Group #32921829 — subgroups

Subgroups — C₇₃×#12425 #32921829

Order 18688.

274 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 86 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	C₇₃×#2207	C₂	1	2	yes	maximal, complemented
2	C₇₃×#1712	C₂	1	1	yes	maximal, characteristic
3	C₈×Q₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₃×#1370	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#1361	C₂	1	2	yes	maximal, complemented
6	C₂×SD₃₂×C₇₃	—	2	2	no	—
7	#493×C₇₃	—	2	1	no	—
8	#493×C₇₃	—	2	1	no	—
9	#493×C₇₃	C₂²	1	2	yes	—
10	C₈×Q₈×C₇₃	—	2	1	no	—
11	C₄×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₁₆×C₇₃	C₂²	1	2	yes	—
13	#357×C₇₃	—	2	2	no	—
14	#335×C₇₃	C₂²	1	1	yes	characteristic
15	Q₈⋊C₈×C₇₃	—	2	1	no	—
16	C₈²×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₇₃	D₄	1	1	yes	characteristic
18	C₂×D₈×C₇₃	—	2	2	no	—
19	C₂×D₈×C₇₃	—	2	2	no	—
20	C₂×D₈×C₇₃	—	4	2	no	—
21	C₂×D₈×C₇₃	—	4	1	no	—
22	C₂×D₈×C₇₃	—	4	1	no	—
23	#127×C₇₃	—	2	2	no	—
24	C₄×Q₈×C₇₃	—	2	1	no	—
25	SD₃₂×C₇₃	—	4	4	no	—
26	C₂×C₁₆×C₇₃	—	2	2	no	—
27	#107×C₇₃	D₄	1	1	yes	characteristic
28	C₄⋊C₈×C₇₃	—	2	1	no	—
29	C₄⋊C₈×C₇₃	—	4	1	no	—
30	#103×C₇₃	—	2	1	no	—
31	C₄×C₈×C₇₃	—	4	1	no	—
32	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
33	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
34	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
35	C₂×Q₈×C₇₃	—	2	1	no	—
36	C₂×D₄×C₇₃	—	4	2	no	—
37	C₂×D₄×C₇₃	—	4	2	no	—
38	C₂×D₄×C₇₃	—	4	2	no	—
39	Q₁₆×C₇₃	—	2	1	no	—
40	Q₁₆×C₇₃	D₈	1	2	yes	complemented
41	D₈×C₇₃	—	4	2	no	—
42	D₈×C₇₃	—	4	2	no	—
43	D₈×C₇₃	—	8	2	no	—
44	D₈×C₇₃	—	8	4	no	—
45	D₈×C₇₃	—	8	2	no	—
46	D₈×C₇₃	—	8	2	no	—
47	C₂×C₈×C₇₃	—	2	1	no	—
48	C₂×C₈×C₇₃	—	4	1	no	—
49	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
50	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
51	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
53	C₄⋊C₄×C₇₃	—	2	1	no	—
54	C₄⋊C₄×C₇₃	—	4	1	no	—
55	C₄²×C₇₃	—	4	1	no	—
56	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
57	C₁₆×C₇₃	—	4	2	no	—
58	C₂³×C₇₃	—	8	2	no	—
59	Q₈×C₇₃	—	2	2	no	—
60	Q₈×C₇₃	—	4	1	no	—
61	D₄×C₇₃	—	8	4	no	—
62	D₄×C₇₃	—	8	2	no	—
63	D₄×C₇₃	—	8	2	no	—
64	D₄×C₇₃	—	8	2	no	—
65	D₄×C₇₃	—	8	2	no	—
66	C₂×C₄×C₇₃	—	4	1	no	—
67	C₂×C₄×C₇₃	—	4	1	no	—
68	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
70	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
71	C₈×C₇₃	—	2	2	no	—
72	C₈×C₇₃	—	2	1	no	—
73	C₈×C₇₃	—	2	1	no	—
74	C₈×C₇₃	—	8	1	no	—
75	C₈×C₇₃	SD₃₂	1	2	yes	complemented
76	C₈×C₇₃	C₂×D₈	1	1	yes	characteristic
77	C₈×C₇₃	#135	1	1	yes	characteristic
78	C₂²×C₇₃	—	16	2	no	—
79	C₂²×C₇₃	—	16	2	no	—
80	C₂²×C₇₃	—	16	2	no	—
81	C₂²×C₇₃	#459	1	1	yes	characteristic
82	C₄×C₇₃	—	2	1	no	—
83	C₄×C₇₃	—	4	2	no	—
84	C₄×C₇₃	—	8	1	no	—
85	C₄×C₇₃	#466	1	1	yes	characteristic
86	C₄×C₇₃	#472	1	1	yes	characteristic
87	C₄×C₇₃	C₂×SD₃₂	1	1	yes	characteristic
88	C₄×C₇₃	#509	1	1	yes	characteristic
