Group #20279704 — subgroups

Subgroups — #20279704

Order 6784.

540 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 189 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×#13133215	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
3	#16810554	C₂	1	4	yes	maximal
4	#16810530	C₂	1	2	yes	maximal
5	#16810519	C₂	1	4	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	1	yes	characteristic
9	#13133215	—	2	8	no	—
10	C₄×Dic₁₀₆	—	2	4	no	—
11	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
13	C₂×#11719919	C₂²	1	8	yes	—
14	C₂×#11719919	C₂²	1	2	yes	—
15	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
16	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
17	#13133124	C₂²	1	8	yes	—
18	#13133124	C₂²	1	2	yes	—
19	C₂²×Q₈×C₅₃	C₂²	1	2	yes	—
20	C₄⋊Q₈×C₅₃	—	2	4	no	—
21	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
23	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×Q₈×C₅₃	—	2	4	no	—
25	C₂×Q₈×C₅₃	—	2	4	no	—
26	C₂×Q₈×C₅₃	—	2	4	no	—
27	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
28	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
31	C₂²×Dic₅₃	C₂³	1	2	yes	—
32	C₂²×Dic₅₃	C₂³	1	2	yes	—
33	C₂²×Dic₅₃	C₂³	1	4	yes	—
34	C₂×Dic₁₀₆	—	2	2	no	—
35	C₂×Dic₁₀₆	—	2	2	no	—
36	C₂×Dic₁₀₆	D₄	1	4	yes	—
37	C₄⋊C₄×C₅₃	—	2	4	no	—
38	C₄⋊C₄×C₅₃	—	2	4	no	—
39	C₄²×C₅₃	D₄	1	4	yes	—
40	C₄⋊Dic₅₃	—	2	2	no	—
41	C₄⋊Dic₅₃	Q₈	1	8	yes	complemented
42	#11719919	—	2	4	no	—
43	#11719919	—	2	16	no	—
44	C₄×Dic₅₃	—	2	4	no	—
45	C₄×Dic₅₃	—	2	4	no	—
46	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
47	Q₈×C₅₃	—	4	8	no	—
48	C₂×C₄×C₅₃	—	2	2	no	—
49	C₂×C₄×C₅₃	—	2	4	no	—
50	C₂×C₄×C₅₃	—	2	4	no	—
51	C₂×C₄×C₅₃	—	2	4	no	—
52	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	—
57	C₂×C₄×C₅₃	#41	1	2	yes	—
58	C₂×Dic₅₃	—	2	2	no	—
59	C₂×Dic₅₃	—	2	4	no	—
60	C₂×Dic₅₃	—	2	2	no	—
61	C₂×Dic₅₃	—	2	4	no	—
62	C₂×Dic₅₃	—	2	2	no	—
63	C₂×Dic₅₃	—	2	4	no	—
64	C₂×Dic₅₃	C₂×D₄	1	4	yes	—
65	C₂×Dic₅₃	C₂×Q₈	1	8	yes	—
66	Dic₁₀₆	—	2	8	no	—
67	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
68	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
69	C₂²×C₅₃	C₂²×Q₈	1	2	yes	—
70	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
71	C₂²×C₅₃	2₋¹⁺⁴	1	2	yes	—
72	C₄×C₅₃	—	2	2	no	—
73	C₄×C₅₃	—	4	4	no	—
74	C₄×C₅₃	#121	1	4	yes	—
75	C₄×C₅₃	C₄⋊Q₈	1	4	yes	complemented
76	Dic₅₃	—	2	4	no	—
77	Dic₅₃	—	2	8	no	—
78	Dic₅₃	—	4	2	no	—
79	#2546	—	53	1	no	maximal, Sylow
80	C₂×C₅₃	C₂×#121	1	1	yes	characteristic
81	C₂×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
82	C₂×C₅₃	#536	1	1	yes	characteristic
83	C₂×C₅₃	D₄×Q₈	1	2	yes	—
