Group #32921898 — subgroups

Subgroups — C₇₃×#12494 #32921898

Order 18688.

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

normal subgroups only

This group has 224 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×#2209	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×#1362	C₂	1	2	yes	maximal
6	C₂×Q₃₂×C₇₃	—	2	2	no	—
7	#500×C₇₃	—	2	1	no	—
8	#500×C₇₃	—	2	1	no	—
9	#494×C₇₃	C₂²	1	2	yes	—
10	#446×C₇₃	—	2	1	no	—
11	C₄×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
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	—	2	2	no	—
18	C₂×Q₁₆×C₇₃	—	2	2	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	Q₃₂×C₇₃	—	4	4	no	—
23	C₂×C₁₆×C₇₃	—	2	2	no	—
24	#107×C₇₃	—	4	2	no	—
25	#107×C₇₃	—	4	1	no	—
26	#107×C₇₃	—	4	1	no	—
27	#107×C₇₃	D₄	1	1	yes	characteristic
28	C₄⋊C₈×C₇₃	—	2	1	no	—
29	C₄⋊C₈×C₇₃	—	4	1	no	—
30	#103×C₇₃	—	2	1	no	—
31	#97×C₇₃	—	4	1	no	—
32	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
33	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
34	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
35	C₂×Q₈×C₇₃	—	2	1	no	—
36	C₂×Q₈×C₇₃	—	4	2	no	—
37	Q₁₆×C₇₃	—	2	2	no	—
38	Q₁₆×C₇₃	—	4	2	no	—
39	Q₁₆×C₇₃	—	4	2	no	—
40	Q₁₆×C₇₃	—	8	2	no	—
41	C₂×C₈×C₇₃	—	2	1	no	—
42	C₂×C₈×C₇₃	—	4	1	no	—
43	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
44	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
45	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
47	C₄⋊C₄×C₇₃	—	2	1	no	—
48	C₄⋊C₄×C₇₃	—	4	1	no	—
49	C₄⋊C₄×C₇₃	—	4	2	no	—
50	C₄⋊C₄×C₇₃	—	4	2	no	—
51	C₄²×C₇₃	—	4	1	no	—
52	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
53	C₁₆×C₇₃	—	4	2	no	—
54	Q₈×C₇₃	—	4	1	no	—
55	Q₈×C₇₃	—	4	1	no	—
56	Q₈×C₇₃	—	8	2	no	—
57	Q₈×C₇₃	—	8	2	no	—
58	C₂×C₄×C₇₃	—	4	1	no	—
59	C₂×C₄×C₇₃	—	4	1	no	—
60	C₂×C₄×C₇₃	—	8	2	no	—
61	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
62	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₈×C₇₃	—	2	2	no	—
65	C₈×C₇₃	—	2	1	no	—
66	C₈×C₇₃	—	2	1	no	—
67	C₈×C₇₃	—	8	1	no	—
68	C₈×C₇₃	Q₃₂	1	2	yes	complemented
69	C₈×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
70	C₂²×C₇₃	#459	1	1	yes	characteristic
71	C₄×C₇₃	—	2	1	no	—
72	C₄×C₇₃	—	8	1	no	—
73	C₄×C₇₃	—	8	1	no	—
74	C₄×C₇₃	—	16	2	no	—
75	C₄×C₇₃	#469	1	1	yes	characteristic
76	C₄×C₇₃	#472	1	1	yes	characteristic
77	C₄×C₇₃	C₂×Q₃₂	1	1	yes	characteristic
78	C₄×C₇₃	#510	1	1	yes	characteristic
79	#12494	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂×C₇₃	#1687	1	1	yes	characteristic
