Group #32923119 — subgroups

Subgroups — C₇₃×#13715 #32923119

Order 18688.

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

normal subgroups only

This group has 176 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×#2216	C₂	1	1	yes	maximal, characteristic
2	C₂×#366×C₇₃	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#1423	C₂	1	2	yes	maximal, complemented
4	C₇₃×#1416	C₂	1	2	yes	maximal, complemented
5	C₇₃×C₂²⋊C₃₂	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₃₂×C₇₃	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
8	#467×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	#358×C₇₃	—	2	1	no	—
15	C₂×Q₁₆×C₇₃	—	2	1	no	—
16	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
17	C₂²⋊Q₈×C₇₃	—	4	1	no	—
18	C₂×C₄⋊C₄×C₇₃	—	4	1	no	—
19	Q₃₂×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	#103×C₇₃	—	4	1	no	—
27	C₃₂×C₇₃	—	2	2	no	—
28	C₂×Q₈×C₇₃	—	4	1	no	—
29	C₂²×C₄×C₇₃	—	8	1	no	—
30	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
31	Q₁₆×C₇₃	—	4	1	no	—
32	Q₁₆×C₇₃	—	4	1	no	—
33	C₂×C₈×C₇₃	—	2	1	no	—
34	C₂×C₈×C₇₃	—	2	1	no	—
35	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
36	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
37	C₄⋊C₄×C₇₃	—	4	1	no	—
38	C₄⋊C₄×C₇₃	—	4	2	no	—
39	C₄⋊C₄×C₇₃	—	8	1	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	D₁₆	1	1	yes	characteristic
45	Q₈×C₇₃	—	8	1	no	—
46	Q₈×C₇₃	—	8	1	no	—
47	C₂×C₄×C₇₃	—	2	1	no	—
48	C₂×C₄×C₇₃	—	2	1	no	—
49	C₂×C₄×C₇₃	—	8	1	no	—
50	C₂×C₄×C₇₃	—	8	1	no	—
51	C₂×C₄×C₇₃	—	8	2	no	—
52	C₂×C₄×C₇₃	—	16	2	no	—
53	C₂×C₄×C₇₃	D₁₆	1	1	yes	characteristic
54	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
55	C₈×C₇₃	—	2	1	no	—
56	C₈×C₇₃	#123	1	1	yes	characteristic
57	C₈×C₇₃	#137	1	1	yes	characteristic
58	C₂²×C₇₃	—	2	1	no	—
59	C₂²×C₇₃	—	2	1	no	—
60	C₂²×C₇₃	Q₆₄	1	2	yes	—
61	C₂²×C₇₃	C₂×D₁₆	1	1	yes	characteristic
62	C₄×C₇₃	—	2	1	no	—
63	C₄×C₇₃	—	16	1	no	—
64	C₄×C₇₃	—	16	1	no	—
65	C₄×C₇₃	—	16	2	no	—
66	C₄×C₇₃	#480	1	1	yes	characteristic
67	C₄×C₇₃	#510	1	1	yes	characteristic
68	#13715	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂×C₇₃	—	2	2	no	—
70	C₂×C₇₃	#2232	1	1	yes	characteristic
71	C₂×C₇₃	C₂×Q₆₄	1	1	yes	characteristic
