Group #22608529 — subgroups

Subgroups — #22608529

Order 9344.

264 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 81 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	#17988644	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#11924012	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988327	C₂	1	4	yes	maximal, complemented
4	C₂×#124×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11924012	—	2	1	no	—
6	C₄⋊#10596831	—	2	1	no	—
7	C₂²⋊#10596831	—	2	1	no	—
8	#14556502	C₂²	1	2	yes	complemented
9	#14556502	C₂²	1	2	yes	complemented
10	C₄×#10596831	—	2	1	no	—
11	C₂²×Q₈×C₇₃	C₄	1	1	yes	characteristic
12	C₂²×D₄×C₇₃	C₄	1	1	yes	characteristic
13	#124×C₇₃	—	2	1	no	—
14	#124×C₇₃	—	2	1	no	—
15	#124×C₇₃	C₄	1	2	yes	—
16	#124×C₇₃	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₄×C₇₃	—	2	1	no	—
18	C₂×C₂²⋊C₄×C₇₃	—	2	1	no	—
19	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
20	#11924012	—	2	4	no	—
21	C₂×#10596831	—	2	2	no	—
22	C₂×#10596831	—	2	2	no	—
23	C₂⁴×C₇₃	—	2	1	no	—
24	C₂×Q₈×C₇₃	—	2	1	no	—
25	C₂×Q₈×C₇₃	—	4	2	no	—
26	C₂×Q₈×C₇₃	C₂×C₄	1	2	yes	—
27	C₂×D₄×C₇₃	—	2	1	no	—
28	C₂×D₄×C₇₃	—	4	2	no	—
29	C₂×D₄×C₇₃	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₇₃	—	2	1	no	—
31	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
34	C₂²⋊C₄×C₇₃	—	2	2	no	—
35	C₂²⋊C₄×C₇₃	—	2	2	no	—
36	C₂²⋊C₄×C₇₃	—	2	2	no	—
37	C₂²⋊C₄×C₇₃	—	2	2	no	—
38	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₇₃	D₄	1	1	yes	characteristic
40	C₄²×C₇₃	D₄	1	1	yes	characteristic
41	C₄²×C₇₃	C₂³	1	1	yes	characteristic
42	C₂³×C₇₃	—	2	2	no	—
43	C₂³×C₇₃	—	4	2	no	—
44	C₂³×C₇₃	—	4	2	no	—
45	C₂³×C₇₃	—	4	2	no	—
46	C₂³×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
47	Q₈×C₇₃	—	4	2	no	—
48	Q₈×C₇₃	—	4	2	no	—
49	D₄×C₇₃	—	4	2	no	—
50	D₄×C₇₃	—	4	2	no	—
51	C₂×C₄×C₇₃	—	2	2	no	—
52	C₂×C₄×C₇₃	—	2	2	no	—
53	C₂×C₄×C₇₃	—	2	2	no	—
54	C₂×C₄×C₇₃	—	4	1	no	—
55	C₂×C₄×C₇₃	—	4	2	no	—
56	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
62	#10596831	—	2	4	no	—
63	#10596831	—	4	2	no	—
64	C₂²×C₇₃	—	2	1	no	—
65	C₂²×C₇₃	—	2	1	no	—
66	C₂²×C₇₃	—	4	2	no	—
67	C₂²×C₇₃	—	4	2	no	—
68	C₂²×C₇₃	—	4	2	no	—
69	C₂²×C₇₃	—	8	4	no	—
70	C₂²×C₇₃	#104	1	2	yes	—
71	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₄×C₇₃	—	2	1	no	—
73	C₄×C₇₃	—	2	1	no	—
74	C₄×C₇₃	—	2	1	no	—
75	C₄×C₇₃	—	2	1	no	—
76	C₄×C₇₃	—	2	1	no	—
77	C₄×C₇₃	—	4	2	no	—
78	C₄×C₇₃	#100	1	2	yes	—
79	C₂×C₇₃	—	2	2	no	—
80	C₂×C₇₃	—	8	2	no	—
