Group #32921902 — subgroups

Subgroups — C₇₃×#12498 #32921902

Order 18688.

212 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 82 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×#2225	C₂	1	2	yes	maximal
2	C₇₃×#1713	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#1591	C₂	1	1	yes	maximal, characteristic
4	C₇₃×#1367	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1364	C₂	1	2	yes	maximal
6	#500×C₇₃	—	2	1	no	—
7	#500×C₇₃	C₂²	1	2	yes	—
8	#494×C₇₃	—	2	1	no	—
9	#446×C₇₃	—	2	1	no	—
10	C₄×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
11	#367×C₇₃	—	2	2	no	—
12	C₄⋊C₁₆×C₇₃	C₂²	1	2	yes	—
13	#358×C₇₃	—	2	2	no	—
14	#335×C₇₃	C₂²	1	1	yes	characteristic
15	Q₈⋊C₈×C₇₃	—	2	1	no	—
16	C₈²×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₇₃	—	4	1	no	—
18	C₂×Q₁₆×C₇₃	—	4	1	no	—
19	C₂×Q₁₆×C₇₃	D₄	1	1	yes	characteristic
20	C₄⋊Q₈×C₇₃	—	2	2	no	—
21	C₄×Q₈×C₇₃	—	2	1	no	—
22	C₂×C₁₆×C₇₃	—	2	2	no	—
23	#107×C₇₃	—	2	2	no	—
24	#107×C₇₃	—	2	2	no	—
25	#107×C₇₃	—	4	2	no	—
26	#107×C₇₃	D₄	1	1	yes	characteristic
27	C₄⋊C₈×C₇₃	—	2	1	no	—
28	C₄⋊C₈×C₇₃	—	4	1	no	—
29	#103×C₇₃	—	2	1	no	—
30	#97×C₇₃	—	4	1	no	—
31	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
33	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
34	C₂×Q₈×C₇₃	—	2	1	no	—
35	C₂×Q₈×C₇₃	—	4	2	no	—
36	Q₁₆×C₇₃	—	2	2	no	—
37	Q₁₆×C₇₃	—	8	2	no	—
38	Q₁₆×C₇₃	—	8	2	no	—
39	C₂×C₈×C₇₃	—	2	1	no	—
40	C₂×C₈×C₇₃	—	4	1	no	—
41	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
42	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
43	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
45	C₄⋊C₄×C₇₃	—	2	1	no	—
46	C₄⋊C₄×C₇₃	—	4	1	no	—
47	C₄⋊C₄×C₇₃	—	4	2	no	—
48	C₄⋊C₄×C₇₃	—	4	2	no	—
49	C₄²×C₇₃	—	4	1	no	—
50	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
51	C₁₆×C₇₃	—	4	2	no	—
52	Q₈×C₇₃	—	4	1	no	—
53	Q₈×C₇₃	—	4	1	no	—
54	Q₈×C₇₃	—	8	2	no	—
55	Q₈×C₇₃	—	8	2	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	AGL(1,ℤ/8)	1	1	yes	characteristic
62	C₈×C₇₃	—	2	2	no	—
63	C₈×C₇₃	—	2	1	no	—
64	C₈×C₇₃	—	2	1	no	—
65	C₈×C₇₃	—	8	1	no	—
66	C₈×C₇₃	SD₃₂	1	2	yes	—
67	C₈×C₇₃	#137	1	2	yes	—
68	C₂²×C₇₃	#459	1	1	yes	characteristic
69	C₄×C₇₃	—	2	1	no	—
70	C₄×C₇₃	—	8	1	no	—
71	C₄×C₇₃	—	8	1	no	—
72	C₄×C₇₃	—	16	2	no	—
73	C₄×C₇₃	#470	1	1	yes	characteristic
74	C₄×C₇₃	#473	1	1	yes	characteristic
75	C₄×C₇₃	C₂×SD₃₂	1	1	yes	characteristic
76	C₄×C₇₃	#509	1	1	yes	characteristic
77	#12498	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂×C₇₃	#1689	1	1	yes	characteristic