72	C₂×C₇₃	#2263	1	1	yes	characteristic
73	#2216	C₂×C₇₃	1	1	yes	characteristic
74	C₂×#366	C₂×C₇₃	1	1	yes	characteristic
75	#1423	C₂×C₇₃	1	2	yes	complemented
76	#1416	C₂×C₇₃	1	2	yes	complemented
77	C₂²⋊C₃₂	C₂×C₇₃	1	1	yes	characteristic
78	C₇₃	#13715	1	1	yes	characteristic, direct product factor, Sylow
79	C₂×Q₃₂	C₂²×C₇₃	1	1	yes	characteristic
80	C₂²×C₁₆	C₂²×C₇₃	1	1	yes	characteristic
81	#467	—	2	1	no	—
82	C₂×#107	—	2	1	no	—
83	C₂×C₃₂	C₂²×C₇₃	1	2	yes	—
84	#366	—	2	1	no	—
85	#366	C₂²×C₇₃	1	2	yes	—
86	#366	C₂²×C₇₃	1	1	yes	characteristic
87	#358	—	2	1	no	—
88	C₂×Q₁₆	—	2	1	no	—
89	C₂²×C₈	D₄×C₇₃	1	1	yes	characteristic
90	C₂²⋊Q₈	—	4	1	no	—
91	C₂×C₄⋊C₄	—	4	1	no	—
92	Q₃₂	—	2	2	no	—
93	C₂×C₁₆	—	2	2	no	—
94	C₂×C₁₆	D₄×C₇₃	1	1	yes	characteristic
95	C₂×C₁₆	C₂³×C₇₃	1	1	yes	characteristic
96	#107	—	2	1	no	—
97	#107	—	2	2	no	—
98	#107	—	4	1	no	—
99	#103	—	4	1	no	—
100	C₃₂	—	2	2	no	—
101	C₂×Q₈	—	4	1	no	—
102	C₂²×C₄	—	8	1	no	—
103	C₂²×C₄	D₈×C₇₃	1	1	yes	characteristic
104	Q₁₆	—	4	1	no	—
105	Q₁₆	—	4	1	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	D₈×C₇₃	1	1	yes	characteristic
109	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
110	C₄⋊C₄	—	4	1	no	—
111	C₄⋊C₄	—	4	2	no	—
112	C₄⋊C₄	—	8	1	no	—
113	C₄⋊C₄	—	8	1	no	—
114	C₂²⋊C₄	—	8	1	no	—
115	C₁₆	—	2	1	no	—
116	C₁₆	#41×C₇₃	1	2	yes	—
117	C₂³	D₁₆×C₇₃	1	1	yes	characteristic
118	Q₈	—	8	1	no	—
119	Q₈	—	8	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	8	1	no	—
123	C₂×C₄	—	8	1	no	—
124	C₂×C₄	—	8	2	no	—
125	C₂×C₄	—	16	2	no	—
126	C₂×C₄	D₁₆×C₇₃	1	1	yes	characteristic
127	C₂×C₄	C₂×D₈×C₇₃	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	#123×C₇₃	1	1	yes	characteristic
130	C₈	#137×C₇₃	1	1	yes	characteristic
131	C₂²	—	2	1	no	—
132	C₂²	—	2	1	no	—
133	C₂²	Q₆₄×C₇₃	1	2	yes	—
134	C₂²	C₂×D₁₆×C₇₃	1	1	yes	characteristic
135	C₄	—	2	1	no	—
136	C₄	—	16	1	no	—
137	C₄	—	16	2	no	—
138	C₄	—	16	1	no	—
139	C₄	#480×C₇₃	1	1	yes	characteristic
140	C₄	#510×C₇₃	1	1	yes	characteristic
141	C₂	—	2	2	no	—
142	C₂	C₇₃×#2232	1	1	yes	characteristic
143	C₂	C₂×Q₆₄×C₇₃	1	1	yes	characteristic
144	C₂	C₇₃×#2263	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃×#2216