81	C₂×C₇₃	C₂×#100	1	1	yes	characteristic
82	C₂×C₇₃	C₂×#104	1	1	yes	characteristic
83	C₂×C₇₃	#421	1	1	yes	characteristic
84	#1527	—	73	1	no	maximal, Sylow
85	C₇₃	#1527	1	1	yes	characteristic, complemented, Sylow
86	C₂×#124	D₇₃	1	1	yes	characteristic
87	#432	—	73	1	no	—
88	C₄×M₁₆	—	73	1	no	—
89	#329	—	73	4	no	—
90	C₂²×Q₈	Dic₇₃	1	1	yes	characteristic
91	C₂²×D₄	Dic₇₃	1	1	yes	characteristic
92	C₂×M₁₆	—	146	1	no	—
93	#124	—	2	1	no	—
94	#124	—	2	1	no	—
95	#124	Dic₇₃	1	2	yes	—
96	#124	C₂×D₇₃	1	2	yes	—
97	C₂×C₂²⋊C₄	—	2	1	no	—
98	C₂×C₂²⋊C₄	—	2	1	no	—
99	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
100	C₄⋊C₈	—	146	1	no	—
101	C₂²⋊C₈	—	146	1	no	—
102	#97	—	73	2	no	—
103	#97	—	73	2	no	—
104	C₄×C₈	—	146	1	no	—
105	C₂⁴	—	2	1	no	—
106	C₂×Q₈	—	2	1	no	—
107	C₂×Q₈	—	4	2	no	—
108	C₂×Q₈	C₂×Dic₇₃	1	2	yes	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	—	4	2	no	—
111	C₂×D₄	C₂×Dic₇₃	1	2	yes	—
112	C₂²×C₄	—	2	1	no	—
113	C₂²×C₄	C₂×Dic₇₃	1	1	yes	characteristic
114	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
115	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
116	M₁₆	—	146	4	no	—
117	C₂×C₈	—	146	2	no	—
118	C₂×C₈	—	146	2	no	—
119	C₂²⋊C₄	—	2	2	no	—
120	C₂²⋊C₄	—	2	2	no	—
121	C₂²⋊C₄	—	2	2	no	—
122	C₂²⋊C₄	—	2	2	no	—
123	C₄²	C₂×Dic₇₃	1	1	yes	characteristic
124	C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
125	C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
126	C₄²	C₂²×D₇₃	1	1	yes	characteristic
127	C₂³	—	2	2	no	—
128	C₂³	—	4	2	no	—
129	C₂³	—	4	2	no	—
130	C₂³	—	4	2	no	—
131	C₂³	C₂²⋊Dic₇₃	1	1	yes	characteristic
132	Q₈	—	4	2	no	—
133	Q₈	—	4	2	no	—
134	D₄	—	4	2	no	—
135	D₄	—	4	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
142	C₂×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
143	C₂×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
144	C₂×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
145	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
147	C₈	—	292	2	no	—
148	C₈	—	146	4	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	4	2	no	—
152	C₂²	—	4	2	no	—
153	C₂²	—	4	2	no	—
154	C₂²	—	8	4	no	—
155	C₂²	#14556518	1	2	yes	—
156	C₂²	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	—	4	2	no	—
163	C₄	#14556508	1	2	yes	—
164	C₂	—	2	2	no	—
165	C₂	—	8	2	no	—
166	C₂	C₂×#14556508	1	1	yes	characteristic
167	C₂	C₂×#14556518	1	1	yes	characteristic
168	C₂	#17988604	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988644

