Group #22605459 — subgroups

Subgroups — C₇₃×#1306 #22605459

Order 9344.

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

normal subgroups only

This group has 192 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	#407×C₇₃	C₂	1	2	yes	maximal
2	#381×C₇₃	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₄×C₇₃	C₂²	1	1	yes	characteristic
4	C₂×M₁₆×C₇₃	—	2	2	no	—
5	C₂×C₂²⋊C₄×C₇₃	C₄	1	2	yes	—
6	C₂²⋊C₈×C₇₃	—	2	2	no	—
7	C₂²⋊C₈×C₇₃	C₄	1	4	yes	complemented
8	#95×C₇₃	—	2	2	no	—
9	C₂⁴×C₇₃	C₂×C₄	1	1	yes	characteristic
10	C₂²×C₄×C₇₃	—	2	2	no	—
11	C₂²×C₄×C₇₃	—	4	2	no	—
12	C₂²×C₄×C₇₃	C₂×C₄	1	2	yes	—
13	C₂²×C₄×C₇₃	D₄	1	2	yes	—
14	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
15	C₂²×C₄×C₇₃	Q₈	1	1	yes	characteristic
16	M₁₆×C₇₃	—	4	4	no	—
17	C₂×C₈×C₇₃	—	2	4	no	—
18	C₂²⋊C₄×C₇₃	—	4	2	no	—
19	C₂³×C₇₃	—	2	2	no	—
20	C₂³×C₇₃	—	4	2	no	—
21	C₂³×C₇₃	C₄²	1	1	yes	characteristic
22	C₂³×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₂³×C₇₃	C₄⋊C₄	1	1	yes	characteristic
24	C₂×C₄×C₇₃	—	2	4	no	—
25	C₂×C₄×C₇₃	—	4	2	no	—
26	C₂×C₄×C₇₃	—	4	1	no	—
27	C₂×C₄×C₇₃	—	4	1	no	—
28	C₂×C₄×C₇₃	—	4	4	no	—
29	C₂×C₄×C₇₃	—	4	2	no	—
30	C₂×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
31	C₂×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
32	C₈×C₇₃	—	4	4	no	—
33	C₂²×C₇₃	—	2	2	no	—
34	C₂²×C₇₃	—	2	1	no	—
35	C₂²×C₇₃	—	2	1	no	—
36	C₂²×C₇₃	—	2	2	no	—
37	C₂²×C₇₃	—	2	2	no	—
38	C₂²×C₇₃	—	4	4	no	—
39	C₂²×C₇₃	#95	1	1	yes	characteristic
40	C₂²×C₇₃	#99	1	2	yes	—
41	C₄×C₇₃	—	2	2	no	—
42	C₄×C₇₃	—	2	2	no	—
43	C₄×C₇₃	—	8	2	no	—
44	C₂×C₇₃	—	2	2	no	—
45	C₂×C₇₃	—	4	1	no	—
46	C₂×C₇₃	—	4	1	no	—
47	C₂×C₇₃	#342	1	1	yes	characteristic
48	C₂×C₇₃	#344	1	2	yes	—
49	#1306	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₇₃	#1306	1	1	yes	characteristic, direct product factor, Sylow
51	#407	C₂×C₇₃	1	2	yes	—
52	#381	C₂×C₇₃	1	1	yes	characteristic
53	C₂³×C₄	C₂²×C₇₃	1	1	yes	characteristic
54	C₂×M₁₆	—	2	2	no	—
55	C₂×C₂²⋊C₄	C₄×C₇₃	1	2	yes	—
56	C₂²⋊C₈	—	2	2	no	—
57	C₂²⋊C₈	C₄×C₇₃	1	4	yes	complemented
58	#95	—	2	2	no	—
59	C₂⁴	C₂×C₄×C₇₃	1	1	yes	characteristic
60	C₂²×C₄	—	2	2	no	—
61	C₂²×C₄	—	4	2	no	—
62	C₂²×C₄	C₂×C₄×C₇₃	1	2	yes	—
63	C₂²×C₄	D₄×C₇₃	1	2	yes	—
64	C₂²×C₄	D₄×C₇₃	1	1	yes	characteristic
65	C₂²×C₄	Q₈×C₇₃	1	1	yes	characteristic
66	M₁₆	—	4	4	no	—
67	C₂×C₈	—	2	4	no	—
68	C₂²⋊C₄	—	4	2	no	—
69	C₂³	—	2	2	no	—
70	C₂³	—	4	2	no	—
71	C₂³	C₄²×C₇₃	1	1	yes	characteristic
72	C₂³	C₂²⋊C₄×C₇₃	1	1	yes	characteristic
73	C₂³	C₄⋊C₄×C₇₃	1	1	yes	characteristic
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	—	4	2	no	—
76	C₂×C₄	—	4	1	no	—
77	C₂×C₄	—	4	1	no	—
78	C₂×C₄	—	4	4	no	—
79	C₂×C₄	—	4	2	no	—
80	C₂×C₄	C₂²⋊C₄×C₇₃	1	2	yes	—
81	C₂×C₄	C₄⋊C₄×C₇₃	1	2	yes	—
82	C₈	—	4	4	no	—
83	C₂²	—	2	2	no	—
84	C₂²	—	2	1	no	—
85	C₂²	—	2	1	no	—
86	C₂²	—	2	2	no	—
87	C₂²	—	2	2	no	—
88	C₂²	—	4	4	no	—
89	C₂²	#95×C₇₃	1	1	yes	characteristic
90	C₂²	#99×C₇₃	1	2	yes	—
91	C₄	—	2	2	no	—
92	C₄	—	2	2	no	—
93	C₄	—	8	2	no	—
94	C₂	—	2	2	no	—
95	C₂	—	4	1	no	—
96	C₂	—	4	1	no	—
97	C₂	#342×C₇₃	1	1	yes	characteristic
98	C₂	#344×C₇₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#407×C₇₃

