Group #20290894 — subgroups

Subgroups — #20290894

Order 6784.

292 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 88 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	C₄×C₄⋊#6157	C₂	1	1	yes	maximal, characteristic
3	#16811496	C₂	1	4	yes	maximal
4	C₄⋊Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×D₅₃⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊#6157	C₂²	1	2	yes	—
7	C₂×C₄×#6157	—	2	1	no	—
8	#13133284	—	2	1	no	—
9	C₄⋊#88324	—	2	1	no	—
10	(C₂×#6157)⋊C₄	—	2	1	no	—
11	C₂×Q₈×D₅₃	—	2	1	no	—
12	#13133217	C₄	1	1	yes	characteristic, complemented
13	#13133216	—	2	1	no	—
14	#13133216	C₄	1	2	yes	complemented
15	#13133215	—	2	1	no	—
16	C₄⋊C₄×D₅₃	—	2	1	no	—
17	C₄⋊C₄×D₅₃	C₂²	1	2	yes	—
18	C₄²×D₅₃	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic, complemented
20	C₂²×#6157	—	2	2	no	—
21	C₂×Q₈×C₅₃	—	2	1	no	—
22	Q₈×D₅₃	—	4	2	no	—
23	Q₈×D₅₃	—	4	2	no	—
24	C₂×C₄×D₅₃	—	2	2	no	—
25	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
26	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
27	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
28	C₂×Dic₁₀₆	—	2	1	no	—
29	C₂×Dic₁₀₆	—	2	2	no	—
30	C₂×#88324	—	2	2	no	—
31	C₄⋊#6157	—	2	2	no	—
32	C₄⋊#6157	Q₈	1	4	yes	complemented
33	C₄×#6157	—	4	2	no	—
34	D₅₃⋊C₈	—	2	4	no	—
35	C₄⋊C₄×C₅₃	—	2	1	no	—
36	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
37	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₄⋊Dic₅₃	—	2	1	no	—
39	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
40	#11719919	—	2	2	no	—
41	#11719919	—	2	2	no	—
42	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄×Dic₅₃	D₄	1	1	yes	characteristic
44	C₄×Dic₅₃	D₄	1	1	yes	characteristic
45	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×#6157	—	2	4	no	—
47	C₂×#6157	—	4	2	no	—
48	C₂×#6157	—	4	2	no	—
49	Q₈×C₅₃	—	4	2	no	—
50	C₂×C₄×C₅₃	—	2	2	no	—
51	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×Dic₅₃	—	2	2	no	—
55	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
58	C₄×D₅₃	—	2	2	no	—
59	C₄×D₅₃	—	4	2	no	—
60	C₄×D₅₃	—	4	2	no	—
61	C₄×D₅₃	D₈	1	2	yes	—
62	C₄×D₅₃	SD₁₆	1	2	yes	—
63	C₄×D₅₃	C₂×Q₈	1	2	yes	—
64	C₄×D₅₃	#41	1	2	yes	—
65	Dic₁₀₆	—	4	2	no	—
66	Dic₁₀₆	—	4	4	no	—
67	#88324	—	4	2	no	—
68	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×D₅₃	C₂²⋊Q₈	1	2	yes	—
70	C₂×D₅₃	C₂×D₈	1	1	yes	characteristic
71	C₂×D₅₃	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×D₅₃	#137	1	2	yes	—
73	#6157	—	4	4	no	—
74	C₄×C₅₃	—	2	1	no	—
75	C₄×C₅₃	—	4	2	no	—
76	C₄×C₅₃	#102	1	2	yes	—
77	C₄×C₅₃	C₄×Q₈	1	2	yes	—
78	Dic₅₃	—	2	1	no	—
79	Dic₅₃	—	4	2	no	—
80	Dic₅₃	#102	1	2	yes	—
