Group #14537053 — subgroups

Subgroups — #14537053

Order 2176.

348 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 133 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	#11762748	C₂	1	3	yes	maximal
2	#11762742	C₂	1	6	yes	maximal
3	#11762729	C₂	1	3	yes	maximal
4	#11762677	C₂	1	2	yes	maximal
5	C₁₇×#384	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₄	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₁₇	C₂²	1	3	yes	—
8	C₂×C₄⋊C₄×C₁₇	C₂²	1	3	yes	—
9	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₇	C₂²	1	3	yes	—
11	C₂×#63225	C₂²	1	6	yes	—
12	#10587502	C₂²	1	6	yes	—
13	#10587502	C₂²	1	3	yes	—
14	#10587502	C₂²	1	3	yes	—
15	#10587502	C₂²	1	6	yes	—
16	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
17	C₂²×C₄×C₁₇	C₂³	1	3	yes	—
18	C₂²×C₄×C₁₇	C₂³	1	3	yes	—
19	C₂²×Dic₁₇	C₂³	1	2	yes	—
20	C₂²×Dic₁₇	C₂³	1	6	yes	—
21	C₂×Dic₃₄	—	4	3	no	—
22	C₄⋊C₄×C₁₇	—	4	3	no	—
23	C₄⋊C₄×C₁₇	—	4	3	no	—
24	C₄²×C₁₇	Q₈	1	4	yes	complemented
25	C₄⋊Dic₁₇	—	4	3	no	—
26	#63225	—	4	6	no	—
27	C₂³×C₁₇	C₂⁴	1	1	yes	characteristic
28	C₂×C₄×C₁₇	—	2	3	no	—
29	C₂×C₄×C₁₇	—	2	3	no	—
30	C₂×C₄×C₁₇	—	2	3	no	—
31	C₂×C₄×C₁₇	—	2	6	no	—
32	C₂×C₄×C₁₇	—	2	3	no	—
33	C₂×C₄×C₁₇	C₂×Q₈	1	6	yes	—
34	C₂×Dic₁₇	—	2	6	no	—
35	C₂×Dic₁₇	—	2	6	no	—
36	C₂×Dic₁₇	—	2	12	no	—
37	Dic₃₄	—	4	4	no	—
38	#2843	—	17	1	no	maximal, Sylow
39	C₂²×C₁₇	C₂²×Q₈	1	1	yes	characteristic
40	C₂²×C₁₇	H(5,𝔽₂)	1	3	yes	—
41	C₂²×C₁₇	2₋¹⁺⁴	1	3	yes	—
42	C₄×C₁₇	—	2	6	no	—
43	C₄×C₁₇	—	4	1	no	—
44	C₄×C₁₇	—	4	3	no	—
45	Dic₁₇	—	4	2	no	—
46	Dic₁₇	—	4	6	no	—
47	#400	—	17	3	no	—
48	#398	—	17	6	no	—
49	#395	—	17	3	no	—
50	#384	D₁₇	1	1	yes	characteristic
51	#383	—	17	2	no	—
52	C₂×C₁₇	#544	1	3	yes	—
53	C₂×C₁₇	#560	1	1	yes	characteristic
54	C₂×C₁₇	#563	1	3	yes	—
55	C₂²×Q₈	—	17	1	no	—
56	C₂×C₄⋊C₄	—	17	6	no	—
57	C₂×C₄⋊C₄	—	17	3	no	—
58	C₂×C₄⋊C₄	C₂×D₁₇	1	3	yes	—
59	C₂×C₄⋊C₄	C₂×D₁₇	1	3	yes	—
60	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
61	#95	—	17	6	no	—
62	#95	—	17	6	no	—
63	#95	—	17	3	no	—
64	#95	—	17	3	no	—
65	C₁₇	#2843	1	1	yes	characteristic, complemented, Sylow
66	C₂×Q₈	—	68	3	no	—
67	C₂²×C₄	—	17	6	no	—
68	C₂²×C₄	—	17	2	no	—
69	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
70	C₂²×C₄	C₂²×D₁₇	1	3	yes	—
71	C₂²×C₄	C₂²×D₁₇	1	3	yes	—
72	C₄⋊C₄	—	4	3	no	—
73	C₄⋊C₄	—	4	3	no	—
74	C₄⋊C₄	—	68	6	no	—
75	C₄⋊C₄	—	68	3	no	—
76	C₄²	Dic₃₄	1	4	yes	complemented
77	C₂³	C₂³×D₁₇	1	1	yes	characteristic
78	Q₈	—	68	4	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	2	6	no	—
82	C₂×C₄	—	2	3	no	—
83	C₂×C₄	—	2	3	no	—
84	C₂×C₄	—	34	6	no	—
85	C₂×C₄	—	34	6	no	—
86	C₂×C₄	—	34	12	no	—
87	C₂×C₄	C₂×Dic₃₄	1	6	yes	—
88	C₂²	C₂²×Dic₃₄	1	1	yes	characteristic
89	C₂²	#10587691	1	3	yes	—
90	C₂²	#10587694	1	3	yes	—
91	C₄	—	2	6	no	—
92	C₄	—	4	1	no	—
93	C₄	—	4	3	no	—
94	C₄	—	68	2	no	—
95	C₄	—	68	6	no	—
96	C₂	#11763440	1	3	yes	—
97	C₂	#11763546	1	1	yes	characteristic
98	C₂	#11763558	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11762748

