Group #22609366 — subgroups

Subgroups — #22609366

Order 9344.

182 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 55 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	#17988933	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988792	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×#10596831	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988339	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988334	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988331	C₂	1	1	yes	maximal, characteristic, complemented
7	#459×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#11924016	—	2	1	no	—
9	C₂²×#10596831	—	2	1	no	—
10	#14556591	C₂²	1	1	yes	characteristic
11	C₄⋊#10596831	C₂²	1	1	yes	characteristic
12	C₄⋊#10596831	C₂²	1	1	yes	characteristic
13	#14556514	—	2	1	no	—
14	#14556512	—	2	1	no	—
15	#14556512	—	2	1	no	—
16	C₂²⋊#10596831	—	2	1	no	—
17	C₄×#10596831	C₂²	1	1	yes	characteristic
18	C₂×D₈×C₇₃	—	2	1	no	—
19	#127×C₇₃	C₂²	1	1	yes	characteristic
20	C₄×D₄×C₇₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
22	#102×C₇₃	—	2	1	no	—
23	C₂×Dic₁₄₆	—	2	1	no	—
24	#11924016	—	4	2	no	—
25	C₂×#10596831	—	2	1	no	—
26	C₂×#10596831	—	2	1	no	—
27	C₂×#10596831	—	4	2	no	—
28	C₂×#10596831	D₄	1	1	yes	characteristic
29	C₂×#10596831	D₄	1	1	yes	characteristic
30	C₄⋊Dic₇₃	—	2	1	no	—
31	C₄⋊Dic₇₃	—	2	1	no	—
32	C₂×D₄×C₇₃	—	2	1	no	—
33	C₂×D₄×C₇₃	—	2	1	no	—
34	C₂×D₄×C₇₃	—	2	1	no	—
35	C₂×D₄×C₇₃	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₇₃	—	2	1	no	—
37	D₈×C₇₃	—	4	2	no	—
38	C₂×C₈×C₇₃	—	2	1	no	—
39	C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
40	C₂²⋊C₄×C₇₃	—	2	1	no	—
41	C₄²×C₇₃	C₂³	1	1	yes	characteristic
42	C₂³×C₇₃	—	2	1	no	—
43	C₂³×C₇₃	—	4	1	no	—
44	D₄×C₇₃	—	2	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	D₄×C₇₃	D₈	1	2	yes	—
51	C₂×C₄×C₇₃	—	2	1	no	—
52	C₂×C₄×C₇₃	—	4	1	no	—
53	C₂×C₄×C₇₃	—	4	1	no	—
54	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
57	C₂×Dic₇₃	—	4	1	no	—
58	Dic₁₄₆	—	4	1	no	—
59	Dic₁₄₆	—	4	1	no	—
60	C₈×C₇₃	—	4	1	no	—
61	#10596831	—	2	1	no	—
62	#10596831	—	2	1	no	—
63	#10596831	—	4	1	no	—
64	#10596831	—	4	1	no	—
65	C₂²×C₇₃	—	2	2	no	—
66	C₂²×C₇₃	—	4	1	no	—
67	C₂²×C₇₃	—	4	1	no	—
68	C₂²×C₇₃	—	8	1	no	—
69	C₂²×C₇₃	—	8	1	no	—
70	C₂²×C₇₃	—	8	1	no	—
71	C₂²×C₇₃	#121	1	1	yes	characteristic
72	C₄×C₇₃	—	2	1	no	—
73	C₄×C₇₃	—	4	1	no	—
74	C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
75	C₄×C₇₃	#135	1	1	yes	characteristic
76	C₄×C₇₃	#135	1	1	yes	characteristic
