Group #32923116 — subgroups

Subgroups — C₇₃×#13712 #32923116

Order 18688.

186 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2215	C₂	1	1	yes	maximal, characteristic
2	C₂×#366×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₃×#1424	C₂	1	2	yes	maximal, complemented
4	C₇₃×#1415	C₂	1	2	yes	maximal, complemented
5	C₇₃×C₂²⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₆×C₇₃	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
8	#466×C₇₃	—	2	1	no	—
9	C₂×#107×C₇₃	—	2	1	no	—
10	C₂×C₃₂×C₇₃	C₂²	1	2	yes	—
11	#366×C₇₃	—	2	1	no	—
12	#366×C₇₃	C₂²	1	2	yes	—
13	#366×C₇₃	C₂²	1	1	yes	characteristic
14	#357×C₇₃	—	2	1	no	—
15	C₂×D₈×C₇₃	—	2	1	no	—
16	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
17	#121×C₇₃	—	4	1	no	—
18	C₂×C₄⋊C₄×C₇₃	—	4	1	no	—
19	D₁₆×C₇₃	—	2	2	no	—
20	C₂×C₁₆×C₇₃	—	2	2	no	—
21	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
23	#107×C₇₃	—	2	2	no	—
24	#107×C₇₃	—	2	1	no	—
25	#107×C₇₃	—	4	1	no	—
26	#102×C₇₃	—	4	1	no	—
27	C₃₂×C₇₃	—	2	2	no	—
28	C₂×D₄×C₇₃	—	4	1	no	—
29	C₂×D₄×C₇₃	—	8	1	no	—
30	C₂²×C₄×C₇₃	—	8	1	no	—
31	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
32	D₈×C₇₃	—	4	1	no	—
33	D₈×C₇₃	—	4	1	no	—
34	C₂×C₈×C₇₃	—	2	1	no	—
35	C₂×C₈×C₇₃	—	2	1	no	—
36	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
37	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
38	C₄⋊C₄×C₇₃	—	4	1	no	—
39	C₄⋊C₄×C₇₃	—	4	2	no	—
40	C₄⋊C₄×C₇₃	—	8	1	no	—
41	C₂²⋊C₄×C₇₃	—	8	1	no	—
42	C₁₆×C₇₃	—	2	1	no	—
43	C₁₆×C₇₃	#41	1	2	yes	—
44	C₂³×C₇₃	—	8	1	no	—
45	C₂³×C₇₃	D₁₆	1	1	yes	characteristic
46	D₄×C₇₃	—	8	1	no	—
47	D₄×C₇₃	—	8	1	no	—
48	D₄×C₇₃	—	16	2	no	—
49	C₂×C₄×C₇₃	—	2	1	no	—
50	C₂×C₄×C₇₃	—	2	1	no	—
51	C₂×C₄×C₇₃	—	8	1	no	—
52	C₂×C₄×C₇₃	—	8	2	no	—
53	C₂×C₄×C₇₃	—	16	2	no	—
54	C₂×C₄×C₇₃	D₁₆	1	1	yes	characteristic
55	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
56	C₈×C₇₃	—	2	1	no	—
57	C₈×C₇₃	#123	1	1	yes	characteristic
58	C₈×C₇₃	#137	1	1	yes	characteristic
59	C₂²×C₇₃	—	2	1	no	—
60	C₂²×C₇₃	—	2	1	no	—
61	C₂²×C₇₃	—	16	1	no	—
62	C₂²×C₇₃	—	16	1	no	—
63	C₂²×C₇₃	—	16	1	no	—
64	C₂²×C₇₃	SD₆₄	1	2	yes	—
65	C₂²×C₇₃	C₂×D₁₆	1	1	yes	characteristic
66	C₄×C₇₃	—	2	1	no	—
67	C₄×C₇₃	—	16	1	no	—
68	C₄×C₇₃	—	16	2	no	—
69	C₄×C₇₃	#480	1	1	yes	characteristic
70	C₄×C₇₃	#510	1	1	yes	characteristic
71	#13712	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂×C₇₃	—	2	2	no	—
