Group #32916702 — subgroups

Subgroups — C₇₃×#7298 #32916702

Order 18688.

216 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2162	C₂	1	1	yes	maximal, characteristic
2	C₄×#108×C₇₃	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#1567	C₂	1	1	yes	maximal, characteristic
4	C₂×#108×C₇₃	—	2	1	no	—
5	#423×C₇₃	—	2	1	no	—
6	C₄×M₁₆×C₇₃	—	2	1	no	—
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	C₄×C₁₆×C₇₃	—	2	1	no	—
11	#341×C₇₃	C₄	1	2	yes	—
12	#335×C₇₃	—	2	1	no	—
13	#335×C₇₃	C₄	1	2	yes	—
14	C₂×M₁₆×C₇₃	—	2	2	no	—
15	C₂×M₁₆×C₇₃	—	4	1	no	—
16	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
18	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
19	C₂×C₁₆×C₇₃	—	2	2	no	—
20	#108×C₇₃	—	2	4	no	—
21	C₄⋊C₈×C₇₃	—	2	2	no	—
22	C₄⋊C₈×C₇₃	—	4	1	no	—
23	#97×C₇₃	—	4	1	no	—
24	C₄×C₈×C₇₃	—	4	1	no	—
25	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
28	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
29	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
31	C₂²×C₄×C₇₃	Q₁₆	1	1	yes	characteristic
32	M₁₆×C₇₃	—	2	4	no	—
33	M₁₆×C₇₃	—	4	2	no	—
34	M₁₆×C₇₃	—	8	2	no	—
35	C₂×C₈×C₇₃	—	2	2	no	—
36	C₂×C₈×C₇₃	—	4	1	no	—
37	C₂×C₈×C₇₃	—	4	2	no	—
38	C₂×C₈×C₇₃	—	4	2	no	—
39	C₂×C₈×C₇₃	C₄²	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	C₂²⋊C₄	1	1	yes	characteristic
44	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₇₃	D₈	1	1	yes	characteristic
46	C₄²×C₇₃	SD₁₆	1	1	yes	characteristic
47	C₁₆×C₇₃	—	4	2	no	—
48	C₂³×C₇₃	#103	1	1	yes	characteristic
49	C₂×C₄×C₇₃	—	2	1	no	—
50	C₂×C₄×C₇₃	—	2	1	no	—
51	C₂×C₄×C₇₃	—	2	1	no	—
52	C₂×C₄×C₇₃	—	2	1	no	—
53	C₂×C₄×C₇₃	—	2	2	no	—
54	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
55	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
56	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
57	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
58	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
59	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
60	C₈×C₇₃	—	2	1	no	—
61	C₈×C₇₃	—	2	1	no	—
62	C₈×C₇₃	—	2	1	no	—
63	C₈×C₇₃	—	4	4	no	—
64	C₈×C₇₃	—	8	2	no	—
65	C₈×C₇₃	#104	1	2	yes	—
66	C₂²×C₇₃	—	2	1	no	—
67	C₂²×C₇₃	—	2	1	no	—
68	C₂²×C₇₃	—	2	1	no	—
69	C₂²×C₇₃	#340	1	1	yes	characteristic
70	C₄×C₇₃	—	2	1	no	—
71	C₄×C₇₃	—	2	1	no	—
72	C₄×C₇₃	—	2	1	no	—
73	C₄×C₇₃	#339	1	1	yes	characteristic
74	C₄×C₇₃	#344	1	1	yes	characteristic
75	C₄×C₇₃	#361	1	1	yes	characteristic
76	C₄×C₇₃	#362	1	1	yes	characteristic
77	C₄×C₇₃	#368	1	2	yes	—
78	#7298	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂×C₇₃	—	4	1	no	—
80	C₂×C₇₃	#1303	1	1	yes	characteristic
