Group #22626240 — subgroups

Subgroups — #22626240

Order 9344.

260 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 64 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	#17989643	C₂	1	1	yes	maximal, characteristic, complemented
2	#17989597	C₂	1	1	yes	maximal, characteristic, complemented
3	#17989499	C₂	1	4	yes	maximal, complemented
4	C₂×D₈×D₇₃	C₂	1	1	yes	maximal, characteristic
5	#14556670	—	2	1	no	—
6	#14556666	—	2	1	no	—
7	#14556665	—	2	1	no	—
8	#14556664	C₂²	1	2	yes	complemented
9	#14556660	C₂²	1	2	yes	complemented
10	#14556659	—	2	1	no	—
11	C₂×D₄×D₇₃	—	2	1	no	—
12	C₂×D₅₈₄	C₄	1	1	yes	characteristic, complemented
13	C₂×#11924013	—	2	1	no	—
14	D₈×D₇₃	—	2	1	no	—
15	D₈×D₇₃	—	2	1	no	—
16	D₈×D₇₃	C₄	1	2	yes	complemented
17	D₈×D₇₃	C₂²	1	2	yes	—
18	C₂×C₈×D₇₃	C₂²	1	1	yes	characteristic
19	C₂×D₈×C₇₃	C₄	1	1	yes	characteristic, complemented
20	C₄⋊#64408	—	2	2	no	—
21	C₂²⋊#64408	—	4	1	no	—
22	C₄×#64408	—	4	1	no	—
23	#11924031	—	2	2	no	—
24	C₂³×D₇₃	—	4	1	no	—
25	C₂×D₂₉₂	—	2	1	no	—
26	C₂×C₂²⋊D₇₃	—	4	1	no	—
27	D₄×D₇₃	—	2	2	no	—
28	D₄×D₇₃	—	2	2	no	—
29	D₄×D₇₃	—	4	1	no	—
30	D₄×D₇₃	—	4	1	no	—
31	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
32	D₅₈₄	—	2	1	no	—
33	D₅₈₄	C₂×C₄	1	2	yes	—
34	#11924013	—	2	2	no	—
35	#11924013	—	2	2	no	—
36	C₂×#10596831	D₄	1	1	yes	characteristic
37	C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
38	C₈×D₇₃	D₄	1	1	yes	characteristic
39	C₈×D₇₃	D₄	1	1	yes	characteristic
40	C₈×D₇₃	C₂³	1	1	yes	characteristic
41	C₂×D₄×C₇₃	—	2	1	no	—
42	D₈×C₇₃	—	2	1	no	—
43	D₈×C₇₃	C₂×C₄	1	2	yes	—
44	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₇₃	—	4	1	no	—
46	C₂²×D₇₃	—	4	1	no	—
47	C₂²×D₇₃	—	4	2	no	—
48	C₂²×D₇₃	—	4	2	no	—
49	C₂²×D₇₃	—	8	4	no	—
50	C₂²×D₇₃	D₈	1	1	yes	characteristic
51	C₂×#64408	—	4	2	no	—
52	D₄×C₇₃	—	2	2	no	—
53	D₄×C₇₃	—	4	1	no	—
54	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²⋊D₇₃	—	4	2	no	—
56	C₂²⋊D₇₃	—	4	2	no	—
57	C₂×Dic₇₃	SD₁₆	1	1	yes	characteristic
58	D₂₉₂	—	2	2	no	—
59	D₂₉₂	—	4	1	no	—
60	C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₄×D₇₃	D₈	1	1	yes	characteristic
62	C₄×D₇₃	SD₁₆	1	1	yes	characteristic
63	C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
64	C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
66	#10596831	C₂²⋊C₄	1	1	yes	characteristic
67	#10596831	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₇₃	—	4	2	no	—
69	C₂²×C₇₃	—	8	2	no	—
70	C₂²×C₇₃	#102	1	1	yes	characteristic
71	C₂×D₇₃	—	2	2	no	—
72	C₂×D₇₃	—	4	2	no	—
73	C₂×D₇₃	—	8	4	no	—
74	C₂×D₇₃	—	8	2	no	—
75	C₂×D₇₃	—	8	4	no	—
76	C₂×D₇₃	#102	1	1	yes	characteristic
