Group #22606396 — subgroups

Subgroups — C₇₃×#2243 #22606396

Order 9344.

142 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 64 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	#500×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#495×C₇₃	C₂	1	1	yes	maximal, characteristic
3	#358×C₇₃	C₂	1	2	yes	maximal, complemented
4	#357×C₇₃	C₂	1	2	yes	maximal
5	#346×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
7	C₂×SD₁₆×C₇₃	—	2	1	no	—
8	C₂×D₈×C₇₃	C₂²	1	1	yes	characteristic
9	C₄⋊Q₈×C₇₃	—	2	1	no	—
10	#124×C₇₃	—	2	1	no	—
11	C₂×C₁₆×C₇₃	C₂²	1	2	yes	—
12	#107×C₇₃	—	2	1	no	—
13	#107×C₇₃	C₂²	1	2	yes	—
14	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₇₃	—	2	1	no	—
16	C₂×Q₈×C₇₃	—	4	1	no	—
17	C₂×D₄×C₇₃	—	2	1	no	—
18	Q₁₆×C₇₃	—	2	2	no	—
19	SD₁₆×C₇₃	—	4	2	no	—
20	D₈×C₇₃	—	2	2	no	—
21	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
22	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₇₃	—	2	2	no	—
24	C₄⋊C₄×C₇₃	—	4	1	no	—
25	C₂²⋊C₄×C₇₃	—	4	1	no	—
26	C₂²⋊C₄×C₇₃	—	4	1	no	—
27	C₄²×C₇₃	D₄	1	1	yes	characteristic
28	C₁₆×C₇₃	—	2	2	no	—
29	C₂³×C₇₃	—	4	1	no	—
30	Q₈×C₇₃	—	4	1	no	—
31	Q₈×C₇₃	—	4	1	no	—
32	Q₈×C₇₃	—	8	2	no	—
33	D₄×C₇₃	—	4	1	no	—
34	D₄×C₇₃	—	4	1	no	—
35	C₂×C₄×C₇₃	—	4	1	no	—
36	C₂×C₄×C₇₃	—	4	2	no	—
37	C₂×C₄×C₇₃	D₈	1	1	yes	characteristic
38	C₂×C₄×C₇₃	D₈	1	1	yes	characteristic
39	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
40	C₈×C₇₃	SD₁₆	1	2	yes	—
41	C₈×C₇₃	#41	1	2	yes	—
42	C₂²×C₇₃	—	8	1	no	—
43	C₂²×C₇₃	—	8	1	no	—
44	C₂²×C₇₃	—	8	1	no	—
45	C₂²×C₇₃	C₂×D₈	1	1	yes	characteristic
46	C₄×C₇₃	—	2	1	no	—
47	C₄×C₇₃	—	2	1	no	—
48	C₄×C₇₃	—	8	1	no	—
49	C₄×C₇₃	—	8	2	no	—
50	C₄×C₇₃	#124	1	1	yes	characteristic
51	C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
52	C₂×C₇₃	—	16	1	no	—
53	C₂×C₇₃	#486	1	1	yes	characteristic
54	C₂×C₇₃	#509	1	1	yes	characteristic
55	C₂×C₇₃	#510	1	1	yes	characteristic
56	#2243	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₇₃	#2243	1	1	yes	characteristic, direct product factor, Sylow
58	#500	C₂×C₇₃	1	1	yes	characteristic, complemented
59	#495	C₂×C₇₃	1	1	yes	characteristic
60	#358	C₂×C₇₃	1	2	yes	complemented
61	#357	C₂×C₇₃	1	2	yes	—
62	#346	C₂×C₇₃	1	1	yes	characteristic, complemented
63	C₂×Q₁₆	C₂²×C₇₃	1	1	yes	characteristic
64	C₂×SD₁₆	—	2	1	no	—
65	C₂×D₈	C₂²×C₇₃	1	1	yes	characteristic
66	C₄⋊Q₈	—	2	1	no	—
67	#124	—	2	1	no	—
68	C₂×C₁₆	C₂²×C₇₃	1	2	yes	—
69	#107	—	2	1	no	—
70	#107	C₂²×C₇₃	1	2	yes	—
71	C₄×C₈	C₂²×C₇₃	1	1	yes	characteristic
72	C₂×Q₈	—	2	1	no	—
73	C₂×Q₈	—	4	1	no	—
74	C₂×D₄	—	2	1	no	—
75	Q₁₆	—	2	2	no	—
76	SD₁₆	—	4	2	no	—
77	D₈	—	2	2	no	—
78	C₂×C₈	D₄×C₇₃	1	1	yes	characteristic
79	C₂×C₈	C₂³×C₇₃	1	1	yes	characteristic
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	4	1	no	—
82	C₂²⋊C₄	—	4	1	no	—
83	C₂²⋊C₄	—	4	1	no	—
84	C₄²	D₄×C₇₃	1	1	yes	characteristic
85	C₁₆	—	2	2	no	—
86	C₂³	—	4	1	no	—
87	Q₈	—	4	1	no	—
88	Q₈	—	4	1	no	—
89	Q₈	—	8	2	no	—
90	D₄	—	4	1	no	—
91	D₄	—	4	1	no	—
92	C₂×C₄	—	4	1	no	—
93	C₂×C₄	—	4	2	no	—
94	C₂×C₄	D₈×C₇₃	1	1	yes	characteristic
95	C₂×C₄	D₈×C₇₃	1	1	yes	characteristic
96	C₂×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
97	C₈	SD₁₆×C₇₃	1	2	yes	—
98	C₈	#41×C₇₃	1	2	yes	—
99	C₂²	—	8	1	no	—
100	C₂²	—	8	1	no	—
101	C₂²	—	8	1	no	—
102	C₂²	C₂×D₈×C₇₃	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	—	8	1	no	—
106	C₄	—	8	2	no	—
107	C₄	#124×C₇₃	1	1	yes	characteristic
108	C₄	C₂×SD₁₆×C₇₃	1	1	yes	characteristic
109	C₂	—	16	1	no	—
110	C₂	#486×C₇₃	1	1	yes	characteristic
111	C₂	#509×C₇₃	1	1	yes	characteristic
112	C₂	#510×C₇₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#500×C₇₃

