Subgroups — C₈×C₁₃×C₁₆×C₁₇ #72205276

Order 28288.

208 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈²×C₁₃×C₁₇	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₃×C₁₆×C₁₇	C₂	1	2	yes	maximal
3	C₄×C₈×C₁₃×C₁₇	C₄	1	2	yes	—
4	C₄×C₈×C₁₃×C₁₇	C₂²	1	1	yes	characteristic
5	C₂×C₁₃×C₁₆×C₁₇	C₄	1	4	yes	—
6	C₄²×C₁₃×C₁₇	C₂×C₄	1	1	yes	characteristic
7	C₂×C₈×C₁₃×C₁₇	C₈	1	4	yes	—
8	C₂×C₈×C₁₃×C₁₇	C₂×C₄	1	2	yes	—
9	C₁₃×C₁₆×C₁₇	C₈	1	8	yes	direct product factor
10	C₈×C₁₆×C₁₇	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂×C₄×C₁₃×C₁₇	C₄²	1	1	yes	characteristic
12	C₂×C₄×C₁₃×C₁₇	C₂×C₈	1	2	yes	—
13	C₈×C₁₃×C₁₇	C₁₆	1	8	yes	direct product factor
14	C₈×C₁₃×C₁₇	C₂×C₈	1	4	yes	—
15	C₈×C₁₃×C₁₆	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₄×C₁₆×C₁₇	C₂×C₁₃	1	2	yes	—
17	C₈²×C₁₇	C₂×C₁₃	1	1	yes	characteristic
18	C₂²×C₁₃×C₁₇	C₄×C₈	1	1	yes	characteristic
19	C₄×C₁₃×C₁₇	C₄×C₈	1	2	yes	—
20	C₄×C₁₃×C₁₇	C₂×C₁₆	1	4	yes	—
21	C₄×C₁₃×C₁₆	C₂×C₁₇	1	2	yes	—
22	C₈²×C₁₃	C₂×C₁₇	1	1	yes	characteristic
23	C₂×C₁₆×C₁₇	C₄×C₁₃	1	4	yes	—
24	C₄×C₈×C₁₇	C₄×C₁₃	1	2	yes	—
25	C₄×C₈×C₁₇	C₂²×C₁₃	1	1	yes	characteristic
26	C₂×C₁₃×C₁₇	C₈²	1	1	yes	characteristic
27	C₂×C₁₃×C₁₇	C₄×C₁₆	1	2	yes	—
28	C₂×C₁₃×C₁₆	C₄×C₁₇	1	4	yes	—
29	C₄×C₈×C₁₃	C₄×C₁₇	1	2	yes	—
30	C₄×C₈×C₁₃	C₂²×C₁₇	1	1	yes	characteristic
31	C₂×C₈×C₁₇	C₈×C₁₃	1	4	yes	—
32	C₂×C₈×C₁₇	C₂×C₄×C₁₃	1	2	yes	—
33	C₄²×C₁₇	C₂×C₄×C₁₃	1	1	yes	characteristic
34	C₁₆×C₁₇	C₈×C₁₃	1	8	yes	direct product factor
35	C₁₃×C₁₇	C₈×C₁₆	1	1	yes	characteristic, direct product factor, Hall
36	C₂×C₈×C₁₃	C₈×C₁₇	1	4	yes	—
37	C₂×C₈×C₁₃	C₂×C₄×C₁₇	1	2	yes	—
38	C₄²×C₁₃	C₂×C₄×C₁₇	1	1	yes	characteristic
39	C₁₃×C₁₆	C₈×C₁₇	1	8	yes	direct product factor
40	C₂×C₄×C₁₇	C₄²×C₁₃	1	1	yes	characteristic
41	C₂×C₄×C₁₇	C₂×C₈×C₁₃	1	2	yes	—
42	C₈×C₁₇	C₁₃×C₁₆	1	8	yes	direct product factor
43	C₈×C₁₇	C₂×C₈×C₁₃	1	4	yes	—
44	C₈×C₁₆	C₁₃×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
45	C₂×C₄×C₁₃	C₄²×C₁₇	1	1	yes	characteristic
46	C₂×C₄×C₁₃	C₂×C₈×C₁₇	1	2	yes	—
47	C₈×C₁₃	C₁₆×C₁₇	1	8	yes	direct product factor
