Group #22611178 — subgroups

Subgroups — #22611178

Order 9344.

298 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 91 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×#14556591	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556516	C₂	1	2	yes	maximal
3	C₂×#14556516	C₂	1	2	yes	maximal
4	#17988485	C₂	1	1	yes	maximal, characteristic
5	#336×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₄₆	C₂²	1	2	yes	—
7	#14556591	—	2	4	no	—
8	C₂×C₄⋊Dic₇₃	—	2	1	no	—
9	C₂×C₄⋊Dic₇₃	—	2	1	no	—
10	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
12	C₂×#11924007	—	2	1	no	—
13	#14556516	—	2	4	no	—
14	#14556516	—	2	4	no	—
15	#14556498	—	2	1	no	—
16	C₂²×C₈×C₇₃	C₂²	1	2	yes	—
17	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
18	C₂×Dic₁₄₆	—	2	2	no	—
19	C₂×Dic₁₄₆	—	2	2	no	—
20	C₂×Dic₁₄₆	—	2	2	no	—
21	C₂×Dic₁₄₆	—	2	2	no	—
22	C₂×Dic₁₄₆	—	2	2	no	—
23	C₂×Dic₁₄₆	D₄	1	4	yes	—
24	C₂²×Dic₇₃	—	2	1	no	—
25	C₂²×Dic₇₃	—	2	2	no	—
26	C₂²×Dic₇₃	—	2	1	no	—
27	C₄⋊Dic₇₃	—	2	4	no	—
28	C₄⋊Dic₇₃	—	2	4	no	—
29	C₄⋊Dic₇₃	—	2	1	no	—
30	C₄⋊Dic₇₃	—	2	2	no	—
31	C₄⋊Dic₇₃	D₄	1	2	yes	—
32	#11924007	—	4	2	no	—
33	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₇₃	—	2	2	no	—
37	C₂×C₈×C₇₃	—	2	2	no	—
38	C₂×C₈×C₇₃	—	2	2	no	—
39	C₄²×C₇₃	—	2	2	no	—
40	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₇₃	—	2	1	no	—
42	C₂×C₄×C₇₃	—	2	1	no	—
43	C₂×C₄×C₇₃	—	2	1	no	—
44	C₂×C₄×C₇₃	—	2	1	no	—
45	C₂×C₄×C₇₃	SD₁₆	1	2	yes	—
46	C₂×C₄×C₇₃	Q₁₆	1	2	yes	—
47	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₇₃	#41	1	2	yes	—
50	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
51	C₂×Dic₇₃	—	2	4	no	—
52	C₂×Dic₇₃	—	2	2	no	—
53	C₂×Dic₇₃	—	4	2	no	—
54	C₂×Dic₇₃	—	4	2	no	—
55	C₂×Dic₇₃	—	4	1	no	—
56	C₂×Dic₇₃	—	4	1	no	—
57	C₂×Dic₇₃	—	4	1	no	—
58	C₂×Dic₇₃	—	4	1	no	—
59	C₂×Dic₇₃	—	4	1	no	—
60	Dic₁₄₆	—	2	4	no	—
61	Dic₁₄₆	—	2	4	no	—
62	Dic₁₄₆	—	4	2	no	—
63	Dic₁₄₆	—	4	2	no	—
64	C₈×C₇₃	—	4	2	no	—
65	C₂²×C₇₃	#120	1	1	yes	characteristic
66	C₂²×C₇₃	#121	1	1	yes	characteristic
67	C₂²×C₇₃	#124	1	1	yes	characteristic
68	C₂²×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
69	C₂²×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
70	C₂²×C₇₃	#137	1	1	yes	characteristic
71	C₂²×C₇₃	#137	1	1	yes	characteristic
72	C₄×C₇₃	—	2	2	no	—
73	C₄×C₇₃	—	2	2	no	—
74	C₄×C₇₃	#121	1	2	yes	—
75	C₄×C₇₃	#123	1	2	yes	—
76	Dic₇₃	—	4	4	no	—
77	Dic₇₃	—	4	2	no	—
78	Dic₇₃	—	8	1	no	—
79	C₂×C₇₃	#394	1	1	yes	characteristic
80	C₂×C₇₃	#448	1	1	yes	characteristic
81	C₂×C₇₃	#451	1	1	yes	characteristic
82	C₂×C₇₃	#461	1	1	yes	characteristic
