Group #32916933 — subgroups

Subgroups — C₇₃×#7529 #32916933

Order 18688.

186 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 30 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×#1712	C₂	1	1	yes	maximal, characteristic
2	C₇₃×#1408	C₂	1	2	yes	maximal, complemented
3	#493×C₇₃	—	2	2	no	—
4	#493×C₇₃	C₄	1	1	yes	characteristic, complemented
5	#493×C₇₃	C₂²	1	1	yes	characteristic
6	#353×C₇₃	—	2	2	no	—
7	C₈²×C₇₃	C₄	1	1	yes	characteristic, complemented
8	C₂×D₈×C₇₃	—	4	1	no	—
9	C₂×D₈×C₇₃	—	4	4	no	—
10	C₂×D₈×C₇₃	—	4	1	no	—
11	#127×C₇₃	—	2	1	no	—
12	#127×C₇₃	D₄	1	2	yes	—
13	#99×C₇₃	—	4	2	no	—
14	C₄×C₈×C₇₃	—	2	1	no	—
15	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
16	C₂×D₄×C₇₃	—	2	2	no	—
17	C₂×D₄×C₇₃	—	4	2	no	—
18	C₂×D₄×C₇₃	—	4	2	no	—
19	C₂×D₄×C₇₃	—	4	1	no	—
20	D₈×C₇₃	—	8	2	no	—
21	D₈×C₇₃	—	8	8	no	—
22	D₈×C₇₃	—	8	2	no	—
23	C₂×C₈×C₇₃	—	2	2	no	—
24	C₂×C₈×C₇₃	—	2	1	no	—
25	C₂²⋊C₄×C₇₃	—	8	2	no	—
26	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂³×C₇₃	—	4	2	no	—
28	C₂³×C₇₃	—	8	1	no	—
29	D₄×C₇₃	—	8	2	no	—
30	D₄×C₇₃	—	8	4	no	—
31	D₄×C₇₃	—	8	4	no	—
32	D₄×C₇₃	—	8	2	no	—
33	C₂×C₄×C₇₃	—	2	1	no	—
34	C₂×C₄×C₇₃	—	16	2	no	—
35	C₂×C₄×C₇₃	#99	1	1	yes	characteristic
36	C₈×C₇₃	—	2	4	no	—
37	C₈×C₇₃	—	2	2	no	—
38	C₂²×C₇₃	—	8	2	no	—
39	C₂²×C₇₃	—	16	2	no	—
40	C₂²×C₇₃	—	16	2	no	—
41	C₂²×C₇₃	—	16	1	no	—
42	C₂²×C₇₃	#353	1	1	yes	characteristic
43	C₄×C₇₃	—	2	2	no	—
44	C₄×C₇₃	—	2	1	no	—
45	C₄×C₇₃	—	32	2	no	—
46	#7529	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
47	C₂×C₇₃	—	2	1	no	—
48	C₂×C₇₃	—	16	2	no	—
49	C₂×C₇₃	—	32	1	no	—
50	C₂×C₇₃	#1408	1	1	yes	characteristic
51	#1712	C₂×C₇₃	1	1	yes	characteristic
52	#1408	C₂×C₇₃	1	2	yes	complemented
53	C₇₃	#7529	1	1	yes	characteristic, direct product factor, Sylow
54	#493	—	2	2	no	—
55	#493	C₄×C₇₃	1	1	yes	characteristic, complemented
56	#493	C₂²×C₇₃	1	1	yes	characteristic
57	#353	—	2	2	no	—
58	C₈²	C₄×C₇₃	1	1	yes	characteristic, complemented
59	C₂×D₈	—	4	1	no	—
60	C₂×D₈	—	4	1	no	—
61	C₂×D₈	—	4	4	no	—
62	#127	—	2	1	no	—
63	#127	D₄×C₇₃	1	2	yes	—
64	#99	—	4	2	no	—
65	C₄×C₈	—	2	1	no	—
66	C₄×C₈	C₂×C₄×C₇₃	1	1	yes	characteristic
67	C₂×D₄	—	2	2	no	—
68	C₂×D₄	—	4	2	no	—
69	C₂×D₄	—	4	2	no	—
70	C₂×D₄	—	4	1	no	—
71	D₈	—	8	2	no	—
72	D₈	—	8	2	no	—
73	D₈	—	8	8	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	2	1	no	—
76	C₂²⋊C₄	—	8	2	no	—
77	C₄²	C₂²⋊C₄×C₇₃	1	1	yes	characteristic
78	C₂³	—	4	2	no	—
79	C₂³	—	8	1	no	—
80	D₄	—	8	2	no	—
81	D₄	—	8	4	no	—
82	D₄	—	8	4	no	—
83	D₄	—	8	2	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	16	2	no	—
86	C₂×C₄	#99×C₇₃	1	1	yes	characteristic
87	C₈	—	2	4	no	—
88	C₈	—	2	2	no	—
89	C₂²	—	8	2	no	—
90	C₂²	—	16	2	no	—
91	C₂²	—	16	2	no	—
92	C₂²	—	16	1	no	—
93	C₂²	#353×C₇₃	1	1	yes	characteristic
94	C₄	—	2	2	no	—
95	C₄	—	2	1	no	—
96	C₄	—	32	2	no	—
97	C₂	—	2	1	no	—
98	C₂	—	16	2	no	—
99	C₂	—	32	1	no	—
100	C₂	C₇₃×#1408	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃×#1712