77	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
78	Dic₇₃	—	8	1	no	—
79	C₂×C₇₃	—	4	2	no	—
80	C₂×C₇₃	—	16	1	no	—
81	C₂×C₇₃	#463	1	1	yes	characteristic
82	C₂×C₇₃	#466	1	1	yes	characteristic
83	C₂×C₇₃	#471	1	1	yes	characteristic
84	#1681	—	73	1	no	maximal, Sylow
85	C₇₃	#1681	1	1	yes	characteristic, complemented, Sylow
86	#486	—	73	1	no	—
87	#461	—	73	1	no	—
88	#459	D₇₃	1	1	yes	characteristic
89	C₈×D₄	—	73	1	no	—
90	#335	—	73	1	no	—
91	#332	—	73	1	no	—
92	#331	—	73	1	no	—
93	C₂×SD₁₆	—	146	1	no	—
94	C₂×D₈	—	2	1	no	—
95	C₂²×C₈	—	146	1	no	—
96	C₄⋊Q₈	—	73	1	no	—
97	#127	C₂×D₇₃	1	1	yes	characteristic
98	C₄×D₄	C₂×D₇₃	1	1	yes	characteristic
99	C₄⋊C₈	—	73	1	no	—
100	C₄⋊C₈	—	73	1	no	—
101	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
102	#103	—	146	1	no	—
103	#102	—	2	1	no	—
104	#102	—	146	1	no	—
105	#102	—	146	1	no	—
106	C₂²⋊C₈	—	146	1	no	—
107	C₄×C₈	—	73	1	no	—
108	C₂×Q₈	—	146	1	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	—	2	1	no	—
111	C₂×D₄	—	2	1	no	—
112	C₂×D₄	C₂²⋊D₇₃	1	1	yes	characteristic
113	C₂²×C₄	—	2	1	no	—
114	SD₁₆	—	292	2	no	—
115	D₈	—	4	2	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	292	2	no	—
118	C₂×C₈	—	73	1	no	—
119	C₂×C₈	—	73	1	no	—
120	C₂×C₈	—	146	1	no	—
121	C₂×C₈	—	146	1	no	—
122	C₄⋊C₄	—	146	1	no	—
123	C₄⋊C₄	—	146	1	no	—
124	C₄⋊C₄	C₂²⋊D₇₃	1	1	yes	characteristic
125	C₂²⋊C₄	—	2	1	no	—
126	C₄²	C₂²×D₇₃	1	1	yes	characteristic
127	C₂³	—	2	1	no	—
128	C₂³	—	4	1	no	—
129	Q₈	—	292	1	no	—
130	Q₈	—	292	1	no	—
131	D₄	—	2	1	no	—
132	D₄	—	4	1	no	—
133	D₄	—	4	1	no	—
134	D₄	—	4	1	no	—
135	D₄	—	4	1	no	—
136	D₄	—	8	1	no	—
137	D₄	#11924013	1	2	yes	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	292	1	no	—
142	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
143	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
144	C₂×C₄	#11924027	1	1	yes	characteristic
145	C₈	—	4	1	no	—
146	C₈	—	292	1	no	—
147	C₈	—	292	1	no	—
148	C₈	—	146	1	no	—
149	C₈	—	146	1	no	—
150	C₂²	—	2	2	no	—
151	C₂²	—	4	1	no	—
152	C₂²	—	4	1	no	—
153	C₂²	—	8	1	no	—
154	C₂²	—	8	1	no	—
155	C₂²	—	8	1	no	—
156	C₂²	#14556563	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	4	1	no	—
159	C₄	—	584	1	no	—
160	C₄	C₂×#11924013	1	1	yes	characteristic
161	C₄	#14556610	1	1	yes	characteristic
162	C₄	#14556611	1	1	yes	characteristic
163	C₄	#14556615	1	1	yes	characteristic
164	C₂	—	4	2	no	—
165	C₂	—	16	1	no	—
166	C₂	#17988800	1	1	yes	characteristic
167	C₂	#17988821	1	1	yes	characteristic
168	C₂	#17988848	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988933