79	C₂×C₇₃	#2207	1	1	yes	characteristic
80	C₂×C₇₃	#2214	1	1	yes	characteristic
81	#2225	C₂×C₇₃	1	2	yes	—
82	#1713	C₂×C₇₃	1	1	yes	characteristic
83	#1591	C₂×C₇₃	1	1	yes	characteristic
84	#1367	C₂×C₇₃	1	1	yes	characteristic
85	#1364	C₂×C₇₃	1	2	yes	—
86	C₇₃	#12498	1	1	yes	characteristic, direct product factor, Sylow
87	#500	—	2	1	no	—
88	#500	C₂²×C₇₃	1	2	yes	—
89	#494	—	2	1	no	—
90	#446	—	2	1	no	—
91	C₄×Q₁₆	C₂²×C₇₃	1	1	yes	characteristic
92	#367	—	2	2	no	—
93	C₄⋊C₁₆	C₂²×C₇₃	1	2	yes	—
94	#358	—	2	2	no	—
95	#335	C₂²×C₇₃	1	1	yes	characteristic
96	Q₈⋊C₈	—	2	1	no	—
97	C₈²	C₂²×C₇₃	1	1	yes	characteristic
98	C₂×Q₁₆	—	4	1	no	—
99	C₂×Q₁₆	—	4	1	no	—
100	C₂×Q₁₆	D₄×C₇₃	1	1	yes	characteristic
101	C₄⋊Q₈	—	2	2	no	—
102	C₄×Q₈	—	2	1	no	—
103	C₂×C₁₆	—	2	2	no	—
104	#107	—	2	2	no	—
105	#107	—	2	2	no	—
106	#107	—	4	2	no	—
107	#107	D₄×C₇₃	1	1	yes	characteristic
108	C₄⋊C₈	—	2	1	no	—
109	C₄⋊C₈	—	4	1	no	—
110	#103	—	2	1	no	—
111	#97	—	4	1	no	—
112	C₄×C₈	D₄×C₇₃	1	1	yes	characteristic
113	C₄×C₈	D₄×C₇₃	1	1	yes	characteristic
114	C₄×C₈	C₂³×C₇₃	1	1	yes	characteristic
115	C₂×Q₈	—	2	1	no	—
116	C₂×Q₈	—	4	2	no	—
117	Q₁₆	—	2	2	no	—
118	Q₁₆	—	8	2	no	—
119	Q₁₆	—	8	2	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	4	1	no	—
122	C₂×C₈	D₈×C₇₃	1	1	yes	characteristic
123	C₂×C₈	D₈×C₇₃	1	1	yes	characteristic
124	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
125	C₂×C₈	#41×C₇₃	1	1	yes	characteristic
126	C₄⋊C₄	—	2	1	no	—
127	C₄⋊C₄	—	4	1	no	—
128	C₄⋊C₄	—	4	2	no	—
129	C₄⋊C₄	—	4	2	no	—
130	C₄²	—	4	1	no	—
131	C₄²	C₂×D₄×C₇₃	1	1	yes	characteristic
132	C₁₆	—	4	2	no	—
133	Q₈	—	4	1	no	—
134	Q₈	—	4	1	no	—
135	Q₈	—	8	2	no	—
136	Q₈	—	8	2	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	—	8	2	no	—
140	C₂×C₄	#121×C₇₃	1	1	yes	characteristic
141	C₂×C₄	C₂×D₈×C₇₃	1	1	yes	characteristic
142	C₂×C₄	AGL(1,ℤ/8)×C₇₃	1	1	yes	characteristic
143	C₈	—	2	2	no	—
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	8	1	no	—
147	C₈	SD₃₂×C₇₃	1	2	yes	—
148	C₈	#137×C₇₃	1	2	yes	—
149	C₂²	#459×C₇₃	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	8	1	no	—
152	C₄	—	8	1	no	—
153	C₄	—	16	2	no	—
154	C₄	#470×C₇₃	1	1	yes	characteristic
155	C₄	#473×C₇₃	1	1	yes	characteristic
156	C₄	C₂×SD₃₂×C₇₃	1	1	yes	characteristic
157	C₄	#509×C₇₃	1	1	yes	characteristic
158	C₂	C₇₃×#1689	1	1	yes	characteristic
159	C₂	C₇₃×#2207	1	1	yes	characteristic
160	C₂	C₇₃×#2214	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃×#2225

