Group #22625432 — subgroups

Subgroups — #22625432

Order 9344.

212 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 68 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	#17989605	C₂	1	1	yes	maximal, characteristic
2	#17989588	C₂	1	1	yes	maximal, characteristic
3	#17989447	C₂	1	2	yes	maximal, complemented
4	#17989446	C₂	1	2	yes	maximal, complemented
5	#17989141	C₂	1	1	yes	maximal, characteristic
6	C₂×#11924036	—	2	1	no	—
7	#14556658	C₂²	1	2	yes	—
8	#14556658	C₂²	1	2	yes	—
9	C₄⋊#10596833	—	2	1	no	—
10	#14556648	—	2	1	no	—
11	C₄×#10596833	—	2	1	no	—
12	#14556590	C₂²	1	1	yes	characteristic
13	#14556589	C₂²	1	1	yes	characteristic
14	#14556581	—	2	1	no	—
15	#14556571	C₄	1	1	yes	characteristic
16	#14556569	—	2	1	no	—
17	#14556567	C₄	1	1	yes	characteristic
18	C₄×Dic₁₄₆	—	2	1	no	—
19	C₄×Dic₁₄₆	C₄	1	1	yes	characteristic
20	#14556548	—	2	1	no	—
21	C₄²×D₇₃	C₂²	1	1	yes	characteristic
22	C₄×Q₈×C₇₃	C₄	1	1	yes	characteristic
23	#11924036	—	4	2	no	—
24	C₂×#10596833	—	2	2	no	—
25	C₂×#10596833	—	2	2	no	—
26	C₂×Dic₁₄₆	—	2	1	no	—
27	C₂×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
30	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
31	C₄⋊Dic₇₃	—	2	1	no	—
32	C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
33	#11924007	—	2	1	no	—
34	#11924007	—	2	1	no	—
35	#11924007	—	2	1	no	—
36	#11924007	—	2	1	no	—
37	#11924007	—	2	1	no	—
38	#11924005	—	2	1	no	—
39	#11924005	—	2	1	no	—
40	C₄×Dic₇₃	—	2	1	no	—
41	C₄×Dic₇₃	D₄	1	1	yes	characteristic
42	C₄×Dic₇₃	D₄	1	1	yes	characteristic
43	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
44	C₂×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
45	C₄⋊C₄×C₇₃	—	2	1	no	—
46	C₄⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
47	C₄²×C₇₃	—	2	1	no	—
48	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
50	Q₈×C₇₃	—	4	1	no	—
51	C₂×C₄×C₇₃	—	2	1	no	—
52	C₂×C₄×C₇₃	—	2	1	no	—
53	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×Dic₇₃	—	2	1	no	—
57	C₂×Dic₇₃	—	2	1	no	—
58	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
60	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
61	C₄×D₇₃	—	2	2	no	—
62	C₄×D₇₃	—	4	1	no	—
63	C₄×D₇₃	SD₁₆	1	2	yes	—
64	C₄×D₇₃	Q₁₆	1	2	yes	—
65	Dic₁₄₆	—	4	2	no	—
66	Dic₁₄₆	—	4	1	no	—
67	#10596833	—	4	2	no	—
68	#10596833	—	4	2	no	—
69	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×D₇₃	—	2	1	no	—
71	C₂×D₇₃	—	2	1	no	—
72	C₂×D₇₃	#103	1	2	yes	—
73	C₄×C₇₃	—	2	1	no	—
74	C₄×C₇₃	—	2	1	no	—
75	C₄×C₇₃	—	4	1	no	—
76	C₄×C₇₃	—	4	1	no	—
77	C₄×C₇₃	#103	1	2	yes	—
78	Dic₇₃	—	2	1	no	—
79	Dic₇₃	—	4	1	no	—
