Group #22609028 — subgroups

Subgroups — #22609028

Order 9344.

210 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 55 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	#17989312	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988935	C₂	1	1	yes	maximal, characteristic
3	#17988788	C₂	1	1	yes	maximal, characteristic
4	#17988788	C₂	1	1	yes	maximal, characteristic
5	#17988317	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988317	C₂	1	1	yes	maximal, characteristic, complemented
7	#332×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11924017	—	2	1	no	—
9	C₂×#11924017	—	2	1	no	—
10	#14556589	—	2	1	no	—
11	#14556588	C₂²	1	1	yes	characteristic
12	#14556582	—	2	1	no	—
13	#14556581	—	2	2	no	—
14	Q₈×Dic₇₃	—	2	1	no	—
15	C₄×Dic₁₄₆	C₂²	1	1	yes	characteristic
16	C₄×Dic₁₄₆	C₂²	1	1	yes	characteristic
17	#14556556	—	2	1	no	—
18	#14556513	—	2	1	no	—
19	#14556513	—	2	1	no	—
20	#14556511	—	2	2	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	D₄	1	1	yes	characteristic
26	C₂×Dic₁₄₆	D₄	1	1	yes	characteristic
27	C₂×D₂₉₂	—	2	1	no	—
28	C₂×D₂₉₂	—	2	1	no	—
29	C₂×D₂₉₂	—	2	1	no	—
30	#11924017	—	4	2	no	—
31	#11924017	—	4	2	no	—
32	C₂×#10596831	—	2	1	no	—
33	C₄⋊Dic₇₃	—	2	2	no	—
34	C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
35	C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
36	#11924007	—	2	2	no	—
37	#11924007	—	2	2	no	—
38	#11924007	—	2	1	no	—
39	#11924007	—	2	1	no	—
40	C₄×Dic₇₃	—	2	2	no	—
41	C₄×Dic₇₃	—	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	—	4	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	—	2	1	no	—
60	D₂₉₂	—	4	2	no	—
61	D₂₉₂	—	4	1	no	—
62	D₂₉₂	—	4	1	no	—
63	D₂₉₂	—	4	1	no	—
64	D₂₉₂	—	4	1	no	—
65	Dic₁₄₆	—	2	1	no	—
66	Dic₁₄₆	—	2	1	no	—
67	Dic₁₄₆	—	2	1	no	—
68	Dic₁₄₆	SD₁₆	1	2	yes	complemented
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	—	8	1	no	—
74	C₂×D₇₃	—	8	1	no	—
75	C₂×D₇₃	—	8	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	#135	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	—	4	1	no	—
86	C₂×C₇₃	#448	1	1	yes	characteristic
87	C₂×C₇₃	#449	1	1	yes	characteristic
88	C₂×C₇₃	#454	1	1	yes	characteristic
89	D₇₃	—	16	1	no	—
90	#1626	—	73	1	no	maximal, Sylow
91	C₇₃	#1626	1	1	yes	characteristic, complemented, Sylow
92	#557	—	73	1	no	—
93	#486	—	73	1	no	—
94	#460	—	73	1	no	—
95	#460	—	73	1	no	—
96	#332	D₇₃	1	1	yes	characteristic, complemented
97	Q₈⋊C₈	—	73	1	no	—
98	Q₈⋊C₈	—	73	1	no	—
99	C₂×SD₁₆	—	146	1	no	—
100	C₂×SD₁₆	—	146	1	no	—
101	C₄⋊Q₈	—	146	1	no	—
102	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
103	#127	—	73	1	no	—
104	#125	—	146	1	no	—
105	#125	—	146	2	no	—
106	C₄×Q₈	—	73	1	no	—
107	C₄×Q₈	—	73	1	no	—
108	C₄×Q₈	—	146	1	no	—
109	C₄×Q₈	—	146	1	no	—
110	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
111	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
112	#102	—	146	1	no	—
113	#102	—	146	1	no	—
114	#102	—	146	2	no	—
115	C₄×C₈	—	73	1	no	—
116	C₂×Q₈	—	2	1	no	—
117	C₂×Q₈	—	73	1	no	—
118	C₂×Q₈	—	73	1	no	—
119	C₂×D₄	—	146	1	no	—
120	C₂×D₄	—	146	1	no	—
121	C₂×D₄	—	146	1	no	—
122	SD₁₆	—	292	2	no	—
123	SD₁₆	—	292	2	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	146	1	no	—
127	C₄⋊C₄	—	2	1	no	—
128	C₄⋊C₄	—	73	1	no	—
129	C₄⋊C₄	—	73	1	no	—
130	C₄⋊C₄	—	146	2	no	—
131	C₄⋊C₄	—	146	2	no	—
132	C₄⋊C₄	—	146	2	no	—
133	C₄⋊C₄	—	146	1	no	—
134	C₄⋊C₄	—	146	1	no	—
135	C₄⋊C₄	D₂₉₂	1	2	yes	—
136	C₄²	—	146	2	no	—
137	C₄²	—	146	1	no	—
138	C₄²	—	146	1	no	—
139	C₄²	C₂²×D₇₃	1	1	yes	characteristic
140	C₂³	—	292	1	no	—
141	Q₈	—	4	1	no	—
142	Q₈	—	4	1	no	—
143	Q₈	—	73	2	no	—
144	Q₈	—	146	1	no	—
145	Q₈	—	146	1	no	—
146	Q₈	—	146	1	no	—
147	D₄	—	292	2	no	—
148	D₄	—	292	1	no	—
149	D₄	—	292	1	no	—
150	D₄	—	292	1	no	—
151	D₄	—	292	1	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	146	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	—	584	1	no	—
164	C₂²	—	584	1	no	—
165	C₂²	—	584	1	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	—	292	1	no	—
172	C₄	—	146	2	no	—
173	C₄	SD₁₆×D₇₃	1	1	yes	characteristic
174	C₄	#14556612	1	1	yes	characteristic
175	C₄	#14556615	1	1	yes	characteristic
176	C₄	#14556623	1	1	yes	characteristic
177	C₂	—	1168	1	no	—
178	C₂	#17988727	1	1	yes	characteristic
179	C₂	#17988733	1	1	yes	characteristic
180	C₂	#17988762	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989312