81	C₂×C₇₃	#2209	1	1	yes	characteristic
82	C₂×C₇₃	#2214	1	1	yes	characteristic
83	#2209	C₂×C₇₃	1	2	yes	—
84	#1713	C₂×C₇₃	1	1	yes	characteristic
85	#1591	C₂×C₇₃	1	1	yes	characteristic
86	#1367	C₂×C₇₃	1	1	yes	characteristic
87	#1362	C₂×C₇₃	1	2	yes	—
88	C₇₃	#12494	1	1	yes	characteristic, direct product factor, Sylow
89	C₂×Q₃₂	—	2	2	no	—
90	#500	—	2	1	no	—
91	#500	—	2	1	no	—
92	#494	C₂²×C₇₃	1	2	yes	—
93	#446	—	2	1	no	—
94	C₄×Q₁₆	C₂²×C₇₃	1	1	yes	characteristic
95	C₄⋊C₁₆	C₂²×C₇₃	1	2	yes	—
96	#358	—	2	2	no	—
97	#335	C₂²×C₇₃	1	1	yes	characteristic
98	Q₈⋊C₈	—	2	1	no	—
99	C₈²	C₂²×C₇₃	1	1	yes	characteristic
100	C₂×Q₁₆	—	2	2	no	—
101	C₂×Q₁₆	—	2	2	no	—
102	C₂×Q₁₆	D₄×C₇₃	1	1	yes	characteristic
103	C₄⋊Q₈	—	2	2	no	—
104	C₄×Q₈	—	2	1	no	—
105	Q₃₂	—	4	4	no	—
106	C₂×C₁₆	—	2	2	no	—
107	#107	—	4	2	no	—
108	#107	—	4	1	no	—
109	#107	—	4	1	no	—
110	#107	D₄×C₇₃	1	1	yes	characteristic
111	C₄⋊C₈	—	2	1	no	—
112	C₄⋊C₈	—	4	1	no	—
113	#103	—	2	1	no	—
114	#97	—	4	1	no	—
115	C₄×C₈	D₄×C₇₃	1	1	yes	characteristic
116	C₄×C₈	D₄×C₇₃	1	1	yes	characteristic
117	C₄×C₈	C₂³×C₇₃	1	1	yes	characteristic
118	C₂×Q₈	—	2	1	no	—
119	C₂×Q₈	—	4	2	no	—
120	Q₁₆	—	2	2	no	—
121	Q₁₆	—	4	2	no	—
122	Q₁₆	—	4	2	no	—
123	Q₁₆	—	8	2	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	4	1	no	—
126	C₂×C₈	D₈×C₇₃	1	1	yes	characteristic
127	C₂×C₈	D₈×C₇₃	1	1	yes	characteristic
128	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
129	C₂×C₈	#41×C₇₃	1	1	yes	characteristic
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	—	4	1	no	—
132	C₄⋊C₄	—	4	2	no	—
133	C₄⋊C₄	—	4	2	no	—
134	C₄²	—	4	1	no	—
135	C₄²	C₂×D₄×C₇₃	1	1	yes	characteristic
136	C₁₆	—	4	2	no	—
137	Q₈	—	4	1	no	—
138	Q₈	—	4	1	no	—
139	Q₈	—	8	2	no	—
140	Q₈	—	8	2	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	8	2	no	—
144	C₂×C₄	#121×C₇₃	1	1	yes	characteristic
145	C₂×C₄	C₂×D₈×C₇₃	1	1	yes	characteristic
146	C₂×C₄	AGL(1,ℤ/8)×C₇₃	1	1	yes	characteristic
147	C₈	—	2	2	no	—
148	C₈	—	2	1	no	—
149	C₈	—	2	1	no	—
150	C₈	—	8	1	no	—
151	C₈	Q₃₂×C₇₃	1	2	yes	complemented
152	C₈	AGL(1,ℤ/8)×C₇₃	1	2	yes	—
153	C₂²	#459×C₇₃	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	8	1	no	—
156	C₄	—	8	1	no	—
157	C₄	—	16	2	no	—
158	C₄	#469×C₇₃	1	1	yes	characteristic
159	C₄	#472×C₇₃	1	1	yes	characteristic
160	C₄	C₂×Q₃₂×C₇₃	1	1	yes	characteristic
161	C₄	#510×C₇₃	1	1	yes	characteristic
162	C₂	C₇₃×#1687	1	1	yes	characteristic
163	C₂	C₇₃×#2209	1	1	yes	characteristic
164	C₂	C₇₃×#2214	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃×#2209