89	#12425	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂×C₇₃	—	32	2	no	—
91	C₂×C₇₃	#1667	1	1	yes	characteristic
92	C₂×C₇₃	#2207	1	1	yes	characteristic
93	C₂×C₇₃	#2213	1	1	yes	characteristic
94	#2207	C₂×C₇₃	1	2	yes	complemented
95	#1712	C₂×C₇₃	1	1	yes	characteristic
96	C₈×Q₁₆	C₂×C₇₃	1	1	yes	characteristic, complemented
97	#1370	C₂×C₇₃	1	1	yes	characteristic, complemented
98	#1361	C₂×C₇₃	1	2	yes	complemented
99	C₇₃	#12425	1	1	yes	characteristic, direct product factor, Sylow
100	C₂×SD₃₂	—	2	2	no	—
101	#493	—	2	1	no	—
102	#493	—	2	1	no	—
103	#493	C₂²×C₇₃	1	2	yes	—
104	C₈×Q₈	—	2	1	no	—
105	C₄×Q₁₆	C₂²×C₇₃	1	1	yes	characteristic
106	C₄⋊C₁₆	C₂²×C₇₃	1	2	yes	—
107	#357	—	2	2	no	—
108	#335	C₂²×C₇₃	1	1	yes	characteristic
109	Q₈⋊C₈	—	2	1	no	—
110	C₈²	C₂²×C₇₃	1	1	yes	characteristic
111	C₂×Q₁₆	D₄×C₇₃	1	1	yes	characteristic
112	C₂×D₈	—	2	2	no	—
113	C₂×D₈	—	2	2	no	—
114	C₂×D₈	—	4	2	no	—
115	C₂×D₈	—	4	1	no	—
116	C₂×D₈	—	4	1	no	—
117	#127	—	2	2	no	—
118	C₄×Q₈	—	2	1	no	—
119	SD₃₂	—	4	4	no	—
120	C₂×C₁₆	—	2	2	no	—
121	#107	D₄×C₇₃	1	1	yes	characteristic
122	C₄⋊C₈	—	2	1	no	—
123	C₄⋊C₈	—	4	1	no	—
124	#103	—	2	1	no	—
125	C₄×C₈	—	4	1	no	—
126	C₄×C₈	D₄×C₇₃	1	1	yes	characteristic
127	C₄×C₈	D₄×C₇₃	1	1	yes	characteristic
128	C₄×C₈	C₂³×C₇₃	1	1	yes	characteristic
129	C₂×Q₈	—	2	1	no	—
130	C₂×D₄	—	4	2	no	—
131	C₂×D₄	—	4	2	no	—
132	C₂×D₄	—	4	2	no	—
133	Q₁₆	—	2	1	no	—
134	Q₁₆	D₈×C₇₃	1	2	yes	complemented
135	D₈	—	4	2	no	—
136	D₈	—	4	2	no	—
137	D₈	—	8	2	no	—
138	D₈	—	8	4	no	—
139	D₈	—	8	2	no	—
140	D₈	—	8	2	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	4	1	no	—
143	C₂×C₈	D₈×C₇₃	1	1	yes	characteristic
144	C₂×C₈	D₈×C₇₃	1	1	yes	characteristic
145	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
146	C₂×C₈	#41×C₇₃	1	1	yes	characteristic
147	C₄⋊C₄	—	2	1	no	—
148	C₄⋊C₄	—	4	1	no	—
149	C₄²	—	4	1	no	—
150	C₄²	C₂×D₄×C₇₃	1	1	yes	characteristic
151	C₁₆	—	4	2	no	—
152	C₂³	—	8	2	no	—
153	Q₈	—	2	2	no	—
154	Q₈	—	4	1	no	—
155	D₄	—	8	4	no	—
156	D₄	—	8	2	no	—
157	D₄	—	8	2	no	—
158	D₄	—	8	2	no	—
159	D₄	—	8	2	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	—	4	1	no	—
162	C₂×C₄	#121×C₇₃	1	1	yes	characteristic
163	C₂×C₄	C₂×D₈×C₇₃	1	1	yes	characteristic
164	C₂×C₄	AGL(1,ℤ/8)×C₇₃	1	1	yes	characteristic
165	C₈	—	2	2	no	—
166	C₈	—	2	1	no	—
167	C₈	—	2	1	no	—
168	C₈	—	8	1	no	—
169	C₈	SD₃₂×C₇₃	1	2	yes	complemented
170	C₈	C₂×D₈×C₇₃	1	1	yes	characteristic
171	C₈	#135×C₇₃	1	1	yes	characteristic
172	C₂²	—	16	2	no	—
173	C₂²	—	16	2	no	—
174	C₂²	—	16	2	no	—
175	C₂²	#459×C₇₃	1	1	yes	characteristic
176	C₄	—	2	1	no	—
177	C₄	—	4	2	no	—
178	C₄	—	8	1	no	—
179	C₄	#466×C₇₃	1	1	yes	characteristic
180	C₄	#472×C₇₃	1	1	yes	characteristic
181	C₄	C₂×SD₃₂×C₇₃	1	1	yes	characteristic
182	C₄	#509×C₇₃	1	1	yes	characteristic
183	C₂	—	32	2	no	—
184	C₂	C₇₃×#1667	1	1	yes	characteristic
185	C₂	C₇₃×#2207	1	1	yes	characteristic
186	C₂	C₇₃×#2213	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃×#2207

