Group #22608608 — subgroups

Subgroups — #22608608

Order 9344.

260 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 97 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	#17988611	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄⋊#10596831	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988333	C₂	1	2	yes	maximal, complemented
4	#17988331	C₂	1	2	yes	maximal, complemented
5	#532×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11924012	—	2	1	no	—
7	C₂²×#10596831	—	2	1	no	—
8	C₄⋊#10596831	—	2	1	no	—
9	C₄⋊#10596831	—	2	1	no	—
10	C₄⋊#10596831	C₂²	1	2	yes	—
11	C₄⋊#10596831	C₂²	1	2	yes	—
12	C₂×#41×C₇₃	—	2	1	no	—
13	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
14	#127×C₇₃	C₂²	1	1	yes	characteristic
15	#124×C₇₃	—	2	1	no	—
16	#121×C₇₃	—	2	1	no	—
17	#121×C₇₃	C₄	1	2	yes	—
18	C₄×D₄×C₇₃	—	2	1	no	—
19	C₄×D₄×C₇₃	C₄	1	2	yes	—
20	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
21	#11924012	—	4	2	no	—
22	C₂×#10596831	—	2	2	no	—
23	C₂×#10596831	—	2	2	no	—
24	C₂×#10596831	—	4	2	no	—
25	#41×C₇₃	—	4	2	no	—
26	#41×C₇₃	—	4	2	no	—
27	C₂×Q₈×C₇₃	—	2	1	no	—
28	C₂×D₄×C₇₃	—	2	2	no	—
29	C₂×D₄×C₇₃	—	2	1	no	—
30	C₂×D₄×C₇₃	—	2	1	no	—
31	C₂×D₄×C₇₃	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₇₃	—	2	2	no	—
33	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₇₃	—	2	1	no	—
37	C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
38	C₂²⋊C₄×C₇₃	—	2	2	no	—
39	C₂²⋊C₄×C₇₃	—	2	2	no	—
40	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
41	C₄²×C₇₃	D₄	1	1	yes	characteristic
42	C₄²×C₇₃	D₄	1	1	yes	characteristic
43	C₄²×C₇₃	C₂³	1	1	yes	characteristic
44	C₂³×C₇₃	—	2	2	no	—
45	C₂³×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
46	Q₈×C₇₃	—	4	1	no	—
47	Q₈×C₇₃	—	4	1	no	—
48	D₄×C₇₃	—	4	2	no	—
49	D₄×C₇₃	—	4	4	no	—
50	D₄×C₇₃	—	4	1	no	—
51	D₄×C₇₃	—	4	1	no	—
52	D₄×C₇₃	—	4	1	no	—
53	D₄×C₇₃	—	4	1	no	—
54	C₂×C₄×C₇₃	—	2	1	no	—
55	C₂×C₄×C₇₃	—	2	1	no	—
56	C₂×C₄×C₇₃	—	2	2	no	—
57	C₂×C₄×C₇₃	—	4	1	no	—
58	C₂×C₄×C₇₃	—	4	4	no	—
59	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₇₃	D₈	1	2	yes	—
63	C₂×C₄×C₇₃	SD₁₆	1	2	yes	—
64	C₂×C₄×C₇₃	C₂²×C₄	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	#10596831	—	4	2	no	—
68	#10596831	—	4	2	no	—
69	C₂²×C₇₃	—	2	1	no	—
70	C₂²×C₇₃	—	2	1	no	—
71	C₂²×C₇₃	—	4	2	no	—
72	C₂²×C₇₃	—	4	2	no	—
73	C₂²×C₇₃	—	4	2	no	—
74	C₂²×C₇₃	#102	1	2	yes	—
75	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₄×C₇₃	—	2	1	no	—
77	C₄×C₇₃	—	2	1	no	—
78	C₄×C₇₃	—	2	1	no	—
79	C₄×C₇₃	—	4	2	no	—
80	C₄×C₇₃	#102	1	2	yes	—
81	C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
82	C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
83	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
84	C₄×C₇₃	#137	1	1	yes	characteristic
85	C₂×C₇₃	—	2	2	no	—
86	C₂×C₇₃	—	8	2	no	—
87	C₂×C₇₃	#413	1	1	yes	characteristic