C₂

yes

maximal, characteristic, complemented

#17988935

C₂

yes

maximal, characteristic

#17988788

C₂

yes

maximal, characteristic

#17988788

C₂

yes

maximal, characteristic

#17988317

C₂

yes

maximal, characteristic, complemented

#17988317

C₂

yes

maximal, characteristic, complemented

#332×C₇₃

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂²

yes

characteristic

#14556582

—

#14556581

—

Q₈×Dic₇₃

—

C₄×Dic₁₄₆

C₂²

yes

characteristic

C₄×Dic₁₄₆

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

D₄

yes

characteristic

C₂×Dic₁₄₆

D₄

yes

characteristic

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

#11924017

—

#11924017

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

D₄

yes

characteristic

C₄⋊Dic₇₃

D₄

yes

—

—

—

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

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

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

SD₁₆

yes

complemented

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

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#1626

—

maximal, Sylow

C₇₃

#1626

yes

characteristic, complemented, Sylow

—

—

—

—

D₇₃

yes

characteristic, complemented

Q₈⋊C₈

—

Q₈⋊C₈

—

C₂×SD₁₆

—

146

—

100

C₂×SD₁₆

—

146

—

101

C₄⋊Q₈

—

146

—

102

C₄⋊Q₈

C₂×D₇₃

yes

characteristic

103

#127

—

104

#125

—

146

—

105

#125

—

146

—

106

C₄×Q₈

—

107

C₄×Q₈

—

108

C₄×Q₈

—

146

—

109

C₄×Q₈

—

146

—

110

C₄⋊C₈

C₂×D₇₃

yes

characteristic

111

C₄⋊C₈

C₂×D₇₃

yes

characteristic

112

#102

—

146

—

113

#102

—

146

—

114

#102

—

146

—

115

C₄×C₈

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×D₄

—

146

—

120

C₂×D₄

—

146

—

121

C₂×D₄

—

146

—

122

SD₁₆

—

292

—

123

SD₁₆

—

292

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

146

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

146

—

131

C₄⋊C₄

—

146

—

132

C₄⋊C₄

—

146

—

133

C₄⋊C₄

—

146

—

134

C₄⋊C₄

—

146

—

135

C₄⋊C₄

D₂₉₂

yes

—

136

C₄²

—

146

—

137

C₄²

—

146

—

138

C₄²

—

146

—

139

C₄²

C₂²×D₇₃

yes

characteristic

140

C₂³

—

292

—

141

Q₈

—

142

Q₈

—

143

Q₈

—

144

Q₈

—

146

—

145

Q₈

—

146

—

146

Q₈

—

146

—

147

D₄

—

292

—

148

D₄

—

292

—

149

D₄

—

292

—

150

D₄

—

292

—

151

D₄

—

292

—

152

C₂×C₄

—

153

C₂×C₄

—

146

—

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

—

584

—

164

C₂²

—

584

—

165

C₂²

—

584

—

166

C₂²

#14556561

yes

characteristic

167

C₄

—

168

C₄

—

169

C₄

—

292

—

170

C₄

—

292

—

171

C₄

—

292

—

172

C₄

—

146

—

173

C₄

SD₁₆×D₇₃

yes

characteristic

174

C₄

#14556612

yes

characteristic

175

C₄

#14556615

yes

characteristic

176

C₄

#14556623

yes

characteristic

177

C₂

—

1168

—

178

C₂

#17988727

yes

characteristic

179

C₂

#17988733

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	#1626
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
1. #17989312	C₂	semidirect
5. #17988317	C₂	semidirect
6. #17988317	C₂	semidirect
7. #332×C₇₃	C₂	semidirect
68. Dic₁₄₆	SD₁₆	semidirect
91. C₇₃	#1626	semidirect
96. #332	D₇₃	semidirect

H ≅ G/N

type

1. #17989312

C₂

semidirect

5. #17988317

C₂

semidirect

6. #17988317

C₂

semidirect

7. #332×C₇₃

C₂

semidirect

68. Dic₁₄₆

SD₁₆

semidirect

91. C₇₃

#1626

semidirect

96. #332

D₇₃

semidirect

Subgroups — #22609028

Conjugacy classes of subgroups

Complements