73	C₂×C₇₃	—	32	1	no	—
74	C₂×C₇₃	#2232	1	1	yes	characteristic
75	C₂×C₇₃	C₂×SD₆₄	1	1	yes	characteristic
76	C₂×C₇₃	#2263	1	1	yes	characteristic
77	#2215	C₂×C₇₃	1	1	yes	characteristic
78	C₂×#366	C₂×C₇₃	1	1	yes	characteristic, complemented
79	#1424	C₂×C₇₃	1	2	yes	complemented
80	#1415	C₂×C₇₃	1	2	yes	complemented
81	C₂²⋊C₃₂	C₂×C₇₃	1	1	yes	characteristic, complemented
82	C₇₃	#13712	1	1	yes	characteristic, direct product factor, Sylow
83	C₂×D₁₆	C₂²×C₇₃	1	1	yes	characteristic
84	C₂²×C₁₆	C₂²×C₇₃	1	1	yes	characteristic
85	#466	—	2	1	no	—
86	C₂×#107	—	2	1	no	—
87	C₂×C₃₂	C₂²×C₇₃	1	2	yes	—
88	#366	—	2	1	no	—
89	#366	C₂²×C₇₃	1	2	yes	—
90	#366	C₂²×C₇₃	1	1	yes	characteristic
91	#357	—	2	1	no	—
92	C₂×D₈	—	2	1	no	—
93	C₂²×C₈	D₄×C₇₃	1	1	yes	characteristic
94	#121	—	4	1	no	—
95	C₂×C₄⋊C₄	—	4	1	no	—
96	D₁₆	—	2	2	no	—
97	C₂×C₁₆	—	2	2	no	—
98	C₂×C₁₆	D₄×C₇₃	1	1	yes	characteristic
99	C₂×C₁₆	C₂³×C₇₃	1	1	yes	characteristic
100	#107	—	2	1	no	—
101	#107	—	2	2	no	—
102	#107	—	4	1	no	—
103	#102	—	4	1	no	—
104	C₃₂	—	2	2	no	—
105	C₂×D₄	—	4	1	no	—
106	C₂×D₄	—	8	1	no	—
107	C₂²×C₄	—	8	1	no	—
108	C₂²×C₄	D₈×C₇₃	1	1	yes	characteristic
109	D₈	—	4	1	no	—
110	D₈	—	4	1	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	D₈×C₇₃	1	1	yes	characteristic
114	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
115	C₄⋊C₄	—	4	1	no	—
116	C₄⋊C₄	—	4	2	no	—
117	C₄⋊C₄	—	8	1	no	—
118	C₂²⋊C₄	—	8	1	no	—
119	C₁₆	—	2	1	no	—
120	C₁₆	#41×C₇₃	1	2	yes	—
121	C₂³	—	8	1	no	—
122	C₂³	D₁₆×C₇₃	1	1	yes	characteristic
123	D₄	—	8	1	no	—
124	D₄	—	8	1	no	—
125	D₄	—	16	2	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	8	1	no	—
129	C₂×C₄	—	8	2	no	—
130	C₂×C₄	—	16	2	no	—
131	C₂×C₄	D₁₆×C₇₃	1	1	yes	characteristic
132	C₂×C₄	C₂×D₈×C₇₃	1	1	yes	characteristic
133	C₈	—	2	1	no	—
134	C₈	#123×C₇₃	1	1	yes	characteristic
135	C₈	#137×C₇₃	1	1	yes	characteristic
136	C₂²	—	2	1	no	—
137	C₂²	—	2	1	no	—
138	C₂²	—	16	1	no	—
139	C₂²	—	16	1	no	—
140	C₂²	—	16	1	no	—
141	C₂²	SD₆₄×C₇₃	1	2	yes	—
142	C₂²	C₂×D₁₆×C₇₃	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	16	1	no	—
145	C₄	—	16	2	no	—
146	C₄	#480×C₇₃	1	1	yes	characteristic
147	C₄	#510×C₇₃	1	1	yes	characteristic
148	C₂	—	2	2	no	—
149	C₂	—	32	1	no	—
150	C₂	C₇₃×#2232	1	1	yes	characteristic
151	C₂	C₂×SD₆₄×C₇₃	1	1	yes	characteristic
152	C₂	C₇₃×#2263	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃×#2215

