Subgroups — C₁₇×#1707 #14524713

Order 2176.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×#500	C₂	1	1	yes	maximal, characteristic
2	C₁₇×#498	C₂	1	2	yes	maximal
3	C₁₇×#494	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#487	C₂	1	2	yes	maximal
5	C₈²×C₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₁₆×C₁₇	—	2	2	no	—
7	C₁₇×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₁₇×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₁₇×#107	—	2	2	no	—
10	C₁₇×#106	—	2	4	no	—
11	C₁₇×#103	—	2	4	no	—
12	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₁₇	—	2	2	no	—
16	C₂×Q₈×C₁₇	—	2	2	no	—
17	Q₁₆×C₁₇	—	4	4	no	—
18	C₂×C₈×C₁₇	D₄	1	2	yes	—
19	C₂×C₈×C₁₇	D₄	1	2	yes	—
20	C₂×C₈×C₁₇	D₄	1	2	yes	—
21	C₄⋊C₄×C₁₇	—	2	2	no	—
22	C₄⋊C₄×C₁₇	—	2	2	no	—
23	C₄⋊C₄×C₁₇	—	2	2	no	—
24	C₄⋊C₄×C₁₇	—	2	2	no	—
25	C₄²×C₁₇	C₂³	1	1	yes	characteristic
26	Q₈×C₁₇	—	4	4	no	—
27	Q₈×C₁₇	—	4	4	no	—
28	C₂×C₄×C₁₇	—	4	2	no	—
29	C₂×C₄×C₁₇	—	4	2	no	—
30	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
33	C₈×C₁₇	—	2	2	no	—
34	C₈×C₁₇	SD₁₆	1	4	yes	—
35	C₈×C₁₇	Q₁₆	1	4	yes	complemented
36	#1707	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂²×C₁₇	#127	1	1	yes	characteristic
38	C₄×C₁₇	—	8	2	no	—
39	C₄×C₁₇	—	8	2	no	—
40	C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
41	C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
42	C₄×C₁₇	#135	1	2	yes	—
43	#500	C₂×C₁₇	1	1	yes	characteristic
44	#498	C₂×C₁₇	1	2	yes	—
45	#494	C₂×C₁₇	1	1	yes	characteristic
46	#487	C₂×C₁₇	1	2	yes	—
47	C₈²	C₂×C₁₇	1	1	yes	characteristic
48	C₂×C₁₇	#492	1	1	yes	characteristic
49	C₂×C₁₇	#494	1	1	yes	characteristic
50	C₂×C₁₇	#495	1	1	yes	characteristic
51	C₂×Q₁₆	—	2	2	no	—
52	C₄⋊Q₈	C₂²×C₁₇	1	2	yes	—
53	C₄⋊Q₈	C₂²×C₁₇	1	2	yes	—
54	#107	—	2	2	no	—
55	#106	—	2	4	no	—
56	#103	—	2	4	no	—
57	C₄×C₈	C₂²×C₁₇	1	1	yes	characteristic
58	C₄×C₈	C₂²×C₁₇	1	1	yes	characteristic
59	C₄×C₈	C₂²×C₁₇	1	1	yes	characteristic
60	C₁₇	#1707	1	1	yes	characteristic, direct product factor, Sylow
61	C₂×Q₈	—	2	2	no	—
62	C₂×Q₈	—	2	2	no	—
63	Q₁₆	—	4	4	no	—
64	C₂×C₈	D₄×C₁₇	1	2	yes	—
65	C₂×C₈	D₄×C₁₇	1	2	yes	—
66	C₂×C₈	D₄×C₁₇	1	2	yes	—
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	—	2	2	no	—
69	C₄⋊C₄	—	2	2	no	—
70	C₄⋊C₄	—	2	2	no	—
71	C₄²	C₂³×C₁₇	1	1	yes	characteristic
72	Q₈	—	4	4	no	—
73	Q₈	—	4	4	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	4	2	no	—
76	C₂×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
77	C₂×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
78	C₂×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
79	C₈	—	2	2	no	—
80	C₈	SD₁₆×C₁₇	1	4	yes	—
81	C₈	Q₁₆×C₁₇	1	4	yes	complemented
82	C₂²	C₁₇×#127	1	1	yes	characteristic
83	C₄	—	8	2	no	—
84	C₄	—	8	2	no	—
85	C₄	C₂×SD₁₆×C₁₇	1	2	yes	—
86	C₄	C₂×Q₁₆×C₁₇	1	2	yes	—
87	C₄	C₁₇×#135	1	2	yes	—
88	C₂	C₁₇×#492	1	1	yes	characteristic
89	C₂	C₁₇×#494	1	1	yes	characteristic
90	C₂	C₁₇×#495	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₇	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₁₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₇ ◁ C₄²×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
35. C₈×C₁₇	Q₁₆	semidirect
36. #1707	C₁₇	direct (N × H)
60. C₁₇	#1707	direct (N × H)
81. C₈	Q₁₆×C₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₇×#500	2
C₁₇×#498	2
C₁₇×#494	2
C₁₇×#487	2
C₈²×C₁₇	2
#1707	17

Subgroups — C17×#1707 #14524713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#1707 #14524713