Group #32916648 — subgroups

Subgroups — C₇₃×#7244 #32916648

Order 18688.

192 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 94 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×#2150	C₂	1	1	yes	maximal, characteristic
2	C₇₃×#1568	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#1566	C₂	1	1	yes	maximal, characteristic
4	C₂×M₃₂×C₇₃	C₄	1	2	yes	—
5	#432×C₇₃	—	2	1	no	—
6	#423×C₇₃	—	2	1	no	—
7	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
8	C₄⋊C₁₆×C₇₃	—	2	1	no	—
9	C₄⋊C₁₆×C₇₃	—	2	1	no	—
10	#335×C₇₃	C₄	1	2	yes	—
11	#334×C₇₃	—	2	1	no	—
12	#334×C₇₃	—	2	1	no	—
13	#334×C₇₃	C₄	1	2	yes	—
14	C₂×M₁₆×C₇₃	—	4	1	no	—
15	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₇₃	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
18	M₃₂×C₇₃	—	4	2	no	—
19	C₂×C₁₆×C₇₃	—	2	2	no	—
20	C₄⋊C₈×C₇₃	—	2	2	no	—
21	C₄⋊C₈×C₇₃	—	2	2	no	—
22	C₄⋊C₈×C₇₃	—	4	1	no	—
23	C₂²⋊C₈×C₇₃	—	4	1	no	—
24	#97×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	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
31	C₂²×C₄×C₇₃	Q₁₆	1	1	yes	characteristic
32	M₁₆×C₇₃	—	8	2	no	—
33	C₂×C₈×C₇₃	—	2	1	no	—
34	C₂×C₈×C₇₃	—	2	1	no	—
35	C₂×C₈×C₇₃	—	4	1	no	—
36	C₂×C₈×C₇₃	—	4	2	no	—
37	C₂×C₈×C₇₃	—	4	2	no	—
38	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
42	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₄²×C₇₃	SD₁₆	1	1	yes	characteristic
45	C₄²×C₇₃	SD₁₆	1	1	yes	characteristic
46	C₁₆×C₇₃	—	4	2	no	—
47	C₂³×C₇₃	#107	1	1	yes	characteristic
48	C₂×C₄×C₇₃	—	2	1	no	—
49	C₂×C₄×C₇₃	—	2	1	no	—
50	C₂×C₄×C₇₃	—	2	2	no	—
51	C₂×C₄×C₇₃	—	4	1	no	—
52	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
53	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
54	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
55	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
56	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
57	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
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	C₈×C₇₃	#100	1	1	yes	characteristic
64	C₈×C₇₃	#101	1	1	yes	characteristic
65	C₂²×C₇₃	—	2	1	no	—
66	C₂²×C₇₃	—	2	1	no	—
67	C₂²×C₇₃	—	2	1	no	—
68	C₂²×C₇₃	#340	1	1	yes	characteristic
69	C₄×C₇₃	—	2	1	no	—
70	C₄×C₇₃	—	2	1	no	—
71	C₄×C₇₃	—	2	1	no	—
72	C₄×C₇₃	—	2	1	no	—
73	C₄×C₇₃	#338	1	1	yes	characteristic
74	C₄×C₇₃	#343	1	1	yes	characteristic
75	C₄×C₇₃	#360	1	2	yes	—
76	#7244	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂×C₇₃	—	4	1	no	—
78	C₂×C₇₃	#1292	1	1	yes	characteristic
79	C₂×C₇₃	#1386	1	1	yes	characteristic
