Group #32921849 — subgroups

Subgroups — C₇₃×#12445 #32921849

Order 18688.

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

normal subgroups only

This group has 234 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×#2224	C₂	1	2	yes	maximal
2	C₇₃×#1713	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×D₈×C₇₃	C₂	1	1	yes	maximal, characteristic
4	C₇₃×#1370	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#1364	C₂	1	2	yes	maximal, complemented
6	#500×C₇₃	—	2	1	no	—
7	#500×C₇₃	C₂²	1	2	yes	—
8	#494×C₇₃	—	2	1	no	—
9	C₄×D₈×C₇₃	C₂²	1	1	yes	characteristic
10	C₈×D₄×C₇₃	—	2	1	no	—
11	#366×C₇₃	—	2	2	no	—
12	C₄⋊C₁₆×C₇₃	C₂²	1	2	yes	—
13	#357×C₇₃	—	2	2	no	—
14	#335×C₇₃	C₂²	1	1	yes	characteristic
15	D₄⋊C₈×C₇₃	—	2	1	no	—
16	C₈²×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₇₃	—	4	1	no	—
18	C₂×Q₁₆×C₇₃	—	4	1	no	—
19	C₂×D₈×C₇₃	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₇₃	—	4	1	no	—
21	C₄⋊Q₈×C₇₃	—	2	2	no	—
22	C₄×D₄×C₇₃	—	2	1	no	—
23	C₂×C₁₆×C₇₃	—	2	2	no	—
24	#107×C₇₃	—	2	2	no	—
25	#107×C₇₃	—	2	2	no	—
26	#107×C₇₃	—	4	2	no	—
27	#107×C₇₃	D₄	1	1	yes	characteristic
28	C₄⋊C₈×C₇₃	—	2	1	no	—
29	#102×C₇₃	—	2	1	no	—
30	C₂²⋊C₈×C₇₃	—	4	1	no	—
31	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
33	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
34	C₂×Q₈×C₇₃	—	4	2	no	—
35	C₂×D₄×C₇₃	—	2	1	no	—
36	C₂²×C₄×C₇₃	—	4	1	no	—
37	Q₁₆×C₇₃	—	8	2	no	—
38	Q₁₆×C₇₃	—	8	2	no	—
39	D₈×C₇₃	—	2	1	no	—
40	D₈×C₇₃	Q₁₆	1	2	yes	complemented
41	C₂×C₈×C₇₃	—	2	1	no	—
42	C₂×C₈×C₇₃	—	4	1	no	—
43	C₂×C₈×C₇₃	—	8	2	no	—
44	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
45	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
46	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
48	C₄⋊C₄×C₇₃	—	2	1	no	—
49	C₄⋊C₄×C₇₃	—	4	2	no	—
50	C₄⋊C₄×C₇₃	—	4	2	no	—
51	C₂²⋊C₄×C₇₃	—	4	1	no	—
52	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
53	C₁₆×C₇₃	—	4	2	no	—
54	C₂³×C₇₃	—	4	1	no	—
55	Q₈×C₇₃	—	8	2	no	—
56	Q₈×C₇₃	—	8	2	no	—
57	D₄×C₇₃	—	2	2	no	—
58	D₄×C₇₃	—	4	1	no	—
59	C₂×C₄×C₇₃	—	4	1	no	—
60	C₂×C₄×C₇₃	—	8	1	no	—
61	C₂×C₄×C₇₃	—	8	1	no	—
62	C₂×C₄×C₇₃	—	8	2	no	—
63	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
64	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₈×C₇₃	—	2	2	no	—
67	C₈×C₇₃	—	2	1	no	—
68	C₈×C₇₃	—	2	1	no	—
69	C₈×C₇₃	—	8	1	no	—
70	C₈×C₇₃	D₁₆	1	2	yes	—
71	C₈×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
72	C₈×C₇₃	#135	1	1	yes	characteristic
73	C₂²×C₇₃	—	4	2	no	—
74	C₂²×C₇₃	—	8	1	no	—
75	C₂²×C₇₃	—	8	1	no	—
76	C₂²×C₇₃	#459	1	1	yes	characteristic
77	C₄×C₇₃	—	2	1	no	—
78	C₄×C₇₃	—	8	1	no	—
79	C₄×C₇₃	—	16	2	no	—
80	C₄×C₇₃	#467	1	1	yes	characteristic
81	C₄×C₇₃	#473	1	1	yes	characteristic
82	C₄×C₇₃	C₂×D₁₆	1	1	yes	characteristic
83	C₄×C₇₃	#509	1	1	yes	characteristic
84	#12445	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂×C₇₃	—	8	2	no	—
