Group #32979749 — subgroups

Subgroups — #32979749

Order 18688.

382 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 163 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	#22610083	C₂	1	1	yes	maximal, characteristic
2	#22609677	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988339	C₂	1	2	yes	maximal
4	C₂×#17988337	C₂	1	2	yes	maximal
5	C₇₃×#1831	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10596831	—	2	2	no	—
7	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
8	C₂×C₄×#10596831	C₂²	1	2	yes	—
9	#17988342	C₄	1	2	yes	—
10	#17988342	C₄	1	2	yes	—
11	#17988339	—	2	4	no	—
12	#17988337	—	2	4	no	—
13	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
15	C₄³×C₇₃	C₂²	1	1	yes	characteristic
16	#336×C₇₃	—	2	2	no	—
17	C₂²×#10596831	—	2	2	no	—
18	C₂²×#10596831	C₂×C₄	1	4	yes	—
19	C₄⋊#10596831	—	2	2	no	—
20	C₄⋊#10596831	—	2	2	no	—
21	C₄⋊#10596831	—	4	4	no	—
22	C₄×#10596831	D₄	1	4	yes	—
23	C₄×#10596831	Q₈	1	4	yes	—
24	C₂²×C₈×C₇₃	—	2	1	no	—
25	C₂²×C₈×C₇₃	—	2	1	no	—
26	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₇₃	D₄	1	2	yes	—
29	C₂×C₄²×C₇₃	Q₈	1	2	yes	—
30	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₇₃	—	2	2	no	—
32	C₄⋊C₈×C₇₃	—	2	2	no	—
33	C₂×#10596831	—	2	4	no	—
34	C₂×#10596831	—	2	4	no	—
35	C₂×#10596831	—	4	2	no	—
36	C₂×#10596831	—	4	4	no	—
37	C₂×#10596831	C₄⋊C₄	1	8	yes	—
38	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₇₃	—	4	1	no	—
44	C₂×C₈×C₇₃	—	4	1	no	—
45	C₂×C₈×C₇₃	—	4	2	no	—
46	C₂×C₈×C₇₃	—	4	2	no	—
47	C₄²×C₇₃	—	2	4	no	—
48	C₄²×C₇₃	—	2	2	no	—
49	C₄²×C₇₃	—	2	2	no	—
50	C₄²×C₇₃	—	2	2	no	—
51	C₄²×C₇₃	—	2	2	no	—
52	C₄²×C₇₃	C₂×D₄	1	2	yes	—
53	C₄²×C₇₃	C₂×Q₈	1	2	yes	—
54	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₇₃	—	2	2	no	—
56	C₂×C₄×C₇₃	—	2	2	no	—
57	C₂×C₄×C₇₃	—	2	2	no	—
58	C₂×C₄×C₇₃	—	2	2	no	—
59	C₂×C₄×C₇₃	—	2	2	no	—
60	C₂×C₄×C₇₃	—	2	2	no	—
61	C₂×C₄×C₇₃	#108	1	4	yes	—
62	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
63	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
64	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
66	C₂×C₄×C₇₃	#131	1	2	yes	—
67	C₂×C₄×C₇₃	#131	1	2	yes	—
68	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
69	C₂×C₄×C₇₃	#137	1	2	yes	—
70	C₈×C₇₃	—	8	1	no	—
71	C₈×C₇₃	—	8	1	no	—
72	#10596831	—	2	8	no	—
73	#10596831	—	8	2	no	—
74	C₂²×C₇₃	#384	1	1	yes	characteristic
75	C₂²×C₇₃	#424	1	1	yes	characteristic
76	C₂²×C₇₃	#424	1	1	yes	characteristic
77	C₂²×C₇₃	#428	1	2	yes	—
78	C₂²×C₇₃	C₂×#108	1	2	yes	—
79	C₄×C₇₃	—	2	4	no	—
80	C₄×C₇₃	—	2	2	no	—
81	C₄×C₇₃	—	4	2	no	—
82	C₄×C₇₃	#424	1	4	yes	—
83	C₄×C₇₃	#496	1	1	yes	characteristic
84	C₄×C₇₃	#497	1	1	yes	characteristic
85	C₄×C₇₃	#501	1	2	yes	—
86	#8812	—	73	1	no	maximal, Sylow
87	C₂×C₇₃	#1573	1	2	yes	—
88	C₂×C₇₃	#1574	1	2	yes	—
