Group #32942281 — subgroups

Subgroups — C₇₃×#32877 #32942281

Order 18688.

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

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 186 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×#2091	C₂	1	4	yes	maximal, complemented
2	C₇₃×#2070	C₂	1	4	yes	maximal, complemented
3	C₇₃×#2069	C₂	1	4	yes	maximal, complemented
4	C₇₃×#1785	C₂	1	3	yes	maximal
5	C₂²×#41×C₇₃	C₂²	1	1	yes	characteristic
6	#430×C₇₃	C₂²	1	12	yes	—
7	C₂×#101×C₇₃	C₂²	1	6	yes	—
8	C₂×#100×C₇₃	C₂²	1	6	yes	—
9	#407×C₇₃	C₂²	1	6	yes	—
10	#338×C₇₃	C₂²	1	4	yes	—
11	C₂×#41×C₇₃	—	4	6	no	—
12	C₂²×Q₈×C₇₃	C₂³	1	1	yes	characteristic
13	C₂²×D₄×C₇₃	Q₈	1	2	yes	—
14	C₂²×D₄×C₇₃	C₂³	1	1	yes	characteristic
15	C₂³×C₄×C₇₃	Q₈	1	2	yes	—
16	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
17	C₂×M₁₆×C₇₃	C₂³	1	12	yes	—
18	#108×C₇₃	—	4	12	no	—
19	#101×C₇₃	—	4	6	no	—
20	#100×C₇₃	—	4	6	no	—
21	C₂²⋊C₈×C₇₃	—	4	6	no	—
22	C₂⁴×C₇₃	C₂×Q₈	1	2	yes	—
23	C₂⁴×C₇₃	C₂×Q₈	1	1	yes	characteristic
24	#41×C₇₃	—	4	16	no	—
25	C₂×Q₈×C₇₃	—	2	6	no	—
26	C₂×D₄×C₇₃	—	2	6	no	—
27	C₂×D₄×C₇₃	—	4	6	no	—
28	C₂²×C₄×C₇₃	—	2	6	no	—
29	C₂²×C₄×C₇₃	—	4	6	no	—
30	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
31	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
32	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
33	M₁₆×C₇₃	—	2	12	no	—
34	M₁₆×C₇₃	—	2	12	no	—
35	C₂×C₈×C₇₃	—	2	12	no	—
36	C₂³×C₇₃	—	2	3	no	—
37	C₂³×C₇₃	—	2	6	no	—
38	C₂³×C₇₃	—	8	1	no	—
39	C₂³×C₇₃	—	8	2	no	—
40	C₂³×C₇₃	C₂²×Q₈	1	1	yes	characteristic
41	Q₈×C₇₃	—	4	4	no	—
42	D₄×C₇₃	—	4	8	no	—
43	D₄×C₇₃	—	4	4	no	—
44	C₂×C₄×C₇₃	—	2	3	no	—
45	C₂×C₄×C₇₃	—	2	6	no	—
46	C₂×C₄×C₇₃	—	4	8	no	—
47	C₂×C₄×C₇₃	—	4	4	no	—
48	C₂×C₄×C₇₃	H(5,𝔽₂)	1	6	yes	—
49	C₈×C₇₃	—	4	12	no	—
50	C₂²×C₇₃	—	4	1	no	—
51	C₂²×C₇₃	—	4	6	no	—
52	C₂²×C₇₃	—	4	1	no	—
53	C₂²×C₇₃	—	4	2	no	—
54	C₂²×C₇₃	—	8	6	no	—
55	C₂²×C₇₃	#543	1	3	yes	—
56	C₄×C₇₃	—	4	1	no	—
57	C₄×C₇₃	—	4	2	no	—
58	C₄×C₇₃	#560	1	4	yes	—
59	#32877	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂×C₇₃	—	2	3	no	—
61	C₂×C₇₃	—	8	1	no	—
62	C₂×C₇₃	—	8	2	no	—
63	C₂×C₇₃	#2842	1	1	yes	characteristic
64	#2091	C₂×C₇₃	1	4	yes	complemented
65	#2070	C₂×C₇₃	1	4	yes	complemented
66	#2069	C₂×C₇₃	1	4	yes	complemented
67	#1785	C₂×C₇₃	1	3	yes	—
68	C₇₃	#32877	1	1	yes	characteristic, direct product factor, Sylow
69	C₂²×#41	C₂²×C₇₃	1	1	yes	characteristic
70	#430	C₂²×C₇₃	1	12	yes	—
71	C₂×#101	C₂²×C₇₃	1	6	yes	—
72	C₂×#100	C₂²×C₇₃	1	6	yes	—
73	#407	C₂²×C₇₃	1	6	yes	—
74	#338	C₂²×C₇₃	1	4	yes	—
75	C₂×#41	—	4	6	no	—
76	C₂²×Q₈	C₂³×C₇₃	1	1	yes	characteristic
77	C₂²×D₄	Q₈×C₇₃	1	2	yes	—
78	C₂²×D₄	C₂³×C₇₃	1	1	yes	characteristic
79	C₂³×C₄	Q₈×C₇₃	1	2	yes	—
80	C₂³×C₄	C₂³×C₇₃	1	1	yes	characteristic
81	C₂×M₁₆	C₂³×C₇₃	1	12	yes	—
82	#108	—	4	12	no	—
83	#101	—	4	6	no	—
84	#100	—	4	6	no	—
85	C₂²⋊C₈	—	4	6	no	—
86	C₂⁴	C₂×Q₈×C₇₃	1	2	yes	—
87	C₂⁴	C₂×Q₈×C₇₃	1	1	yes	characteristic
88	#41	—	4	16	no	—
89	C₂×Q₈	—	2	6	no	—
90	C₂×D₄	—	2	6	no	—
91	C₂×D₄	—	4	6	no	—
92	C₂²×C₄	—	2	6	no	—
93	C₂²×C₄	—	4	6	no	—
94	C₂²×C₄	C₂×Q₈×C₇₃	1	1	yes	characteristic
95	C₂²×C₄	C₂×Q₈×C₇₃	1	2	yes	—
96	C₂²×C₄	C₂⁴×C₇₃	1	1	yes	characteristic
97	M₁₆	—	2	12	no	—
98	M₁₆	—	2	12	no	—
99	C₂×C₈	—	2	12	no	—
100	C₂³	—	2	3	no	—
101	C₂³	—	2	6	no	—
102	C₂³	—	8	1	no	—
103	C₂³	—	8	2	no	—
104	C₂³	C₂²×Q₈×C₇₃	1	1	yes	characteristic
105	Q₈	—	4	4	no	—
106	D₄	—	4	8	no	—
107	D₄	—	4	4	no	—
108	C₂×C₄	—	2	3	no	—
109	C₂×C₄	—	2	6	no	—
110	C₂×C₄	—	4	8	no	—
111	C₂×C₄	—	4	4	no	—
112	C₂×C₄	H(5,𝔽₂)×C₇₃	1	6	yes	—
113	C₈	—	4	12	no	—
114	C₂²	—	4	1	no	—
115	C₂²	—	4	6	no	—
116	C₂²	—	4	1	no	—
117	C₂²	—	4	2	no	—
118	C₂²	—	8	6	no	—
119	C₂²	#543×C₇₃	1	3	yes	—
120	C₄	—	4	1	no	—
121	C₄	—	4	2	no	—
122	C₄	#560×C₇₃	1	4	yes	—
123	C₂	—	2	3	no	—
124	C₂	—	8	1	no	—
125	C₂	—	8	2	no	—
126	C₂	C₇₃×#2842	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃×#2091

