Group #17949592 — subgroups

Subgroups — #17949592

Order 4352.

186 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 22 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	#14543916	C₂	1	2	yes	maximal, complemented
2	#14528398	C₂	1	1	yes	maximal, characteristic
3	#11763714	—	2	2	no	—
4	#11763186	C₂²	1	1	yes	characteristic
5	#11763185	—	2	2	no	—
6	C₈×#3622	C₄	1	1	yes	characteristic, complemented
7	C₁₇×#493	C₄	1	1	yes	characteristic, complemented
8	C₂×D₈×C₁₇	—	4	1	no	—
9	C₁₇×#127	—	2	1	no	—
10	C₂×#63228	—	4	4	no	—
11	C₂×D₁₃₆	—	4	1	no	—
12	#10587560	D₄	1	2	yes	—
13	#10587539	—	4	2	no	—
14	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
15	C₄×#3622	—	2	1	no	—
16	C₂×D₄×C₁₇	—	4	2	no	—
17	C₂×D₄×C₁₇	—	4	1	no	—
18	C₂×D₆₈	—	2	2	no	—
19	C₂×D₆₈	—	4	2	no	—
20	C₂²⋊#595	—	8	2	no	—
21	D₈×C₁₇	—	8	2	no	—
22	C₂×C₈×C₁₇	—	2	1	no	—
23	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
24	#63228	—	8	8	no	—
25	C₂×#3622	—	2	2	no	—
26	D₁₃₆	—	8	2	no	—
27	#7529	—	17	1	no	maximal, Sylow
28	C₂³×C₁₇	—	8	1	no	—
29	C₂²×D₁₇	—	4	2	no	—
30	C₂×#595	—	16	2	no	—
31	D₄×C₁₇	—	8	2	no	—
32	D₄×C₁₇	—	8	4	no	—
33	C₂×C₄×C₁₇	—	2	1	no	—
34	C₂×C₄×C₁₇	#99	1	1	yes	characteristic
35	D₆₈	—	8	2	no	—
36	D₆₈	—	8	4	no	—
37	C₈×C₁₇	—	2	2	no	—
38	#3622	—	2	4	no	—
39	#1712	—	17	1	no	—
40	#1408	—	17	2	no	—
41	C₂²×C₁₇	—	16	2	no	—
42	C₂²×C₁₇	—	16	1	no	—
43	C₂²×C₁₇	#353	1	1	yes	characteristic
44	C₂×D₁₇	—	8	2	no	—
45	C₂×D₁₇	—	16	2	no	—
46	#595	—	32	2	no	—
47	C₄×C₁₇	—	2	2	no	—
48	C₄×C₁₇	—	2	1	no	—
49	#493	—	17	1	no	—
50	#493	—	34	2	no	—
51	#493	#595	1	1	yes	characteristic, complemented
52	#353	—	34	2	no	—
53	C₈²	—	17	1	no	—
54	C₂×C₁₇	—	2	1	no	—
55	C₂×C₁₇	—	32	1	no	—
56	C₂×C₁₇	#1408	1	1	yes	characteristic
57	D₁₇	—	16	2	no	—
58	C₂×D₈	—	4	1	no	—
59	C₂×D₈	—	68	1	no	—
60	C₂×D₈	—	68	4	no	—
61	#127	—	2	1	no	—
62	#127	—	17	2	no	—
63	#99	—	68	2	no	—
64	C₄×C₈	—	34	1	no	—
65	C₄×C₈	C₂×#595	1	1	yes	characteristic
66	C₁₇	#7529	1	1	yes	characteristic, complemented, Sylow
67	C₂×D₄	—	4	2	no	—
68	C₂×D₄	—	4	1	no	—
69	C₂×D₄	—	34	2	no	—
70	C₂×D₄	—	68	2	no	—
71	D₈	—	8	2	no	—
72	D₈	—	136	2	no	—
73	D₈	—	136	8	no	—
74	C₂×C₈	—	2	1	no	—
75	C₂×C₈	—	34	2	no	—
76	C₂²⋊C₄	—	136	2	no	—
77	C₄²	C₂²⋊#595	1	1	yes	characteristic
78	C₂³	—	8	1	no	—
79	C₂³	—	68	2	no	—
80	D₄	—	8	2	no	—
81	D₄	—	8	4	no	—
82	D₄	—	136	4	no	—
83	D₄	—	136	2	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	272	2	no	—
86	C₂×C₄	#10587551	1	1	yes	characteristic
87	C₈	—	2	2	no	—
88	C₈	—	34	4	no	—
89	C₂²	—	16	2	no	—
90	C₂²	—	16	1	no	—
91	C₂²	—	272	2	no	—
92	C₂²	—	136	2	no	—
93	C₂²	#11763713	1	1	yes	characteristic
94	C₄	—	2	2	no	—
95	C₄	—	2	1	no	—
96	C₄	—	544	2	no	—
97	C₂	—	2	1	no	—
98	C₂	—	272	2	no	—
99	C₂	—	32	1	no	—
100	C₂	#14543915	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14543916

