Group #32916569 — subgroups

Subgroups — C₇₃×#7165 #32916569

Order 18688.

222 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2163	C₂	1	1	yes	maximal, characteristic
2	C₂×#346×C₇₃	C₂	1	1	yes	maximal, characteristic
3	C₈×M₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₆×C₇₃	—	2	1	no	—
5	#431×C₇₃	C₄	1	1	yes	characteristic
6	C₄×M₁₆×C₇₃	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
8	C₄⋊C₁₆×C₇₃	C₄	1	2	yes	—
9	C₂²⋊C₁₆×C₇₃	—	2	1	no	—
10	#346×C₇₃	—	2	1	no	—
11	#346×C₇₃	—	2	1	no	—
12	#346×C₇₃	C₄	1	2	yes	—
13	#322×C₇₃	—	2	1	no	—
14	C₈²×C₇₃	—	2	1	no	—
15	C₂×M₁₆×C₇₃	—	2	1	no	—
16	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₇₃	Q₈	1	1	yes	characteristic
18	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆×C₇₃	—	2	4	no	—
20	C₂×C₁₆×C₇₃	—	2	2	no	—
21	C₂×C₁₆×C₇₃	—	2	2	no	—
22	C₄⋊C₈×C₇₃	—	2	1	no	—
23	C₂²⋊C₈×C₇₃	C₈	1	2	yes	—
24	#97×C₇₃	C₈	1	2	yes	—
25	C₄×C₈×C₇₃	—	2	1	no	—
26	C₄×C₈×C₇₃	—	2	1	no	—
27	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
34	M₁₆×C₇₃	—	2	4	no	—
35	C₂×C₈×C₇₃	—	2	2	no	—
36	C₂×C₈×C₇₃	—	2	2	no	—
37	C₂×C₈×C₇₃	—	2	2	no	—
38	C₂×C₈×C₇₃	—	2	2	no	—
39	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₇₃	C₄²	1	1	yes	characteristic
44	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
46	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
47	C₁₆×C₇₃	—	2	4	no	—
48	C₁₆×C₇₃	—	4	2	no	—
49	C₂³×C₇₃	C₄⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₇₃	—	2	2	no	—
51	C₂×C₄×C₇₃	—	2	1	no	—
52	C₂×C₄×C₇₃	—	2	1	no	—
53	C₂×C₄×C₇₃	—	4	1	no	—
54	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
55	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
56	C₂×C₄×C₇₃	#97	1	1	yes	characteristic
57	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
58	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
60	C₈×C₇₃	—	2	4	no	—
61	C₈×C₇₃	—	2	1	no	—
62	C₈×C₇₃	—	2	1	no	—
63	C₈×C₇₃	—	4	2	no	—
64	C₈×C₇₃	#104	1	2	yes	—
65	C₈×C₇₃	#108	1	2	yes	—
66	C₂²×C₇₃	—	2	1	no	—
67	C₂²×C₇₃	—	2	1	no	—
68	C₂²×C₇₃	#336	1	1	yes	characteristic
69	C₂²×C₇₃	#364	1	2	yes	—
70	C₄×C₇₃	—	2	1	no	—
71	C₄×C₇₃	—	2	1	no	—
72	C₄×C₇₃	—	2	1	no	—
73	C₄×C₇₃	—	2	1	no	—
74	C₄×C₇₃	—	2	1	no	—
75	C₄×C₇₃	#339	1	1	yes	characteristic
76	C₄×C₇₃	#341	1	1	yes	characteristic
77	#7165	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂×C₇₃	—	2	2	no	—
79	C₂×C₇₃	#1278	1	1	yes	characteristic
80	C₂×C₇₃	#1376	1	1	yes	characteristic
81	C₂×C₇₃	#1378	1	1	yes	characteristic