77	C₂×D₇₃	C₂×D₈	1	1	yes	characteristic
78	#64408	—	8	2	no	—
79	C₄×C₇₃	#102	1	1	yes	characteristic
80	C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	Dic₇₃	#102	1	1	yes	characteristic
82	Dic₇₃	C₂×SD₁₆	1	1	yes	characteristic
83	C₂×C₇₃	—	2	1	no	—
84	C₂×C₇₃	—	8	2	no	—
85	C₂×C₇₃	C₂×#102	1	1	yes	characteristic
86	D₇₃	—	2	1	no	—
87	D₇₃	—	8	2	no	—
88	D₇₃	#509	1	2	yes	—
89	#2140	—	73	1	no	maximal, Sylow
90	C₇₃	#2140	1	1	yes	characteristic, complemented, Sylow
91	C₂²×D₈	—	73	1	no	—
92	C₂×M₃₂	—	73	1	no	—
93	#427	—	73	1	no	—
94	#357	—	73	4	no	—
95	C₂²×D₄	—	146	1	no	—
96	C₂×D₈	—	73	2	no	—
97	C₂×D₈	—	73	2	no	—
98	C₂×D₈	—	73	1	no	—
99	C₂×D₈	—	146	1	no	—
100	C₂×D₈	—	146	1	no	—
101	C₂×D₈	—	146	1	no	—
102	C₂×D₈	#64408	1	1	yes	characteristic, complemented
103	C₂²×C₈	—	73	1	no	—
104	#117	—	146	1	no	—
105	M₃₂	—	146	1	no	—
106	M₃₂	—	146	1	no	—
107	C₂×C₁₆	—	73	2	no	—
108	#107	—	73	2	no	—
109	#106	—	146	1	no	—
110	C₂⁴	—	292	1	no	—
111	C₂×D₄	—	2	1	no	—
112	C₂×D₄	—	292	1	no	—
113	C₂×D₄	—	292	1	no	—
114	C₂×D₄	—	292	1	no	—
115	C₂×D₄	—	146	2	no	—
116	C₂×D₄	—	146	2	no	—
117	C₂×D₄	—	146	1	no	—
118	C₂²×C₄	—	73	1	no	—
119	D₈	—	2	1	no	—
120	D₈	—	73	2	no	—
121	D₈	—	146	2	no	—
122	D₈	—	146	2	no	—
123	D₈	—	146	1	no	—
124	D₈	C₂×#64408	1	2	yes	—
125	C₂×C₈	—	73	1	no	—
126	C₂×C₈	—	73	1	no	—
127	C₂×C₈	—	73	1	no	—
128	C₂×C₈	—	73	1	no	—
129	C₂×C₈	—	73	1	no	—
130	C₂×C₈	C₂×#64408	1	1	yes	characteristic
131	C₄⋊C₄	—	146	2	no	—
132	C₂²⋊C₄	—	292	1	no	—
133	C₄²	—	292	1	no	—
134	C₁₆	—	146	2	no	—
135	C₂³	—	4	1	no	—
136	C₂³	—	292	2	no	—
137	C₂³	—	292	2	no	—
138	C₂³	—	292	1	no	—
139	C₂³	—	584	4	no	—
140	C₂³	—	73	1	no	—
141	D₄	—	2	2	no	—
142	D₄	—	4	1	no	—
143	D₄	—	292	2	no	—
144	D₄	—	292	2	no	—
145	D₄	—	292	1	no	—
146	D₄	—	146	2	no	—
147	C₂×C₄	—	292	2	no	—
148	C₂×C₄	—	73	1	no	—
149	C₂×C₄	—	73	1	no	—
150	C₂×C₄	—	73	1	no	—
151	C₂×C₄	—	73	1	no	—
152	C₂×C₄	—	73	1	no	—
153	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
154	C₈	—	73	1	no	—
155	C₈	—	73	1	no	—
156	C₈	C₂²⋊#64408	1	1	yes	characteristic
157	C₈	C₂²×#64408	1	1	yes	characteristic
158	C₂²	—	4	2	no	—
159	C₂²	—	8	2	no	—
160	C₂²	—	292	2	no	—
161	C₂²	—	584	2	no	—
162	C₂²	—	584	4	no	—
163	C₂²	—	584	4	no	—
164	C₂²	—	73	1	no	—
165	C₂²	—	73	1	no	—
166	C₂²	—	146	2	no	—
167	C₂²	#14556653	1	1	yes	characteristic
168	C₄	—	584	2	no	—
169	C₄	—	73	1	no	—
170	C₄	—	73	1	no	—
171	C₄	#14556653	1	1	yes	characteristic
172	C₄	C₂×C₂²⋊#64408	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	8	2	no	—
175	C₂	—	584	2	no	—
176	C₂	—	73	2	no	—
177	C₂	—	146	1	no	—
178	C₂	C₂×#14556653	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989643

