Group #22611014 — subgroups

Subgroups — #22611014

Order 9344.

406 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 99 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	C₂²×#11924015	C₂	1	2	yes	maximal
2	C₂×#14556515	C₂	1	2	yes	maximal, complemented
3	#17988489	C₂	1	1	yes	maximal, characteristic
4	#17988342	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊Q₈×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₉₂	C₂²	1	1	yes	characteristic
7	C₂×#11924015	—	2	4	no	—
8	C₂×#11924015	—	2	4	no	—
9	C₂×#11924015	—	2	4	no	—
10	C₂²×#10596831	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
12	C₂×#11924005	—	2	2	no	—
13	#14556515	—	2	4	no	—
14	C₂²×Q₈×C₇₃	C₂²	1	2	yes	—
15	C₄⋊Q₈×C₇₃	—	2	4	no	—
16	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
17	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
18	C₂³×D₇₃	—	2	1	no	—
19	C₂×D₂₉₂	—	2	2	no	—
20	C₂×D₂₉₂	—	2	2	no	—
21	C₂×D₂₉₂	—	2	1	no	—
22	C₂×D₂₉₂	D₄	1	2	yes	—
23	C₂²×Dic₇₃	—	2	1	no	—
24	#11924015	—	2	16	no	—
25	C₂×#10596831	—	2	2	no	—
26	C₂×#10596831	—	2	2	no	—
27	C₂×#10596831	D₄	1	4	yes	—
28	C₄⋊Dic₇₃	—	2	2	no	—
29	#11924005	—	4	4	no	—
30	C₂×Q₈×C₇₃	—	2	2	no	—
31	C₂×Q₈×C₇₃	—	2	2	no	—
32	C₂×Q₈×C₇₃	—	2	2	no	—
33	C₂×Q₈×C₇₃	—	2	2	no	—
34	C₂×Q₈×C₇₃	—	2	2	no	—
35	C₂×Q₈×C₇₃	D₄	1	4	yes	—
36	C₂²×C₄×C₇₃	—	2	2	no	—
37	C₂²×C₄×C₇₃	D₄	1	2	yes	—
38	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₇₃	—	2	8	no	—
40	C₄²×C₇₃	—	2	2	no	—
41	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×D₇₃	—	2	2	no	—
43	C₂²×D₇₃	—	4	1	no	—
44	C₂²×D₇₃	—	4	1	no	—
45	C₂²×D₇₃	—	4	2	no	—
46	C₂²×D₇₃	—	4	2	no	—
47	Q₈×C₇₃	—	2	4	no	—
48	Q₈×C₇₃	—	2	4	no	—
49	Q₈×C₇₃	—	4	2	no	—
50	Q₈×C₇₃	—	4	2	no	—
51	C₂×C₄×C₇₃	—	2	2	no	—
52	C₂×C₄×C₇₃	—	2	2	no	—
53	C₂×C₄×C₇₃	—	2	4	no	—
54	C₂×C₄×C₇₃	—	4	2	no	—
55	C₂×C₄×C₇₃	—	4	2	no	—
56	C₂×C₄×C₇₃	SD₁₆	1	4	yes	complemented
57	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
61	C₂×Dic₇₃	—	4	1	no	—
62	C₂×Dic₇₃	—	4	2	no	—
63	D₂₉₂	—	2	4	no	—
64	D₂₉₂	—	4	2	no	—
65	#10596831	—	2	4	no	—
66	C₂²×C₇₃	#120	1	1	yes	characteristic
67	C₂²×C₇₃	#121	1	1	yes	characteristic
68	C₂²×C₇₃	#127	1	1	yes	characteristic
69	C₂²×C₇₃	C₂×SD₁₆	1	2	yes	—
70	C₂²×C₇₃	#137	1	2	yes	—
71	C₂×D₇₃	—	4	2	no	—
72	C₂×D₇₃	—	4	4	no	—
73	C₂×D₇₃	—	8	2	no	—
74	C₂×D₇₃	—	8	1	no	—
75	C₂×D₇₃	—	8	1	no	—
76	C₄×C₇₃	—	2	4	no	—
77	C₄×C₇₃	—	4	4	no	—
78	C₄×C₇₃	#120	1	2	yes	—
79	C₄×C₇₃	#121	1	2	yes	—
80	Dic₇₃	—	8	1	no	—
81	C₂×C₇₃	#392	1	1	yes	characteristic
82	C₂×C₇₃	#448	1	2	yes	—
83	C₂×C₇₃	#461	1	2	yes	—