84	C₂×C₅₃	#557	1	2	yes	—
85	C₂×C₄⋊Q₈	—	53	2	no	—
86	C₂×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
87	C₂×C₄×Q₈	—	53	1	no	—
88	#398	—	53	4	no	—
89	#391	—	53	2	no	—
90	#389	—	53	4	no	—
91	C₄×C₄⋊C₄	—	53	1	no	—
92	C₅₃	#2546	1	1	yes	characteristic, complemented, Sylow
93	C₂²×Q₈	—	53	1	no	—
94	C₂²×Q₈	C₂×D₅₃	1	2	yes	—
95	C₄⋊Q₈	—	2	4	no	—
96	C₄⋊Q₈	—	106	8	no	—
97	C₄×Q₈	—	106	4	no	—
98	C₂×C₄⋊C₄	—	53	2	no	—
99	C₂×C₄⋊C₄	—	53	2	no	—
100	C₂×C₄⋊C₄	—	53	1	no	—
101	C₂×C₄⋊C₄	—	53	8	no	—
102	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
103	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
104	C₂×C₄²	—	53	2	no	—
105	C₂×C₄²	—	53	2	no	—
106	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
107	#95	—	53	2	no	—
108	#95	—	53	8	no	—
109	C₂×Q₈	—	2	4	no	—
110	C₂×Q₈	—	2	4	no	—
111	C₂×Q₈	—	2	4	no	—
112	C₂×Q₈	—	53	4	no	—
113	C₂×Q₈	—	106	2	no	—
114	C₂×Q₈	—	106	2	no	—
115	C₂²×C₄	—	53	4	no	—
116	C₂²×C₄	—	53	2	no	—
117	C₂²×C₄	—	53	2	no	—
118	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
119	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
120	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
121	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
122	C₄⋊C₄	—	2	4	no	—
123	C₄⋊C₄	—	2	4	no	—
124	C₄⋊C₄	—	53	8	no	—
125	C₄⋊C₄	—	106	16	no	—
126	C₄⋊C₄	—	106	4	no	—
127	C₄⋊C₄	—	106	2	no	—
128	C₄²	—	106	4	no	—
129	C₄²	—	106	4	no	—
130	C₄²	C₂²⋊D₅₃	1	4	yes	—
131	C₂³	C₂³×D₅₃	1	1	yes	characteristic
132	Q₈	—	4	8	no	—
133	Q₈	—	106	8	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	53	4	no	—
139	C₂×C₄	—	53	8	no	—
140	C₂×C₄	—	106	4	no	—
141	C₂×C₄	—	106	2	no	—
142	C₂×C₄	—	106	2	no	—
143	C₂×C₄	—	106	4	no	—
144	C₂×C₄	—	106	2	no	—
145	C₂×C₄	—	106	4	no	—
146	C₂×C₄	D₄×D₅₃	1	2	yes	—
147	C₂×C₄	#11719947	1	2	yes	—
148	C₂×C₄	Q₈×D₅₃	1	4	yes	—
149	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
150	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
151	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
152	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
153	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
154	C₂²	C₂×Q₈×D₅₃	1	2	yes	—
155	C₂²	C₂×#11719947	1	1	yes	characteristic
156	C₂²	#13133265	1	2	yes	—
157	C₄	—	2	2	no	—
158	C₄	—	4	4	no	—
159	C₄	—	106	8	no	—
160	C₄	—	106	4	no	—
161	C₄	—	212	2	no	—
162	C₄	#13133188	1	4	yes	—
163	C₄	#13133215	1	4	yes	complemented
164	C₂	C₂×#13133188	1	1	yes	characteristic
165	C₂	C₂×#13133215	1	1	yes	characteristic
166	C₂	#16811197	1	1	yes	characteristic
167	C₂	#16811288	1	2	yes	—
168	C₂	#16811348	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133215

