Group #32923046 — subgroups

Subgroups — C₇₃×#13642 #32923046

Order 18688.

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

normal subgroups only

This group has 170 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	C₄×M₃₂×C₇₃	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	C₄	1	1	yes	characteristic
6	C₂×M₃₂×C₇₃	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₈×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	C₂²×C₈×C₇₃	C₈	1	1	yes	characteristic
14	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₇₃	C₈	1	1	yes	characteristic
16	M₃₂×C₇₃	—	2	2	no	—
17	M₃₂×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	C₄×C₈×C₇₃	C₈	1	1	yes	characteristic
23	C₄×C₈×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	—	2	1	no	—
28	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
29	C₂×C₈×C₇₃	—	2	2	no	—
30	C₂×C₈×C₇₃	—	2	1	no	—
31	C₂×C₈×C₇₃	—	2	1	no	—
32	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
36	C₄²×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	—	4	2	no	—
49	C₂×C₄×C₇₃	M₃₂	1	1	yes	characteristic
50	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
51	C₈×C₇₃	—	2	1	no	—
52	C₈×C₇₃	—	2	1	no	—
53	C₈×C₇₃	M₃₂	1	2	yes	—
54	C₈×C₇₃	#117	1	1	yes	characteristic
55	C₈×C₇₃	C₂×M₁₆	1	1	yes	characteristic
56	C₂²×C₇₃	—	2	1	no	—
57	C₂²×C₇₃	—	4	1	no	—
58	C₂²×C₇₃	C₂×M₃₂	1	1	yes	characteristic
59	C₄×C₇₃	—	2	2	no	—
60	C₄×C₇₃	—	2	1	no	—
61	C₄×C₇₃	—	4	1	no	—
62	C₄×C₇₃	#431	1	1	yes	characteristic
63	C₄×C₇₃	C₂×M₃₂	1	1	yes	characteristic
64	#13642	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂×C₇₃	—	2	1	no	—
66	C₂×C₇₃	—	4	1	no	—
67	C₂×C₇₃	#2163	1	1	yes	characteristic
68	C₄×M₃₂	C₂×C₇₃	1	1	yes	characteristic
69	#1422	C₂×C₇₃	1	2	yes	complemented
70	#1400	C₂×C₇₃	1	2	yes	—
71	#1398	C₂×C₇₃	1	2	yes	complemented
72	C₇₃	#13642	1	1	yes	characteristic, direct product factor, Sylow
73	C₂×M₃₂	C₄×C₇₃	1	1	yes	characteristic
74	C₂×M₃₂	C₂²×C₇₃	1	1	yes	characteristic
75	C₂×C₄×C₈	C₄×C₇₃	1	1	yes	characteristic
76	M₆₄	C₂²×C₇₃	1	4	yes	—
77	#346	C₄×C₇₃	1	1	yes	characteristic
78	#346	C₂²×C₇₃	1	1	yes	characteristic
79	C₄×C₁₆	C₄×C₇₃	1	1	yes	characteristic
80	C₄×C₁₆	C₂²×C₇₃	1	1	yes	characteristic
81	C₂²×C₈	C₈×C₇₃	1	1	yes	characteristic
82	C₂²×C₈	C₂×C₄×C₇₃	1	1	yes	characteristic
83	C₂×C₄²	C₈×C₇₃	1	1	yes	characteristic
84	M₃₂	—	2	2	no	—
85	M₃₂	—	2	2	no	—
86	C₂×C₁₆	C₂×C₄×C₇₃	1	1	yes	characteristic
87	C₂×C₁₆	C₂×C₄×C₇₃	1	1	yes	characteristic
88	C₂×C₁₆	C₂×C₄×C₇₃	1	1	yes	characteristic
89	C₂×C₁₆	C₂³×C₇₃	1	1	yes	characteristic
90	C₄×C₈	C₈×C₇₃	1	1	yes	characteristic
91	C₄×C₈	C₈×C₇₃	1	1	yes	characteristic
92	C₄×C₈	C₂×C₄×C₇₃	1	1	yes	characteristic
93	C₄×C₈	C₂×C₄×C₇₃	1	1	yes	characteristic
94	C₃₂	—	2	4	no	—
95	C₂²×C₄	—	2	1	no	—
96	C₂²×C₄	C₂×C₈×C₇₃	1	1	yes	characteristic
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	C₂×C₈×C₇₃	1	1	yes	characteristic
101	C₂×C₈	C₂×C₈×C₇₃	1	1	yes	characteristic
102	C₂×C₈	C₂×C₈×C₇₃	1	1	yes	characteristic
103	C₂×C₈	C₂²×C₄×C₇₃	1	1	yes	characteristic
104	C₄²	—	2	1	no	—
105	C₄²	C₂×C₈×C₇₃	1	1	yes	characteristic
106	C₄²	C₂×C₈×C₇₃	1	1	yes	characteristic
107	C₁₆	—	2	1	no	—
108	C₁₆	—	2	1	no	—
109	C₁₆	M₁₆×C₇₃	1	2	yes	—
110	C₁₆	#41×C₇₃	1	2	yes	—
111	C₂³	M₃₂×C₇₃	1	1	yes	characteristic
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	M₃₂×C₇₃	1	1	yes	characteristic
118	C₂×C₄	C₂²×C₈×C₇₃	1	1	yes	characteristic
119	C₈	—	2	1	no	—
120	C₈	—	2	1	no	—
121	C₈	M₃₂×C₇₃	1	2	yes	—
122	C₈	#117×C₇₃	1	1	yes	characteristic
123	C₈	C₂×M₁₆×C₇₃	1	1	yes	characteristic
124	C₂²	—	2	1	no	—
125	C₂²	—	4	1	no	—
126	C₂²	C₂×M₃₂×C₇₃	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	2	1	no	—
129	C₄	—	4	1	no	—
130	C₄	#431×C₇₃	1	1	yes	characteristic
131	C₄	C₂×M₃₂×C₇₃	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	4	1	no	—
134	C₂	C₇₃×#2163	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×M₃₂×C₇₃