84	C₂×C₇₃	#492	1	1	yes	characteristic
85	C₂×C₇₃	#497	1	1	yes	characteristic
86	D₇₃	—	8	2	no	—
87	#2013	—	73	1	no	maximal, Sylow
88	C₇₃	#2013	1	1	yes	characteristic, complemented, Sylow
89	C₂²×SD₁₆	—	73	2	no	—
90	C₂×C₄⋊Q₈	D₇₃	1	1	yes	characteristic, complemented
91	C₂×#103	—	73	2	no	—
92	#390	—	73	1	no	—
93	#336	—	73	1	no	—
94	C₂²×Q₈	C₂×D₇₃	1	2	yes	—
95	C₂²×D₄	—	73	1	no	—
96	C₂×SD₁₆	—	146	4	no	—
97	C₂×SD₁₆	—	146	4	no	—
98	C₂×SD₁₆	—	146	4	no	—
99	C₂²×C₈	—	73	2	no	—
100	C₄⋊Q₈	—	2	4	no	—
101	C₂×C₄⋊C₄	—	2	2	no	—
102	C₂×C₄⋊C₄	—	73	1	no	—
103	C₂×C₂²⋊C₄	—	146	2	no	—
104	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
105	#103	—	146	4	no	—
106	C₂⁴	—	146	1	no	—
107	C₂×Q₈	—	2	2	no	—
108	C₂×Q₈	—	2	2	no	—
109	C₂×Q₈	—	2	2	no	—
110	C₂×Q₈	—	2	2	no	—
111	C₂×Q₈	—	2	2	no	—
112	C₂×Q₈	C₂²⋊D₇₃	1	4	yes	—
113	C₂×D₄	—	73	2	no	—
114	C₂×D₄	—	146	2	no	—
115	C₂×D₄	—	146	2	no	—
116	C₂×D₄	—	146	1	no	—
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	—	146	1	no	—
119	C₂²×C₄	C₂²⋊D₇₃	1	2	yes	—
120	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
121	SD₁₆	—	146	16	no	—
122	C₂×C₈	—	73	4	no	—
123	C₂×C₈	—	146	2	no	—
124	C₂×C₈	—	146	2	no	—
125	C₄⋊C₄	—	2	8	no	—
126	C₄⋊C₄	—	146	2	no	—
127	C₂²⋊C₄	—	292	4	no	—
128	C₄²	—	2	2	no	—
129	C₂³	—	292	1	no	—
130	C₂³	—	292	1	no	—
131	C₂³	—	292	2	no	—
132	C₂³	—	292	2	no	—
133	C₂³	—	146	2	no	—
134	C₂³	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
135	Q₈	—	2	4	no	—
136	Q₈	—	2	4	no	—
137	Q₈	—	4	2	no	—
138	Q₈	—	4	2	no	—
139	D₄	—	292	2	no	—
140	D₄	—	146	4	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	292	1	no	—
147	C₂×C₄	—	292	2	no	—
148	C₂×C₄	#11924015	1	4	yes	complemented
149	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
150	C₂×C₄	D₄×D₇₃	1	2	yes	—
151	C₂×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
152	C₂×C₄	#11924027	1	1	yes	characteristic
153	C₈	—	146	4	no	—
154	C₂²	—	292	2	no	—
155	C₂²	—	292	4	no	—
156	C₂²	—	584	1	no	—
157	C₂²	—	584	1	no	—
158	C₂²	—	584	2	no	—
159	C₂²	#14556560	1	1	yes	characteristic
160	C₂²	#14556562	1	1	yes	characteristic
161	C₂²	#14556587	1	1	yes	characteristic
162	C₂²	C₂×#11924015	1	2	yes	—
163	C₂²	#14556622	1	2	yes	—
164	C₄	—	2	4	no	—
165	C₄	—	4	4	no	—
166	C₄	—	584	1	no	—
167	C₄	#14556559	1	2	yes	—
168	C₄	#14556563	1	2	yes	—
169	C₂	—	584	2	no	—
170	C₂	#17988495	1	1	yes	characteristic
171	C₂	#17988726	1	2	yes	—
172	C₂	#17988789	1	2	yes	—
173	C₂	#17988961	1	1	yes	characteristic
174	C₂	#17988979	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11924015

