Group #22609290 — subgroups

Subgroups — #22609290

Order 9344.

188 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 53 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	#17988963	C₂	1	1	yes	maximal, characteristic
2	#17988788	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988784	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988711	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988339	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988330	C₂	1	1	yes	maximal, characteristic, complemented
7	#331×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11924017	—	2	1	no	—
9	C₂×#11924016	—	2	1	no	—
10	C₂×#11924013	—	2	1	no	—
11	C₂×#11924013	—	2	1	no	—
12	#14556588	C₂²	1	1	yes	characteristic
13	C₄×Dic₁₄₆	C₂²	1	1	yes	characteristic
14	C₄⋊#10596831	C₂²	1	1	yes	characteristic
15	#14556520	—	2	1	no	—
16	#14556513	—	2	1	no	—
17	#14556512	—	2	1	no	—
18	C₄×#10596831	C₂²	1	1	yes	characteristic
19	C₈×Dic₇₃	—	2	1	no	—
20	#127×C₇₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₄₆	D₄	1	1	yes	characteristic
24	C₂×D₂₉₂	—	2	1	no	—
25	C₂×D₂₉₂	—	2	1	no	—
26	C₂×D₂₉₂	—	2	1	no	—
27	#11924017	—	4	2	no	—
28	#11924016	—	4	2	no	—
29	#11924013	—	4	2	no	—
30	#11924013	—	4	2	no	—
31	C₂×#10596831	—	2	1	no	—
32	C₂×#10596831	D₄	1	1	yes	characteristic
33	C₂×#10596831	D₄	1	1	yes	characteristic
34	C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
35	#11924007	—	2	1	no	—
36	C₄×Dic₇₃	—	2	1	no	—
37	C₂×D₄×C₇₃	—	2	1	no	—
38	C₂×D₄×C₇₃	—	2	1	no	—
39	C₂×D₄×C₇₃	—	2	1	no	—
40	C₂×C₈×C₇₃	—	2	1	no	—
41	C₂×C₈×C₇₃	—	2	1	no	—
42	C₄²×C₇₃	C₂³	1	1	yes	characteristic
43	C₂³×C₇₃	—	4	1	no	—
44	C₂²×D₇₃	—	4	1	no	—
45	D₄×C₇₃	—	4	1	no	—
46	D₄×C₇₃	—	4	1	no	—
47	D₄×C₇₃	—	4	1	no	—
48	D₄×C₇₃	—	4	1	no	—
49	D₄×C₇₃	—	8	1	no	—
50	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
53	C₂×Dic₇₃	—	2	1	no	—
54	C₂×Dic₇₃	—	2	1	no	—
55	D₂₉₂	—	4	1	no	—
56	D₂₉₂	—	4	1	no	—
57	D₂₉₂	—	4	1	no	—
58	D₂₉₂	—	4	1	no	—
59	D₂₉₂	—	8	1	no	—
60	Dic₁₄₆	—	2	1	no	—
61	Dic₁₄₆	D₈	1	2	yes	complemented
62	C₈×C₇₃	—	4	1	no	—
63	C₈×C₇₃	—	4	1	no	—
64	#10596831	—	2	1	no	—
65	#10596831	—	4	1	no	—
66	#10596831	SD₁₆	1	2	yes	complemented
67	C₂²×C₇₃	—	8	1	no	—
68	C₂²×C₇₃	—	8	1	no	—
69	C₂²×C₇₃	—	8	1	no	—
70	C₂²×C₇₃	#121	1	1	yes	characteristic
71	C₂×D₇₃	—	8	1	no	—
72	C₂×D₇₃	—	8	1	no	—
73	C₂×D₇₃	—	8	1	no	—
74	C₄×C₇₃	—	2	1	no	—
75	C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
76	C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
77	C₄×C₇₃	#135	1	1	yes	characteristic
78	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
