Group #22609199 — subgroups

Subgroups — #22609199

Order 9344.

200 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 53 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	#17989310	C₂	1	1	yes	maximal, characteristic
2	#17988983	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988864	C₂	1	1	yes	maximal, characteristic
4	#17988332	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈⋊#10596831	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988314	C₂	1	1	yes	maximal, characteristic
7	#462×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#14556591	C₂²	1	1	yes	characteristic
9	#14556586	—	2	2	no	—
10	#14556579	—	2	1	no	—
11	#14556567	—	2	1	no	—
12	C₄×Dic₁₄₆	—	2	1	no	—
13	C₄×D₂₉₂	C₂²	1	1	yes	characteristic
14	#14556549	—	2	1	no	—
15	#14556524	—	2	1	no	—
16	#14556522	—	2	2	no	—
17	C₄⋊#10596831	C₂²	1	1	yes	characteristic
18	#14556511	—	2	1	no	—
19	C₄×#10596831	C₂²	1	1	yes	characteristic
20	C₂×Q₁₆×C₇₃	—	2	1	no	—
21	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
22	C₄×Q₈×C₇₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
24	#103×C₇₃	—	2	1	no	—
25	C₂×Dic₁₄₆	—	2	1	no	—
26	C₂×D₂₉₂	D₄	1	1	yes	characteristic
27	C₂×C₄×D₇₃	—	2	1	no	—
28	C₂×#10596831	—	2	1	no	—
29	C₂×#10596831	—	2	1	no	—
30	C₄⋊Dic₇₃	—	2	1	no	—
31	C₄⋊Dic₇₃	D₄	1	2	yes	—
32	C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
33	#11924007	—	2	2	no	—
34	#11924005	—	2	2	no	—
35	#11924005	—	2	2	no	—
36	#11924005	—	2	1	no	—
37	C₄×Dic₇₃	—	2	2	no	—
38	C₂×Q₈×C₇₃	—	2	1	no	—
39	C₂×Q₈×C₇₃	D₄	1	1	yes	characteristic
40	Q₁₆×C₇₃	—	4	2	no	—
41	C₂×C₈×C₇₃	—	2	1	no	—
42	C₄⋊C₄×C₇₃	—	2	1	no	—
43	C₄⋊C₄×C₇₃	—	2	1	no	—
44	C₄⋊C₄×C₇₃	—	2	1	no	—
45	C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
46	C₄²×C₇₃	—	2	1	no	—
47	C₄²×C₇₃	C₂³	1	1	yes	characteristic
48	C₂²×D₇₃	—	2	1	no	—
49	Q₈×C₇₃	—	2	1	no	—
50	Q₈×C₇₃	—	2	1	no	—
51	Q₈×C₇₃	—	4	1	no	—
52	Q₈×C₇₃	—	4	1	no	—
53	C₂×C₄×C₇₃	—	2	1	no	—
54	C₂×C₄×C₇₃	—	2	1	no	—
55	C₂×C₄×C₇₃	—	4	1	no	—
56	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
59	C₂×Dic₇₃	—	2	1	no	—
60	C₂×Dic₇₃	—	2	2	no	—
61	D₂₉₂	—	2	1	no	—
62	D₂₉₂	—	2	1	no	—
63	C₄×D₇₃	—	4	1	no	—
64	C₄×D₇₃	—	4	1	no	—
65	Dic₁₄₆	—	4	1	no	—
66	Dic₁₄₆	—	4	1	no	—
67	C₈×C₇₃	—	4	1	no	—
68	#10596831	—	2	2	no	—
69	#10596831	—	4	1	no	—
70	C₂²×C₇₃	#120	1	1	yes	characteristic
71	C₂×D₇₃	—	4	1	no	—
72	C₂×D₇₃	—	4	1	no	—
73	C₂×D₇₃	—	4	1	no	—
74	C₄×C₇₃	—	2	1	no	—
75	C₄×C₇₃	—	4	1	no	—
76	C₄×C₇₃	—	4	1	no	—
77	C₄×C₇₃	—	8	1	no	—
78	C₄×C₇₃	#135	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	—	4	1	no	—
83	Dic₇₃	—	4	2	no	—
