Group #22609492 — subgroups

Subgroups — #22609492

Order 9344.

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

normal subgroups only

This group has 176 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	#17988934	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988792	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988782	C₂	1	1	yes	maximal, characteristic
4	#17988338	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	#436×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11924016	—	2	1	no	—
9	C₂×#11924013	—	2	1	no	—
10	#14556591	C₂²	1	1	yes	characteristic
11	#14556588	C₂²	1	1	yes	characteristic
12	C₄⋊#10596831	C₂²	1	1	yes	characteristic
13	C₄⋊#10596831	C₂²	1	1	yes	characteristic
14	#14556514	—	2	1	no	—
15	#14556513	—	2	1	no	—
16	#14556513	—	2	1	no	—
17	#14556511	—	2	1	no	—
18	C₂×M₁₆×C₇₃	—	2	1	no	—
19	C₄×D₄×C₇₃	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
21	C₂²⋊C₈×C₇₃	—	2	1	no	—
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	#11924016	—	4	2	no	—
28	#11924013	—	4	2	no	—
29	C₂×#10596831	—	2	1	no	—
30	C₂×#10596831	—	2	1	no	—
31	C₄⋊Dic₇₃	—	2	1	no	—
32	C₄⋊Dic₇₃	—	2	1	no	—
33	C₂×D₄×C₇₃	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₃	—	2	1	no	—
35	M₁₆×C₇₃	—	4	2	no	—
36	C₂×C₈×C₇₃	—	2	1	no	—
37	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
38	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
39	C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
40	C₂²⋊C₄×C₇₃	—	2	1	no	—
41	C₄²×C₇₃	C₂³	1	1	yes	characteristic
42	C₂³×C₇₃	—	2	1	no	—
43	C₂²×D₇₃	—	4	1	no	—
44	D₄×C₇₃	—	2	1	no	—
45	D₄×C₇₃	—	2	1	no	—
46	C₂×C₄×C₇₃	—	2	1	no	—
47	C₂×C₄×C₇₃	—	4	1	no	—
48	C₂×C₄×C₇₃	—	4	1	no	—
49	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
52	C₂×Dic₇₃	—	4	1	no	—
53	D₂₉₂	—	4	1	no	—
54	D₂₉₂	—	4	1	no	—
55	D₂₉₂	—	4	1	no	—
56	D₂₉₂	—	4	1	no	—
57	D₂₉₂	—	8	1	no	—
58	Dic₁₄₆	—	4	1	no	—
59	Dic₁₄₆	—	4	1	no	—
60	C₈×C₇₃	—	2	1	no	—
61	C₈×C₇₃	—	2	1	no	—
62	C₈×C₇₃	—	4	1	no	—
63	#10596831	—	4	1	no	—
64	#10596831	—	4	1	no	—
65	C₂²×C₇₃	—	4	1	no	—
66	C₂²×C₇₃	—	4	1	no	—
67	C₂²×C₇₃	—	4	1	no	—
68	C₂²×C₇₃	#121	1	1	yes	characteristic
69	C₂×D₇₃	—	8	1	no	—
70	C₂×D₇₃	—	8	1	no	—
71	C₂×D₇₃	—	8	1	no	—
72	C₄×C₇₃	—	2	1	no	—
73	C₄×C₇₃	—	4	1	no	—
74	C₄×C₇₃	#135	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	AGL(1,ℤ/8)	1	1	yes	characteristic
78	Dic₇₃	—	8	1	no	—
79	C₂×C₇₃	—	8	1	no	—
80	C₂×C₇₃	#463	1	1	yes	characteristic
