Group #32923092 — subgroups

Subgroups — C₇₃×#13688 #32923092

Order 18688.

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

normal subgroups only

This group has 200 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×Q₆₄×C₇₃	C₂	1	2	yes	maximal
2	C₇₃×#2247	C₂	1	1	yes	maximal, characteristic
3	C₄×Q₃₂×C₇₃	C₂	1	1	yes	maximal, characteristic
4	C₇₃×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1416	C₂	1	2	yes	maximal
6	C₂×Q₃₂×C₇₃	—	2	1	no	—
7	C₂×Q₃₂×C₇₃	C₂²	1	2	yes	—
8	C₂×Q₃₂×C₇₃	C₂²	1	1	yes	characteristic
9	#494×C₇₃	—	2	1	no	—
10	C₄×Q₁₆×C₇₃	—	2	1	no	—
11	Q₆₄×C₇₃	—	2	4	no	—
12	C₂×C₃₂×C₇₃	C₂²	1	2	yes	—
13	#366×C₇₃	C₂²	1	1	yes	characteristic
14	#358×C₇₃	—	2	1	no	—
15	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₇₃	—	2	1	no	—
17	C₂×Q₁₆×C₇₃	—	2	2	no	—
18	C₂×Q₁₆×C₇₃	—	4	1	no	—
19	C₄⋊Q₈×C₇₃	—	4	1	no	—
20	C₄×Q₈×C₇₃	—	4	1	no	—
21	Q₃₂×C₇₃	—	2	2	no	—
22	Q₃₂×C₇₃	—	2	1	no	—
23	Q₃₂×C₇₃	—	2	2	no	—
24	Q₃₂×C₇₃	—	4	2	no	—
25	Q₃₂×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	#103×C₇₃	—	4	1	no	—
30	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
31	C₃₂×C₇₃	—	2	2	no	—
32	C₂×Q₈×C₇₃	—	4	1	no	—
33	C₂×Q₈×C₇₃	—	4	2	no	—
34	Q₁₆×C₇₃	—	2	2	no	—
35	Q₁₆×C₇₃	—	4	2	no	—
36	Q₁₆×C₇₃	—	4	1	no	—
37	Q₁₆×C₇₃	—	4	2	no	—
38	Q₁₆×C₇₃	—	8	2	no	—
39	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
40	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
41	C₄⋊C₄×C₇₃	—	4	1	no	—
42	C₄⋊C₄×C₇₃	—	8	1	no	—
43	C₄⋊C₄×C₇₃	—	8	1	no	—
44	C₄⋊C₄×C₇₃	—	8	1	no	—
45	C₄²×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	Q₈×C₇₃	—	4	2	no	—
51	Q₈×C₇₃	—	8	1	no	—
52	Q₈×C₇₃	—	8	2	no	—
53	Q₈×C₇₃	—	8	2	no	—
54	C₂×C₄×C₇₃	—	8	1	no	—
55	C₂×C₄×C₇₃	—	8	1	no	—
56	C₂×C₄×C₇₃	—	8	2	no	—
57	C₂×C₄×C₇₃	D₁₆	1	1	yes	characteristic
58	C₂×C₄×C₇₃	D₁₆	1	1	yes	characteristic
59	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
60	C₈×C₇₃	—	2	1	no	—
61	C₈×C₇₃	#121	1	1	yes	characteristic
62	C₈×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
63	C₂²×C₇₃	C₂×D₁₆	1	1	yes	characteristic
64	C₄×C₇₃	—	2	1	no	—
65	C₄×C₇₃	—	8	2	no	—
66	C₄×C₇₃	—	16	2	no	—
67	C₄×C₇₃	—	16	1	no	—
68	C₄×C₇₃	Q₆₄	1	2	yes	complemented
69	C₄×C₇₃	#459	1	1	yes	characteristic
70	C₄×C₇₃	#509	1	1	yes	characteristic
71	#13688	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂×C₇₃	#2206	1	1	yes	characteristic
73	C₂×C₇₃	C₂×Q₆₄	1	1	yes	characteristic
74	C₂×C₇₃	#2264	1	1	yes	characteristic
75	C₂×Q₆₄	C₂×C₇₃	1	2	yes	—
76	#2247	C₂×C₇₃	1	1	yes	characteristic
77	C₄×Q₃₂	C₂×C₇₃	1	1	yes	characteristic
78	C₄⋊C₃₂	C₂×C₇₃	1	1	yes	characteristic
79	#1416	C₂×C₇₃	1	2	yes	—
80	C₇₃	#13688	1	1	yes	characteristic, direct product factor, Sylow
81	C₂×Q₃₂	—	2	1	no	—
82	C₂×Q₃₂	C₂²×C₇₃	1	2	yes	—
83	C₂×Q₃₂	C₂²×C₇₃	1	1	yes	characteristic
84	#494	—	2	1	no	—
85	C₄×Q₁₆	—	2	1	no	—
86	Q₆₄	—	2	4	no	—
87	C₂×C₃₂	C₂²×C₇₃	1	2	yes	—
88	#366	C₂²×C₇₃	1	1	yes	characteristic
89	#358	—	2	1	no	—
90	C₄×C₁₆	C₂²×C₇₃	1	1	yes	characteristic
91	C₂×Q₁₆	—	2	1	no	—
92	C₂×Q₁₆	—	2	2	no	—
93	C₂×Q₁₆	—	4	1	no	—
94	C₄⋊Q₈	—	4	1	no	—
95	C₄×Q₈	—	4	1	no	—
96	Q₃₂	—	2	2	no	—
97	Q₃₂	—	2	1	no	—
98	Q₃₂	—	2	2	no	—
99	Q₃₂	—	4	2	no	—
100	Q₃₂	D₄×C₇₃	1	2	yes	—
101	C₂×C₁₆	D₄×C₇₃	1	1	yes	characteristic
102	C₂×C₁₆	C₂³×C₇₃	1	1	yes	characteristic
103	#107	—	2	1	no	—
104	#103	—	4	1	no	—
105	C₄×C₈	D₄×C₇₃	1	1	yes	characteristic
106	C₃₂	—	2	2	no	—
107	C₂×Q₈	—	4	1	no	—
108	C₂×Q₈	—	4	2	no	—
109	Q₁₆	—	2	2	no	—
110	Q₁₆	—	4	2	no	—
111	Q₁₆	—	4	1	no	—
112	Q₁₆	—	4	2	no	—
113	Q₁₆	—	8	2	no	—
114	C₂×C₈	D₈×C₇₃	1	1	yes	characteristic
115	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
116	C₄⋊C₄	—	4	1	no	—
117	C₄⋊C₄	—	8	1	no	—
118	C₄⋊C₄	—	8	1	no	—
119	C₄⋊C₄	—	8	1	no	—
120	C₄²	—	8	1	no	—
121	C₄²	D₈×C₇₃	1	1	yes	characteristic
122	C₁₆	—	2	1	no	—
123	C₁₆	C₂×D₄×C₇₃	1	1	yes	characteristic
124	C₁₆	#41×C₇₃	1	1	yes	characteristic
125	Q₈	—	4	2	no	—
126	Q₈	—	8	1	no	—
127	Q₈	—	8	2	no	—
128	Q₈	—	8	2	no	—
129	C₂×C₄	—	8	1	no	—
130	C₂×C₄	—	8	1	no	—
131	C₂×C₄	—	8	2	no	—
132	C₂×C₄	D₁₆×C₇₃	1	1	yes	characteristic
133	C₂×C₄	D₁₆×C₇₃	1	1	yes	characteristic
134	C₂×C₄	C₂×D₈×C₇₃	1	1	yes	characteristic
135	C₈	—	2	1	no	—
136	C₈	#121×C₇₃	1	1	yes	characteristic
137	C₈	AGL(1,ℤ/8)×C₇₃	1	1	yes	characteristic
138	C₂²	C₂×D₁₆×C₇₃	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	—	8	2	no	—
141	C₄	—	16	2	no	—
142	C₄	—	16	1	no	—
143	C₄	Q₆₄×C₇₃	1	2	yes	complemented
144	C₄	#459×C₇₃	1	1	yes	characteristic
145	C₄	#509×C₇₃	1	1	yes	characteristic
146	C₂	C₇₃×#2206	1	1	yes	characteristic
147	C₂	C₂×Q₆₄×C₇₃	1	1	yes	characteristic
148	C₂	C₇₃×#2264	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₆₄×C₇₃