80	C₂×C₇₃	#1388	1	1	yes	characteristic
81	#2150	C₂×C₇₃	1	1	yes	characteristic
82	#1568	C₂×C₇₃	1	1	yes	characteristic
83	#1566	C₂×C₇₃	1	1	yes	characteristic
84	C₇₃	#7244	1	1	yes	characteristic, direct product factor, Sylow
85	C₂×M₃₂	C₄×C₇₃	1	2	yes	—
86	#432	—	2	1	no	—
87	#423	—	2	1	no	—
88	C₂×C₄×C₈	C₂²×C₇₃	1	1	yes	characteristic
89	C₄⋊C₁₆	—	2	1	no	—
90	C₄⋊C₁₆	—	2	1	no	—
91	#335	C₄×C₇₃	1	2	yes	—
92	#334	—	2	1	no	—
93	#334	—	2	1	no	—
94	#334	C₄×C₇₃	1	2	yes	—
95	C₂×M₁₆	—	4	1	no	—
96	C₂²×C₈	C₂×C₄×C₇₃	1	1	yes	characteristic
97	C₂²×C₈	Q₈×C₇₃	1	1	yes	characteristic
98	C₂×C₄²	D₄×C₇₃	1	1	yes	characteristic
99	M₃₂	—	4	2	no	—
100	C₂×C₁₆	—	2	2	no	—
101	C₄⋊C₈	—	2	2	no	—
102	C₄⋊C₈	—	2	2	no	—
103	C₄⋊C₈	—	4	1	no	—
104	C₂²⋊C₈	—	4	1	no	—
105	#97	—	4	1	no	—
106	C₄×C₈	C₂×C₄×C₇₃	1	1	yes	characteristic
107	C₄×C₈	C₂×C₄×C₇₃	1	1	yes	characteristic
108	C₄×C₈	D₄×C₇₃	1	1	yes	characteristic
109	C₄×C₈	D₄×C₇₃	1	1	yes	characteristic
110	C₂²×C₄	C₄⋊C₄×C₇₃	1	1	yes	characteristic
111	C₂²×C₄	D₈×C₇₃	1	1	yes	characteristic
112	C₂²×C₄	Q₁₆×C₇₃	1	1	yes	characteristic
113	M₁₆	—	8	2	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	4	1	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	C₄²×C₇₃	1	1	yes	characteristic
120	C₂×C₈	C₂²⋊C₄×C₇₃	1	1	yes	characteristic
121	C₂×C₈	C₄⋊C₄×C₇₃	1	1	yes	characteristic
122	C₂×C₈	C₄⋊C₄×C₇₃	1	1	yes	characteristic
123	C₄²	C₂²⋊C₄×C₇₃	1	1	yes	characteristic
124	C₄²	C₂²⋊C₄×C₇₃	1	1	yes	characteristic
125	C₄²	SD₁₆×C₇₃	1	1	yes	characteristic
126	C₄²	SD₁₆×C₇₃	1	1	yes	characteristic
127	C₁₆	—	4	2	no	—
128	C₂³	#107×C₇₃	1	1	yes	characteristic
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	#95×C₇₃	1	1	yes	characteristic
134	C₂×C₄	#102×C₇₃	1	1	yes	characteristic
135	C₂×C₄	#102×C₇₃	1	1	yes	characteristic
136	C₂×C₄	#103×C₇₃	1	1	yes	characteristic
137	C₂×C₄	#103×C₇₃	1	1	yes	characteristic
138	C₂×C₄	#106×C₇₃	1	1	yes	characteristic
139	C₈	—	2	1	no	—
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	—	8	2	no	—
143	C₈	—	8	2	no	—
144	C₈	#100×C₇₃	1	1	yes	characteristic
145	C₈	#101×C₇₃	1	1	yes	characteristic
146	C₂²	—	2	1	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	#340×C₇₃	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	#338×C₇₃	1	1	yes	characteristic
155	C₄	#343×C₇₃	1	1	yes	characteristic
156	C₄	#360×C₇₃	1	2	yes	—
157	C₂	—	4	1	no	—
158	C₂	C₇₃×#1292	1	1	yes	characteristic
159	C₂	C₇₃×#1386	1	1	yes	characteristic
160	C₂	C₇₃×#1388	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃×#2150

