Group #22611242 — subgroups

Subgroups — #22611242

Order 9344.

360 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 89 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	C₂²×D₅₈₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556513	C₂	1	2	yes	maximal, complemented
3	#17988489	C₂	1	2	yes	maximal, complemented
4	#17988350	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊C₈×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₉₂	C₂²	1	2	yes	—
7	C₂×D₅₈₄	—	2	4	no	—
8	C₂×D₅₈₄	—	2	2	no	—
9	C₂×C₄⋊Dic₇₃	C₂²	1	2	yes	—
10	C₂×#11924005	—	2	2	no	—
11	C₂×#11924005	—	2	2	no	—
12	#14556513	—	2	4	no	—
13	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₇₃	—	2	2	no	—
17	C₂³×D₇₃	—	2	2	no	—
18	C₂×D₂₉₂	—	2	2	no	—
19	C₂×D₂₉₂	—	2	2	no	—
20	C₂×D₂₉₂	—	2	2	no	—
21	C₂×D₂₉₂	—	2	2	no	—
22	C₂×D₂₉₂	—	2	2	no	—
23	C₂×D₂₉₂	D₄	1	4	yes	—
24	C₂²×Dic₇₃	—	2	2	no	—
25	D₅₈₄	—	2	8	no	—
26	C₄⋊Dic₇₃	—	2	2	no	—
27	C₄⋊Dic₇₃	—	2	2	no	—
28	#11924005	—	4	4	no	—
29	#11924005	—	4	4	no	—
30	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₇₃	—	2	2	no	—
34	C₂×C₈×C₇₃	—	2	1	no	—
35	C₂×C₈×C₇₃	—	2	2	no	—
36	C₂×C₈×C₇₃	D₄	1	2	yes	—
37	C₄²×C₇₃	—	2	1	no	—
38	C₄²×C₇₃	—	2	1	no	—
39	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×D₇₃	—	2	4	no	—
41	C₂²×D₇₃	—	4	2	no	—
42	C₂²×D₇₃	—	4	2	no	—
43	C₂²×D₇₃	—	4	2	no	—
44	C₂²×D₇₃	—	4	2	no	—
45	C₂²×D₇₃	—	4	2	no	—
46	C₂²×D₇₃	—	4	2	no	—
47	C₂×C₄×C₇₃	—	2	1	no	—
48	C₂×C₄×C₇₃	—	2	1	no	—
49	C₂×C₄×C₇₃	—	2	1	no	—
50	C₂×C₄×C₇₃	—	2	2	no	—
51	C₂×C₄×C₇₃	D₈	1	2	yes	complemented
52	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
55	C₂×C₄×C₇₃	#41	1	2	yes	—
56	C₂×Dic₇₃	—	4	2	no	—
57	C₂×Dic₇₃	—	4	2	no	—
58	C₂×Dic₇₃	—	4	2	no	—
59	D₂₉₂	—	2	4	no	—
60	D₂₉₂	—	2	4	no	—
61	D₂₉₂	—	4	2	no	—
62	D₂₉₂	—	4	2	no	—
63	C₈×C₇₃	—	2	2	no	—
64	C₈×C₇₃	—	4	1	no	—
65	C₂²×C₇₃	#121	1	1	yes	characteristic
66	C₂²×C₇₃	#121	1	2	yes	—
67	C₂²×C₇₃	C₂×D₈	1	1	yes	characteristic
68	C₂²×C₇₃	#135	1	1	yes	characteristic
69	C₂²×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
70	C₂×D₇₃	—	4	4	no	—
71	C₂×D₇₃	—	4	4	no	—
72	C₂×D₇₃	—	4	4	no	—
73	C₂×D₇₃	—	8	2	no	—
74	C₂×D₇₃	—	8	2	no	—
75	C₂×D₇₃	—	8	2	no	—
76	C₂×D₇₃	—	8	2	no	—
77	C₄×C₇₃	—	2	2	no	—
78	C₄×C₇₃	—	4	1	no	—
79	C₄×C₇₃	#120	1	1	yes	characteristic
80	C₄×C₇₃	#123	1	2	yes	—
81	C₄×C₇₃	#124	1	1	yes	characteristic
82	Dic₇₃	—	8	2	no	—
83	C₂×C₇₃	#394	1	1	yes	characteristic
84	C₂×C₇₃	#459	1	2	yes	—
85	C₂×C₇₃	#463	1	2	yes	—
86	C₂×C₇₃	#466	1	1	yes	characteristic
87	C₂×C₇₃	#469	1	1	yes	characteristic
