Group #32923047 — subgroups

Subgroups — C₇₃×#13643 #32923047

Order 18688.

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

normal subgroups only

This group has 164 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×#2108	C₂	1	1	yes	maximal, characteristic
2	C₇₃×#1422	C₂	1	2	yes	maximal, complemented
3	C₇₃×#1400	C₂	1	2	yes	maximal
4	C₇₃×#1398	C₂	1	2	yes	maximal, complemented
5	C₂×M₃₂×C₇₃	C₂²	1	1	yes	characteristic
6	C₂²×C₁₆×C₇₃	C₄	1	1	yes	characteristic
7	#405×C₇₃	C₄	1	1	yes	characteristic
8	M₆₄×C₇₃	C₂²	1	4	yes	—
9	#346×C₇₃	C₄	1	1	yes	characteristic
10	#346×C₇₃	C₂²	1	1	yes	characteristic
11	C₄×C₁₆×C₇₃	C₄	1	1	yes	characteristic
12	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₇₃	C₈	1	1	yes	characteristic
14	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
15	#117×C₇₃	C₈	1	1	yes	characteristic
16	M₃₂×C₇₃	—	2	2	no	—
17	C₂×C₁₆×C₇₃	—	2	2	no	—
18	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
22	#97×C₇₃	C₈	1	1	yes	characteristic
23	#97×C₇₃	C₈	1	1	yes	characteristic
24	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
26	C₃₂×C₇₃	—	2	4	no	—
27	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
28	M₁₆×C₇₃	—	2	2	no	—
29	C₂×C₈×C₇₃	—	2	1	no	—
30	C₂×C₈×C₇₃	—	2	1	no	—
31	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
35	C₄⋊C₄×C₇₃	—	2	1	no	—
36	C₂²⋊C₄×C₇₃	—	2	1	no	—
37	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
38	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
39	C₁₆×C₇₃	—	2	1	no	—
40	C₁₆×C₇₃	—	2	1	no	—
41	C₁₆×C₇₃	M₁₆	1	2	yes	—
42	C₁₆×C₇₃	#41	1	2	yes	—
43	C₂³×C₇₃	M₃₂	1	1	yes	characteristic
44	C₂×C₄×C₇₃	—	2	1	no	—
45	C₂×C₄×C₇₃	—	2	1	no	—
46	C₂×C₄×C₇₃	—	2	1	no	—
47	C₂×C₄×C₇₃	—	2	1	no	—
48	C₂×C₄×C₇₃	M₃₂	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
50	C₈×C₇₃	—	2	1	no	—
51	C₈×C₇₃	—	2	1	no	—
52	C₈×C₇₃	M₃₂	1	2	yes	—
53	C₈×C₇₃	#117	1	1	yes	characteristic
54	C₈×C₇₃	C₂×M₁₆	1	1	yes	characteristic
55	C₂²×C₇₃	—	2	1	no	—
56	C₂²×C₇₃	—	4	1	no	—
57	C₂²×C₇₃	C₂×M₃₂	1	1	yes	characteristic
58	C₄×C₇₃	—	2	1	no	—
59	C₄×C₇₃	—	4	1	no	—
60	C₄×C₇₃	—	4	1	no	—
61	C₄×C₇₃	#431	1	1	yes	characteristic
62	C₄×C₇₃	C₂×M₃₂	1	1	yes	characteristic
63	#13643	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂×C₇₃	—	2	1	no	—
