Group #22609397 — subgroups

Subgroups — #22609397

Order 9344.

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

normal subgroups only

This group has 182 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	#17988960	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988868	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988783	C₂	1	1	yes	maximal, characteristic
4	#17988666	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988332	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988330	C₂	1	1	yes	maximal, characteristic, complemented
7	#335×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11924017	—	2	1	no	—
9	C₂×#11924017	—	2	1	no	—
10	C₂×D₅₈₄	—	2	1	no	—
11	C₂×#11924011	—	2	1	no	—
12	#14556591	C₂²	1	1	yes	characteristic
13	#14556588	C₂²	1	1	yes	characteristic
14	C₄×D₂₉₂	C₂²	1	1	yes	characteristic
15	#14556525	—	2	1	no	—
16	C₄⋊#10596831	C₂²	1	1	yes	characteristic
17	#14556513	—	2	1	no	—
18	#14556513	—	2	1	no	—
19	#14556503	—	2	1	no	—
20	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₄₆	—	2	1	no	—
24	C₂×D₂₉₂	—	2	1	no	—
25	C₂×D₂₉₂	—	2	1	no	—
26	C₂×D₂₉₂	—	2	1	no	—
27	C₂×D₂₉₂	D₄	1	1	yes	characteristic
28	C₂×C₄×D₇₃	—	2	1	no	—
29	#11924017	—	4	2	no	—
30	#11924017	—	4	2	no	—
31	D₅₈₄	—	4	2	no	—
32	#11924011	—	4	2	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	#11924005	—	2	1	no	—
38	C₂×C₈×C₇₃	—	2	1	no	—
39	C₂×C₈×C₇₃	—	2	1	no	—
40	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
41	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
42	C₄²×C₇₃	C₂³	1	1	yes	characteristic
43	C₂²×D₇₃	—	2	1	no	—
44	C₂²×D₇₃	—	4	1	no	—
45	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
48	C₂×Dic₇₃	—	2	1	no	—
49	C₂×Dic₇₃	—	4	1	no	—
50	D₂₉₂	—	2	1	no	—
51	D₂₉₂	—	2	1	no	—
52	D₂₉₂	—	4	1	no	—
53	D₂₉₂	—	4	1	no	—
54	D₂₉₂	—	4	1	no	—
55	D₂₉₂	—	4	1	no	—
56	D₂₉₂	—	8	1	no	—
57	C₄×D₇₃	—	4	1	no	—
58	C₄×D₇₃	—	4	1	no	—
59	Dic₁₄₆	—	4	1	no	—
60	Dic₁₄₆	—	4	1	no	—
61	C₈×C₇₃	—	2	1	no	—
62	C₈×C₇₃	—	4	1	no	—
63	C₈×C₇₃	—	4	1	no	—
64	C₈×C₇₃	D₈	1	2	yes	complemented
65	#10596831	—	4	1	no	—
66	C₂²×C₇₃	#121	1	1	yes	characteristic
67	C₂×D₇₃	—	4	1	no	—
68	C₂×D₇₃	—	4	1	no	—
69	C₂×D₇₃	—	4	1	no	—
70	C₂×D₇₃	—	8	1	no	—
71	C₂×D₇₃	—	8	1	no	—
72	C₂×D₇₃	—	8	1	no	—
73	C₄×C₇₃	—	2	1	no	—
74	C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
75	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
76	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₄×C₇₃	#137	1	1	yes	characteristic
78	Dic₇₃	—	4	1	no	—
79	Dic₇₃	—	8	1	no	—
80	C₂×C₇₃	#459	1	1	yes	characteristic
