Group #22608610 — subgroups

Subgroups — #22608610

Order 9344.

240 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 75 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	C₄⋊Q₈×D₇₃	C₂	1	1	yes	maximal, characteristic
2	#17988639	C₂	1	1	yes	maximal, characteristic
3	#17988609	C₂	1	1	yes	maximal, characteristic
4	#17988336	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988335	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988335	C₂	1	1	yes	maximal, characteristic, complemented
7	#333×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₇₃	C₄	1	2	yes	—
9	C₂×#11924011	—	2	1	no	—
10	#14556591	C₂²	1	1	yes	characteristic
11	#14556590	—	2	1	no	—
12	#14556590	—	2	1	no	—
13	#14556589	C₄	1	2	yes	—
14	C₄⋊C₄×D₇₃	—	2	1	no	—
15	C₄⋊C₄×D₇₃	—	2	1	no	—
16	C₄²×D₇₃	C₂²	1	1	yes	characteristic
17	C₄⋊#10596831	C₂²	1	1	yes	characteristic
18	C₄⋊#10596831	C₂²	1	1	yes	characteristic
19	#14556520	—	2	1	no	—
20	#14556503	—	2	1	no	—
21	C₈×Dic₇₃	—	2	1	no	—
22	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
25	C₂×Dic₁₄₆	—	2	2	no	—
26	C₂×Dic₁₄₆	C₂×C₄	1	2	yes	—
27	Q₈×D₇₃	—	4	4	no	—
28	C₂×C₄×D₇₃	—	2	2	no	—
29	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
31	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
32	#11924011	—	4	2	no	—
33	C₂×#10596831	—	2	1	no	—
34	C₂×#10596831	—	2	1	no	—
35	C₄⋊Dic₇₃	—	2	1	no	—
36	C₄⋊Dic₇₃	—	2	1	no	—
37	#11924007	—	2	2	no	—
38	#11924007	—	2	2	no	—
39	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
40	C₄×Dic₇₃	D₄	1	1	yes	characteristic
41	C₄×Dic₇₃	D₄	1	1	yes	characteristic
42	C₂×Q₈×C₇₃	C₂×C₄	1	2	yes	—
43	C₂×C₈×C₇₃	—	2	1	no	—
44	C₂×C₈×C₇₃	—	2	1	no	—
45	C₄⋊C₄×C₇₃	—	2	1	no	—
46	C₄⋊C₄×C₇₃	—	2	1	no	—
47	C₄²×C₇₃	C₂³	1	1	yes	characteristic
48	C₂²×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
49	Q₈×C₇₃	—	4	2	no	—
50	C₂×C₄×C₇₃	—	2	2	no	—
51	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₇₃	—	2	2	no	—
55	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₄×D₇₃	—	2	1	no	—
59	C₄×D₇₃	—	2	1	no	—
60	C₄×D₇₃	—	2	2	no	—
61	C₄×D₇₃	—	4	4	no	—
62	C₄×D₇₃	SD₁₆	1	2	yes	—
63	C₄×D₇₃	Q₁₆	1	2	yes	—
64	Dic₁₄₆	—	4	4	no	—
65	Dic₁₄₆	—	4	2	no	—
66	C₈×C₇₃	—	4	1	no	—
67	C₈×C₇₃	—	4	1	no	—
68	#10596831	—	4	1	no	—
69	#10596831	—	4	1	no	—
70	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×D₇₃	—	2	1	no	—
72	C₂×D₇₃	—	2	1	no	—
73	C₂×D₇₃	#103	1	2	yes	—
74	C₄×C₇₃	—	2	1	no	—
75	C₄×C₇₃	—	4	2	no	—
76	C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
77	C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
78	C₄×C₇₃	#137	1	1	yes	characteristic
79	C₄×C₇₃	#137	1	1	yes	characteristic
80	Dic₇₃	—	2	1	no	—
81	Dic₇₃	—	4	2	no	—
82	Dic₇₃	#100	1	2	yes	—
