Group #22609096 — subgroups

Subgroups — #22609096

Order 9344.

228 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 65 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	#17988939	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988882	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988877	C₂	1	1	yes	maximal, characteristic
4	#17988333	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈⋊#10596831	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄⋊#10596831	C₂	1	1	yes	maximal, characteristic
7	D₄×Q₈×C₇₃	C₂	1	1	yes	maximal, characteristic
8	#14556591	C₂²	1	1	yes	characteristic
9	#14556524	—	2	1	no	—
10	#14556522	—	2	1	no	—
11	C₄⋊#10596831	C₂²	1	1	yes	characteristic
12	C₄⋊#10596831	C₂²	1	1	yes	characteristic
13	#14556515	—	2	1	no	—
14	#14556514	—	2	2	no	—
15	#14556512	—	2	1	no	—
16	C₄×#10596831	C₂²	1	1	yes	characteristic
17	C₂²×Q₈×C₇₃	—	2	1	no	—
18	C₄⋊Q₈×C₇₃	—	2	1	no	—
19	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
20	C₂²⋊Q₈×C₇₃	—	2	1	no	—
21	C₂²⋊Q₈×C₇₃	—	2	2	no	—
22	C₄×Q₈×C₇₃	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₇₃	—	2	1	no	—
24	C₄×D₄×C₇₃	C₂²	1	1	yes	characteristic
25	C₂×Dic₁₄₆	—	2	1	no	—
26	C₂×#10596831	—	2	1	no	—
27	C₂×#10596831	—	2	1	no	—
28	C₂×#10596831	—	2	1	no	—
29	C₄⋊Dic₇₃	—	2	1	no	—
30	C₄⋊Dic₇₃	—	2	1	no	—
31	C₂×Q₈×C₇₃	—	2	2	no	—
32	C₂×Q₈×C₇₃	—	2	1	no	—
33	C₂×Q₈×C₇₃	—	4	2	no	—
34	C₂×Q₈×C₇₃	—	4	2	no	—
35	C₂×Q₈×C₇₃	D₄	1	1	yes	characteristic
36	C₂×D₄×C₇₃	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₇₃	—	2	2	no	—
38	C₂²×C₄×C₇₃	—	2	1	no	—
39	C₄⋊C₄×C₇₃	—	2	2	no	—
40	C₄⋊C₄×C₇₃	—	2	1	no	—
41	C₄⋊C₄×C₇₃	—	2	1	no	—
42	C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
43	C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
44	C₄⋊C₄×C₇₃	D₄	1	2	yes	—
45	C₂²⋊C₄×C₇₃	—	2	2	no	—
46	C₂²⋊C₄×C₇₃	—	2	1	no	—
47	C₄²×C₇₃	—	2	1	no	—
48	C₄²×C₇₃	C₂³	1	1	yes	characteristic
49	C₂³×C₇₃	—	2	1	no	—
50	Q₈×C₇₃	—	2	1	no	—
51	Q₈×C₇₃	—	4	4	no	—
52	Q₈×C₇₃	—	4	1	no	—
53	Q₈×C₇₃	—	4	1	no	—
54	Q₈×C₇₃	SD₁₆	1	2	yes	—
55	D₄×C₇₃	—	2	1	no	—
56	D₄×C₇₃	Q₁₆	1	2	yes	—
57	C₂×C₄×C₇₃	—	2	1	no	—
58	C₂×C₄×C₇₃	—	2	1	no	—
59	C₂×C₄×C₇₃	—	2	2	no	—
60	C₂×C₄×C₇₃	—	2	1	no	—
61	C₂×C₄×C₇₃	—	4	4	no	—
62	C₂×C₄×C₇₃	—	4	1	no	—
63	C₂×C₄×C₇₃	—	4	1	no	—
64	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
67	C₂×Dic₇₃	—	4	1	no	—
68	Dic₁₄₆	—	4	2	no	—
69	#10596831	—	4	1	no	—
70	#10596831	—	4	1	no	—
71	#10596831	—	4	1	no	—
72	C₂²×C₇₃	—	2	2	no	—
73	C₂²×C₇₃	—	4	1	no	—
74	C₂²×C₇₃	—	4	1	no	—
75	C₂²×C₇₃	#120	1	1	yes	characteristic
76	C₄×C₇₃	—	2	2	no	—
77	C₄×C₇₃	—	2	1	no	—
78	C₄×C₇₃	—	4	2	no	—
79	C₄×C₇₃	—	4	1	no	—
80	C₄×C₇₃	—	4	1	no	—
81	C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
82	C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
