Subgroups — #17949598

Order 4352.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14543922	C₂	1	2	yes	maximal
2	#14528400	C₂	1	1	yes	maximal, characteristic
3	#11763720	—	2	2	no	—
4	#11763212	—	2	2	no	—
5	#11763189	C₂²	1	1	yes	characteristic
6	C₈×#3622	C₄	1	1	yes	characteristic
7	C₁₇×#494	C₄	1	1	yes	characteristic
8	C₂×Q₁₆×C₁₇	—	4	1	no	—
9	C₁₇×C₄⋊Q₈	—	2	1	no	—
10	C₂×Dic₆₈	—	4	1	no	—
11	#10587555	D₄	1	2	yes	—
12	#10587545	—	4	2	no	—
13	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
14	#10587478	—	4	4	no	—
15	C₄×#3622	—	2	1	no	—
16	C₂×Q₈×C₁₇	—	4	1	no	—
17	C₂×Dic₃₄	—	2	2	no	—
18	#63247	—	8	2	no	—
19	Q₁₆×C₁₇	—	8	2	no	—
20	C₂×C₈×C₁₇	—	2	1	no	—
21	C₄⋊C₄×C₁₇	—	4	2	no	—
22	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₄⋊Dic₁₇	—	4	2	no	—
24	C₂×#3622	—	2	2	no	—
25	Dic₆₈	—	8	2	no	—
26	#7532	—	17	1	no	maximal, Sylow
27	Q₈×C₁₇	—	8	2	no	—
28	C₂×C₄×C₁₇	—	2	1	no	—
29	C₂×C₄×C₁₇	—	8	1	no	—
30	C₂×C₄×C₁₇	#99	1	1	yes	characteristic
31	C₂×Dic₁₇	—	4	2	no	—
32	Dic₃₄	—	8	2	no	—
33	#3624	—	16	2	no	—
34	C₈×C₁₇	—	2	2	no	—
35	#3622	—	2	4	no	—
36	#1713	—	17	1	no	—
37	#1411	—	17	2	no	—
38	C₂²×C₁₇	#353	1	1	yes	characteristic
39	C₄×C₁₇	—	2	2	no	—
40	C₄×C₁₇	—	2	1	no	—
41	C₄×C₁₇	—	16	1	no	—
42	Dic₁₇	—	8	2	no	—
43	#500	—	34	2	no	—
44	#494	—	17	1	no	—
45	#494	#595	1	1	yes	characteristic
46	#356	—	34	2	no	—
47	C₈²	—	17	1	no	—
48	C₂×C₁₇	—	2	1	no	—
49	C₂×C₁₇	#1408	1	1	yes	characteristic
50	C₂×Q₁₆	—	4	1	no	—
51	C₂×Q₁₆	—	68	1	no	—
52	C₄⋊Q₈	—	2	1	no	—
53	C₄⋊Q₈	—	17	2	no	—
54	#107	—	68	4	no	—
55	#101	—	68	2	no	—
56	C₄×C₈	—	34	1	no	—
57	C₄×C₈	C₂×#595	1	1	yes	characteristic
58	C₁₇	#7532	1	1	yes	characteristic, complemented, Sylow
59	C₂×Q₈	—	4	1	no	—
60	C₂×Q₈	—	34	2	no	—
61	Q₁₆	—	8	2	no	—
62	Q₁₆	—	136	2	no	—
63	M₁₆	—	136	2	no	—
64	C₂×C₈	—	2	1	no	—
65	C₂×C₈	—	34	2	no	—
66	C₄⋊C₄	—	4	2	no	—
67	C₄⋊C₄	—	68	2	no	—
68	C₄²	C₂²⋊#595	1	1	yes	characteristic
69	Q₈	—	8	2	no	—
70	Q₈	—	136	2	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	8	1	no	—
73	C₂×C₄	—	68	2	no	—
74	C₂×C₄	#10587551	1	1	yes	characteristic
75	C₈	—	2	2	no	—
76	C₈	—	272	2	no	—
77	C₈	—	34	4	no	—
78	C₂²	#11763713	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	2	1	no	—
81	C₄	—	16	1	no	—
82	C₄	—	136	2	no	—
83	C₂	—	2	1	no	—
84	C₂	#14543915	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#14543915
commutator	C₄×C₈×C₁₇	C₂×C₄
Frattini	C₄×C₈	C₂×#595
Fitting	C₁₇×#494	C₄
socle	C₂×C₁₇	#1408
prime radical	C₂²×C₁₇	#353
cosocle	#11763189	C₂²
hypercenter	#494	#595
hypocenter	C₁₇	#7532
LCS	G ▷ C₄×C₈×C₁₇ ▷ C₄²×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #494

Complements

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

N	H ≅ G/N	type
58. C₁₇	#7532	semidirect

Maximal subgroups

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

subgroup	index
#14543922	2
#14528400	2
#7532	17