C₂

yes

maximal

C₂×#14556515

C₂

yes

maximal, complemented

#17988489

C₂

yes

maximal, characteristic

#17988342

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊Q₈×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₉₂

C₂²

yes

—

—

—

C₂²

yes

—

C₂×C₄⋊Dic₇₃

C₂²

yes

characteristic

C₂×#11924005

—

#14556515

—

C₂²×Q₈×C₇₃

C₂²

yes

—

C₄⋊Q₈×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₂³×D₇₃

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

D₄

yes

—

C₂²×Dic₇₃

—

—

—

—

D₄

yes

—

C₄⋊Dic₇₃

—

#11924005

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

D₄

yes

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

D₄

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

SD₁₆

yes

complemented

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

#41

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

#10596831

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

C₂×SD₁₆

yes

—

C₂²×C₇₃

#137

yes

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#120

yes

—

C₄×C₇₃

#121

yes

—

Dic₇₃

—

C₂×C₇₃

#392

yes

characteristic

C₂×C₇₃

#448

yes

—

C₂×C₇₃

#461

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#2013

—

maximal, Sylow

C₇₃

#2013

yes

characteristic, complemented, Sylow

C₂²×SD₁₆

—

C₂×C₄⋊Q₈

D₇₃

yes

characteristic, complemented

C₂×#103

—

#390

—

#336

—

C₂²×Q₈

C₂×D₇₃

yes

—

C₂²×D₄

—

C₂×SD₁₆

—

146

—

C₂×SD₁₆

—

146

—

C₂×SD₁₆

—

146

—

C₂²×C₈

—

100

C₄⋊Q₈

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

—

103

C₂×C₂²⋊C₄

—

146

—

104

C₂×C₄²

C₂×D₇₃

yes

characteristic

105

#103

—

146

—

106

C₂⁴

—

146

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

C₂²⋊D₇₃

yes

—

113

C₂×D₄

—

114

C₂×D₄

—

146

—

115

C₂×D₄

—

146

—

116

C₂×D₄

—

146

—

117

C₂²×C₄

—

118

C₂²×C₄

—

146

—

119

C₂²×C₄

C₂²⋊D₇₃

yes

—

120

C₂²×C₄

C₂²×D₇₃

yes

characteristic

121

SD₁₆

—

146

—

122

C₂×C₈

—

123

C₂×C₈

—

146

—

124

C₂×C₈

—

146

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

146

—

127

C₂²⋊C₄

—

292

—

128

C₄²

—

129

C₂³

—

292

—

130

C₂³

—

292

—

131

C₂³

—

292

—

132

C₂³

—

292

—

133

C₂³

—

146

—

134

C₂³

C₂×C₂²⋊D₇₃

yes

characteristic

135

Q₈

—

136

Q₈

—

137

Q₈

—

138

Q₈

—

139

D₄

—

292

—

140

D₄

—

146

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

292

—

147

C₂×C₄

—

292

—

148

C₂×C₄

#11924015

yes

complemented

149

C₂×C₄

D₄×D₇₃

yes

characteristic

150

C₂×C₄

D₄×D₇₃

yes

—

151

C₂×C₄

C₂×C₂²⋊D₇₃

yes

—

152

C₂×C₄

#11924027

yes

characteristic

153

C₈

—

146

—

154

C₂²

—

292

—

155

C₂²

—

292

—

156

C₂²

—

584

—

157

C₂²

—

584

—

158

C₂²

—

584

—

159

C₂²

#14556560

yes

characteristic

160

C₂²

#14556562

yes

characteristic

161

C₂²

#14556587

yes

characteristic

162

C₂²

C₂×#11924015

yes

—

163

C₂²

#14556622

yes

—

164

C₄

—

165

C₄

—

166

C₄

—

584

—

167

C₄

#14556559

yes

—

168

C₄

#14556563

yes

—

169

C₂

—

584

—

170

C₂

#17988495

yes

characteristic

171

C₂

#17988726

yes

—

172

C₂

#17988789

yes

—

173

C₂

#17988961

yes

characteristic

174

C₂

#17988979

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₃
commutator	C₂²×C₄×C₇₃	C₂³
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	C₂×C₄⋊Q₈×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₇₃
hypocenter	C₇₃	#2013
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₃

commutator

C₂²×C₄×C₇₃

C₂³

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

C₂×C₄⋊Q₈×C₇₃

C₂

socle

C₂³×C₇₃

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₇₃

hypocenter

C₇₃

#2013

LCS

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. C₂×#14556515	C₂	semidirect
4. #17988342	C₂	semidirect
5. C₂×C₄⋊Q₈×C₇₃	C₂	semidirect
56. C₂×C₄×C₇₃	SD₁₆	semidirect
88. C₇₃	#2013	semidirect
90. C₂×C₄⋊Q₈	D₇₃	semidirect
148. C₂×C₄	#11924015	semidirect

H ≅ G/N

type

2. C₂×#14556515

C₂

semidirect

4. #17988342

C₂

semidirect

5. C₂×C₄⋊Q₈×C₇₃

C₂

semidirect

56. C₂×C₄×C₇₃

SD₁₆

semidirect

88. C₇₃

#2013

semidirect

90. C₂×C₄⋊Q₈

D₇₃

semidirect

148. C₂×C₄

#11924015

semidirect

Subgroups — #22611014

Conjugacy classes of subgroups

Complements