C₂

yes

maximal, characteristic

C₂×#366×C₇₃

C₂

yes

maximal, characteristic

C₇₃×#1423

C₂

yes

maximal, complemented

C₇₃×#1416

C₂

yes

maximal, complemented

C₇₃×C₂²⋊C₃₂

C₂

yes

maximal, characteristic

C₂×Q₃₂×C₇₃

C₂²

yes

characteristic

C₂²×C₁₆×C₇₃

C₂²

yes

characteristic

#467×C₇₃

—

C₂×#107×C₇₃

—

C₂×C₃₂×C₇₃

C₂²

yes

—

#366×C₇₃

—

#366×C₇₃

C₂²

yes

—

#366×C₇₃

C₂²

yes

characteristic

#358×C₇₃

—

C₂×Q₁₆×C₇₃

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²⋊Q₈×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

Q₃₂×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂³

yes

characteristic

#107×C₇₃

—

#107×C₇₃

—

#107×C₇₃

—

#103×C₇₃

—

C₃₂×C₇₃

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

D₈

yes

characteristic

Q₁₆×C₇₃

—

Q₁₆×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

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₁₆×C₇₃

—

C₁₆×C₇₃

#41

yes

—

C₂³×C₇₃

D₁₆

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×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

Q₆₄

yes

—

C₂²×C₇₃

C₂×D₁₆

yes

characteristic

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

#2232

yes

characteristic

C₂×C₇₃

C₂×Q₆₄

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

characteristic

C₂×#366

C₂×C₇₃

yes

characteristic

#1423

C₂×C₇₃

yes

complemented

#1416

C₂×C₇₃

yes

complemented

C₂²⋊C₃₂

C₂×C₇₃

yes

characteristic

C₇₃

#13715

yes

characteristic, direct product factor, Sylow

C₂×Q₃₂

C₂²×C₇₃

yes

characteristic

C₂²×C₁₆

C₂²×C₇₃

yes

characteristic

#467

—

C₂×#107

—

C₂×C₃₂

C₂²×C₇₃

yes

—

#366

—

#366

C₂²×C₇₃

yes

—

#366

C₂²×C₇₃

yes

characteristic

#358

—

C₂×Q₁₆

—

C₂²×C₈

D₄×C₇₃

yes

characteristic

C₂²⋊Q₈

—

C₂×C₄⋊C₄

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

D₄×C₇₃

yes

characteristic

C₂×C₁₆

C₂³×C₇₃

yes

—

—

—

—

100

C₃₂

—

101

C₂×Q₈

—

102

C₂²×C₄

—

103

C₂²×C₄

D₈×C₇₃

yes

characteristic

104

Q₁₆

—

105

Q₁₆

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

D₈×C₇₃

yes

characteristic

109

C₂×C₈

C₂×D₄×C₇₃

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₂²⋊C₄

—

115

C₁₆

—

116

C₁₆

#41×C₇₃

yes

—

117

C₂³

D₁₆×C₇₃

yes

characteristic

118

Q₈

—

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

D₁₆×C₇₃

yes

characteristic

127

C₂×C₄

C₂×D₈×C₇₃

yes

characteristic

128

C₈

—

129

C₈

#123×C₇₃

yes

characteristic

130

C₈

#137×C₇₃

yes

characteristic

131

C₂²

—

132

C₂²

—

133

C₂²

Q₆₄×C₇₃

yes

—

134

C₂²

C₂×D₁₆×C₇₃

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

#480×C₇₃

yes

characteristic

140

C₄

#510×C₇₃

yes

characteristic

141

C₂

—

142

C₂

C₇₃×#2232

yes

characteristic

143

C₂

C₂×Q₆₄×C₇₃

yes

characteristic

144

C₂

C₇₃×#2263

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₇₃	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₇₃
prime radical	C₂³×C₇₃	D₁₆
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

C₂³×C₇₃

D₁₆

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
3. C₇₃×#1423	C₂	semidirect
4. C₇₃×#1416	C₂	semidirect
68. #13715	C₇₃	direct (N × H)
75. #1423	C₂×C₇₃	semidirect
76. #1416	C₂×C₇₃	semidirect
78. C₇₃	#13715	direct (N × H)

H ≅ G/N

type

3. C₇₃×#1423

C₂

semidirect

4. C₇₃×#1416

C₂

semidirect

68. #13715

C₇₃

direct (N × H)

75. #1423

C₂×C₇₃

semidirect

76. #1416

C₂×C₇₃

semidirect

78. C₇₃

#13715

direct (N × H)

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

subgroup	index
C₇₃×#2216	2
C₂×#366×C₇₃	2
C₇₃×#1423	2
C₇₃×#1416	2
C₇₃×C₂²⋊C₃₂	2
#13715	73

subgroup

index

C₇₃×#2216

C₂×#366×C₇₃

C₇₃×#1423

C₇₃×#1416

C₇₃×C₂²⋊C₃₂

#13715

Subgroups — C₇₃×#13715 #32923119

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#13715 #32923119

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#13715 #32923119