65	C₂×C₇₃	—	4	1	no	—
66	C₂×C₇₃	#2163	1	1	yes	characteristic
67	#2108	C₂×C₇₃	1	1	yes	characteristic
68	#1422	C₂×C₇₃	1	2	yes	complemented
69	#1400	C₂×C₇₃	1	2	yes	—
70	#1398	C₂×C₇₃	1	2	yes	complemented
71	C₇₃	#13643	1	1	yes	characteristic, direct product factor, Sylow
72	C₂×M₃₂	C₂²×C₇₃	1	1	yes	characteristic
73	C₂²×C₁₆	C₄×C₇₃	1	1	yes	characteristic
74	#405	C₄×C₇₃	1	1	yes	characteristic
75	M₆₄	C₂²×C₇₃	1	4	yes	—
76	#346	C₄×C₇₃	1	1	yes	characteristic
77	#346	C₂²×C₇₃	1	1	yes	characteristic
78	C₄×C₁₆	C₄×C₇₃	1	1	yes	characteristic
79	C₄×C₁₆	C₂²×C₇₃	1	1	yes	characteristic
80	C₂×M₁₆	C₈×C₇₃	1	1	yes	characteristic
81	C₂²×C₈	C₂×C₄×C₇₃	1	1	yes	characteristic
82	#117	C₈×C₇₃	1	1	yes	characteristic
83	M₃₂	—	2	2	no	—
84	C₂×C₁₆	—	2	2	no	—
85	C₂×C₁₆	C₂×C₄×C₇₃	1	1	yes	characteristic
86	C₂×C₁₆	C₂×C₄×C₇₃	1	1	yes	characteristic
87	C₂×C₁₆	C₂×C₄×C₇₃	1	1	yes	characteristic
88	C₂×C₁₆	C₂³×C₇₃	1	1	yes	characteristic
89	#97	C₈×C₇₃	1	1	yes	characteristic
90	#97	C₈×C₇₃	1	1	yes	characteristic
91	C₄×C₈	C₂×C₄×C₇₃	1	1	yes	characteristic
92	C₄×C₈	C₂×C₄×C₇₃	1	1	yes	characteristic
93	C₃₂	—	2	4	no	—
94	C₂²×C₄	C₂×C₈×C₇₃	1	1	yes	characteristic
95	M₁₆	—	2	2	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	C₂×C₈×C₇₃	1	1	yes	characteristic
99	C₂×C₈	C₂×C₈×C₇₃	1	1	yes	characteristic
100	C₂×C₈	C₂×C₈×C₇₃	1	1	yes	characteristic
101	C₂×C₈	C₂²×C₄×C₇₃	1	1	yes	characteristic
102	C₄⋊C₄	—	2	1	no	—
103	C₂²⋊C₄	—	2	1	no	—
104	C₄²	C₂×C₈×C₇₃	1	1	yes	characteristic
105	C₄²	C₂×C₈×C₇₃	1	1	yes	characteristic
106	C₁₆	—	2	1	no	—
107	C₁₆	—	2	1	no	—
108	C₁₆	M₁₆×C₇₃	1	2	yes	—
109	C₁₆	#41×C₇₃	1	2	yes	—
110	C₂³	M₃₂×C₇₃	1	1	yes	characteristic
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	M₃₂×C₇₃	1	1	yes	characteristic
116	C₂×C₄	C₂²×C₈×C₇₃	1	1	yes	characteristic
117	C₈	—	2	1	no	—
118	C₈	—	2	1	no	—
119	C₈	M₃₂×C₇₃	1	2	yes	—
120	C₈	#117×C₇₃	1	1	yes	characteristic
121	C₈	C₂×M₁₆×C₇₃	1	1	yes	characteristic
122	C₂²	—	2	1	no	—
123	C₂²	—	4	1	no	—
124	C₂²	C₂×M₃₂×C₇₃	1	1	yes	characteristic
125	C₄	—	2	1	no	—
126	C₄	—	4	1	no	—
127	C₄	—	4	1	no	—
128	C₄	#431×C₇₃	1	1	yes	characteristic
129	C₄	C₂×M₃₂×C₇₃	1	1	yes	characteristic
130	C₂	—	2	1	no	—
131	C₂	—	4	1	no	—
132	C₂	C₇₃×#2163	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃×#2108

