Group #32916789 — subgroups

Subgroups — C₇₃×#7385 #32916789

Order 18688.

136 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 58 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×D₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
2	C₇₃×#1397	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₃×#1372	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₆×C₇₃	C₄	1	1	yes	characteristic
5	C₄×D₈×C₇₃	—	2	1	no	—
6	C₂×C₃₂×C₇₃	—	2	1	no	—
7	#366×C₇₃	C₄	1	1	yes	characteristic
8	#357×C₇₃	—	2	1	no	—
9	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
10	#335×C₇₃	—	2	1	no	—
11	C₂×D₈×C₇₃	—	2	1	no	—
12	C₄×D₄×C₇₃	—	4	1	no	—
13	D₁₆×C₇₃	—	2	1	no	—
14	D₁₆×C₇₃	C₈	1	2	yes	complemented
15	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
17	#107×C₇₃	—	2	1	no	—
18	C₄⋊C₈×C₇₃	—	4	1	no	—
19	#102×C₇₃	—	4	1	no	—
20	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
21	C₃₂×C₇₃	—	2	2	no	—
22	C₂×D₄×C₇₃	—	4	1	no	—
23	C₂²×C₄×C₇₃	—	8	1	no	—
24	D₈×C₇₃	—	2	2	no	—
25	D₈×C₇₃	—	4	1	no	—
26	C₂×C₈×C₇₃	—	8	1	no	—
27	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₈×C₇₃	SD₁₆	1	1	yes	characteristic
29	C₄⋊C₄×C₇₃	—	4	1	no	—
30	C₂²⋊C₄×C₇₃	—	8	1	no	—
31	C₄²×C₇₃	D₈	1	1	yes	characteristic
32	C₁₆×C₇₃	—	2	1	no	—
33	C₁₆×C₇₃	C₂×C₈	1	1	yes	characteristic
34	C₁₆×C₇₃	M₁₆	1	1	yes	characteristic
35	C₂³×C₇₃	—	8	1	no	—
36	D₄×C₇₃	—	4	2	no	—
37	D₄×C₇₃	—	8	1	no	—
38	C₂×C₄×C₇₃	—	8	1	no	—
39	C₂×C₄×C₇₃	—	16	2	no	—
40	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
41	C₂×C₄×C₇₃	D₁₆	1	1	yes	characteristic
42	C₂×C₄×C₇₃	SD₃₂	1	1	yes	characteristic
43	C₈×C₇₃	—	2	1	no	—
44	C₈×C₇₃	—	16	1	no	—
45	C₈×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
46	C₈×C₇₃	#104	1	1	yes	characteristic
47	C₂²×C₇₃	—	8	2	no	—
48	C₂²×C₇₃	—	16	1	no	—
49	C₂²×C₇₃	—	16	1	no	—
50	C₂²×C₇₃	#357	1	1	yes	characteristic
51	C₄×C₇₃	—	2	1	no	—
52	C₄×C₇₃	—	2	1	no	—
53	C₄×C₇₃	—	16	1	no	—
54	C₄×C₇₃	D₄⋊C₈	1	1	yes	characteristic
55	C₄×C₇₃	#359	1	1	yes	characteristic
