Group #32983504 — subgroups

Subgroups — #32983504

Order 18688.

384 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 95 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	#22620313	C₂	1	1	yes	maximal, characteristic
2	#22619037	C₂	1	1	yes	maximal, characteristic
3	#22608647	C₂	1	2	yes	maximal
4	#22607804	C₂	1	2	yes	maximal
5	C₇₃×#1537	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556591	C₂²	1	1	yes	characteristic
7	#17988957	—	2	2	no	—
8	#17988939	C₄	1	4	yes	—
9	#17988718	—	2	4	no	—
10	#17988648	—	2	2	no	—
11	#17988644	C₂²	1	2	yes	—
12	#17988595	—	2	2	no	—
13	C₄×#11924012	C₂²	1	1	yes	characteristic
14	#17988332	—	2	4	no	—
15	#533×C₇₃	C₂²	1	1	yes	characteristic
16	#432×C₇₃	C₂²	1	2	yes	—
17	#330×C₇₃	—	2	2	no	—
18	C₂²×Dic₁₄₆	—	2	1	no	—
19	C₂×#11924012	—	2	1	no	—
20	#14556591	—	2	2	no	—
21	#14556591	—	2	1	no	—
22	#14556591	C₂×C₄	1	2	yes	—
23	C₂×C₄⋊Dic₇₃	—	2	2	no	—
24	C₄⋊#10596831	—	2	4	no	—
25	C₄⋊#10596831	D₄	1	4	yes	—
26	#14556514	—	2	8	no	—
27	C₂²⋊#10596831	—	2	2	no	—
28	#14556502	—	2	2	no	—
29	#14556502	—	2	2	no	—
30	#14556502	—	2	1	no	—
31	C₄×#10596831	—	2	2	no	—
32	C₄×#10596831	C₂×C₄	1	2	yes	—
33	C₄⋊Q₈×C₇₃	C₂×C₄	1	2	yes	—
34	#125×C₇₃	D₄	1	2	yes	—
35	C₂²⋊Q₈×C₇₃	—	2	2	no	—
36	C₄×Q₈×C₇₃	—	2	2	no	—
37	#117×C₇₃	D₄	1	2	yes	—
38	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₇₃	—	2	2	no	—
40	C₂²⋊C₈×C₇₃	—	2	2	no	—
41	#97×C₇₃	—	2	2	no	—
42	C₂×Dic₁₄₆	—	2	2	no	—
43	C₂×Dic₁₄₆	—	4	4	no	—
44	C₂×Dic₁₄₆	—	4	1	no	—
45	C₂²×Dic₇₃	—	4	1	no	—
46	#11924012	—	4	2	no	—
47	C₂×#10596831	—	2	2	no	—
48	C₂×#10596831	—	2	4	no	—
49	C₄⋊Dic₇₃	—	2	4	no	—
50	C₄⋊Dic₇₃	—	4	2	no	—
51	C₂×Q₈×C₇₃	—	2	2	no	—
52	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₇₃	#41	1	2	yes	—
54	C₂×C₈×C₇₃	—	4	2	no	—
55	C₄⋊C₄×C₇₃	—	2	2	no	—
56	C₄⋊C₄×C₇₃	—	2	4	no	—
57	C₄⋊C₄×C₇₃	C₂²⋊C₄	1	4	yes	—
58	C₂²⋊C₄×C₇₃	—	2	2	no	—
59	C₄²×C₇₃	—	2	2	no	—
60	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
61	C₄²×C₇₃	C₂×D₄	1	2	yes	—
62	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
63	C₂³×C₇₃	#123	1	1	yes	characteristic
64	Q₈×C₇₃	—	8	2	no	—
65	C₂×C₄×C₇₃	—	2	2	no	—
66	C₂×C₄×C₇₃	—	2	4	no	—
67	C₂×C₄×C₇₃	—	4	2	no	—
68	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
70	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
71	C₂×C₄×C₇₃	#121	1	2	yes	—
72	C₂×Dic₇₃	—	4	2	no	—
73	C₂×Dic₇₃	—	8	2	no	—
74	Dic₁₄₆	—	4	4	no	—
75	Dic₁₄₆	—	8	2	no	—
76	C₈×C₇₃	—	8	2	no	—
77	#10596831	—	4	2	no	—
78	#10596831	—	4	4	no	—
79	C₂²×C₇₃	—	2	1	no	—
80	C₂²×C₇₃	—	2	2	no	—
81	C₂²×C₇₃	#386	1	1	yes	characteristic
82	C₄×C₇₃	—	2	1	no	—
83	C₄×C₇₃	—	2	2	no	—
84	C₄×C₇₃	—	2	2	no	—