79	Dic₇₃	—	2	2	no	—
80	Dic₇₃	—	4	1	no	—
81	C₂×C₇₃	—	16	1	no	—
82	C₂×C₇₃	#460	1	1	yes	characteristic
83	C₂×C₇₃	#466	1	1	yes	characteristic
84	C₂×C₇₃	#472	1	1	yes	characteristic
85	D₇₃	—	16	1	no	—
86	#1668	—	73	1	no	maximal, Sylow
87	C₇₃	#1668	1	1	yes	characteristic, complemented, Sylow
88	#493	—	73	1	no	—
89	#460	—	73	1	no	—
90	#460	—	73	1	no	—
91	C₈×Q₈	—	73	1	no	—
92	#335	—	73	1	no	—
93	#331	—	73	1	no	—
94	#331	D₇₃	1	1	yes	characteristic, complemented
95	C₂×SD₁₆	—	146	1	no	—
96	C₂×SD₁₆	—	146	1	no	—
97	C₂×D₈	—	146	1	no	—
98	C₂×D₈	—	146	1	no	—
99	#127	—	73	1	no	—
100	#127	C₂×D₇₃	1	1	yes	characteristic
101	C₄×Q₈	—	73	1	no	—
102	C₄⋊C₈	—	73	1	no	—
103	C₄⋊C₈	—	146	1	no	—
104	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
105	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
106	#102	—	146	1	no	—
107	#102	—	146	1	no	—
108	C₄×C₈	—	73	1	no	—
109	C₄×C₈	—	146	1	no	—
110	C₂×Q₈	—	73	1	no	—
111	C₂×D₄	—	2	1	no	—
112	C₂×D₄	—	2	1	no	—
113	C₂×D₄	—	2	1	no	—
114	C₂×D₄	—	146	1	no	—
115	C₂×D₄	—	146	1	no	—
116	C₂×D₄	—	146	1	no	—
117	SD₁₆	—	292	2	no	—
118	SD₁₆	—	292	2	no	—
119	D₈	—	292	2	no	—
120	D₈	—	292	2	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	73	1	no	—
124	C₂×C₈	—	73	1	no	—
125	C₂×C₈	—	146	1	no	—
126	C₄⋊C₄	—	73	1	no	—
127	C₄⋊C₄	—	146	1	no	—
128	C₄²	—	146	1	no	—
129	C₄²	C₂²×D₇₃	1	1	yes	characteristic
130	C₂³	—	4	1	no	—
131	C₂³	—	292	1	no	—
132	Q₈	—	73	2	no	—
133	Q₈	—	146	1	no	—
134	D₄	—	4	1	no	—
135	D₄	—	4	1	no	—
136	D₄	—	4	1	no	—
137	D₄	—	4	1	no	—
138	D₄	—	8	1	no	—
139	D₄	—	292	1	no	—
140	D₄	—	292	1	no	—
141	D₄	—	292	1	no	—
142	D₄	—	292	1	no	—
143	D₄	—	584	1	no	—
144	C₂×C₄	—	146	1	no	—
145	C₂×C₄	—	146	1	no	—
146	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
147	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
148	C₂×C₄	#11924028	1	1	yes	characteristic
149	C₈	—	4	1	no	—
150	C₈	—	4	1	no	—
151	C₈	—	292	1	no	—
152	C₈	—	73	2	no	—
153	C₈	—	146	1	no	—
154	C₂²	—	8	1	no	—
155	C₂²	—	8	1	no	—
156	C₂²	—	8	1	no	—
157	C₂²	—	584	1	no	—
158	C₂²	—	584	1	no	—
159	C₂²	—	584	1	no	—
160	C₂²	#14556564	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	292	1	no	—
163	C₄	—	146	2	no	—
164	C₄	D₈×D₇₃	1	1	yes	characteristic
165	C₄	SD₁₆×D₇₃	1	1	yes	characteristic
166	C₄	#14556612	1	1	yes	characteristic
167	C₄	#14556615	1	1	yes	characteristic
168	C₂	—	16	1	no	—
169	C₂	—	1168	1	no	—
170	C₂	#17988786	1	1	yes	characteristic
171	C₂	#17988822	1	1	yes	characteristic
172	C₂	#17988856	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988963

