Group #32990763 — subgroups

Subgroups — #32990763

Order 18688.

310 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 85 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	#22610763	C₂	1	1	yes	maximal, characteristic
2	#22608581	C₂	1	2	yes	maximal
3	#22607828	C₂	1	2	yes	maximal, complemented
4	#22607760	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#1832	C₂	1	1	yes	maximal, characteristic, complemented
6	#17989137	C₂²	1	1	yes	characteristic
7	#17988640	C₂²	1	2	yes	—
8	#17988611	C₂²	1	1	yes	characteristic
9	C₂×#14556519	—	2	2	no	—
10	#17988334	—	2	2	no	—
11	#17988330	—	2	2	no	—
12	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
13	#376×C₇₃	C₂²	1	1	yes	characteristic
14	#336×C₇₃	—	2	2	no	—
15	C₂×#11924028	—	2	1	no	—
16	C₂×#11924012	—	2	1	no	—
17	#14556591	D₄	1	1	yes	characteristic
18	#14556588	D₄	1	1	yes	characteristic
19	#14556579	—	2	1	no	—
20	#14556565	—	2	2	no	—
21	C₄×D₂₉₂	—	2	2	no	—
22	C₄⋊#10596831	—	2	2	no	—
23	C₄⋊#10596831	—	2	2	no	—
24	#14556519	—	4	4	no	—
25	C₂²⋊#10596831	—	2	2	no	—
26	C₄×#10596831	—	2	2	no	—
27	C₂²×C₈×C₇₃	—	2	2	no	—
28	C₂×C₄²×C₇₃	D₄	1	2	yes	—
29	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₇₃	—	2	2	no	—
31	C₄⋊C₈×C₇₃	—	2	2	no	—
32	#95×C₇₃	—	2	2	no	—
33	#11924028	—	4	2	no	—
34	#11924028	—	4	2	no	—
35	C₂×Dic₁₄₆	—	2	1	no	—
36	C₂×D₂₉₂	—	2	1	no	—
37	C₂×D₂₉₂	—	2	2	no	—
38	C₂×C₂²⋊D₇₃	—	4	1	no	—
39	C₂×C₄×D₇₃	—	4	1	no	—
40	#11924012	—	4	2	no	—
41	C₂×#10596831	—	4	2	no	—
42	C₂×#10596831	—	4	1	no	—
43	C₄⋊Dic₇₃	—	2	2	no	—
44	#11924005	—	4	2	no	—
45	C₂²×C₄×C₇₃	—	2	2	no	—
46	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₇₃	#41	1	2	yes	—
48	C₂×C₈×C₇₃	—	4	2	no	—
49	C₂×C₈×C₇₃	—	4	2	no	—
50	C₂×C₈×C₇₃	—	4	2	no	—
51	C₄²×C₇₃	—	2	2	no	—
52	C₄²×C₇₃	—	2	2	no	—
53	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₇₃	#41	1	1	yes	characteristic
55	C₄²×C₇₃	#41	1	2	yes	—
56	C₂³×C₇₃	#123	1	1	yes	characteristic
57	C₂²×D₇₃	—	4	1	no	—
58	C₂×C₄×C₇₃	—	4	2	no	—
59	C₂×C₄×C₇₃	—	4	4	no	—
60	C₂×C₄×C₇₃	—	4	2	no	—
61	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
62	C₂×C₄×C₇₃	#123	1	2	yes	—
63	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
64	C₂×C₄×C₇₃	#126	1	2	yes	—
65	C₂×C₄×C₇₃	#135	1	2	yes	—
66	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
67	C₂²⋊D₇₃	—	8	2	no	—
68	C₂×Dic₇₃	—	4	1	no	—
69	D₂₉₂	—	4	2	no	—
70	D₂₉₂	—	8	2	no	—
71	C₄×D₇₃	—	8	1	no	—
72	C₄×D₇₃	—	8	1	no	—
73	Dic₁₄₆	—	4	2	no	—
74	C₈×C₇₃	—	8	2	no	—
75	#10596831	—	8	2	no	—
76	#10596831	—	8	1	no	—
77	C₂²×C₇₃	—	2	2	no	—
78	C₂²×C₇₃	#397	1	1	yes	characteristic
79	C₂²×C₇₃	#482	1	2	yes	—
80	C₂×D₇₃	—	8	1	no	—
81	C₂×D₇₃	—	8	2	no	—
82	C₄×C₇₃	—	2	2	no	—
83	C₄×C₇₃	—	2	2	no	—
84	C₄×C₇₃	—	8	2	no	—
85	C₄×C₇₃	#463	1	2	yes	—
