Group #32923091 — subgroups

Subgroups — C₇₃×#13687 #32923091

Order 18688.

212 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 78 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	C₂×SD₆₄×C₇₃	C₂	1	2	yes	maximal
2	C₇₃×#2247	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
4	C₇₃×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#1416	C₂	1	2	yes	maximal, complemented
6	C₂×Q₃₂×C₇₃	—	2	1	no	—
7	C₂×Q₃₂×C₇₃	C₂²	1	2	yes	—
8	C₂×D₁₆×C₇₃	C₂²	1	1	yes	characteristic
9	#494×C₇₃	—	2	1	no	—
10	C₄×D₈×C₇₃	—	2	1	no	—
11	SD₆₄×C₇₃	—	2	4	no	—
12	C₂×C₃₂×C₇₃	C₂²	1	2	yes	—
13	#366×C₇₃	C₂²	1	1	yes	characteristic
14	#357×C₇₃	—	2	1	no	—
15	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₇₃	—	2	2	no	—
17	C₂×Q₁₆×C₇₃	—	4	1	no	—
18	C₂×D₈×C₇₃	—	2	1	no	—
19	C₄⋊Q₈×C₇₃	—	4	1	no	—
20	C₄×D₄×C₇₃	—	4	1	no	—
21	Q₃₂×C₇₃	—	2	2	no	—
22	Q₃₂×C₇₃	—	2	2	no	—
23	Q₃₂×C₇₃	—	4	2	no	—
24	D₁₆×C₇₃	—	2	1	no	—
25	D₁₆×C₇₃	D₄	1	2	yes	—
26	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
28	#107×C₇₃	—	2	1	no	—
29	#102×C₇₃	—	4	1	no	—
30	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
31	C₃₂×C₇₃	—	2	2	no	—
32	C₂×Q₈×C₇₃	—	4	2	no	—
33	C₂×D₄×C₇₃	—	4	1	no	—
34	C₂²×C₄×C₇₃	—	8	1	no	—
35	Q₁₆×C₇₃	—	4	2	no	—
36	Q₁₆×C₇₃	—	4	2	no	—
37	Q₁₆×C₇₃	—	8	2	no	—
38	D₈×C₇₃	—	2	2	no	—
39	D₈×C₇₃	—	4	1	no	—
40	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
41	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
42	C₄⋊C₄×C₇₃	—	4	1	no	—
43	C₄⋊C₄×C₇₃	—	8	1	no	—
44	C₄⋊C₄×C₇₃	—	8	1	no	—
45	C₂²⋊C₄×C₇₃	—	8	1	no	—
46	C₄²×C₇₃	D₈	1	1	yes	characteristic
47	C₁₆×C₇₃	—	2	1	no	—
48	C₁₆×C₇₃	C₂×D₄	1	1	yes	characteristic
49	C₁₆×C₇₃	#41	1	1	yes	characteristic
50	C₂³×C₇₃	—	8	1	no	—
51	Q₈×C₇₃	—	8	2	no	—
52	Q₈×C₇₃	—	8	2	no	—
53	D₄×C₇₃	—	4	2	no	—
54	D₄×C₇₃	—	8	1	no	—
55	C₂×C₄×C₇₃	—	8	1	no	—
56	C₂×C₄×C₇₃	—	8	2	no	—
57	C₂×C₄×C₇₃	—	16	2	no	—
58	C₂×C₄×C₇₃	D₁₆	1	1	yes	characteristic
59	C₂×C₄×C₇₃	D₁₆	1	1	yes	characteristic
60	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
61	C₈×C₇₃	—	2	1	no	—
62	C₈×C₇₃	#121	1	1	yes	characteristic
63	C₈×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₂²×C₇₃	—	8	2	no	—
65	C₂²×C₇₃	—	16	1	no	—
66	C₂²×C₇₃	—	16	1	no	—
67	C₂²×C₇₃	C₂×D₁₆	1	1	yes	characteristic
68	C₄×C₇₃	—	2	1	no	—
69	C₄×C₇₃	—	16	2	no	—
70	C₄×C₇₃	—	16	1	no	—
71	C₄×C₇₃	SD₆₄	1	2	yes	complemented
72	C₄×C₇₃	#459	1	1	yes	characteristic
73	C₄×C₇₃	#509	1	1	yes	characteristic
74	#13687	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂×C₇₃	—	16	2	no	—
