Subgroups — C₂³×C₄⋊C₄×C₁₇ #14526426

Order 2176.

2488 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄×C₁₇	C₂	1	2	yes	maximal
2	C₂⁴×C₄×C₁₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊C₄×C₁₇	C₂	1	28	yes	maximal, direct product factor
4	C₂⁵×C₁₇	C₂²	1	1	yes	characteristic
5	C₂³×C₄×C₁₇	—	2	16	no	—
6	C₂³×C₄×C₁₇	C₄	1	16	yes	complemented
7	C₂³×C₄×C₁₇	C₂²	1	28	yes	—
8	C₂³×C₄×C₁₇	C₂²	1	14	yes	—
9	C₂×C₄⋊C₄×C₁₇	C₂²	1	112	yes	direct product factor
10	C₂⁴×C₁₇	C₂×C₄	1	8	yes	—
11	C₂⁴×C₁₇	D₄	1	8	yes	—
12	C₂⁴×C₁₇	Q₈	1	8	yes	—
13	C₂⁴×C₁₇	C₂³	1	7	yes	—
14	C₂²×C₄×C₁₇	—	2	112	no	—
15	C₂²×C₄×C₁₇	C₂×C₄	1	112	yes	complemented
16	C₂²×C₄×C₁₇	C₂³	1	56	yes	—
17	C₂²×C₄×C₁₇	C₂³	1	28	yes	—
18	C₄⋊C₄×C₁₇	C₂³	1	64	yes	direct product factor
19	C₂³×C₁₇	C₄⋊C₄	1	64	yes	direct product factor
20	C₂³×C₁₇	C₂²×C₄	1	28	yes	—
21	C₂³×C₁₇	C₂×D₄	1	28	yes	—
22	C₂³×C₁₇	C₂×Q₈	1	28	yes	—
23	C₂³×C₁₇	C₂⁴	1	7	yes	—
24	C₂×C₄×C₁₇	—	2	112	no	—
25	C₂×C₄×C₁₇	C₂²×C₄	1	112	yes	complemented
26	C₂×C₄×C₁₇	C₂⁴	1	16	yes	—
27	C₂×C₄×C₁₇	C₂⁴	1	8	yes	—
28	C₂³×C₄⋊C₄	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₂²×C₁₇	C₂×C₄⋊C₄	1	112	yes	direct product factor
30	C₂²×C₁₇	C₂³×C₄	1	14	yes	—
31	C₂²×C₁₇	C₂²×D₄	1	14	yes	—
32	C₂²×C₁₇	C₂²×Q₈	1	14	yes	—
33	C₂²×C₁₇	C₂⁵	1	1	yes	characteristic
34	C₄×C₁₇	—	2	16	no	—
35	C₄×C₁₇	C₂³×C₄	1	16	yes	complemented
36	C₂⁴×C₄	C₂×C₁₇	1	2	yes	—
37	C₂⁴×C₄	C₂×C₁₇	1	1	yes	characteristic
38	C₂²×C₄⋊C₄	C₂×C₁₇	1	28	yes	direct product factor
39	C₂×C₁₇	C₂²×C₄⋊C₄	1	28	yes	direct product factor
40	C₂×C₁₇	C₂⁴×C₄	1	1	yes	characteristic
41	C₂×C₁₇	C₂³×D₄	1	1	yes	characteristic
42	C₂×C₁₇	C₂³×Q₈	1	1	yes	characteristic
43	C₂⁵	C₂²×C₁₇	1	1	yes	characteristic
44	C₂³×C₄	—	2	16	no	—
45	C₂³×C₄	C₄×C₁₇	1	16	yes	complemented
46	C₂³×C₄	C₂²×C₁₇	1	28	yes	—
47	C₂³×C₄	C₂²×C₁₇	1	14	yes	—
48	C₂×C₄⋊C₄	C₂²×C₁₇	1	112	yes	direct product factor
49	C₁₇	C₂³×C₄⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
50	C₂⁴	C₂×C₄×C₁₇	1	8	yes	—
51	C₂⁴	D₄×C₁₇	1	8	yes	—
52	C₂⁴	Q₈×C₁₇	1	8	yes	—
