Group #32917949 — subgroups

Subgroups — C₇₃×#8545 #32917949

Order 18688.

910 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 150 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	C₂²×#126×C₇₃	C₂	1	1	yes	maximal, characteristic
2	C₇₃×#1819	C₂	1	6	yes	maximal
3	C₂×#126×C₇₃	—	2	12	no	—
4	C₂²×C₄⋊C₄×C₇₃	C₄	1	3	yes	complemented
5	C₂²×C₂²⋊C₄×C₇₃	C₂²	1	3	yes	—
6	C₂²×C₄²×C₇₃	C₄	1	1	yes	characteristic, complemented
7	#379×C₇₃	C₂²	1	4	yes	—
8	#342×C₇₃	—	2	12	no	—
9	C₂⁵×C₇₃	C₂³	1	1	yes	characteristic
10	C₂³×C₄×C₇₃	C₂×C₄	1	3	yes	—
11	C₂³×C₄×C₇₃	C₂×C₄	1	3	yes	—
12	#126×C₇₃	—	2	32	no	—
13	C₂×C₄⋊C₄×C₇₃	—	2	6	no	—
14	C₂×C₄⋊C₄×C₇₃	—	2	6	no	—
15	C₂×C₄⋊C₄×C₇₃	—	2	6	no	—
16	C₂×C₂²⋊C₄×C₇₃	—	2	6	no	—
17	C₂×C₂²⋊C₄×C₇₃	—	2	6	no	—
18	C₂×C₂²⋊C₄×C₇₃	—	2	12	no	—
19	C₂×C₂²⋊C₄×C₇₃	D₄	1	12	yes	—
20	C₂×C₄²×C₇₃	—	2	6	no	—
21	C₂⁴×C₇₃	—	4	6	no	—
22	C₂⁴×C₇₃	C₂²×C₄	1	1	yes	characteristic
23	C₂⁴×C₇₃	C₂×D₄	1	6	yes	—
24	C₂²×C₄×C₇₃	—	2	3	no	—
25	C₂²×C₄×C₇₃	—	2	3	no	—
26	C₂²×C₄×C₇₃	—	2	3	no	—
27	C₂²×C₄×C₇₃	—	2	3	no	—
28	C₂²×C₄×C₇₃	—	4	6	no	—
29	C₂²×C₄×C₇₃	—	4	6	no	—
30	C₂²×C₄×C₇₃	—	4	4	no	—
31	C₂²×C₄×C₇₃	C₂²⋊C₄	1	6	yes	—
32	C₂²×C₄×C₇₃	C₂²⋊C₄	1	6	yes	—
33	C₄⋊C₄×C₇₃	—	2	24	no	—
34	C₂²⋊C₄×C₇₃	—	2	24	no	—
35	C₂²⋊C₄×C₇₃	—	4	24	no	—
36	C₄²×C₇₃	—	2	8	no	—
37	C₂³×C₇₃	—	2	6	no	—
38	C₂³×C₇₃	—	2	12	no	—
39	C₂³×C₇₃	—	4	6	no	—
40	C₂³×C₇₃	—	4	6	no	—
41	C₂³×C₇₃	—	8	8	no	—
42	C₂³×C₇₃	C₂×C₂²⋊C₄	1	3	yes	—
43	C₂³×C₇₃	#120	1	4	yes	—
44	C₂×C₄×C₇₃	—	2	6	no	—
45	C₂×C₄×C₇₃	—	2	6	no	—
46	C₂×C₄×C₇₃	—	4	6	no	—
47	C₂×C₄×C₇₃	—	4	6	no	—
48	C₂×C₄×C₇₃	—	4	6	no	—
49	C₂×C₄×C₇₃	—	4	6	no	—
50	C₂×C₄×C₇₃	—	4	6	no	—
51	C₂×C₄×C₇₃	—	4	6	no	—
52	C₂×C₄×C₇₃	—	8	12	no	—
53	C₂²×C₇₃	—	2	8	no	—
54	C₂²×C₇₃	—	2	3	no	—
55	C₂²×C₇₃	—	2	3	no	—
56	C₂²×C₇₃	—	2	12	no	—
57	C₂²×C₇₃	—	8	12	no	—
58	C₂²×C₇₃	#379	1	1	yes	characteristic
59	C₂²×C₇₃	#410	1	6	yes	—
60	C₄×C₇₃	—	4	6	no	—
61	C₄×C₇₃	—	4	6	no	—