C₂

yes

maximal, characteristic

C₇₃×#1568

C₂

yes

maximal, characteristic

C₇₃×#1566

C₂

yes

maximal, characteristic

C₂×M₃₂×C₇₃

C₄

yes

—

#432×C₇₃

—

#423×C₇₃

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₃

—

C₄⋊C₁₆×C₇₃

—

#335×C₇₃

C₄

yes

—

#334×C₇₃

—

#334×C₇₃

—

#334×C₇₃

C₄

yes

—

C₂×M₁₆×C₇₃

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₃

Q₈

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

characteristic

M₃₂×C₇₃

—

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

D₄

yes

characteristic

C₂²×C₄×C₇₃

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

D₈

yes

characteristic

C₂²×C₄×C₇₃

Q₁₆

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×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

C₂²⋊C₄

yes

characteristic

C₄²×C₇₃

SD₁₆

yes

characteristic

C₄²×C₇₃

SD₁₆

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

#107

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

yes

C₈×C₇₃

yes

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#340

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

#7244

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

characteristic

#1568

C₂×C₇₃

yes

characteristic

#1566

C₂×C₇₃

yes

characteristic

C₇₃

#7244

yes

characteristic, direct product factor, Sylow

C₂×M₃₂

C₄×C₇₃

yes

—

#432

—

#423

—

C₂×C₄×C₈

C₂²×C₇₃

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#335

C₄×C₇₃

yes

—

#334

—

#334

—

#334

C₄×C₇₃

yes

—

C₂×M₁₆

—

C₂²×C₈

C₂×C₄×C₇₃

yes

characteristic

C₂²×C₈

Q₈×C₇₃

yes

characteristic

C₂×C₄²

D₄×C₇₃

yes

characteristic

M₃₂

—

100

C₂×C₁₆

—

101

C₄⋊C₈

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₂²⋊C₈

—

105

#97

—

106

C₄×C₈

C₂×C₄×C₇₃

yes

characteristic

107

C₄×C₈

C₂×C₄×C₇₃

yes

characteristic

108

C₄×C₈

D₄×C₇₃

yes

characteristic

109

C₄×C₈

D₄×C₇₃

yes

characteristic

110

C₂²×C₄

C₄⋊C₄×C₇₃

yes

characteristic

111

C₂²×C₄

D₈×C₇₃

yes

characteristic

112

C₂²×C₄

Q₁₆×C₇₃

yes

characteristic

113

M₁₆

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

C₄²×C₇₃

yes

characteristic

120

C₂×C₈

C₂²⋊C₄×C₇₃

yes

characteristic

121

C₂×C₈

C₄⋊C₄×C₇₃

yes

characteristic

122

C₂×C₈

C₄⋊C₄×C₇₃

yes

characteristic

123

C₄²

C₂²⋊C₄×C₇₃

yes

characteristic

124

C₄²

C₂²⋊C₄×C₇₃

yes

characteristic

125

C₄²

SD₁₆×C₇₃

yes

characteristic

126

C₄²

SD₁₆×C₇₃

yes

characteristic

127

C₁₆

—

128

C₂³

#107×C₇₃

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

#95×C₇₃

yes

characteristic

134

C₂×C₄

#102×C₇₃

yes

characteristic

135

C₂×C₄

#102×C₇₃

yes

characteristic

136

C₂×C₄

#103×C₇₃

yes

characteristic

137

C₂×C₄

#103×C₇₃

yes

characteristic

138

C₂×C₄

#106×C₇₃

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

#100×C₇₃

yes

characteristic

145

C₈

#101×C₇₃

yes

characteristic

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

#340×C₇₃

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

#338×C₇₃

yes

characteristic

155

C₄

#343×C₇₃

yes

characteristic

156

C₄

#360×C₇₃

yes

—

157

C₂

—

158

C₂

C₇₃×#1292

yes

characteristic

159

C₂

C₇₃×#1386

yes

characteristic

160

C₂

C₇₃×#1388

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	#107
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

#107

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

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

N	H ≅ G/N	type
76. #7244	C₇₃	direct (N × H)
84. C₇₃	#7244	direct (N × H)

H ≅ G/N

type

76. #7244

C₇₃

direct (N × H)

84. C₇₃

#7244

direct (N × H)

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

subgroup	index
C₇₃×#2150	2
C₇₃×#1568	2
C₇₃×#1566	2
#7244	73

subgroup

index

C₇₃×#2150

C₇₃×#1568

C₇₃×#1566

#7244

Subgroups — C₇₃×#7244 #32916648

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#7244 #32916648

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#7244 #32916648