53	C₂⁴	C₂³×C₁₇	1	7	yes	—
54	C₂²×C₄	—	2	112	no	—
55	C₂²×C₄	C₂×C₄×C₁₇	1	112	yes	complemented
56	C₂²×C₄	C₂³×C₁₇	1	56	yes	—
57	C₂²×C₄	C₂³×C₁₇	1	28	yes	—
58	C₄⋊C₄	C₂³×C₁₇	1	64	yes	direct product factor
59	C₂³	C₄⋊C₄×C₁₇	1	64	yes	direct product factor
60	C₂³	C₂²×C₄×C₁₇	1	28	yes	—
61	C₂³	C₂×D₄×C₁₇	1	28	yes	—
62	C₂³	C₂×Q₈×C₁₇	1	28	yes	—
63	C₂³	C₂⁴×C₁₇	1	7	yes	—
64	C₂×C₄	—	2	112	no	—
65	C₂×C₄	C₂²×C₄×C₁₇	1	112	yes	complemented
66	C₂×C₄	C₂⁴×C₁₇	1	16	yes	—
67	C₂×C₄	C₂⁴×C₁₇	1	8	yes	—
68	C₂²	C₂×C₄⋊C₄×C₁₇	1	112	yes	direct product factor
69	C₂²	C₂³×C₄×C₁₇	1	14	yes	—
70	C₂²	C₂²×D₄×C₁₇	1	14	yes	—
71	C₂²	C₂²×Q₈×C₁₇	1	14	yes	—
72	C₂²	C₂⁵×C₁₇	1	1	yes	characteristic
73	C₄	—	2	16	no	—
74	C₄	C₂³×C₄×C₁₇	1	16	yes	complemented
75	C₂	C₂²×C₄⋊C₄×C₁₇	1	28	yes	direct product factor
76	C₂	C₂⁴×C₄×C₁₇	1	1	yes	characteristic
77	C₂	C₂³×D₄×C₁₇	1	1	yes	characteristic
78	C₂	C₂³×Q₈×C₁₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁵×C₁₇	C₂²
commutator	C₂	C₂⁴×C₄×C₁₇
Frattini, cosocle	C₂²	C₂⁵×C₁₇
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂²×C₄⋊C₄×C₁₇	C₂	direct (N × H)
6. C₂³×C₄×C₁₇	C₄	semidirect
9. C₂×C₄⋊C₄×C₁₇	C₂²	direct (N × H)
15. C₂²×C₄×C₁₇	C₂×C₄	semidirect
18. C₄⋊C₄×C₁₇	C₂³	direct (N × H)
19. C₂³×C₁₇	C₄⋊C₄	direct (N × H)
25. C₂×C₄×C₁₇	C₂²×C₄	semidirect
28. C₂³×C₄⋊C₄	C₁₇	direct (N × H)
29. C₂²×C₁₇	C₂×C₄⋊C₄	direct (N × H)
35. C₄×C₁₇	C₂³×C₄	semidirect
38. C₂²×C₄⋊C₄	C₂×C₁₇	direct (N × H)
39. C₂×C₁₇	C₂²×C₄⋊C₄	direct (N × H)
45. C₂³×C₄	C₄×C₁₇	semidirect
48. C₂×C₄⋊C₄	C₂²×C₁₇	direct (N × H)
49. C₁₇	C₂³×C₄⋊C₄	direct (N × H)
55. C₂²×C₄	C₂×C₄×C₁₇	semidirect
58. C₄⋊C₄	C₂³×C₁₇	direct (N × H)
59. C₂³	C₄⋊C₄×C₁₇	direct (N × H)
65. C₂×C₄	C₂²×C₄×C₁₇	semidirect
68. C₂²	C₂×C₄⋊C₄×C₁₇	direct (N × H)
74. C₄	C₂³×C₄×C₁₇	semidirect
75. C₂	C₂²×C₄⋊C₄×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
C₂⁴×C₄×C₁₇	2
C₂²×C₄⋊C₄×C₁₇	2
C₂³×C₄⋊C₄	17

Subgroups — C23×C4⋊C4×C17 #14526426

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄⋊C₄×C₁₇ #14526426