89	C₂×C₇₃	#1843	1	1	yes	characteristic
90	C₂×C₇₃	#1847	1	1	yes	characteristic
91	C₂×C₇₃	#1848	1	1	yes	characteristic
92	#1842	—	73	1	no	—
93	#1831	D₇₃	1	1	yes	characteristic
94	#1744	—	73	1	no	—
95	C₂×#335	—	73	2	no	—
96	C₂×#334	—	73	2	no	—
97	C₇₃	#8812	1	1	yes	characteristic, complemented, Sylow
98	C₂×C₄⋊C₈	—	73	2	no	—
99	C₂×C₄⋊C₈	—	146	2	no	—
100	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
101	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
102	C₂×C₄×C₈	—	73	2	no	—
103	C₄³	C₂×D₇₃	1	1	yes	characteristic
104	#336	—	2	2	no	—
105	#336	—	73	2	no	—
106	#336	—	73	2	no	—
107	#335	—	146	4	no	—
108	#334	—	146	4	no	—
109	C₂²×C₈	—	2	1	no	—
110	C₂²×C₈	—	2	1	no	—
111	C₂²×C₈	—	73	4	no	—
112	C₂²×C₈	—	146	2	no	—
113	C₂×C₄²	Dic₁₄₆	1	2	yes	—
114	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
115	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
116	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
117	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	292	4	no	—
121	C₄⋊C₈	—	146	2	no	—
122	C₄⋊C₈	—	146	2	no	—
123	C₄×C₈	—	73	4	no	—
124	C₄×C₈	—	73	4	no	—
125	C₂²×C₄	#11924007	1	2	yes	—
126	C₂²×C₄	#11924007	1	2	yes	—
127	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
128	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
129	C₂²×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
130	C₂×C₈	—	4	1	no	—
131	C₂×C₈	—	4	1	no	—
132	C₂×C₈	—	4	2	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	—	292	2	no	—
135	C₂×C₈	—	292	4	no	—
136	C₂×C₈	—	73	8	no	—
137	C₂×C₈	—	146	4	no	—
138	C₂×C₈	—	146	4	no	—
139	C₄²	—	2	4	no	—
140	C₄²	—	2	2	no	—
141	C₄²	—	2	2	no	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	2	no	—
144	C₄²	D₄×D₇₃	1	2	yes	—
145	C₄²	Q₈×D₇₃	1	2	yes	—
146	C₂³	C₂×#11924007	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	#14556526	1	4	yes	—
154	C₂×C₄	C₄⋊C₄×D₇₃	1	2	yes	—
155	C₂×C₄	#14556587	1	1	yes	characteristic
156	C₂×C₄	#14556589	1	1	yes	characteristic
157	C₂×C₄	#14556590	1	2	yes	—
158	C₂×C₄	#14556597	1	2	yes	—
159	C₂×C₄	#14556599	1	2	yes	—
160	C₂×C₄	#14556616	1	2	yes	—
161	C₂×C₄	#14556621	1	2	yes	—
162	C₈	—	8	1	no	—
163	C₈	—	8	1	no	—
164	C₈	—	584	2	no	—
165	C₈	—	146	8	no	—
166	C₂²	#17988456	1	1	yes	characteristic
167	C₂²	#17988613	1	1	yes	characteristic
168	C₂²	#17988615	1	1	yes	characteristic
169	C₂²	#17988630	1	2	yes	—
170	C₂²	C₂×#14556526	1	2	yes	—
171	C₄	—	2	4	no	—
172	C₄	—	2	2	no	—
173	C₄	—	4	2	no	—
174	C₄	#17988612	1	4	yes	—
175	C₄	#17988977	1	1	yes	characteristic
176	C₄	#17988982	1	1	yes	characteristic
177	C₄	#17988997	1	2	yes	—
178	C₂	#22608717	1	2	yes	—
179	C₂	#22608719	1	2	yes	—
180	C₂	#22610084	1	1	yes	characteristic
181	C₂	#22610100	1	1	yes	characteristic
182	C₂	#22610102	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22610083