C₂

yes

maximal, characteristic

C₇₃×#1422

C₂

yes

maximal, complemented

C₇₃×#1400

C₂

yes

maximal

C₇₃×#1398

C₂

yes

maximal, complemented

C₂×M₃₂×C₇₃

C₂²

yes

characteristic

C₂²×C₁₆×C₇₃

C₄

yes

characteristic

#405×C₇₃

C₄

yes

characteristic

M₆₄×C₇₃

C₂²

yes

—

#346×C₇₃

C₄

yes

characteristic

#346×C₇₃

C₂²

yes

characteristic

C₄×C₁₆×C₇₃

C₄

yes

characteristic

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×M₁₆×C₇₃

C₈

yes

characteristic

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

#117×C₇₃

C₈

yes

characteristic

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂³

yes

characteristic

#97×C₇₃

C₈

yes

characteristic

#97×C₇₃

C₈

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₃₂×C₇₃

—

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

—

C₁₆×C₇₃

M₁₆

yes

—

C₁₆×C₇₃

#41

yes

—

C₂³×C₇₃

M₃₂

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

M₃₂

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

M₃₂

yes

—

C₈×C₇₃

#117

yes

characteristic

C₈×C₇₃

C₂×M₁₆

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×M₃₂

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#431

yes

characteristic

C₄×C₇₃

C₂×M₃₂

yes

characteristic

#13643

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#2163

yes

characteristic

#2108

C₂×C₇₃

yes

characteristic

#1422

C₂×C₇₃

yes

complemented

#1400

C₂×C₇₃

yes

—

#1398

C₂×C₇₃

yes

complemented

C₇₃

#13643

yes

characteristic, direct product factor, Sylow

C₂×M₃₂

C₂²×C₇₃

yes

characteristic

C₂²×C₁₆

C₄×C₇₃

yes

characteristic

#405

C₄×C₇₃

yes

characteristic

M₆₄

C₂²×C₇₃

yes

—

#346

C₄×C₇₃

yes

characteristic

#346

C₂²×C₇₃

yes

characteristic

C₄×C₁₆

C₄×C₇₃

yes

characteristic

C₄×C₁₆

C₂²×C₇₃

yes

characteristic

C₂×M₁₆

C₈×C₇₃

yes

characteristic

C₂²×C₈

C₂×C₄×C₇₃

yes

characteristic

#117

C₈×C₇₃

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄×C₇₃

yes

characteristic

C₂×C₁₆

C₂×C₄×C₇₃

yes

characteristic

C₂×C₁₆

C₂×C₄×C₇₃

yes

characteristic

C₂×C₁₆

C₂³×C₇₃

yes

characteristic

#97

C₈×C₇₃

yes

characteristic

#97

C₈×C₇₃

yes

characteristic

C₄×C₈

C₂×C₄×C₇₃

yes

characteristic

C₄×C₈

C₂×C₄×C₇₃

yes

characteristic

C₃₂

—

C₂²×C₄

C₂×C₈×C₇₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₈×C₇₃

yes

characteristic

C₂×C₈

C₂×C₈×C₇₃

yes

characteristic

100

C₂×C₈

C₂×C₈×C₇₃

yes

characteristic

101

C₂×C₈

C₂²×C₄×C₇₃

yes

characteristic

102

C₄⋊C₄

—

103

C₂²⋊C₄

—

104

C₄²

C₂×C₈×C₇₃

yes

characteristic

105

C₄²

C₂×C₈×C₇₃

yes

characteristic

106

C₁₆

—

107

C₁₆

—

108

C₁₆

M₁₆×C₇₃

yes

—

109

C₁₆

#41×C₇₃

yes

—

110

C₂³

M₃₂×C₇₃

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

M₃₂×C₇₃

yes

characteristic

116

C₂×C₄

C₂²×C₈×C₇₃

yes

characteristic

117

C₈

—

118

C₈

—

119

C₈

M₃₂×C₇₃

yes

—

120

C₈

#117×C₇₃

yes

characteristic

121

C₈

C₂×M₁₆×C₇₃

yes

characteristic

122

C₂²

—

123

C₂²

—

124

C₂²

C₂×M₃₂×C₇₃

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

#431×C₇₃

yes

characteristic

129

C₄

C₂×M₃₂×C₇₃

yes

characteristic

130

C₂

—

131

C₂

—

132

C₂

C₇₃×#2163

yes

characteristic

	subgroup	G/H
center	C₈×C₇₃	#117
commutator	C₂×C₄	C₂²×C₈×C₇₃
Frattini, cosocle	C₂×C₁₆	C₂³×C₇₃
socle	C₂×C₇₃	#2163
prime radical	C₂³×C₇₃	M₃₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₇₃ ◁ #346×C₇₃ ◁ G

subgroup

G/H

center

C₈×C₇₃

#117

commutator

C₂×C₄

C₂²×C₈×C₇₃

Frattini, cosocle

C₂×C₁₆

C₂³×C₇₃

socle

C₂×C₇₃

#2163

prime radical

C₂³×C₇₃

M₃₂

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₈×C₇₃ ◁ #346×C₇₃ ◁ G

N	H ≅ G/N	type
2. C₇₃×#1422	C₂	semidirect
4. C₇₃×#1398	C₂	semidirect
63. #13643	C₇₃	direct (N × H)
68. #1422	C₂×C₇₃	semidirect
70. #1398	C₂×C₇₃	semidirect
71. C₇₃	#13643	direct (N × H)

H ≅ G/N

type

2. C₇₃×#1422

C₂

semidirect

4. C₇₃×#1398

C₂

semidirect

63. #13643

C₇₃

direct (N × H)

68. #1422

C₂×C₇₃

semidirect

70. #1398

C₂×C₇₃

semidirect

71. C₇₃

#13643

direct (N × H)

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

subgroup	index
C₇₃×#2108	2
C₇₃×#1422	2
C₇₃×#1400	2
C₇₃×#1398	2
#13643	73

subgroup

index

C₇₃×#2108

C₇₃×#1422

C₇₃×#1400

C₇₃×#1398

#13643

Subgroups — C₇₃×#13643 #32923047

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#13643 #32923047

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#13643 #32923047