C₂

yes

maximal, complemented

C₇₃×#1712

C₂

yes

maximal, characteristic

C₈×Q₁₆×C₇₃

C₂

yes

maximal, characteristic, complemented

C₇₃×#1370

C₂

yes

maximal, characteristic, complemented

C₇₃×#1361

C₂

yes

maximal, complemented

C₂×SD₃₂×C₇₃

—

#493×C₇₃

—

#493×C₇₃

—

#493×C₇₃

C₂²

yes

—

C₈×Q₈×C₇₃

—

C₄×Q₁₆×C₇₃

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₃

C₂²

yes

—

#357×C₇₃

—

#335×C₇₃

C₂²

yes

characteristic

Q₈⋊C₈×C₇₃

—

C₈²×C₇₃

C₂²

yes

characteristic

C₂×Q₁₆×C₇₃

D₄

yes

characteristic

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

#127×C₇₃

—

C₄×Q₈×C₇₃

—

SD₃₂×C₇₃

—

C₂×C₁₆×C₇₃

—

#107×C₇₃

D₄

yes

characteristic

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

—

#103×C₇₃

—

C₄×C₈×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×Q₈×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

D₈

yes

complemented

D₈×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

characteristic

C₂×C₈×C₇₃

D₈

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

#41

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#121

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

SD₃₂

yes

complemented

C₈×C₇₃

C₂×D₈

yes

characteristic

C₈×C₇₃

#135

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#459

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

C₂×SD₃₂

yes

C₄×C₇₃

yes

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

complemented

#1712

C₂×C₇₃

yes

characteristic

C₈×Q₁₆

C₂×C₇₃

yes

characteristic, complemented

#1370

C₂×C₇₃

yes

characteristic, complemented

#1361

C₂×C₇₃

yes

complemented

C₇₃

#12425

yes

characteristic, direct product factor, Sylow

100

C₂×SD₃₂

—

101

#493

—

102

#493

—

103

#493

C₂²×C₇₃

yes

—

104

C₈×Q₈

—

105

C₄×Q₁₆

C₂²×C₇₃

yes

characteristic

106

C₄⋊C₁₆

C₂²×C₇₃

yes

—

107

#357

—

108

#335

C₂²×C₇₃

yes

characteristic

109

Q₈⋊C₈

—

110

C₈²

C₂²×C₇₃

yes

characteristic

111

C₂×Q₁₆

D₄×C₇₃

yes

characteristic

112

C₂×D₈

—

113

C₂×D₈

—

114

C₂×D₈

—

115

C₂×D₈

—

116

C₂×D₈

—

117

#127

—

118

C₄×Q₈

—

119

SD₃₂

—

120

C₂×C₁₆

—

121

#107

D₄×C₇₃

yes

characteristic

122

C₄⋊C₈

—

123

C₄⋊C₈

—

124

#103

—

125

C₄×C₈

—

126

C₄×C₈

D₄×C₇₃

yes

characteristic

127

C₄×C₈

D₄×C₇₃

yes

characteristic

128

C₄×C₈

C₂³×C₇₃

yes

characteristic

129

C₂×Q₈

—

130

C₂×D₄

—

131

C₂×D₄

—

132

C₂×D₄

—

133

Q₁₆

—

134

Q₁₆

D₈×C₇₃

yes

complemented

135

D₈

—

136

D₈

—

137

D₈

—

138

D₈

—

139

D₈

—

140

D₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

D₈×C₇₃

yes

characteristic

144

C₂×C₈

D₈×C₇₃

yes

characteristic

145

C₂×C₈

C₂×D₄×C₇₃

yes

