Group #14545328 — subgroups

Subgroups — #14545328

Order 2176.

648 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 152 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	#11763757	C₂	1	12	yes	maximal, complemented
2	#11763754	C₂	1	2	yes	maximal, complemented
3	#127×D₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×#595	C₂²	1	4	yes	—
5	C₂×D₄×D₁₇	C₂²	1	6	yes	—
6	C₂×C₂²⋊#595	C₂²	1	12	yes	—
7	C₁₇×#127	C₄	1	1	yes	characteristic, complemented
8	#10587626	C₄	1	6	yes	complemented
9	#10587560	C₄	1	1	yes	characteristic, complemented
10	C₄²×D₁₇	C₂²	1	1	yes	characteristic
11	(C₂×#595)⋊C₄	C₂²	1	12	yes	—
12	C₂³×D₁₇	C₂³	1	4	yes	—
13	C₂²×#595	—	4	12	no	—
14	C₂²×#595	C₂³	1	8	yes	—
15	C₂×D₄×C₁₇	C₂×C₄	1	6	yes	complemented
16	C₂×C₂²⋊D₁₇	C₂×C₄	1	12	yes	complemented
17	D₄×D₁₇	—	4	6	no	—
18	D₄×D₁₇	—	4	6	no	—
19	C₂×D₆₈	C₂×C₄	1	6	yes	complemented
20	C₂×C₄×D₁₇	C₂³	1	3	yes	—
21	C₂²⋊#595	—	4	12	no	—
22	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic, complemented
23	C₄×Dic₁₇	C₂×C₄	1	3	yes	complemented
24	C₂³×C₁₇	C₂²×C₄	1	4	yes	—
25	C₂²×D₁₇	—	2	12	no	—
26	C₂²×D₁₇	—	2	12	no	—
27	C₂²×D₁₇	C₂²×C₄	1	4	yes	—
28	C₂²×D₁₇	C₂⁴	1	1	yes	characteristic
29	C₂×#595	—	2	24	no	—
30	C₂×#595	—	4	16	no	—
31	D₄×C₁₇	—	4	6	no	—
32	C₂×C₄×C₁₇	C₂²×C₄	1	3	yes	—
33	C₂²⋊D₁₇	—	4	12	no	—
34	C₂×Dic₁₇	C₂²×C₄	1	3	yes	—
35	D₆₈	—	4	6	no	—
36	C₄×D₁₇	—	2	6	no	—
37	#2380	—	17	1	no	maximal, Sylow
38	C₂²×C₁₇	—	2	12	no	—
39	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
40	C₂×D₁₇	—	2	12	no	—
41	C₂×D₁₇	—	4	4	no	—
42	C₂×D₁₇	—	4	12	no	—
43	C₂×D₁₇	H(5,𝔽₂)	1	6	yes	—
44	#595	—	4	8	no	—
45	C₄×C₁₇	—	2	3	no	—
46	Dic₁₇	—	2	3	no	—
47	C₂×#127	—	17	1	no	—
48	#386	—	17	12	no	—
49	#383	—	17	2	no	—
50	C₂×C₁₇	—	4	4	no	—
51	C₂×C₁₇	#518	1	3	yes	—
52	D₁₇	—	4	4	no	—
53	D₁₇	#560	1	4	yes	—
54	C₂²×D₄	—	17	6	no	—
55	C₂³×C₄	—	17	4	no	—
56	#127	—	17	6	no	—
57	#127	—	17	1	no	—
58	#127	#595	1	1	yes	characteristic, complemented
59	C₂×C₂²⋊C₄	—	17	12	no	—
60	C₂×C₄²	—	17	1	no	—
61	#95	—	17	12	no	—
62	C₁₇	#2380	1	1	yes	characteristic, complemented, Sylow
63	C₂⁴	—	17	4	no	—
64	C₂×D₄	—	17	12	no	—
65	C₂×D₄	—	17	6	no	—
66	C₂×D₄	—	68	6	no	—
67	C₂×D₄	—	68	6	no	—
68	C₂×D₄	C₂×#595	1	6	yes	complemented
69	C₂²×C₄	—	17	3	no	—
70	C₂²×C₄	—	17	8	no	—
71	C₂²×C₄	—	68	12	no	—
72	C₂²⋊C₄	—	68	12	no	—
73	C₄²	—	17	3	no	—
74	C₄²	C₂×#595	1	1	yes	characteristic, complemented
75	C₂³	—	17	1	no	—
76	C₂³	—	17	4	no	—
77	C₂³	—	34	12	no	—
78	C₂³	—	34	12	no	—
79	C₂³	C₂²×#595	1	4	yes	—
80	D₄	—	4	6	no	—
81	D₄	—	68	6	no	—
82	D₄	—	68	12	no	—
83	C₂×C₄	—	17	3	no	—
84	C₂×C₄	—	34	6	no	—
85	C₂×C₄	—	34	24	no	—
86	C₂×C₄	—	68	16	no	—
87	C₂×C₄	C₂²×#595	1	3	yes	—
88	C₂²	—	2	12	no	—
89	C₂²	—	17	6	no	—
90	C₂²	—	34	12	no	—
91	C₂²	—	68	4	no	—
92	C₂²	—	68	12	no	—
93	C₂²	C₂³×#595	1	1	yes	characteristic
94	C₄	—	2	3	no	—
95	C₄	—	34	3	no	—
96	C₄	—	68	8	no	—
97	C₂	—	4	4	no	—
98	C₂	—	17	4	no	—
99	C₂	—	68	4	no	—
100	C₂	#11763876	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11763757