C₂

yes

maximal, complemented

C₇₃×#2070

C₂

yes

maximal, complemented

C₇₃×#2069

C₂

yes

maximal, complemented

C₇₃×#1785

C₂

yes

maximal

C₂²×#41×C₇₃

C₂²

yes

characteristic

#430×C₇₃

C₂²

yes

—

C₂×#101×C₇₃

C₂²

yes

—

C₂×#100×C₇₃

C₂²

yes

—

#407×C₇₃

C₂²

yes

—

#338×C₇₃

C₂²

yes

—

C₂×#41×C₇₃

—

C₂²×Q₈×C₇₃

C₂³

yes

characteristic

C₂²×D₄×C₇₃

Q₈

yes

—

C₂²×D₄×C₇₃

C₂³

yes

characteristic

C₂³×C₄×C₇₃

Q₈

yes

—

C₂³×C₄×C₇₃

C₂³

yes

characteristic

C₂×M₁₆×C₇₃

C₂³

yes

—

#108×C₇₃

—

#101×C₇₃

—

#100×C₇₃

—

C₂²⋊C₈×C₇₃

—

C₂⁴×C₇₃

C₂×Q₈

yes

—

C₂⁴×C₇₃

C₂×Q₈

yes

characteristic

#41×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₇₃

C₂×Q₈

yes

—

C₂²×C₄×C₇₃

C₂⁴

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

C₂²×Q₈

yes

characteristic

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

H(5,𝔽₂)