C₂

yes

maximal

C₇₃×#2247

C₂

yes

maximal, characteristic

C₄×Q₃₂×C₇₃

C₂

yes

maximal, characteristic

C₇₃×C₄⋊C₃₂

C₂

yes

maximal, characteristic

C₇₃×#1416

C₂

yes

maximal

C₂×Q₃₂×C₇₃

—

C₂×Q₃₂×C₇₃

C₂²

yes

—

C₂×Q₃₂×C₇₃

C₂²

yes

characteristic

#494×C₇₃

—

C₄×Q₁₆×C₇₃

—

Q₆₄×C₇₃

—

C₂×C₃₂×C₇₃

C₂²

yes

—

#366×C₇₃

C₂²

yes

characteristic

#358×C₇₃

—

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×Q₁₆×C₇₃

—

C₂×Q₁₆×C₇₃

—

C₂×Q₁₆×C₇₃

—

C₄⋊Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

Q₃₂×C₇₃

—

Q₃₂×C₇₃

—

Q₃₂×C₇₃

—

Q₃₂×C₇₃

—

Q₃₂×C₇₃

D₄

yes

—

C₂×C₁₆×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂³

yes

characteristic

#107×C₇₃

—

#103×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

characteristic

C₃₂×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

D₈

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

#41

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×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

