Group #32916628 — subgroups

Subgroups — C₇₃×#7224 #32916628

Order 18688.

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

normal subgroups only

This group has 184 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×#2163	C₂	1	1	yes	maximal, characteristic
2	C₇₃×#1837	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#1568	C₂	1	1	yes	maximal, characteristic
4	#432×C₇₃	—	2	1	no	—
5	#423×C₇₃	—	2	1	no	—
6	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
7	C₄⋊C₁₆×C₇₃	—	2	1	no	—
8	C₂²⋊C₁₆×C₇₃	—	2	1	no	—
9	#346×C₇₃	C₄	1	2	yes	—
10	#341×C₇₃	—	2	1	no	—
11	#341×C₇₃	C₄	1	2	yes	—
12	#335×C₇₃	—	2	1	no	—
13	#334×C₇₃	C₄	1	2	yes	—
14	C₂×M₁₆×C₇₃	—	2	2	no	—
15	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
17	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₇₃	—	2	2	no	—
19	C₄⋊C₈×C₇₃	—	2	2	no	—
20	C₄⋊C₈×C₇₃	—	4	1	no	—
21	C₄⋊C₈×C₇₃	—	4	1	no	—
22	C₂²⋊C₈×C₇₃	—	4	1	no	—
23	#97×C₇₃	—	4	1	no	—
24	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
27	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
28	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
30	C₂²×C₄×C₇₃	Q₁₆	1	1	yes	characteristic
31	M₁₆×C₇₃	—	4	2	no	—
32	M₁₆×C₇₃	—	4	2	no	—
33	C₂×C₈×C₇₃	—	4	1	no	—
34	C₂×C₈×C₇₃	—	4	1	no	—
35	C₂×C₈×C₇₃	—	4	2	no	—
36	C₂×C₈×C₇₃	—	4	2	no	—
37	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
41	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₇₃	D₈	1	1	yes	characteristic
44	C₄²×C₇₃	SD₁₆	1	1	yes	characteristic
45	C₁₆×C₇₃	—	4	2	no	—
46	C₂³×C₇₃	#103	1	1	yes	characteristic
47	C₂×C₄×C₇₃	—	2	1	no	—
48	C₂×C₄×C₇₃	—	2	1	no	—
49	C₂×C₄×C₇₃	—	4	1	no	—
50	C₂×C₄×C₇₃	—	4	1	no	—
51	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
52	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
53	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
54	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
55	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
56	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
57	C₈×C₇₃	—	2	1	no	—
58	C₈×C₇₃	—	2	1	no	—
59	C₈×C₇₃	—	2	1	no	—
60	C₈×C₇₃	—	2	1	no	—
61	C₈×C₇₃	—	8	2	no	—
62	C₈×C₇₃	—	8	2	no	—
63	C₂²×C₇₃	—	2	1	no	—
64	C₂²×C₇₃	—	2	1	no	—
65	C₂²×C₇₃	—	2	1	no	—
66	C₂²×C₇₃	#340	1	1	yes	characteristic
67	C₄×C₇₃	—	2	1	no	—
68	C₄×C₇₃	—	2	1	no	—
69	C₄×C₇₃	—	2	1	no	—
70	C₄×C₇₃	—	2	1	no	—
71	C₄×C₇₃	—	2	1	no	—
72	C₄×C₇₃	H(3,ℤ/4)	1	1	yes	characteristic
73	C₄×C₇₃	#344	1	1	yes	characteristic
74	#7224	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂×C₇₃	—	4	1	no	—
