Group #22609271 — subgroups

Subgroups — #22609271

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	#17988959	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988868	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988779	C₂	1	1	yes	maximal, characteristic
4	#17988652	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988337	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988334	C₂	1	1	yes	maximal, characteristic, complemented
7	#331×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11924016	—	2	1	no	—
9	C₂×#11924016	—	2	1	no	—
10	C₂×#11924013	—	2	1	no	—
11	C₂×C₈×D₇₃	—	2	1	no	—
12	#14556591	C₂²	1	1	yes	characteristic
13	C₄×D₂₉₂	C₂²	1	1	yes	characteristic
14	#14556525	—	2	1	no	—
15	C₄⋊#10596831	C₂²	1	1	yes	characteristic
16	#14556513	—	2	1	no	—
17	#14556512	—	2	1	no	—
18	#14556503	—	2	1	no	—
19	C₄×#10596831	C₂²	1	1	yes	characteristic
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	—	2	1	no	—
24	C₂×D₂₉₂	D₄	1	1	yes	characteristic
25	C₂×C₄×D₇₃	—	2	1	no	—
26	#11924016	—	4	2	no	—
27	#11924016	—	4	2	no	—
28	#11924013	—	4	2	no	—
29	C₂×#10596831	—	2	1	no	—
30	C₂×#10596831	D₄	1	1	yes	characteristic
31	C₂×#10596831	D₄	1	1	yes	characteristic
32	C₈×D₇₃	—	4	2	no	—
33	C₄⋊Dic₇₃	—	2	1	no	—
34	C₄⋊Dic₇₃	—	2	1	no	—
35	C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
36	#11924005	—	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	—	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	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	—	4	1	no	—
55	D₂₉₂	—	2	1	no	—
56	D₂₉₂	SD₁₆	1	2	yes	complemented
57	C₄×D₇₃	—	4	1	no	—
58	C₄×D₇₃	—	4	1	no	—
59	Dic₁₄₆	—	4	1	no	—
60	Dic₁₄₆	—	4	1	no	—
61	C₈×C₇₃	—	4	1	no	—
62	C₈×C₇₃	—	4	1	no	—
63	#10596831	—	2	1	no	—
64	#10596831	—	4	1	no	—
65	#10596831	D₈	1	2	yes	complemented
66	C₂²×C₇₃	—	8	1	no	—
67	C₂²×C₇₃	—	8	1	no	—
68	C₂²×C₇₃	—	8	1	no	—
69	C₂²×C₇₃	#121	1	1	yes	characteristic
70	C₂×D₇₃	—	2	2	no	—
71	C₂×D₇₃	—	4	1	no	—
72	C₂×D₇₃	—	4	1	no	—
73	C₄×C₇₃	—	2	1	no	—
74	C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
75	C₄×C₇₃	C₂×SD₁₆	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	—	4	1	no	—
79	Dic₇₃	—	8	1	no	—
80	C₂×C₇₃	—	16	1	no	—
81	C₂×C₇₃	#459	1	1	yes	characteristic
82	C₂×C₇₃	#465	1	1	yes	characteristic
83	C₂×C₇₃	#471	1	1	yes	characteristic
84	D₇₃	—	4	2	no	—
