Group #32921855 — subgroups

Subgroups — C₇₃×#12451 #32921855

Order 18688.

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

normal subgroups only

This group has 224 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×#2252	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#1673	C₂	1	2	yes	maximal
4	C₇₃×#1371	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#1364	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₆×C₇₃	—	2	1	no	—
7	#500×C₇₃	C₂²	1	2	yes	—
8	#461×C₇₃	—	2	2	no	—
9	C₈×D₄×C₇₃	C₂²	1	1	yes	characteristic
10	#366×C₇₃	—	2	1	no	—
11	#366×C₇₃	—	2	1	no	—
12	C₄⋊C₁₆×C₇₃	C₂²	1	1	yes	characteristic
13	C₂²⋊C₁₆×C₇₃	—	2	1	no	—
14	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
15	#335×C₇₃	C₂²	1	2	yes	—
16	#332×C₇₃	—	2	2	no	—
17	C₂×SD₁₆×C₇₃	—	4	2	no	—
18	C₂²×C₈×C₇₃	—	2	1	no	—
19	C₄⋊Q₈×C₇₃	—	2	2	no	—
20	C₄×D₄×C₇₃	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₇₃	—	2	1	no	—
22	C₂×C₁₆×C₇₃	—	4	2	no	—
23	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
25	#107×C₇₃	—	2	2	no	—
26	#107×C₇₃	—	2	2	no	—
27	C₄⋊C₈×C₇₃	—	2	2	no	—
28	C₄⋊C₈×C₇₃	D₄	1	1	yes	characteristic
29	#103×C₇₃	—	4	2	no	—
30	C₂²⋊C₈×C₇₃	—	2	1	no	—
31	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
32	C₂×Q₈×C₇₃	—	4	2	no	—
33	C₂×D₄×C₇₃	D₈	1	1	yes	characteristic
34	C₂²×C₄×C₇₃	—	2	1	no	—
35	SD₁₆×C₇₃	—	8	4	no	—
36	C₂×C₈×C₇₃	—	2	1	no	—
37	C₂×C₈×C₇₃	—	4	1	no	—
38	C₂×C₈×C₇₃	—	4	1	no	—
39	C₂×C₈×C₇₃	—	4	2	no	—
40	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
42	C₄⋊C₄×C₇₃	—	4	2	no	—
43	C₄⋊C₄×C₇₃	—	4	2	no	—
44	C₄⋊C₄×C₇₃	D₈	1	1	yes	characteristic
45	C₂²⋊C₄×C₇₃	—	2	1	no	—
46	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
47	C₁₆×C₇₃	—	2	1	no	—
48	C₁₆×C₇₃	—	4	1	no	—
49	C₁₆×C₇₃	SD₁₆	1	2	yes	complemented
50	C₂³×C₇₃	—	2	1	no	—
51	Q₈×C₇₃	—	8	2	no	—
52	Q₈×C₇₃	—	8	2	no	—
53	D₄×C₇₃	—	2	1	no	—
54	D₄×C₇₃	D₁₆	1	2	yes	—
55	C₂×C₄×C₇₃	—	2	1	no	—
56	C₂×C₄×C₇₃	—	4	1	no	—
57	C₂×C₄×C₇₃	—	4	1	no	—
58	C₂×C₄×C₇₃	—	8	2	no	—
59	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
60	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
62	C₈×C₇₃	—	2	1	no	—
63	C₈×C₇₃	—	4	1	no	—
64	C₈×C₇₃	—	8	2	no	—
65	C₈×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
66	C₈×C₇₃	#137	1	1	yes	characteristic
67	C₂²×C₇₃	—	2	2	no	—
68	C₂²×C₇₃	—	4	1	no	—
69	C₂²×C₇₃	—	4	1	no	—
70	C₂²×C₇₃	#466	1	1	yes	characteristic
71	C₄×C₇₃	—	2	1	no	—
72	C₄×C₇₃	—	4	1	no	—
73	C₄×C₇₃	—	16	2	no	—
74	C₄×C₇₃	#461	1	1	yes	characteristic
75	C₄×C₇₃	#473	1	1	yes	characteristic
76	C₄×C₇₃	C₂×D₁₆	1	1	yes	characteristic
77	C₄×C₇₃	#508	1	1	yes	characteristic
78	#12451	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂×C₇₃	—	4	2	no	—