C₂

yes

maximal

#11762742

C₂

yes

maximal

#11762729

C₂

yes

maximal

#11762677

C₂

yes

maximal

C₁₇×#384

C₂

yes

maximal, characteristic

C₂²×Dic₃₄

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₇

C₂²

yes

—

C₂×C₄⋊C₄×C₁₇

C₂²

yes

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇

C₂²

yes

—

C₂×#63225

C₂²

yes

—

#10587502

C₂²

yes

—

#10587502

C₂²

yes

—

#10587502

C₂²

yes

—

#10587502

C₂²

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×C₄×C₁₇

C₂³

yes

—

C₂²×C₄×C₁₇

C₂³

yes

—

C₂²×Dic₁₇

C₂³

yes

—

C₂²×Dic₁₇

C₂³

yes

—

C₂×Dic₃₄

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

Q₈

yes

complemented

C₄⋊Dic₁₇

—

#63225

—

C₂³×C₁₇

C₂⁴

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×Q₈

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

#2843

—

maximal, Sylow

C₂²×C₁₇

C₂²×Q₈

yes

characteristic

C₂²×C₁₇

H(5,𝔽₂)

yes

—

C₂²×C₁₇

2₋¹⁺⁴

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

Dic₁₇

—

Dic₁₇

—

—

—

—

D₁₇

yes

characteristic

#383

—

C₂×C₁₇

#544

yes

—

C₂×C₁₇

#560

yes

characteristic

C₂×C₁₇

#563

yes

—

C₂²×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₇

yes

—

C₂×C₄⋊C₄

C₂×D₁₇

yes

—

C₂×C₄²

C₂×D₁₇

yes

—

—

—

—

C₁₇

#2843

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

C₂²×C₄

C₂²×D₁₇

yes

—

C₂²×C₄

C₂²×D₁₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Dic₃₄

yes

complemented

C₂³

C₂³×D₁₇

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×Dic₃₄

yes

—

C₂²

C₂²×Dic₃₄

yes

characteristic

C₂²

#10587691

yes

—

C₂²

#10587694

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₂

#11763440

yes

—

C₂

#11763546

yes

characteristic

C₂

#11763558

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₇
commutator, socle, prime radical, cosocle	C₂³×C₁₇	C₂⁴
Fitting	C₁₇×#384	C₂
hypercenter	#384	D₁₇
hypocenter	C₁₇	#2843
LCS	G ▷ C₂³×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₇

commutator, socle, prime radical, cosocle

C₂³×C₁₇

C₂⁴

Fitting

C₁₇×#384

C₂

D₁₇

C₁₇

G ▷ C₂³×C₁₇ ▷ C₁₇

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
24. C₄²×C₁₇	Q₈	semidirect
65. C₁₇	#2843	semidirect
76. C₄²	Dic₃₄	semidirect

H ≅ G/N

type

24. C₄²×C₁₇

Q₈

semidirect

65. C₁₇

#2843

semidirect

76. C₄²

Dic₃₄

semidirect

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

subgroup	index
#11762748	2
#11762742	2
#11762729	2
#11762677	2
C₁₇×#384	2
#2843	17

subgroup

index

C₁₇×#384

Subgroups — #14537053

Conjugacy classes of subgroups

Complements

Maximal subgroups