82	#2163	C₂×C₇₃	1	1	yes	characteristic
83	C₂×#346	C₂×C₇₃	1	1	yes	characteristic
84	C₈×M₁₆	C₂×C₇₃	1	1	yes	characteristic
85	C₇₃	#7165	1	1	yes	characteristic, direct product factor, Sylow
86	C₂²×C₁₆	—	2	1	no	—
87	#431	C₄×C₇₃	1	1	yes	characteristic
88	C₄×M₁₆	C₄×C₇₃	1	1	yes	characteristic
89	C₂×C₄×C₈	C₂²×C₇₃	1	1	yes	characteristic
90	C₄⋊C₁₆	C₄×C₇₃	1	2	yes	—
91	C₂²⋊C₁₆	—	2	1	no	—
92	#346	—	2	1	no	—
93	#346	—	2	1	no	—
94	#346	C₄×C₇₃	1	2	yes	—
95	#322	—	2	1	no	—
96	C₈²	—	2	1	no	—
97	C₂×M₁₆	—	2	1	no	—
98	C₂²×C₈	D₄×C₇₃	1	1	yes	characteristic
99	C₂²×C₈	Q₈×C₇₃	1	1	yes	characteristic
100	C₂×C₄²	C₂×C₄×C₇₃	1	1	yes	characteristic
101	C₂×C₁₆	—	2	4	no	—
102	C₂×C₁₆	—	2	2	no	—
103	C₂×C₁₆	—	2	2	no	—
104	C₄⋊C₈	—	2	1	no	—
105	C₂²⋊C₈	C₈×C₇₃	1	2	yes	—
106	#97	C₈×C₇₃	1	2	yes	—
107	C₄×C₈	—	2	1	no	—
108	C₄×C₈	—	2	1	no	—
109	C₄×C₈	C₂×C₄×C₇₃	1	1	yes	characteristic
110	C₄×C₈	C₂×C₄×C₇₃	1	1	yes	characteristic
111	C₄×C₈	D₄×C₇₃	1	1	yes	characteristic
112	C₄×C₈	D₄×C₇₃	1	1	yes	characteristic
113	C₂²×C₄	C₄⋊C₄×C₇₃	1	1	yes	characteristic
114	C₂²×C₄	C₂×C₈×C₇₃	1	1	yes	characteristic
115	C₂²×C₄	M₁₆×C₇₃	1	1	yes	characteristic
116	M₁₆	—	2	4	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	C₂²⋊C₄×C₇₃	1	1	yes	characteristic
122	C₂×C₈	C₂²⋊C₄×C₇₃	1	1	yes	characteristic
123	C₂×C₈	C₄⋊C₄×C₇₃	1	1	yes	characteristic
124	C₂×C₈	C₄⋊C₄×C₇₃	1	1	yes	characteristic
125	C₄²	C₄²×C₇₃	1	1	yes	characteristic
126	C₄²	C₂²⋊C₄×C₇₃	1	1	yes	characteristic
127	C₄²	C₂×C₈×C₇₃	1	1	yes	characteristic
128	C₄²	M₁₆×C₇₃	1	1	yes	characteristic
129	C₁₆	—	2	4	no	—
130	C₁₆	—	4	2	no	—
131	C₂³	C₄⋊C₈×C₇₃	1	1	yes	characteristic
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	#95×C₇₃	1	1	yes	characteristic
137	C₂×C₄	C₄×C₈×C₇₃	1	1	yes	characteristic
138	C₂×C₄	#97×C₇₃	1	1	yes	characteristic
139	C₂×C₄	C₂²⋊C₈×C₇₃	1	1	yes	characteristic
140	C₂×C₄	C₂²⋊C₈×C₇₃	1	1	yes	characteristic
141	C₂×C₄	C₄⋊C₈×C₇₃	1	1	yes	characteristic
142	C₈	—	2	4	no	—
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	4	2	no	—
146	C₈	#104×C₇₃	1	2	yes	—
147	C₈	#108×C₇₃	1	2	yes	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	#336×C₇₃	1	1	yes	characteristic
151	C₂²	#364×C₇₃	1	2	yes	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	#339×C₇₃	1	1	yes	characteristic
158	C₄	#341×C₇₃	1	1	yes	characteristic
159	C₂	—	2	2	no	—
160	C₂	C₇₃×#1278	1	1	yes	characteristic
161	C₂	C₇₃×#1376	1	1	yes	characteristic
162	C₂	C₇₃×#1378	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃×#2163