80	Dic₇₃	—	4	1	no	—
81	Dic₇₃	C₂×SD₁₆	1	1	yes	characteristic
82	Dic₇₃	C₂×Q₁₆	1	1	yes	characteristic
83	Dic₇₃	#135	1	2	yes	—
84	C₂×C₇₃	#413	1	1	yes	characteristic
85	C₂×C₇₃	C₂×#103	1	1	yes	characteristic
86	C₂×C₇₃	#416	1	1	yes	characteristic
87	D₇₃	—	2	2	no	—
88	#1542	—	73	1	no	maximal, Sylow
89	C₇₃	#1542	1	1	yes	characteristic, complemented, Sylow
90	#533	—	73	1	no	—
91	#431	—	73	1	no	—
92	C₂×C₄⋊C₈	—	73	1	no	—
93	#333	—	73	2	no	—
94	#332	—	73	2	no	—
95	C₂²×C₈	—	146	1	no	—
96	C₄⋊Q₈	—	73	1	no	—
97	C₄⋊Q₈	—	73	1	no	—
98	#125	—	146	1	no	—
99	C₂²⋊Q₈	—	73	1	no	—
100	C₂²⋊Q₈	—	73	1	no	—
101	C₂²⋊Q₈	—	146	1	no	—
102	C₄×Q₈	—	73	1	no	—
103	C₄×Q₈	—	146	1	no	—
104	C₄×Q₈	#64408	1	1	yes	characteristic
105	#117	—	146	1	no	—
106	C₂×C₄²	—	73	1	no	—
107	C₄⋊C₈	—	73	2	no	—
108	C₄⋊C₈	—	73	2	no	—
109	C₄⋊C₈	—	146	1	no	—
110	C₂²⋊C₈	—	146	1	no	—
111	C₄×C₈	—	146	1	no	—
112	C₂×Q₈	—	73	1	no	—
113	C₂×Q₈	—	146	1	no	—
114	C₂×Q₈	C₂×#64408	1	1	yes	characteristic
115	C₂²×C₄	—	73	1	no	—
116	C₂²×C₄	—	73	1	no	—
117	C₂²×C₄	—	73	1	no	—
118	C₂×C₈	—	292	2	no	—
119	C₂×C₈	—	146	2	no	—
120	C₂×C₈	—	146	2	no	—
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	73	1	no	—
123	C₄⋊C₄	—	146	1	no	—
124	C₄⋊C₄	—	146	1	no	—
125	C₄⋊C₄	—	146	1	no	—
126	C₄⋊C₄	—	146	1	no	—
127	C₄⋊C₄	—	146	1	no	—
128	C₄⋊C₄	—	146	1	no	—
129	C₄⋊C₄	C₂×#64408	1	1	yes	characteristic
130	C₂²⋊C₄	—	146	1	no	—
131	C₂²⋊C₄	—	146	1	no	—
132	C₄²	—	2	1	no	—
133	C₄²	—	73	1	no	—
134	C₄²	—	73	1	no	—
135	C₄²	—	73	1	no	—
136	C₄²	—	146	1	no	—
137	C₄²	C₂×#64408	1	1	yes	characteristic
138	C₂³	—	73	1	no	—
139	Q₈	—	4	1	no	—
140	Q₈	—	292	1	no	—
141	Q₈	—	292	2	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	292	1	no	—
145	C₂×C₄	—	73	1	no	—
146	C₂×C₄	—	73	1	no	—
147	C₂×C₄	—	73	2	no	—
148	C₂×C₄	—	73	2	no	—
149	C₂×C₄	—	73	1	no	—
150	C₂×C₄	—	146	2	no	—
151	C₂×C₄	—	146	1	no	—
152	C₂×C₄	—	146	1	no	—
153	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
154	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
155	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
156	C₈	—	292	2	no	—
157	C₈	—	292	2	no	—
158	C₂²	—	73	2	no	—
159	C₂²	—	146	1	no	—
160	C₂²	—	146	1	no	—
161	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	4	1	no	—
165	C₄	—	4	1	no	—
166	C₄	—	292	1	no	—
167	C₄	—	292	1	no	—
168	C₄	—	73	1	no	—
169	C₄	—	73	1	no	—
170	C₄	—	73	2	no	—
171	C₄	—	146	1	no	—
172	C₄	#14556654	1	2	yes	—
173	C₂	—	146	2	no	—
174	C₂	#17989573	1	1	yes	characteristic
175	C₂	C₂×#14556654	1	1	yes	characteristic
176	C₂	#17989577	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989605