83	Dic₇₃	#103	1	2	yes	—
84	C₂×C₇₃	C₂×#100	1	1	yes	characteristic
85	C₂×C₇₃	C₂×#103	1	1	yes	characteristic
86	C₂×C₇₃	#419	1	1	yes	characteristic
87	D₇₃	—	2	2	no	—
88	#1548	—	73	1	no	maximal, Sylow
89	C₇₃	#1548	1	1	yes	characteristic, complemented, Sylow
90	C₂×C₄⋊Q₈	—	73	1	no	—
91	#431	—	73	1	no	—
92	#423	—	73	1	no	—
93	#333	—	73	1	no	—
94	#333	—	73	1	no	—
95	#333	—	73	1	no	—
96	#333	D₇₃	1	1	yes	characteristic, complemented
97	C₂²×Q₈	—	73	2	no	—
98	C₂×M₁₆	—	146	1	no	—
99	C₄⋊Q₈	—	73	2	no	—
100	C₄⋊Q₈	—	73	1	no	—
101	C₄⋊Q₈	—	146	1	no	—
102	C₄⋊Q₈	—	146	1	no	—
103	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
104	C₂×C₄⋊C₄	—	146	1	no	—
105	C₂×C₄⋊C₄	—	146	1	no	—
106	C₂×C₄²	—	73	1	no	—
107	C₄⋊C₈	—	73	1	no	—
108	C₄⋊C₈	—	73	1	no	—
109	C₄⋊C₈	—	146	1	no	—
110	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
111	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
112	C₂²⋊C₈	—	146	1	no	—
113	C₄×C₈	—	146	1	no	—
114	C₂×Q₈	—	292	4	no	—
115	C₂×Q₈	—	73	2	no	—
116	C₂×Q₈	—	146	2	no	—
117	C₂×Q₈	C₄×D₇₃	1	2	yes	—
118	C₂²×C₄	—	73	1	no	—
119	C₂²×C₄	—	73	1	no	—
120	C₂²×C₄	—	73	1	no	—
121	C₂²×C₄	—	146	2	no	—
122	M₁₆	—	292	2	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	146	1	no	—
126	C₂×C₈	—	146	1	no	—
127	C₄⋊C₄	—	2	1	no	—
128	C₄⋊C₄	—	2	1	no	—
129	C₄⋊C₄	—	146	2	no	—
130	C₄⋊C₄	—	146	2	no	—
131	C₄⋊C₄	—	146	1	no	—
132	C₄⋊C₄	—	146	1	no	—
133	C₄²	—	73	1	no	—
134	C₄²	—	73	1	no	—
135	C₄²	—	73	1	no	—
136	C₄²	C₂²×D₇₃	1	1	yes	characteristic
137	C₂³	—	73	1	no	—
138	Q₈	—	4	2	no	—
139	Q₈	—	292	2	no	—
140	Q₈	—	292	4	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	292	4	no	—
143	C₂×C₄	—	73	1	no	—
144	C₂×C₄	—	73	1	no	—
145	C₂×C₄	—	73	2	no	—
146	C₂×C₄	—	73	2	no	—
147	C₂×C₄	—	73	1	no	—
148	C₂×C₄	—	146	2	no	—
149	C₂×C₄	—	146	2	no	—
150	C₂×C₄	—	146	1	no	—
151	C₂×C₄	—	146	1	no	—
152	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
153	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
154	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
155	C₈	—	4	1	no	—
156	C₈	—	4	1	no	—
157	C₈	—	292	1	no	—
158	C₈	—	292	1	no	—
159	C₂²	—	73	2	no	—
160	C₂²	—	146	1	no	—
161	C₂²	—	146	1	no	—
162	C₂²	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	4	2	no	—
165	C₄	—	292	2	no	—
166	C₄	—	73	2	no	—
167	C₄	—	73	2	no	—
168	C₄	—	146	1	no	—
169	C₄	SD₁₆×D₇₃	1	1	yes	characteristic
170	C₄	Q₁₆×D₇₃	1	1	yes	characteristic
171	C₄	#14556620	1	1	yes	characteristic
172	C₄	#14556623	1	1	yes	characteristic
173	C₂	—	146	2	no	—
174	C₂	#100×D₇₃	1	1	yes	characteristic
175	C₂	#103×D₇₃	1	1	yes	characteristic
176	C₂	#17988594	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊Q₈×D₇₃