48	C₈×C₁₃	C₂×C₈×C₁₇	1	4	yes	—
49	C₂²×C₁₇	C₄×C₈×C₁₃	1	1	yes	characteristic
50	C₄×C₁₇	C₄×C₈×C₁₃	1	2	yes	—
51	C₄×C₁₇	C₂×C₁₃×C₁₆	1	4	yes	—
52	C₄×C₁₆	C₂×C₁₃×C₁₇	1	2	yes	—
53	C₈²	C₂×C₁₃×C₁₇	1	1	yes	characteristic
54	C₂²×C₁₃	C₄×C₈×C₁₇	1	1	yes	characteristic
55	C₄×C₁₃	C₄×C₈×C₁₇	1	2	yes	—
56	C₄×C₁₃	C₂×C₁₆×C₁₇	1	4	yes	—
57	C₂×C₁₇	C₈²×C₁₃	1	1	yes	characteristic
58	C₂×C₁₇	C₄×C₁₃×C₁₆	1	2	yes	—
59	C₂×C₁₆	C₄×C₁₃×C₁₇	1	4	yes	—
60	C₄×C₈	C₄×C₁₃×C₁₇	1	2	yes	—
61	C₄×C₈	C₂²×C₁₃×C₁₇	1	1	yes	characteristic
62	C₂×C₁₃	C₈²×C₁₇	1	1	yes	characteristic
63	C₂×C₁₃	C₄×C₁₆×C₁₇	1	2	yes	—
64	C₁₇	C₈×C₁₃×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
65	C₂×C₈	C₈×C₁₃×C₁₇	1	4	yes	—
66	C₂×C₈	C₂×C₄×C₁₃×C₁₇	1	2	yes	—
67	C₄²	C₂×C₄×C₁₃×C₁₇	1	1	yes	characteristic
68	C₁₆	C₈×C₁₃×C₁₇	1	8	yes	direct product factor
69	C₁₃	C₈×C₁₆×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
70	C₂×C₄	C₂×C₈×C₁₃×C₁₇	1	2	yes	—
71	C₂×C₄	C₄²×C₁₃×C₁₇	1	1	yes	characteristic
72	C₈	C₁₃×C₁₆×C₁₇	1	8	yes	direct product factor
73	C₈	C₂×C₈×C₁₃×C₁₇	1	4	yes	—
74	C₂²	C₄×C₈×C₁₃×C₁₇	1	1	yes	characteristic
75	C₄	C₂×C₁₃×C₁₆×C₁₇	1	4	yes	—
76	C₄	C₄×C₈×C₁₃×C₁₇	1	2	yes	—
77	C₂	C₄×C₁₃×C₁₆×C₁₇	1	2	yes	—
78	C₂	C₈²×C₁₃×C₁₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄×C₈	C₂²×C₁₃×C₁₇
socle, prime radical	C₂²×C₁₃×C₁₇	C₄×C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₁₃×C₁₆×C₁₇	C₈	direct (N × H)
10. C₈×C₁₆×C₁₇	C₁₃	direct (N × H)
13. C₈×C₁₃×C₁₇	C₁₆	direct (N × H)
15. C₈×C₁₃×C₁₆	C₁₇	direct (N × H)
34. C₁₆×C₁₇	C₈×C₁₃	direct (N × H)
35. C₁₃×C₁₇	C₈×C₁₆	direct (N × H)
39. C₁₃×C₁₆	C₈×C₁₇	direct (N × H)
42. C₈×C₁₇	C₁₃×C₁₆	direct (N × H)
44. C₈×C₁₆	C₁₃×C₁₇	direct (N × H)
47. C₈×C₁₃	C₁₆×C₁₇	direct (N × H)
64. C₁₇	C₈×C₁₃×C₁₆	direct (N × H)
68. C₁₆	C₈×C₁₃×C₁₇	direct (N × H)
69. C₁₃	C₈×C₁₆×C₁₇	direct (N × H)
72. C₈	C₁₃×C₁₆×C₁₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈²×C₁₃×C₁₇	2
C₄×C₁₃×C₁₆×C₁₇	2
C₈×C₁₆×C₁₇	13
C₈×C₁₃×C₁₆	17

Subgroups — C8×C13×C16×C17 #72205276

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×C₁₃×C₁₆×C₁₇ #72205276