83	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
84	C₄×C₇₃	#137	1	1	yes	characteristic
85	Dic₇₃	—	8	1	no	—
86	C₂×C₇₃	—	4	2	no	—
87	C₂×C₇₃	#450	1	1	yes	characteristic
88	C₂×C₇₃	#451	1	1	yes	characteristic
89	C₂×C₇₃	#454	1	1	yes	characteristic
90	#1637	—	73	1	no	maximal, Sylow
91	C₇₃	#1637	1	1	yes	characteristic, complemented, Sylow
92	D₄×Q₈	D₇₃	1	1	yes	characteristic
93	#487	—	73	1	no	—
94	#477	—	73	1	no	—
95	#476	—	73	1	no	—
96	#332	—	73	1	no	—
97	Q₈⋊C₈	—	73	1	no	—
98	D₄⋊C₈	—	73	1	no	—
99	C₂²×Q₈	—	2	1	no	—
100	C₄⋊Q₈	—	2	1	no	—
101	C₄⋊Q₈	—	73	1	no	—
102	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
103	C₂²⋊Q₈	—	2	1	no	—
104	C₂²⋊Q₈	—	2	2	no	—
105	C₄×Q₈	C₂×D₇₃	1	1	yes	characteristic
106	C₄×D₄	—	2	1	no	—
107	C₄×D₄	C₂×D₇₃	1	1	yes	characteristic
108	#107	—	146	1	no	—
109	#106	—	146	1	no	—
110	C₄⋊C₈	—	73	1	no	—
111	C₄⋊C₈	—	73	1	no	—
112	#103	—	146	1	no	—
113	#103	—	146	2	no	—
114	#102	—	146	1	no	—
115	C₄×C₈	—	73	1	no	—
116	C₂×Q₈	—	2	2	no	—
117	C₂×Q₈	—	2	1	no	—
118	C₂×Q₈	—	4	2	no	—
119	C₂×Q₈	—	4	2	no	—
120	C₂×Q₈	—	146	1	no	—
121	C₂×Q₈	C₂²⋊D₇₃	1	1	yes	characteristic
122	C₂×D₄	C₂²⋊D₇₃	1	1	yes	characteristic
123	C₂²×C₄	—	2	2	no	—
124	C₂²×C₄	—	2	1	no	—
125	C₂×C₈	—	146	1	no	—
126	C₂×C₈	—	146	1	no	—
127	C₂×C₈	—	146	1	no	—
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	2	1	no	—
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	—	146	1	no	—
132	C₄⋊C₄	—	146	1	no	—
133	C₄⋊C₄	C₂²⋊D₇₃	1	2	yes	—
134	C₄⋊C₄	C₂²⋊D₇₃	1	1	yes	characteristic
135	C₄⋊C₄	C₂²⋊D₇₃	1	1	yes	characteristic
136	C₂²⋊C₄	—	2	2	no	—
137	C₂²⋊C₄	—	2	1	no	—
138	C₄²	—	2	1	no	—
139	C₄²	C₂²×D₇₃	1	1	yes	characteristic
140	C₂³	—	2	1	no	—
141	Q₈	—	2	1	no	—
142	Q₈	—	4	4	no	—
143	Q₈	—	4	1	no	—
144	Q₈	—	4	1	no	—
145	Q₈	—	292	2	no	—
146	Q₈	#11924016	1	2	yes	—
147	D₄	—	2	1	no	—
148	D₄	#11924018	1	2	yes	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	4	4	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	292	1	no	—
157	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
158	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
159	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
160	C₈	—	292	1	no	—
161	C₈	—	292	1	no	—
162	C₈	—	292	1	no	—
163	C₂²	—	2	2	no	—
164	C₂²	—	4	1	no	—
165	C₂²	—	4	1	no	—
166	C₂²	#14556560	1	1	yes	characteristic
167	C₄	—	2	2	no	—
168	C₄	—	2	1	no	—
169	C₄	—	4	2	no	—
170	C₄	—	4	1	no	—
171	C₄	—	4	1	no	—
172	C₄	—	584	1	no	—
173	C₄	C₂×#11924016	1	1	yes	characteristic
174	C₄	C₂×#11924018	1	1	yes	characteristic
175	C₄	#14556617	1	1	yes	characteristic
176	C₄	#14556622	1	1	yes	characteristic
177	C₂	—	4	2	no	—
178	C₂	#17988738	1	1	yes	characteristic
179	C₂	#17988743	1	1	yes	characteristic
180	C₂	#17988760	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988939