83	C₂×C₇₃	#462	1	1	yes	characteristic
84	C₂×C₇₃	#487	1	1	yes	characteristic
85	C₂×C₇₃	#491	1	1	yes	characteristic
86	#2043	—	73	1	no	maximal, Sylow
87	C₇₃	#2043	1	1	yes	characteristic, complemented, Sylow
88	C₂×C₄⋊Q₈	—	73	1	no	—
89	C₂×#103	—	73	2	no	—
90	C₂×#103	—	73	2	no	—
91	#389	—	73	1	no	—
92	#336	D₇₃	1	1	yes	characteristic
93	C₂²×Q₈	—	73	2	no	—
94	C₂²×C₈	C₂×D₇₃	1	2	yes	—
95	C₄⋊Q₈	—	146	4	no	—
96	C₂×C₄⋊C₄	—	73	1	no	—
97	C₂×C₄⋊C₄	—	73	1	no	—
98	C₂×C₄⋊C₄	—	146	1	no	—
99	C₂×C₄⋊C₄	—	146	1	no	—
100	C₂×C₄⋊C₄	—	146	1	no	—
101	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
102	#103	—	146	4	no	—
103	#103	—	146	4	no	—
104	#95	—	146	1	no	—
105	C₂×Q₈	—	73	4	no	—
106	C₂×Q₈	—	146	2	no	—
107	C₂×Q₈	—	146	2	no	—
108	C₂×Q₈	—	146	2	no	—
109	C₂×Q₈	—	146	2	no	—
110	C₂×Q₈	—	146	2	no	—
111	C₂²×C₄	—	146	1	no	—
112	C₂²×C₄	—	146	1	no	—
113	C₂²×C₄	—	146	2	no	—
114	C₂²×C₄	D₂₉₂	1	1	yes	characteristic
115	C₂²×C₄	D₂₉₂	1	1	yes	characteristic
116	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₄⋊C₄	—	292	2	no	—
121	C₄⋊C₄	—	73	2	no	—
122	C₄⋊C₄	—	146	4	no	—
123	C₄⋊C₄	—	146	4	no	—
124	C₄⋊C₄	—	146	2	no	—
125	C₄⋊C₄	—	146	1	no	—
126	C₄²	—	2	2	no	—
127	C₂³	C₂×D₂₉₂	1	1	yes	characteristic
128	Q₈	—	292	2	no	—
129	Q₈	—	292	2	no	—
130	Q₈	—	146	4	no	—
131	Q₈	—	146	4	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	292	2	no	—
137	C₂×C₄	—	292	2	no	—
138	C₂×C₄	—	292	1	no	—
139	C₂×C₄	—	292	1	no	—
140	C₂×C₄	—	292	1	no	—
141	C₂×C₄	—	292	1	no	—
142	C₂×C₄	—	292	1	no	—
143	C₂×C₄	—	146	4	no	—
144	C₂×C₄	—	146	2	no	—
145	C₂×C₄	#11924017	1	2	yes	—
146	C₂×C₄	Dic₂₉₂	1	2	yes	—
147	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
148	C₂×C₄	D₄×D₇₃	1	2	yes	—
149	C₂×C₄	#11924028	1	2	yes	—
150	C₂×C₄	#11924029	1	1	yes	characteristic
151	C₈	—	4	2	no	—
152	C₂²	#14556561	1	1	yes	characteristic
153	C₂²	#14556566	1	1	yes	characteristic
154	C₂²	#14556579	1	1	yes	characteristic
155	C₂²	C₂×#11924017	1	1	yes	characteristic
156	C₂²	C₂×Dic₂₉₂	1	1	yes	characteristic
157	C₂²	#14556624	1	1	yes	characteristic
158	C₂²	#14556624	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	292	4	no	—
162	C₄	—	292	2	no	—
163	C₄	—	584	1	no	—
164	C₄	#14556564	1	2	yes	—
165	C₄	#14556576	1	2	yes	—
166	C₂	#17988505	1	1	yes	characteristic
167	C₂	#17988728	1	1	yes	characteristic
168	C₂	#17988745	1	1	yes	characteristic
169	C₂	#17988793	1	1	yes	characteristic
170	C₂	#17988798	1	1	yes	characteristic
171	C₂	#17988938	1	1	yes	characteristic
172	C₂	#17988956	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556591