C₂

yes

maximal

C₂×C₄×Dic₁₀₆

C₂

yes

maximal, characteristic

#16810554

C₂

yes

maximal

#16810530

C₂

yes

maximal

#16810519

C₂

yes

maximal

C₄×C₄⋊Dic₅₃

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

characteristic

#13133215

—

C₄×Dic₁₀₆

—

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719919

C₂²

yes

—

C₂×#11719919

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

#13133124

C₂²

yes

—

#13133124

C₂²

yes

—

C₂²×Q₈×C₅₃

C₂²

yes

—

C₄⋊Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

D₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

Q₈

yes

complemented

#11719919

—

#11719919

—

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×D₄

yes

—

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

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

C₂×Q₈

yes

—

Dic₁₀₆

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂²×C₅₃

C₂²×Q₈

yes

—

C₂²×C₅₃

C₂×#41

yes

characteristic

C₂²×C₅₃

2₋¹⁺⁴

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#121

yes

—

C₄×C₅₃

C₄⋊Q₈

yes

complemented

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₅₃

#536

yes

characteristic

C₂×C₅₃

D₄×Q₈

yes

—

C₂×C₅₃

#557

yes

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

D₅₃

yes

characteristic

C₂×C₄×Q₈

—

#398

—

#391

—

#389

—

C₄×C₄⋊C₄

—

C₅₃

#2546

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

C₂×D₅₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

C₄×Q₈

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

100

C₂×C₄⋊C₄

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

103

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

104

C₂×C₄²

—

105

C₂×C₄²

—

106

C₂×C₄²

C₂×D₅₃

yes

characteristic

107

#95

—

108

#95

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×Q₈

—

106

—

114

C₂×Q₈

—

106

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

C₂²×D₅₃

yes

—

119

C₂²×C₄

C₂²×D₅₃

yes

characteristic

120

C₂²×C₄

C₂²×D₅₃

yes

characteristic

121

C₂²×C₄

C₂²×D₅₃

yes

characteristic

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

106

—

126

C₄⋊C₄

—

106

—

127

C₄⋊C₄

—

106

—

128

C₄²

—

106

—

129

C₄²

—

106

—

130

C₄²

C₂²⋊D₅₃

yes

—

131

C₂³

C₂³×D₅₃

yes

characteristic

132

Q₈

—

133

Q₈

—

106

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

106

—

141

C₂×C₄

—

106

—

142

C₂×C₄

—

106

—

143

C₂×C₄

—

106

—

144

C₂×C₄

—

106

—

145

C₂×C₄

—

106

—

146

C₂×C₄

D₄×D₅₃

yes

—

147

C₂×C₄

#11719947

yes

—

148

C₂×C₄

Q₈×D₅₃

yes

—

149

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

150

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

151

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

152

C₂²

C₂×D₄×D₅₃

yes

characteristic

153

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

154

C₂²

C₂×Q₈×D₅₃

yes

—

155

C₂²

C₂×#11719947

yes

characteristic

156

C₂²

#13133265

yes

—

157

C₄

—

158

C₄

—

159

C₄

—

106

—

160

C₄

—

106

—

161

C₄

—

212

—

162

C₄

#13133188

yes

—

163

C₄

#13133215

yes

complemented

164

C₂

C₂×#13133188

yes

characteristic

165

C₂

C₂×#13133215

yes

characteristic

166

C₂

#16811197

yes

characteristic

167

C₂

#16811288

yes

—

168

C₂

#16811348

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	#2546
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
41. C₄⋊Dic₅₃	Q₈	semidirect
75. C₄×C₅₃	C₄⋊Q₈	semidirect
92. C₅₃	#2546	semidirect
163. C₄	#13133215	semidirect

H ≅ G/N

type

41. C₄⋊Dic₅₃

Q₈

semidirect

75. C₄×C₅₃

C₄⋊Q₈

semidirect

92. C₅₃

#2546

semidirect

163. C₄

#13133215

semidirect

Subgroups — #20279704

Conjugacy classes of subgroups

Complements