62	C₄×C₇₃	—	16	4	no	—
63	#8545	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂×C₇₃	—	2	6	no	—
65	C₂×C₇₃	—	4	4	no	—
66	C₂×C₇₃	#1782	1	3	yes	—
67	C₂²×#126	C₂×C₇₃	1	1	yes	characteristic
68	#1819	C₂×C₇₃	1	6	yes	—
69	C₇₃	#8545	1	1	yes	characteristic, direct product factor, Sylow
70	C₂×#126	—	2	12	no	—
71	C₂²×C₄⋊C₄	C₄×C₇₃	1	3	yes	complemented
72	C₂²×C₂²⋊C₄	C₂²×C₇₃	1	3	yes	—
73	C₂²×C₄²	C₄×C₇₃	1	1	yes	characteristic, complemented
74	#379	C₂²×C₇₃	1	4	yes	—
75	#342	—	2	12	no	—
76	C₂⁵	C₂³×C₇₃	1	1	yes	characteristic
77	C₂³×C₄	C₂×C₄×C₇₃	1	3	yes	—
78	C₂³×C₄	C₂×C₄×C₇₃	1	3	yes	—
79	#126	—	2	32	no	—
80	C₂×C₄⋊C₄	—	2	6	no	—
81	C₂×C₄⋊C₄	—	2	6	no	—
82	C₂×C₄⋊C₄	—	2	6	no	—
83	C₂×C₂²⋊C₄	—	2	6	no	—
84	C₂×C₂²⋊C₄	—	2	6	no	—
85	C₂×C₂²⋊C₄	—	2	12	no	—
86	C₂×C₂²⋊C₄	D₄×C₇₃	1	12	yes	—
87	C₂×C₄²	—	2	6	no	—
88	C₂⁴	—	4	6	no	—
89	C₂⁴	C₂²×C₄×C₇₃	1	1	yes	characteristic
90	C₂⁴	C₂×D₄×C₇₃	1	6	yes	—
91	C₂²×C₄	—	2	3	no	—
92	C₂²×C₄	—	2	3	no	—
93	C₂²×C₄	—	2	3	no	—
94	C₂²×C₄	—	2	3	no	—
95	C₂²×C₄	—	4	6	no	—
96	C₂²×C₄	—	4	6	no	—
97	C₂²×C₄	—	4	4	no	—
98	C₂²×C₄	C₂²⋊C₄×C₇₃	1	6	yes	—
99	C₂²×C₄	C₂²⋊C₄×C₇₃	1	6	yes	—
100	C₄⋊C₄	—	2	24	no	—
101	C₂²⋊C₄	—	2	24	no	—
102	C₂²⋊C₄	—	4	24	no	—
103	C₄²	—	2	8	no	—
104	C₂³	—	2	6	no	—
105	C₂³	—	2	12	no	—
106	C₂³	—	4	6	no	—
107	C₂³	—	4	6	no	—
108	C₂³	—	8	8	no	—
109	C₂³	C₂×C₂²⋊C₄×C₇₃	1	3	yes	—
110	C₂³	#120×C₇₃	1	4	yes	—
111	C₂×C₄	—	2	6	no	—
112	C₂×C₄	—	2	6	no	—
113	C₂×C₄	—	4	6	no	—
114	C₂×C₄	—	4	6	no	—
115	C₂×C₄	—	4	6	no	—
116	C₂×C₄	—	4	6	no	—
117	C₂×C₄	—	4	6	no	—
118	C₂×C₄	—	4	6	no	—
119	C₂×C₄	—	8	12	no	—
120	C₂²	—	2	8	no	—
121	C₂²	—	2	3	no	—
122	C₂²	—	2	3	no	—
123	C₂²	—	2	12	no	—
124	C₂²	—	8	12	no	—
125	C₂²	#379×C₇₃	1	1	yes	characteristic
126	C₂²	#410×C₇₃	1	6	yes	—
127	C₄	—	4	6	no	—
128	C₄	—	4	6	no	—
129	C₄	—	16	4	no	—
130	C₂	—	2	6	no	—
131	C₂	—	4	4	no	—
132	C₂	C₇₃×#1782	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#126×C₇₃