86	C₂×C₇₃	#1669	1	1	yes	characteristic
87	C₂×C₇₃	#2206	1	1	yes	characteristic
88	C₂×C₇₃	#2214	1	1	yes	characteristic
89	#2224	C₂×C₇₃	1	2	yes	—
90	#1713	C₂×C₇₃	1	1	yes	characteristic, complemented
91	C₈×D₈	C₂×C₇₃	1	1	yes	characteristic
92	#1370	C₂×C₇₃	1	1	yes	characteristic, complemented
93	#1364	C₂×C₇₃	1	2	yes	complemented
94	C₇₃	#12445	1	1	yes	characteristic, direct product factor, Sylow
95	#500	—	2	1	no	—
96	#500	C₂²×C₇₃	1	2	yes	—
97	#494	—	2	1	no	—
98	C₄×D₈	C₂²×C₇₃	1	1	yes	characteristic
99	C₈×D₄	—	2	1	no	—
100	#366	—	2	2	no	—
101	C₄⋊C₁₆	C₂²×C₇₃	1	2	yes	—
102	#357	—	2	2	no	—
103	#335	C₂²×C₇₃	1	1	yes	characteristic
104	D₄⋊C₈	—	2	1	no	—
105	C₈²	C₂²×C₇₃	1	1	yes	characteristic
106	C₂×Q₁₆	—	4	1	no	—
107	C₂×Q₁₆	—	4	1	no	—
108	C₂×D₈	D₄×C₇₃	1	1	yes	characteristic
109	C₂²×C₈	—	4	1	no	—
110	C₄⋊Q₈	—	2	2	no	—
111	C₄×D₄	—	2	1	no	—
112	C₂×C₁₆	—	2	2	no	—
113	#107	—	2	2	no	—
114	#107	—	2	2	no	—
115	#107	—	4	2	no	—
116	#107	D₄×C₇₃	1	1	yes	characteristic
117	C₄⋊C₈	—	2	1	no	—
118	#102	—	2	1	no	—
119	C₂²⋊C₈	—	4	1	no	—
120	C₄×C₈	D₄×C₇₃	1	1	yes	characteristic
121	C₄×C₈	D₄×C₇₃	1	1	yes	characteristic
122	C₄×C₈	C₂³×C₇₃	1	1	yes	characteristic
123	C₂×Q₈	—	4	2	no	—
124	C₂×D₄	—	2	1	no	—
125	C₂²×C₄	—	4	1	no	—
126	Q₁₆	—	8	2	no	—
127	Q₁₆	—	8	2	no	—
128	D₈	—	2	1	no	—
129	D₈	Q₁₆×C₇₃	1	2	yes	complemented
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	4	1	no	—
132	C₂×C₈	—	8	2	no	—
133	C₂×C₈	D₈×C₇₃	1	1	yes	characteristic
134	C₂×C₈	D₈×C₇₃	1	1	yes	characteristic
135	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
136	C₂×C₈	#41×C₇₃	1	1	yes	characteristic
137	C₄⋊C₄	—	2	1	no	—
138	C₄⋊C₄	—	4	2	no	—
139	C₄⋊C₄	—	4	2	no	—
140	C₂²⋊C₄	—	4	1	no	—
141	C₄²	C₂×D₄×C₇₃	1	1	yes	characteristic
142	C₁₆	—	4	2	no	—
143	C₂³	—	4	1	no	—
144	Q₈	—	8	2	no	—
145	Q₈	—	8	2	no	—
146	D₄	—	2	2	no	—
147	D₄	—	4	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	8	1	no	—
150	C₂×C₄	—	8	1	no	—
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	#121×C₇₃	1	1	yes	characteristic
153	C₂×C₄	C₂×D₈×C₇₃	1	1	yes	characteristic
154	C₂×C₄	AGL(1,ℤ/8)×C₇₃	1	1	yes	characteristic
155	C₈	—	2	2	no	—
156	C₈	—	2	1	no	—
157	C₈	—	2	1	no	—
158	C₈	—	8	1	no	—
159	C₈	D₁₆×C₇₃	1	2	yes	—
160	C₈	C₂×Q₁₆×C₇₃	1	1	yes	characteristic
161	C₈	#135×C₇₃	1	1	yes	characteristic
162	C₂²	—	4	2	no	—
163	C₂²	—	8	1	no	—
164	C₂²	—	8	1	no	—
165	C₂²	#459×C₇₃	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	8	1	no	—
168	C₄	—	16	2	no	—
169	C₄	#467×C₇₃	1	1	yes	characteristic
170	C₄	#473×C₇₃	1	1	yes	characteristic
171	C₄	C₂×D₁₆×C₇₃	1	1	yes	characteristic
172	C₄	#509×C₇₃	1	1	yes	characteristic
173	C₂	—	8	2	no	—
174	C₂	C₇₃×#1669	1	1	yes	characteristic
175	C₂	C₇₃×#2206	1	1	yes	characteristic
176	C₂	C₇₃×#2214	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃×#2224