yes

—

C₈×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#543

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#560

yes

—

#32877

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#2842

yes

characteristic

#2091

C₂×C₇₃

yes

complemented

#2070

C₂×C₇₃

yes

complemented

#2069

C₂×C₇₃

yes

complemented

#1785

C₂×C₇₃

yes

—

C₇₃

#32877

yes

characteristic, direct product factor, Sylow

C₂²×#41

C₂²×C₇₃

yes

characteristic

#430

C₂²×C₇₃

yes

—

C₂×#101

C₂²×C₇₃

yes

—

C₂×#100

C₂²×C₇₃

yes

—

#407

C₂²×C₇₃

yes

—

#338

C₂²×C₇₃

yes

—

C₂×#41

—

C₂²×Q₈

C₂³×C₇₃

yes

characteristic

C₂²×D₄

Q₈×C₇₃

yes

—

C₂²×D₄

C₂³×C₇₃

yes

characteristic

C₂³×C₄

Q₈×C₇₃

yes

—

C₂³×C₄

C₂³×C₇₃

yes

characteristic

C₂×M₁₆

C₂³×C₇₃

yes

—

#108

—

#101

—

#100

—

C₂²⋊C₈

—

C₂⁴

C₂×Q₈×C₇₃

yes

—

C₂⁴

C₂×Q₈×C₇₃

yes

characteristic

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Q₈×C₇₃

yes

characteristic

C₂²×C₄

C₂×Q₈×C₇₃

yes

—

C₂²×C₄

C₂⁴×C₇₃

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

100

C₂³

—

101

C₂³

—

102

C₂³

—

103

C₂³

—

104

C₂³

C₂²×Q₈×C₇₃

yes

characteristic

105

Q₈

—

106

D₄

—

107

D₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

H(5,𝔽₂)×C₇₃

yes

—

113

C₈

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

#543×C₇₃

yes

—

120

C₄

—

121

C₄

—

122

C₄

#560×C₇₃

yes

—

123

C₂

—

124

C₂

—

125

C₂

—

126

C₂

C₇₃×#2842

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₇₃	#2842
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₃
prime radical	C₂²×#41×C₇₃	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ G

subgroup

G/H

center, socle

C₂×C₇₃

#2842

commutator, Frattini, cosocle

C₂²×C₄

C₂⁴×C₇₃

prime radical

C₂²×#41×C₇₃

C₂²

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ G

N	H ≅ G/N	type
1. C₇₃×#2091	C₂	semidirect
2. C₇₃×#2070	C₂	semidirect
3. C₇₃×#2069	C₂	semidirect
59. #32877	C₇₃	direct (N × H)
64. #2091	C₂×C₇₃	semidirect
65. #2070	C₂×C₇₃	semidirect
66. #2069	C₂×C₇₃	semidirect
68. C₇₃	#32877	direct (N × H)

H ≅ G/N

type

1. C₇₃×#2091

C₂

semidirect

2. C₇₃×#2070

C₂

semidirect

3. C₇₃×#2069

C₂

semidirect

59. #32877

C₇₃

direct (N × H)

64. #2091

C₂×C₇₃

semidirect

65. #2070

C₂×C₇₃

semidirect

66. #2069

C₂×C₇₃

semidirect

68. C₇₃

#32877

direct (N × H)

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

subgroup	index
C₇₃×#2091	2
C₇₃×#2070	2
C₇₃×#2069	2
C₇₃×#1785	2
#32877	73

subgroup

index

C₇₃×#2091

C₇₃×#2070

C₇₃×#2069

C₇₃×#1785

#32877

Subgroups — C₇₃×#32877 #32942281

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#32877 #32942281

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#32877 #32942281