characteristic

146

C₂×C₈

#41×C₇₃

yes

characteristic

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄²

—

150

C₄²

C₂×D₄×C₇₃

yes

characteristic

151

C₁₆

—

152

C₂³

—

153

Q₈

—

154

Q₈

—

155

D₄

—

156

D₄

—

157

D₄

—

158

D₄

—

159

D₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

#121×C₇₃

yes

characteristic

163

C₂×C₄

C₂×D₈×C₇₃

yes

characteristic

164

C₂×C₄

AGL(1,ℤ/8)×C₇₃

yes

characteristic

165

C₈

—

166

C₈

—

167

C₈

—

168

C₈

—

169

C₈

SD₃₂×C₇₃

yes

complemented

170

C₈

C₂×D₈×C₇₃

yes

characteristic

171

C₈

#135×C₇₃

yes

characteristic

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

#459×C₇₃

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

#466×C₇₃

yes

characteristic

180

C₄

#472×C₇₃

yes

characteristic

181

C₄

C₂×SD₃₂×C₇₃

yes

characteristic

182

C₄

#509×C₇₃

yes

characteristic

183

C₂

—

184

C₂

C₇₃×#1667

yes

characteristic

185

C₂

C₇₃×#2207

yes

characteristic

186

C₂

C₇₃×#2213

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₇₃	#459
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇₃
prime radical	C₇₃×#1712	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₄²×C₇₃ ◁ C₈²×C₇₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₃

#459

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₇₃

prime radical

C₇₃×#1712

C₂

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₇₃ ◁ C₄²×C₇₃ ◁ C₈²×C₇₃ ◁ G

N	H ≅ G/N	type
1. C₇₃×#2207	C₂	semidirect
3. C₈×Q₁₆×C₇₃	C₂	semidirect
4. C₇₃×#1370	C₂	semidirect
5. C₇₃×#1361	C₂	semidirect
40. Q₁₆×C₇₃	D₈	semidirect
75. C₈×C₇₃	SD₃₂	semidirect
89. #12425	C₇₃	direct (N × H)
94. #2207	C₂×C₇₃	semidirect
96. C₈×Q₁₆	C₂×C₇₃	semidirect
97. #1370	C₂×C₇₃	semidirect
98. #1361	C₂×C₇₃	semidirect
99. C₇₃	#12425	direct (N × H)
134. Q₁₆	D₈×C₇₃	semidirect
169. C₈	SD₃₂×C₇₃	semidirect

H ≅ G/N

type

1. C₇₃×#2207

C₂

semidirect

3. C₈×Q₁₆×C₇₃

C₂

semidirect

4. C₇₃×#1370

C₂

semidirect

5. C₇₃×#1361

C₂

semidirect

40. Q₁₆×C₇₃

D₈

semidirect

75. C₈×C₇₃

SD₃₂

semidirect

89. #12425

C₇₃

direct (N × H)

94. #2207

C₂×C₇₃

semidirect

96. C₈×Q₁₆

C₂×C₇₃

semidirect

97. #1370

C₂×C₇₃

semidirect

98. #1361

C₂×C₇₃

semidirect

99. C₇₃

#12425

direct (N × H)

134. Q₁₆

D₈×C₇₃

semidirect

169. C₈

SD₃₂×C₇₃

semidirect

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

subgroup	index
C₇₃×#2207	2
C₇₃×#1712	2
C₈×Q₁₆×C₇₃	2
C₇₃×#1370	2
C₇₃×#1361	2
#12425	73

subgroup

index

C₇₃×#2207

C₇₃×#1712

C₈×Q₁₆×C₇₃

C₇₃×#1370

C₇₃×#1361

#12425

Subgroups — C₇₃×#12425 #32921829

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#12425 #32921829

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#12425 #32921829