C₂

yes

maximal

C₇₃×#1713

C₂

yes

maximal, characteristic, complemented

C₈×D₈×C₇₃

C₂

yes

maximal, characteristic

C₇₃×#1370

C₂

yes

maximal, characteristic, complemented

C₇₃×#1364

C₂

yes

maximal, complemented

#500×C₇₃

—

#500×C₇₃

C₂²

yes

—

#494×C₇₃

—

C₄×D₈×C₇₃

C₂²

yes

characteristic

C₈×D₄×C₇₃

—

#366×C₇₃

—

C₄⋊C₁₆×C₇₃

C₂²

yes

—

#357×C₇₃

—

#335×C₇₃

C₂²

yes

characteristic

D₄⋊C₈×C₇₃

—

C₈²×C₇₃

C₂²

yes

characteristic

C₂×Q₁₆×C₇₃

—

C₂×Q₁₆×C₇₃

—

C₂×D₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

—

C₄⋊Q₈×C₇₃

—

C₄×D₄×C₇₃

—

C₂×C₁₆×C₇₃

—

#107×C₇₃

—

#107×C₇₃

—

#107×C₇₃

—

#107×C₇₃

D₄

yes

characteristic

C₄⋊C₈×C₇₃

—

#102×C₇₃

—

C₂²⋊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

—

C₂²×C₄×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

Q₁₆

yes

complemented

C₂×C₈×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

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×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