86	C₄×C₇₃	#488	1	1	yes	characteristic
87	C₄×C₇₃	#490	1	1	yes	characteristic
88	Dic₇₃	—	8	1	no	—
89	#10629	—	73	1	no	maximal, Sylow
90	C₂×C₇₃	—	2	2	no	—
91	C₂×C₇₃	#2073	1	1	yes	characteristic
92	C₂×C₇₃	#2080	1	2	yes	—
93	D₇₃	—	16	1	no	—
94	#1963	—	73	1	no	—
95	#1832	D₇₃	1	1	yes	characteristic, complemented
96	#1540	—	73	2	no	—
97	#1301	—	73	2	no	—
98	#1274	—	73	1	no	—
99	C₇₃	#10629	1	1	yes	characteristic, complemented, Sylow
100	#532	—	73	1	no	—
101	#431	—	73	2	no	—
102	#423	—	73	1	no	—
103	C₂×C₄⋊C₈	C₂×D₇₃	1	2	yes	—
104	C₂×#104	—	146	2	no	—
105	#376	C₂×D₇₃	1	1	yes	characteristic
106	#336	—	2	2	no	—
107	#332	—	146	2	no	—
108	#331	—	146	2	no	—
109	C₂×#41	—	146	1	no	—
110	C₂×M₁₆	—	146	1	no	—
111	C₂²×C₈	—	2	2	no	—
112	C₄⋊Q₈	—	73	1	no	—
113	#127	—	73	1	no	—
114	#124	—	146	1	no	—
115	#121	—	146	2	no	—
116	C₄×D₄	—	146	2	no	—
117	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
118	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	146	2	no	—
122	C₄⋊C₈	—	146	2	no	—
123	#104	—	292	4	no	—
124	C₂²⋊C₈	—	146	2	no	—
125	C₄×C₈	—	146	2	no	—
126	#95	—	2	2	no	—
127	#41	—	292	2	no	—
128	#41	—	292	2	no	—
129	C₂×Q₈	—	146	1	no	—
130	C₂×D₄	—	292	1	no	—
131	C₂×D₄	—	146	1	no	—
132	C₂×D₄	—	146	2	no	—
133	C₂²×C₄	—	2	2	no	—
134	C₂²×C₄	—	292	1	no	—
135	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
136	C₂²×C₄	#11924028	1	2	yes	—
137	M₁₆	—	292	2	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	292	2	no	—
142	C₂×C₈	—	292	1	no	—
143	C₄⋊C₄	—	146	2	no	—
144	C₂²⋊C₄	—	292	2	no	—
145	C₄²	—	2	2	no	—
146	C₄²	—	2	2	no	—
147	C₄²	D₄×D₇₃	1	1	yes	characteristic
148	C₄²	#11924027	1	1	yes	characteristic
149	C₄²	#11924029	1	2	yes	—
150	C₂³	—	292	1	no	—
151	C₂³	#14556574	1	1	yes	characteristic
152	Q₈	—	292	2	no	—
153	D₄	—	292	2	no	—
154	D₄	—	584	2	no	—
155	D₄	—	584	2	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	292	1	no	—
160	C₂×C₄	—	584	1	no	—
161	C₂×C₄	—	584	1	no	—
162	C₂×C₄	#14556562	1	1	yes	characteristic
163	C₂×C₄	#14556576	1	2	yes	—
164	C₂×C₄	#14556580	1	1	yes	characteristic
165	C₂×C₄	#14556586	1	2	yes	—
166	C₂×C₄	#14556611	1	2	yes	—
167	C₂×C₄	#14556616	1	2	yes	—
168	C₈	—	8	2	no	—
169	C₈	—	584	2	no	—
170	C₈	—	584	1	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	584	1	no	—
173	C₂²	—	584	2	no	—
174	C₂²	#17988513	1	1	yes	characteristic
175	C₂²	#17988911	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	8	2	no	—
179	C₄	—	584	1	no	—
180	C₄	#17988799	1	2	yes	—
181	C₄	#17988944	1	1	yes	characteristic
182	C₄	#17988953	1	1	yes	characteristic
183	C₂	—	2	2	no	—
184	C₂	—	1168	1	no	—
185	C₂	#22611345	1	1	yes	characteristic
186	C₂	#22611382	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22610763