76	C₂×C₇₃	#2206	1	1	yes	characteristic
77	C₂×C₇₃	C₂×SD₆₄	1	1	yes	characteristic
78	C₂×C₇₃	#2264	1	1	yes	characteristic
79	C₂×SD₆₄	C₂×C₇₃	1	2	yes	—
80	#2247	C₂×C₇₃	1	1	yes	characteristic, complemented
81	C₄×D₁₆	C₂×C₇₃	1	1	yes	characteristic
82	C₄⋊C₃₂	C₂×C₇₃	1	1	yes	characteristic, complemented
83	#1416	C₂×C₇₃	1	2	yes	complemented
84	C₇₃	#13687	1	1	yes	characteristic, direct product factor, Sylow
85	C₂×Q₃₂	—	2	1	no	—
86	C₂×Q₃₂	C₂²×C₇₃	1	2	yes	—
87	C₂×D₁₆	C₂²×C₇₃	1	1	yes	characteristic
88	#494	—	2	1	no	—
89	C₄×D₈	—	2	1	no	—
90	SD₆₄	—	2	4	no	—
91	C₂×C₃₂	C₂²×C₇₃	1	2	yes	—
92	#366	C₂²×C₇₃	1	1	yes	characteristic
93	#357	—	2	1	no	—
94	C₄×C₁₆	C₂²×C₇₃	1	1	yes	characteristic
95	C₂×Q₁₆	—	2	2	no	—
96	C₂×Q₁₆	—	4	1	no	—
97	C₂×D₈	—	2	1	no	—
98	C₄⋊Q₈	—	4	1	no	—
99	C₄×D₄	—	4	1	no	—
100	Q₃₂	—	2	2	no	—
101	Q₃₂	—	2	2	no	—
102	Q₃₂	—	4	2	no	—
103	D₁₆	—	2	1	no	—
104	D₁₆	D₄×C₇₃	1	2	yes	—
105	C₂×C₁₆	D₄×C₇₃	1	1	yes	characteristic
106	C₂×C₁₆	C₂³×C₇₃	1	1	yes	characteristic
107	#107	—	2	1	no	—
108	#102	—	4	1	no	—
109	C₄×C₈	D₄×C₇₃	1	1	yes	characteristic
110	C₃₂	—	2	2	no	—
111	C₂×Q₈	—	4	2	no	—
112	C₂×D₄	—	4	1	no	—
113	C₂²×C₄	—	8	1	no	—
114	Q₁₆	—	4	2	no	—
115	Q₁₆	—	4	2	no	—
116	Q₁₆	—	8	2	no	—
117	D₈	—	2	2	no	—
118	D₈	—	4	1	no	—
119	C₂×C₈	D₈×C₇₃	1	1	yes	characteristic
120	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
121	C₄⋊C₄	—	4	1	no	—
122	C₄⋊C₄	—	8	1	no	—
123	C₄⋊C₄	—	8	1	no	—
124	C₂²⋊C₄	—	8	1	no	—
125	C₄²	D₈×C₇₃	1	1	yes	characteristic
126	C₁₆	—	2	1	no	—
127	C₁₆	C₂×D₄×C₇₃	1	1	yes	characteristic
128	C₁₆	#41×C₇₃	1	1	yes	characteristic
129	C₂³	—	8	1	no	—
130	Q₈	—	8	2	no	—
131	Q₈	—	8	2	no	—
132	D₄	—	4	2	no	—
133	D₄	—	8	1	no	—
134	C₂×C₄	—	8	1	no	—
135	C₂×C₄	—	8	2	no	—
136	C₂×C₄	—	16	2	no	—
137	C₂×C₄	D₁₆×C₇₃	1	1	yes	characteristic
138	C₂×C₄	D₁₆×C₇₃	1	1	yes	characteristic
139	C₂×C₄	C₂×D₈×C₇₃	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	#121×C₇₃	1	1	yes	characteristic
142	C₈	AGL(1,ℤ/8)×C₇₃	1	1	yes	characteristic
143	C₂²	—	8	2	no	—
144	C₂²	—	16	1	no	—
145	C₂²	—	16	1	no	—
146	C₂²	C₂×D₁₆×C₇₃	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	16	2	no	—
149	C₄	—	16	1	no	—
150	C₄	SD₆₄×C₇₃	1	2	yes	complemented
151	C₄	#459×C₇₃	1	1	yes	characteristic
152	C₄	#509×C₇₃	1	1	yes	characteristic
153	C₂	—	16	2	no	—
154	C₂	C₇₃×#2206	1	1	yes	characteristic
155	C₂	C₂×SD₆₄×C₇₃	1	1	yes	characteristic
156	C₂	C₇₃×#2264	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×SD₆₄×C₇₃

