Group #32916629 — subgroups

Subgroups — C₇₃×#7225 #32916629

Order 18688.

192 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 66 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×#2165	C₂	1	1	yes	maximal, characteristic
2	C₇₃×#1841	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#1455	C₂	1	1	yes	maximal, characteristic
4	C₄×M₁₆×C₇₃	—	2	1	no	—
5	C₄×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
6	#364×C₇₃	—	2	1	no	—
7	#349×C₇₃	—	2	1	no	—
8	#347×C₇₃	C₄	1	2	yes	complemented
9	#344×C₇₃	—	2	1	no	—
10	#339×C₇₃	—	2	1	no	—
11	#338×C₇₃	C₄	1	1	yes	characteristic
12	H(3,ℤ/4)×C₇₃	C₄	1	1	yes	characteristic
13	#322×C₇₃	—	2	1	no	—
14	#322×C₇₃	C₄	1	1	yes	characteristic, complemented
15	C₈²×C₇₃	C₄	1	1	yes	characteristic, complemented
16	C₂×M₁₆×C₇₃	—	2	2	no	—
17	C₂×M₁₆×C₇₃	—	2	1	no	—
18	C₂×M₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
19	C₂×M₁₆×C₇₃	D₄	1	1	yes	characteristic
20	#117×C₇₃	—	2	1	no	—
21	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
22	M₃₂×C₇₃	—	2	2	no	—
23	#97×C₇₃	—	2	2	no	—
24	#97×C₇₃	C₂×C₄	1	1	yes	characteristic
25	#97×C₇₃	Q₈	1	1	yes	characteristic
26	C₄×C₈×C₇₃	—	2	1	no	—
27	C₄×C₈×C₇₃	—	2	1	no	—
28	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₇₃	—	2	1	no	—
31	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
32	M₁₆×C₇₃	—	4	1	no	—
33	M₁₆×C₇₃	—	4	1	no	—
34	M₁₆×C₇₃	—	4	4	no	—
35	M₁₆×C₇₃	—	4	1	no	—
36	M₁₆×C₇₃	—	4	1	no	—
37	C₂×C₈×C₇₃	—	2	2	no	—
38	C₂×C₈×C₇₃	—	2	2	no	—
39	C₂×C₈×C₇₃	—	4	1	no	—
40	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
44	C₄⋊C₄×C₇₃	—	4	1	no	—
45	C₂²⋊C₄×C₇₃	—	4	1	no	—
46	C₄²×C₇₃	—	2	1	no	—
47	C₄²×C₇₃	—	4	1	no	—
48	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
50	C₁₆×C₇₃	—	4	2	no	—
51	C₂³×C₇₃	#104	1	1	yes	characteristic
52	C₂×C₄×C₇₃	—	2	1	no	—
53	C₂×C₄×C₇₃	—	2	1	no	—
54	C₂×C₄×C₇₃	—	2	1	no	—
55	C₂×C₄×C₇₃	—	2	1	no	—
56	C₂×C₄×C₇₃	—	4	2	no	—
57	C₂×C₄×C₇₃	—	8	1	no	—
58	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
59	C₂×C₄×C₇₃	#104	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	—	2	1	no	—
64	C₈×C₇₃	—	2	1	no	—
65	C₈×C₇₃	—	4	2	no	—
66	C₈×C₇₃	—	8	1	no	—
67	C₂²×C₇₃	—	2	1	no	—
68	C₂²×C₇₃	—	4	1	no	—
69	C₂²×C₇₃	#339	1	1	yes	characteristic
70	C₄×C₇₃	—	2	2	no	—
71	C₄×C₇₃	—	2	1	no	—
72	C₄×C₇₃	—	4	1	no	—
73	C₄×C₇₃	—	16	1	no	—
74	C₄×C₇₃	H(3,ℤ/4)	1	1	yes	characteristic
75	C₄×C₇₃	#344	1	1	yes	characteristic