C₂

yes

maximal

C₇₃×#1713

C₂

yes

maximal, characteristic

C₇₃×#1591

C₂

yes

maximal, characteristic

C₇₃×#1367

C₂

yes

maximal, characteristic

C₇₃×#1362

C₂

yes

maximal

C₂×Q₃₂×C₇₃

—

#500×C₇₃

—

#500×C₇₃

—

#494×C₇₃

C₂²

yes

—

#446×C₇₃

—

C₄×Q₁₆×C₇₃

C₂²

yes

characteristic

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

—

Q₃₂×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

—

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

Q₃₂

yes

complemented

C₈×C₇₃

AGL(1,ℤ/8)

yes

—

C₂²×C₇₃

#459

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

C₂×Q₃₂

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

#1362

C₂×C₇₃

yes

—

C₇₃

#12494

yes

characteristic, direct product factor, Sylow

C₂×Q₃₂

—

#500

—

#500

—

#494

C₂²×C₇₃

yes

—

#446

—

C₄×Q₁₆

C₂²×C₇₃

yes

characteristic

C₄⋊C₁₆

C₂²×C₇₃

yes

—

#358

—

#335

C₂²×C₇₃

yes

characteristic

Q₈⋊C₈

—

C₈²

C₂²×C₇₃

yes

characteristic

100

C₂×Q₁₆

—

101

C₂×Q₁₆

—

102

C₂×Q₁₆

D₄×C₇₃

yes

characteristic

103

C₄⋊Q₈

—

104

C₄×Q₈

—

105

Q₃₂

—

106

C₂×C₁₆

—

107

#107

—

108

#107

—

109

#107

—

110

#107

D₄×C₇₃

yes

characteristic

111

C₄⋊C₈

—

112

C₄⋊C₈

—

113

#103

—

114

#97

—

115

C₄×C₈

D₄×C₇₃

yes

characteristic

116

C₄×C₈

D₄×C₇₃

yes

characteristic

117

C₄×C₈

C₂³×C₇₃

yes

characteristic

118

C₂×Q₈

—

119

C₂×Q₈

—

120

Q₁₆

—

121

Q₁₆

—

122

Q₁₆

—

123

Q₁₆

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

D₈×C₇₃

yes

characteristic

127

C₂×C₈

D₈×C₇₃

yes

characteristic

128

C₂×C₈

C₂×D₄×C₇₃

yes

characteristic

129

C₂×C₈

#41×C₇₃

yes

characteristic

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄²

—

135

C₄²

C₂×D₄×C₇₃

yes

characteristic

136

C₁₆

—

137

Q₈

—

138

Q₈

—

139

Q₈

—

140

Q₈

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

#121×C₇₃

yes

characteristic

145

C₂×C₄

C₂×D₈×C₇₃

yes

characteristic

146

C₂×C₄

AGL(1,ℤ/8)×C₇₃

yes

characteristic

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

Q₃₂×C₇₃

yes

complemented

152

C₈

AGL(1,ℤ/8)×C₇₃

yes

—

153

C₂²

#459×C₇₃

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

#469×C₇₃

yes

characteristic

159

C₄

#472×C₇₃

yes

characteristic

160

C₄

C₂×Q₃₂×C₇₃

yes

characteristic

161

C₄

#510×C₇₃

yes

characteristic

162

C₂

C₇₃×#1687

yes

characteristic

163

C₂

C₇₃×#2209

yes

characteristic

164

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
68. C₈×C₇₃	Q₃₂	semidirect
79. #12494	C₇₃	direct (N × H)
88. C₇₃	#12494	direct (N × H)
151. C₈	Q₃₂×C₇₃	semidirect

H ≅ G/N

type

68. C₈×C₇₃

Q₃₂

semidirect

79. #12494

C₇₃

direct (N × H)

88. C₇₃

#12494

direct (N × H)

151. C₈

Q₃₂×C₇₃

semidirect

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

subgroup	index
C₇₃×#2209	2
C₇₃×#1713	2
C₇₃×#1591	2
C₇₃×#1367	2
C₇₃×#1362	2
#12494	73

subgroup

index

C₇₃×#2209

C₇₃×#1713

C₇₃×#1591

C₇₃×#1367

C₇₃×#1362

#12494

Subgroups — C₇₃×#12494 #32921898

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#12494 #32921898

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#12494 #32921898