81	C₂×C₇₃	#1381	1	1	yes	characteristic
82	C₂×C₇₃	#1387	1	1	yes	characteristic
83	#2162	C₂×C₇₃	1	1	yes	characteristic
84	C₄×#108	C₂×C₇₃	1	1	yes	characteristic
85	#1567	C₂×C₇₃	1	1	yes	characteristic
86	C₇₃	#7298	1	1	yes	characteristic, direct product factor, Sylow
87	C₂×#108	—	2	1	no	—
88	#423	—	2	1	no	—
89	C₄×M₁₆	—	2	1	no	—
90	C₂×C₄×C₈	C₂²×C₇₃	1	1	yes	characteristic
91	C₄⋊C₁₆	C₄×C₇₃	1	2	yes	—
92	C₂²⋊C₁₆	—	2	1	no	—
93	C₄×C₁₆	—	2	1	no	—
94	#341	C₄×C₇₃	1	2	yes	—
95	#335	—	2	1	no	—
96	#335	C₄×C₇₃	1	2	yes	—
97	C₂×M₁₆	—	2	2	no	—
98	C₂×M₁₆	—	4	1	no	—
99	C₂²×C₈	C₂×C₄×C₇₃	1	1	yes	characteristic
100	C₂²×C₈	D₄×C₇₃	1	1	yes	characteristic
101	C₂×C₄²	D₄×C₇₃	1	1	yes	characteristic
102	C₂×C₁₆	—	2	2	no	—
103	#108	—	2	4	no	—
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	4	1	no	—
106	#97	—	4	1	no	—
107	C₄×C₈	—	4	1	no	—
108	C₄×C₈	C₂×C₄×C₇₃	1	1	yes	characteristic
109	C₄×C₈	C₂×C₄×C₇₃	1	1	yes	characteristic
110	C₄×C₈	D₄×C₇₃	1	1	yes	characteristic
111	C₄×C₈	Q₈×C₇₃	1	1	yes	characteristic
112	C₂²×C₄	C₂²⋊C₄×C₇₃	1	1	yes	characteristic
113	C₂²×C₄	SD₁₆×C₇₃	1	1	yes	characteristic
114	C₂²×C₄	Q₁₆×C₇₃	1	1	yes	characteristic
115	M₁₆	—	2	4	no	—
116	M₁₆	—	4	2	no	—
117	M₁₆	—	8	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	4	1	no	—
120	C₂×C₈	—	4	2	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	C₄²×C₇₃	1	1	yes	characteristic
123	C₂×C₈	C₂²⋊C₄×C₇₃	1	1	yes	characteristic
124	C₂×C₈	C₄⋊C₄×C₇₃	1	1	yes	characteristic
125	C₂×C₈	C₄⋊C₄×C₇₃	1	1	yes	characteristic
126	C₄²	C₂²⋊C₄×C₇₃	1	1	yes	characteristic
127	C₄²	C₄⋊C₄×C₇₃	1	1	yes	characteristic
128	C₄²	D₈×C₇₃	1	1	yes	characteristic
129	C₄²	SD₁₆×C₇₃	1	1	yes	characteristic
130	C₁₆	—	4	2	no	—
131	C₂³	#103×C₇₃	1	1	yes	characteristic
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	#95×C₇₃	1	1	yes	characteristic
138	C₂×C₄	#102×C₇₃	1	1	yes	characteristic
139	C₂×C₄	#102×C₇₃	1	1	yes	characteristic
140	C₂×C₄	#103×C₇₃	1	1	yes	characteristic
141	C₂×C₄	#106×C₇₃	1	1	yes	characteristic
142	C₂×C₄	#107×C₇₃	1	1	yes	characteristic
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	4	4	no	—
147	C₈	—	8	2	no	—
148	C₈	#104×C₇₃	1	2	yes	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	#340×C₇₃	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	#339×C₇₃	1	1	yes	characteristic
157	C₄	#344×C₇₃	1	1	yes	characteristic
158	C₄	#361×C₇₃	1	1	yes	characteristic
159	C₄	#362×C₇₃	1	1	yes	characteristic
160	C₄	#368×C₇₃	1	2	yes	—
161	C₂	—	4	1	no	—
162	C₂	C₇₃×#1303	1	1	yes	characteristic
163	C₂	C₇₃×#1381	1	1	yes	characteristic
164	C₂	C₇₃×#1387	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃×#2162

