Group #22608658 — subgroups

Subgroups — #22608658

Order 9344.

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

normal subgroups only

This group has 202 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	#17989142	C₂	1	1	yes	maximal, characteristic
2	#17988643	C₂	1	1	yes	maximal, characteristic
3	#17988609	C₂	1	1	yes	maximal, characteristic
4	#17988336	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988334	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988332	C₂	1	1	yes	maximal, characteristic, complemented
7	#333×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11924011	—	2	1	no	—
9	#14556591	C₂²	1	1	yes	characteristic
10	#14556582	—	2	1	no	—
11	#14556571	—	2	1	no	—
12	#14556567	C₄	1	2	yes	—
13	Q₈×Dic₇₃	C₄	1	2	yes	—
14	#14556556	—	2	1	no	—
15	#14556549	—	2	1	no	—
16	C₄²×D₇₃	C₂²	1	1	yes	characteristic
17	C₄⋊#10596831	C₂²	1	1	yes	characteristic
18	C₄⋊#10596831	C₂²	1	1	yes	characteristic
19	#14556520	—	2	1	no	—
20	#14556503	—	2	1	no	—
21	#14556501	—	2	1	no	—
22	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
25	C₂×Dic₁₄₆	—	2	1	no	—
26	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
28	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
29	#11924011	—	4	2	no	—
30	C₂×#10596831	—	2	1	no	—
31	C₂×#10596831	—	2	1	no	—
32	C₄⋊Dic₇₃	—	2	1	no	—
33	C₄⋊Dic₇₃	—	2	2	no	—
34	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
35	#11924007	—	2	2	no	—
36	#11924005	—	2	2	no	—
37	C₄×Dic₇₃	—	2	2	no	—
38	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
39	C₄×Dic₇₃	D₄	1	1	yes	characteristic
40	C₄×Dic₇₃	D₄	1	1	yes	characteristic
41	C₂×Q₈×C₇₃	C₂×C₄	1	2	yes	—
42	C₂×C₈×C₇₃	—	2	1	no	—
43	C₂×C₈×C₇₃	—	2	1	no	—
44	C₄⋊C₄×C₇₃	—	2	1	no	—
45	C₄⋊C₄×C₇₃	—	2	1	no	—
46	C₄²×C₇₃	C₂³	1	1	yes	characteristic
47	C₂²×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
48	Q₈×C₇₃	—	4	2	no	—
49	C₂×C₄×C₇₃	—	2	2	no	—
50	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
53	C₂×Dic₇₃	—	2	2	no	—
54	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₄×D₇₃	—	2	1	no	—
58	C₄×D₇₃	—	2	1	no	—
59	C₄×D₇₃	—	2	1	no	—
60	C₄×D₇₃	—	2	1	no	—
61	C₄×D₇₃	—	4	1	no	—
62	Dic₁₄₆	—	4	1	no	—
63	Dic₁₄₆	—	4	1	no	—
64	C₈×C₇₃	—	4	1	no	—
65	C₈×C₇₃	—	4	1	no	—
66	#10596831	—	4	1	no	—
67	#10596831	—	4	1	no	—
68	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×D₇₃	—	2	1	no	—
70	C₂×D₇₃	—	2	1	no	—
71	C₂×D₇₃	—	2	1	no	—
72	C₄×C₇₃	—	2	1	no	—
73	C₄×C₇₃	—	4	2	no	—
74	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
75	C₄×C₇₃	#137	1	1	yes	characteristic
76	C₄×C₇₃	#137	1	1	yes	characteristic
77	C₄×C₇₃	#137	1	1	yes	characteristic
78	Dic₇₃	—	2	1	no	—