C₂

yes

maximal

#381×C₇₃

C₂

yes

maximal, characteristic

C₂³×C₄×C₇₃

C₂²

yes

characteristic

C₂×M₁₆×C₇₃

—

C₂×C₂²⋊C₄×C₇₃

C₄

yes

—

C₂²⋊C₈×C₇₃

—

C₂²⋊C₈×C₇₃

C₄

yes

complemented

#95×C₇₃

—

C₂⁴×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×C₄

yes

—

C₂²×C₄×C₇₃

D₄

yes

—

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

Q₈

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

C₄²

yes

characteristic

C₂³×C₇₃

C₂²⋊C₄

yes

characteristic

C₂³×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂×C₄×C₇₃

C₄⋊C₄

yes

—

C₈×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#95

yes

characteristic

C₂²×C₇₃

#99

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#342

yes

characteristic

C₂×C₇₃

#344

yes

—

#1306

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₇₃

#1306

yes

characteristic, direct product factor, Sylow

#407

C₂×C₇₃

yes

—

#381

C₂×C₇₃

yes

characteristic

C₂³×C₄

C₂²×C₇₃

yes

characteristic

C₂×M₁₆

—

C₂×C₂²⋊C₄

C₄×C₇₃

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

C₄×C₇₃

yes

complemented

#95

—

C₂⁴

C₂×C₄×C₇₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×C₇₃

yes

—

C₂²×C₄

D₄×C₇₃

yes

—

C₂²×C₄

D₄×C₇₃

yes

characteristic

C₂²×C₄

Q₈×C₇₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂²⋊C₄

—

C₂³

—

C₂³

—

C₂³

C₄²×C₇₃

yes

characteristic

C₂³

C₂²⋊C₄×C₇₃

yes

characteristic

C₂³

C₄⋊C₄×C₇₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₄×C₇₃

yes

—

C₂×C₄

C₄⋊C₄×C₇₃

yes

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#95×C₇₃

yes

characteristic

C₂²

#99×C₇₃

yes

—

C₄

—

C₄

—

C₄

—

C₂

—

C₂

—

C₂

—

C₂

#342×C₇₃

yes

characteristic

C₂

#344×C₇₃

yes

—

	subgroup	G/H
center, socle	C₂²×C₇₃	#95
commutator	C₂³	C₄²×C₇₃
Frattini, cosocle	C₂³×C₄	C₂²×C₇₃
prime radical	C₂⁴×C₇₃	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂³×C₄×C₇₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₃

#95

commutator

C₂³

C₄²×C₇₃

Frattini, cosocle

C₂³×C₄

C₂²×C₇₃

prime radical

C₂⁴×C₇₃

C₂×C₄

LCS

G ▷ C₂³ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₇₃ ◁ C₂³×C₄×C₇₃ ◁ G

N	H ≅ G/N	type
7. C₂²⋊C₈×C₇₃	C₄	semidirect
49. #1306	C₇₃	direct (N × H)
50. C₇₃	#1306	direct (N × H)
57. C₂²⋊C₈	C₄×C₇₃	semidirect

H ≅ G/N

type

7. C₂²⋊C₈×C₇₃

C₄

semidirect

49. #1306

C₇₃

direct (N × H)

50. C₇₃

#1306

direct (N × H)

57. C₂²⋊C₈

C₄×C₇₃

semidirect

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

subgroup	index
#407×C₇₃	2
#381×C₇₃	2
#1306	73

subgroup

index

#407×C₇₃

#381×C₇₃

#1306

Subgroups — C₇₃×#1306 #22605459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#1306 #22605459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#1306 #22605459