C₂

yes

maximal, characteristic

#22609677

C₂

yes

maximal, characteristic

C₂×#17988339

C₂

yes

maximal

C₂×#17988337

C₂

yes

maximal

C₇₃×#1831

C₂

yes

maximal, characteristic

C₂×C₄⋊#10596831

—

C₂×C₄⋊#10596831

C₂²

yes

—

C₂×C₄×#10596831

C₂²

yes

—

#17988342

C₄

yes

—

#17988342

C₄

yes

—

#17988339

—

#17988337

—

C₂×C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₄³×C₇₃

C₂²

yes

characteristic

#336×C₇₃

—

C₂²×#10596831

—

C₂²×#10596831

C₂×C₄

yes

—

—

—

—

D₄

yes

—

C₄×#10596831

Q₈

yes

—

C₂²×C₈×C₇₃

—

C₂²×C₈×C₇₃

—

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

Q₈

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

—

—

—

—

—

C₄⋊C₄

yes

—

C₂²×C₄×C₇₃

C₄⋊C₄

yes

—

C₂²×C₄×C₇₃

C₄⋊C₄

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

—

C₄²×C₇₃

C₂×Q₈

yes

—

C₂³×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#108

yes

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

C₄⋊Q₈

yes

—

C₂×C₄×C₇₃

#131

yes

—

C₂×C₄×C₇₃

#131

yes

—

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇₃

#137

yes

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₂²×C₇₃

C₂×#108

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#424

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

#8812

—

maximal, Sylow

C₂×C₇₃

#1573

yes

—

C₂×C₇₃

#1574

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

yes

—

—

—

C₇₃

#8812

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

146

—

100

C₂×C₄⋊C₈

C₂×D₇₃

yes

characteristic

101

C₂×C₄⋊C₈

C₂×D₇₃

yes

characteristic

102

C₂×C₄×C₈

—

103

C₄³

C₂×D₇₃

yes

characteristic

104

#336

—

105

#336

—

106

#336

—

107

#335

—

146

—

108

#334

—

146

—

109

C₂²×C₈

—

110

C₂²×C₈

—

111

C₂²×C₈

—

112

C₂²×C₈

—

146

—

113

C₂×C₄²

Dic₁₄₆

yes

—

114

C₂×C₄²

C₄×D₇₃

yes

characteristic

115

C₂×C₄²

C₄×D₇₃

yes

characteristic

116

C₂×C₄²

C₂²⋊D₇₃

yes

—

117

C₂×C₄²

C₂²×D₇₃

yes

characteristic

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₄⋊C₈

—

292

—

121

C₄⋊C₈

—

146

—

122

C₄⋊C₈

—

146

—

123

C₄×C₈

—

124

C₄×C₈

—

125

C₂²×C₄

#11924007

yes

—

126

C₂²×C₄

#11924007

yes

—

127

C₂²×C₄

C₂×C₄×D₇₃

yes

characteristic

128

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

129

C₂²×C₄

C₂×Dic₁₄₆

yes

characteristic

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

292

—

135

C₂×C₈

—

292

—

136

C₂×C₈

—

137

C₂×C₈

—

146

—

138

C₂×C₈

—

146

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

142

C₄²

—

143

C₄²

—

144

C₄²

D₄×D₇₃

yes

—

145

C₄²

Q₈×D₇₃

yes

—

146

C₂³

C₂×#11924007

yes

characteristic

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

#14556526

yes

—

154

C₂×C₄

C₄⋊C₄×D₇₃

yes

—

155

C₂×C₄

#14556587

yes

characteristic

156

C₂×C₄

#14556589

yes

characteristic

157

C₂×C₄

#14556590

yes

—

158

C₂×C₄

#14556597

yes

—

159

C₂×C₄

#14556599

yes

—

160

C₂×C₄

#14556616

yes

—

161

C₂×C₄

#14556621

yes

—

162

C₈

—

163

C₈

—

164

C₈

—

584

—

165

C₈

—

146

—

166

C₂²

#17988456

yes

characteristic

167

C₂²

#17988613

yes

characteristic

168

C₂²

#17988615

yes

characteristic

169

C₂²

#17988630

yes

—

170

C₂²

C₂×#14556526

yes

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

#17988612

yes

—

175

C₄

#17988977

yes

characteristic

176

C₄

#17988982

yes

characteristic

177

C₄

#17988997

yes

—

178

C₂

#22608717

yes

—

179

C₂

#22608719

yes

—

180

C₂

#22610084

yes

characteristic

181

C₂

#22610100

yes

characteristic

182

C₂

#22610102

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₃
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1831	C₂
socle, prime radical	C₂³×C₇₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1831	D₇₃
hypocenter	C₇₃	#8812
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂²×C₄

C₂×C₂²⋊D₇₃

commutator

C₂²×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1831

C₂

socle, prime radical

C₂³×C₇₃

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

N	H ≅ G/N	type
97. C₇₃	#8812	semidirect

H ≅ G/N

type

97. C₇₃

#8812

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22610083	2
#22609677	2
C₂×#17988339	2
C₂×#17988337	2
C₇₃×#1831	2
#8812	73

subgroup

index

C₇₃×#1831

Subgroups — #32979749

Conjugacy classes of subgroups

Complements

Maximal subgroups