79	Dic₇₃	—	2	1	no	—
80	Dic₇₃	—	2	1	no	—
81	Dic₇₃	—	4	2	no	—
82	C₂×C₇₃	#413	1	1	yes	characteristic
83	C₂×C₇₃	#417	1	1	yes	characteristic
84	C₂×C₇₃	#419	1	1	yes	characteristic
85	D₇₃	—	4	1	no	—
86	#1559	—	73	1	no	maximal, Sylow
87	C₇₃	#1559	1	1	yes	characteristic, complemented, Sylow
88	#533	—	73	1	no	—
89	#432	—	73	1	no	—
90	#423	—	73	1	no	—
91	#333	—	73	1	no	—
92	#333	D₇₃	1	1	yes	characteristic, complemented
93	#332	—	73	1	no	—
94	#332	—	73	1	no	—
95	C₂×M₁₆	—	146	1	no	—
96	C₄⋊Q₈	—	73	1	no	—
97	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
98	#125	—	146	1	no	—
99	C₂²⋊Q₈	—	73	2	no	—
100	C₂²⋊Q₈	—	146	1	no	—
101	C₄×Q₈	—	73	2	no	—
102	C₄×Q₈	—	146	1	no	—
103	#117	—	146	1	no	—
104	C₂×C₄²	—	73	1	no	—
105	C₄⋊C₈	—	73	1	no	—
106	C₄⋊C₈	—	73	1	no	—
107	C₄⋊C₈	—	146	1	no	—
108	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
109	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
110	C₂²⋊C₈	—	146	1	no	—
111	#97	—	146	1	no	—
112	C₂×Q₈	—	146	1	no	—
113	C₂×Q₈	C₄×D₇₃	1	2	yes	—
114	C₂²×C₄	—	73	1	no	—
115	C₂²×C₄	—	73	1	no	—
116	C₂²×C₄	—	73	1	no	—
117	M₁₆	—	292	2	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	146	1	no	—
121	C₂×C₈	—	146	1	no	—
122	C₄⋊C₄	—	2	1	no	—
123	C₄⋊C₄	—	2	1	no	—
124	C₄⋊C₄	—	73	2	no	—
125	C₄⋊C₄	—	146	2	no	—
126	C₄⋊C₄	—	146	2	no	—
127	C₄⋊C₄	—	146	1	no	—
128	C₂²⋊C₄	—	146	2	no	—
129	C₄²	—	73	1	no	—
130	C₄²	—	73	1	no	—
131	C₄²	—	73	1	no	—
132	C₄²	—	146	2	no	—
133	C₄²	C₂²×D₇₃	1	1	yes	characteristic
134	C₂³	—	73	1	no	—
135	Q₈	—	4	2	no	—
136	Q₈	—	292	1	no	—
137	Q₈	—	292	1	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	292	1	no	—
140	C₂×C₄	—	73	1	no	—
141	C₂×C₄	—	73	1	no	—
142	C₂×C₄	—	73	1	no	—
143	C₂×C₄	—	146	2	no	—
144	C₂×C₄	—	146	1	no	—
145	C₂×C₄	—	146	1	no	—
146	C₂×C₄	—	146	1	no	—
147	C₂×C₄	—	146	1	no	—
148	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
149	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
150	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
151	C₈	—	4	1	no	—
152	C₈	—	4	1	no	—
153	C₈	—	292	1	no	—
154	C₈	—	292	1	no	—
155	C₂²	—	146	1	no	—
156	C₂²	—	146	1	no	—
157	C₂²	—	146	1	no	—
158	C₂²	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	4	2	no	—
161	C₄	—	292	2	no	—
162	C₄	—	146	1	no	—
163	C₄	—	146	1	no	—
164	C₄	—	146	1	no	—
165	C₄	#14556618	1	1	yes	characteristic
166	C₄	#14556620	1	1	yes	characteristic
167	C₄	#14556623	1	1	yes	characteristic
168	C₄	#14556623	1	1	yes	characteristic
169	C₂	—	292	1	no	—
170	C₂	#17988568	1	1	yes	characteristic
171	C₂	#17988587	1	1	yes	characteristic
172	C₂	#17988594	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989142