85	#1665	—	73	1	no	maximal, Sylow
86	C₇₃	#1665	1	1	yes	characteristic, complemented, Sylow
87	#492	—	73	1	no	—
88	#474	—	73	1	no	—
89	#459	—	73	1	no	—
90	C₈×D₄	—	73	1	no	—
91	#334	—	73	1	no	—
92	#332	—	73	1	no	—
93	#331	D₇₃	1	1	yes	characteristic, complemented
94	C₂×SD₁₆	—	146	1	no	—
95	C₂×SD₁₆	—	146	1	no	—
96	C₂×D₈	—	146	1	no	—
97	C₂²×C₈	—	146	1	no	—
98	C₄⋊Q₈	—	73	1	no	—
99	#127	C₂×D₇₃	1	1	yes	characteristic
100	C₄×D₄	—	73	1	no	—
101	#107	—	146	1	no	—
102	C₄⋊C₈	—	73	1	no	—
103	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
104	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
105	#102	—	146	1	no	—
106	#102	—	146	1	no	—
107	C₂²⋊C₈	—	146	1	no	—
108	C₄×C₈	—	73	1	no	—
109	C₂×Q₈	—	146	1	no	—
110	C₂×D₄	—	2	1	no	—
111	C₂×D₄	—	2	1	no	—
112	C₂×D₄	—	2	1	no	—
113	C₂×D₄	—	73	1	no	—
114	C₂²×C₄	—	146	1	no	—
115	SD₁₆	—	292	2	no	—
116	SD₁₆	—	292	2	no	—
117	D₈	—	292	2	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	292	2	no	—
121	C₂×C₈	—	73	1	no	—
122	C₂×C₈	—	73	1	no	—
123	C₂×C₈	—	146	1	no	—
124	C₄⋊C₄	—	73	1	no	—
125	C₄⋊C₄	—	146	1	no	—
126	C₄⋊C₄	—	146	1	no	—
127	C₂²⋊C₄	—	146	1	no	—
128	C₄²	C₂²×D₇₃	1	1	yes	characteristic
129	C₂³	—	4	1	no	—
130	C₂³	—	146	1	no	—
131	Q₈	—	292	1	no	—
132	Q₈	—	292	1	no	—
133	D₄	—	4	1	no	—
134	D₄	—	4	1	no	—
135	D₄	—	4	1	no	—
136	D₄	—	4	1	no	—
137	D₄	—	8	1	no	—
138	D₄	—	73	2	no	—
139	D₄	—	146	1	no	—
140	C₂×C₄	—	292	1	no	—
141	C₂×C₄	—	292	1	no	—
142	C₂×C₄	—	292	1	no	—
143	C₂×C₄	—	146	1	no	—
144	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
145	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
146	C₂×C₄	#11924028	1	1	yes	characteristic
147	C₈	—	4	1	no	—
148	C₈	—	4	1	no	—
149	C₈	—	292	1	no	—
150	C₈	—	73	2	no	—
151	C₈	—	146	1	no	—
152	C₂²	—	8	1	no	—
153	C₂²	—	8	1	no	—
154	C₂²	—	8	1	no	—
155	C₂²	—	292	1	no	—
156	C₂²	—	292	1	no	—
157	C₂²	—	146	2	no	—
158	C₂²	#14556564	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	292	1	no	—
161	C₄	—	584	1	no	—
162	C₄	D₈×D₇₃	1	1	yes	characteristic
163	C₄	SD₁₆×D₇₃	1	1	yes	characteristic
164	C₄	#14556610	1	1	yes	characteristic
165	C₄	#14556618	1	1	yes	characteristic
166	C₂	—	16	1	no	—
167	C₂	—	292	2	no	—
168	C₂	#17988781	1	1	yes	characteristic
169	C₂	#17988815	1	1	yes	characteristic
170	C₂	#17988849	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988959