80	C₂×C₇₃	#1673	1	1	yes	characteristic
81	C₂×C₇₃	#2215	1	1	yes	characteristic
82	C₂×C₇₃	#2222	1	1	yes	characteristic
83	#2252	C₂×C₇₃	1	1	yes	characteristic, complemented
84	D₄×C₁₆	C₂×C₇₃	1	1	yes	characteristic
85	#1673	C₂×C₇₃	1	2	yes	—
86	#1371	C₂×C₇₃	1	1	yes	characteristic, complemented
87	#1364	C₂×C₇₃	1	2	yes	complemented
88	C₇₃	#12451	1	1	yes	characteristic, direct product factor, Sylow
89	C₂²×C₁₆	—	2	1	no	—
90	#500	C₂²×C₇₃	1	2	yes	—
91	#461	—	2	2	no	—
92	C₈×D₄	C₂²×C₇₃	1	1	yes	characteristic
93	#366	—	2	1	no	—
94	#366	—	2	1	no	—
95	C₄⋊C₁₆	C₂²×C₇₃	1	1	yes	characteristic
96	C₂²⋊C₁₆	—	2	1	no	—
97	C₄×C₁₆	C₂²×C₇₃	1	1	yes	characteristic
98	#335	C₂²×C₇₃	1	2	yes	—
99	#332	—	2	2	no	—
100	C₂×SD₁₆	—	4	2	no	—
101	C₂²×C₈	—	2	1	no	—
102	C₄⋊Q₈	—	2	2	no	—
103	C₄×D₄	D₄×C₇₃	1	1	yes	characteristic
104	C₂×C₁₆	—	2	1	no	—
105	C₂×C₁₆	—	4	2	no	—
106	C₂×C₁₆	D₄×C₇₃	1	1	yes	characteristic
107	C₂×C₁₆	D₄×C₇₃	1	1	yes	characteristic
108	#107	—	2	2	no	—
109	#107	—	2	2	no	—
110	C₄⋊C₈	—	2	2	no	—
111	C₄⋊C₈	D₄×C₇₃	1	1	yes	characteristic
112	#103	—	4	2	no	—
113	C₂²⋊C₈	—	2	1	no	—
114	C₄×C₈	C₂³×C₇₃	1	1	yes	characteristic
115	C₂×Q₈	—	4	2	no	—
116	C₂×D₄	D₈×C₇₃	1	1	yes	characteristic
117	C₂²×C₄	—	2	1	no	—
118	SD₁₆	—	8	4	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	4	1	no	—
121	C₂×C₈	—	4	1	no	—
122	C₂×C₈	—	4	2	no	—
123	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
124	C₂×C₈	#41×C₇₃	1	1	yes	characteristic
125	C₄⋊C₄	—	4	2	no	—
126	C₄⋊C₄	—	4	2	no	—
127	C₄⋊C₄	D₈×C₇₃	1	1	yes	characteristic
128	C₂²⋊C₄	—	2	1	no	—
129	C₄²	C₂×D₄×C₇₃	1	1	yes	characteristic
130	C₁₆	—	2	1	no	—
131	C₁₆	—	4	1	no	—
132	C₁₆	SD₁₆×C₇₃	1	2	yes	complemented
133	C₂³	—	2	1	no	—
134	Q₈	—	8	2	no	—
135	Q₈	—	8	2	no	—
136	D₄	—	2	1	no	—
137	D₄	D₁₆×C₇₃	1	2	yes	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	8	2	no	—
142	C₂×C₄	#121×C₇₃	1	1	yes	characteristic
143	C₂×C₄	C₂×D₈×C₇₃	1	1	yes	characteristic
144	C₂×C₄	#135×C₇₃	1	1	yes	characteristic
145	C₈	—	2	1	no	—
146	C₈	—	4	1	no	—
147	C₈	—	8	2	no	—
148	C₈	C₂×SD₁₆×C₇₃	1	1	yes	characteristic
149	C₈	#137×C₇₃	1	1	yes	characteristic
150	C₂²	—	2	2	no	—
151	C₂²	—	4	1	no	—
152	C₂²	—	4	1	no	—
153	C₂²	#466×C₇₃	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	4	1	no	—
156	C₄	—	16	2	no	—
157	C₄	#461×C₇₃	1	1	yes	characteristic
158	C₄	#473×C₇₃	1	1	yes	characteristic
159	C₄	C₂×D₁₆×C₇₃	1	1	yes	characteristic
160	C₄	#508×C₇₃	1	1	yes	characteristic
161	C₂	—	4	2	no	—
162	C₂	C₇₃×#1673	1	1	yes	characteristic
163	C₂	C₇₃×#2215	1	1	yes	characteristic
164	C₂	C₇₃×#2222	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃×#2252