C₂

yes

maximal

C₇₃×#2247

C₂

yes

maximal, characteristic, complemented

C₄×D₁₆×C₇₃

C₂

yes

maximal, characteristic

C₇₃×C₄⋊C₃₂

C₂

yes

maximal, characteristic, complemented

C₇₃×#1416

C₂

yes

maximal, complemented

C₂×Q₃₂×C₇₃

—

C₂×Q₃₂×C₇₃

C₂²

yes

—

C₂×D₁₆×C₇₃

C₂²

yes

characteristic

#494×C₇₃

—

C₄×D₈×C₇₃

—

SD₆₄×C₇₃

—

C₂×C₃₂×C₇₃

C₂²

yes

—

#366×C₇₃

C₂²

yes

characteristic

#357×C₇₃

—

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×Q₁₆×C₇₃

—

C₂×Q₁₆×C₇₃

—

C₂×D₈×C₇₃

—

C₄⋊Q₈×C₇₃

—

C₄×D₄×C₇₃

—

Q₃₂×C₇₃

—

Q₃₂×C₇₃

—

Q₃₂×C₇₃

—

D₁₆×C₇₃

—

D₁₆×C₇₃

D₄

yes

—

C₂×C₁₆×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂³

yes

characteristic

#107×C₇₃

—

#102×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

characteristic

C₃₂×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×D₄×C₇₃

—

C₂²×C₄×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

D₈

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

#41

yes

characteristic

C₂³×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

D₁₆

yes

characteristic

C₂×C₄×C₇₃

D₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

yes

C₈×C₇₃

yes

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×D₁₆

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

SD₆₄

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₃

—

C₂×C₇₃

#2206

yes

characteristic

C₂×C₇₃

C₂×SD₆₄

yes

characteristic

C₂×C₇₃

#2264

yes

characteristic

C₂×SD₆₄

C₂×C₇₃

yes

—

#2247

C₂×C₇₃

yes

characteristic, complemented

C₄×D₁₆

C₂×C₇₃

yes

characteristic

C₄⋊C₃₂

C₂×C₇₃

yes

characteristic, complemented

#1416

C₂×C₇₃

yes

complemented

C₇₃

#13687

yes

characteristic, direct product factor, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