C₂

yes

maximal, characteristic

C₇₃×#1819

C₂

yes

maximal

C₂×#126×C₇₃

—

C₂²×C₄⋊C₄×C₇₃

C₄

yes

complemented

C₂²×C₂²⋊C₄×C₇₃

C₂²

yes

—

C₂²×C₄²×C₇₃

C₄

yes

characteristic, complemented

#379×C₇₃

C₂²

yes

—

#342×C₇₃

—

C₂⁵×C₇₃

C₂³

yes

characteristic

C₂³×C₄×C₇₃

C₂×C₄

yes

—

C₂³×C₄×C₇₃

C₂×C₄

yes

—

#126×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₂²⋊C₄×C₇₃

—

C₂×C₂²⋊C₄×C₇₃

—

C₂×C₂²⋊C₄×C₇₃

—

C₂×C₂²⋊C₄×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

—

C₂⁴×C₇₃

—

C₂⁴×C₇₃

C₂²×C₄

yes

characteristic

C₂⁴×C₇₃

C₂×D₄

yes

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

C₂×C₂²⋊C₄

yes

—

C₂³×C₇₃

#120

yes

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#379

yes

characteristic

C₂²×C₇₃

#410

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

#8545

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#1782

yes

—

C₂²×#126

C₂×C₇₃

yes

characteristic

#1819

C₂×C₇₃

yes

—

C₇₃

#8545

yes

characteristic, direct product factor, Sylow

C₂×#126

—

C₂²×C₄⋊C₄

C₄×C₇₃

yes

complemented

C₂²×C₂²⋊C₄

C₂²×C₇₃

yes

—

C₂²×C₄²

C₄×C₇₃

yes

characteristic, complemented

#379

C₂²×C₇₃

yes

—

#342

—

C₂⁵

C₂³×C₇₃

yes

characteristic

C₂³×C₄

C₂×C₄×C₇₃

yes

—

C₂³×C₄

C₂×C₄×C₇₃

yes

—

#126

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

D₄×C₇₃

yes

—

C₂×C₄²

—

C₂⁴

—

C₂⁴

C₂²×C₄×C₇₃

yes

characteristic

C₂⁴

C₂×D₄×C₇₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₄×C₇₃

yes

—

C₂²×C₄

C₂²⋊C₄×C₇₃

yes

—

100

C₄⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₂³

—

105

C₂³

—

106

C₂³

—

107

C₂³

—

108

C₂³

—

109

C₂³

C₂×C₂²⋊C₄×C₇₃

yes

—

110

C₂³

#120×C₇₃

yes

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

#379×C₇₃

yes

characteristic

126

C₂²

#410×C₇₃

yes

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₂

—

131

C₂

—

132

C₂

C₇₃×#1782

yes

—

	subgroup	G/H
center, socle	C₂²×C₇₃	#379
commutator	C₂⁴	C₂²×C₄×C₇₃
Frattini, cosocle	C₂⁵	C₂³×C₇₃
prime radical	C₂⁵×C₇₃	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂⁵×C₇₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₃

#379

commutator

C₂⁴

C₂²×C₄×C₇₃

Frattini, cosocle

C₂⁵

C₂³×C₇₃

prime radical

C₂⁵×C₇₃

C₂³

LCS

G ▷ C₂⁴ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₇₃ ◁ C₂⁵×C₇₃ ◁ G

N	H ≅ G/N	type
4. C₂²×C₄⋊C₄×C₇₃	C₄	semidirect
6. C₂²×C₄²×C₇₃	C₄	semidirect
63. #8545	C₇₃	direct (N × H)
69. C₇₃	#8545	direct (N × H)
71. C₂²×C₄⋊C₄	C₄×C₇₃	semidirect
73. C₂²×C₄²	C₄×C₇₃	semidirect

H ≅ G/N

type

4. C₂²×C₄⋊C₄×C₇₃

C₄

semidirect

6. C₂²×C₄²×C₇₃

C₄

semidirect

63. #8545

C₇₃

direct (N × H)

69. C₇₃

#8545

direct (N × H)

71. C₂²×C₄⋊C₄

C₄×C₇₃

semidirect

73. C₂²×C₄²

C₄×C₇₃

semidirect

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

subgroup	index
C₂²×#126×C₇₃	2
C₇₃×#1819	2
#8545	73

subgroup

index

C₂²×#126×C₇₃

C₇₃×#1819

#8545

Subgroups — C₇₃×#8545 #32917949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#8545 #32917949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#8545 #32917949