Group #32916652 — subgroups

Subgroups — C₇₃×#7248 #32916652

Order 18688.

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

normal subgroups only

This group has 244 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	C₇₃×#1984	C₂	1	1	yes	maximal, characteristic
2	C₇₃×#1467	C₂	1	2	yes	maximal
3	C₂×C₄×Q₈×C₇₃	C₂²	1	1	yes	characteristic
4	#432×C₇₃	—	2	2	no	—
5	C₂×C₄⋊C₈×C₇₃	C₄	1	1	yes	characteristic
6	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
7	#336×C₇₃	—	2	2	no	—
8	#324×C₇₃	—	2	2	no	—
9	#324×C₇₃	C₄	1	4	yes	—
10	C₂²×Q₈×C₇₃	Q₈	1	1	yes	characteristic
11	C₂²×C₈×C₇₃	—	2	1	no	—
12	C₂²×C₈×C₇₃	—	2	1	no	—
13	C₄×Q₈×C₇₃	—	4	1	no	—
14	C₄×Q₈×C₇₃	—	4	1	no	—
15	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
16	C₂×C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
17	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₇₃	D₄	1	2	yes	—
19	C₄⋊C₈×C₇₃	—	4	2	no	—
20	C₄⋊C₈×C₇₃	—	4	1	no	—
21	C₂²⋊C₈×C₇₃	—	2	4	no	—
22	#97×C₇₃	—	4	2	no	—
23	C₄×C₈×C₇₃	—	4	1	no	—
24	C₂×Q₈×C₇₃	—	2	2	no	—
25	C₂×Q₈×C₇₃	—	4	1	no	—
26	C₂×Q₈×C₇₃	—	4	1	no	—
27	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
28	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
29	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
31	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₇₃	—	2	2	no	—
33	C₂×C₈×C₇₃	—	4	1	no	—
34	C₂×C₈×C₇₃	—	4	1	no	—
35	C₂×C₈×C₇₃	—	4	4	no	—
36	C₂×C₈×C₇₃	—	4	2	no	—
37	C₂×C₈×C₇₃	—	4	1	no	—
38	C₄⋊C₄×C₇₃	—	4	2	no	—
39	C₄⋊C₄×C₇₃	—	4	1	no	—
40	C₄²×C₇₃	—	2	1	no	—
41	C₄²×C₇₃	—	2	2	no	—
42	C₄²×C₇₃	—	2	1	no	—
43	C₄²×C₇₃	—	2	2	no	—
44	C₂³×C₇₃	#95	1	1	yes	characteristic
45	Q₈×C₇₃	—	4	4	no	—
46	C₂×C₄×C₇₃	—	2	1	no	—
47	C₂×C₄×C₇₃	—	2	1	no	—
48	C₂×C₄×C₇₃	—	2	1	no	—
49	C₂×C₄×C₇₃	—	2	1	no	—
50	C₂×C₄×C₇₃	—	2	2	no	—
51	C₂×C₄×C₇₃	—	2	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	—	4	1	no	—
56	C₂×C₄×C₇₃	—	4	2	no	—
57	C₂×C₄×C₇₃	#99	1	1	yes	characteristic
58	C₂×C₄×C₇₃	#99	1	1	yes	characteristic
59	C₂×C₄×C₇₃	#108	1	2	yes	—
60	C₈×C₇₃	—	4	2	no	—
61	C₈×C₇₃	—	8	4	no	—
62	C₈×C₇₃	—	8	1	no	—
63	C₂²×C₇₃	—	2	1	no	—
64	C₂²×C₇₃	—	2	1	no	—
65	C₂²×C₇₃	#338	1	1	yes	characteristic
66	C₂²×C₇₃	#341	1	1	yes	characteristic
67	C₂²×C₇₃	#342	1	1	yes	characteristic
68	C₄×C₇₃	—	2	2	no	—
69	C₄×C₇₃	—	2	1	no	—
70	C₄×C₇₃	—	4	2	no	—
71	C₄×C₇₃	—	4	2	no	—
72	C₄×C₇₃	—	4	1	no	—
73	C₄×C₇₃	#355	1	1	yes	characteristic
