Group #22611482 — subgroups

Subgroups — #22611482

Order 9344.

238 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 83 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	C₂×#14556524	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556522	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556514	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556514	C₂	1	1	yes	maximal, characteristic
5	#17988493	C₂	1	1	yes	maximal, characteristic
6	#17988342	C₂	1	1	yes	maximal, characteristic
7	#382×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₄₆	C₂²	1	1	yes	characteristic
9	C₂²×#10596831	C₂²	1	1	yes	characteristic
10	C₂²×#10596831	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
12	#14556524	—	2	2	no	—
13	#14556522	—	2	2	no	—
14	#14556514	—	2	2	no	—
15	#14556514	—	2	2	no	—
16	#14556498	—	2	2	no	—
17	C₂×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
20	#95×C₇₃	—	2	2	no	—
21	C₂×Dic₁₄₆	—	2	1	no	—
22	C₂×Dic₁₄₆	—	2	1	no	—
23	C₂×Dic₁₄₆	—	2	1	no	—
24	C₂×Dic₁₄₆	—	2	1	no	—
25	C₂×Dic₁₄₆	—	2	1	no	—
26	C₂×Dic₁₄₆	Q₈	1	2	yes	—
27	C₂²×Dic₇₃	—	2	1	no	—
28	C₂²×Dic₇₃	—	2	1	no	—
29	C₂×#10596831	—	2	1	no	—
30	C₂×#10596831	—	2	1	no	—
31	C₂×#10596831	—	2	1	no	—
32	C₂×#10596831	—	2	1	no	—
33	C₂×#10596831	D₄	1	2	yes	—
34	C₂×#10596831	D₄	1	2	yes	—
35	C₄⋊Dic₇₃	—	2	1	no	—
36	C₄⋊Dic₇₃	—	2	1	no	—
37	C₂²×C₄×C₇₃	—	2	1	no	—
38	C₂²×C₄×C₇₃	—	2	1	no	—
39	C₂²×C₄×C₇₃	D₄	1	2	yes	—
40	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₇₃	—	2	1	no	—
42	C₄⋊C₄×C₇₃	—	2	1	no	—
43	C₄⋊C₄×C₇₃	—	2	1	no	—
44	C₄⋊C₄×C₇₃	—	2	1	no	—
45	C₄²×C₇₃	—	2	1	no	—
46	C₄²×C₇₃	—	2	1	no	—
47	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₇₃	—	2	2	no	—
49	C₂×C₄×C₇₃	—	2	2	no	—
50	C₂×C₄×C₇₃	—	2	2	no	—
51	C₂×C₄×C₇₃	—	4	1	no	—
52	C₂×C₄×C₇₃	—	4	1	no	—
53	C₂×C₄×C₇₃	—	4	2	no	—
54	C₂×C₄×C₇₃	—	4	2	no	—
55	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
59	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
60	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
61	C₂×Dic₇₃	—	2	2	no	—
62	C₂×Dic₇₃	—	4	1	no	—
63	C₂×Dic₇₃	—	4	1	no	—
64	C₂×Dic₇₃	—	4	1	no	—
65	C₂×Dic₇₃	—	4	2	no	—
66	Dic₁₄₆	—	2	2	no	—
67	Dic₁₄₆	—	2	2	no	—
68	Dic₁₄₆	—	4	1	no	—
69	Dic₁₄₆	—	4	1	no	—
70	#10596831	—	2	2	no	—
71	#10596831	—	2	2	no	—
72	C₂²×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
73	C₂²×C₇₃	#123	1	1	yes	characteristic
74	C₂²×C₇₃	#127	1	1	yes	characteristic
75	C₂²×C₇₃	#135	1	2	yes	—
76	C₂²×C₇₃	#137	1	2	yes	—
77	C₄×C₇₃	—	4	2	no	—
78	C₄×C₇₃	—	8	1	no	—
79	C₄×C₇₃	—	8	1	no	—
80	C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
81	C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
82	C₄×C₇₃	#123	1	1	yes	characteristic
83	C₄×C₇₃	#123	1	1	yes	characteristic
84	Dic₇₃	—	4	2	no	—
85	Dic₇₃	—	8	1	no	—