56	#7385	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂×C₇₃	—	16	2	no	—
58	C₂×C₇₃	#1331	1	1	yes	characteristic
59	C₂×C₇₃	#1418	1	1	yes	characteristic
60	C₂×C₇₃	#1419	1	1	yes	characteristic
61	C₄×D₁₆	C₂×C₇₃	1	1	yes	characteristic
62	#1397	C₂×C₇₃	1	1	yes	characteristic, complemented
63	#1372	C₂×C₇₃	1	1	yes	characteristic, complemented
64	C₇₃	#7385	1	1	yes	characteristic, direct product factor, Sylow
65	C₂×D₁₆	C₄×C₇₃	1	1	yes	characteristic
66	C₄×D₈	—	2	1	no	—
67	C₂×C₃₂	—	2	1	no	—
68	#366	C₄×C₇₃	1	1	yes	characteristic
69	#357	—	2	1	no	—
70	C₄×C₁₆	C₂²×C₇₃	1	1	yes	characteristic
71	#335	—	2	1	no	—
72	C₂×D₈	—	2	1	no	—
73	C₄×D₄	—	4	1	no	—
74	D₁₆	—	2	1	no	—
75	D₁₆	C₈×C₇₃	1	2	yes	complemented
76	C₂×C₁₆	C₂×C₄×C₇₃	1	1	yes	characteristic
77	C₂×C₁₆	D₄×C₇₃	1	1	yes	characteristic
78	#107	—	2	1	no	—
79	C₄⋊C₈	—	4	1	no	—
80	#102	—	4	1	no	—
81	C₄×C₈	D₄×C₇₃	1	1	yes	characteristic
82	C₃₂	—	2	2	no	—
83	C₂×D₄	—	4	1	no	—
84	C₂²×C₄	—	8	1	no	—
85	D₈	—	2	2	no	—
86	D₈	—	4	1	no	—
87	C₂×C₈	—	8	1	no	—
88	C₂×C₈	C₂²⋊C₄×C₇₃	1	1	yes	characteristic
89	C₂×C₈	SD₁₆×C₇₃	1	1	yes	characteristic
90	C₄⋊C₄	—	4	1	no	—
91	C₂²⋊C₄	—	8	1	no	—
92	C₄²	D₈×C₇₃	1	1	yes	characteristic
93	C₁₆	—	2	1	no	—
94	C₁₆	C₂×C₈×C₇₃	1	1	yes	characteristic
95	C₁₆	M₁₆×C₇₃	1	1	yes	characteristic
96	C₂³	—	8	1	no	—
97	D₄	—	4	2	no	—
98	D₄	—	8	1	no	—
99	C₂×C₄	—	8	1	no	—
100	C₂×C₄	—	16	2	no	—
101	C₂×C₄	#102×C₇₃	1	1	yes	characteristic
102	C₂×C₄	D₁₆×C₇₃	1	1	yes	characteristic
103	C₂×C₄	SD₃₂×C₇₃	1	1	yes	characteristic
104	C₈	—	2	1	no	—
105	C₈	—	16	1	no	—
106	C₈	C₂²⋊C₈×C₇₃	1	1	yes	characteristic
107	C₈	#104×C₇₃	1	1	yes	characteristic
108	C₂²	—	8	2	no	—
109	C₂²	—	16	1	no	—
110	C₂²	—	16	1	no	—
111	C₂²	#357×C₇₃	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	—	16	1	no	—
115	C₄	D₄⋊C₈×C₇₃	1	1	yes	characteristic
116	C₄	#359×C₇₃	1	1	yes	characteristic
117	C₂	—	16	2	no	—
118	C₂	C₇₃×#1331	1	1	yes	characteristic
119	C₂	C₇₃×#1418	1	1	yes	characteristic
120	C₂	C₇₃×#1419	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₁₆×C₇₃