C₂

yes

maximal

C₇₃×#1713

C₂

yes

maximal, characteristic

C₇₃×#1591

C₂

yes

maximal, characteristic

C₇₃×#1367

C₂

yes

maximal, characteristic

C₇₃×#1364

C₂

yes

maximal

#500×C₇₃

—

#500×C₇₃

C₂²

yes

—

#494×C₇₃

—

#446×C₇₃

—

C₄×Q₁₆×C₇₃

C₂²

yes

characteristic

#367×C₇₃

—

C₄⋊C₁₆×C₇₃

C₂²

yes

—

#358×C₇₃

—

#335×C₇₃

C₂²

yes

characteristic

Q₈⋊C₈×C₇₃

—

C₈²×C₇₃

C₂²

yes

characteristic

C₂×Q₁₆×C₇₃

—

C₂×Q₁₆×C₇₃

—

C₂×Q₁₆×C₇₃

D₄

yes

characteristic

C₄⋊Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

C₂×C₁₆×C₇₃

—

#107×C₇₃

—

#107×C₇₃

—

#107×C₇₃

—

#107×C₇₃

D₄

yes

characteristic

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

—

#103×C₇₃

—

#97×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

characteristic

C₂×C₈×C₇₃

D₈

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

#41

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×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

AGL(1,ℤ/8)

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

SD₃₂

yes

—

C₈×C₇₃

#137

yes

—

C₂²×C₇₃

#459

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

C₂×SD₃₂

yes

C₄×C₇₃

yes

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

#1713

C₂×C₇₃

yes

characteristic

#1591

C₂×C₇₃

yes

characteristic

#1367

C₂×C₇₃

yes

characteristic

#1364

C₂×C₇₃

yes

—

C₇₃

#12498

yes

characteristic, direct product factor, Sylow

#500

—

#500

C₂²×C₇₃

yes

—

#494

—

#446

—

C₄×Q₁₆

C₂²×C₇₃

yes

characteristic

#367

—

C₄⋊C₁₆

C₂²×C₇₃

yes

—

#358

—

#335

C₂²×C₇₃

yes

characteristic

Q₈⋊C₈

—

C₈²

C₂²×C₇₃

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

100

C₂×Q₁₆

D₄×C₇₃

yes

characteristic

101

C₄⋊Q₈

—

102

C₄×Q₈

—

103

C₂×C₁₆

—

104

#107

—

105

#107

—

106

#107

—

107

#107

D₄×C₇₃

yes

characteristic

108

C₄⋊C₈

—

109

C₄⋊C₈

—

110

#103

—

111

#97

—

112

C₄×C₈

D₄×C₇₃

yes

characteristic

113

C₄×C₈

D₄×C₇₃

yes

characteristic

114

C₄×C₈

C₂³×C₇₃

yes

characteristic

115

C₂×Q₈

—

116

C₂×Q₈

—

117

Q₁₆

—

118

Q₁₆

—

119

Q₁₆

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

D₈×C₇₃

yes

characteristic

123

C₂×C₈

D₈×C₇₃

yes

characteristic

124

C₂×C₈

C₂×D₄×C₇₃

yes

characteristic

125

C₂×C₈

#41×C₇₃

yes

characteristic

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄²

—

131

C₄²

C₂×D₄×C₇₃

yes

characteristic

132

C₁₆

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

136

Q₈

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

#121×C₇₃

yes

characteristic

141

C₂×C₄

C₂×D₈×C₇₃

yes

characteristic

142

C₂×C₄

AGL(1,ℤ/8)×C₇₃

yes

characteristic

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

SD₃₂×C₇₃

yes

—

148

C₈

#137×C₇₃

yes

—

149

C₂²

#459×C₇₃

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

#470×C₇₃

yes

characteristic

155

C₄

#473×C₇₃

yes

characteristic

156

C₄

C₂×SD₃₂×C₇₃

yes

characteristic

157

C₄

#509×C₇₃

yes

characteristic

158

C₂

C₇₃×#1689

yes

characteristic

159

C₂

C₇₃×#2207

yes

characteristic

160

C₂

C₇₃×#2214

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₇₃	#459
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇₃
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₄²×C₇₃ ◁ C₈²×C₇₃ ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₇₃

#459

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₇₃

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₇₃ ◁ C₄²×C₇₃ ◁ C₈²×C₇₃ ◁ G

N	H ≅ G/N	type
77. #12498	C₇₃	direct (N × H)
86. C₇₃	#12498	direct (N × H)

H ≅ G/N

type

77. #12498

C₇₃

direct (N × H)

86. C₇₃

#12498

direct (N × H)

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

subgroup	index
C₇₃×#2225	2
C₇₃×#1713	2
C₇₃×#1591	2
C₇₃×#1367	2
C₇₃×#1364	2
#12498	73

subgroup

index

C₇₃×#2225

C₇₃×#1713

C₇₃×#1591

C₇₃×#1367

C₇₃×#1364

#12498

Subgroups — C₇₃×#12498 #32921902

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#12498 #32921902

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#12498 #32921902