88	D₇₃	—	8	4	no	—
89	#2054	—	73	1	no	maximal, Sylow
90	C₇₃	#2054	1	1	yes	characteristic, complemented, Sylow
91	C₂²×D₈	—	73	1	no	—
92	C₂×C₄⋊C₈	D₇₃	1	1	yes	characteristic, complemented
93	C₂×#102	—	73	2	no	—
94	#390	—	73	2	no	—
95	#340	—	73	1	no	—
96	C₂²×D₄	—	73	2	no	—
97	C₂×D₈	—	146	4	no	—
98	C₂×D₈	—	146	2	no	—
99	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
100	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
101	C₂×C₄⋊C₄	—	73	2	no	—
102	C₂×C₂²⋊C₄	—	146	2	no	—
103	C₂×C₂²⋊C₄	—	146	2	no	—
104	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
105	C₄⋊C₈	—	2	2	no	—
106	#102	—	146	4	no	—
107	C₂⁴	—	146	2	no	—
108	C₂×D₄	—	73	4	no	—
109	C₂×D₄	—	146	2	no	—
110	C₂×D₄	—	146	2	no	—
111	C₂×D₄	—	146	2	no	—
112	C₂×D₄	—	146	2	no	—
113	C₂×D₄	—	146	2	no	—
114	C₂²×C₄	—	146	2	no	—
115	C₂²×C₄	D₂₉₂	1	1	yes	characteristic
116	C₂²×C₄	D₂₉₂	1	1	yes	characteristic
117	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
118	D₈	—	146	8	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	C₂²⋊D₇₃	1	2	yes	—
123	C₄⋊C₄	—	146	2	no	—
124	C₄⋊C₄	—	146	2	no	—
125	C₂²⋊C₄	—	292	4	no	—
126	C₂²⋊C₄	—	292	4	no	—
127	C₄²	—	2	1	no	—
128	C₄²	—	2	1	no	—
129	C₂³	—	292	2	no	—
130	C₂³	—	292	2	no	—
131	C₂³	—	292	2	no	—
132	C₂³	—	292	2	no	—
133	C₂³	—	292	2	no	—
134	C₂³	—	292	2	no	—
135	C₂³	—	146	4	no	—
136	C₂³	C₂×D₂₉₂	1	1	yes	characteristic
137	D₄	—	292	2	no	—
138	D₄	—	292	2	no	—
139	D₄	—	146	4	no	—
140	D₄	—	146	4	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	292	2	no	—
146	C₂×C₄	—	292	2	no	—
147	C₂×C₄	—	292	2	no	—
148	C₂×C₄	D₅₈₄	1	2	yes	complemented
149	C₂×C₄	D₄×D₇₃	1	2	yes	—
150	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
151	C₂×C₄	#11924028	1	1	yes	characteristic
152	C₂×C₄	#11924029	1	2	yes	—
153	C₈	—	2	2	no	—
154	C₈	—	4	1	no	—
155	C₂²	—	292	4	no	—
156	C₂²	—	292	4	no	—
157	C₂²	—	292	4	no	—
158	C₂²	—	584	2	no	—
159	C₂²	—	584	2	no	—
160	C₂²	—	584	2	no	—
161	C₂²	—	584	2	no	—
162	C₂²	#14556565	1	1	yes	characteristic
163	C₂²	#14556566	1	2	yes	—
164	C₂²	C₂×D₅₈₄	1	1	yes	characteristic
165	C₂²	#14556613	1	1	yes	characteristic
166	C₂²	#14556619	1	2	yes	—
167	C₄	—	2	2	no	—
168	C₄	—	4	1	no	—
169	C₄	—	584	2	no	—
170	C₄	#14556559	1	1	yes	characteristic
171	C₄	#14556576	1	2	yes	—
172	C₄	#14556580	1	1	yes	characteristic
173	C₂	—	584	4	no	—
174	C₂	#17988504	1	1	yes	characteristic
175	C₂	#17988783	1	2	yes	—
176	C₂	#17988803	1	2	yes	—
177	C₂	#17988823	1	1	yes	characteristic
178	C₂	#17988840	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₅₈₄