C₂×D₁₆

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

Q₆₄

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₃

#2206

yes

characteristic

C₂×C₇₃

C₂×Q₆₄

yes

characteristic

C₂×C₇₃

#2264

yes

characteristic

C₂×Q₆₄

C₂×C₇₃

yes

—

#2247

C₂×C₇₃

yes

characteristic

C₄×Q₃₂

C₂×C₇₃

yes

characteristic

C₄⋊C₃₂

C₂×C₇₃

yes

characteristic

#1416

C₂×C₇₃

yes

—

C₇₃

#13688

yes

characteristic, direct product factor, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

C₂²×C₇₃

yes

—

C₂×Q₃₂

C₂²×C₇₃

yes

characteristic

#494

—

C₄×Q₁₆

—

Q₆₄

—

C₂×C₃₂

C₂²×C₇₃

yes

—

#366

C₂²×C₇₃

yes

characteristic

#358

—

C₄×C₁₆

C₂²×C₇₃

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

100

Q₃₂

D₄×C₇₃

yes

—

101

C₂×C₁₆

D₄×C₇₃

yes

characteristic

102

C₂×C₁₆

C₂³×C₇₃

yes

characteristic

103

#107

—

104

#103

—

105

C₄×C₈

D₄×C₇₃

yes

characteristic

106

C₃₂

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

Q₁₆

—

110

Q₁₆

—

111

Q₁₆

—

112

Q₁₆

—

113

Q₁₆

—

114

C₂×C₈

D₈×C₇₃

yes

characteristic

115

C₂×C₈

C₂×D₄×C₇₃

yes

characteristic

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄²

—

121

C₄²

D₈×C₇₃

yes

characteristic

122

C₁₆

—

123

C₁₆

C₂×D₄×C₇₃

yes

characteristic

124

C₁₆

#41×C₇₃

yes

characteristic

125

Q₈

—

126

Q₈

—

127

Q₈

—

128

Q₈

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

D₁₆×C₇₃

yes

characteristic

133

C₂×C₄

D₁₆×C₇₃

yes

characteristic

134

C₂×C₄

C₂×D₈×C₇₃

yes

characteristic

135

C₈

—

136

C₈

#121×C₇₃

yes

characteristic

137

C₈

AGL(1,ℤ/8)×C₇₃

yes

characteristic

138

C₂²

C₂×D₁₆×C₇₃

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

Q₆₄×C₇₃

yes

complemented

144

C₄

#459×C₇₃

yes

characteristic

145

C₄

#509×C₇₃

yes

characteristic

146

C₂

C₇₃×#2206

yes

characteristic

147

C₂

C₂×Q₆₄×C₇₃

yes

characteristic

148

C₂

C₇₃×#2264

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₇₃	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₇₃
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, prime radical

C₂²×C₇₃

C₂×D₁₆

commutator, Frattini, cosocle

C₂×C₁₆

C₂³×C₇₃

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
68. C₄×C₇₃	Q₆₄	semidirect
71. #13688	C₇₃	direct (N × H)
80. C₇₃	#13688	direct (N × H)
143. C₄	Q₆₄×C₇₃	semidirect

H ≅ G/N

type

68. C₄×C₇₃

Q₆₄

semidirect

71. #13688

C₇₃

direct (N × H)

80. C₇₃

#13688

direct (N × H)

143. C₄

Q₆₄×C₇₃

semidirect

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

subgroup	index
C₂×Q₆₄×C₇₃	2
C₇₃×#2247	2
C₄×Q₃₂×C₇₃	2
C₇₃×C₄⋊C₃₂	2
C₇₃×#1416	2
#13688	73

subgroup

index

C₂×Q₆₄×C₇₃

C₇₃×#2247

C₄×Q₃₂×C₇₃

C₇₃×C₄⋊C₃₂

C₇₃×#1416

#13688

Subgroups — C₇₃×#13688 #32923092

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#13688 #32923092

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#13688 #32923092