Group #22609097 — subgroups

Subgroups — #22609097

Order 9344.

228 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 57 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	#17989252	C₂	1	1	yes	maximal, characteristic
2	#17988939	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988883	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988878	C₂	1	1	yes	maximal, characteristic
5	#17988317	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988314	C₂	1	1	yes	maximal, characteristic
7	#332×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₇₃	—	2	1	no	—
9	#14556591	C₂²	1	1	yes	characteristic
10	#14556590	—	2	1	no	—
11	#14556569	—	2	2	no	—
12	#14556567	—	2	1	no	—
13	C₄×Dic₁₄₆	C₂²	1	1	yes	characteristic
14	C₄×D₂₉₂	C₂²	1	1	yes	characteristic
15	#14556553	—	2	1	no	—
16	#14556525	—	2	1	no	—
17	#14556523	—	2	1	no	—
18	#14556516	—	2	1	no	—
19	#14556514	—	2	2	no	—
20	#14556513	—	2	1	no	—
21	C₄×#10596831	C₂²	1	1	yes	characteristic
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	—	2	1	no	—
27	C₂×Dic₁₄₆	D₄	1	1	yes	characteristic
28	Q₈×D₇₃	—	4	2	no	—
29	Q₈×D₇₃	—	4	2	no	—
30	C₂×D₂₉₂	D₄	1	1	yes	characteristic
31	C₂×C₄×D₇₃	—	2	2	no	—
32	C₂×C₄×D₇₃	—	2	1	no	—
33	C₂×#10596831	—	2	1	no	—
34	C₄⋊Dic₇₃	—	2	1	no	—
35	C₄⋊Dic₇₃	—	2	1	no	—
36	C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
37	C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
38	#11924007	—	2	2	no	—
39	#11924007	—	2	1	no	—
40	#11924005	—	2	2	no	—
41	#11924005	—	2	1	no	—
42	C₄×Dic₇₃	—	2	1	no	—
43	C₂×Q₈×C₇₃	—	2	1	no	—
44	C₂×C₈×C₇₃	—	2	1	no	—
45	C₂×C₈×C₇₃	—	2	1	no	—
46	C₄⋊C₄×C₇₃	—	2	1	no	—
47	C₄⋊C₄×C₇₃	D₄	1	2	yes	—
48	C₄²×C₇₃	C₂³	1	1	yes	characteristic
49	C₂²×D₇₃	—	2	1	no	—
50	Q₈×C₇₃	—	4	1	no	—
51	Q₈×C₇₃	—	4	1	no	—
52	C₂×C₄×C₇₃	—	2	2	no	—
53	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
56	C₂×Dic₇₃	—	2	1	no	—
57	C₂×Dic₇₃	—	2	1	no	—
58	C₂×Dic₇₃	—	2	1	no	—
59	C₂×Dic₇₃	—	4	1	no	—
60	D₂₉₂	—	2	1	no	—
61	D₂₉₂	Q₁₆	1	2	yes	—
62	C₄×D₇₃	—	4	4	no	—
63	C₄×D₇₃	—	4	1	no	—
64	C₄×D₇₃	—	4	1	no	—
65	Dic₁₄₆	—	2	1	no	—
66	Dic₁₄₆	—	4	4	no	—
67	Dic₁₄₆	—	4	2	no	—
68	Dic₁₄₆	SD₁₆	1	2	yes	—
69	C₈×C₇₃	—	4	1	no	—
70	C₈×C₇₃	—	4	1	no	—
71	#10596831	—	4	1	no	—
72	C₂²×C₇₃	#120	1	1	yes	characteristic
73	C₂×D₇₃	—	2	2	no	—
74	C₂×D₇₃	—	4	1	no	—
75	C₂×D₇₃	—	4	1	no	—
76	C₄×C₇₃	—	2	1	no	—
77	C₄×C₇₃	—	4	2	no	—
78	C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
79	C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
80	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₄×C₇₃	#137	1	1	yes	characteristic
82	Dic₇₃	—	2	2	no	—
83	Dic₇₃	—	4	1	no	—
84	Dic₇₃	—	4	1	no	—
85	Dic₇₃	—	8	1	no	—
