Group #20290211 — subgroups

Subgroups — D₅₃⋊#332 #20290211

Order 6784.

256 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 84 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	D₅₃⋊(C₄⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	D₅₃⋊(C₄⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#16811482	C₂	1	4	yes	maximal, complemented
4	C₄⋊Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×D₅₃⋊C₈	—	2	1	no	—
6	C₂×D₅₃⋊C₈	—	2	1	no	—
7	#13133284	C₂²	1	2	yes	—
8	#13133284	C₂²	1	2	yes	—
9	C₄⋊#88324	—	2	1	no	—
10	C₄⋊#88324	—	2	1	no	—
11	C₂×Q₈×D₅₃	—	2	1	no	—
12	#13133217	C₄	1	1	yes	characteristic
13	#13133216	—	2	1	no	—
14	#13133216	C₂²	1	2	yes	—
15	#13133215	—	2	1	no	—
16	C₄⋊C₄×D₅₃	—	2	1	no	—
17	C₄⋊C₄×D₅₃	C₄	1	2	yes	—
18	C₄²×D₅₃	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
20	C₂×Q₈×C₅₃	—	2	1	no	—
21	Q₈×D₅₃	—	4	2	no	—
22	Q₈×D₅₃	—	4	2	no	—
23	C₂×C₄×D₅₃	—	2	2	no	—
24	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
26	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
27	C₂×Dic₁₀₆	—	2	1	no	—
28	C₂×Dic₁₀₆	—	2	2	no	—
29	C₂×#88324	—	2	2	no	—
30	C₂×#88324	—	2	2	no	—
31	D₅₃⋊C₈	—	4	2	no	—
32	D₅₃⋊C₈	—	4	2	no	—
33	C₄⋊C₄×C₅₃	—	2	1	no	—
34	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
35	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄⋊Dic₅₃	—	2	1	no	—
37	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
38	#11719919	—	2	2	no	—
39	#11719919	—	2	2	no	—
40	C₄×Dic₅₃	D₄	1	1	yes	characteristic
41	C₄×Dic₅₃	D₄	1	1	yes	characteristic
42	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
43	C₂²×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
44	Q₈×C₅₃	—	4	2	no	—
45	C₂×C₄×C₅₃	—	2	2	no	—
46	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×Dic₅₃	—	2	2	no	—
50	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
53	C₄×D₅₃	—	2	2	no	—
54	C₄×D₅₃	—	4	4	no	—
55	C₄×D₅₃	D₈	1	2	yes	—
56	C₄×D₅₃	SD₁₆	1	2	yes	—
57	C₄×D₅₃	SD₁₆	1	2	yes	—
58	C₄×D₅₃	Q₁₆	1	2	yes	—
59	Dic₁₀₆	—	4	2	no	—
60	Dic₁₀₆	—	4	2	no	—
61	Dic₁₀₆	—	4	2	no	—
62	#88324	—	4	2	no	—
63	#88324	—	4	2	no	—
64	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×D₅₃	#101	1	2	yes	—
66	C₂×D₅₃	#102	1	2	yes	—
67	C₂×D₅₃	#103	1	2	yes	—
68	C₄×C₅₃	—	2	1	no	—
69	C₄×C₅₃	—	4	2	no	—
70	C₄×C₅₃	#102	1	2	yes	—
71	C₄×C₅₃	#103	1	2	yes	—
72	Dic₅₃	—	2	1	no	—
73	Dic₅₃	—	4	2	no	—
74	Dic₅₃	C₂×D₈	1	1	yes	characteristic
75	Dic₅₃	C₂×SD₁₆	1	1	yes	characteristic
76	Dic₅₃	C₂×SD₁₆	1	1	yes	characteristic