D₁₆

yes

—

C₈×C₇₃

C₂×Q₁₆

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×D₁₆

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

—

#1713

C₂×C₇₃

yes

characteristic, complemented

C₈×D₈

C₂×C₇₃

yes

characteristic

#1370

C₂×C₇₃

yes

characteristic, complemented

#1364

C₂×C₇₃

yes

complemented

C₇₃

#12445

yes

characteristic, direct product factor, Sylow

#500

—

#500

C₂²×C₇₃

yes

—

#494

—

C₄×D₈

C₂²×C₇₃

yes

characteristic

C₈×D₄

—

100

#366

—

101

C₄⋊C₁₆

C₂²×C₇₃

yes

—

102

#357

—

103

#335

C₂²×C₇₃

yes

characteristic

104

D₄⋊C₈

—

105

C₈²

C₂²×C₇₃

yes

characteristic

106

C₂×Q₁₆

—

107

C₂×Q₁₆

—

108

C₂×D₈

D₄×C₇₃

yes

characteristic

109

C₂²×C₈

—

110

C₄⋊Q₈

—

111

C₄×D₄

—

112

C₂×C₁₆

—

113

#107

—

114

#107

—

115

#107

—

116

#107

D₄×C₇₃

yes

characteristic

117

C₄⋊C₈

—

118

#102

—

119

C₂²⋊C₈

—

120

C₄×C₈

D₄×C₇₃

yes

characteristic

121

C₄×C₈

D₄×C₇₃

yes

characteristic

122

C₄×C₈

C₂³×C₇₃

yes

characteristic

123

C₂×Q₈

—

124

C₂×D₄

—

125

C₂²×C₄

—

126

Q₁₆

—

127

Q₁₆

—

128

D₈

—

129

D₈

Q₁₆×C₇₃

yes

complemented

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

D₈×C₇₃

yes

characteristic

134

C₂×C₈

D₈×C₇₃

yes

characteristic

135

C₂×C₈

C₂×D₄×C₇₃

yes

characteristic

136

C₂×C₈

#41×C₇₃

yes

characteristic

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₂²⋊C₄

—

141

C₄²

C₂×D₄×C₇₃

yes

characteristic

142

C₁₆

—

143

C₂³

—

144

Q₈

—

145

Q₈

—

146

D₄

—

147

D₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

#121×C₇₃

yes

characteristic

153

C₂×C₄

C₂×D₈×C₇₃

yes

characteristic

154

C₂×C₄

AGL(1,ℤ/8)×C₇₃

yes

characteristic

155

C₈

—

156

C₈

—

157

C₈

—

158

C₈

—

159

C₈

D₁₆×C₇₃

yes

—

160

C₈

C₂×Q₁₆×C₇₃

yes

characteristic

161

C₈

#135×C₇₃

yes

characteristic

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

#459×C₇₃

yes

characteristic

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

#467×C₇₃

yes

characteristic

170

C₄

#473×C₇₃

yes

characteristic

171

C₄

C₂×D₁₆×C₇₃

yes

characteristic

172

C₄

#509×C₇₃

yes

characteristic

173

C₂

—

174

C₂

C₇₃×#1669

yes

characteristic

175

C₂

C₇₃×#2206

yes

characteristic

176

C₂

C₇₃×#2214

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₇₃	#459
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇₃
prime radical	C₂×D₈×C₇₃	D₄
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

C₂×D₈×C₇₃

D₄

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₇₃ ◁ C₄²×C₇₃ ◁ C₈²×C₇₃ ◁ G

N	H ≅ G/N	type
2. C₇₃×#1713	C₂	semidirect
4. C₇₃×#1370	C₂	semidirect
5. C₇₃×#1364	C₂	semidirect
40. D₈×C₇₃	Q₁₆	semidirect
84. #12445	C₇₃	direct (N × H)
90. #1713	C₂×C₇₃	semidirect
92. #1370	C₂×C₇₃	semidirect
93. #1364	C₂×C₇₃	semidirect
94. C₇₃	#12445	direct (N × H)
129. D₈	Q₁₆×C₇₃	semidirect

H ≅ G/N

type

2. C₇₃×#1713

C₂

semidirect

4. C₇₃×#1370

C₂

semidirect

5. C₇₃×#1364

C₂

semidirect

40. D₈×C₇₃

Q₁₆

semidirect

84. #12445

C₇₃

direct (N × H)

90. #1713

C₂×C₇₃

semidirect

92. #1370

C₂×C₇₃

semidirect

93. #1364

C₂×C₇₃

semidirect

94. C₇₃

#12445

direct (N × H)

129. D₈

Q₁₆×C₇₃

semidirect

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

subgroup	index
C₇₃×#2224	2
C₇₃×#1713	2
C₈×D₈×C₇₃	2
C₇₃×#1370	2
C₇₃×#1364	2
#12445	73

subgroup

index

C₇₃×#2224

C₇₃×#1713

C₈×D₈×C₇₃

C₇₃×#1370

C₇₃×#1364

#12445

Subgroups — C₇₃×#12445 #32921849

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#12445 #32921849

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#12445 #32921849