86	C₂×C₇₃	#450	1	1	yes	characteristic
87	C₂×C₇₃	#451	1	1	yes	characteristic
88	C₂×C₇₃	#454	1	1	yes	characteristic
89	D₇₃	—	4	2	no	—
90	#1637	—	73	1	no	maximal, Sylow
91	C₇₃	#1637	1	1	yes	characteristic, complemented, Sylow
92	D₄×Q₈	—	73	1	no	—
93	#487	—	73	1	no	—
94	#477	—	73	1	no	—
95	#476	—	73	1	no	—
96	#332	D₇₃	1	1	yes	characteristic, complemented
97	Q₈⋊C₈	—	73	1	no	—
98	D₄⋊C₈	—	73	1	no	—
99	C₂²×Q₈	—	146	1	no	—
100	C₄⋊Q₈	—	73	1	no	—
101	C₄⋊Q₈	—	146	1	no	—
102	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
103	C₂²⋊Q₈	—	146	1	no	—
104	C₂²⋊Q₈	—	146	2	no	—
105	C₄×Q₈	—	73	1	no	—
106	C₄×D₄	—	73	1	no	—
107	C₄×D₄	—	146	1	no	—
108	#107	—	146	1	no	—
109	#106	—	146	1	no	—
110	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
111	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
112	#103	—	146	1	no	—
113	#103	—	146	2	no	—
114	#102	—	146	1	no	—
115	C₄×C₈	—	73	1	no	—
116	C₂×Q₈	—	2	1	no	—
117	C₂×Q₈	—	292	2	no	—
118	C₂×Q₈	—	292	2	no	—
119	C₂×Q₈	—	73	1	no	—
120	C₂×Q₈	—	146	2	no	—
121	C₂×Q₈	—	146	1	no	—
122	C₂×D₄	—	73	1	no	—
123	C₂²×C₄	—	146	2	no	—
124	C₂²×C₄	—	146	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	146	1	no	—
128	C₄⋊C₄	—	2	1	no	—
129	C₄⋊C₄	—	73	1	no	—
130	C₄⋊C₄	—	73	1	no	—
131	C₄⋊C₄	—	146	2	no	—
132	C₄⋊C₄	—	146	1	no	—
133	C₄⋊C₄	—	146	1	no	—
134	C₄⋊C₄	—	146	1	no	—
135	C₄⋊C₄	D₂₉₂	1	2	yes	—
136	C₂²⋊C₄	—	146	2	no	—
137	C₂²⋊C₄	—	146	1	no	—
138	C₄²	—	146	1	no	—
139	C₄²	C₂²×D₇₃	1	1	yes	characteristic
140	C₂³	—	146	1	no	—
141	Q₈	—	4	1	no	—
142	Q₈	—	4	1	no	—
143	Q₈	—	292	2	no	—
144	Q₈	—	292	4	no	—
145	Q₈	—	73	2	no	—
146	Q₈	—	146	1	no	—
147	D₄	—	73	2	no	—
148	D₄	—	146	1	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	292	4	no	—
151	C₂×C₄	—	292	1	no	—
152	C₂×C₄	—	292	1	no	—
153	C₂×C₄	—	292	1	no	—
154	C₂×C₄	—	146	1	no	—
155	C₂×C₄	—	146	1	no	—
156	C₂×C₄	—	146	1	no	—
157	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
158	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
159	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
160	C₈	—	4	1	no	—
161	C₈	—	4	1	no	—
162	C₈	—	292	1	no	—
163	C₂²	—	292	1	no	—
164	C₂²	—	292	1	no	—
165	C₂²	—	146	2	no	—
166	C₂²	#14556561	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	4	2	no	—
169	C₄	—	292	1	no	—
170	C₄	—	292	1	no	—
171	C₄	—	584	1	no	—
172	C₄	—	146	2	no	—
173	C₄	SD₁₆×D₇₃	1	1	yes	characteristic
174	C₄	Q₁₆×D₇₃	1	1	yes	characteristic
175	C₄	#14556615	1	1	yes	characteristic
176	C₄	#14556620	1	1	yes	characteristic
177	C₂	—	292	2	no	—
178	C₂	#17988739	1	1	yes	characteristic
179	C₂	#17988744	1	1	yes	characteristic
180	C₂	#17988762	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989252