77	Dic₅₃	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×#332	—	53	1	no	maximal, Sylow
79	C₂×C₅₃	C₂×#101	1	1	yes	characteristic
80	C₂×C₅₃	C₂×#102	1	1	yes	characteristic
81	C₂×C₅₃	C₂×#103	1	1	yes	characteristic
82	D₅₃	#332	1	4	yes	complemented
83	C₂×C₄⋊Q₈	—	53	1	no	—
84	C₂×C₄⋊C₈	—	53	1	no	—
85	C₂×C₄⋊C₈	—	53	1	no	—
86	#332	—	53	4	no	—
87	C₅₃	C₂×#332	1	1	yes	characteristic, complemented, Sylow
88	C₂²×Q₈	—	106	1	no	—
89	C₂²×C₈	—	106	1	no	—
90	C₂²×C₈	—	106	1	no	—
91	C₄⋊Q₈	—	53	2	no	—
92	C₄⋊Q₈	—	53	1	no	—
93	C₄⋊Q₈	—	106	1	no	—
94	C₄⋊Q₈	—	106	1	no	—
95	C₄⋊Q₈	#6157	1	1	yes	characteristic
96	C₂×C₄⋊C₄	—	53	2	no	—
97	C₂×C₄⋊C₄	—	106	1	no	—
98	C₂×C₄²	—	53	1	no	—
99	C₄⋊C₈	—	53	2	no	—
100	C₄⋊C₈	—	53	2	no	—
101	C₄⋊C₈	—	106	1	no	—
102	C₄⋊C₈	—	106	1	no	—
103	C₂×Q₈	—	2	1	no	—
104	C₂×Q₈	—	106	2	no	—
105	C₂×Q₈	—	106	1	no	—
106	C₂×Q₈	—	212	2	no	—
107	C₂×Q₈	—	212	2	no	—
108	C₂²×C₄	—	53	1	no	—
109	C₂²×C₄	—	53	1	no	—
110	C₂²×C₄	—	53	1	no	—
111	C₂²×C₄	—	106	2	no	—
112	C₂×C₈	—	106	2	no	—
113	C₂×C₈	—	106	2	no	—
114	C₂×C₈	—	212	2	no	—
115	C₂×C₈	—	212	2	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	53	2	no	—
118	C₄⋊C₄	—	106	2	no	—
119	C₄⋊C₄	—	106	2	no	—
120	C₄⋊C₄	—	106	1	no	—
121	C₄⋊C₄	C₂×#6157	1	2	yes	—
122	C₄²	—	53	1	no	—
123	C₄²	—	53	1	no	—
124	C₄²	—	53	1	no	—
125	C₄²	C₂×#6157	1	1	yes	characteristic
126	C₂³	—	53	1	no	—
127	Q₈	—	4	2	no	—
128	Q₈	—	212	2	no	—
129	Q₈	—	212	2	no	—
130	Q₈	—	212	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	53	1	no	—
133	C₂×C₄	—	53	1	no	—
134	C₂×C₄	—	53	2	no	—
135	C₂×C₄	—	53	2	no	—
136	C₂×C₄	—	53	2	no	—
137	C₂×C₄	—	53	2	no	—
138	C₂×C₄	—	53	1	no	—
139	C₂×C₄	—	106	2	no	—
140	C₂×C₄	—	106	2	no	—
141	C₂×C₄	—	212	4	no	—
142	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
143	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
144	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
145	C₈	—	212	2	no	—
146	C₈	—	212	2	no	—
147	C₂²	—	53	2	no	—
148	C₂²	—	53	2	no	—
149	C₂²	—	53	2	no	—
150	C₂²	C₂×C₂²⋊#6157	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	4	2	no	—
153	C₄	—	53	1	no	—
154	C₄	—	53	1	no	—
155	C₄	—	53	1	no	—
156	C₄	—	53	1	no	—
157	C₄	—	106	1	no	—
158	C₄	—	212	2	no	—
159	C₄	#13133279	1	2	yes	—
160	C₄	#13133280	1	2	yes	—
161	C₂	—	53	4	no	—
162	C₂	D₅₃⋊#101	1	1	yes	characteristic
163	C₂	C₂×#13133279	1	1	yes	characteristic
164	C₂	C₂×#13133280	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₅₃⋊(C₄⋊C₈)

