Group #32916616 — subgroups

Subgroups — C₇₃×#7212 #32916616

Order 18688.

212 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2162	C₂	1	1	yes	maximal, characteristic
2	C₇₃×#2150	C₂	1	1	yes	maximal, characteristic
3	C₈×M₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×M₃₂×C₇₃	C₄	1	2	yes	—
5	#431×C₇₃	C₄	1	1	yes	characteristic
6	C₄×M₁₆×C₇₃	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
8	C₄⋊C₁₆×C₇₃	—	2	1	no	—
9	C₄⋊C₁₆×C₇₃	—	2	1	no	—
10	C₄⋊C₁₆×C₇₃	—	2	1	no	—
11	C₂²⋊C₁₆×C₇₃	C₄	1	2	yes	—
12	C₄×C₁₆×C₇₃	—	2	1	no	—
13	#322×C₇₃	—	2	1	no	—
14	C₈²×C₇₃	—	2	1	no	—
15	C₂×M₁₆×C₇₃	—	2	1	no	—
16	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
19	M₃₂×C₇₃	—	4	2	no	—
20	C₂×C₁₆×C₇₃	—	2	2	no	—
21	C₂×C₁₆×C₇₃	—	2	2	no	—
22	C₄⋊C₈×C₇₃	C₈	1	2	yes	—
23	C₂²⋊C₈×C₇₃	—	2	1	no	—
24	#97×C₇₃	C₈	1	2	yes	—
25	C₄×C₈×C₇₃	—	2	1	no	—
26	C₄×C₈×C₇₃	—	2	1	no	—
27	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
28	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
31	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
32	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
33	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
34	M₁₆×C₇₃	—	2	4	no	—
35	C₂×C₈×C₇₃	—	2	2	no	—
36	C₂×C₈×C₇₃	—	2	1	no	—
37	C₂×C₈×C₇₃	—	2	1	no	—
38	C₂×C₈×C₇₃	—	2	2	no	—
39	C₂×C₈×C₇₃	—	4	1	no	—
40	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
46	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
47	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
48	C₁₆×C₇₃	—	4	2	no	—
49	C₁₆×C₇₃	—	4	2	no	—
50	C₂³×C₇₃	#97	1	1	yes	characteristic
51	C₂×C₄×C₇₃	—	2	1	no	—
52	C₂×C₄×C₇₃	—	2	1	no	—
53	C₂×C₄×C₇₃	—	2	2	no	—
54	C₂×C₄×C₇₃	—	2	2	no	—
55	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
56	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
57	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
58	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
61	C₈×C₇₃	—	2	4	no	—
62	C₈×C₇₃	—	2	1	no	—
63	C₈×C₇₃	—	2	1	no	—
64	C₈×C₇₃	—	2	1	no	—
65	C₈×C₇₃	—	4	2	no	—
66	C₈×C₇₃	#100	1	1	yes	characteristic
67	C₈×C₇₃	#101	1	1	yes	characteristic
68	C₂²×C₇₃	—	2	1	no	—
69	C₂²×C₇₃	—	2	1	no	—
70	C₂²×C₇₃	—	2	1	no	—
71	C₂²×C₇₃	#336	1	1	yes	characteristic
72	C₄×C₇₃	—	2	1	no	—
73	C₄×C₇₃	—	2	1	no	—
74	C₄×C₇₃	—	2	1	no	—
75	C₄×C₇₃	#343	1	1	yes	characteristic
76	C₄×C₇₃	#344	1	1	yes	characteristic
77	C₄×C₇₃	#350	1	2	yes	—
78	C₄×C₇₃	#364	1	2	yes	—
79	#7212	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂×C₇₃	—	4	1	no	—
81	C₂×C₇₃	#1283	1	1	yes	characteristic
82	C₂×C₇₃	#1376	1	1	yes	characteristic
83	C₂×C₇₃	#1379	1	1	yes	characteristic