C₂

yes

maximal, characteristic

#17989588

C₂

yes

maximal, characteristic

#17989447

C₂

yes

maximal, complemented

#17989446

C₂

yes

maximal, complemented

#17989141

C₂

yes

maximal, characteristic

C₂×#11924036

—

#14556658

C₂²

yes

—

#14556658

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

#14556589

C₂²

yes

characteristic

#14556581

—

#14556571

C₄

yes

characteristic

#14556569

—

#14556567

C₄

yes

characteristic

C₄×Dic₁₄₆

—

C₄×Dic₁₄₆

C₄

yes

characteristic

#14556548

—

C₄²×D₇₃

C₂²

yes

characteristic

C₄×Q₈×C₇₃

C₄

yes

—

—

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

C₂×C₄

yes

characteristic

C₂×C₄×D₇₃

C₂×C₄

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

—

—

—

—

—

—

C₄×Dic₇₃

—

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

C₂³

yes

characteristic

C₂×Q₈×C₇₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂×C₄

yes

characteristic

C₄²×C₇₃

—

C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂²×D₇₃

C₂²⋊C₄

yes

characteristic

Q₈×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×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

SD₁₆

yes

—

C₄×D₇₃

Q₁₆

yes

—

Dic₁₄₆

—

Dic₁₄₆

—

#10596833

—

#10596833

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

#103

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#103

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

C₂×SD₁₆

yes

characteristic

Dic₇₃

C₂×Q₁₆

yes

characteristic

Dic₇₃

#135

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#1542

—

maximal, Sylow

C₇₃

#1542

yes

characteristic, complemented, Sylow

#533

—

#431

—

C₂×C₄⋊C₈

—

#333

—

#332

—

C₂²×C₈

—

146

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

146

—

C₂²⋊Q₈

—

100

C₂²⋊Q₈

—

101

C₂²⋊Q₈

—

146

—

102

C₄×Q₈

—

103

C₄×Q₈

—

146

—

104

C₄×Q₈

#64408

yes

characteristic

105

#117

—

146

—

106

C₂×C₄²

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

146

—

110

C₂²⋊C₈

—

146

—

111

C₄×C₈

—

146

—

112

C₂×Q₈

—

113

C₂×Q₈

—

146

—

114

C₂×Q₈

C₂×#64408

yes

characteristic

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂×C₈

—

292

—

119

C₂×C₈

—

146

—

120

C₂×C₈

—

146

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

146

—

124

C₄⋊C₄

—

146

—

125

C₄⋊C₄

—

146

—

126

C₄⋊C₄

—

146

—

127

C₄⋊C₄

—

146

—

128

C₄⋊C₄

—

146

—

129

C₄⋊C₄

C₂×#64408

yes

characteristic

130

C₂²⋊C₄

—

146

—

131

C₂²⋊C₄

—

146

—

132

C₄²

—

133

C₄²

—

134

C₄²

—

135

C₄²

—

136

C₄²

—

146

—

137

C₄²

C₂×#64408

yes

characteristic

138

C₂³

—

139

Q₈

—

140

Q₈

—

292

—

141

Q₈

—

292

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

292

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

146

—

151

C₂×C₄

—

146

—

152

C₂×C₄

—

146

—

153

C₂×C₄

C₂²⋊#64408

yes

characteristic

154

C₂×C₄

C₂²⋊#64408

yes

characteristic

155

C₂×C₄

C₂²×#64408

yes

characteristic

156

C₈

—

292

—

157

C₈

—

292

—

158

C₂²

—

159

C₂²

—

146

—

160

C₂²

—

146

—

161

C₂²

C₂×C₂²⋊#64408

yes

characteristic

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

292

—

167

C₄

—

292

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

146

—

172

C₄

#14556654

yes

—

173

C₂

—

146

—

174

C₂

#17989573

yes

characteristic

175

C₂

C₂×#14556654

yes

characteristic

176

C₂

#17989577

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊#64408
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#64408
Fitting	C₄×Q₈×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₂²⋊C₄
hypercenter	C₄×Q₈	#64408
hypocenter	C₇₃	#1542
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₈

subgroup

G/H

center

C₂×C₄

C₂²⋊#64408

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#64408

Fitting

C₄×Q₈×C₇₃

C₄

socle

C₂²×C₇₃

C₂×C₂²⋊C₄

C₄×Q₈

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₈

N	H ≅ G/N	type
3. #17989447	C₂	semidirect
4. #17989446	C₂	semidirect
89. C₇₃	#1542	semidirect

H ≅ G/N

type

3. #17989447

C₂

semidirect

4. #17989446

C₂

semidirect

89. C₇₃

#1542

semidirect

Subgroups — #22625432

Conjugacy classes of subgroups

Complements