C₂

yes

maximal, characteristic

C₂×#346×C₇₃

C₂

yes

maximal, characteristic

C₈×M₁₆×C₇₃

C₂

yes

maximal, characteristic

C₂²×C₁₆×C₇₃

—

#431×C₇₃

C₄

yes

characteristic

C₄×M₁₆×C₇₃

C₄

yes

characteristic

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₃

C₄

yes

—

C₂²⋊C₁₆×C₇₃

—

#346×C₇₃

—

#346×C₇₃

—

#346×C₇₃

C₄

yes

—

#322×C₇₃

—

C₈²×C₇₃

—

C₂×M₁₆×C₇₃

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

Q₈

yes

characteristic

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

—

C₄⋊C₈×C₇₃

—

C₂²⋊C₈×C₇₃

C₈

yes

—

#97×C₇₃

C₈

yes

—

C₄×C₈×C₇₃

—

C₄×C₈×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₃

M₁₆

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₄²×C₇₃

C₄²

yes

characteristic

C₄²×C₇₃

C₂²⋊C₄

yes

characteristic

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₄²×C₇₃

M₁₆

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

—

C₂³×C₇₃

C₄⋊C₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#95

yes

characteristic

C₂×C₄×C₇₃

C₄×C₈

yes

characteristic

C₂×C₄×C₇₃

#97

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₃

C₄⋊C₈

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

#104

yes

—

C₈×C₇₃

#108

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#336

yes

characteristic

C₂²×C₇₃

#364

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

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

characteristic

C₂×#346

C₂×C₇₃

yes

characteristic

C₈×M₁₆

C₂×C₇₃

yes

characteristic

C₇₃

#7165

yes

characteristic, direct product factor, Sylow

C₂²×C₁₆

—

#431

C₄×C₇₃

yes

characteristic

C₄×M₁₆

C₄×C₇₃

yes

characteristic

C₂×C₄×C₈

C₂²×C₇₃

yes

characteristic

C₄⋊C₁₆

C₄×C₇₃

yes

—

C₂²⋊C₁₆

—

#346

—

#346

—

#346

C₄×C₇₃

yes

—

#322

—

C₈²

—

C₂×M₁₆

—

C₂²×C₈

D₄×C₇₃

yes

characteristic

C₂²×C₈

Q₈×C₇₃

yes

characteristic

100

C₂×C₄²

C₂×C₄×C₇₃

yes

characteristic

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

C₂×C₁₆

—

104

C₄⋊C₈

—

105

C₂²⋊C₈

C₈×C₇₃

yes

—

106

#97

C₈×C₇₃

yes

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

C₂×C₄×C₇₃

yes

characteristic

110

C₄×C₈

C₂×C₄×C₇₃

yes

characteristic

111

C₄×C₈

D₄×C₇₃

yes

characteristic

112

C₄×C₈

D₄×C₇₃

yes

characteristic

113

C₂²×C₄

C₄⋊C₄×C₇₃

yes

characteristic

114

C₂²×C₄

C₂×C₈×C₇₃

yes

characteristic

115

C₂²×C₄

M₁₆×C₇₃

yes

characteristic

116

M₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

C₂²⋊C₄×C₇₃

yes

characteristic

122

C₂×C₈

C₂²⋊C₄×C₇₃

yes

characteristic

123

C₂×C₈

C₄⋊C₄×C₇₃

yes

characteristic

124

C₂×C₈

C₄⋊C₄×C₇₃

yes

characteristic

125

C₄²

C₄²×C₇₃

yes

characteristic

126

C₄²

C₂²⋊C₄×C₇₃

yes

characteristic

127

C₄²

C₂×C₈×C₇₃

yes

characteristic

128

C₄²

M₁₆×C₇₃

yes

characteristic

129

C₁₆

—

130

C₁₆

—

131

C₂³

C₄⋊C₈×C₇₃

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

#95×C₇₃

yes

characteristic

137

C₂×C₄

C₄×C₈×C₇₃

yes

characteristic

138

C₂×C₄

#97×C₇₃

yes

characteristic

139

C₂×C₄

C₂²⋊C₈×C₇₃

yes

characteristic

140

C₂×C₄

C₂²⋊C₈×C₇₃

yes

characteristic

141

C₂×C₄

C₄⋊C₈×C₇₃

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

#104×C₇₃

yes

—

147

C₈

#108×C₇₃

yes

—

148

C₂²

—

149

C₂²

—

150

C₂²

#336×C₇₃

yes

characteristic

151

C₂²

#364×C₇₃

yes

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

#339×C₇₃

yes

characteristic

158

C₄

#341×C₇₃

yes

characteristic

159

C₂

—

160

C₂

C₇₃×#1278

yes

characteristic

161

C₂

C₇₃×#1376

yes

characteristic

162

C₂

C₇₃×#1378

yes

characteristic

	subgroup	G/H
center	C₂×C₈×C₇₃	C₂²⋊C₄
commutator	C₂×C₄	C₄×C₈×C₇₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₇₃
socle	C₂²×C₇₃	#336
prime radical	C₂³×C₇₃	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₇₃ ◁ C₂×C₄×C₈×C₇₃ ◁ G

subgroup

G/H

center

C₂×C₈×C₇₃

C₂²⋊C₄

commutator

C₂×C₄

C₄×C₈×C₇₃

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₇₃

socle

C₂²×C₇₃

#336

prime radical

C₂³×C₇₃

C₄⋊C₈

LCS

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₈×C₇₃ ◁ C₂×C₄×C₈×C₇₃ ◁ G

N	H ≅ G/N	type
77. #7165	C₇₃	direct (N × H)
85. C₇₃	#7165	direct (N × H)

H ≅ G/N

type

77. #7165

C₇₃

direct (N × H)

85. C₇₃

#7165

direct (N × H)

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

subgroup	index
C₇₃×#2163	2
C₂×#346×C₇₃	2
C₈×M₁₆×C₇₃	2
#7165	73

subgroup

index

C₇₃×#2163

C₂×#346×C₇₃

C₈×M₁₆×C₇₃

#7165

Subgroups — C₇₃×#7165 #32916569

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#7165 #32916569

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#7165 #32916569