84	C₂×C₇₃	#449	1	1	yes	characteristic
85	C₂×C₇₃	#452	1	1	yes	characteristic
86	C₂×C₇₃	#456	1	1	yes	characteristic
87	D₇₃	—	8	1	no	—
88	#1652	—	73	1	no	maximal, Sylow
89	C₇₃	#1652	1	1	yes	characteristic, complemented, Sylow
90	#556	—	73	1	no	—
91	#498	—	73	1	no	—
92	#474	—	73	1	no	—
93	#462	D₇₃	1	1	yes	characteristic, complemented
94	#332	—	73	1	no	—
95	Q₈⋊C₈	—	73	1	no	—
96	D₄⋊C₈	—	73	1	no	—
97	C₂×Q₁₆	—	2	1	no	—
98	C₄⋊Q₈	—	73	1	no	—
99	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
100	#126	—	146	2	no	—
101	#124	—	146	1	no	—
102	C₂²⋊Q₈	—	146	1	no	—
103	C₄×Q₈	—	146	1	no	—
104	C₄×Q₈	C₂×D₇₃	1	1	yes	characteristic
105	C₄×D₄	—	73	1	no	—
106	#117	—	146	1	no	—
107	#107	—	146	1	no	—
108	#106	—	146	2	no	—
109	C₄⋊C₈	—	73	1	no	—
110	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
111	#103	—	2	1	no	—
112	#102	—	146	1	no	—
113	C₄×C₈	—	73	1	no	—
114	C₂×Q₈	—	2	1	no	—
115	C₂×Q₈	—	146	1	no	—
116	C₂×Q₈	C₂²⋊D₇₃	1	1	yes	characteristic
117	C₂×D₄	—	73	1	no	—
118	C₂²×C₄	—	146	1	no	—
119	Q₁₆	—	4	2	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	146	1	no	—
122	C₂×C₈	—	146	1	no	—
123	C₄⋊C₄	—	2	1	no	—
124	C₄⋊C₄	—	2	1	no	—
125	C₄⋊C₄	—	2	1	no	—
126	C₄⋊C₄	—	73	2	no	—
127	C₄⋊C₄	—	73	1	no	—
128	C₄⋊C₄	—	146	2	no	—
129	C₄⋊C₄	—	146	1	no	—
130	C₄⋊C₄	C₂²⋊D₇₃	1	1	yes	characteristic
131	C₂²⋊C₄	—	146	2	no	—
132	C₂²⋊C₄	—	146	2	no	—
133	C₂²⋊C₄	—	146	1	no	—
134	C₄²	—	2	1	no	—
135	C₄²	—	146	2	no	—
136	C₄²	C₂²×D₇₃	1	1	yes	characteristic
137	C₂³	—	146	1	no	—
138	Q₈	—	2	1	no	—
139	Q₈	—	2	1	no	—
140	Q₈	—	4	1	no	—
141	Q₈	—	4	1	no	—
142	Q₈	—	292	1	no	—
143	Q₈	—	292	1	no	—
144	D₄	—	146	1	no	—
145	D₄	—	146	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	292	1	no	—
150	C₂×C₄	—	292	1	no	—
151	C₂×C₄	—	146	1	no	—
152	C₂×C₄	—	146	2	no	—
153	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
154	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
155	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
156	C₈	—	4	1	no	—
157	C₈	—	292	1	no	—
158	C₈	—	146	2	no	—
159	C₂²	—	292	1	no	—
160	C₂²	—	292	1	no	—
161	C₂²	—	292	1	no	—
162	C₂²	#14556559	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	4	1	no	—
165	C₄	—	4	1	no	—
166	C₄	—	8	1	no	—
167	C₄	—	292	1	no	—
168	C₄	—	292	2	no	—
169	C₄	#14556611	1	1	yes	characteristic
170	C₄	#14556614	1	1	yes	characteristic
171	C₄	#14556617	1	1	yes	characteristic
172	C₄	#14556623	1	1	yes	characteristic
173	C₂	—	584	1	no	—
174	C₂	#17988730	1	1	yes	characteristic
175	C₂	#17988748	1	1	yes	characteristic
176	C₂	#17988768	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989310