C₂

yes

maximal, characteristic

#17988788

C₂

yes

maximal, characteristic, complemented

#17988784

C₂

yes

maximal, characteristic, complemented

#17988711

C₂

yes

maximal, characteristic, complemented

#17988339

C₂

yes

maximal, characteristic, complemented

#17988330

C₂

yes

maximal, characteristic, complemented

#331×C₇₃

C₂

yes

maximal, characteristic, complemented

—

—

—

—

C₂²

yes

characteristic

C₄×Dic₁₄₆

C₂²

yes

characteristic

C₄⋊#10596831

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₈×Dic₇₃

—

#127×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

D₄

yes

characteristic

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

—

—

—

—

—

D₄

yes

characteristic

C₂×#10596831

D₄

yes

characteristic

C₄⋊Dic₇₃

D₄

yes

characteristic

#11924007

—

C₄×Dic₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

—

C₂²×D₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

#41

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

Dic₁₄₆

—

Dic₁₄₆

D₈

yes

complemented

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

#10596831

SD₁₆

yes

complemented

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#121

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×D₈

yes

characteristic

C₄×C₇₃

C₂×SD₁₆

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#1668

—

maximal, Sylow

C₇₃

#1668

yes

characteristic, complemented, Sylow

#493

—

#460

—

#460

—

C₈×Q₈

—

#335

—

#331

—

#331

D₇₃

yes

characteristic, complemented

C₂×SD₁₆

—

146

—

C₂×SD₁₆

—

146

—

C₂×D₈

—

146

—

C₂×D₈

—

146

—

#127

—

100

#127

C₂×D₇₃

yes

characteristic

101

C₄×Q₈

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

146

—

104

C₄⋊C₈

C₂×D₇₃

yes

characteristic

105

C₄⋊C₈

C₂×D₇₃

yes

characteristic

106

#102

—

146

—

107

#102

—

146

—

108

C₄×C₈

—

109

C₄×C₈

—

146

—

110

C₂×Q₈

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂×D₄

—

146

—

115

C₂×D₄

—

146

—

116

C₂×D₄

—

146

—

117

SD₁₆

—

292

—

118

SD₁₆

—

292

—

119

D₈

—

292

—

120

D₈

—

292

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

146

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

146

—

128

C₄²

—

146

—

129

C₄²

C₂²×D₇₃

yes

characteristic

130

C₂³

—

131

C₂³

—

292

—

132

Q₈

—

133

Q₈

—

146

—

134

D₄

—

135

D₄

—

136

D₄

—

137

D₄

—

138

D₄

—

139

D₄

—

292

—

140

D₄

—

292

—

141

D₄

—

292

—

142

D₄

—

292

—

143

D₄

—

584

—

144

C₂×C₄

—

146

—

145

C₂×C₄

—

146

—

146

C₂×C₄

D₄×D₇₃

yes

characteristic

147

C₂×C₄

D₄×D₇₃

yes

characteristic

148

C₂×C₄

#11924028

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

—

292

—

152

C₈

—

153

C₈

—

146

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

584

—

158

C₂²

—

584

—

159

C₂²

—

584

—

160

C₂²

#14556564

yes

characteristic

161

C₄

—

162

C₄

—

292

—

163

C₄

—

146

—

164

C₄

D₈×D₇₃

yes

characteristic

165

C₄

SD₁₆×D₇₃

yes

characteristic

166

C₄

#14556612

yes

characteristic

167

C₄

#14556615

yes

characteristic

168

C₂

—

169

C₂

—

1168

—

170

C₂

#17988786

yes

characteristic

171

C₂

#17988822

yes

characteristic

172

C₂

#17988856

yes

characteristic

	subgroup	G/H
center	C₂²	#14556564
commutator	C₄²×C₇₃	C₂³
Frattini	C₄²	C₂²×D₇₃
Fitting	#331×C₇₃	C₂
socle	C₂²×C₇₃	#121
hypercenter	#331	D₇₃
hypocenter	C₇₃	#1668
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

#14556564

commutator

C₄²×C₇₃

C₂³

Frattini

C₄²

C₂²×D₇₃

Fitting

#331×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
2. #17988788	C₂	semidirect
3. #17988784	C₂	semidirect
4. #17988711	C₂	semidirect
5. #17988339	C₂	semidirect
6. #17988330	C₂	semidirect
7. #331×C₇₃	C₂	semidirect
61. Dic₁₄₆	D₈	semidirect
66. #10596831	SD₁₆	semidirect
87. C₇₃	#1668	semidirect
94. #331	D₇₃	semidirect

H ≅ G/N

type

2. #17988788

C₂

semidirect

3. #17988784

C₂

semidirect

4. #17988711

C₂

semidirect

5. #17988339

C₂

semidirect

6. #17988330

C₂

semidirect

7. #331×C₇₃

C₂

semidirect

61. Dic₁₄₆

D₈

semidirect

66. #10596831

SD₁₆

semidirect

87. C₇₃

#1668

semidirect

94. #331

D₇₃

semidirect

Subgroups — #22609290

Conjugacy classes of subgroups

Complements