Group #32918148 — subgroups

Subgroups — C₇₃×#8744 #32918148

Order 18688.

384 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 110 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×#2298	C₂	1	1	yes	maximal, characteristic
2	C₇₃×#1464	C₂	1	2	yes	maximal
3	C₇₃×#1285	C₂	1	4	yes	maximal, complemented
4	C₂×#117×C₇₃	C₂²	1	1	yes	characteristic
5	#432×C₇₃	C₂²	1	4	yes	—
6	#388×C₇₃	—	2	2	no	—
7	#387×C₇₃	—	2	2	no	—
8	C₄×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
9	C₄×C₂²⋊C₄×C₇₃	C₄	1	2	yes	—
10	#376×C₇₃	C₄	1	1	yes	characteristic
11	C₄³×C₇₃	C₄	1	1	yes	characteristic
12	#324×C₇₃	—	2	2	no	—
13	#323×C₇₃	—	2	2	no	—
14	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
15	#117×C₇₃	—	2	4	no	—
16	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
17	C₂×C₄⋊C₄×C₇₃	D₄	1	2	yes	—
18	C₂×C₂²⋊C₄×C₇₃	—	2	2	no	—
19	C₂×C₂²⋊C₄×C₇₃	D₄	1	2	yes	—
20	C₂×C₄²×C₇₃	—	2	2	no	—
21	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
25	C₄⋊C₈×C₇₃	—	2	4	no	—
26	C₂²⋊C₈×C₇₃	—	2	4	no	—
27	#97×C₇₃	—	2	4	no	—
28	#95×C₇₃	—	2	2	no	—
29	#95×C₇₃	—	2	2	no	—
30	#95×C₇₃	—	2	2	no	—
31	C₂⁴×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₇₃	—	2	2	no	—
33	C₂²×C₄×C₇₃	—	2	2	no	—
34	C₂²×C₄×C₇₃	—	2	2	no	—
35	C₂²×C₄×C₇₃	—	4	2	no	—
36	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
38	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
40	C₂×C₈×C₇₃	—	4	4	no	—
41	C₄⋊C₄×C₇₃	—	2	4	no	—
42	C₄⋊C₄×C₇₃	—	4	4	no	—
43	C₂²⋊C₄×C₇₃	—	2	4	no	—
44	C₂²⋊C₄×C₇₃	—	4	4	no	—
45	C₄²×C₇₃	—	2	2	no	—
46	C₄²×C₇₃	—	4	2	no	—
47	C₄²×C₇₃	—	4	2	no	—
48	C₄²×C₇₃	—	4	4	no	—
49	C₄²×C₇₃	#41	1	4	yes	—
50	C₂³×C₇₃	—	2	1	no	—
51	C₂³×C₇₃	—	2	2	no	—
52	C₂³×C₇₃	—	4	2	no	—
53	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₇₃	—	2	2	no	—
55	C₂×C₄×C₇₃	—	2	4	no	—
56	C₂×C₄×C₇₃	—	2	2	no	—
57	C₂×C₄×C₇₃	—	2	1	no	—
58	C₂×C₄×C₇₃	—	2	2	no	—
59	C₂×C₄×C₇₃	—	4	2	no	—
60	C₂×C₄×C₇₃	—	4	4	no	—
61	C₂×C₄×C₇₃	—	4	2	no	—
62	C₂×C₄×C₇₃	—	4	2	no	—
63	C₂×C₄×C₇₃	—	4	2	no	—
64	C₂×C₄×C₇₃	—	4	2	no	—
65	C₂×C₄×C₇₃	—	4	2	no	—
66	C₂×C₄×C₇₃	—	4	2	no	—
67	C₂×C₄×C₇₃	#117	1	2	yes	—
68	C₂×C₄×C₇₃	#123	1	4	yes	—
69	C₈×C₇₃	—	8	4	no	—
70	C₂²×C₇₃	—	2	2	no	—
71	C₂²×C₇₃	—	4	1	no	—
72	C₂²×C₇₃	—	4	2	no	—
73	C₂²×C₇₃	—	8	2	no	—
74	C₂²×C₇₃	#381	1	1	yes	characteristic
75	C₂²×C₇₃	#413	1	2	yes	—
76	C₄×C₇₃	—	2	2	no	—
77	C₄×C₇₃	—	2	4	no	—
78	C₄×C₇₃	—	4	4	no	—
79	C₄×C₇₃	—	4	1	no	—