74	C₄×C₇₃	#356	1	1	yes	characteristic
75	#7248	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂×C₇₃	—	2	1	no	—
77	C₂×C₇₃	—	2	1	no	—
78	C₂×C₇₃	#1293	1	1	yes	characteristic
79	C₂×C₇₃	#1391	1	1	yes	characteristic
80	C₂×C₇₃	#1394	1	1	yes	characteristic
81	#1984	C₂×C₇₃	1	1	yes	characteristic
82	#1467	C₂×C₇₃	1	2	yes	—
83	C₇₃	#7248	1	1	yes	characteristic, direct product factor, Sylow
84	C₂×C₄×Q₈	C₂²×C₇₃	1	1	yes	characteristic
85	#432	—	2	2	no	—
86	C₂×C₄⋊C₈	C₄×C₇₃	1	1	yes	characteristic
87	C₂×C₄×C₈	C₄×C₇₃	1	1	yes	characteristic
88	#336	—	2	2	no	—
89	#324	—	2	2	no	—
90	#324	C₄×C₇₃	1	4	yes	—
91	C₂²×Q₈	Q₈×C₇₃	1	1	yes	characteristic
92	C₂²×C₈	—	2	1	no	—
93	C₂²×C₈	—	2	1	no	—
94	C₄×Q₈	—	4	1	no	—
95	C₄×Q₈	—	4	1	no	—
96	C₂×C₄⋊C₄	C₂×C₄×C₇₃	1	2	yes	—
97	C₂×C₄⋊C₄	D₄×C₇₃	1	1	yes	characteristic
98	C₂×C₄²	C₂×C₄×C₇₃	1	1	yes	characteristic
99	C₂×C₄²	D₄×C₇₃	1	2	yes	—
100	C₄⋊C₈	—	4	1	no	—
101	C₄⋊C₈	—	4	2	no	—
102	C₂²⋊C₈	—	2	4	no	—
103	#97	—	4	2	no	—
104	C₄×C₈	—	4	1	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	4	1	no	—
107	C₂×Q₈	—	4	1	no	—
108	C₂²×C₄	C₄²×C₇₃	1	1	yes	characteristic
109	C₂²×C₄	C₂²⋊C₄×C₇₃	1	1	yes	characteristic
110	C₂²×C₄	C₂²⋊C₄×C₇₃	1	2	yes	—
111	C₂²×C₄	C₄⋊C₄×C₇₃	1	2	yes	—
112	C₂²×C₄	C₄⋊C₄×C₇₃	1	1	yes	characteristic
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	4	1	no	—
115	C₂×C₈	—	4	1	no	—
116	C₂×C₈	—	4	4	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	4	1	no	—
119	C₄⋊C₄	—	4	2	no	—
120	C₄⋊C₄	—	4	1	no	—
121	C₄²	—	2	1	no	—
122	C₄²	—	2	2	no	—
123	C₄²	—	2	1	no	—
124	C₄²	—	2	2	no	—
125	C₂³	#95×C₇₃	1	1	yes	characteristic
126	Q₈	—	4	4	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	#99×C₇₃	1	1	yes	characteristic
139	C₂×C₄	#99×C₇₃	1	1	yes	characteristic
140	C₂×C₄	#108×C₇₃	1	2	yes	—
141	C₈	—	4	2	no	—
142	C₈	—	8	4	no	—
143	C₈	—	8	1	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	#338×C₇₃	1	1	yes	characteristic
147	C₂²	#341×C₇₃	1	1	yes	characteristic
148	C₂²	#342×C₇₃	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	1	no	—
151	C₄	—	4	2	no	—
152	C₄	—	4	2	no	—
153	C₄	—	4	1	no	—
154	C₄	#355×C₇₃	1	1	yes	characteristic
155	C₄	#356×C₇₃	1	1	yes	characteristic
156	C₂	—	2	1	no	—
157	C₂	—	2	1	no	—
158	C₂	C₇₃×#1293	1	1	yes	characteristic
159	C₂	C₇₃×#1391	1	1	yes	characteristic
160	C₂	C₇₃×#1394	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃×#1984