81	Dic₅₃	#124	1	1	yes	characteristic
82	Dic₅₃	C₄⋊Q₈	1	1	yes	characteristic
83	#1988	—	53	1	no	maximal, Sylow
84	C₂×C₅₃	#391	1	1	yes	characteristic
85	C₂×C₅₃	C₂×#102	1	1	yes	characteristic
86	C₂×C₅₃	#419	1	1	yes	characteristic
87	D₅₃	#474	1	2	yes	—
88	D₅₃	#475	1	2	yes	—
89	C₂×C₄⋊Q₈	—	53	1	no	—
90	C₂×C₄⋊C₈	—	53	1	no	—
91	C₄×C₄⋊C₄	—	53	1	no	—
92	#340	—	53	4	no	—
93	C₅₃	#1988	1	1	yes	characteristic, complemented, Sylow
94	C₂²×Q₈	—	106	1	no	—
95	C₂²×C₈	—	53	2	no	—
96	C₄⋊Q₈	—	53	2	no	—
97	C₄⋊Q₈	—	53	1	no	—
98	C₄⋊Q₈	—	106	1	no	—
99	C₄⋊Q₈	—	106	1	no	—
100	C₄⋊Q₈	#6157	1	1	yes	characteristic, complemented
101	C₂×C₄⋊C₄	—	53	2	no	—
102	C₂×C₄⋊C₄	—	53	2	no	—
103	C₂×C₄⋊C₄	—	106	1	no	—
104	C₂×C₄²	—	53	1	no	—
105	C₂×C₄²	—	106	1	no	—
106	C₄⋊C₈	—	106	1	no	—
107	C₄⋊C₈	—	106	1	no	—
108	#95	—	106	1	no	—
109	C₂×Q₈	—	2	1	no	—
110	C₂×Q₈	—	106	2	no	—
111	C₂×Q₈	—	106	1	no	—
112	C₂×Q₈	—	212	2	no	—
113	C₂×Q₈	—	212	2	no	—
114	C₂²×C₄	—	53	1	no	—
115	C₂²×C₄	—	53	1	no	—
116	C₂²×C₄	—	53	1	no	—
117	C₂²×C₄	—	106	2	no	—
118	C₂²×C₄	—	106	2	no	—
119	C₂×C₈	—	106	2	no	—
120	C₂×C₈	—	106	4	no	—
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	53	4	no	—
123	C₄⋊C₄	—	53	2	no	—
124	C₄⋊C₄	—	106	2	no	—
125	C₄⋊C₄	—	106	2	no	—
126	C₄⋊C₄	—	106	2	no	—
127	C₄⋊C₄	—	106	1	no	—
128	C₄⋊C₄	C₂×#6157	1	2	yes	—
129	C₄²	—	53	1	no	—
130	C₄²	—	53	1	no	—
131	C₄²	—	53	1	no	—
132	C₄²	—	212	2	no	—
133	C₄²	C₂×#6157	1	1	yes	characteristic
134	C₂³	—	53	1	no	—
135	Q₈	—	4	2	no	—
136	Q₈	—	212	4	no	—
137	Q₈	—	212	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	53	1	no	—
140	C₂×C₄	—	53	1	no	—
141	C₂×C₄	—	53	2	no	—
142	C₂×C₄	—	53	2	no	—
143	C₂×C₄	—	53	2	no	—
144	C₂×C₄	—	53	2	no	—
145	C₂×C₄	—	53	1	no	—
146	C₂×C₄	—	106	2	no	—
147	C₂×C₄	—	106	4	no	—
148	C₂×C₄	—	106	2	no	—
149	C₂×C₄	—	212	2	no	—
150	C₂×C₄	—	212	2	no	—
151	C₂×C₄	—	212	2	no	—
152	C₂×C₄	—	212	2	no	—
153	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
154	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
155	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
156	C₈	—	212	2	no	—
157	C₂²	—	53	2	no	—
158	C₂²	—	53	1	no	—
159	C₂²	—	53	1	no	—
160	C₂²	—	53	2	no	—
161	C₂²	C₂×C₂²⋊#6157	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	4	2	no	—
164	C₄	—	53	1	no	—
165	C₄	—	53	1	no	—
166	C₄	—	53	2	no	—
167	C₄	—	106	1	no	—
168	C₄	—	212	4	no	—
169	C₄	—	212	2	no	—
170	C₄	#13133279	1	2	yes	—
171	C₄	Q₈×#6157	1	2	yes	—
172	C₂	—	53	2	no	—
173	C₂	—	53	2	no	—
174	C₂	(C₂×#6157)⋊Q₈	1	1	yes	characteristic
175	C₂	C₂×#13133279	1	1	yes	characteristic
176	C₂	#16811616	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₅₃⋊(C₄⋊C₈)