C₂

yes

maximal, characteristic, complemented

#17988792

C₂

yes

maximal, characteristic, complemented

D₄×#10596831

C₂

yes

maximal, characteristic, complemented

#17988339

C₂

yes

maximal, characteristic, complemented

#17988334

C₂

yes

maximal, characteristic, complemented

#17988331

C₂

yes

maximal, characteristic, complemented

#459×C₇₃

C₂

yes

maximal, characteristic

C₂×#11924016

—

C₂²×#10596831

—

#14556591

C₂²

yes

characteristic

C₄⋊#10596831

C₂²

yes

characteristic

C₄⋊#10596831

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₂×D₈×C₇₃

—

#127×C₇₃

C₂²

yes

characteristic

C₄×D₄×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

#102×C₇₃

—

C₂×Dic₁₄₆

—

—

—

—

—

D₄

yes

characteristic

C₂×#10596831

D₄

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

D₄

yes

characteristic

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

—

C₂³×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

D₈

yes

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

#41

yes

characteristic

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

—

—

—

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#121

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×D₈

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

#486

—

#461

—

#459

D₇₃

yes

characteristic

C₈×D₄

—

#335

—

#332

—

#331

—

C₂×SD₁₆

—

146

—

C₂×D₈

—

C₂²×C₈

—

146

—

C₄⋊Q₈

—

#127

C₂×D₇₃

yes

characteristic

C₄×D₄

C₂×D₇₃

yes

characteristic

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₄⋊C₈

C₂×D₇₃

yes

characteristic

102

#103

—

146

—

103

#102

—

104

#102

—

146

—

105

#102

—

146

—

106

C₂²⋊C₈

—

146

—

107

C₄×C₈

—

108

C₂×Q₈

—

146

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

C₂²⋊D₇₃

yes

characteristic

113

C₂²×C₄

—

114

SD₁₆

—

292

—

115

D₈

—

116

C₂×C₈

—

117

C₂×C₈

—

292

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

146

—

121

C₂×C₈

—

146

—

122

C₄⋊C₄

—

146

—

123

C₄⋊C₄

—

146

—

124

C₄⋊C₄

C₂²⋊D₇₃

yes

characteristic

125

C₂²⋊C₄

—

126

C₄²

C₂²×D₇₃

yes

characteristic

127

C₂³

—

128

C₂³

—

129

Q₈

—

292

—

130

Q₈

—

292

—

131

D₄

—

132

D₄

—

133

D₄

—

134

D₄

—

135

D₄

—

136

D₄

—

137

D₄

#11924013

yes

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

292

—

142

C₂×C₄

D₄×D₇₃

yes

characteristic

143

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

144

C₂×C₄

#11924027

yes

characteristic

145

C₈

—

146

C₈

—

292

—

147

C₈

—

292

—

148

C₈

—

146

—

149

C₈

—

146

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

#14556563

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

584

—

160

C₄

C₂×#11924013

yes

characteristic

161

C₄

#14556610

yes

characteristic

162

C₄

#14556611

yes

characteristic

163

C₄

#14556615

yes

characteristic

164

C₂

—

165

C₂

—

166

C₂

#17988800

yes

characteristic

167

C₂

#17988821

yes

characteristic

168

C₂

#17988848

yes

characteristic

	subgroup	G/H
center	C₂²	#14556563
commutator	C₄²×C₇₃	C₂³
Frattini	C₄²	C₂²×D₇₃
Fitting	#459×C₇₃	C₂
socle	C₂²×C₇₃	#121
hypercenter	#459	D₇₃
hypocenter	C₇₃	#1681
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #459

subgroup

G/H

center

C₂²

#14556563

commutator

C₄²×C₇₃

C₂³

Frattini

C₄²

C₂²×D₇₃

Fitting

#459×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #459

N	H ≅ G/N	type
1. #17988933	C₂	semidirect
2. #17988792	C₂	semidirect
3. D₄×#10596831	C₂	semidirect
4. #17988339	C₂	semidirect
5. #17988334	C₂	semidirect
6. #17988331	C₂	semidirect
85. C₇₃	#1681	semidirect

H ≅ G/N

type

1. #17988933

C₂

semidirect

2. #17988792

C₂

semidirect

3. D₄×#10596831

C₂

semidirect

4. #17988339

C₂

semidirect

5. #17988334

C₂

semidirect

6. #17988331

C₂

semidirect

85. C₇₃

#1681

semidirect

Subgroups — #22609366

Conjugacy classes of subgroups

Complements