81	C₂×C₇₃	#469	1	1	yes	characteristic
82	C₂×C₇₃	#471	1	1	yes	characteristic
83	D₇₃	—	16	1	no	—
84	#1701	—	73	1	no	maximal, Sylow
85	C₇₃	#1701	1	1	yes	characteristic, complemented, Sylow
86	#486	—	73	1	no	—
87	#461	—	73	1	no	—
88	#459	—	73	1	no	—
89	#436	D₇₃	1	1	yes	characteristic, complemented
90	#334	—	73	1	no	—
91	#332	—	73	1	no	—
92	#331	—	73	1	no	—
93	C₂×SD₁₆	—	146	1	no	—
94	C₂×D₈	—	146	1	no	—
95	C₂×M₁₆	—	2	1	no	—
96	C₄⋊Q₈	—	73	1	no	—
97	#127	—	73	1	no	—
98	C₄×D₄	C₂×D₇₃	1	1	yes	characteristic
99	C₄⋊C₈	—	73	1	no	—
100	C₄⋊C₈	—	73	1	no	—
101	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
102	#103	—	146	1	no	—
103	#102	—	146	1	no	—
104	#102	—	146	1	no	—
105	#102	—	146	1	no	—
106	C₂²⋊C₈	—	2	1	no	—
107	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
108	C₂×Q₈	—	146	1	no	—
109	C₂×D₄	—	146	1	no	—
110	C₂×D₄	—	146	1	no	—
111	C₂×D₄	—	146	1	no	—
112	C₂×D₄	D₂₉₂	1	1	yes	characteristic
113	C₂²×C₄	—	2	1	no	—
114	SD₁₆	—	292	2	no	—
115	D₈	—	292	2	no	—
116	M₁₆	—	4	2	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	146	1	no	—
119	C₂×C₈	—	146	1	no	—
120	C₂×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
121	C₂×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
122	C₄⋊C₄	—	146	1	no	—
123	C₄⋊C₄	—	146	1	no	—
124	C₄⋊C₄	D₂₉₂	1	1	yes	characteristic
125	C₂²⋊C₄	—	2	1	no	—
126	C₄²	C₂²×D₇₃	1	1	yes	characteristic
127	C₂³	—	2	1	no	—
128	C₂³	—	292	1	no	—
129	Q₈	—	292	1	no	—
130	Q₈	—	292	1	no	—
131	D₄	—	2	1	no	—
132	D₄	—	2	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	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	292	1	no	—
142	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
143	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
144	C₂×C₄	#11924028	1	1	yes	characteristic
145	C₈	—	2	1	no	—
146	C₈	—	2	1	no	—
147	C₈	—	4	1	no	—
148	C₈	—	292	1	no	—
149	C₈	—	292	1	no	—
150	C₂²	—	4	1	no	—
151	C₂²	—	4	1	no	—
152	C₂²	—	4	1	no	—
153	C₂²	—	584	1	no	—
154	C₂²	—	584	1	no	—
155	C₂²	—	584	1	no	—
156	C₂²	#14556565	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	4	1	no	—
159	C₄	—	584	1	no	—
160	C₄	#14556611	1	1	yes	characteristic
161	C₄	#14556617	1	1	yes	characteristic
162	C₄	#14556619	1	1	yes	characteristic
163	C₄	#14556619	1	1	yes	characteristic
164	C₂	—	8	1	no	—
165	C₂	—	1168	1	no	—
166	C₂	#17988802	1	1	yes	characteristic
167	C₂	#17988840	1	1	yes	characteristic
168	C₂	#17988850	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988934

