Subgroups — #11763142

Order 1088.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₁₇	C₂	1	1	yes	maximal, characteristic
3	#10587506	C₂	1	2	yes	maximal, complemented
4	C₂²⋊#3622	C₂	1	1	yes	maximal, characteristic
5	#10587478	C₂	1	2	yes	maximal, complemented
6	C₂×Q₈×C₁₇	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₁₇	C₂²	1	1	yes	characteristic
8	C₂²×Dic₁₇	—	2	1	no	—
9	C₄⋊C₄×C₁₇	—	2	1	no	—
10	C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
11	C₂²⋊C₄×C₁₇	—	2	1	no	—
12	C₄⋊Dic₁₇	—	2	1	no	—
13	C₄⋊Dic₁₇	C₂²	1	2	yes	—
14	C₂×#3622	C₂²	1	2	yes	—
15	C₂³×C₁₇	D₄	1	1	yes	characteristic
16	Q₈×C₁₇	—	2	2	no	—
17	C₂×C₄×C₁₇	—	2	2	no	—
18	C₂×C₄×C₁₇	—	2	1	no	—
19	C₂×C₄×C₁₇	—	2	1	no	—
20	C₂×C₄×C₁₇	D₄	1	1	yes	characteristic
21	C₂×C₄×C₁₇	C₂³	1	1	yes	characteristic
22	C₂×Dic₁₇	—	2	2	no	—
23	C₂×Dic₁₇	—	4	2	no	—
24	#3622	—	2	2	no	—
25	C₂²×C₁₇	—	2	1	no	—
26	C₂²×C₁₇	—	2	1	no	—
27	C₂²×C₁₇	Q₁₆	1	2	yes	—
28	C₂²×C₁₇	C₂×D₄	1	1	yes	characteristic
29	C₄×C₁₇	—	2	1	no	—
30	C₄×C₁₇	—	4	1	no	—
31	C₄×C₁₇	—	4	1	no	—
32	C₄×C₁₇	#41	1	2	yes	—
33	Dic₁₇	—	4	2	no	—
34	#484	—	17	1	no	maximal, Sylow
35	C₂×C₁₇	—	2	2	no	—
36	C₂×C₁₇	#123	1	1	yes	characteristic
37	C₂×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
38	C₂×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
39	C₂²⋊Q₈	D₁₇	1	1	yes	characteristic
40	C₂×C₄⋊C₄	—	17	1	no	—
41	#107	—	17	2	no	—
42	#103	—	17	2	no	—
43	C₂²⋊C₈	—	17	1	no	—
44	C₁₇	#484	1	1	yes	characteristic, complemented, Sylow
45	C₂×Q₈	C₂×D₁₇	1	1	yes	characteristic
46	C₂²×C₄	—	34	1	no	—
47	C₂²×C₄	C₂×D₁₇	1	1	yes	characteristic
48	C₂×C₈	—	17	2	no	—
49	C₄⋊C₄	—	2	1	no	—
50	C₄⋊C₄	—	17	2	no	—
51	C₄⋊C₄	—	34	1	no	—
52	C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
53	C₂²⋊C₄	—	2	1	no	—
54	C₂³	C₂²⋊D₁₇	1	1	yes	characteristic
55	Q₈	—	2	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	34	2	no	—
60	C₂×C₄	—	68	2	no	—
61	C₂×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₁₇	1	1	yes	characteristic
63	C₈	—	34	2	no	—
64	C₂²	—	2	1	no	—
65	C₂²	—	2	1	no	—
66	C₂²	#63231	1	2	yes	—
67	C₂²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	4	1	no	—
70	C₄	—	4	1	no	—
71	C₄	—	68	2	no	—
72	C₄	#63254	1	2	yes	—
73	C₂	—	2	2	no	—
74	C₂	#10587621	1	1	yes	characteristic
75	C₂	C₂×#63231	1	1	yes	characteristic
76	C₂	#10587635	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₇
Fitting	C₁₇×C₂²⋊Q₈	C₂
socle	C₂²×C₁₇	C₂×D₄
prime radical	C₂³×C₁₇	D₄
hypercenter	C₂²⋊Q₈	D₁₇
hypocenter	C₁₇	#484
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊Q₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #10587506	C₂	semidirect
5. #10587478	C₂	semidirect
44. C₁₇	#484	semidirect

Maximal subgroups

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

subgroup	index
C₁₇×C₂²⋊Q₈	2
C₂×C₄⋊Dic₁₇	2
#10587506	2
C₂²⋊#3622	2
#10587478	2
#484	17