Subgroups — C₁₇×#1716 #14524722

Order 2176.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×#498	C₂	1	2	yes	maximal
2	C₁₇×#498	C₂	1	1	yes	maximal, characteristic
3	C₁₇×#494	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#487	C₂	1	2	yes	maximal
5	C₁₇×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₁₆×C₁₇	—	2	2	no	—
7	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₁₇×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₁₇×#106	—	2	4	no	—
11	C₁₇×#106	—	2	2	no	—
12	C₁₇×#103	—	2	4	no	—
13	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₇	C₂²	1	2	yes	—
15	C₂×Q₈×C₁₇	—	2	1	no	—
16	C₂×Q₈×C₁₇	—	2	1	no	—
17	C₂×Q₈×C₁₇	—	2	2	no	—
18	Q₁₆×C₁₇	—	4	4	no	—
19	C₂×C₈×C₁₇	D₄	1	4	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	2	no	—
27	Q₈×C₁₇	—	4	2	no	—
28	Q₈×C₁₇	—	4	4	no	—
29	C₂×C₄×C₁₇	—	4	1	no	—
30	C₂×C₄×C₁₇	—	4	1	no	—
31	C₂×C₄×C₁₇	—	4	2	no	—
32	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
34	C₈×C₁₇	—	2	4	no	—
35	C₈×C₁₇	SD₁₆	1	4	yes	—
36	#1716	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂²×C₁₇	#127	1	1	yes	characteristic
38	C₄×C₁₇	—	8	1	no	—
39	C₄×C₁₇	—	8	1	no	—
40	C₄×C₁₇	—	8	2	no	—
41	C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
42	C₄×C₁₇	#137	1	4	yes	—
43	#498	C₂×C₁₇	1	2	yes	—
44	#498	C₂×C₁₇	1	1	yes	characteristic
45	#494	C₂×C₁₇	1	1	yes	characteristic
46	#487	C₂×C₁₇	1	2	yes	—
47	#322	C₂×C₁₇	1	1	yes	characteristic
48	C₂×C₁₇	#492	1	1	yes	characteristic
49	C₂×C₁₇	#497	1	2	yes	—
50	C₂×Q₁₆	—	2	2	no	—
51	C₄⋊Q₈	C₂²×C₁₇	1	1	yes	characteristic
52	C₄⋊Q₈	C₂²×C₁₇	1	2	yes	—
53	C₄⋊Q₈	C₂²×C₁₇	1	1	yes	characteristic
54	#106	—	2	4	no	—
55	#106	—	2	2	no	—
56	#103	—	2	4	no	—
57	C₄×C₈	C₂²×C₁₇	1	2	yes	—
58	C₄×C₈	C₂²×C₁₇	1	1	yes	characteristic
59	C₁₇	#1716	1	1	yes	characteristic, direct product factor, Sylow
60	C₂×Q₈	—	2	2	no	—
61	C₂×Q₈	—	2	1	no	—
62	C₂×Q₈	—	2	1	no	—
63	Q₁₆	—	4	4	no	—
64	C₂×C₈	D₄×C₁₇	1	4	yes	—
65	C₂×C₈	D₄×C₁₇	1	2	yes	—
66	C₄⋊C₄	—	2	2	no	—
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	—	2	2	no	—
69	C₄⋊C₄	—	2	2	no	—
70	C₄²	C₂³×C₁₇	1	1	yes	characteristic
71	Q₈	—	4	2	no	—
72	Q₈	—	4	2	no	—
73	Q₈	—	4	4	no	—
74	C₂×C₄	—	4	1	no	—
75	C₂×C₄	—	4	1	no	—
76	C₂×C₄	—	4	2	no	—
77	C₂×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
78	C₂×C₄	C₂×D₄×C₁₇	1	2	yes	—
79	C₈	—	2	4	no	—
80	C₈	SD₁₆×C₁₇	1	4	yes	—
81	C₂²	C₁₇×#127	1	1	yes	characteristic
82	C₄	—	8	1	no	—
83	C₄	—	8	1	no	—
84	C₄	—	8	2	no	—
85	C₄	C₂×SD₁₆×C₁₇	1	2	yes	—
86	C₄	C₁₇×#137	1	4	yes	—
87	C₂	C₁₇×#492	1	1	yes	characteristic
88	C₂	C₁₇×#497	1	2	yes	—

	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
36. #1716	C₁₇	direct (N × H)
59. C₁₇	#1716	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₇×#498	2
C₁₇×#494	2
C₁₇×#487	2
C₁₇×#322	2
#1716	17

Subgroups — C17×#1716 #14524722

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#1716 #14524722