C₂

yes

maximal, characteristic, complemented

#17988882

C₂

yes

maximal, characteristic, complemented

#17988877

C₂

yes

maximal, characteristic

#17988333

C₂

yes

maximal, characteristic, complemented

Q₈⋊#10596831

C₂

yes

maximal, characteristic, complemented

D₄⋊#10596831

C₂

yes

maximal, characteristic

D₄×Q₈×C₇₃

C₂

yes

maximal, characteristic

#14556591

C₂²

yes

—

—

C₂²

yes

characteristic

C₄⋊#10596831

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₂²×Q₈×C₇₃

—

C₄⋊Q₈×C₇₃

—

C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

C₂²⋊Q₈×C₇₃

—

C₂²⋊Q₈×C₇₃

—

C₄×Q₈×C₇₃

C₂²

yes

characteristic

C₄×D₄×C₇₃

—

C₄×D₄×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

D₄

yes

characteristic

C₂×D₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

D₄

yes

characteristic

C₄⋊C₄×C₇₃

D₄

yes

characteristic

C₄⋊C₄×C₇₃

D₄

yes

—

C₂²⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

SD₁₆

yes

—

D₄×C₇₃

—

D₄×C₇₃

Q₁₆

yes

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×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

—

Dic₁₄₆

—

#10596831

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#120

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×SD₁₆

yes

characteristic

C₄×C₇₃

C₂×Q₁₆

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

D₄×Q₈

D₇₃

yes

—

—

—

—

Q₈⋊C₈

—

D₄⋊C₈

—

C₂²×Q₈

—

100

C₄⋊Q₈

—

101

C₄⋊Q₈

—

102

C₄⋊Q₈

C₂×D₇₃

yes

characteristic

103

C₂²⋊Q₈

—

104

C₂²⋊Q₈

—

105

C₄×Q₈

C₂×D₇₃

yes

characteristic

106

C₄×D₄

—

107

C₄×D₄

C₂×D₇₃

yes

characteristic

108

#107

—

146

—

109

#106

—

146

—

110

C₄⋊C₈

—

111

C₄⋊C₈

—

112

#103

—

146

—

113

#103

—

146

—

114

#102

—

146

—

115

C₄×C₈

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂×Q₈

—

146

—

121

C₂×Q₈

C₂²⋊D₇₃

yes

characteristic

122

C₂×D₄

C₂²⋊D₇₃

yes

characteristic

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂×C₈

—

146

—

126

C₂×C₈

—

146

—

127

C₂×C₈

—

146

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

146

—

132

C₄⋊C₄

—

146

—

133

C₄⋊C₄

C₂²⋊D₇₃

yes

—

134

C₄⋊C₄

C₂²⋊D₇₃

yes

characteristic

135

C₄⋊C₄

C₂²⋊D₇₃

yes

characteristic

136

C₂²⋊C₄

—

137

C₂²⋊C₄

—

138

C₄²

—

139

C₄²

C₂²×D₇₃

yes

characteristic

140

C₂³

—

141

Q₈

—

142

Q₈

—

143

Q₈

—

144

Q₈

—

145

Q₈

—

292

—

146

Q₈

#11924016

yes

—

147

D₄

—

148

D₄

#11924018

yes

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

292

—

157

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

158

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

159

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

160

C₈

—

292

—

161

C₈

—

292

—

162

C₈

—

292

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

#14556560

yes

characteristic

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

584

—

173

C₄

C₂×#11924016

yes

characteristic

174

C₄

C₂×#11924018

yes

characteristic

175

C₄

#14556617

yes

characteristic

176

C₄

#14556622

yes

characteristic

177

C₂

—

178

C₂

#17988738

yes

characteristic

179

C₂

#17988743

yes

characteristic

180

C₂

#17988760

yes

characteristic

	subgroup	G/H
center	C₂²	#14556560
commutator	C₄²×C₇₃	C₂³
Frattini	C₄²	C₂²×D₇₃
Fitting	D₄×Q₈×C₇₃	C₂
socle	C₂²×C₇₃	#120
hypercenter	D₄×Q₈	D₇₃
hypocenter	C₇₃	#1637
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ D₄×Q₈

subgroup

G/H

center

C₂²

#14556560

commutator

C₄²×C₇₃

C₂³

Frattini

C₄²

C₂²×D₇₃

Fitting

D₄×Q₈×C₇₃

C₂

socle

C₂²×C₇₃

#120

hypercenter

D₄×Q₈

D₇₃

hypocenter

C₇₃

#1637

LCS

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ D₄×Q₈

N	H ≅ G/N	type
1. #17988939	C₂	semidirect
2. #17988882	C₂	semidirect
4. #17988333	C₂	semidirect
5. Q₈⋊#10596831	C₂	semidirect
91. C₇₃	#1637	semidirect

H ≅ G/N

type

1. #17988939

C₂

semidirect

2. #17988882

C₂

semidirect

4. #17988333

C₂

semidirect

5. Q₈⋊#10596831

C₂

semidirect

91. C₇₃

#1637

semidirect

Subgroups — #22609096

Conjugacy classes of subgroups

Complements