76	C₂×C₇₃	#1286	1	1	yes	characteristic
77	C₂×C₇₃	#1388	1	1	yes	characteristic
78	C₂×C₇₃	#1389	1	1	yes	characteristic
79	#2163	C₂×C₇₃	1	1	yes	characteristic
80	#1837	C₂×C₇₃	1	1	yes	characteristic
81	#1568	C₂×C₇₃	1	1	yes	characteristic
82	C₇₃	#7224	1	1	yes	characteristic, direct product factor, Sylow
83	#432	—	2	1	no	—
84	#423	—	2	1	no	—
85	C₂×C₄×C₈	C₂²×C₇₃	1	1	yes	characteristic
86	C₄⋊C₁₆	—	2	1	no	—
87	C₂²⋊C₁₆	—	2	1	no	—
88	#346	C₄×C₇₃	1	2	yes	—
89	#341	—	2	1	no	—
90	#341	C₄×C₇₃	1	2	yes	—
91	#335	—	2	1	no	—
92	#334	C₄×C₇₃	1	2	yes	—
93	C₂×M₁₆	—	2	2	no	—
94	C₂²×C₈	C₂×C₄×C₇₃	1	1	yes	characteristic
95	C₂²×C₈	D₄×C₇₃	1	1	yes	characteristic
96	C₂×C₄²	D₄×C₇₃	1	1	yes	characteristic
97	C₂×C₁₆	—	2	2	no	—
98	C₄⋊C₈	—	2	2	no	—
99	C₄⋊C₈	—	4	1	no	—
100	C₄⋊C₈	—	4	1	no	—
101	C₂²⋊C₈	—	4	1	no	—
102	#97	—	4	1	no	—
103	C₄×C₈	C₂×C₄×C₇₃	1	1	yes	characteristic
104	C₄×C₈	C₂×C₄×C₇₃	1	1	yes	characteristic
105	C₄×C₈	D₄×C₇₃	1	1	yes	characteristic
106	C₄×C₈	Q₈×C₇₃	1	1	yes	characteristic
107	C₂²×C₄	C₂²⋊C₄×C₇₃	1	1	yes	characteristic
108	C₂²×C₄	SD₁₆×C₇₃	1	1	yes	characteristic
109	C₂²×C₄	Q₁₆×C₇₃	1	1	yes	characteristic
110	M₁₆	—	4	2	no	—
111	M₁₆	—	4	2	no	—
112	C₂×C₈	—	4	1	no	—
113	C₂×C₈	—	4	1	no	—
114	C₂×C₈	—	4	2	no	—
115	C₂×C₈	—	4	2	no	—
116	C₂×C₈	C₄²×C₇₃	1	1	yes	characteristic
117	C₂×C₈	C₂²⋊C₄×C₇₃	1	1	yes	characteristic
118	C₂×C₈	C₄⋊C₄×C₇₃	1	1	yes	characteristic
119	C₂×C₈	C₄⋊C₄×C₇₃	1	1	yes	characteristic
120	C₄²	C₂²⋊C₄×C₇₃	1	1	yes	characteristic
121	C₄²	C₄⋊C₄×C₇₃	1	1	yes	characteristic
122	C₄²	D₈×C₇₃	1	1	yes	characteristic
123	C₄²	SD₁₆×C₇₃	1	1	yes	characteristic
124	C₁₆	—	4	2	no	—
125	C₂³	#103×C₇₃	1	1	yes	characteristic
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	#95×C₇₃	1	1	yes	characteristic
131	C₂×C₄	#102×C₇₃	1	1	yes	characteristic
132	C₂×C₄	#102×C₇₃	1	1	yes	characteristic
133	C₂×C₄	#103×C₇₃	1	1	yes	characteristic
134	C₂×C₄	#106×C₇₃	1	1	yes	characteristic
135	C₂×C₄	#107×C₇₃	1	1	yes	characteristic
136	C₈	—	2	1	no	—
137	C₈	—	2	1	no	—
138	C₈	—	2	1	no	—
139	C₈	—	2	1	no	—
140	C₈	—	8	2	no	—
141	C₈	—	8	2	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	1	no	—
144	C₂²	—	2	1	no	—
145	C₂²	#340×C₇₃	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	H(3,ℤ/4)×C₇₃	1	1	yes	characteristic
152	C₄	#344×C₇₃	1	1	yes	characteristic
153	C₂	—	4	1	no	—
154	C₂	C₇₃×#1286	1	1	yes	characteristic
155	C₂	C₇₃×#1388	1	1	yes	characteristic
156	C₂	C₇₃×#1389	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃×#2163