C₂

yes

maximal, characteristic

C₇₃×#1422

C₂

yes

maximal, complemented

C₇₃×#1400

C₂

yes

maximal

C₇₃×#1398

C₂

yes

maximal, complemented

C₂×M₃₂×C₇₃

C₄

yes

characteristic

C₂×M₃₂×C₇₃

C₂²

yes

characteristic

C₂×C₄×C₈×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

C₂²×C₈×C₇₃

C₈

yes

characteristic

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₃

C₈

yes

characteristic

M₃₂×C₇₃

—

M₃₂×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

C₄×C₈×C₇₃

C₈

yes

characteristic

C₄×C₈×C₇₃

C₈

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₃₂×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×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

C₄²×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

—

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

#13642

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#2163

yes

characteristic

C₄×M₃₂

C₂×C₇₃

yes

characteristic

#1422

C₂×C₇₃

yes

complemented

#1400

C₂×C₇₃

yes

—

#1398

C₂×C₇₃

yes

complemented

C₇₃

#13642

yes

characteristic, direct product factor, Sylow

C₂×M₃₂

C₄×C₇₃

yes

characteristic

C₂×M₃₂

C₂²×C₇₃

yes

characteristic

C₂×C₄×C₈

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

C₂²×C₈

C₈×C₇₃

yes

characteristic

C₂²×C₈

C₂×C₄×C₇₃

yes

characteristic

C₂×C₄²

C₈×C₇₃

yes

characteristic

M₃₂

—

M₃₂

—

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

C₄×C₈

C₈×C₇₃

yes

characteristic

C₄×C₈

C₈×C₇₃

yes

characteristic

C₄×C₈

C₂×C₄×C₇₃

yes

characteristic

C₄×C₈

C₂×C₄×C₇₃

yes

characteristic

C₃₂

—

C₂²×C₄

—

C₂²×C₄

C₂×C₈×C₇₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

C₂×C₈×C₇₃

yes

characteristic

101

C₂×C₈

C₂×C₈×C₇₃

yes

characteristic

102

C₂×C₈

C₂×C₈×C₇₃

yes

characteristic

103

C₂×C₈

C₂²×C₄×C₇₃

yes

characteristic

104

C₄²

—

105

C₄²

C₂×C₈×C₇₃

yes

characteristic

106

C₄²

C₂×C₈×C₇₃

yes

characteristic

107

C₁₆

—

108

C₁₆

—

109

C₁₆

M₁₆×C₇₃

yes

—

110

C₁₆

#41×C₇₃

yes

—

111

C₂³

M₃₂×C₇₃

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

M₃₂×C₇₃

yes

characteristic

118

C₂×C₄

C₂²×C₈×C₇₃

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

M₃₂×C₇₃

yes

—

122

C₈

#117×C₇₃

yes

characteristic

123

C₈

C₂×M₁₆×C₇₃

yes

characteristic

124

C₂²

—

125

C₂²

—

126

C₂²

C₂×M₃₂×C₇₃

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#431×C₇₃

yes

characteristic

131

C₄

C₂×M₃₂×C₇₃

yes

characteristic

132

C₂

—

133

C₂

—

134

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
64. #13642	C₇₃	direct (N × H)
69. #1422	C₂×C₇₃	semidirect
71. #1398	C₂×C₇₃	semidirect
72. C₇₃	#13642	direct (N × H)

H ≅ G/N

type

2. C₇₃×#1422

C₂

semidirect

4. C₇₃×#1398

C₂

semidirect

64. #13642

C₇₃

direct (N × H)

69. #1422

C₂×C₇₃

semidirect

71. #1398

C₂×C₇₃

semidirect

72. C₇₃

#13642

direct (N × H)

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

subgroup	index
C₄×M₃₂×C₇₃	2
C₇₃×#1422	2
C₇₃×#1400	2
C₇₃×#1398	2
#13642	73

subgroup

index

C₄×M₃₂×C₇₃

C₇₃×#1422

C₇₃×#1400

C₇₃×#1398

#13642

Subgroups — C₇₃×#13642 #32923046

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#13642 #32923046

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#13642 #32923046