C₂

yes

maximal, characteristic

C₄×C₄⋊#6157

C₂

yes

maximal, characteristic

#16811496

C₂

yes

maximal

C₄⋊Q₈×D₅₃

C₂

yes

maximal, characteristic

C₂×D₅₃⋊C₈

C₂²

yes

—

C₂×C₄⋊#6157

C₂²

yes

—

C₂×C₄×#6157

—

#13133284

—

C₄⋊#88324

—

(C₂×#6157)⋊C₄

—

C₂×Q₈×D₅₃

—

#13133217

C₄

yes

characteristic, complemented

#13133216

—

#13133216

C₄

yes

complemented

#13133215

—

C₄⋊C₄×D₅₃

—

C₄⋊C₄×D₅₃

C₂²

yes

—

C₄²×D₅₃

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₄

yes

characteristic, complemented

C₂²×#6157

—

C₂×Q₈×C₅₃

—

Q₈×D₅₃

—

Q₈×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×#88324

—

C₄⋊#6157

—

C₄⋊#6157

Q₈

yes

complemented

C₄×#6157

—

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

C₂×C₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₂²×D₅₃

C₂×D₄

yes

—

—

—

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

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

D₈

yes

—

C₄×D₅₃

SD₁₆

yes

—

C₄×D₅₃

C₂×Q₈

yes

—

C₄×D₅₃

#41

yes

—

Dic₁₀₆

—

Dic₁₀₆

—

#88324

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅₃

C₂²⋊Q₈

yes

—

C₂×D₅₃

C₂×D₈

yes

characteristic

C₂×D₅₃

C₂×SD₁₆

yes

characteristic

C₂×D₅₃

#137

yes

—

#6157

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#102

yes

—

C₄×C₅₃

C₄×Q₈

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#102

yes

—

Dic₅₃

#124

yes

characteristic

Dic₅₃

C₄⋊Q₈

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

—

D₅₃

#475

yes

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

—

C₄×C₄⋊C₄

—

#340

—

C₅₃

#1988

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

106

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

C₄⋊Q₈

—

106

—

100

C₄⋊Q₈

#6157

yes

characteristic, complemented

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

106

—

104

C₂×C₄²

—

105

C₂×C₄²

—

106

—

106

C₄⋊C₈

—

106

—

107

C₄⋊C₈

—

106

—

108

#95

—

106

—

109

C₂×Q₈

—

110

C₂×Q₈

—

106

—

111

C₂×Q₈

—

106

—

112

C₂×Q₈

—

212

—

113

C₂×Q₈

—

212

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

106

—

118

C₂²×C₄

—

106

—

119

C₂×C₈

—

106

—

120

C₂×C₈

—

106

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

106

—

125

C₄⋊C₄

—

106

—

126

C₄⋊C₄

—

106

—

127

C₄⋊C₄

—

106

—

128

C₄⋊C₄

C₂×#6157

yes

—

129

C₄²

—

130

C₄²

—

131

C₄²

—

132

C₄²

—

212

—

133

C₄²

C₂×#6157

yes

characteristic

134

C₂³

—

135

Q₈

—

136

Q₈

—

212

—

137

Q₈

—

212

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

106

—

147

C₂×C₄

—

106

—

148

C₂×C₄

—

106

—

149

C₂×C₄

—

212

—

150

C₂×C₄

—

212

—

151

C₂×C₄

—

212

—

152

C₂×C₄

—

212

—

153

C₂×C₄

C₂²⋊#6157

yes

characteristic

154

C₂×C₄

C₂²⋊#6157

yes

characteristic

155

C₂×C₄

C₂²×#6157

yes

characteristic

156

C₈

—

212

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

C₂×C₂²⋊#6157

yes

characteristic

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

—

106

—

168

C₄

—

212

—

169

C₄

—

212

—

170

C₄

#13133279

yes

—

171

C₄

Q₈×#6157

yes

—

172

C₂

—

173

C₂

—

174

C₂

(C₂×#6157)⋊Q₈

yes

characteristic

175

C₂

C₂×#13133279

yes

characteristic

176

C₂

#16811616

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	#1988
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×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 ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
12. #13133217	C₄	semidirect
14. #13133216	C₄	semidirect
19. C₄⋊Q₈×C₅₃	C₄	semidirect
32. C₄⋊#6157	Q₈	semidirect
93. C₅₃	#1988	semidirect
100. C₄⋊Q₈	#6157	semidirect

H ≅ G/N

type

12. #13133217

C₄

semidirect

14. #13133216

C₄

semidirect

19. C₄⋊Q₈×C₅₃

C₄

semidirect

32. C₄⋊#6157

Q₈

semidirect

93. C₅₃

#1988

semidirect

100. C₄⋊Q₈

#6157

semidirect

Subgroups — #20290894

Conjugacy classes of subgroups

Complements