C₂

yes

maximal, complemented

#11763754

C₂

yes

maximal, complemented

#127×D₁₇

C₂

yes

maximal, characteristic

C₂³×#595

C₂²

yes

—

C₂×D₄×D₁₇

C₂²

yes

—

C₂×C₂²⋊#595

C₂²

yes

—

C₁₇×#127

C₄

yes

characteristic, complemented

#10587626

C₄

yes

complemented

#10587560

C₄

yes

characteristic, complemented

C₄²×D₁₇

C₂²

yes

characteristic

(C₂×#595)⋊C₄

C₂²

yes

—

C₂³×D₁₇

C₂³

yes

—

C₂²×#595

—

C₂²×#595

C₂³

yes

—

C₂×D₄×C₁₇

C₂×C₄

yes

complemented

C₂×C₂²⋊D₁₇

C₂×C₄

yes

complemented

D₄×D₁₇

—

D₄×D₁₇

—

C₂×D₆₈

C₂×C₄

yes

complemented

C₂×C₄×D₁₇

C₂³

yes

—

C₂²⋊#595

—

C₄²×C₁₇

C₂×C₄

yes

characteristic, complemented

C₄×Dic₁₇

C₂×C₄

yes

complemented

C₂³×C₁₇

C₂²×C₄

yes

—

C₂²×D₁₇

—

C₂²×D₁₇

—

C₂²×D₁₇

C₂²×C₄

yes

—

C₂²×D₁₇

C₂⁴

yes

characteristic

C₂×#595

—

C₂×#595

—

D₄×C₁₇

—

C₂×C₄×C₁₇

C₂²×C₄

yes

—

C₂²⋊D₁₇

—

C₂×Dic₁₇

C₂²×C₄

yes

—

D₆₈

—

C₄×D₁₇

—

#2380

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

C₂³×C₄

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

H(5,𝔽₂)

yes

—

#595

—

C₄×C₁₇

—

Dic₁₇

—

C₂×#127

—

#386

—

#383

—

C₂×C₁₇

—

C₂×C₁₇

#518

yes

—

D₁₇

—

D₁₇

#560

yes

—

C₂²×D₄

—

C₂³×C₄

—

—

—

yes

characteristic, complemented

C₂×C₂²⋊C₄

—

C₂×C₄²

—

#95

—

C₁₇

#2380

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×#595

yes

complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×#595

yes

characteristic, complemented

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²×#595

yes

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×#595

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂³×#595

yes

characteristic

C₄

—

C₄

—

C₄

—

C₂

—

C₂

—

C₂

—

100

C₂

#11763876

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×#595
commutator, socle	C₂²×C₁₇	C₂³×C₄
Fitting	C₁₇×#127	C₄
prime radical	#127×D₁₇	C₂
cosocle	C₂²×D₁₇	C₂⁴
hypercenter	#127	#595
hypocenter	C₁₇	#2380
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ #127

subgroup

G/H

center, Frattini

C₂²

C₂³×#595

commutator, socle

C₂²×C₁₇

C₂³×C₄

Fitting

C₁₇×#127

C₄

prime radical

#127×D₁₇

C₂

cosocle

C₂²×D₁₇

C₂⁴

C₁₇

G ▷ C₂²×C₁₇ ▷ C₁₇

C₁ ◁ C₂² ◁ #127

N	H ≅ G/N	type
1. #11763757	C₂	semidirect
2. #11763754	C₂	semidirect
7. C₁₇×#127	C₄	semidirect
8. #10587626	C₄	semidirect
9. #10587560	C₄	semidirect
15. C₂×D₄×C₁₇	C₂×C₄	semidirect
16. C₂×C₂²⋊D₁₇	C₂×C₄	semidirect
19. C₂×D₆₈	C₂×C₄	semidirect
22. C₄²×C₁₇	C₂×C₄	semidirect
23. C₄×Dic₁₇	C₂×C₄	semidirect
58. #127	#595	semidirect
62. C₁₇	#2380	semidirect
68. C₂×D₄	C₂×#595	semidirect
74. C₄²	C₂×#595	semidirect

H ≅ G/N

type

1. #11763757

C₂

semidirect

2. #11763754

C₂

semidirect

7. C₁₇×#127

C₄

semidirect

8. #10587626

C₄

semidirect

9. #10587560

C₄

semidirect

15. C₂×D₄×C₁₇

C₂×C₄

semidirect

16. C₂×C₂²⋊D₁₇

C₂×C₄

semidirect

19. C₂×D₆₈

C₂×C₄

semidirect

22. C₄²×C₁₇

C₂×C₄

semidirect

23. C₄×Dic₁₇

C₂×C₄

semidirect

58. #127

#595

semidirect

62. C₁₇

#2380

semidirect

68. C₂×D₄

C₂×#595

semidirect

74. C₄²

C₂×#595

semidirect

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

subgroup	index
#11763757	2
#11763754	2
#127×D₁₇	2
#2380	17

subgroup

index

#11763757

#11763754

#127×D₁₇

#2380

Subgroups — #14545328

Conjugacy classes of subgroups

Complements

Maximal subgroups