C₂

yes

maximal, characteristic, complemented

C₄×#11924012

C₂

yes

maximal, characteristic, complemented

#17988327

C₂

yes

maximal, complemented

C₂×#124×C₇₃

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

complemented

#14556502

C₂²

yes

complemented

C₄×#10596831

—

C₂²×Q₈×C₇₃

C₄

yes

characteristic

C₂²×D₄×C₇₃

C₄

yes

characteristic

#124×C₇₃

—

#124×C₇₃

—

#124×C₇₃

C₄

yes

—

#124×C₇₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₇₃

—

C₂×C₂²⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

C₂²

yes

—

—

—

C₂⁴×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

C₂×C₄

yes

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

C₂×C₄

yes

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×C₄

yes

characteristic

C₄²×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

C₂²⋊C₄

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#104

yes

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#100

yes

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×#124

D₇₃

yes

characteristic

#432

—

C₄×M₁₆

—

#329

—

C₂²×Q₈

Dic₇₃

yes

characteristic

C₂²×D₄

Dic₇₃

yes

characteristic

C₂×M₁₆

—

146

—

#124

—

#124

—

#124

Dic₇₃

yes

—

#124

C₂×D₇₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₇₃

yes

characteristic

100

C₄⋊C₈

—

146

—

101

C₂²⋊C₈

—

146

—

102

#97

—

103

#97

—

104

C₄×C₈

—

146

—

105

C₂⁴

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

C₂×Dic₇₃

yes

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

C₂×Dic₇₃

yes

—

112

C₂²×C₄

—

113

C₂²×C₄

C₂×Dic₇₃

yes

characteristic

114

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

115

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

116

M₁₆

—

146

—

117

C₂×C₈

—

146

—

118

C₂×C₈

—

146

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

121

C₂²⋊C₄

—

122

C₂²⋊C₄

—

123

C₄²

C₂×Dic₇₃

yes

characteristic

124

C₄²

C₂²⋊D₇₃

yes

characteristic

125

C₄²

C₂²⋊D₇₃

yes

characteristic

126

C₄²

C₂²×D₇₃

yes

characteristic

127

C₂³

—

128

C₂³

—

129

C₂³

—

130

C₂³

—

131

C₂³

C₂²⋊Dic₇₃

yes

characteristic

132

Q₈

—

133

Q₈

—

134

D₄

—

135

D₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

C₂²⋊Dic₇₃

yes

characteristic

142

C₂×C₄

C₂²⋊Dic₇₃

yes

characteristic

143

C₂×C₄

C₂²⋊Dic₇₃

yes

characteristic

144

C₂×C₄

C₂²×Dic₇₃

yes

characteristic

145

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

146

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

147

C₈

—

292

—

148

C₈

—

146

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

#14556518

yes

—

156

C₂²

C₂×C₂²⋊Dic₇₃

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

#14556508

yes

—

164

C₂

—

165

C₂

—

166

C₂

C₂×#14556508

yes

characteristic

167

C₂

C₂×#14556518

yes

characteristic

168

C₂

#17988604

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₇₃
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₄²	C₂²×D₇₃
Fitting	C₂×#124×C₇₃	C₂
socle	C₂²×C₇₃	C₂×C₂²⋊C₄
hypercenter	C₂×#124	D₇₃
hypocenter	C₇₃	#1527
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#124

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₇₃

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₄²

C₂²×D₇₃

Fitting

C₂×#124×C₇₃

C₂

socle

C₂²×C₇₃

C₂×C₂²⋊C₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#124

N	H ≅ G/N	type
1. #17988644	C₂	semidirect
2. C₄×#11924012	C₂	semidirect
3. #17988327	C₂	semidirect
8. #14556502	C₂²	semidirect
9. #14556502	C₂²	semidirect
85. C₇₃	#1527	semidirect

H ≅ G/N

type

1. #17988644

C₂

semidirect

2. C₄×#11924012

C₂

semidirect

3. #17988327

C₂

semidirect

8. #14556502

C₂²

semidirect

9. #14556502

C₂²

semidirect

85. C₇₃

#1527

semidirect

Subgroups — #22608529

Conjugacy classes of subgroups

Complements