Group #17979222 — subgroups

Subgroups — #564×#595 #17979222

Order 4352.

1800 conjugacy classes of subgroups; only the 776 normal subgroups are shown.

normal subgroups only

This group has 1800 conjugacy classes of subgroups — too many to draw. Only its 776 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	#14545330	C₂	1	15	yes	maximal
2	#125×#595	C₂	1	15	yes	maximal
3	D₁₇×#564	C₂	1	1	yes	maximal, characteristic
4	#11763760	C₂²	1	30	yes	—
5	#11763758	C₂²	1	60	yes	—
6	#11763753	C₂²	1	15	yes	—
7	C₄⋊C₄×#595	C₂²	1	30	yes	—
8	C₄²×#595	C₂²	1	5	yes	—
9	C₁₇⋊#564	C₄	1	15	yes	complemented
10	#125×D₁₇	C₂²	1	15	yes	—
11	C₁₇×#564	C₄	1	1	yes	characteristic, complemented
12	C₂×C₄⋊#595	C₂³	1	30	yes	—
13	C₂×C₄×#595	C₂³	1	15	yes	—
14	C₂×C₄×#595	C₂³	1	15	yes	—
15	C₁₇×#125	C₂×C₄	1	15	yes	—
16	C₄⋊C₄×D₁₇	C₂³	1	30	yes	—
17	#10587576	C₂×C₄	1	30	yes	—
18	#10587575	C₂×C₄	1	60	yes	—
19	C₄²×D₁₇	C₂³	1	5	yes	—
20	#10587556	C₂×C₄	1	15	yes	—
21	(C₂×#595)⋊C₄	C₂³	1	60	yes	—
22	C₂²×#595	C₂⁴	1	15	yes	—
23	C₂²×#595	C₂⁴	1	1	yes	characteristic
24	C₂×C₄×D₁₇	C₂⁴	1	15	yes	—
25	C₄⋊#595	—	4	30	no	—
26	C₄×#595	—	2	30	no	—
27	C₄×#595	—	4	15	no	—
28	C₄⋊C₄×C₁₇	C₂²×C₄	1	30	yes	—
29	C₄²×C₁₇	C₂²×C₄	1	5	yes	—
30	C₄⋊Dic₁₇	C₂²×C₄	1	30	yes	—
31	#63225	C₂²×C₄	1	60	yes	—
32	C₄×Dic₁₇	C₂²×C₄	1	15	yes	—
33	C₄×#564	—	17	1	no	maximal, Sylow
34	C₂²×D₁₇	C₂⁵	1	1	yes	characteristic
35	C₂×#595	—	2	15	no	—
36	C₂×#595	—	2	30	no	—
37	C₂×#595	2₋¹⁺⁴	1	6	yes	—
38	C₂×C₄×C₁₇	C₂³×C₄	1	15	yes	—
39	C₂×Dic₁₇	C₂³×C₄	1	15	yes	—
40	C₄×D₁₇	—	2	15	no	—
41	C₄×D₁₇	—	2	15	no	—
42	C₂×#564	—	17	1	no	—
43	#2382	—	17	15	no	—
44	C₄×#125	—	17	15	no	—
45	C₂²×C₁₇	C₂⁴×C₄	1	1	yes	characteristic
46	C₂×D₁₇	C₂×2₋¹⁺⁴	1	3	yes	—
47	C₂×D₁₇	#585	1	3	yes	—
48	#595	—	4	15	no	—
49	#595	#564	1	4	yes	direct product factor
50	C₄×C₁₇	—	2	15	no	—
51	Dic₁₇	—	2	15	no	—
52	#564	—	17	15	no	—
53	#564	#595	1	1	yes	characteristic, direct product factor
54	C₂×#125	—	17	15	no	—
55	#389	—	17	30	no	—
56	#387	—	17	60	no	—
57	#382	—	17	15	no	—
58	C₄×C₄⋊C₄	—	17	30	no	—
59	C₄³	—	17	5	no	—
60	C₂×C₁₇	C₄×2₋¹⁺⁴	1	3	yes	—
61	D₁₇	C₂×#564	1	1	yes	characteristic
62	D₁₇	#3531	1	3	yes	—
63	#125	—	17	60	no	—
64	#125	—	17	30	no	—
65	#125	—	17	15	no	—
66	#125	C₂×#595	1	15	yes	—
67	C₂×C₄⋊C₄	—	17	30	no	—
68	C₂×C₄⋊C₄	—	17	30	no	—
69	C₂×C₄²	—	17	5	no	—
70	C₂×C₄²	—	17	15	no	—
71	C₂×C₄²	—	17	15	no	—
72	#95	—	17	60	no	—
73	C₁₇	C₄×#564	1	1	yes	characteristic, complemented, Sylow
74	C₂²×C₄	—	17	1	no	—
75	C₂²×C₄	—	17	15	no	—
76	C₂²×C₄	—	17	15	no	—
77	C₄⋊C₄	—	17	60	no	—
78	C₄⋊C₄	—	17	30	no	—
79	C₄⋊C₄	—	68	30	no	—
80	C₄⋊C₄	C₂²×#595	1	30	yes	—
81	C₄²	—	17	15	no	—
82	C₄²	—	34	30	no	—
83	C₄²	—	68	15	no	—
84	C₄²	C₂²×#595	1	5	yes	—
85	C₂³	—	17	1	no	—
86	C₂×C₄	—	17	6	no	—
87	C₂×C₄	—	17	15	no	—
88	C₂×C₄	—	34	15	no	—
89	C₂×C₄	—	34	15	no	—
90	C₂×C₄	—	34	15	no	—
91	C₂×C₄	—	34	30	no	—
92	C₂×C₄	C₂³×#595	1	15	yes	—
93	C₂²	—	17	3	no	—
94	C₂²	—	17	3	no	—
95	C₂²	C₂⁴×#595	1	1	yes	characteristic
96	C₄	—	2	15	no	—
97	C₄	—	17	4	no	—
98	C₄	—	34	15	no	—
99	C₄	—	68	15	no	—
100	C₂	—	17	1	no	—
101	C₂	—	17	3	no	—
102	C₂	2₋¹⁺⁴×#595	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14545330