C₂

yes

maximal, complemented

#14528398

C₂

yes

maximal, characteristic

#11763714

—

#11763186

C₂²

yes

characteristic

#11763185

—

C₈×#3622

C₄

yes

characteristic, complemented

C₁₇×#493

C₄

yes

characteristic, complemented

C₂×D₈×C₁₇

—

C₁₇×#127

—

C₂×#63228

—

C₂×D₁₃₆

—

#10587560

D₄

yes

—

#10587539

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×#3622

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂²⋊#595

—

D₈×C₁₇

—

C₂×C₈×C₁₇

—

C₄²×C₁₇

C₂²⋊C₄

yes

characteristic

#63228

—

C₂×#3622

—

D₁₃₆

—

#7529

—

maximal, Sylow

C₂³×C₁₇

—

C₂²×D₁₇

—

C₂×#595

—

D₄×C₁₇

—

D₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#99

yes

characteristic

D₆₈

—

D₆₈

—

C₈×C₁₇

—

#3622

—

#1712

—

#1408

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#353

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

#595

—

C₄×C₁₇

—

C₄×C₁₇

—

—

—

yes

characteristic, complemented

#353

—

C₈²

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

#1408

yes

characteristic

D₁₇

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

#127

—

#127

—

#99

—

C₄×C₈

—

C₄×C₈

C₂×#595

yes

characteristic

C₁₇

#7529

yes

characteristic, complemented, Sylow

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

D₈

—

D₈

—

136

—

D₈

—

136

—

C₂×C₈

—

C₂×C₈

—

C₂²⋊C₄

—

136

—

C₄²

C₂²⋊#595

yes

characteristic

C₂³

—

C₂³

—

D₄

—

D₄

—

D₄

—

136

—

D₄

—

136

—

C₂×C₄

—

C₂×C₄

—

272

—

C₂×C₄

#10587551

yes

characteristic

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

272

—

C₂²

—

136

—

C₂²

#11763713

yes

characteristic

C₄

—

C₄

—

C₄

—

544

—

C₂

—

C₂

—

272

—

C₂

—

100

C₂

#14543915

yes

characteristic

	subgroup	G/H
center	C₂	#14543915
commutator	C₄×C₈×C₁₇	C₂×C₄
Frattini	C₄×C₈	C₂×#595
Fitting	C₁₇×#493	C₄
socle	C₂×C₁₇	#1408
prime radical	#14528398	C₂
cosocle	#11763186	C₂²
hypercenter	#493	#595
hypocenter	C₁₇	#7529
LCS	G ▷ C₄×C₈×C₁₇ ▷ C₄²×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #493

subgroup

G/H

center

C₂

#14543915

commutator

C₄×C₈×C₁₇

C₂×C₄

Frattini

C₄×C₈

C₂×#595

Fitting

C₁₇×#493

C₄

C₂×C₁₇

C₂

C₂²

C₁₇

G ▷ C₄×C₈×C₁₇ ▷ C₄²×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #493

N	H ≅ G/N	type
1. #14543916	C₂	semidirect
6. C₈×#3622	C₄	semidirect
7. C₁₇×#493	C₄	semidirect
51. #493	#595	semidirect
66. C₁₇	#7529	semidirect

H ≅ G/N

type

1. #14543916

C₂

semidirect

6. C₈×#3622

C₄

semidirect

7. C₁₇×#493

C₄

semidirect

51. #493

#595

semidirect

66. C₁₇

#7529

semidirect

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

subgroup	index
#14543916	2
#14528398	2
#7529	17

subgroup

index

#14543916

#14528398

#7529

Subgroups — #17949592

Conjugacy classes of subgroups

Complements

Maximal subgroups