C₂

yes

maximal, characteristic

#17988643

C₂

yes

maximal, characteristic

#17988609

C₂

yes

maximal, characteristic

#17988336

C₂

yes

maximal, characteristic, complemented

#17988334

C₂

yes

maximal, characteristic, complemented

#17988332

C₂

yes

maximal, characteristic, complemented

#333×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#11924011

—

#14556591

C₂²

yes

—

—

C₄

yes

—

Q₈×Dic₇₃

C₄

yes

—

#14556556

—

#14556549

—

C₄²×D₇₃

C₂²

yes

characteristic

C₄⋊#10596831

C₂²

yes

characteristic

C₄⋊#10596831

C₂²

yes

—

—

—

C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×C₄×D₇₃

C₂×C₄

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

—

—

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

#11924007

—

#11924005

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₂×Q₈×C₇₃

C₂×C₄

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

C₂²×D₇₃

C₂²⋊C₄

yes

characteristic

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

C₂×C₂²⋊C₄

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

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#1559

—

maximal, Sylow

C₇₃

#1559

yes

characteristic, complemented, Sylow

—

—

—

—

D₇₃

yes

characteristic, complemented

#332

—

#332

—

C₂×M₁₆

—

146

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₃

yes

characteristic

#125

—

146

—

C₂²⋊Q₈

—

100

C₂²⋊Q₈

—

146

—

101

C₄×Q₈

—

102

C₄×Q₈

—

146

—

103

#117

—

146

—

104

C₂×C₄²

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

146

—

108

C₄⋊C₈

C₂×D₇₃

yes

characteristic

109

C₄⋊C₈

C₂×D₇₃

yes

characteristic

110

C₂²⋊C₈

—

146

—

111

#97

—

146

—

112

C₂×Q₈

—

146

—

113

C₂×Q₈

C₄×D₇₃

yes

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

M₁₆

—

292

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

146

—

121

C₂×C₈

—

146

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

146

—

126

C₄⋊C₄

—

146

—

127

C₄⋊C₄

—

146

—

128

C₂²⋊C₄

—

146

—

129

C₄²

—

130

C₄²

—

131

C₄²

—

132

C₄²

—

146

—

133

C₄²

C₂²×D₇₃

yes

characteristic

134

C₂³

—

135

Q₈

—

136

Q₈

—

292

—

137

Q₈

—

292

—

138

C₂×C₄

—

139

C₂×C₄

—

292

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

146

—

144

C₂×C₄

—

146

—

145

C₂×C₄

—

146

—

146

C₂×C₄

—

146

—

147

C₂×C₄

—

146

—

148

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

149

C₂×C₄

D₄×D₇₃

yes

characteristic

150

C₂×C₄

D₄×D₇₃

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

292

—

154

C₈

—

292

—

155

C₂²

—

146

—

156

C₂²

—

146

—

157

C₂²

—

146

—

158

C₂²

C₂²⋊C₄×D₇₃

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

292

—

162

C₄

—

146

—

163

C₄

—

146

—

164

C₄

—

146

—

165

C₄

#14556618

yes

characteristic

166

C₄

#14556620

yes

characteristic

167

C₄

#14556623

yes

characteristic

168

C₄

#14556623

yes

characteristic

169

C₂

—

292

—

170

C₂

#17988568

yes

characteristic

171

C₂

#17988587

yes

characteristic

172

C₂

#17988594

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₄²	C₂²×D₇₃
Fitting	#333×C₇₃	C₂
socle	C₂²×C₇₃	C₂×C₂²⋊C₄
hypercenter	#333	D₇₃
hypocenter	C₇₃	#1559
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₇₃

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₄²

C₂²×D₇₃

Fitting

#333×C₇₃

C₂

socle

C₂²×C₇₃

C₂×C₂²⋊C₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
4. #17988336	C₂	semidirect
5. #17988334	C₂	semidirect
6. #17988332	C₂	semidirect
7. #333×C₇₃	C₂	semidirect
87. C₇₃	#1559	semidirect
92. #333	D₇₃	semidirect

H ≅ G/N

type

4. #17988336

C₂

semidirect

5. #17988334

C₂

semidirect

6. #17988332

C₂

semidirect

7. #333×C₇₃

C₂

semidirect

87. C₇₃

#1559

semidirect

92. #333

D₇₃

semidirect

Subgroups — #22608658

Conjugacy classes of subgroups

Complements