76	#7225	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂×C₇₃	—	2	1	no	—
78	C₂×C₇₃	—	4	1	no	—
79	C₂×C₇₃	#1288	1	1	yes	characteristic
80	#2165	C₂×C₇₃	1	1	yes	characteristic
81	#1841	C₂×C₇₃	1	1	yes	characteristic
82	#1455	C₂×C₇₃	1	1	yes	characteristic
83	C₇₃	#7225	1	1	yes	characteristic, direct product factor, Sylow
84	C₄×M₁₆	—	2	1	no	—
85	C₄×M₁₆	C₂²×C₇₃	1	1	yes	characteristic
86	#364	—	2	1	no	—
87	#349	—	2	1	no	—
88	#347	C₄×C₇₃	1	2	yes	complemented
89	#344	—	2	1	no	—
90	#339	—	2	1	no	—
91	#338	C₄×C₇₃	1	1	yes	characteristic
92	H(3,ℤ/4)	C₄×C₇₃	1	1	yes	characteristic
93	#322	—	2	1	no	—
94	#322	C₄×C₇₃	1	1	yes	characteristic, complemented
95	C₈²	C₄×C₇₃	1	1	yes	characteristic, complemented
96	C₂×M₁₆	—	2	2	no	—
97	C₂×M₁₆	—	2	1	no	—
98	C₂×M₁₆	C₂×C₄×C₇₃	1	1	yes	characteristic
99	C₂×M₁₆	D₄×C₇₃	1	1	yes	characteristic
100	#117	—	2	1	no	—
101	C₂×C₄²	D₄×C₇₃	1	1	yes	characteristic
102	M₃₂	—	2	2	no	—
103	#97	—	2	2	no	—
104	#97	C₂×C₄×C₇₃	1	1	yes	characteristic
105	#97	Q₈×C₇₃	1	1	yes	characteristic
106	C₄×C₈	—	2	1	no	—
107	C₄×C₈	—	2	1	no	—
108	C₄×C₈	C₂×C₄×C₇₃	1	1	yes	characteristic
109	C₄×C₈	D₄×C₇₃	1	1	yes	characteristic
110	C₂²×C₄	—	2	1	no	—
111	C₂²×C₄	C₂²⋊C₄×C₇₃	1	1	yes	characteristic
112	M₁₆	—	4	1	no	—
113	M₁₆	—	4	1	no	—
114	M₁₆	—	4	4	no	—
115	M₁₆	—	4	1	no	—
116	M₁₆	—	4	1	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	4	1	no	—
120	C₂×C₈	C₄²×C₇₃	1	1	yes	characteristic
121	C₂×C₈	C₂²⋊C₄×C₇₃	1	1	yes	characteristic
122	C₂×C₈	C₄⋊C₄×C₇₃	1	1	yes	characteristic
123	C₂×C₈	C₄⋊C₄×C₇₃	1	1	yes	characteristic
124	C₄⋊C₄	—	4	1	no	—
125	C₂²⋊C₄	—	4	1	no	—
126	C₄²	—	2	1	no	—
127	C₄²	—	4	1	no	—
128	C₄²	C₂²⋊C₄×C₇₃	1	1	yes	characteristic
129	C₄²	C₄⋊C₄×C₇₃	1	1	yes	characteristic
130	C₁₆	—	4	2	no	—
131	C₂³	#104×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	—	4	2	no	—
137	C₂×C₄	—	8	1	no	—
138	C₂×C₄	#95×C₇₃	1	1	yes	characteristic
139	C₂×C₄	#104×C₇₃	1	1	yes	characteristic
140	C₈	—	2	4	no	—
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	4	2	no	—
146	C₈	—	8	1	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	4	1	no	—
149	C₂²	#339×C₇₃	1	1	yes	characteristic
150	C₄	—	2	2	no	—
151	C₄	—	2	1	no	—
152	C₄	—	4	1	no	—
153	C₄	—	16	1	no	—
154	C₄	H(3,ℤ/4)×C₇₃	1	1	yes	characteristic
155	C₄	#344×C₇₃	1	1	yes	characteristic
156	C₂	—	2	1	no	—
157	C₂	—	4	1	no	—
158	C₂	C₇₃×#1288	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃×#2165