80	C₄×C₇₃	—	4	2	no	—
81	C₄×C₇₃	—	8	2	no	—
82	#8744	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂×C₇₃	—	2	2	no	—
84	C₂×C₇₃	—	8	1	no	—
85	C₂×C₇₃	#1822	1	1	yes	characteristic
86	C₂×C₇₃	#1827	1	2	yes	—
87	#2298	C₂×C₇₃	1	1	yes	characteristic
88	#1464	C₂×C₇₃	1	2	yes	—
89	#1285	C₂×C₇₃	1	4	yes	complemented
90	C₇₃	#8744	1	1	yes	characteristic, direct product factor, Sylow
91	C₂×#117	C₂²×C₇₃	1	1	yes	characteristic
92	#432	C₂²×C₇₃	1	4	yes	—
93	#388	—	2	2	no	—
94	#387	—	2	2	no	—
95	C₄×C₄⋊C₄	C₂²×C₇₃	1	2	yes	—
96	C₄×C₂²⋊C₄	C₄×C₇₃	1	2	yes	—
97	#376	C₄×C₇₃	1	1	yes	characteristic
98	C₄³	C₄×C₇₃	1	1	yes	characteristic
99	#324	—	2	2	no	—
100	#323	—	2	2	no	—
101	C₂³×C₄	C₂×C₄×C₇₃	1	1	yes	characteristic
102	#117	—	2	4	no	—
103	C₂×C₄⋊C₄	—	2	2	no	—
104	C₂×C₄⋊C₄	D₄×C₇₃	1	2	yes	—
105	C₂×C₂²⋊C₄	—	2	2	no	—
106	C₂×C₂²⋊C₄	D₄×C₇₃	1	2	yes	—
107	C₂×C₄²	—	2	2	no	—
108	C₂×C₄²	C₂×C₄×C₇₃	1	1	yes	characteristic
109	C₂×C₄²	C₂×C₄×C₇₃	1	2	yes	—
110	C₂×C₄²	C₂×C₄×C₇₃	1	2	yes	—
111	C₂×C₄²	C₂³×C₇₃	1	1	yes	characteristic
112	C₄⋊C₈	—	2	4	no	—
113	C₂²⋊C₈	—	2	4	no	—
114	#97	—	2	4	no	—
115	#95	—	2	2	no	—
116	#95	—	2	2	no	—
117	#95	—	2	2	no	—
118	C₂⁴	C₂²⋊C₄×C₇₃	1	1	yes	characteristic
119	C₂²×C₄	—	2	2	no	—
120	C₂²×C₄	—	2	2	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	4	2	no	—
123	C₂²×C₄	C₂²⋊C₄×C₇₃	1	1	yes	characteristic
124	C₂²×C₄	C₂²⋊C₄×C₇₃	1	2	yes	—
125	C₂²×C₄	C₂²×C₄×C₇₃	1	1	yes	characteristic
126	C₂²×C₄	C₂×D₄×C₇₃	1	2	yes	—
127	C₂×C₈	—	4	4	no	—
128	C₄⋊C₄	—	2	4	no	—
129	C₄⋊C₄	—	4	4	no	—
130	C₂²⋊C₄	—	2	4	no	—
131	C₂²⋊C₄	—	4	4	no	—
132	C₄²	—	2	2	no	—
133	C₄²	—	4	2	no	—
134	C₄²	—	4	2	no	—
135	C₄²	—	4	4	no	—
136	C₄²	#41×C₇₃	1	4	yes	—
137	C₂³	—	2	1	no	—
138	C₂³	—	2	2	no	—
139	C₂³	—	4	2	no	—
140	C₂³	C₂×C₂²⋊C₄×C₇₃	1	1	yes	characteristic
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	4	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	#117×C₇₃	1	2	yes	—
155	C₂×C₄	#123×C₇₃	1	4	yes	—
156	C₈	—	8	4	no	—
157	C₂²	—	2	2	no	—
158	C₂²	—	4	1	no	—
159	C₂²	—	4	2	no	—
160	C₂²	—	8	2	no	—
161	C₂²	#381×C₇₃	1	1	yes	characteristic
162	C₂²	#413×C₇₃	1	2	yes	—
163	C₄	—	2	2	no	—
164	C₄	—	2	4	no	—
165	C₄	—	4	4	no	—
166	C₄	—	4	1	no	—
167	C₄	—	4	2	no	—
168	C₄	—	8	2	no	—
169	C₂	—	2	2	no	—
170	C₂	—	8	1	no	—
171	C₂	C₇₃×#1822	1	1	yes	characteristic
172	C₂	C₇₃×#1827	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃×#2298