C₂

yes

maximal, characteristic

C₂×#14556513

C₂

yes

maximal, complemented

#17988489

C₂

yes

maximal, complemented

#17988350

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊C₈×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₉₂

C₂²

yes

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

C₂×C₄⋊Dic₇₃

C₂²

yes

—

C₂×#11924005

—

C₂×#11924005

—

#14556513

—

C₂²×C₈×C₇₃

C₂²

yes

characteristic

C₂²×C₈×C₇₃

C₂²

yes

characteristic

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

—

C₂³×D₇₃

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

D₄

yes

—

C₂²×Dic₇₃

—

D₅₈₄

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924005

—

#11924005

—

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

D₈

yes

complemented

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₈×C₇₃

—

C₈×C₇₃

—

C₂²×C₇₃

#121

yes

characteristic

C₂²×C₇₃

#121

yes

—

C₂²×C₇₃

C₂×D₈

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#120

yes

characteristic

C₄×C₇₃

#123

yes

—

C₄×C₇₃

#124

yes

characteristic

Dic₇₃

—

C₂×C₇₃

#394

yes

characteristic

C₂×C₇₃

#459

yes

—

C₂×C₇₃

#463

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#2054

—

maximal, Sylow

C₇₃

#2054

yes

characteristic, complemented, Sylow

C₂²×D₈

—

C₂×C₄⋊C₈

D₇₃

yes

characteristic, complemented

C₂×#102

—

#390

—

#340

—

C₂²×D₄

—

C₂×D₈

—

146

—

C₂×D₈

—

146

—

C₂²×C₈

C₂×D₇₃

yes

characteristic

100

C₂²×C₈

C₂×D₇₃

yes

characteristic

101

C₂×C₄⋊C₄

—

102

C₂×C₂²⋊C₄

—

146

—

103

C₂×C₂²⋊C₄

—

146

—

104

C₂×C₄²

C₂×D₇₃

yes

characteristic

105

C₄⋊C₈

—

106

#102

—

146

—

107

C₂⁴

—

146

—

108

C₂×D₄

—

109

C₂×D₄

—

146

—

110

C₂×D₄

—

146

—

111

C₂×D₄

—

146

—

112

C₂×D₄

—

146

—

113

C₂×D₄

—

146

—

114

C₂²×C₄

—

146

—

115

C₂²×C₄

D₂₉₂

yes

characteristic

116

C₂²×C₄

D₂₉₂

yes

characteristic

117

C₂²×C₄

C₂²×D₇₃

yes

characteristic

118

D₈

—

146

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

C₂²⋊D₇₃

yes

—

123

C₄⋊C₄

—

146

—

124

C₄⋊C₄

—

146

—

125

C₂²⋊C₄

—

292

—

126

C₂²⋊C₄

—

292

—

127

C₄²

—

128

C₄²

—

129

C₂³

—

292

—

130

C₂³

—

292

—

131

C₂³

—

292

—

132

C₂³

—

292

—

133

C₂³

—

292

—

134

C₂³

—

292

—

135

C₂³

—

146

—

136

C₂³

C₂×D₂₉₂

yes

characteristic

137

D₄

—

292

—

138

D₄

—

292

—

139

D₄

—

146

—

140

D₄

—

146

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

292

—

146

C₂×C₄

—

292

—

147

C₂×C₄

—

292

—

148

C₂×C₄

D₅₈₄

yes

complemented

149

C₂×C₄

D₄×D₇₃

yes

—

150

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

151

C₂×C₄

#11924028

yes

characteristic

152

C₂×C₄

#11924029

yes

—

153

C₈

—

154

C₈

—

155

C₂²

—

292

—

156

C₂²

—

292

—

157

C₂²

—

292

—

158

C₂²

—

584

—

159

C₂²

—

584

—

160

C₂²

—

584

—

161

C₂²

—

584

—

162

C₂²

#14556565

yes

characteristic

163

C₂²

#14556566

yes

—

164

C₂²

C₂×D₅₈₄

yes

characteristic

165

C₂²

#14556613

yes

characteristic

166

C₂²

#14556619

yes

—

167

C₄

—

168

C₄

—

169

C₄

—

584

—

170

C₄

#14556559

yes

characteristic

171

C₄

#14556576

yes

—

172

C₄

#14556580

yes

characteristic

173

C₂

—

584

—

174

C₂

#17988504

yes

characteristic

175

C₂

#17988783

yes

—

176

C₂

#17988803

yes

—

177

C₂

#17988823

yes

characteristic

178

C₂

#17988840

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₉₂
commutator	C₂²×C₄×C₇₃	C₂³
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	C₂×C₄⋊C₈×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₇₃
hypocenter	C₇₃	#2054
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₂₉₂

commutator

C₂²×C₄×C₇₃

C₂³

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

C₂×C₄⋊C₈×C₇₃

C₂

socle

C₂³×C₇₃

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₇₃

hypocenter

C₇₃

#2054

LCS

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
2. C₂×#14556513	C₂	semidirect
3. #17988489	C₂	semidirect
4. #17988350	C₂	semidirect
5. C₂×C₄⋊C₈×C₇₃	C₂	semidirect
51. C₂×C₄×C₇₃	D₈	semidirect
90. C₇₃	#2054	semidirect
92. C₂×C₄⋊C₈	D₇₃	semidirect
148. C₂×C₄	D₅₈₄	semidirect

H ≅ G/N

type

2. C₂×#14556513

C₂

semidirect

3. #17988489

C₂

semidirect

4. #17988350

C₂

semidirect

5. C₂×C₄⋊C₈×C₇₃

C₂

semidirect

51. C₂×C₄×C₇₃

D₈

semidirect

90. C₇₃

#2054

semidirect

92. C₂×C₄⋊C₈

D₇₃

semidirect

148. C₂×C₄

D₅₈₄

semidirect

Subgroups — #22611242

Conjugacy classes of subgroups

Complements