C₂

yes

maximal, characteristic, complemented

#17989597

C₂

yes

maximal, characteristic, complemented

#17989499

C₂

yes

maximal, complemented

C₂×D₈×D₇₃

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

complemented

#14556660

C₂²

yes

complemented

#14556659

—

C₂×D₄×D₇₃

—

C₂×D₅₈₄

C₄

yes

characteristic, complemented

C₂×#11924013

—

D₈×D₇₃

—

D₈×D₇₃

—

D₈×D₇₃

C₄

yes

complemented

D₈×D₇₃

C₂²

yes

—

C₂×C₈×D₇₃

C₂²

yes

characteristic

C₂×D₈×C₇₃

C₄

yes

characteristic, complemented

—

—

—

—

C₂³×D₇₃

—

C₂×D₂₉₂

—

C₂×C₂²⋊D₇₃

—

D₄×D₇₃

—

D₄×D₇₃

—

D₄×D₇₃

—

D₄×D₇₃

—

C₂×C₄×D₇₃

D₄

yes

characteristic

D₅₈₄

—

D₅₈₄

C₂×C₄

yes

—

#11924013

—

#11924013

—

C₂×#10596831

D₄

yes

characteristic

C₈×D₇₃

C₂×C₄

yes

characteristic

C₈×D₇₃

D₄

yes

characteristic

C₈×D₇₃

D₄

yes

characteristic

C₈×D₇₃

C₂³

yes

characteristic

C₂×D₄×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

C₂×C₄

yes

—

C₂×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂³×C₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