C₂

yes

maximal, characteristic, complemented

D₄×C₁₆×C₇₃

C₂

yes

maximal, characteristic

C₇₃×#1673

C₂

yes

maximal

C₇₃×#1371

C₂

yes

maximal, characteristic, complemented

C₇₃×#1364

C₂

yes

maximal, complemented

C₂²×C₁₆×C₇₃

—

#500×C₇₃

C₂²

yes

—

#461×C₇₃

—

C₈×D₄×C₇₃

C₂²

yes

characteristic

#366×C₇₃

—

#366×C₇₃

—

C₄⋊C₁₆×C₇₃

C₂²

yes

characteristic

C₂²⋊C₁₆×C₇₃

—

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

#335×C₇₃

C₂²

yes

—

#332×C₇₃

—

C₂×SD₁₆×C₇₃

—

C₂²×C₈×C₇₃

—

C₄⋊Q₈×C₇₃

—

C₄×D₄×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

D₄

yes

characteristic

#107×C₇₃

—

#107×C₇₃

—

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

D₄

yes

characteristic

#103×C₇₃

—

C₂²⋊C₈×C₇₃

—

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×Q₈×C₇₃

—

C₂×D₄×C₇₃

D₈

yes

characteristic

C₂²×C₄×C₇₃

—

SD₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

#41

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

D₈

yes

characteristic

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

—

C₁₆×C₇₃

SD₁₆

yes

complemented

C₂³×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

D₁₆

yes

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#121

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

#135

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

C₂×SD₁₆

yes

characteristic

C₈×C₇₃

#137

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#466

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

C₂×D₁₆

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, complemented

D₄×C₁₆

C₂×C₇₃

yes

characteristic

#1673

C₂×C₇₃

yes

—

#1371

C₂×C₇₃

yes

characteristic, complemented

#1364

C₂×C₇₃

yes

complemented

C₇₃

#12451

yes

characteristic, direct product factor, Sylow

C₂²×C₁₆

—

#500

C₂²×C₇₃

yes

—

#461

—

C₈×D₄

C₂²×C₇₃

yes

characteristic

#366

—

#366

—

C₄⋊C₁₆

C₂²×C₇₃

yes

characteristic

C₂²⋊C₁₆

—

C₄×C₁₆

C₂²×C₇₃

yes

characteristic

#335

C₂²×C₇₃

yes

—

#332

—

100

C₂×SD₁₆

—

101

C₂²×C₈

—

102

C₄⋊Q₈

—

103

C₄×D₄

D₄×C₇₃

yes

characteristic

104

C₂×C₁₆

—

105

C₂×C₁₆

—

106

C₂×C₁₆

D₄×C₇₃

yes

characteristic

107

C₂×C₁₆

D₄×C₇₃

yes

characteristic

108

#107

—

109

#107

—

110

C₄⋊C₈

—

111

C₄⋊C₈

D₄×C₇₃

yes

characteristic

112

#103

—

113

C₂²⋊C₈

—

114

C₄×C₈

C₂³×C₇₃

yes

characteristic

115

C₂×Q₈

—

116

C₂×D₄

D₈×C₇₃

yes

characteristic

117

C₂²×C₄

—

118

SD₁₆

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