85	C₄×C₇₃	—	4	4	no	—
86	C₄×C₇₃	#454	1	2	yes	—
87	Dic₇₃	—	8	2	no	—
88	#9422	—	73	1	no	maximal, Sylow
89	C₂×C₇₃	—	4	1	no	—
90	C₂×C₇₃	#1888	1	1	yes	characteristic
91	C₂×C₇₃	#1902	1	2	yes	—
92	#3139	—	73	1	no	—
93	#2987	—	73	1	no	—
94	#1556	—	73	2	no	—
95	#1537	D₇₃	1	1	yes	characteristic
96	#1291	—	73	2	no	—
97	C₇₃	#9422	1	1	yes	characteristic, complemented, Sylow
98	#533	C₂×D₇₃	1	1	yes	characteristic
99	C₂×C₄⋊Q₈	—	73	1	no	—
100	#491	—	146	2	no	—
101	#487	—	73	4	no	—
102	#446	—	146	4	no	—
103	#433	—	146	2	no	—
104	#432	—	73	2	no	—
105	#432	C₂×D₇₃	1	2	yes	—
106	#419	—	146	2	no	—
107	C₄×M₁₆	—	73	1	no	—
108	#332	—	146	4	no	—
109	#330	—	2	2	no	—
110	C₂²×Q₈	—	146	1	no	—
111	C₂×M₁₆	—	146	1	no	—
112	C₄⋊Q₈	—	73	2	no	—
113	C₄⋊Q₈	—	146	2	no	—
114	C₄⋊Q₈	—	146	1	no	—
115	C₄⋊Q₈	C₄×D₇₃	1	2	yes	—
116	#125	C₂²⋊D₇₃	1	2	yes	—
117	C₂²⋊Q₈	—	2	2	no	—
118	C₄×Q₈	—	2	2	no	—
119	#117	D₂₉₂	1	2	yes	—
120	C₂×C₄⋊C₄	—	146	2	no	—
121	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
122	C₄⋊C₈	—	2	2	no	—
123	C₄⋊C₈	—	73	4	no	—
124	C₄⋊C₈	—	146	4	no	—
125	#103	—	146	8	no	—
126	C₂²⋊C₈	—	2	2	no	—
127	C₂²⋊C₈	—	146	2	no	—
128	#97	—	2	2	no	—
129	#97	—	146	2	no	—
130	#97	—	146	2	no	—
131	#97	—	146	1	no	—
132	C₄×C₈	—	73	2	no	—
133	C₄×C₈	—	146	2	no	—
134	C₂×Q₈	—	2	2	no	—
135	C₂×Q₈	—	292	4	no	—
136	C₂×Q₈	—	292	1	no	—
137	C₂×Q₈	—	146	2	no	—
138	C₂²×C₄	—	292	1	no	—
139	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
140	C₂²×C₄	#11924027	1	2	yes	—
141	M₁₆	—	292	2	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	146	2	no	—
144	C₂×C₈	—	146	4	no	—
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	292	2	no	—
148	C₄⋊C₄	—	146	4	no	—
149	C₄⋊C₄	#11924005	1	4	yes	—
150	C₂²⋊C₄	—	2	2	no	—
151	C₄²	—	2	2	no	—
152	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
153	C₄²	D₄×D₇₃	1	2	yes	—
154	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
155	C₂³	#14556575	1	1	yes	characteristic
156	Q₈	—	8	2	no	—
157	Q₈	—	292	4	no	—
158	Q₈	—	584	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	292	2	no	—
163	C₂×C₄	—	584	2	no	—
164	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
165	C₂×C₄	#14556551	1	2	yes	—
166	C₂×C₄	#14556561	1	1	yes	characteristic
167	C₂×C₄	#14556563	1	2	yes	—
168	C₈	—	8	2	no	—
169	C₈	—	292	2	no	—
170	C₈	—	292	4	no	—
171	C₂²	—	2	1	no	—
172	C₂²	—	2	2	no	—
173	C₂²	#17988464	1	1	yes	characteristic
174	C₄	—	2	1	no	—
175	C₄	—	2	2	no	—
176	C₄	—	2	2	no	—
177	C₄	—	4	4	no	—
178	C₄	—	584	2	no	—
179	C₄	#17988762	1	2	yes	—
180	C₂	—	4	1	no	—
181	C₂	#22610332	1	1	yes	characteristic
182	C₂	#22610402	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22620313