C₂

yes

maximal, characteristic

#17988939

C₂

yes

maximal, characteristic, complemented

#17988883

C₂

yes

maximal, characteristic, complemented

#17988878

C₂

yes

maximal, characteristic

#17988317

C₂

yes

maximal, characteristic, complemented

#17988314

C₂

yes

maximal, characteristic

#332×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₇₃

—

#14556591

C₂²

yes

—

—

—

C₄×Dic₁₄₆

C₂²

yes

characteristic

C₄×D₂₉₂

C₂²

yes

—

—

—

—

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

D₄

yes

characteristic

Q₈×D₇₃

—

Q₈×D₇₃

—

C₂×D₂₉₂

D₄

yes

characteristic

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

D₄

yes

characteristic

C₄⋊Dic₇₃

D₄

yes

—

—

—

—

C₄×Dic₇₃

—

C₂×Q₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

D₄

yes

—

C₄²×C₇₃

C₂³

yes

characteristic

C₂²×D₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

Q₁₆

yes

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

SD₁₆

yes

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

#120

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×SD₁₆

yes

characteristic

C₄×C₇₃

C₂×Q₁₆

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#1637

—

maximal, Sylow

C₇₃

#1637

yes

characteristic, complemented, Sylow

D₄×Q₈

—

—

—

—

D₇₃

yes

characteristic, complemented

Q₈⋊C₈

—

D₄⋊C₈

—

C₂²×Q₈

—

146

—

100

C₄⋊Q₈

—

101

C₄⋊Q₈

—

146

—

102

C₄⋊Q₈

C₂×D₇₃

yes

characteristic

103

C₂²⋊Q₈

—

146

—

104

C₂²⋊Q₈

—

146

—

105

C₄×Q₈

—

106

C₄×D₄

—

107

C₄×D₄

—

146

—

108

#107

—

146

—

109

#106

—

146

—

110

C₄⋊C₈

C₂×D₇₃

yes

characteristic

111

C₄⋊C₈

C₂×D₇₃

yes

characteristic

112

#103

—

146

—

113

#103

—

146

—

114

#102

—

146

—

115

C₄×C₈

—

116

C₂×Q₈

—

117

C₂×Q₈

—

292

—

118

C₂×Q₈

—

292

—

119

C₂×Q₈

—

120

C₂×Q₈

—

146

—

121

C₂×Q₈

—

146

—

122

C₂×D₄

—

123

C₂²×C₄

—

146

—

124

C₂²×C₄

—

146

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

146

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

146

—

132

C₄⋊C₄

—

146

—

133

C₄⋊C₄

—

146

—

134

C₄⋊C₄

—

146

—

135

C₄⋊C₄

D₂₉₂

yes

—

136

C₂²⋊C₄

—

146

—

137

C₂²⋊C₄

—

146

—

138

C₄²

—

146

—

139

C₄²

C₂²×D₇₃

yes

characteristic

140

C₂³

—

146

—

141

Q₈

—

142

Q₈

—

143

Q₈

—

292

—

144

Q₈

—

292

—

145

Q₈

—

146

Q₈

—

146

—

147

D₄

—

148

D₄

—

146

—

149

C₂×C₄

—

150

C₂×C₄

—

292

—

151

C₂×C₄

—

292

—

152

C₂×C₄

—

292

—

153

C₂×C₄

—

292

—

154

C₂×C₄

—

146

—

155

C₂×C₄

—

146

—

156

C₂×C₄

—

146

—

157

C₂×C₄

D₄×D₇₃

yes

characteristic

158

C₂×C₄

D₄×D₇₃

yes

characteristic

159

C₂×C₄

C₂×D₂₉₂

yes

characteristic

160

C₈

—

161

C₈

—

162

C₈

—

292

—

163

C₂²

—

292

—

164

C₂²

—

292

—

165

C₂²

—

146

—

166

C₂²

#14556561

yes

characteristic

167

C₄

—

168

C₄

—

169

C₄

—

292

—

170

C₄

—

292

—

171

C₄

—

584

—

172

C₄

—

146

—

173

C₄

SD₁₆×D₇₃

yes

characteristic

174

C₄

Q₁₆×D₇₃

yes

characteristic

175

C₄

#14556615

yes

characteristic

176

C₄

#14556620

yes

characteristic

177

C₂

—

292

—

178

C₂

#17988739

yes

characteristic

179

C₂

#17988744

yes

characteristic

180

C₂

#17988762

yes

characteristic

	subgroup	G/H
center	C₂²	#14556561
commutator	C₄²×C₇₃	C₂³
Frattini	C₄²	C₂²×D₇₃
Fitting	#332×C₇₃	C₂
socle	C₂²×C₇₃	#120
hypercenter	#332	D₇₃
hypocenter	C₇₃	#1637
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

#14556561

commutator

C₄²×C₇₃

C₂³

Frattini

C₄²

C₂²×D₇₃

Fitting

#332×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
2. #17988939	C₂	semidirect
3. #17988883	C₂	semidirect
5. #17988317	C₂	semidirect
7. #332×C₇₃	C₂	semidirect
91. C₇₃	#1637	semidirect
96. #332	D₇₃	semidirect

H ≅ G/N

type

2. #17988939

C₂

semidirect

3. #17988883

C₂

semidirect

5. #17988317

C₂

semidirect

7. #332×C₇₃

C₂

semidirect

91. C₇₃

#1637

semidirect

96. #332

D₇₃

semidirect

Subgroups — #22609097

Conjugacy classes of subgroups

Complements