C₂

yes

maximal, characteristic

#22608581

C₂

yes

maximal

#22607828

C₂

yes

maximal, complemented

#22607760

C₂

yes

maximal, characteristic, complemented

C₇₃×#1832

C₂

yes

maximal, characteristic, complemented

#17989137

C₂²

yes

characteristic

#17988640

C₂²

yes

—

#17988611

C₂²

yes

—

—

—

C₂×C₄⋊C₈×C₇₃

C₂²

yes

—

#376×C₇₃

C₂²

yes

characteristic

#336×C₇₃

—

C₂×#11924028

—

C₂×#11924012

—

#14556591

D₄

yes

characteristic

#14556588

D₄

yes

characteristic

#14556579

—

#14556565

—

C₄×D₂₉₂

—

—

—

—

—

—

C₂²×C₈×C₇₃

—

C₂×C₄²×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

—

#95×C₇₃

—

#11924028

—

#11924028

—

C₂×Dic₁₄₆

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×C₂²⋊D₇₃

—

C₂×C₄×D₇₃

—

#11924012

—

C₂×#10596831

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

#11924005

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

#41

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

C₄²×C₇₃

yes

C₄²×C₇₃

yes

—

C₂³×C₇₃

#123

yes

characteristic

C₂²×D₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#121

yes

characteristic

C₂×C₄×C₇₃

#123

yes

—

C₂×C₄×C₇₃

#124

yes

characteristic

C₂×C₄×C₇₃

#126

yes

—

C₂×C₄×C₇₃

#135

yes

—

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

#397

yes

characteristic

C₂²×C₇₃

#482

yes

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#463

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

#10629

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#2073

yes

characteristic

C₂×C₇₃

#2080

yes

—

D₇₃

—

#1963

—

#1832

D₇₃

yes

characteristic, complemented

#1540

—

#1301

—

#1274

—

C₇₃

#10629

yes

characteristic, complemented, Sylow

100

#532

—

101

#431

—

102

#423

—

103

C₂×C₄⋊C₈

C₂×D₇₃

yes

—

104

C₂×#104

—

146

—

105

#376

C₂×D₇₃

yes

characteristic

106

#336

—

107

#332

—

146

—

108

#331

—

146

—

109

C₂×#41

—

146

—

110

C₂×M₁₆

—

146

—

111

C₂²×C₈

—

112

C₄⋊Q₈

—

113

#127

—

114

#124

—

146

—

115

#121

—

146

—

116

C₄×D₄

—

146

—

117

C₂×C₄²

C₂²⋊D₇₃

yes

—

118

C₂×C₄²

C₂²×D₇₃

yes

characteristic

119

C₄⋊C₈

—

120

C₄⋊C₈

—

121

C₄⋊C₈

—

146

—

122

C₄⋊C₈

—

146

—

123

#104

—

292

—

124

C₂²⋊C₈

—

146

—

125

C₄×C₈

—

146

—

126

#95

—

127

#41

—

292

—

128

#41

—

292

—

129

C₂×Q₈

—

146

—

130

C₂×D₄

—

292

—

131

C₂×D₄

—

146

—

132

C₂×D₄

—

146

—

133

C₂²×C₄

—

134

C₂²×C₄

—

292

—

135

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

136

C₂²×C₄

#11924028

yes

—

137

M₁₆

—

292

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

292

—

142

C₂×C₈

—

292

—

143

C₄⋊C₄

—

146

—

144

C₂²⋊C₄

—

292

—

145

C₄²

—

146

C₄²

—

147

C₄²

D₄×D₇₃

yes

characteristic

148

C₄²

#11924027

yes

characteristic

149

C₄²

#11924029

yes

—

150

C₂³

—

292

—

151

C₂³

#14556574

yes

characteristic

152

Q₈

—

292

—

153

D₄

—

292

—

154

D₄

—

584

—

155

D₄

—

584

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

292

—

160

C₂×C₄

—

584

—

161

C₂×C₄

—

584

—

162

C₂×C₄

#14556562

yes

characteristic

163

C₂×C₄

#14556576

yes

—

164

C₂×C₄

#14556580

yes

characteristic

165

C₂×C₄

#14556586

yes

—

166

C₂×C₄

#14556611

yes

—

167

C₂×C₄

#14556616

yes

—

168

C₈

—

169

C₈

—

584

—

170

C₈

—

584

—

171

C₂²

—

172

C₂²

—

584

—

173

C₂²

—

584

—

174

C₂²

#17988513

yes

characteristic

175

C₂²

#17988911

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

584

—

180

C₄

#17988799

yes

—

181

C₄

#17988944

yes

characteristic

182

C₄

#17988953

yes

characteristic

183

C₂

—

184

C₂

—

1168

—

185

C₂

#22611345

yes

characteristic

186

C₂

#22611382

yes

—

	subgroup	G/H
center	C₂×C₄	#14556580
commutator, cosocle	C₂×C₄²×C₇₃	C₂³
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1832	C₂
socle	C₂²×C₇₃	#397
prime radical	#17989137	C₂²
hypercenter	#1832	D₇₃
hypocenter	C₇₃	#10629
LCS	G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1832

subgroup

G/H

center

C₂×C₄

#14556580

commutator, cosocle

C₂×C₄²×C₇₃

C₂³

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1832

C₂

C₂²×C₇₃

C₂²

D₇₃

C₇₃

G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1832

N	H ≅ G/N	type
3. #22607828	C₂	semidirect
4. #22607760	C₂	semidirect
5. C₇₃×#1832	C₂	semidirect
95. #1832	D₇₃	semidirect
99. C₇₃	#10629	semidirect

H ≅ G/N

type

3. #22607828

C₂

semidirect

4. #22607760

C₂

semidirect

5. C₇₃×#1832

C₂

semidirect

95. #1832

D₇₃

semidirect

99. C₇₃

#10629

semidirect

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

subgroup	index
#22610763	2
#22608581	2
#22607828	2
#22607760	2
C₇₃×#1832	2
#10629	73

subgroup

index

C₇₃×#1832

Subgroups — #32990763

Conjugacy classes of subgroups

Complements

Maximal subgroups