86	C₂×C₇₃	#399	1	1	yes	characteristic
87	C₂×C₇₃	#479	1	2	yes	—
88	C₂×C₇₃	#485	1	2	yes	—
89	C₂×C₇₃	#495	1	1	yes	characteristic
90	C₂×C₇₃	#497	1	1	yes	characteristic
91	#2097	—	73	1	no	maximal, Sylow
92	C₇₃	#2097	1	1	yes	characteristic, complemented, Sylow
93	C₂×#107	—	73	1	no	—
94	C₂×#106	—	73	1	no	—
95	C₂×#103	—	73	1	no	—
96	C₂×#103	—	73	1	no	—
97	#391	—	73	1	no	—
98	#382	D₇₃	1	1	yes	characteristic
99	#336	—	73	1	no	—
100	C₂²×Q₈	—	73	1	no	—
101	C₂²×C₈	—	73	1	no	—
102	C₂²×C₈	—	73	1	no	—
103	C₂×C₄⋊C₄	—	73	1	no	—
104	C₂×C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
105	C₂×C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
106	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
107	#107	—	146	2	no	—
108	#106	—	146	2	no	—
109	#103	—	146	2	no	—
110	#103	—	146	2	no	—
111	#95	—	2	2	no	—
112	#95	—	146	2	no	—
113	C₂×Q₈	—	73	2	no	—
114	C₂×Q₈	—	146	1	no	—
115	C₂×Q₈	—	146	1	no	—
116	C₂×Q₈	—	146	1	no	—
117	C₂×Q₈	—	146	1	no	—
118	C₂×Q₈	—	146	1	no	—
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	—	146	1	no	—
122	C₂²×C₄	—	146	1	no	—
123	C₂²×C₄	C₂²⋊D₇₃	1	2	yes	—
124	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
125	C₂×C₈	—	73	2	no	—
126	C₂×C₈	—	73	2	no	—
127	C₂×C₈	—	146	1	no	—
128	C₂×C₈	—	146	1	no	—
129	C₂×C₈	—	146	1	no	—
130	C₂×C₈	—	146	1	no	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	2	1	no	—
134	C₄⋊C₄	—	2	1	no	—
135	C₄⋊C₄	—	146	1	no	—
136	C₄⋊C₄	—	146	1	no	—
137	C₄²	—	2	1	no	—
138	C₄²	—	2	1	no	—
139	C₂³	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
140	Q₈	—	292	1	no	—
141	Q₈	—	292	1	no	—
142	Q₈	—	146	2	no	—
143	Q₈	—	146	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	—	292	1	no	—
152	C₂×C₄	—	292	1	no	—
153	C₂×C₄	—	292	1	no	—
154	C₂×C₄	—	292	2	no	—
155	C₂×C₄	—	146	2	no	—
156	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
157	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
158	C₂×C₄	Q₈×D₇₃	1	1	yes	characteristic
159	C₂×C₄	#11924028	1	1	yes	characteristic
160	C₂×C₄	#11924028	1	1	yes	characteristic
161	C₂×C₄	#11924029	1	1	yes	characteristic
162	C₈	—	146	2	no	—
163	C₈	—	146	2	no	—
164	C₂²	#14556567	1	1	yes	characteristic
165	C₂²	#14556574	1	1	yes	characteristic
166	C₂²	#14556587	1	1	yes	characteristic
167	C₂²	#14556611	1	2	yes	—
168	C₂²	#14556621	1	2	yes	—
169	C₄	—	4	2	no	—
170	C₄	—	8	1	no	—
171	C₄	—	8	1	no	—
172	C₄	—	292	2	no	—
173	C₄	—	584	1	no	—
174	C₄	#14556569	1	1	yes	characteristic
175	C₄	#14556569	1	1	yes	characteristic
176	C₄	#14556576	1	1	yes	characteristic
177	C₄	#14556576	1	1	yes	characteristic
178	C₂	#17988522	1	1	yes	characteristic
179	C₂	#17988891	1	2	yes	—
180	C₂	#17988928	1	2	yes	—
181	C₂	#17988969	1	1	yes	characteristic
182	C₂	#17988982	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556524