C₂

yes

maximal, characteristic

C₇₃×#1408

C₂

yes

maximal, complemented

#493×C₇₃

—

#493×C₇₃

C₄

yes

characteristic, complemented

#493×C₇₃

C₂²

yes

characteristic

#353×C₇₃

—

C₈²×C₇₃

C₄

yes

characteristic, complemented

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

#127×C₇₃

—

#127×C₇₃

D₄

yes

—

#99×C₇₃

—

C₄×C₈×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

C₂²⋊C₄

yes

characteristic

C₂³×C₇₃

—

C₂³×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#99

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#353

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

#7529

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#1408

yes

characteristic

#1712

C₂×C₇₃

yes

characteristic

#1408

C₂×C₇₃

yes

complemented

C₇₃

#7529

yes

characteristic, direct product factor, Sylow

#493

—

#493

C₄×C₇₃

yes

characteristic, complemented

#493

C₂²×C₇₃

yes

characteristic

#353

—

C₈²

C₄×C₇₃

yes

characteristic, complemented

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

#127

—

#127

D₄×C₇₃

yes

—

#99

—

C₄×C₈

—

C₄×C₈

C₂×C₄×C₇₃

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

D₈

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₂²⋊C₄

—

C₄²

C₂²⋊C₄×C₇₃

yes

characteristic

C₂³

—

C₂³

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#99×C₇₃

yes

characteristic

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#353×C₇₃

yes

characteristic

C₄

—

C₄

—

C₄

—

C₂

—

C₂

—

C₂

—

100

C₂

C₇₃×#1408

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₇₃	#1408
commutator	C₄×C₈	C₂×C₄×C₇₃
Frattini, cosocle	#493	C₂²×C₇₃
prime radical	C₇₃×#1712	C₂
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₃ ◁ C₂²×C₇₃ ◁ C₂×C₄×C₇₃ ◁ C₄²×C₇₃ ◁ #493×C₇₃ ◁ G

subgroup

G/H

center, socle

C₂×C₇₃

#1408

commutator

C₄×C₈

C₂×C₄×C₇₃

Frattini, cosocle

#493

C₂²×C₇₃

prime radical

C₇₃×#1712

C₂

LCS

G ▷ C₄×C₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₇₃ ◁ C₂²×C₇₃ ◁ C₂×C₄×C₇₃ ◁ C₄²×C₇₃ ◁ #493×C₇₃ ◁ G

N	H ≅ G/N	type
2. C₇₃×#1408	C₂	semidirect
4. #493×C₇₃	C₄	semidirect
7. C₈²×C₇₃	C₄	semidirect
46. #7529	C₇₃	direct (N × H)
52. #1408	C₂×C₇₃	semidirect
53. C₇₃	#7529	direct (N × H)
55. #493	C₄×C₇₃	semidirect
58. C₈²	C₄×C₇₃	semidirect

H ≅ G/N

type

2. C₇₃×#1408

C₂

semidirect

4. #493×C₇₃

C₄

semidirect

7. C₈²×C₇₃

C₄

semidirect

46. #7529

C₇₃

direct (N × H)

52. #1408

C₂×C₇₃

semidirect

53. C₇₃

#7529

direct (N × H)

55. #493

C₄×C₇₃

semidirect

58. C₈²

C₄×C₇₃

semidirect

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

subgroup	index
C₇₃×#1712	2
C₇₃×#1408	2
#7529	73

subgroup

index

C₇₃×#1712

C₇₃×#1408

#7529

Subgroups — C₇₃×#7529 #32916933

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#7529 #32916933

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#7529 #32916933