C₂

yes

maximal, characteristic

C₂×#366×C₇₃

C₂

yes

maximal, characteristic, complemented

C₇₃×#1424

C₂

yes

maximal, complemented

C₇₃×#1415

C₂

yes

maximal, complemented

C₇₃×C₂²⋊C₃₂

C₂

yes

maximal, characteristic, complemented

C₂×D₁₆×C₇₃

C₂²

yes

characteristic

C₂²×C₁₆×C₇₃

C₂²

yes

characteristic

#466×C₇₃

—

C₂×#107×C₇₃

—

C₂×C₃₂×C₇₃

C₂²

yes

—

#366×C₇₃

—

#366×C₇₃

C₂²

yes

—

#366×C₇₃

C₂²

yes

characteristic

#357×C₇₃

—

C₂×D₈×C₇₃

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

#121×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

D₁₆×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂³

yes

characteristic

#107×C₇₃

—

#107×C₇₃

—

#107×C₇₃

—

#102×C₇₃

—

C₃₂×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

D₈

yes

characteristic

D₈×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₁₆×C₇₃

—

C₁₆×C₇₃

#41

yes

—

C₂³×C₇₃

—

C₂³×C₇₃

D₁₆

yes

characteristic

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

D₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

yes

C₈×C₇₃

yes

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

SD₆₄

yes

—

C₂²×C₇₃

C₂×D₁₆

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

#2232

yes

characteristic

C₂×C₇₃

C₂×SD₆₄

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

characteristic

C₂×#366

C₂×C₇₃

yes

characteristic, complemented

#1424

C₂×C₇₃

yes

complemented

#1415

C₂×C₇₃

yes

complemented

C₂²⋊C₃₂

C₂×C₇₃

yes

characteristic, complemented

C₇₃

#13712

yes

characteristic, direct product factor, Sylow

C₂×D₁₆

C₂²×C₇₃

yes

characteristic

C₂²×C₁₆

C₂²×C₇₃

yes

characteristic

#466

—

C₂×#107

—

C₂×C₃₂

C₂²×C₇₃

yes

—

#366

—

#366

C₂²×C₇₃

yes

—

#366

C₂²×C₇₃

yes

characteristic

#357

—

C₂×D₈

—

C₂²×C₈

D₄×C₇₃

yes

characteristic

#121

—

C₂×C₄⋊C₄

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₄×C₇₃

yes

characteristic

C₂×C₁₆

C₂³×C₇₃

yes

characteristic

100

#107

—

101

#107

—

102

#107

—

103

#102

—

104

C₃₂

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂²×C₄

—

108

C₂²×C₄

D₈×C₇₃

yes

characteristic

109

D₈

—

110

D₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

D₈×C₇₃

yes

characteristic

114

C₂×C₈

C₂×D₄×C₇₃

yes

characteristic

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₂²⋊C₄

—

119

C₁₆

—

120

C₁₆

#41×C₇₃

yes

—

121

C₂³

—

122

C₂³

D₁₆×C₇₃

yes

characteristic

123

D₄

—

124

D₄

—

125

D₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

D₁₆×C₇₃

yes

characteristic

132

C₂×C₄

C₂×D₈×C₇₃

yes

characteristic

133

C₈

—

134

C₈

#123×C₇₃

yes

characteristic

135

C₈

#137×C₇₃

yes

characteristic

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

SD₆₄×C₇₃

yes

—

142

C₂²

C₂×D₁₆×C₇₃

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

#480×C₇₃

yes

characteristic

147

C₄

#510×C₇₃

yes

characteristic

148

C₂

—

149

C₂

—

150

C₂

C₇₃×#2232

yes

characteristic

151

C₂

C₂×SD₆₄×C₇₃

yes

characteristic

152

C₂

C₇₃×#2263

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₇₃	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₇₃
prime radical	C₇₃×#2215	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ C₂²×C₈×C₇₃ ◁ C₂²×C₁₆×C₇₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₃

C₂×D₁₆

commutator, Frattini, cosocle

C₂×C₁₆

C₂³×C₇₃

prime radical

C₇₃×#2215

C₂

LCS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ C₂²×C₈×C₇₃ ◁ C₂²×C₁₆×C₇₃ ◁ G

N	H ≅ G/N	type
2. C₂×#366×C₇₃	C₂	semidirect
3. C₇₃×#1424	C₂	semidirect
4. C₇₃×#1415	C₂	semidirect
5. C₇₃×C₂²⋊C₃₂	C₂	semidirect
71. #13712	C₇₃	direct (N × H)
78. C₂×#366	C₂×C₇₃	semidirect
79. #1424	C₂×C₇₃	semidirect
80. #1415	C₂×C₇₃	semidirect
81. C₂²⋊C₃₂	C₂×C₇₃	semidirect
82. C₇₃	#13712	direct (N × H)

H ≅ G/N

type

2. C₂×#366×C₇₃

C₂

semidirect

3. C₇₃×#1424

C₂

semidirect

4. C₇₃×#1415

C₂

semidirect

5. C₇₃×C₂²⋊C₃₂

C₂

semidirect

71. #13712

C₇₃

direct (N × H)

78. C₂×#366

C₂×C₇₃

semidirect

79. #1424

C₂×C₇₃

semidirect

80. #1415

C₂×C₇₃

semidirect

81. C₂²⋊C₃₂

C₂×C₇₃

semidirect

82. C₇₃

#13712

direct (N × H)

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

subgroup	index
C₇₃×#2215	2
C₂×#366×C₇₃	2
C₇₃×#1424	2
C₇₃×#1415	2
C₇₃×C₂²⋊C₃₂	2
#13712	73

subgroup

index

C₇₃×#2215

C₂×#366×C₇₃

C₇₃×#1424

C₇₃×#1415

C₇₃×C₂²⋊C₃₂

#13712

Subgroups — C₇₃×#13712 #32923116

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#13712 #32923116

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#13712 #32923116