C₂

yes

maximal, characteristic, complemented

#17988792

C₂

yes

maximal, characteristic, complemented

#17988782

C₂

yes

maximal, characteristic

#17988338

C₂

yes

maximal, characteristic, complemented

#17988332

C₂

yes

maximal, characteristic, complemented

#17988330

C₂

yes

maximal, characteristic, complemented

#436×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#11924016

—

C₂×#11924013

—

#14556591

C₂²

yes

characteristic

#14556588

C₂²

yes

characteristic

C₄⋊#10596831

C₂²

yes

characteristic

C₄⋊#10596831

C₂²

yes

—

—

—

—

C₂×M₁₆×C₇₃

—

C₄×D₄×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₂²⋊C₈×C₇₃

—

C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

—

—

—

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂×D₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

D₄

yes

characteristic

C₄⋊C₄×C₇₃

D₄

yes

characteristic

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

—

C₂²×D₇₃

—

D₄×C₇₃

—

D₄×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

#41

yes

characteristic

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#121

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#1701

—

maximal, Sylow

C₇₃

#1701

yes

characteristic, complemented, Sylow

—

—

—

D₇₃

yes

characteristic, complemented

#334

—

#332

—

#331

—

C₂×SD₁₆

—

146

—

C₂×D₈

—

146

—

C₂×M₁₆

—

C₄⋊Q₈

—

#127

—

C₄×D₄

C₂×D₇₃

yes

characteristic

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₄⋊C₈

C₂×D₇₃

yes

characteristic

102

#103

—

146

—

103

#102

—

146

—

104

#102

—

146

—

105

#102

—

146

—

106

C₂²⋊C₈

—

107

C₄×C₈

C₂×D₇₃

yes

characteristic

108

C₂×Q₈

—

146

—

109

C₂×D₄

—

146

—

110

C₂×D₄

—

146

—

111

C₂×D₄

—

146

—

112

C₂×D₄

D₂₉₂

yes

characteristic

113

C₂²×C₄

—

114

SD₁₆

—

292

—

115

D₈

—

292

—

116

M₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

146

—

119

C₂×C₈

—

146

—

120

C₂×C₈

C₂²⋊D₇₃

yes

characteristic

121

C₂×C₈

C₂²⋊D₇₃

yes

characteristic

122

C₄⋊C₄

—

146

—

123

C₄⋊C₄

—

146

—

124

C₄⋊C₄

D₂₉₂

yes

characteristic

125

C₂²⋊C₄

—

126

C₄²

C₂²×D₇₃

yes

characteristic

127

C₂³

—

128

C₂³

—

292

—

129

Q₈

—

292

—

130

Q₈

—

292

—

131

D₄

—

132

D₄

—

133

D₄

—

292

—

134

D₄

—

292

—

135

D₄

—

292

—

136

D₄

—

292

—

137

D₄

—

584

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

292

—

142

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

143

C₂×C₄

C₂×D₂₉₂

yes

characteristic

144

C₂×C₄

#11924028

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

—

292

—

149

C₈

—

292

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

584

—

154

C₂²

—

584

—

155

C₂²

—

584

—

156

C₂²

#14556565

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

584

—

160

C₄

#14556611

yes

characteristic

161

C₄

#14556617

yes

characteristic

162

C₄

#14556619

yes

characteristic

163

C₄

#14556619

yes

characteristic

164

C₂

—

165

C₂

—

1168

—

166

C₂

#17988802

yes

characteristic

167

C₂

#17988840

yes

characteristic

168

C₂

#17988850

yes

characteristic

	subgroup	G/H
center	C₂²	#14556565
commutator	C₄²×C₇₃	C₂³
Frattini	C₄²	C₂²×D₇₃
Fitting	#436×C₇₃	C₂
socle	C₂²×C₇₃	#121
hypercenter	#436	D₇₃
hypocenter	C₇₃	#1701
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #436

subgroup

G/H

center

C₂²

#14556565

commutator

C₄²×C₇₃

C₂³

Frattini

C₄²

C₂²×D₇₃

Fitting

#436×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #436

N	H ≅ G/N	type
1. #17988934	C₂	semidirect
2. #17988792	C₂	semidirect
4. #17988338	C₂	semidirect
5. #17988332	C₂	semidirect
6. #17988330	C₂	semidirect
7. #436×C₇₃	C₂	semidirect
85. C₇₃	#1701	semidirect
89. #436	D₇₃	semidirect

H ≅ G/N

type

1. #17988934

C₂

semidirect

2. #17988792

C₂

semidirect

4. #17988338

C₂

semidirect

5. #17988332

C₂

semidirect

6. #17988330

C₂

semidirect

7. #436×C₇₃

C₂

semidirect

85. C₇₃

#1701

semidirect

89. #436

D₇₃

semidirect

Subgroups — #22609492

Conjugacy classes of subgroups

Complements