C₂

yes

maximal, characteristic

C₇₃×#1837

C₂

yes

maximal, characteristic

C₇₃×#1568

C₂

yes

maximal, characteristic

#432×C₇₃

—

#423×C₇₃

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₃

—

C₂²⋊C₁₆×C₇₃

—

#346×C₇₃

C₄

yes

—

#341×C₇₃

—

#341×C₇₃

C₄

yes

—

#335×C₇₃

—

#334×C₇₃

C₄

yes

—

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

—

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

—

C₂²⋊C₈×C₇₃

—

#97×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

—

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

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

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#340

yes

characteristic

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

#1837

C₂×C₇₃

yes

characteristic

#1568

C₂×C₇₃

yes

characteristic

C₇₃

#7224

yes

characteristic, direct product factor, Sylow

#432

—

#423

—

C₂×C₄×C₈

C₂²×C₇₃

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

C₄×C₇₃

yes

—

#341

—

#341

C₄×C₇₃

yes

—

#335

—

#334

C₄×C₇₃

yes

—

C₂×M₁₆

—

C₂²×C₈

C₂×C₄×C₇₃

yes

characteristic

C₂²×C₈

D₄×C₇₃

yes

characteristic

C₂×C₄²

D₄×C₇₃

yes

characteristic

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₂²⋊C₈

—

102

#97

—

103

C₄×C₈

C₂×C₄×C₇₃

yes

characteristic

104

C₄×C₈

C₂×C₄×C₇₃

yes

characteristic

105

C₄×C₈

D₄×C₇₃

yes

characteristic

106

C₄×C₈

Q₈×C₇₃

yes

characteristic

107

C₂²×C₄

C₂²⋊C₄×C₇₃

yes

characteristic

108

C₂²×C₄

SD₁₆×C₇₃

yes

characteristic

109

C₂²×C₄

Q₁₆×C₇₃

yes

characteristic

110

M₁₆

—

111

M₁₆

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

C₄²×C₇₃

yes

characteristic

117

C₂×C₈

C₂²⋊C₄×C₇₃

yes

characteristic

118

C₂×C₈

C₄⋊C₄×C₇₃

yes

characteristic

119

C₂×C₈

C₄⋊C₄×C₇₃

yes

characteristic

120

C₄²

C₂²⋊C₄×C₇₃

yes

characteristic

121

C₄²

C₄⋊C₄×C₇₃

yes

characteristic

122

C₄²

D₈×C₇₃

yes

characteristic

123

C₄²

SD₁₆×C₇₃

yes

characteristic

124

C₁₆

—

125

C₂³

#103×C₇₃

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

#95×C₇₃

yes

characteristic

131

C₂×C₄

#102×C₇₃

yes

characteristic

132

C₂×C₄

#102×C₇₃

yes

characteristic

133

C₂×C₄

#103×C₇₃

yes

characteristic

134

C₂×C₄

#106×C₇₃

yes

characteristic

135

C₂×C₄

#107×C₇₃

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

#340×C₇₃

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

H(3,ℤ/4)×C₇₃

yes

characteristic

152

C₄

#344×C₇₃

yes

characteristic

153

C₂

—

154

C₂

C₇₃×#1286

yes

characteristic

155

C₂

C₇₃×#1388

yes

characteristic

156

C₂

C₇₃×#1389

yes

characteristic

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

subgroup

G/H

center, socle

C₂²×C₇₃

#340

commutator

C₂×C₈

C₄²×C₇₃

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₇₃

prime radical

C₂³×C₇₃

#103

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ C₂×C₄×C₈×C₇₃ ◁ G

N	H ≅ G/N	type
74. #7224	C₇₃	direct (N × H)
82. C₇₃	#7224	direct (N × H)

H ≅ G/N

type

74. #7224

C₇₃

direct (N × H)

82. C₇₃

#7224

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
C₇₃×#1837	2
C₇₃×#1568	2
#7224	73

subgroup

index

C₇₃×#2163

C₇₃×#1837

C₇₃×#1568

#7224

Subgroups — C₇₃×#7224 #32916628

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#7224 #32916628

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#7224 #32916628