Group #32916632 — subgroups

Subgroups — C₇₃×#7228 #32916632

Order 18688.

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

normal subgroups only

This group has 252 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×#2113	C₂	1	1	yes	maximal, characteristic
2	C₇₃×#1812	C₂	1	2	yes	maximal
3	C₂×#131×C₇₃	C₂²	1	1	yes	characteristic
4	C₂×M₃₂×C₇₃	C₄	1	1	yes	characteristic, complemented
5	C₂²×C₁₆×C₇₃	C₄	1	1	yes	characteristic, complemented
6	#427×C₇₃	—	2	2	no	—
7	#410×C₇₃	—	2	2	no	—
8	C₂²⋊C₁₆×C₇₃	—	2	2	no	—
9	H(3,ℤ/4)×C₇₃	C₄	1	4	yes	complemented
10	C₂×#41×C₇₃	D₄	1	1	yes	characteristic
11	#131×C₇₃	—	4	1	no	—
12	#131×C₇₃	—	4	1	no	—
13	C₂×M₁₆×C₇₃	C₂×C₄	1	2	yes	—
14	C₂×M₁₆×C₇₃	Q₈	1	1	yes	characteristic
15	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₇₃	D₄	1	2	yes	—
17	#117×C₇₃	—	2	4	no	—
18	M₃₂×C₇₃	—	4	1	no	—
19	C₂×C₁₆×C₇₃	—	2	1	no	—
20	C₂×C₁₆×C₇₃	—	2	1	no	—
21	C₂×C₁₆×C₇₃	—	4	1	no	—
22	#107×C₇₃	—	4	2	no	—
23	#106×C₇₃	—	4	2	no	—
24	#99×C₇₃	—	4	2	no	—
25	#99×C₇₃	—	4	2	no	—
26	#41×C₇₃	—	4	1	no	—
27	#41×C₇₃	—	4	1	no	—
28	C₂×Q₈×C₇₃	Q₁₆	1	1	yes	characteristic
29	C₂×D₄×C₇₃	D₈	1	1	yes	characteristic
30	C₂×D₄×C₇₃	SD₁₆	1	2	yes	—
31	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
32	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
33	M₁₆×C₇₃	—	4	2	no	—
34	M₁₆×C₇₃	—	4	1	no	—
35	C₂×C₈×C₇₃	—	2	2	no	—
36	C₂×C₈×C₇₃	—	2	2	no	—
37	C₂×C₈×C₇₃	—	2	1	no	—
38	C₂×C₈×C₇₃	—	2	1	no	—
39	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
42	C₄⋊C₄×C₇₃	—	4	4	no	—
43	C₂²⋊C₄×C₇₃	—	4	4	no	—
44	C₄²×C₇₃	—	4	4	no	—
45	C₁₆×C₇₃	—	4	1	no	—
46	C₁₆×C₇₃	—	4	1	no	—
47	C₂³×C₇₃	#102	1	2	yes	—
48	C₂³×C₇₃	#106	1	1	yes	characteristic
49	Q₈×C₇₃	—	4	1	no	—
50	D₄×C₇₃	—	4	1	no	—
51	D₄×C₇₃	—	4	2	no	—
52	C₂×C₄×C₇₃	—	2	2	no	—
53	C₂×C₄×C₇₃	—	2	1	no	—
54	C₂×C₄×C₇₃	—	2	2	no	—
55	C₂×C₄×C₇₃	—	2	1	no	—
56	C₂×C₄×C₇₃	—	8	4	no	—
57	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
58	C₂×C₄×C₇₃	#103	1	2	yes	—
59	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
60	C₈×C₇₃	—	2	1	no	—
61	C₈×C₇₃	—	2	2	no	—
62	C₈×C₇₃	#99	1	1	yes	characteristic
63	C₈×C₇₃	#99	1	1	yes	characteristic
64	C₂²×C₇₃	—	2	2	no	—
65	C₂²×C₇₃	—	2	1	no	—
66	C₂²×C₇₃	—	4	2	no	—
67	C₂²×C₇₃	—	4	1	no	—
68	C₂²×C₇₃	#340	1	1	yes	characteristic
69	C₄×C₇₃	—	2	2	no	—
70	C₄×C₇₃	—	2	1	no	—
71	C₄×C₇₃	—	16	4	no	—
72	C₄×C₇₃	H(3,ℤ/4)	1	1	yes	characteristic
