle

C₂³×C₅₃

C₂⁴

Fitting

C₂×C₄×Q₈×C₅₃

C₂

hypercenter

C₂×C₄×Q₈

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
25. C₂×Q₈×C₅₃	Q₈	semidirect
88. C₅₃	#2568	semidirect
108. C₂×Q₈	Dic₁₀₆	semidirect

H ≅ G/N

type

25. C₂×Q₈×C₅₃

Q₈

semidirect

88. C₅₃

#2568

semidirect

108. C₂×Q₈

Dic₁₀₆

semidirect

Subgroups — #20279927

Conjugacy classes of subgroups

Complements