C₂

yes

maximal, characteristic

C₇₃×#1467

C₂

yes

maximal

C₂×C₄×Q₈×C₇₃

C₂²

yes

characteristic

#432×C₇₃

—

C₂×C₄⋊C₈×C₇₃

C₄

yes

characteristic

C₂×C₄×C₈×C₇₃

C₄

yes

characteristic

#336×C₇₃

—

#324×C₇₃

—

#324×C₇₃

C₄

yes

—

C₂²×Q₈×C₇₃

Q₈

yes

characteristic

C₂²×C₈×C₇₃

—

C₂²×C₈×C₇₃

—

C₄×Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

C₂×C₄⋊C₄×C₇₃

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₇₃

D₄

yes

characteristic

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

—

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

—

C₂²⋊C₈×C₇₃

—

#97×C₇₃

—

C₄×C₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

C₄²

yes

characteristic

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

C₄⋊C₄

yes

—

C₂²×C₄×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

#95

yes

characteristic

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

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

C₂×C₄×C₇₃

yes

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

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

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

characteristic

#1467

C₂×C₇₃

yes

—

C₇₃

#7248

yes

characteristic, direct product factor, Sylow

C₂×C₄×Q₈

C₂²×C₇₃

yes

characteristic

#432

—

C₂×C₄⋊C₈

C₄×C₇₃

yes

characteristic

C₂×C₄×C₈

C₄×C₇₃

yes

—

—

C₄×C₇₃

yes

—

C₂²×Q₈

Q₈×C₇₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

C₂×C₄×C₇₃

yes

—

C₂×C₄⋊C₄

D₄×C₇₃

yes

characteristic

C₂×C₄²

C₂×C₄×C₇₃

yes

characteristic

C₂×C₄²

D₄×C₇₃

yes

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

C₂²⋊C₈

—

103

#97

—

104

C₄×C₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂²×C₄

C₄²×C₇₃

yes

characteristic

109

C₂²×C₄

C₂²⋊C₄×C₇₃

yes

characteristic

110

C₂²×C₄

C₂²⋊C₄×C₇₃

yes

—

111

C₂²×C₄

C₄⋊C₄×C₇₃

yes

—

112

C₂²×C₄

C₄⋊C₄×C₇₃

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄²

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

125

C₂³

#95×C₇₃

yes

characteristic

126

Q₈

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

#99×C₇₃

yes

characteristic

139

C₂×C₄

#99×C₇₃

yes

characteristic

140

C₂×C₄

#108×C₇₃

yes

—

141

C₈

—

142

C₈

—

143

C₈

—

144

C₂²

—

145

C₂²

—

146

C₂²

#338×C₇₃

yes

characteristic

147

C₂²

#341×C₇₃

yes

characteristic

148

C₂²

#342×C₇₃

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

#355×C₇₃

yes

characteristic

155

C₄

#356×C₇₃

yes

characteristic

156

C₂

—

157

C₂

—

158

C₂

C₇₃×#1293

yes

characteristic

159

C₂

C₇₃×#1391

yes

characteristic

160

C₂

C₇₃×#1394

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₇₃	#342
commutator	C₂²×C₄	C₄²×C₇₃
Frattini, cosocle	C₂×C₄×Q₈	C₂²×C₇₃
prime radical	C₂³×C₇₃	#95
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ C₂×C₄×Q₈×C₇₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₃

#342

commutator

C₂²×C₄

C₄²×C₇₃

Frattini, cosocle

C₂×C₄×Q₈

C₂²×C₇₃

prime radical

C₂³×C₇₃

#95

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ C₂×C₄×Q₈×C₇₃ ◁ G

N	H ≅ G/N	type
75. #7248	C₇₃	direct (N × H)
83. C₇₃	#7248	direct (N × H)

H ≅ G/N

type

75. #7248

C₇₃

direct (N × H)

83. C₇₃

#7248

direct (N × H)

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

subgroup	index
C₇₃×#1984	2
C₇₃×#1467	2
#7248	73

subgroup

index

C₇₃×#1984

C₇₃×#1467

#7248

Subgroups — C₇₃×#7248 #32916652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#7248 #32916652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#7248 #32916652