C₂

yes

maximal, characteristic

C₇₃×#1464

C₂

yes

maximal

C₇₃×#1285

C₂

yes

maximal, complemented

C₂×#117×C₇₃

C₂²

yes

characteristic

#432×C₇₃

C₂²

yes

—

#388×C₇₃

—

#387×C₇₃

—

C₄×C₄⋊C₄×C₇₃

C₂²

yes

—

C₄×C₂²⋊C₄×C₇₃

C₄

yes

—

#376×C₇₃

C₄

yes

characteristic

C₄³×C₇₃

C₄

yes

characteristic

#324×C₇₃

—

#323×C₇₃

—

C₂³×C₄×C₇₃

C₂×C₄

yes

characteristic

#117×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

D₄

yes

—

C₂×C₂²⋊C₄×C₇₃

—

C₂×C₂²⋊C₄×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

—

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄⋊C₈×C₇₃

—

C₂²⋊C₈×C₇₃

—

#97×C₇₃

—

#95×C₇₃

—

#95×C₇₃

—

#95×C₇₃

—

C₂⁴×C₇₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

#41

yes

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#117

yes

—

C₂×C₄×C₇₃

#123

yes

—

C₈×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#381

yes

characteristic

C₂²×C₇₃

#413

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

#8744

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#1822

yes

characteristic

C₂×C₇₃

#1827

yes

—

#2298

C₂×C₇₃

yes

characteristic

#1464

C₂×C₇₃

yes

—

#1285

C₂×C₇₃

yes

complemented

C₇₃

#8744

yes

characteristic, direct product factor, Sylow

C₂×#117

C₂²×C₇₃

yes

characteristic

#432

C₂²×C₇₃

yes

—

#388

—

#387

—

C₄×C₄⋊C₄

C₂²×C₇₃

yes

—

C₄×C₂²⋊C₄

C₄×C₇₃

yes

—

#376

C₄×C₇₃

yes

characteristic

C₄³

C₄×C₇₃

yes

characteristic

#324

—

100

#323

—

101

C₂³×C₄

C₂×C₄×C₇₃

yes

characteristic

102

#117

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

D₄×C₇₃

yes

—

105

C₂×C₂²⋊C₄

—

106

C₂×C₂²⋊C₄

D₄×C₇₃

yes

—

107

C₂×C₄²

—

108

C₂×C₄²

C₂×C₄×C₇₃

yes

characteristic

109

C₂×C₄²

C₂×C₄×C₇₃

yes

—

110

C₂×C₄²

C₂×C₄×C₇₃

yes

—

111

C₂×C₄²

C₂³×C₇₃

yes

characteristic

112

C₄⋊C₈

—

113

C₂²⋊C₈

—

114

#97

—

115

#95

—

116

#95

—

117

#95

—

118

C₂⁴

C₂²⋊C₄×C₇₃

yes

characteristic

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

C₂²⋊C₄×C₇₃

yes

characteristic

124

C₂²×C₄

C₂²⋊C₄×C₇₃

yes

—

125

C₂²×C₄

C₂²×C₄×C₇₃

yes

characteristic

126

C₂²×C₄

C₂×D₄×C₇₃

yes

—

127

C₂×C₈

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₂²⋊C₄

—

131

C₂²⋊C₄

—

132

C₄²

—

133

C₄²

—

134

C₄²

—

135

C₄²

—

136

C₄²

#41×C₇₃

yes

—

137

C₂³

—

138

C₂³

—

139

C₂³

—

140

C₂³

C₂×C₂²⋊C₄×C₇₃

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

#117×C₇₃

yes

—

155

C₂×C₄

#123×C₇₃

yes

—

156

C₈

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

#381×C₇₃

yes

characteristic

162

C₂²

#413×C₇₃

yes

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₂

—

170

C₂

—

171

C₂

C₇₃×#1822

yes

characteristic

172

C₂

C₇₃×#1827

yes

—

	subgroup	G/H
center, socle	C₂²×C₇₃	#381
commutator	C₂²×C₄	C₂²×C₄×C₇₃
Frattini, cosocle	C₂×C₄²	C₂³×C₇₃
prime radical	C₂⁴×C₇₃	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₃

#381

commutator

C₂²×C₄

C₂²×C₄×C₇₃

Frattini, cosocle

C₂×C₄²

C₂³×C₇₃

prime radical

C₂⁴×C₇₃

C₂²⋊C₄

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ G

N	H ≅ G/N	type
3. C₇₃×#1285	C₂	semidirect
82. #8744	C₇₃	direct (N × H)
89. #1285	C₂×C₇₃	semidirect
90. C₇₃	#8744	direct (N × H)

H ≅ G/N

type

3. C₇₃×#1285

C₂

semidirect

82. #8744

C₇₃

direct (N × H)

89. #1285

C₂×C₇₃

semidirect

90. C₇₃

#8744

direct (N × H)

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

subgroup	index
C₇₃×#2298	2
C₇₃×#1464	2
C₇₃×#1285	2
#8744	73

subgroup

index

C₇₃×#2298

C₇₃×#1464

C₇₃×#1285

#8744

Subgroups — C₇₃×#8744 #32918148

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#8744 #32918148

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#8744 #32918148