C₂

yes

maximal, characteristic

C₂×#14556516

C₂

yes

maximal

C₂×#14556516

C₂

yes

maximal

#17988485

C₂

yes

maximal, characteristic

#336×C₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₆

C₂²

yes

—

#14556591

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₃

C₂²

yes

—

—

—

—

C₂²×C₈×C₇₃

C₂²

yes

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

D₄

yes

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

D₄

yes

—

#11924007

—

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

SD₁₆

yes

—

C₂×C₄×C₇₃

Q₁₆

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

#41

yes

—

C₂×C₄×C₇₃

#41

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

C₂×SD₁₆

yes

characteristic

C₂²×C₇₃

C₂×Q₁₆

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#121

yes

—

C₄×C₇₃

#123

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

—

—

—

D₇₃

yes

characteristic

C₂²×Q₈

—

C₂²×C₈

C₂×D₇₃

yes

—

C₄⋊Q₈

—

146

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄⋊C₄

—

146

—

100

C₂×C₄⋊C₄

—

146

—

101

C₂×C₄²

C₂×D₇₃

yes

characteristic

102

#103

—

146

—

103

#103

—

146

—

104

#95

—

146

—

105

C₂×Q₈

—

106

C₂×Q₈

—

146

—

107

C₂×Q₈

—

146

—

108

C₂×Q₈

—

146

—

109

C₂×Q₈

—

146

—

110

C₂×Q₈

—

146

—

111

C₂²×C₄

—

146

—

112

C₂²×C₄

—

146

—

113

C₂²×C₄

—

146

—

114

C₂²×C₄

D₂₉₂

yes

characteristic

115

C₂²×C₄

D₂₉₂

yes

characteristic

116

C₂²×C₄

C₂²×D₇₃

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₄⋊C₄

—

292

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

146

—

123

C₄⋊C₄

—

146

—

124

C₄⋊C₄

—

146

—

125

C₄⋊C₄

—

146

—

126

C₄²

—

127

C₂³

C₂×D₂₉₂

yes

characteristic

128

Q₈

—

292

—

129

Q₈

—

292

—

130

Q₈

—

146

—

131

Q₈

—

146

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

292

—

137

C₂×C₄

—

292

—

138

C₂×C₄

—

292

—

139

C₂×C₄

—

292

—

140

C₂×C₄

—

292

—

141

C₂×C₄

—

292

—

142

C₂×C₄

—

292

—

143

C₂×C₄

—

146

—

144

C₂×C₄

—

146

—

145

C₂×C₄

#11924017

yes

—

146

C₂×C₄

Dic₂₉₂

yes

—

147

C₂×C₄

D₄×D₇₃

yes

characteristic

148

C₂×C₄

D₄×D₇₃

yes

—

149

C₂×C₄

#11924028

yes

—

150

C₂×C₄

#11924029

yes

characteristic

151

C₈

—

152

C₂²

#14556561

yes

characteristic

153

C₂²

#14556566

yes

characteristic

154

C₂²

#14556579

yes

characteristic

155

C₂²

C₂×#11924017

yes

characteristic

156

C₂²

C₂×Dic₂₉₂

yes

characteristic

157

C₂²

#14556624

yes

characteristic

158

C₂²

#14556624

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

292

—

162

C₄

—

292

—

163

C₄

—

584

—

164

C₄

#14556564

yes

—

165

C₄

#14556576

yes

—

166

C₂

#17988505

yes

characteristic

167

C₂

#17988728

yes

characteristic

168

C₂

#17988745

yes

characteristic

169

C₂

#17988793

yes

characteristic

170

C₂

#17988798

yes

characteristic

171

C₂

#17988938

yes

characteristic

172

C₂

#17988956

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₉₂
commutator	C₂²×C₄×C₇₃	C₂³
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	#336×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₄
hypercenter	#336	D₇₃
hypocenter	C₇₃	#2043
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₂₉₂

commutator

C₂²×C₄×C₇₃

C₂³

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

#336×C₇₃

C₂

socle

C₂³×C₇₃

C₂×D₄

D₇₃

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
87. C₇₃	#2043	semidirect

H ≅ G/N

type

87. C₇₃

#2043

semidirect

Subgroups — #22611178

Conjugacy classes of subgroups

Complements