C₂

yes

maximal

#125×#595

C₂

yes

maximal

D₁₇×#564

C₂

yes

maximal, characteristic

#11763760

C₂²

yes

—

#11763758

C₂²

yes

—

#11763753

C₂²

yes

—

C₄⋊C₄×#595

C₂²

yes

—

C₄²×#595

C₂²

yes

—

C₁₇⋊#564

C₄

yes

complemented

#125×D₁₇

C₂²

yes

—

C₁₇×#564

C₄

yes

characteristic, complemented

C₂×C₄⋊#595

C₂³

yes

—

C₂×C₄×#595

C₂³

yes

—

C₂×C₄×#595

C₂³

yes

—

C₁₇×#125

C₂×C₄

yes

—

C₄⋊C₄×D₁₇

C₂³

yes

—

#10587576

C₂×C₄

yes

—

#10587575

C₂×C₄

yes

—

C₄²×D₁₇

C₂³

yes

—

#10587556

C₂×C₄

yes

—

(C₂×#595)⋊C₄

C₂³

yes

—

C₂²×#595

C₂⁴

yes

—

C₂²×#595

C₂⁴

yes

characteristic

C₂×C₄×D₁₇

C₂⁴

yes

—

C₄⋊#595

—

C₄×#595

—

C₄×#595

—

C₄⋊C₄×C₁₇

C₂²×C₄

yes

—

C₄²×C₁₇

C₂²×C₄

yes

—

C₄⋊Dic₁₇

C₂²×C₄

yes

—

#63225

C₂²×C₄

yes

—

C₄×Dic₁₇

C₂²×C₄

yes

—

C₄×#564

—

maximal, Sylow

C₂²×D₁₇

C₂⁵

yes

—

—

2₋¹⁺⁴

yes

—

C₂×C₄×C₁₇

C₂³×C₄

yes

—

C₂×Dic₁₇

C₂³×C₄

yes

—

C₄×D₁₇

—

C₄×D₁₇

—

C₂×#564

—

#2382

—

C₄×#125

—

C₂²×C₁₇

C₂⁴×C₄

yes

characteristic

C₂×D₁₇

C₂×2₋¹⁺⁴

yes

—

C₂×D₁₇

#585

yes

—

#595

—

#595

#564

yes

direct product factor

C₄×C₁₇

—

Dic₁₇

—

#564

—

#564

#595

yes

characteristic, direct product factor

—

—

—

—

C₄×C₄⋊C₄

—

C₄³

—

C₂×C₁₇

C₄×2₋¹⁺⁴

yes

—

D₁₇

C₂×#564

yes

characteristic

D₁₇

#3531

yes

—

—

—

—

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

#95

—

C₁₇

C₄×#564

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×#595

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²×#595

yes

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂³×#595

yes

—

C₂²

—

C₂²

—

C₂²

C₂⁴×#595

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

100

C₂

—

101

C₂

—

102

C₂

2₋¹⁺⁴×#595

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×#595
commutator, socle	C₂²×C₁₇	C₂⁴×C₄
Fitting	C₁₇×#564	C₄
prime radical, cosocle	C₂²×D₁₇	C₂⁵
hypercenter	#564	#595
hypocenter	C₁₇	C₄×#564
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ #564

subgroup

G/H

center, Frattini

C₂²

C₂⁴×#595

commutator, socle

C₂²×C₁₇

C₂⁴×C₄

Fitting

C₁₇×#564

C₄

prime radical, cosocle

C₂²×D₁₇

C₂⁵

C₁₇

G ▷ C₂²×C₁₇ ▷ C₁₇

C₁ ◁ C₂² ◁ #564

N	H ≅ G/N	type
9. C₁₇⋊#564	C₄	semidirect
11. C₁₇×#564	C₄	semidirect
49. #595	#564	direct (N × H)
53. #564	#595	direct (N × H)
73. C₁₇	C₄×#564	semidirect

H ≅ G/N

type

9. C₁₇⋊#564

C₄

semidirect

11. C₁₇×#564

C₄

semidirect

49. #595

#564

direct (N × H)

53. #564

#595

direct (N × H)

73. C₁₇

C₄×#564

semidirect

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

subgroup	index
#14545330	2
#125×#595	2
D₁₇×#564	2
C₄×#564	17

subgroup

index

#14545330

#125×#595

D₁₇×#564

C₄×#564

Subgroups — #564×#595 #17979222

Conjugacy classes of subgroups

Complements

Maximal subgroups