C₂

yes

maximal, characteristic

C₇₃×#1841

C₂

yes

maximal, characteristic

C₇₃×#1455

C₂

yes

maximal, characteristic

C₄×M₁₆×C₇₃

—

C₄×M₁₆×C₇₃

C₂²

yes

characteristic

#364×C₇₃

—

#349×C₇₃

—

#347×C₇₃

C₄

yes

complemented

#344×C₇₃

—

#339×C₇₃

—

#338×C₇₃

C₄

yes

characteristic

H(3,ℤ/4)×C₇₃

C₄

yes

characteristic

#322×C₇₃

—

#322×C₇₃

C₄

yes

characteristic, complemented

C₈²×C₇₃

C₄

yes

characteristic, complemented

C₂×M₁₆×C₇₃

—

C₂×M₁₆×C₇₃

—

C₂×M₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×M₁₆×C₇₃

D₄

yes

characteristic

#117×C₇₃

—

C₂×C₄²×C₇₃

D₄

yes

characteristic

M₃₂×C₇₃

—

#97×C₇₃

—

#97×C₇₃

C₂×C₄

yes

characteristic

#97×C₇₃

Q₈

yes

characteristic

C₄×C₈×C₇₃

—

C₄×C₈×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

M₁₆×C₇₃

—

M₁₆×C₇₃

—

M₁₆×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

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂²⋊C₄

yes

characteristic

C₄²×C₇₃

C₄⋊C₄

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

#104

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#339

yes

characteristic

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

—

C₂×C₇₃

#1288

yes

characteristic

#2165

C₂×C₇₃

yes

characteristic

#1841

C₂×C₇₃

yes

characteristic

#1455

C₂×C₇₃

yes

characteristic

C₇₃

#7225

yes

characteristic, direct product factor, Sylow

C₄×M₁₆

—

C₄×M₁₆

C₂²×C₇₃

yes

—

—

C₄×C₇₃

yes

—

—

C₄×C₇₃

yes

characteristic

H(3,ℤ/4)

C₄×C₇₃

yes

characteristic

#322

—

#322

C₄×C₇₃

yes

characteristic, complemented

C₈²

C₄×C₇₃

yes

characteristic, complemented

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×C₄×C₇₃

yes

characteristic

C₂×M₁₆

D₄×C₇₃

yes

characteristic

100

#117

—

101

C₂×C₄²

D₄×C₇₃

yes

characteristic

102

M₃₂

—

103

#97

—

104

#97

C₂×C₄×C₇₃

yes

characteristic

105

#97

Q₈×C₇₃

yes

characteristic

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

C₂×C₄×C₇₃

yes

characteristic

109

C₄×C₈

D₄×C₇₃

yes

characteristic

110

C₂²×C₄

—

111

C₂²×C₄

C₂²⋊C₄×C₇₃

yes

characteristic

112

M₁₆

—

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

116

M₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

C₄²×C₇₃

yes

characteristic

121

C₂×C₈

C₂²⋊C₄×C₇₃

yes

characteristic

122

C₂×C₈

C₄⋊C₄×C₇₃

yes

characteristic

123

C₂×C₈

C₄⋊C₄×C₇₃

yes

characteristic

124

C₄⋊C₄

—

125

C₂²⋊C₄

—

126

C₄²

—

127

C₄²

—

128

C₄²

C₂²⋊C₄×C₇₃

yes

characteristic

129

C₄²

C₄⋊C₄×C₇₃

yes

characteristic

130

C₁₆

—

131

C₂³

#104×C₇₃

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

#95×C₇₃

yes

characteristic

139

C₂×C₄

#104×C₇₃

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₂²

—

148

C₂²

—

149

C₂²

#339×C₇₃

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

H(3,ℤ/4)×C₇₃

yes

characteristic

155

C₄

#344×C₇₃

yes

characteristic

156

C₂

—

157

C₂

—

158

C₂

C₇₃×#1288

yes

characteristic

	subgroup	G/H
center	C₄×C₇₃	H(3,ℤ/4)
commutator	C₂×C₈	C₄²×C₇₃
Frattini, cosocle	C₄×M₁₆	C₂²×C₇₃
socle	C₂×C₇₃	#1288
prime radical	C₂³×C₇₃	#104
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₇₃ ◁ C₂×C₈×C₇₃ ◁ G

subgroup

G/H

center

C₄×C₇₃

H(3,ℤ/4)

commutator

C₂×C₈

C₄²×C₇₃

Frattini, cosocle

C₄×M₁₆

C₂²×C₇₃

C₂×C₇₃

C₂³×C₇₃

G ▷ C₂×C₈ ▷ C₄ ▷ C₁

C₁ ◁ C₄×C₇₃ ◁ C₂×C₈×C₇₃ ◁ G

N	H ≅ G/N	type
8. #347×C₇₃	C₄	semidirect
14. #322×C₇₃	C₄	semidirect
15. C₈²×C₇₃	C₄	semidirect
76. #7225	C₇₃	direct (N × H)
83. C₇₃	#7225	direct (N × H)
88. #347	C₄×C₇₃	semidirect
94. #322	C₄×C₇₃	semidirect
95. C₈²	C₄×C₇₃	semidirect

H ≅ G/N

type

8. #347×C₇₃

C₄

semidirect

14. #322×C₇₃

C₄

semidirect

15. C₈²×C₇₃

C₄

semidirect

76. #7225

C₇₃

direct (N × H)

83. C₇₃

#7225

direct (N × H)

88. #347

C₄×C₇₃

semidirect

94. #322

C₄×C₇₃

semidirect

95. C₈²

C₄×C₇₃

semidirect

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

subgroup	index
C₇₃×#2165	2
C₇₃×#1841	2
C₇₃×#1455	2
#7225	73

subgroup

index

C₇₃×#2165

C₇₃×#1841

C₇₃×#1455

#7225

Subgroups — C₇₃×#7225 #32916629

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#7225 #32916629

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#7225 #32916629