C₂²×C₇₃

yes

—

C₂×D₁₆

C₂²×C₇₃

yes

characteristic

#494

—

C₄×D₈

—

SD₆₄

—

C₂×C₃₂

C₂²×C₇₃

yes

—

#366

C₂²×C₇₃

yes

characteristic

#357

—

C₄×C₁₆

C₂²×C₇₃

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄×D₄

—

100

Q₃₂

—

101

Q₃₂

—

102

Q₃₂

—

103

D₁₆

—

104

D₁₆

D₄×C₇₃

yes

—

105

C₂×C₁₆

D₄×C₇₃

yes

characteristic

106

C₂×C₁₆

C₂³×C₇₃

yes

characteristic

107

#107

—

108

#102

—

109

C₄×C₈

D₄×C₇₃

yes

characteristic

110

C₃₂

—

111

C₂×Q₈

—

112

C₂×D₄

—

113

C₂²×C₄

—

114

Q₁₆

—

115

Q₁₆

—

116

Q₁₆

—

117

D₈

—

118

D₈

—

119

C₂×C₈

D₈×C₇₃

yes

characteristic

120

C₂×C₈

C₂×D₄×C₇₃

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

D₈×C₇₃

yes

characteristic

126

C₁₆

—

127

C₁₆

C₂×D₄×C₇₃

yes

characteristic

128

C₁₆

#41×C₇₃

yes

characteristic

129

C₂³

—

130

Q₈

—

131

Q₈

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

D₁₆×C₇₃

yes

characteristic

138

C₂×C₄

D₁₆×C₇₃

yes

characteristic

139

C₂×C₄

C₂×D₈×C₇₃

yes

characteristic

140

C₈

—

141

C₈

#121×C₇₃

yes

characteristic

142

C₈

AGL(1,ℤ/8)×C₇₃

yes

characteristic

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₂×D₁₆×C₇₃

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

SD₆₄×C₇₃

yes

complemented

151

C₄

#459×C₇₃

yes

characteristic

152

C₄

#509×C₇₃

yes

characteristic

153

C₂

—

154

C₂

C₇₃×#2206

yes

characteristic

155

C₂

C₂×SD₆₄×C₇₃

yes

characteristic

156

C₂

C₇₃×#2264

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₇₃	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₇₃
prime radical	C₂×D₁₆×C₇₃	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₄²×C₇₃ ◁ C₄×C₈×C₇₃ ◁ C₄×C₁₆×C₇₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₃

C₂×D₁₆

commutator, Frattini, cosocle

C₂×C₁₆

C₂³×C₇₃

prime radical

C₂×D₁₆×C₇₃

C₂²

LCS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₇₃ ◁ C₄²×C₇₃ ◁ C₄×C₈×C₇₃ ◁ C₄×C₁₆×C₇₃ ◁ G

N	H ≅ G/N	type
2. C₇₃×#2247	C₂	semidirect
4. C₇₃×C₄⋊C₃₂	C₂	semidirect
5. C₇₃×#1416	C₂	semidirect
71. C₄×C₇₃	SD₆₄	semidirect
74. #13687	C₇₃	direct (N × H)
80. #2247	C₂×C₇₃	semidirect
82. C₄⋊C₃₂	C₂×C₇₃	semidirect
83. #1416	C₂×C₇₃	semidirect
84. C₇₃	#13687	direct (N × H)
150. C₄	SD₆₄×C₇₃	semidirect

H ≅ G/N

type

2. C₇₃×#2247

C₂

semidirect

4. C₇₃×C₄⋊C₃₂

C₂

semidirect

5. C₇₃×#1416

C₂

semidirect

71. C₄×C₇₃

SD₆₄

semidirect

74. #13687

C₇₃

direct (N × H)

80. #2247

C₂×C₇₃

semidirect

82. C₄⋊C₃₂

C₂×C₇₃

semidirect

83. #1416

C₂×C₇₃

semidirect

84. C₇₃

#13687

direct (N × H)

150. C₄

SD₆₄×C₇₃

semidirect

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

subgroup	index
C₂×SD₆₄×C₇₃	2
C₇₃×#2247	2
C₄×D₁₆×C₇₃	2
C₇₃×C₄⋊C₃₂	2
C₇₃×#1416	2
#13687	73

subgroup

index

C₂×SD₆₄×C₇₃

C₇₃×#2247

C₄×D₁₆×C₇₃

C₇₃×C₄⋊C₃₂

C₇₃×#1416

#13687

Subgroups — C₇₃×#13687 #32923091

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#13687 #32923091

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#13687 #32923091