D₈

yes

characteristic

C₂×#64408

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂²⋊D₇₃

—

C₂²⋊D₇₃

—

C₂×Dic₇₃

SD₁₆

yes

characteristic

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

C₂²⋊C₄

yes

characteristic

C₄×D₇₃

D₈

yes

characteristic

C₄×D₇₃

SD₁₆

yes

characteristic

C₄×D₇₃

C₂×D₄

yes

characteristic

C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₈×C₇₃

C₂²×C₄

yes

characteristic

#10596831

C₂²⋊C₄

yes

characteristic

#10596831

C₂×D₄

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#102

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

#102

yes

characteristic

C₂×D₇₃

C₂×D₈

yes

characteristic

#64408

—

C₄×C₇₃

#102

yes

characteristic

C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

Dic₇₃

#102

yes

characteristic

Dic₇₃

C₂×SD₁₆

yes

characteristic

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

C₂×#102

yes

characteristic

D₇₃

—

D₇₃

—

D₇₃

#509

yes

—

#2140

—

maximal, Sylow

C₇₃

#2140

yes

characteristic, complemented, Sylow

C₂²×D₈

—

C₂×M₃₂

—

#427

—

#357

—

C₂²×D₄

—

146

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

146

—

100

C₂×D₈

—

146

—

101

C₂×D₈

—

146

—

102

C₂×D₈

#64408

yes

characteristic, complemented

103

C₂²×C₈

—

104

#117

—

146

—

105

M₃₂

—

146

—

106

M₃₂

—

146

—

107

C₂×C₁₆

—

108

#107

—

109

#106

—

146

—

110

C₂⁴

—

292

—

111

C₂×D₄

—

112

C₂×D₄

—

292

—

113

C₂×D₄

—

292

—

114

C₂×D₄

—

292

—

115

C₂×D₄

—

146

—

116

C₂×D₄

—

146

—

117

C₂×D₄

—

146

—

118

C₂²×C₄

—

119

D₈

—

120

D₈

—

121

D₈

—

146

—

122

D₈

—

146

—

123

D₈

—

146

—

124

D₈

C₂×#64408

yes

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

C₂×#64408

yes

characteristic

131

C₄⋊C₄

—

146

—

132

C₂²⋊C₄

—

292

—

133

C₄²

—

292

—

134

C₁₆

—

146

—

135

C₂³

—

136

C₂³

—

292

—

137

C₂³

—

292

—

138

C₂³

—

292

—

139

C₂³

—

584

—

140

C₂³

—

141

D₄

—

142

D₄

—

143

D₄

—

292

—

144

D₄

—

292

—

145

D₄

—

292

—

146

D₄

—

146

—

147

C₂×C₄

—

292

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

C₂²⋊#64408

yes

characteristic

154

C₈

—

155

C₈

—

156

C₈

C₂²⋊#64408

yes

characteristic

157

C₈

C₂²×#64408

yes

characteristic

158

C₂²

—

159

C₂²

—

160

C₂²

—

292

—

161

C₂²

—

584

—

162

C₂²

—

584

—

163

C₂²

—

584

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

146

—

167

C₂²

#14556653

yes

characteristic

168

C₄

—

584

—

169

C₄

—

170

C₄

—

171

C₄

#14556653

yes

characteristic

172

C₄

C₂×C₂²⋊#64408

yes

characteristic

173

C₂

—

174

C₂

—

175

C₂

—

584

—

176

C₂

—

177

C₂

—

146

—

178

C₂

C₂×#14556653

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#14556653
commutator	C₈×C₇₃	C₂²×C₄
Frattini	C₈	C₂²×#64408
Fitting	C₂×D₈×C₇₃	C₄
socle	C₂×C₇₃	C₂×#102
hypercenter	C₂×D₈	#64408
hypocenter	C₇₃	#2140
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₈

subgroup

G/H

center

C₂

C₂×#14556653

commutator

C₈×C₇₃

C₂²×C₄

Frattini

C₈

C₂²×#64408

Fitting

C₂×D₈×C₇₃

C₄

C₂×C₇₃

C₂×D₈

C₇₃

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₈

N	H ≅ G/N	type
1. #17989643	C₂	semidirect
2. #17989597	C₂	semidirect
3. #17989499	C₂	semidirect
8. #14556664	C₂²	semidirect
9. #14556660	C₂²	semidirect
12. C₂×D₅₈₄	C₄	semidirect
16. D₈×D₇₃	C₄	semidirect
19. C₂×D₈×C₇₃	C₄	semidirect
90. C₇₃	#2140	semidirect
102. C₂×D₈	#64408	semidirect

H ≅ G/N

type

1. #17989643

C₂

semidirect

2. #17989597

C₂

semidirect

3. #17989499

C₂

semidirect

8. #14556664

C₂²

semidirect

9. #14556660

C₂²

semidirect

12. C₂×D₅₈₄

C₄

semidirect

16. D₈×D₇₃

C₄

semidirect

19. C₂×D₈×C₇₃

C₄

semidirect

90. C₇₃

#2140

semidirect

102. C₂×D₈

#64408

semidirect

Subgroups — #22626240

Conjugacy classes of subgroups

Complements