88	C₂×C₇₃	C₂×#102	1	1	yes	characteristic
89	C₂×C₇₃	#417	1	1	yes	characteristic
90	#1547	—	73	1	no	maximal, Sylow
91	C₇₃	#1547	1	1	yes	characteristic, complemented, Sylow
92	#532	D₇₃	1	1	yes	characteristic
93	#423	—	73	1	no	—
94	C₂×C₄⋊C₈	—	73	1	no	—
95	#332	—	73	2	no	—
96	#331	—	73	2	no	—
97	C₂×#41	—	2	1	no	—
98	C₂×M₁₆	—	146	1	no	—
99	C₂²×C₈	—	146	1	no	—
100	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
101	#127	C₂×D₇₃	1	1	yes	characteristic
102	#124	—	2	1	no	—
103	#121	—	2	1	no	—
104	#121	Dic₇₃	1	2	yes	—
105	C₄×D₄	—	2	1	no	—
106	C₄×D₄	Dic₇₃	1	2	yes	—
107	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
108	C₄⋊C₈	—	73	2	no	—
109	C₄⋊C₈	—	73	2	no	—
110	C₄⋊C₈	—	146	1	no	—
111	C₄⋊C₈	—	146	1	no	—
112	#41	—	4	2	no	—
113	#41	—	4	2	no	—
114	C₂×Q₈	—	2	1	no	—
115	C₂×D₄	—	2	2	no	—
116	C₂×D₄	—	2	1	no	—
117	C₂×D₄	—	2	1	no	—
118	C₂×D₄	C₂×Dic₇₃	1	2	yes	—
119	C₂²×C₄	—	2	2	no	—
120	C₂²×C₄	C₂×Dic₇₃	1	1	yes	characteristic
121	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
122	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
123	M₁₆	—	292	2	no	—
124	C₂×C₈	—	292	2	no	—
125	C₂×C₈	—	146	2	no	—
126	C₂×C₈	—	146	2	no	—
127	C₄⋊C₄	—	2	1	no	—
128	C₄⋊C₄	C₂×Dic₇₃	1	2	yes	—
129	C₂²⋊C₄	—	2	2	no	—
130	C₂²⋊C₄	—	2	2	no	—
131	C₄²	C₂×Dic₇₃	1	1	yes	characteristic
132	C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
133	C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
134	C₄²	C₂²×D₇₃	1	1	yes	characteristic
135	C₂³	—	2	2	no	—
136	C₂³	C₂²⋊Dic₇₃	1	1	yes	characteristic
137	Q₈	—	4	1	no	—
138	Q₈	—	4	1	no	—
139	D₄	—	4	2	no	—
140	D₄	—	4	4	no	—
141	D₄	—	4	1	no	—
142	D₄	—	4	1	no	—
143	D₄	—	4	1	no	—
144	D₄	—	4	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	4	no	—
150	C₂×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
151	C₂×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
152	C₂×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
153	C₂×C₄	#11924013	1	2	yes	—
154	C₂×C₄	#11924016	1	2	yes	—
155	C₂×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
156	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
157	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
158	C₈	—	292	2	no	—
159	C₈	—	292	2	no	—
160	C₂²	—	2	1	no	—
161	C₂²	—	2	1	no	—
162	C₂²	—	4	2	no	—
163	C₂²	—	4	2	no	—
164	C₂²	—	4	2	no	—
165	C₂²	#14556512	1	2	yes	—
166	C₂²	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	4	2	no	—
171	C₄	#14556512	1	2	yes	—
172	C₄	C₂×#11924013	1	1	yes	characteristic
173	C₄	C₂×#11924016	1	1	yes	characteristic
174	C₄	#14556616	1	1	yes	characteristic
175	C₄	#14556621	1	1	yes	characteristic
176	C₂	—	2	2	no	—
177	C₂	—	8	2	no	—
178	C₂	#17988571	1	1	yes	characteristic
179	C₂	C₂×#14556512	1	1	yes	characteristic
180	C₂	#17988588	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988611