C₂

yes

maximal, characteristic

#17988983

C₂

yes

maximal, characteristic, complemented

#17988864

C₂

yes

maximal, characteristic

#17988332

C₂

yes

maximal, characteristic, complemented

Q₈⋊#10596831

C₂

yes

maximal, characteristic, complemented

#17988314

C₂

yes

maximal, characteristic

#462×C₇₃

C₂

yes

maximal, characteristic, complemented

#14556591

C₂²

yes

—

—

—

C₄×Dic₁₄₆

—

C₄×D₂₉₂

C₂²

yes

—

—

—

C₂²

yes

characteristic

#14556511

—

C₄×#10596831

C₂²

yes

characteristic

C₂×Q₁₆×C₇₃

—

C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

C₄×Q₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

#103×C₇₃

—

C₂×Dic₁₄₆

—

C₂×D₂₉₂

D₄

yes

characteristic

C₂×C₄×D₇₃

—

C₂×#10596831

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

D₄

yes

—

C₄⋊Dic₇₃

D₄

yes

—

—

—

—

C₄×Dic₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

D₄

yes

characteristic

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

C₂²×D₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×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

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

#120

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#1652

—

maximal, Sylow

C₇₃

#1652

yes

characteristic, complemented, Sylow

—

—

—

D₇₃

yes

characteristic, complemented

#332

—

Q₈⋊C₈

—

D₄⋊C₈

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₃

yes

characteristic

100

#126

—

146

—

101

#124

—

146

—

102

C₂²⋊Q₈

—

146

—

103

C₄×Q₈

—

146

—

104

C₄×Q₈

C₂×D₇₃

yes

characteristic

105

C₄×D₄

—

106

#117

—

146

—

107

#107

—

146

—

108

#106

—

146

—

109

C₄⋊C₈

—

110

C₄⋊C₈

C₂×D₇₃

yes

characteristic

111

#103

—

112

#102

—

146

—

113

C₄×C₈

—

114

C₂×Q₈

—

115

C₂×Q₈

—

146

—

116

C₂×Q₈

C₂²⋊D₇₃

yes

characteristic

117

C₂×D₄

—

118

C₂²×C₄

—

146

—

119

Q₁₆

—

120

C₂×C₈

—

121

C₂×C₈

—

146

—

122

C₂×C₈

—

146

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

146

—

129

C₄⋊C₄

—

146

—

130

C₄⋊C₄

C₂²⋊D₇₃

yes

characteristic

131

C₂²⋊C₄

—

146

—

132

C₂²⋊C₄

—

146

—

133

C₂²⋊C₄

—

146

—

134

C₄²

—

135

C₄²

—

146

—

136

C₄²

C₂²×D₇₃

yes

characteristic

137

C₂³

—

146

—

138

Q₈

—

139

Q₈

—

140

Q₈

—

141

Q₈

—

142

Q₈

—

292

—

143

Q₈

—

292

—

144

D₄

—

146

—

145

D₄

—

146

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

292

—

150

C₂×C₄

—

292

—

151

C₂×C₄

—

146

—

152

C₂×C₄

—

146

—

153

C₂×C₄

D₄×D₇₃

yes

characteristic

154

C₂×C₄

D₄×D₇₃

yes

characteristic

155

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

156

C₈

—

157

C₈

—

292

—

158

C₈

—

146

—

159

C₂²

—

292

—

160

C₂²

—

292

—

161

C₂²

—

292

—

162

C₂²

#14556559

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

—

292

—

168

C₄

—

292

—

169

C₄

#14556611

yes

characteristic

170

C₄

#14556614

yes

characteristic

171

C₄

#14556617

yes

characteristic

172

C₄

#14556623

yes

characteristic

173

C₂

—

584

—

174

C₂

#17988730

yes

characteristic

175

C₂

#17988748

yes

characteristic

176

C₂

#17988768

yes

characteristic

	subgroup	G/H
center	C₂²	#14556559
commutator	C₄²×C₇₃	C₂³
Frattini	C₄²	C₂²×D₇₃
Fitting	#462×C₇₃	C₂
socle	C₂²×C₇₃	#120
hypercenter	#462	D₇₃
hypocenter	C₇₃	#1652
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #462

subgroup

G/H

center

C₂²

#14556559

commutator

C₄²×C₇₃

C₂³

Frattini

C₄²

C₂²×D₇₃

Fitting

#462×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #462

N	H ≅ G/N	type
2. #17988983	C₂	semidirect
4. #17988332	C₂	semidirect
5. Q₈⋊#10596831	C₂	semidirect
7. #462×C₇₃	C₂	semidirect
89. C₇₃	#1652	semidirect
93. #462	D₇₃	semidirect

H ≅ G/N

type

2. #17988983

C₂

semidirect

4. #17988332

C₂

semidirect

5. Q₈⋊#10596831

C₂

semidirect

7. #462×C₇₃

C₂

semidirect

89. C₇₃

#1652

semidirect

93. #462

D₇₃

semidirect

Subgroups — #22609199

Conjugacy classes of subgroups

Complements