73	C₄×C₇₃	#342	1	1	yes	characteristic
74	#7228	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂×C₇₃	—	2	1	no	—
76	C₂×C₇₃	—	4	2	no	—
77	C₂×C₇₃	—	4	1	no	—
78	C₂×C₇₃	#1289	1	1	yes	characteristic
79	#2113	C₂×C₇₃	1	1	yes	characteristic
80	#1812	C₂×C₇₃	1	2	yes	—
81	C₇₃	#7228	1	1	yes	characteristic, direct product factor, Sylow
82	C₂×#131	C₂²×C₇₃	1	1	yes	characteristic
83	C₂×M₃₂	C₄×C₇₃	1	1	yes	characteristic, complemented
84	C₂²×C₁₆	C₄×C₇₃	1	1	yes	characteristic, complemented
85	#427	—	2	2	no	—
86	#410	—	2	2	no	—
87	C₂²⋊C₁₆	—	2	2	no	—
88	H(3,ℤ/4)	C₄×C₇₃	1	4	yes	complemented
89	C₂×#41	D₄×C₇₃	1	1	yes	characteristic
90	#131	—	4	1	no	—
91	#131	—	4	1	no	—
92	C₂×M₁₆	C₂×C₄×C₇₃	1	2	yes	—
93	C₂×M₁₆	Q₈×C₇₃	1	1	yes	characteristic
94	C₂²×C₈	C₂×C₄×C₇₃	1	1	yes	characteristic
95	C₂²×C₈	D₄×C₇₃	1	2	yes	—
96	#117	—	2	4	no	—
97	M₃₂	—	4	1	no	—
98	C₂×C₁₆	—	2	1	no	—
99	C₂×C₁₆	—	2	1	no	—
100	C₂×C₁₆	—	4	1	no	—
101	#107	—	4	2	no	—
102	#106	—	4	2	no	—
103	#99	—	4	2	no	—
104	#99	—	4	2	no	—
105	#41	—	4	1	no	—
106	#41	—	4	1	no	—
107	C₂×Q₈	Q₁₆×C₇₃	1	1	yes	characteristic
108	C₂×D₄	D₈×C₇₃	1	1	yes	characteristic
109	C₂×D₄	SD₁₆×C₇₃	1	2	yes	—
110	C₂²×C₄	C₂²⋊C₄×C₇₃	1	2	yes	—
111	C₂²×C₄	C₄⋊C₄×C₇₃	1	1	yes	characteristic
112	M₁₆	—	4	2	no	—
113	M₁₆	—	4	1	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	C₄²×C₇₃	1	1	yes	characteristic
119	C₂×C₈	C₂²⋊C₄×C₇₃	1	1	yes	characteristic
120	C₂×C₈	C₄⋊C₄×C₇₃	1	2	yes	—
121	C₄⋊C₄	—	4	4	no	—
122	C₂²⋊C₄	—	4	4	no	—
123	C₄²	—	4	4	no	—
124	C₁₆	—	4	1	no	—
125	C₁₆	—	4	1	no	—
126	C₂³	#102×C₇₃	1	2	yes	—
127	C₂³	#106×C₇₃	1	1	yes	characteristic
128	Q₈	—	4	1	no	—
129	D₄	—	4	1	no	—
130	D₄	—	4	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	8	4	no	—
136	C₂×C₄	#95×C₇₃	1	1	yes	characteristic
137	C₂×C₄	#103×C₇₃	1	2	yes	—
138	C₂×C₄	#107×C₇₃	1	1	yes	characteristic
139	C₈	—	2	1	no	—
140	C₈	—	2	2	no	—
141	C₈	#99×C₇₃	1	1	yes	characteristic
142	C₈	#99×C₇₃	1	1	yes	characteristic
143	C₂²	—	2	2	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	4	2	no	—
146	C₂²	—	4	1	no	—
147	C₂²	#340×C₇₃	1	1	yes	characteristic
148	C₄	—	2	2	no	—
149	C₄	—	2	1	no	—
150	C₄	—	16	4	no	—
151	C₄	H(3,ℤ/4)×C₇₃	1	1	yes	characteristic
152	C₄	#342×C₇₃	1	1	yes	characteristic
153	C₂	—	2	1	no	—
154	C₂	—	4	2	no	—
155	C₂	—	4	1	no	—
156	C₂	C₇₃×#1289	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃×#2113