84	#2162	C₂×C₇₃	1	1	yes	characteristic
85	#2150	C₂×C₇₃	1	1	yes	characteristic
86	C₈×M₁₆	C₂×C₇₃	1	1	yes	characteristic
87	C₇₃	#7212	1	1	yes	characteristic, direct product factor, Sylow
88	C₂×M₃₂	C₄×C₇₃	1	2	yes	—
89	#431	C₄×C₇₃	1	1	yes	characteristic
90	C₄×M₁₆	C₄×C₇₃	1	1	yes	characteristic
91	C₂×C₄×C₈	C₂²×C₇₃	1	1	yes	characteristic
92	C₄⋊C₁₆	—	2	1	no	—
93	C₄⋊C₁₆	—	2	1	no	—
94	C₄⋊C₁₆	—	2	1	no	—
95	C₂²⋊C₁₆	C₄×C₇₃	1	2	yes	—
96	C₄×C₁₆	—	2	1	no	—
97	#322	—	2	1	no	—
98	C₈²	—	2	1	no	—
99	C₂×M₁₆	—	2	1	no	—
100	C₂²×C₈	C₂×C₄×C₇₃	1	1	yes	characteristic
101	C₂²×C₈	C₂×C₄×C₇₃	1	1	yes	characteristic
102	C₂×C₄²	C₂×C₄×C₇₃	1	1	yes	characteristic
103	M₃₂	—	4	2	no	—
104	C₂×C₁₆	—	2	2	no	—
105	C₂×C₁₆	—	2	2	no	—
106	C₄⋊C₈	C₈×C₇₃	1	2	yes	—
107	C₂²⋊C₈	—	2	1	no	—
108	#97	C₈×C₇₃	1	2	yes	—
109	C₄×C₈	—	2	1	no	—
110	C₄×C₈	—	2	1	no	—
111	C₄×C₈	D₄×C₇₃	1	1	yes	characteristic
112	C₄×C₈	D₄×C₇₃	1	1	yes	characteristic
113	C₄×C₈	D₄×C₇₃	1	1	yes	characteristic
114	C₄×C₈	Q₈×C₇₃	1	1	yes	characteristic
115	C₂²×C₄	C₄²×C₇₃	1	1	yes	characteristic
116	C₂²×C₄	M₁₆×C₇₃	1	1	yes	characteristic
117	C₂²×C₄	M₁₆×C₇₃	1	1	yes	characteristic
118	M₁₆	—	2	4	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	4	1	no	—
124	C₂×C₈	C₂²⋊C₄×C₇₃	1	1	yes	characteristic
125	C₂×C₈	C₂²⋊C₄×C₇₃	1	1	yes	characteristic
126	C₂×C₈	C₄⋊C₄×C₇₃	1	1	yes	characteristic
127	C₂×C₈	C₄⋊C₄×C₇₃	1	1	yes	characteristic
128	C₄²	C₂²⋊C₄×C₇₃	1	1	yes	characteristic
129	C₄²	C₄⋊C₄×C₇₃	1	1	yes	characteristic
130	C₄²	C₂×C₈×C₇₃	1	1	yes	characteristic
131	C₄²	C₂×C₈×C₇₃	1	1	yes	characteristic
132	C₁₆	—	4	2	no	—
133	C₁₆	—	4	2	no	—
134	C₂³	#97×C₇₃	1	1	yes	characteristic
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	#95×C₇₃	1	1	yes	characteristic
140	C₂×C₄	C₄×C₈×C₇₃	1	1	yes	characteristic
141	C₂×C₄	C₂²⋊C₈×C₇₃	1	1	yes	characteristic
142	C₂×C₄	C₂²⋊C₈×C₇₃	1	1	yes	characteristic
143	C₂×C₄	C₄⋊C₈×C₇₃	1	1	yes	characteristic
144	C₂×C₄	C₄⋊C₈×C₇₃	1	1	yes	characteristic
145	C₈	—	2	4	no	—
146	C₈	—	2	1	no	—
147	C₈	—	2	1	no	—
148	C₈	—	2	1	no	—
149	C₈	—	4	2	no	—
150	C₈	#100×C₇₃	1	1	yes	characteristic
151	C₈	#101×C₇₃	1	1	yes	characteristic
152	C₂²	—	2	1	no	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	#336×C₇₃	1	1	yes	characteristic
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	#343×C₇₃	1	1	yes	characteristic
160	C₄	#344×C₇₃	1	1	yes	characteristic
161	C₄	#350×C₇₃	1	2	yes	—
162	C₄	#364×C₇₃	1	2	yes	—
163	C₂	—	4	1	no	—
164	C₂	C₇₃×#1283	1	1	yes	characteristic
165	C₂	C₇₃×#1376	1	1	yes	characteristic
166	C₂	C₇₃×#1379	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃×#2162