81	C₂×C₇₃	#468	1	1	yes	characteristic
82	C₂×C₇₃	#471	1	1	yes	characteristic
83	D₇₃	—	8	1	no	—
84	D₇₃	—	16	1	no	—
85	#1685	—	73	1	no	maximal, Sylow
86	C₇₃	#1685	1	1	yes	characteristic, complemented, Sylow
87	#492	—	73	1	no	—
88	#474	—	73	1	no	—
89	#459	—	73	1	no	—
90	#436	—	73	1	no	—
91	#335	D₇₃	1	1	yes	characteristic, complemented
92	#332	—	73	1	no	—
93	#331	—	73	1	no	—
94	C₂×SD₁₆	—	146	1	no	—
95	C₂×SD₁₆	—	146	1	no	—
96	C₂×D₈	—	146	1	no	—
97	C₂×M₁₆	—	146	1	no	—
98	C₄⋊Q₈	—	73	1	no	—
99	#127	—	73	1	no	—
100	C₄×D₄	—	73	1	no	—
101	#107	—	146	1	no	—
102	C₄⋊C₈	—	73	1	no	—
103	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
104	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
105	#102	—	146	1	no	—
106	#102	—	146	1	no	—
107	C₂²⋊C₈	—	146	1	no	—
108	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
109	C₂×Q₈	—	146	1	no	—
110	C₂×D₄	—	73	1	no	—
111	C₂×D₄	—	146	1	no	—
112	C₂×D₄	—	146	1	no	—
113	C₂×D₄	—	146	1	no	—
114	C₂²×C₄	—	146	1	no	—
115	SD₁₆	—	292	2	no	—
116	SD₁₆	—	292	2	no	—
117	D₈	—	292	2	no	—
118	M₁₆	—	292	2	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	146	1	no	—
122	C₂×C₈	D₂₉₂	1	1	yes	characteristic
123	C₂×C₈	D₂₉₂	1	1	yes	characteristic
124	C₄⋊C₄	—	73	1	no	—
125	C₄⋊C₄	—	146	1	no	—
126	C₄⋊C₄	—	146	1	no	—
127	C₂²⋊C₄	—	146	1	no	—
128	C₄²	C₂²×D₇₃	1	1	yes	characteristic
129	C₂³	—	292	1	no	—
130	C₂³	—	146	1	no	—
131	Q₈	—	292	1	no	—
132	Q₈	—	292	1	no	—
133	D₄	—	292	1	no	—
134	D₄	—	292	1	no	—
135	D₄	—	292	1	no	—
136	D₄	—	292	1	no	—
137	D₄	—	584	1	no	—
138	D₄	—	146	1	no	—
139	D₄	—	146	1	no	—
140	C₂×C₄	—	292	1	no	—
141	C₂×C₄	—	292	1	no	—
142	C₂×C₄	—	292	1	no	—
143	C₂×C₄	—	146	1	no	—
144	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
145	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
146	C₂×C₄	#11924029	1	1	yes	characteristic
147	C₈	—	2	1	no	—
148	C₈	—	4	1	no	—
149	C₈	—	4	1	no	—
150	C₈	—	292	1	no	—
151	C₈	D₅₈₄	1	2	yes	complemented
152	C₂²	—	292	1	no	—
153	C₂²	—	292	1	no	—
154	C₂²	—	292	1	no	—
155	C₂²	—	584	1	no	—
156	C₂²	—	584	1	no	—
157	C₂²	—	584	1	no	—
158	C₂²	#14556566	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	292	1	no	—
161	C₄	—	584	1	no	—
162	C₄	C₂×D₅₈₄	1	1	yes	characteristic
163	C₄	#14556615	1	1	yes	characteristic
164	C₄	#14556619	1	1	yes	characteristic
165	C₄	#14556623	1	1	yes	characteristic
166	C₂	—	584	1	no	—
167	C₂	—	1168	1	no	—
168	C₂	#17988783	1	1	yes	characteristic
169	C₂	#17988834	1	1	yes	characteristic
170	C₂	#17988851	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988960