C₂

yes

maximal, characteristic

C₇₃×#1812

C₂

yes

maximal

C₂×#131×C₇₃

C₂²

yes

characteristic

C₂×M₃₂×C₇₃

C₄

yes

characteristic, complemented

C₂²×C₁₆×C₇₃

C₄

yes

characteristic, complemented

#427×C₇₃

—

#410×C₇₃

—

C₂²⋊C₁₆×C₇₃

—

H(3,ℤ/4)×C₇₃

C₄

yes

complemented

C₂×#41×C₇₃

D₄

yes

characteristic

#131×C₇₃

—

#131×C₇₃

—

C₂×M₁₆×C₇₃

C₂×C₄

yes

—

C₂×M₁₆×C₇₃

Q₈

yes

characteristic

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₃

D₄

yes

—

#117×C₇₃

—

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

—

#107×C₇₃

—

#106×C₇₃

—

#99×C₇₃

—

#99×C₇₃

—

#41×C₇₃

—

#41×C₇₃

—

C₂×Q₈×C₇₃

Q₁₆

yes

characteristic

C₂×D₄×C₇₃

D₈

yes

characteristic

C₂×D₄×C₇₃

SD₁₆

yes

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₃

C₄⋊C₄

yes

characteristic

M₁₆×C₇₃

—

M₁₆×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

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₁₆×C₇₃

—

C₁₆×C₇₃

—

C₂³×C₇₃

#102

yes

—

C₂³×C₇₃

#106

yes

characteristic

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#95

yes

characteristic

C₂×C₄×C₇₃

#103

yes

—

C₂×C₄×C₇₃

#107

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

yes

C₈×C₇₃

yes

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#340

yes

characteristic

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

—

C₂×C₇₃

#1289

yes

characteristic

#2113

C₂×C₇₃

yes

characteristic

#1812

C₂×C₇₃

yes

—

C₇₃

#7228

yes

characteristic, direct product factor, Sylow

C₂×#131

C₂²×C₇₃

yes

characteristic

C₂×M₃₂

C₄×C₇₃

yes

characteristic, complemented

C₂²×C₁₆

C₄×C₇₃

yes

characteristic, complemented

#427

—

#410

—

C₂²⋊C₁₆

—

H(3,ℤ/4)

C₄×C₇₃

yes

complemented

C₂×#41

D₄×C₇₃

yes

characteristic

#131

—

#131

—

C₂×M₁₆

C₂×C₄×C₇₃

yes

—

C₂×M₁₆

Q₈×C₇₃

yes

characteristic

C₂²×C₈

C₂×C₄×C₇₃

yes

characteristic

C₂²×C₈

D₄×C₇₃

yes

—

#117

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

#107

—

102

#106

—

103

#99

—

104

#99

—

105

#41

—

106

#41

—

107

C₂×Q₈

Q₁₆×C₇₃

yes

characteristic

108

C₂×D₄

D₈×C₇₃

yes

characteristic

109

C₂×D₄

SD₁₆×C₇₃

yes

—

110

C₂²×C₄

C₂²⋊C₄×C₇₃

yes

—

111

C₂²×C₄

C₄⋊C₄×C₇₃

yes

characteristic

112

M₁₆

—

113

M₁₆

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

C₄²×C₇₃

yes

characteristic

119

C₂×C₈

C₂²⋊C₄×C₇₃

yes

characteristic

120

C₂×C₈

C₄⋊C₄×C₇₃

yes

—

121

C₄⋊C₄

—

122

C₂²⋊C₄

—

123

C₄²

—

124

C₁₆

—

125

C₁₆

—

126

C₂³

#102×C₇₃

yes

—

127

C₂³

#106×C₇₃

yes

characteristic

128

Q₈

—

129

D₄

—

130

D₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

#95×C₇₃

yes

characteristic

137

C₂×C₄

#103×C₇₃

yes

—

138

C₂×C₄

#107×C₇₃

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

#99×C₇₃

yes

characteristic

142

C₈

#99×C₇₃

yes

characteristic

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

#340×C₇₃

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

H(3,ℤ/4)×C₇₃

yes

characteristic

152

C₄

#342×C₇₃

yes

characteristic

153

C₂

—

154

C₂

—

155

C₂

—

156

C₂

C₇₃×#1289

yes

characteristic

	subgroup	G/H
center	C₄×C₇₃	H(3,ℤ/4)
commutator	C₂×C₈	C₄²×C₇₃
Frattini, cosocle	C₂×#131	C₂²×C₇₃
socle	C₂×C₇₃	#1289
prime radical	C₂×#41×C₇₃	D₄
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

C₂²×C₇₃

C₂×C₇₃

C₂×#41×C₇₃

D₄

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₁

UCS

C₁ ◁ C₄×C₇₃ ◁ C₂×C₈×C₇₃ ◁ G

N	H ≅ G/N	type
4. C₂×M₃₂×C₇₃	C₄	semidirect
5. C₂²×C₁₆×C₇₃	C₄	semidirect
9. H(3,ℤ/4)×C₇₃	C₄	semidirect
74. #7228	C₇₃	direct (N × H)
81. C₇₃	#7228	direct (N × H)
83. C₂×M₃₂	C₄×C₇₃	semidirect
84. C₂²×C₁₆	C₄×C₇₃	semidirect
88. H(3,ℤ/4)	C₄×C₇₃	semidirect

H ≅ G/N

type

4. C₂×M₃₂×C₇₃

C₄

semidirect

5. C₂²×C₁₆×C₇₃

C₄

semidirect

9. H(3,ℤ/4)×C₇₃

C₄

semidirect

74. #7228

C₇₃

direct (N × H)

81. C₇₃

#7228

direct (N × H)

83. C₂×M₃₂

C₄×C₇₃

semidirect

84. C₂²×C₁₆

C₄×C₇₃

semidirect

88. H(3,ℤ/4)

C₄×C₇₃

semidirect

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

subgroup	index
C₇₃×#2113	2
C₇₃×#1812	2
#7228	73

subgroup

index

C₇₃×#2113

C₇₃×#1812

#7228

Subgroups — C₇₃×#7228 #32916632

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#7228 #32916632

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#7228 #32916632