C₂

yes

maximal, characteristic

C₇₃×#1397

C₂

yes

maximal, characteristic, complemented

C₇₃×#1372

C₂

yes

maximal, characteristic, complemented

C₂×D₁₆×C₇₃

C₄

yes

characteristic

C₄×D₈×C₇₃

—

C₂×C₃₂×C₇₃

—

#366×C₇₃

C₄

yes

characteristic

#357×C₇₃

—

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

#335×C₇₃

—

C₂×D₈×C₇₃

—

C₄×D₄×C₇₃

—

D₁₆×C₇₃

—

D₁₆×C₇₃

C₈

yes

complemented

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

D₄

yes

characteristic

#107×C₇₃

—

C₄⋊C₈×C₇₃

—

#102×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

characteristic

C₃₂×C₇₃

—

C₂×D₄×C₇₃

—

C₂²×C₄×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₃

SD₁₆

yes

characteristic

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

D₈

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

C₂×C₈

yes

characteristic

C₁₆×C₇₃

M₁₆

yes

characteristic

C₂³×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#102

yes

characteristic

C₂×C₄×C₇₃

D₁₆

yes

characteristic

C₂×C₄×C₇₃

SD₃₂

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

C₂²⋊C₈

yes

characteristic

C₈×C₇₃

#104

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#357

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

D₄⋊C₈

yes

C₄×C₇₃

yes

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₄×D₁₆

C₂×C₇₃

yes

characteristic

#1397

C₂×C₇₃

yes

characteristic, complemented

#1372

C₂×C₇₃

yes

characteristic, complemented

C₇₃

#7385

yes

characteristic, direct product factor, Sylow

C₂×D₁₆

C₄×C₇₃

yes

characteristic

C₄×D₈

—

C₂×C₃₂

—

#366

C₄×C₇₃

yes

characteristic

#357

—

C₄×C₁₆

C₂²×C₇₃

yes

characteristic

#335

—

C₂×D₈

—

C₄×D₄

—

D₁₆

—

D₁₆

C₈×C₇₃

yes

complemented

C₂×C₁₆

C₂×C₄×C₇₃

yes

characteristic

C₂×C₁₆

D₄×C₇₃

yes

characteristic

#107

—

C₄⋊C₈

—

#102

—

C₄×C₈

D₄×C₇₃

yes

characteristic

C₃₂

—

C₂×D₄

—

C₂²×C₄

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄×C₇₃

yes

characteristic

C₂×C₈

SD₁₆×C₇₃

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

D₈×C₇₃

yes

characteristic

C₁₆

—

C₁₆

C₂×C₈×C₇₃

yes

characteristic

C₁₆

M₁₆×C₇₃

yes

characteristic

C₂³

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

#102×C₇₃

yes

characteristic

102

C₂×C₄

D₁₆×C₇₃

yes

characteristic

103

C₂×C₄

SD₃₂×C₇₃

yes

characteristic

104

C₈

—

105

C₈

—

106

C₈

C₂²⋊C₈×C₇₃

yes

characteristic

107

C₈

#104×C₇₃

yes

characteristic

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

#357×C₇₃

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

D₄⋊C₈×C₇₃

yes

characteristic

116

C₄

#359×C₇₃

yes

characteristic

117

C₂

—

118

C₂

C₇₃×#1331

yes

characteristic

119

C₂

C₇₃×#1418

yes

characteristic

120

C₂

C₇₃×#1419

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₇₃	#357
commutator	C₁₆	C₂×C₈×C₇₃
Frattini, cosocle	C₄×C₁₆	C₂²×C₇₃
prime radical	C₂×D₁₆×C₇₃	C₄
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₄²×C₇₃ ◁ C₄×C₈×C₇₃ ◁ C₄×C₁₆×C₇₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₃

#357

commutator

C₁₆

C₂×C₈×C₇₃

Frattini, cosocle

C₄×C₁₆

C₂²×C₇₃

prime radical

C₂×D₁₆×C₇₃

C₄

LCS

G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₇₃ ◁ C₄²×C₇₃ ◁ C₄×C₈×C₇₃ ◁ C₄×C₁₆×C₇₃ ◁ G

N	H ≅ G/N	type
2. C₇₃×#1397	C₂	semidirect
3. C₇₃×#1372	C₂	semidirect
14. D₁₆×C₇₃	C₈	semidirect
56. #7385	C₇₃	direct (N × H)
62. #1397	C₂×C₇₃	semidirect
63. #1372	C₂×C₇₃	semidirect
64. C₇₃	#7385	direct (N × H)
75. D₁₆	C₈×C₇₃	semidirect

H ≅ G/N

type

2. C₇₃×#1397

C₂

semidirect

3. C₇₃×#1372

C₂

semidirect

14. D₁₆×C₇₃

C₈

semidirect

56. #7385

C₇₃

direct (N × H)

62. #1397

C₂×C₇₃

semidirect

63. #1372

C₂×C₇₃

semidirect

64. C₇₃

#7385

direct (N × H)

75. D₁₆

C₈×C₇₃

semidirect

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

subgroup	index
C₄×D₁₆×C₇₃	2
C₇₃×#1397	2
C₇₃×#1372	2
#7385	73

subgroup

index

C₄×D₁₆×C₇₃

C₇₃×#1397

C₇₃×#1372

#7385

Subgroups — C₇₃×#7385 #32916789

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#7385 #32916789

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#7385 #32916789