C₂

yes

maximal, characteristic

C₄×#108×C₇₃

C₂

yes

maximal, characteristic

C₇₃×#1567

C₂

yes

maximal, characteristic

C₂×#108×C₇₃

—

#423×C₇₃

—

C₄×M₁₆×C₇₃

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₃

C₄

yes

—

C₂²⋊C₁₆×C₇₃

—

C₄×C₁₆×C₇₃

—

#341×C₇₃

C₄

yes

—

#335×C₇₃

—

#335×C₇₃

C₄

yes

—

C₂×M₁₆×C₇₃

—

C₂×M₁₆×C₇₃

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

—

#108×C₇₃

—

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

—

#97×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

Q₈

yes

characteristic

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

SD₁₆

yes

characteristic

C₂²×C₄×C₇₃

Q₁₆

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄²

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

C₂²⋊C₄

yes

characteristic

C₄²×C₇₃

C₄⋊C₄

yes

characteristic

C₄²×C₇₃

D₈

yes

characteristic

C₄²×C₇₃

SD₁₆

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

#103

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

#104

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#340

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

#7298

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

characteristic

C₄×#108

C₂×C₇₃

yes

characteristic

#1567

C₂×C₇₃

yes

characteristic

C₇₃

#7298

yes

characteristic, direct product factor, Sylow

C₂×#108

—

#423

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂²×C₇₃

yes

characteristic

C₄⋊C₁₆

C₄×C₇₃

yes

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

#341

C₄×C₇₃

yes

—

#335

—

#335

C₄×C₇₃

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×C₄×C₇₃

yes

characteristic

100

C₂²×C₈

D₄×C₇₃

yes

characteristic

101

C₂×C₄²

D₄×C₇₃

yes

characteristic

102

C₂×C₁₆

—

103

#108

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

#97

—

107

C₄×C₈

—

108

C₄×C₈

C₂×C₄×C₇₃

yes

characteristic

109

C₄×C₈

C₂×C₄×C₇₃

yes

characteristic

110

C₄×C₈

D₄×C₇₃

yes

characteristic

111

C₄×C₈

Q₈×C₇₃

yes

characteristic

112

C₂²×C₄

C₂²⋊C₄×C₇₃

yes

characteristic

113

C₂²×C₄

SD₁₆×C₇₃

yes

characteristic

114

C₂²×C₄

Q₁₆×C₇₃

yes

characteristic

115

M₁₆

—

116

M₁₆

—

117

M₁₆

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

C₄²×C₇₃

yes

characteristic

123

C₂×C₈

C₂²⋊C₄×C₇₃

yes

characteristic

124

C₂×C₈

C₄⋊C₄×C₇₃

yes

characteristic

125

C₂×C₈

C₄⋊C₄×C₇₃

yes

characteristic

126

C₄²

C₂²⋊C₄×C₇₃

yes

characteristic

127

C₄²

C₄⋊C₄×C₇₃

yes

characteristic

128

C₄²

D₈×C₇₃

yes

characteristic

129

C₄²

SD₁₆×C₇₃

yes

characteristic

130

C₁₆

—

131

C₂³

#103×C₇₃

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#95×C₇₃

yes

characteristic

138

C₂×C₄

#102×C₇₃

yes

characteristic

139

C₂×C₄

#102×C₇₃

yes

characteristic

140

C₂×C₄

#103×C₇₃

yes

characteristic

141

C₂×C₄

#106×C₇₃

yes

characteristic

142

C₂×C₄

#107×C₇₃

yes

characteristic

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

#104×C₇₃

yes

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

#340×C₇₃

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

#339×C₇₃

yes

characteristic

157

C₄

#344×C₇₃

yes

characteristic

158

C₄

#361×C₇₃

yes

characteristic

159

C₄

#362×C₇₃

yes

characteristic

160

C₄

#368×C₇₃

yes

—

161

C₂

—

162

C₂

C₇₃×#1303

yes

characteristic

163

C₂

C₇₃×#1381

yes

characteristic

164

C₂

C₇₃×#1387

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₇₃	#102
commutator	C₂×C₈	C₄²×C₇₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₇₃
socle	C₂²×C₇₃	#340
prime radical	C₂³×C₇₃	#103
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ C₂×C₄×C₈×C₇₃ ◁ G

subgroup

G/H

center

C₂×C₄×C₇₃

#102

commutator

C₂×C₈

C₄²×C₇₃

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₇₃

C₂²×C₇₃

C₂³×C₇₃

G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ C₂×C₄×C₈×C₇₃ ◁ G

N	H ≅ G/N	type
78. #7298	C₇₃	direct (N × H)
86. C₇₃	#7298	direct (N × H)

H ≅ G/N

type

78. #7298

C₇₃

direct (N × H)

86. C₇₃

#7298

direct (N × H)

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

subgroup	index
C₇₃×#2162	2
C₄×#108×C₇₃	2
C₇₃×#1567	2
#7298	73

subgroup

index

C₇₃×#2162

C₄×#108×C₇₃

C₇₃×#1567

#7298

Subgroups — C₇₃×#7298 #32916702

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#7298 #32916702

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#7298 #32916702