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊#10596831

C₂

yes

maximal, characteristic, complemented

#17988333

C₂

yes

maximal, complemented

#17988331

C₂

yes

maximal, complemented

#532×C₇₃

C₂

yes

maximal, characteristic

—

—

—

—

C₂²

yes

—

C₄⋊#10596831

C₂²

yes

—

C₂×#41×C₇₃

—

C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

#127×C₇₃

C₂²

yes

characteristic

#124×C₇₃

—

#121×C₇₃

—

#121×C₇₃

C₄

yes

—

C₄×D₄×C₇₃

—

C₄×D₄×C₇₃

C₄

yes

—

C₂×C₄²×C₇₃

C₂²

yes

—

—

—

—

#41×C₇₃

—

#41×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

C₂×C₄

yes

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂×C₄

yes

—

C₂²⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×C₄

yes

characteristic

C₄²×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

—

C₂³×C₇₃

C₂²⋊C₄

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

D₈

yes

—

C₂×C₄×C₇₃

SD₁₆

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#102

yes

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#102

yes

—

C₄×C₇₃

C₂×D₈

yes

characteristic

C₄×C₇₃

C₂×SD₁₆

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

#532

D₇₃

yes

characteristic

#423

—

C₂×C₄⋊C₈

—

#332

—

#331

—

C₂×#41

—

C₂×M₁₆

—

146

—

C₂²×C₈

—

146

—

100

C₄⋊Q₈

C₂×D₇₃

yes

characteristic

101

#127

C₂×D₇₃

yes

characteristic

102

#124

—

103

#121

—

104

#121

Dic₇₃

yes

—

105

C₄×D₄

—

106

C₄×D₄

Dic₇₃

yes

—

107

C₂×C₄²

C₂×D₇₃

yes

characteristic

108

C₄⋊C₈

—

109

C₄⋊C₈

—

110

C₄⋊C₈

—

146

—

111

C₄⋊C₈

—

146

—

112

#41

—

113

#41

—

114

C₂×Q₈

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂×D₄

—

118

C₂×D₄

C₂×Dic₇₃

yes

—

119

C₂²×C₄

—

120

C₂²×C₄

C₂×Dic₇₃

yes

characteristic

121

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

122

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

123

M₁₆

—

292

—

124

C₂×C₈

—

292

—

125

C₂×C₈

—

146

—

126

C₂×C₈

—

146

—

127

C₄⋊C₄

—

128

C₄⋊C₄

C₂×Dic₇₃

yes

—

129

C₂²⋊C₄

—

130

C₂²⋊C₄

—

131

C₄²

C₂×Dic₇₃

yes

characteristic

132

C₄²

C₂²⋊D₇₃

yes

characteristic

133

C₄²

C₂²⋊D₇₃

yes

characteristic

134

C₄²

C₂²×D₇₃

yes

characteristic

135

C₂³

—

136

C₂³

C₂²⋊Dic₇₃

yes

characteristic

137

Q₈

—

138

Q₈

—

139

D₄

—

140

D₄

—

141

D₄

—

142

D₄

—

143

D₄

—

144

D₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

C₂²⋊Dic₇₃

yes

characteristic

151

C₂×C₄

C₂²⋊Dic₇₃

yes

characteristic

152

C₂×C₄

C₂²⋊Dic₇₃

yes

characteristic

153

C₂×C₄

#11924013

yes

—

154

C₂×C₄

#11924016

yes

—

155

C₂×C₄

C₂²×Dic₇₃

yes

characteristic

156

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

157

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

158

C₈

—

292

—

159

C₈

—

292

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

#14556512

yes

—

166

C₂²

C₂×C₂²⋊Dic₇₃

yes

characteristic

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

#14556512

yes

—

172

C₄

C₂×#11924013

yes

characteristic

173

C₄

C₂×#11924016

yes

characteristic

174

C₄

#14556616

yes

characteristic

175

C₄

#14556621

yes

characteristic

176

C₂

—

177

C₂

—

178

C₂

#17988571

yes

characteristic

179

C₂

C₂×#14556512

yes

characteristic

180

C₂

#17988588

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₇₃
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₄²	C₂²×D₇₃
Fitting	#532×C₇₃	C₂
socle	C₂²×C₇₃	C₂×C₂²⋊C₄
hypercenter	#532	D₇₃
hypocenter	C₇₃	#1547
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₇₃

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₄²

C₂²×D₇₃

Fitting

#532×C₇₃

C₂

socle

C₂²×C₇₃

C₂×C₂²⋊C₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #17988611	C₂	semidirect
2. C₂×C₄⋊#10596831	C₂	semidirect
3. #17988333	C₂	semidirect
4. #17988331	C₂	semidirect
91. C₇₃	#1547	semidirect

H ≅ G/N

type

1. #17988611

C₂

semidirect

2. C₂×C₄⋊#10596831

C₂

semidirect

3. #17988333

C₂

semidirect

4. #17988331

C₂

semidirect

91. C₇₃

#1547

semidirect

Subgroups — #22608608

Conjugacy classes of subgroups

Complements