C₂

yes

maximal, characteristic

C₂×#14556522

C₂

yes

maximal, characteristic

C₂×#14556514

C₂

yes

maximal, characteristic

C₂×#14556514

C₂

yes

maximal, characteristic

#17988493

C₂

yes

maximal, characteristic

#17988342

C₂

yes

maximal, characteristic

#382×C₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₆

C₂²

yes

characteristic

C₂²×#10596831

C₂²

yes

characteristic

C₂²×#10596831

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₃

C₂²

yes

—

—

—

—

—

C₂×C₄⋊C₄×C₇₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₃

C₂²

yes

characteristic

C₂×C₄²×C₇₃

C₂²

yes

characteristic

#95×C₇₃

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

Q₈

yes

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

—

—

—

—

D₄

yes

—

C₂×#10596831

D₄

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

D₄

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

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×Q₈

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

#10596831

—

#10596831

—

C₂²×C₇₃

C₂²⋊Q₈

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₂²×C₇₃

#137

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

C₄×C₇₃

C₂²⋊Q₈

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

#399

yes

characteristic

C₂×C₇₃

#479

yes

—

C₂×C₇₃

#485

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₇₃

yes

characteristic

#336

—

100

C₂²×Q₈

—

101

C₂²×C₈

—

102

C₂²×C₈

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

C₂×D₇₃

yes

characteristic

105

C₂×C₄⋊C₄

C₂×D₇₃

yes

characteristic

106

C₂×C₄²

C₂×D₇₃

yes

characteristic

107

#107

—

146

—

108

#106

—

146

—

109

#103

—

146

—

110

#103

—

146

—

111

#95

—

112

#95

—

146

—

113

C₂×Q₈

—

114

C₂×Q₈

—

146

—

115

C₂×Q₈

—

146

—

116

C₂×Q₈

—

146

—

117

C₂×Q₈

—

146

—

118

C₂×Q₈

—

146

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

146

—

122

C₂²×C₄

—

146

—

123

C₂²×C₄

C₂²⋊D₇₃

yes

—

124

C₂²×C₄

C₂²×D₇₃

yes

characteristic

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

146

—

128

C₂×C₈

—

146

—

129

C₂×C₈

—

146

—

130

C₂×C₈

—

146

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

146

—

136

C₄⋊C₄

—

146

—

137

C₄²

—

138

C₄²

—

139

C₂³

C₂×C₂²⋊D₇₃

yes

characteristic

140

Q₈

—

292

—

141

Q₈

—

292

—

142

Q₈

—

146

—

143

Q₈

—

146

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

292

—

152

C₂×C₄

—

292

—

153

C₂×C₄

—

292

—

154

C₂×C₄

—

292

—

155

C₂×C₄

—

146

—

156

C₂×C₄

D₄×D₇₃

yes

characteristic

157

C₂×C₄

D₄×D₇₃

yes

characteristic

158

C₂×C₄

Q₈×D₇₃

yes

characteristic

159

C₂×C₄

#11924028

yes

characteristic

160

C₂×C₄

#11924028

yes

characteristic

161

C₂×C₄

#11924029

yes

characteristic

162

C₈

—

146

—

163

C₈

—

146

—

164

C₂²

#14556567

yes

characteristic

165

C₂²

#14556574

yes

characteristic

166

C₂²

#14556587

yes

characteristic

167

C₂²

#14556611

yes

—

168

C₂²

#14556621

yes

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

292

—

173

C₄

—

584

—

174

C₄

#14556569

yes

characteristic

175

C₄

#14556569

yes

characteristic

176

C₄

#14556576

yes

characteristic

177

C₄

#14556576

yes

characteristic

178

C₂

#17988522

yes

characteristic

179

C₂

#17988891

yes

—

180

C₂

#17988928

yes

—

181

C₂

#17988969

yes

characteristic

182

C₂

#17988982

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₃
commutator	C₂²×C₄×C₇₃	C₂³
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	#382×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₄
hypercenter	#382	D₇₃
hypocenter	C₇₃	#2097
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₃

commutator

C₂²×C₄×C₇₃

C₂³

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

#382×C₇₃

C₂

socle

C₂³×C₇₃

C₂×D₄

D₇₃

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
92. C₇₃	#2097	semidirect

H ≅ G/N

type

92. C₇₃

#2097

semidirect

Subgroups — #22611482

Conjugacy classes of subgroups

Complements