C₂

yes

maximal, characteristic, complemented

#17988868

C₂

yes

maximal, characteristic, complemented

#17988779

C₂

yes

maximal, characteristic

#17988652

C₂

yes

maximal, characteristic, complemented

#17988337

C₂

yes

maximal, characteristic, complemented

#17988334

C₂

yes

maximal, characteristic, complemented

#331×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#11924016

—

C₂×#11924016

—

C₂×#11924013

—

C₂×C₈×D₇₃

—

#14556591

C₂²

yes

characteristic

C₄×D₂₉₂

C₂²

yes

characteristic

#14556525

—

C₄⋊#10596831

C₂²

yes

—

—

—

C₂²

yes

characteristic

#127×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×D₂₉₂

D₄

yes

characteristic

C₂×C₄×D₇₃

—

—

—

—

—

D₄

yes

characteristic

C₂×#10596831

D₄

yes

characteristic

C₈×D₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

D₄

yes

characteristic

#11924005

—

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

SD₁₆

yes

complemented

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

#10596831

D₈

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

—

#1665

—

maximal, Sylow

C₇₃

#1665

yes

characteristic, complemented, Sylow

#492

—

#474

—

#459

—

C₈×D₄

—

#334

—

#332

—

#331

D₇₃

yes

characteristic, complemented

C₂×SD₁₆

—

146

—

C₂×SD₁₆

—

146

—

C₂×D₈

—

146

—

C₂²×C₈

—

146

—

C₄⋊Q₈

—

#127

C₂×D₇₃

yes

characteristic

100

C₄×D₄

—

101

#107

—

146

—

102

C₄⋊C₈

—

103

C₄⋊C₈

C₂×D₇₃

yes

characteristic

104

C₄⋊C₈

C₂×D₇₃

yes

characteristic

105

#102

—

146

—

106

#102

—

146

—

107

C₂²⋊C₈

—

146

—

108

C₄×C₈

—

109

C₂×Q₈

—

146

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂²×C₄

—

146

—

115

SD₁₆

—

292

—

116

SD₁₆

—

292

—

117

D₈

—

292

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

292

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

146

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

146

—

126

C₄⋊C₄

—

146

—

127

C₂²⋊C₄

—

146

—

128

C₄²

C₂²×D₇₃

yes

characteristic

129

C₂³

—

130

C₂³

—

146

—

131

Q₈

—

292

—

132

Q₈

—

292

—

133

D₄

—

134

D₄

—

135

D₄

—

136

D₄

—

137

D₄

—

138

D₄

—

139

D₄

—

146

—

140

C₂×C₄

—

292

—

141

C₂×C₄

—

292

—

142

C₂×C₄

—

292

—

143

C₂×C₄

—

146

—

144

C₂×C₄

D₄×D₇₃

yes

characteristic

145

C₂×C₄

D₄×D₇₃

yes

characteristic

146

C₂×C₄

#11924028

yes

characteristic

147

C₈

—

148

C₈

—

149

C₈

—

292

—

150

C₈

—

151

C₈

—

146

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

292

—

156

C₂²

—

292

—

157

C₂²

—

146

—

158

C₂²

#14556564

yes

characteristic

159

C₄

—

160

C₄

—

292

—

161

C₄

—

584

—

162

C₄

D₈×D₇₃

yes

characteristic

163

C₄

SD₁₆×D₇₃

yes

characteristic

164

C₄

#14556610

yes

characteristic

165

C₄

#14556618

yes

characteristic

166

C₂

—

167

C₂

—

292

—

168

C₂

#17988781

yes

characteristic

169

C₂

#17988815

yes

characteristic

170

C₂

#17988849

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	#1665
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
1. #17988959	C₂	semidirect
2. #17988868	C₂	semidirect
4. #17988652	C₂	semidirect
5. #17988337	C₂	semidirect
6. #17988334	C₂	semidirect
7. #331×C₇₃	C₂	semidirect
56. D₂₉₂	SD₁₆	semidirect
65. #10596831	D₈	semidirect
86. C₇₃	#1665	semidirect
93. #331	D₇₃	semidirect

H ≅ G/N

type

1. #17988959

C₂

semidirect

2. #17988868

C₂

semidirect

4. #17988652

C₂

semidirect

5. #17988337

C₂

semidirect

6. #17988334

C₂

semidirect

7. #331×C₇₃

C₂

semidirect

56. D₂₉₂

SD₁₆

semidirect

65. #10596831

D₈

semidirect

86. C₇₃

#1665

semidirect

93. #331

D₇₃

semidirect

Subgroups — #22609271

Conjugacy classes of subgroups

Complements