C₂

yes

maximal, characteristic

#22619037

C₂

yes

maximal, characteristic

#22608647

C₂

yes

maximal

#22607804

C₂

yes

maximal

C₇₃×#1537

C₂

yes

maximal, characteristic

C₂×#14556591

C₂²

yes

characteristic

#17988957

—

#17988939

C₄

yes

—

#17988718

—

#17988648

—

#17988644

C₂²

yes

—

#17988595

—

C₄×#11924012

C₂²

yes

characteristic

#17988332

—

#533×C₇₃

C₂²

yes

characteristic

#432×C₇₃

C₂²

yes

—

#330×C₇₃

—

C₂²×Dic₁₄₆

—

—

—

—

C₂×C₄

yes

—

C₂×C₄⋊Dic₇₃

—

C₄⋊#10596831

—

C₄⋊#10596831

D₄

yes

—

—

—

—

—

—

—

C₂×C₄

yes

—

C₄⋊Q₈×C₇₃

C₂×C₄

yes

—

#125×C₇₃

D₄

yes

—

C₂²⋊Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

#117×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄⋊C₈×C₇₃

—

C₂²⋊C₈×C₇₃

—

#97×C₇₃

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

—

#11924012

—

C₂×#10596831

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

#41

yes

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂²⋊C₄

yes

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₂³×C₇₃

#123

yes

characteristic

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₄×D₄

yes

—

C₂×C₄×C₇₃

#120

yes

characteristic

C₂×C₄×C₇₃

#121

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#386

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#454

yes

—

Dic₇₃

—

#9422

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1888

yes

characteristic

C₂×C₇₃

#1902

yes

—

—

—

—

D₇₃

yes

characteristic

#1291

—

C₇₃

#9422

yes

characteristic, complemented, Sylow

#533

C₂×D₇₃

yes

characteristic

C₂×C₄⋊Q₈

—

100

#491

—

146

—

101

#487

—

102

#446

—

146

—

103

#433

—

146

—

104

#432

—

105

#432

C₂×D₇₃

yes

—

106

#419

—

146

—

107

C₄×M₁₆

—

108

#332

—

146

—

109

#330

—

110

C₂²×Q₈

—

146

—

111

C₂×M₁₆

—

146

—

112

C₄⋊Q₈

—

113

C₄⋊Q₈

—

146

—

114

C₄⋊Q₈

—

146

—

115

C₄⋊Q₈

C₄×D₇₃

yes

—

116

#125

C₂²⋊D₇₃

yes

—

117

C₂²⋊Q₈

—

118

C₄×Q₈

—

119

#117

D₂₉₂

yes

—

120

C₂×C₄⋊C₄

—

146

—

121

C₂×C₄²

C₂²×D₇₃

yes

characteristic

122

C₄⋊C₈

—

123

C₄⋊C₈

—

124

C₄⋊C₈

—

146

—

125

#103

—

146

—

126

C₂²⋊C₈

—

127

C₂²⋊C₈

—

146

—

128

#97

—

129

#97

—

146

—

130

#97

—

146

—

131

#97

—

146

—

132

C₄×C₈

—

133

C₄×C₈

—

146

—

134

C₂×Q₈

—

135

C₂×Q₈

—

292

—

136

C₂×Q₈

—

292

—

137

C₂×Q₈

—

146

—

138

C₂²×C₄

—

292

—

139

C₂²×C₄

C₂×D₂₉₂

yes

characteristic

140

C₂²×C₄

#11924027

yes

—

141

M₁₆

—

292

—

142

C₂×C₈

—

143

C₂×C₈

—

146

—

144

C₂×C₈

—

146

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

292

—

148

C₄⋊C₄

—

146

—

149

C₄⋊C₄

#11924005

yes

—

150

C₂²⋊C₄

—

151

C₄²

—

152

C₄²

C₂×C₄×D₇₃

yes

characteristic

153

C₄²

D₄×D₇₃

yes

—

154

C₄²

C₂×C₂²⋊D₇₃

yes

characteristic

155

C₂³

#14556575

yes

characteristic

156

Q₈

—

157

Q₈

—

292

—

158

Q₈

—

584

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

292

—

163

C₂×C₄

—

584

—

164

C₂×C₄

C₂×#11924005

yes

characteristic

165

C₂×C₄

#14556551

yes

—

166

C₂×C₄

#14556561

yes

characteristic

167

C₂×C₄

#14556563

yes

—

168

C₈

—

169

C₈

—

292

—

170

C₈

—

292

—

171

C₂²

—

172

C₂²

—

173

C₂²

#17988464

yes

characteristic

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

—

584

—

179

C₄

#17988762

yes

—

180

C₂

—

181

C₂

#22610332

yes

characteristic

182

C₂

#22610402

yes

—

	subgroup	G/H
center	C₂²	#17988464
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1537	C₂
socle	C₂²×C₇₃	#386
prime radical	C₂³×C₇₃	#123
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1537	D₇₃
hypocenter	C₇₃	#9422
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1537

subgroup

G/H

center

C₂²

#17988464

commutator

C₄²×C₇₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1537

C₂

C₂²×C₇₃

C₂³×C₇₃

C₂×C₄²×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1537

N	H ≅ G/N	type
97. C₇₃	#9422	semidirect

H ≅ G/N

type

97. C₇₃

#9422

semidirect

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

subgroup	index
#22620313	2
#22619037	2
#22608647	2
#22607804	2
C₇₃×#1537	2
#9422	73

subgroup

index

C₇₃×#1537

Subgroups — #32983504

Conjugacy classes of subgroups

Complements

Maximal subgroups