C₂

yes

maximal, characteristic

#17988639

C₂

yes

maximal, characteristic

#17988609

C₂

yes

maximal, characteristic

#17988336

C₂

yes

maximal, characteristic, complemented

#17988335

C₂

yes

maximal, characteristic, complemented

#17988335

C₂

yes

maximal, characteristic, complemented

#333×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₇₃

C₄

yes

—

C₂×#11924011

—

#14556591

C₂²

yes

—

—

C₄

yes

—

C₄⋊C₄×D₇₃

—

C₄⋊C₄×D₇₃

—

C₄²×D₇₃

C₂²

yes

characteristic

C₄⋊#10596831

C₂²

yes

characteristic

C₄⋊#10596831

C₂²

yes

characteristic

#14556520

—

#14556503

—

C₈×Dic₇₃

—

C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

C₂×C₄

yes

—

Q₈×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

C₂×C₄

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

—

—

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

#11924007

—

C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₂×Q₈×C₇₃

C₂×C₄

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

C₂²×D₇₃

C₂²⋊C₄

yes

characteristic

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

SD₁₆

yes

—

C₄×D₇₃

Q₁₆

yes

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

#103

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×SD₁₆

yes

characteristic

C₄×C₇₃

C₂×Q₁₆

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

#100

yes

—

Dic₇₃

#103

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#1548

—

maximal, Sylow

C₇₃

#1548

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

—

—

—

—

—

D₇₃

yes

characteristic, complemented

C₂²×Q₈

—

C₂×M₁₆

—

146

—

C₄⋊Q₈

—

100

C₄⋊Q₈

—

101

C₄⋊Q₈

—

146

—

102

C₄⋊Q₈

—

146

—

103

C₄⋊Q₈

C₂×D₇₃

yes

characteristic

104

C₂×C₄⋊C₄

—

146

—

105

C₂×C₄⋊C₄

—

146

—

106

C₂×C₄²

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

146

—

110

C₄⋊C₈

C₂×D₇₃

yes

characteristic

111

C₄⋊C₈

C₂×D₇₃

yes

characteristic

112

C₂²⋊C₈

—

146

—

113

C₄×C₈

—

146

—

114

C₂×Q₈

—

292

—

115

C₂×Q₈

—

116

C₂×Q₈

—

146

—

117

C₂×Q₈

C₄×D₇₃

yes

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

146

—

122

M₁₆

—

292

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

146

—

126

C₂×C₈

—

146

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

146

—

130

C₄⋊C₄

—

146

—

131

C₄⋊C₄

—

146

—

132

C₄⋊C₄

—

146

—

133

C₄²

—

134

C₄²

—

135

C₄²

—

136

C₄²

C₂²×D₇₃

yes

characteristic

137

C₂³

—

138

Q₈

—

139

Q₈

—

292

—

140

Q₈

—

292

—

141

C₂×C₄

—

142

C₂×C₄

—

292

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

146

—

149

C₂×C₄

—

146

—

150

C₂×C₄

—

146

—

151

C₂×C₄

—

146

—

152

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

153

C₂×C₄

D₄×D₇₃

yes

characteristic

154

C₂×C₄

D₄×D₇₃

yes

characteristic

155

C₈

—

156

C₈

—

157

C₈

—

292

—

158

C₈

—

292

—

159

C₂²

—

160

C₂²

—

146

—

161

C₂²

—

146

—

162

C₂²

C₂²⋊C₄×D₇₃

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

—

292

—

166

C₄

—

167

C₄

—

168

C₄

—

146

—

169

C₄

SD₁₆×D₇₃

yes

characteristic

170

C₄

Q₁₆×D₇₃

yes

characteristic

171

C₄

#14556620

yes

characteristic

172

C₄

#14556623

yes

characteristic

173

C₂

—

146

—

174

C₂

#100×D₇₃

yes

characteristic

175

C₂

#103×D₇₃

yes

characteristic

176

C₂

#17988594

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₄²	C₂²×D₇₃
Fitting	#333×C₇₃	C₂
socle	C₂²×C₇₃	C₂×C₂²⋊C₄
hypercenter	#333	D₇₃
hypocenter	C₇₃	#1548
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₇₃

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₄²

C₂²×D₇₃

Fitting

#333×C₇₃

C₂

socle

C₂²×C₇₃

C₂×C₂²⋊C₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
4. #17988336	C₂	semidirect
5. #17988335	C₂	semidirect
6. #17988335	C₂	semidirect
7. #333×C₇₃	C₂	semidirect
89. C₇₃	#1548	semidirect
96. #333	D₇₃	semidirect

H ≅ G/N

type

4. #17988336

C₂

semidirect

5. #17988335

C₂

semidirect

6. #17988335

C₂

semidirect

7. #333×C₇₃

C₂

semidirect

89. C₇₃

#1548

semidirect

96. #333

D₇₃

semidirect

Subgroups — #22608610

Conjugacy classes of subgroups

Complements