C₂

yes

maximal, characteristic

C₇₃×#2150

C₂

yes

maximal, characteristic

C₈×M₁₆×C₇₃

C₂

yes

maximal, characteristic

C₂×M₃₂×C₇₃

C₄

yes

—

#431×C₇₃

C₄

yes

characteristic

C₄×M₁₆×C₇₃

C₄

yes

characteristic

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₃

—

C₄⋊C₁₆×C₇₃

—

C₄⋊C₁₆×C₇₃

—

C₂²⋊C₁₆×C₇₃

C₄

yes

—

C₄×C₁₆×C₇₃

—

#322×C₇₃

—

C₈²×C₇₃

—

C₂×M₁₆×C₇₃

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

—

C₄⋊C₈×C₇₃

C₈

yes

—

C₂²⋊C₈×C₇₃

—

#97×C₇₃

C₈

yes

—

C₄×C₈×C₇₃

—

C₄×C₈×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

Q₈

yes

characteristic

C₂²×C₄×C₇₃

C₄²

yes

characteristic

C₂²×C₄×C₇₃

M₁₆

yes

characteristic

C₂²×C₄×C₇₃

M₁₆

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₄²×C₇₃

C₂²⋊C₄

yes

characteristic

C₄²×C₇₃

C₄⋊C₄

yes

characteristic

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

—

C₂³×C₇₃

#97

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#95

yes

characteristic

C₂×C₄×C₇₃

C₄×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₃

C₄⋊C₈

yes

characteristic

C₂×C₄×C₇₃

C₄⋊C₈

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

yes

C₈×C₇₃

yes

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#336

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

C₄×C₇₃

#364

yes

—

#7212

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

characteristic

#2150

C₂×C₇₃

yes

characteristic

C₈×M₁₆

C₂×C₇₃

yes

characteristic

C₇₃

#7212

yes

characteristic, direct product factor, Sylow

C₂×M₃₂

C₄×C₇₃

yes

—

#431

C₄×C₇₃

yes

characteristic

C₄×M₁₆

C₄×C₇₃

yes

characteristic

C₂×C₄×C₈

C₂²×C₇₃

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

C₄×C₇₃

yes

—

C₄×C₁₆

—

#322

—

C₈²

—

C₂×M₁₆

—

100

C₂²×C₈

C₂×C₄×C₇₃

yes

characteristic

101

C₂²×C₈

C₂×C₄×C₇₃

yes

characteristic

102

C₂×C₄²

C₂×C₄×C₇₃

yes

characteristic

103

M₃₂

—

104

C₂×C₁₆

—

105

C₂×C₁₆

—

106

C₄⋊C₈

C₈×C₇₃

yes

—

107

C₂²⋊C₈

—

108

#97

C₈×C₇₃

yes

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

D₄×C₇₃

yes

characteristic

112

C₄×C₈

D₄×C₇₃

yes

characteristic

113

C₄×C₈

D₄×C₇₃

yes

characteristic

114

C₄×C₈

Q₈×C₇₃

yes

characteristic

115

C₂²×C₄

C₄²×C₇₃

yes

characteristic

116

C₂²×C₄

M₁₆×C₇₃

yes

characteristic

117

C₂²×C₄

M₁₆×C₇₃

yes

characteristic

118

M₁₆

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

C₂²⋊C₄×C₇₃

yes

characteristic

125

C₂×C₈

C₂²⋊C₄×C₇₃

yes

characteristic

126

C₂×C₈

C₄⋊C₄×C₇₃

yes

characteristic

127

C₂×C₈

C₄⋊C₄×C₇₃

yes

characteristic

128

C₄²

C₂²⋊C₄×C₇₃

yes

characteristic

129

C₄²

C₄⋊C₄×C₇₃

yes

characteristic

130

C₄²

C₂×C₈×C₇₃

yes

characteristic

131

C₄²

C₂×C₈×C₇₃

yes

characteristic

132

C₁₆

—

133

C₁₆

—

134

C₂³

#97×C₇₃

yes

characteristic

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

#95×C₇₃

yes

characteristic

140

C₂×C₄

C₄×C₈×C₇₃

yes

characteristic

141

C₂×C₄

C₂²⋊C₈×C₇₃

yes

characteristic

142

C₂×C₄

C₂²⋊C₈×C₇₃

yes

characteristic

143

C₂×C₄

C₄⋊C₈×C₇₃

yes

characteristic

144

C₂×C₄

C₄⋊C₈×C₇₃

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

#100×C₇₃

yes

characteristic

151

C₈

#101×C₇₃

yes

characteristic

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

#336×C₇₃

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

#343×C₇₃

yes

characteristic

160

C₄

#344×C₇₃

yes

characteristic

161

C₄

#350×C₇₃

yes

—

162

C₄

#364×C₇₃

yes

—

163

C₂

—

164

C₂

C₇₃×#1283

yes

characteristic

165

C₂

C₇₃×#1376

yes

characteristic

166

C₂

C₇₃×#1379

yes

characteristic

	subgroup	G/H
center	C₂×C₈×C₇₃	C₂²⋊C₄
commutator	C₂×C₄	C₄×C₈×C₇₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₇₃
socle	C₂²×C₇₃	#336
prime radical	C₂³×C₇₃	#97
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₇₃ ◁ C₂×C₄×C₈×C₇₃ ◁ G

subgroup

G/H

center

C₂×C₈×C₇₃

C₂²⋊C₄

commutator

C₂×C₄

C₄×C₈×C₇₃

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₇₃

C₂²×C₇₃

C₂³×C₇₃

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₈×C₇₃ ◁ C₂×C₄×C₈×C₇₃ ◁ G

N	H ≅ G/N	type
79. #7212	C₇₃	direct (N × H)
87. C₇₃	#7212	direct (N × H)

H ≅ G/N

type

79. #7212

C₇₃

direct (N × H)

87. C₇₃

#7212

direct (N × H)

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

subgroup	index
C₇₃×#2162	2
C₇₃×#2150	2
C₈×M₁₆×C₇₃	2
#7212	73

subgroup

index

C₇₃×#2162

C₇₃×#2150

C₈×M₁₆×C₇₃

#7212

Subgroups — C₇₃×#7212 #32916616

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#7212 #32916616

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#7212 #32916616