C₂×D₄×C₇₃

yes

characteristic

124

C₂×C₈

#41×C₇₃

yes

characteristic

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

D₈×C₇₃

yes

characteristic

128

C₂²⋊C₄

—

129

C₄²

C₂×D₄×C₇₃

yes

characteristic

130

C₁₆

—

131

C₁₆

—

132

C₁₆

SD₁₆×C₇₃

yes

complemented

133

C₂³

—

134

Q₈

—

135

Q₈

—

136

D₄

—

137

D₄

D₁₆×C₇₃

yes

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

#121×C₇₃

yes

characteristic

143

C₂×C₄

C₂×D₈×C₇₃

yes

characteristic

144

C₂×C₄

#135×C₇₃

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

C₂×SD₁₆×C₇₃

yes

characteristic

149

C₈

#137×C₇₃

yes

characteristic

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

#466×C₇₃

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

#461×C₇₃

yes

characteristic

158

C₄

#473×C₇₃

yes

characteristic

159

C₄

C₂×D₁₆×C₇₃

yes

characteristic

160

C₄

#508×C₇₃

yes

characteristic

161

C₂

—

162

C₂

C₇₃×#1673

yes

characteristic

163

C₂

C₇₃×#2215

yes

characteristic

164

C₂

C₇₃×#2222

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₇₃	#466
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇₃
prime radical	C₂×D₄×C₇₃	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₄²×C₇₃ ◁ C₈×D₄×C₇₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₃

#466

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₇₃

prime radical

C₂×D₄×C₇₃

D₈

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₇₃ ◁ C₄²×C₇₃ ◁ C₈×D₄×C₇₃ ◁ G

N	H ≅ G/N	type
1. C₇₃×#2252	C₂	semidirect
4. C₇₃×#1371	C₂	semidirect
5. C₇₃×#1364	C₂	semidirect
49. C₁₆×C₇₃	SD₁₆	semidirect
78. #12451	C₇₃	direct (N × H)
83. #2252	C₂×C₇₃	semidirect
86. #1371	C₂×C₇₃	semidirect
87. #1364	C₂×C₇₃	semidirect
88. C₇₃	#12451	direct (N × H)
132. C₁₆	SD₁₆×C₇₃	semidirect

H ≅ G/N

type

1. C₇₃×#2252

C₂

semidirect

4. C₇₃×#1371

C₂

semidirect

5. C₇₃×#1364

C₂

semidirect

49. C₁₆×C₇₃

SD₁₆

semidirect

78. #12451

C₇₃

direct (N × H)

83. #2252

C₂×C₇₃

semidirect

86. #1371

C₂×C₇₃

semidirect

87. #1364

C₂×C₇₃

semidirect

88. C₇₃

#12451

direct (N × H)

132. C₁₆

SD₁₆×C₇₃

semidirect

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

subgroup	index
C₇₃×#2252	2
D₄×C₁₆×C₇₃	2
C₇₃×#1673	2
C₇₃×#1371	2
C₇₃×#1364	2
#12451	73

subgroup

index

C₇₃×#2252

D₄×C₁₆×C₇₃

C₇₃×#1673

C₇₃×#1371

C₇₃×#1364

#12451

Subgroups — C₇₃×#12451 #32921855

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#12451 #32921855

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#12451 #32921855