C₂

yes

maximal, characteristic, complemented

#495×C₇₃

C₂

yes

maximal, characteristic

#358×C₇₃

C₂

yes

maximal, complemented

#357×C₇₃

C₂

yes

maximal

#346×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×Q₁₆×C₇₃

C₂²

yes

characteristic

C₂×SD₁₆×C₇₃

—

C₂×D₈×C₇₃

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

—

#124×C₇₃

—

C₂×C₁₆×C₇₃

C₂²

yes

—

#107×C₇₃

—

#107×C₇₃

C₂²

yes

—

C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×D₄×C₇₃

—

Q₁₆×C₇₃

—

SD₁₆×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

D₄

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

D₈

yes

characteristic

C₂×C₄×C₇₃

D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₈×C₇₃

SD₁₆

yes

—

C₈×C₇₃

#41

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×D₈

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#124

yes

characteristic

C₄×C₇₃

C₂×SD₁₆

yes

characteristic

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₇₃

#2243

yes

characteristic, direct product factor, Sylow

#500

C₂×C₇₃

yes

characteristic, complemented

#495

C₂×C₇₃

yes

characteristic

#358

C₂×C₇₃

yes

complemented

#357

C₂×C₇₃

yes

—

#346

C₂×C₇₃

yes

characteristic, complemented

C₂×Q₁₆

C₂²×C₇₃

yes

characteristic

C₂×SD₁₆

—

C₂×D₈

C₂²×C₇₃

yes

characteristic

C₄⋊Q₈

—

#124

—

C₂×C₁₆

C₂²×C₇₃

yes

—

#107

—

#107

C₂²×C₇₃

yes

—

C₄×C₈

C₂²×C₇₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

Q₁₆

—

SD₁₆

—

D₈

—

C₂×C₈

D₄×C₇₃

yes

characteristic

C₂×C₈

C₂³×C₇₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

D₄×C₇₃

yes

characteristic

C₁₆

—

C₂³

—

Q₈

—

Q₈

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₈×C₇₃

yes

characteristic

C₂×C₄

D₈×C₇₃

yes

characteristic

C₂×C₄

C₂×D₄×C₇₃

yes

characteristic

C₈

SD₁₆×C₇₃

yes

—

C₈

#41×C₇₃

yes

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

C₂×D₈×C₇₃

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

#124×C₇₃

yes

characteristic

108

C₄

C₂×SD₁₆×C₇₃

yes

characteristic

109

C₂

—

110

C₂

#486×C₇₃

yes

characteristic

111

C₂

#509×C₇₃

yes

characteristic

112

C₂

#510×C₇₃

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₇₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₇₃
prime radical	C₂×D₈×C₇₃	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₄²×C₇₃ ◁ C₄×C₈×C₇₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₃

C₂×D₈

commutator, Frattini, cosocle

C₂×C₈

C₂³×C₇₃

prime radical

C₂×D₈×C₇₃

C₂²

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

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

N	H ≅ G/N	type
1. #500×C₇₃	C₂	semidirect
3. #358×C₇₃	C₂	semidirect
5. #346×C₇₃	C₂	semidirect
56. #2243	C₇₃	direct (N × H)
57. C₇₃	#2243	direct (N × H)
58. #500	C₂×C₇₃	semidirect
60. #358	C₂×C₇₃	semidirect
62. #346	C₂×C₇₃	semidirect

H ≅ G/N

type

1. #500×C₇₃

C₂

semidirect

3. #358×C₇₃

C₂

semidirect

5. #346×C₇₃

C₂

semidirect

56. #2243

C₇₃

direct (N × H)

57. C₇₃

#2243

direct (N × H)

58. #500

C₂×C₇₃

semidirect

60. #358

C₂×C₇₃

semidirect

62. #346

C₂×C₇₃

semidirect

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

subgroup	index
#500×C₇₃	2
#495×C₇₃	2
#358×C₇₃	2
#357×C₇₃	2
#346×C₇₃	2
#2243	73

subgroup

index

#500×C₇₃

#495×C₇₃

#358×C₇₃

#357×C₇₃

#346×C₇₃

#2243

Subgroups — C₇₃×#2243 #22606396

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#2243 #22606396

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#2243 #22606396