C₂

yes

maximal, characteristic, complemented

#17988868

C₂

yes

maximal, characteristic, complemented

#17988783

C₂

yes

maximal, characteristic

#17988666

C₂

yes

maximal, characteristic, complemented

#17988332

C₂

yes

maximal, characteristic, complemented

#17988330

C₂

yes

maximal, characteristic, complemented

#335×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#11924017

—

C₂×#11924017

—

C₂×D₅₈₄

—

C₂×#11924011

—

#14556591

C₂²

yes

characteristic

#14556588

C₂²

yes

characteristic

C₄×D₂₉₂

C₂²

yes

characteristic

#14556525

—

C₄⋊#10596831

C₂²

yes

—

—

—

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

D₄

yes

characteristic

C₂×C₄×D₇₃

—

#11924017

—

#11924017

—

D₅₈₄

—

#11924011

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

D₄

yes

characteristic

#11924005

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

C₂³

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

#41

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

D₈

yes

complemented

#10596831

—

C₂²×C₇₃

#121

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×D₈

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

D₇₃

—

#1685

—

maximal, Sylow

C₇₃

#1685

yes

characteristic, complemented, Sylow

—

—

—

—

D₇₃

yes

characteristic, complemented

#332

—

#331

—

C₂×SD₁₆

—

146

—

C₂×SD₁₆

—

146

—

C₂×D₈

—

146

—

C₂×M₁₆

—

146

—

C₄⋊Q₈

—

#127

—

100

C₄×D₄

—

101

#107

—

146

—

102

C₄⋊C₈

—

103

C₄⋊C₈

C₂×D₇₃

yes

characteristic

104

C₄⋊C₈

C₂×D₇₃

yes

characteristic

105

#102

—

146

—

106

#102

—

146

—

107

C₂²⋊C₈

—

146

—

108

C₄×C₈

C₂×D₇₃

yes

characteristic

109

C₂×Q₈

—

146

—

110

C₂×D₄

—

111

C₂×D₄

—

146

—

112

C₂×D₄

—

146

—

113

C₂×D₄

—

146

—

114

C₂²×C₄

—

146

—

115

SD₁₆

—

292

—

116

SD₁₆

—

292

—

117

D₈

—

292

—

118

M₁₆

—

292

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

146

—

122

C₂×C₈

D₂₉₂

yes

characteristic

123

C₂×C₈

D₂₉₂

yes

characteristic

124

C₄⋊C₄

—

125

C₄⋊C₄

—

146

—

126

C₄⋊C₄

—

146

—

127

C₂²⋊C₄

—

146

—

128

C₄²

C₂²×D₇₃

yes

characteristic

129

C₂³

—

292

—

130

C₂³

—

146

—

131

Q₈

—

292

—

132

Q₈

—

292

—

133

D₄

—

292

—

134

D₄

—

292

—

135

D₄

—

292

—

136

D₄

—

292

—

137

D₄

—

584

—

138

D₄

—

146

—

139

D₄

—

146

—

140

C₂×C₄

—

292

—

141

C₂×C₄

—

292

—

142

C₂×C₄

—

292

—

143

C₂×C₄

—

146

—

144

C₂×C₄

D₄×D₇₃

yes

characteristic

145

C₂×C₄

C₂×D₂₉₂

yes

characteristic

146

C₂×C₄

#11924029

yes

characteristic

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

—

292

—

151

C₈

D₅₈₄

yes

complemented

152

C₂²

—

292

—

153

C₂²

—

292

—

154

C₂²

—

292

—

155

C₂²

—

584

—

156

C₂²

—

584

—

157

C₂²

—

584

—

158

C₂²

#14556566

yes

characteristic

159

C₄

—

160

C₄

—

292

—

161

C₄

—

584

—

162

C₄

C₂×D₅₈₄

yes

characteristic

163

C₄

#14556615

yes

characteristic

164

C₄

#14556619

yes

characteristic

165

C₄

#14556623

yes

characteristic

166

C₂

—

584

—

167

C₂

—

1168

—

168

C₂

#17988783

yes

characteristic

169

C₂

#17988834

yes

characteristic

170

C₂

#17988851

yes

characteristic

	subgroup	G/H
center	C₂²	#14556566
commutator	C₄²×C₇₃	C₂³
Frattini	C₄²	C₂²×D₇₃
Fitting	#335×C₇₃	C₂
socle	C₂²×C₇₃	#121
hypercenter	#335	D₇₃
hypocenter	C₇₃	#1685
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

subgroup

G/H

center

C₂²

#14556566

commutator

C₄²×C₇₃

C₂³

Frattini

C₄²

C₂²×D₇₃

Fitting

#335×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #335

N	H ≅ G/N	type
1. #17988960	C₂	semidirect
2. #17988868	C₂	semidirect
4. #17988666	C₂	semidirect
5. #17988332	C₂	semidirect
6. #17988330	C₂	semidirect
7. #335×C₇₃	C₂	semidirect
64. C₈×C₇₃	D₈	semidirect
86. C₇₃	#1685	semidirect
91. #335	D₇₃	semidirect
151. C₈	D₅₈₄	semidirect

H ≅ G/N

type

1. #17988960

C₂

semidirect

2. #17988868

C₂

semidirect

4. #17988666

C₂

semidirect

5. #17988332

C₂

semidirect

6. #17988330

C₂

semidirect

7. #335×C₇₃

C₂

semidirect

64. C₈×C₇₃

D₈

semidirect

86. C₇₃

#1685

semidirect

91. #335

D₇₃

semidirect

151. C₈

D₅₈₄

semidirect

Subgroups — #22609397

Conjugacy classes of subgroups

Complements