C₂

yes

maximal, characteristic

D₅₃⋊(C₄⋊C₈)

C₂

yes

maximal, characteristic

#16811482

C₂

yes

maximal, complemented

C₄⋊Q₈×D₅₃

C₂

yes

maximal, characteristic

C₂×D₅₃⋊C₈

—

C₂×D₅₃⋊C₈

—

#13133284

C₂²

yes

—

#13133284

C₂²

yes

—

C₄⋊#88324

—

C₄⋊#88324

—

C₂×Q₈×D₅₃

—

#13133217

C₄

yes

characteristic

#13133216

—

#13133216

C₂²

yes

—

#13133215

—

C₄⋊C₄×D₅₃

—

C₄⋊C₄×D₅₃

C₄

yes

—

C₄²×D₅₃

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₄

yes

characteristic

C₂×Q₈×C₅₃

—

Q₈×D₅₃

—

Q₈×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×#88324

—

C₂×#88324

—

D₅₃⋊C₈

—

D₅₃⋊C₈

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

#11719919

—

#11719919

—

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

C₂³

yes

characteristic

C₂²×D₅₃

C₂²⋊C₄

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

D₈

yes

—

C₄×D₅₃

SD₁₆

yes

—

C₄×D₅₃

SD₁₆

yes

—

C₄×D₅₃

Q₁₆

yes

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

#88324

—

#88324

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅₃

#101

yes

—

C₂×D₅₃

#102

yes

—

C₂×D₅₃

#103

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#102

yes

—

C₄×C₅₃

#103

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₂×D₈

yes

characteristic

Dic₅₃

C₂×SD₁₆

yes

characteristic

Dic₅₃

C₂×SD₁₆

yes

characteristic

Dic₅₃

C₂×Q₁₆

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

#332

—

C₅₃

C₂×#332

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

106

—

C₂²×C₈

—

106

—

C₂²×C₈

—

106

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

C₄⋊Q₈

—

106

—

C₄⋊Q₈

#6157

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

—

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

106

—

102

C₄⋊C₈

—

106

—

103

C₂×Q₈

—

104

C₂×Q₈

—

106

—

105

C₂×Q₈

—

106

—

106

C₂×Q₈

—

212

—

107

C₂×Q₈

—

212

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

106

—

112

C₂×C₈

—

106

—

113

C₂×C₈

—

106

—

114

C₂×C₈

—

212

—

115

C₂×C₈

—

212

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

106

—

119

C₄⋊C₄

—

106

—

120

C₄⋊C₄

—

106

—

121

C₄⋊C₄

C₂×#6157

yes

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

125

C₄²

C₂×#6157

yes

characteristic

126

C₂³

—

127

Q₈

—

128

Q₈

—

212

—

129

Q₈

—

212

—

130

Q₈

—

212

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

106

—

140

C₂×C₄

—

106

—

141

C₂×C₄

—

212

—

142

C₂×C₄

C₂²⋊#6157

yes

characteristic

143

C₂×C₄

C₂²⋊#6157

yes

characteristic

144

C₂×C₄

C₂²×#6157

yes

characteristic

145

C₈

—

212

—

146

C₈

—

212

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

C₂×C₂²⋊#6157

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

106

—

158

C₄

—

212

—

159

C₄

#13133279

yes

—

160

C₄

#13133280

yes

—

161

C₂

—

162

C₂

D₅₃⋊#101

yes

characteristic

163

C₂

C₂×#13133279

yes

characteristic

164

C₂

C₂×#13133280

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₄⋊Q₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	C₄⋊Q₈	#6157
hypocenter	C₅₃	C₂×#332
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#6157

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6157

Fitting

C₄⋊Q₈×C₅₃

C₄

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

C₄⋊Q₈

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
3. #16811482	C₂	semidirect
82. D₅₃	#332	semidirect
87. C₅₃	C₂×#332	semidirect

H ≅ G/N

type

3. #16811482

C₂

semidirect

82. D₅₃

#332

semidirect

87. C₅₃

C₂×#332

semidirect

Subgroups — D₅₃⋊#332 #20290211

Conjugacy classes of subgroups

Complements

Subgroups — D53⋊#332 #20290211

Conjugacy classes of subgroups

Complements

Subgroups — D₅₃⋊#332 #20290211