Subgroups — C₁₇×#3458 #14526464

Order 2176.

996 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×#544	C₂	1	12	yes	maximal, complemented
2	C₁₇×#543	C₂	1	4	yes	maximal
3	C₁₇×#533	C₂	1	12	yes	maximal, complemented
4	C₂×C₁₇×#117	C₂	1	3	yes	maximal
5	C₂³×C₄×C₁₇	C₂²	1	1	yes	characteristic
6	C₁₇×C₄⋊Q₈	C₂²	1	24	yes	complemented
7	C₁₇×#125	C₂²	1	24	yes	complemented
8	C₁₇×C₂²⋊Q₈	C₂²	1	48	yes	—
9	C₄×Q₈×C₁₇	C₂²	1	16	yes	complemented
10	C₁₇×#117	C₂²	1	12	yes	—
11	C₁₇×#117	C₂²	1	12	yes	—
12	C₂×C₄⋊C₄×C₁₇	C₂²	1	6	yes	—
13	C₂×C₂²⋊C₄×C₁₇	C₂²	1	6	yes	—
14	C₂×C₄²×C₁₇	C₂²	1	6	yes	—
15	C₂⁴×C₁₇	C₂³	1	1	yes	characteristic
16	C₂×Q₈×C₁₇	C₂³	1	16	yes	—
17	C₂²×C₄×C₁₇	Q₈	1	8	yes	complemented
18	C₂²×C₄×C₁₇	C₂³	1	12	yes	—
19	C₂²×C₄×C₁₇	C₂³	1	3	yes	—
20	C₂²×C₄×C₁₇	C₂³	1	3	yes	—
21	C₄⋊C₄×C₁₇	C₂³	1	48	yes	—
22	C₄⋊C₄×C₁₇	C₂³	1	24	yes	—
23	C₂²⋊C₄×C₁₇	C₂³	1	24	yes	—
24	C₄²×C₁₇	C₂³	1	12	yes	—
25	C₄²×C₁₇	C₂³	1	12	yes	—
26	C₂³×C₁₇	—	4	2	no	—
27	C₂³×C₁₇	C₂×Q₈	1	4	yes	—
28	C₂³×C₁₇	C₂⁴	1	3	yes	—
29	Q₈×C₁₇	—	4	16	no	—
30	C₂×C₄×C₁₇	—	4	12	no	—
31	C₂×C₄×C₁₇	C₂×Q₈	1	12	yes	—
32	C₂×C₄×C₁₇	C₂×Q₈	1	12	yes	—
33	C₂×C₄×C₁₇	C₂⁴	1	24	yes	—
34	C₂×C₄×C₁₇	C₂⁴	1	3	yes	—
35	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
36	#3458	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂²×C₁₇	—	2	6	no	—
38	C₂²×C₁₇	—	4	4	no	—
39	C₂²×C₁₇	C₂²×Q₈	1	6	yes	—
40	C₂²×C₁₇	C₂⁵	1	1	yes	characteristic
41	C₄×C₁₇	—	2	24	no	—
42	C₄×C₁₇	C₂²×Q₈	1	6	yes	—
43	C₄×C₁₇	C₂²×Q₈	1	2	yes	—
44	#544	C₂×C₁₇	1	12	yes	complemented
45	#543	C₂×C₁₇	1	4	yes	—
46	#533	C₂×C₁₇	1	12	yes	complemented
47	C₂×#117	C₂×C₁₇	1	3	yes	—
48	C₂×C₁₇	—	2	6	no	—
49	C₂×C₁₇	C₂³×Q₈	1	1	yes	characteristic
50	C₂×C₁₇	#585	1	2	yes	—
51	C₂³×C₄	C₂²×C₁₇	1	1	yes	characteristic
52	C₄⋊Q₈	C₂²×C₁₇	1	24	yes	complemented
53	#125	C₂²×C₁₇	1	24	yes	complemented
54	C₂²⋊Q₈	C₂²×C₁₇	1	48	yes	—
55	C₄×Q₈	C₂²×C₁₇	1	16	yes	complemented
56	#117	C₂²×C₁₇	1	12	yes	—
57	#117	C₂²×C₁₇	1	12	yes	—
58	C₂×C₄⋊C₄	C₂²×C₁₇	1	6	yes	—
59	C₂×C₂²⋊C₄	C₂²×C₁₇	1	6	yes	—
60	C₂×C₄²	C₂²×C₁₇	1	6	yes	—
61	C₁₇	#3458	1	1	yes	characteristic, direct product factor, Sylow
62	C₂⁴	C₂³×C₁₇	1	1	yes	characteristic
63	C₂×Q₈	C₂³×C₁₇	1	16	yes	—
64	C₂²×C₄	Q₈×C₁₇	1	8	yes	complemented
65	C₂²×C₄	C₂³×C₁₇	1	12	yes	—
66	C₂²×C₄	C₂³×C₁₇	1	3	yes	—
67	C₂²×C₄	C₂³×C₁₇	1	3	yes	—
68	C₄⋊C₄	C₂³×C₁₇	1	24	yes	—
69	C₄⋊C₄	C₂³×C₁₇	1	48	yes	—
70	C₂²⋊C₄	C₂³×C₁₇	1	24	yes	—
71	C₄²	C₂³×C₁₇	1	12	yes	—
72	C₄²	C₂³×C₁₇	1	12	yes	—
73	C₂³	—	4	2	no	—
74	C₂³	C₂×Q₈×C₁₇	1	4	yes	—
75	C₂³	C₂⁴×C₁₇	1	3	yes	—
76	Q₈	—	4	16	no	—
77	C₂×C₄	—	4	12	no	—
78	C₂×C₄	C₂×Q₈×C₁₇	1	12	yes	—
79	C₂×C₄	C₂×Q₈×C₁₇	1	12	yes	—
80	C₂×C₄	C₂⁴×C₁₇	1	24	yes	—
81	C₂×C₄	C₂⁴×C₁₇	1	1	yes	characteristic
82	C₂×C₄	C₂⁴×C₁₇	1	3	yes	—
83	C₂²	—	2	6	no	—
84	C₂²	—	4	4	no	—
85	C₂²	C₂²×Q₈×C₁₇	1	6	yes	—
86	C₂²	C₂⁵×C₁₇	1	1	yes	characteristic
87	C₄	—	2	24	no	—
88	C₄	C₂²×Q₈×C₁₇	1	2	yes	—
89	C₄	C₂²×Q₈×C₁₇	1	6	yes	—
90	C₂	—	2	6	no	—
91	C₂	C₂³×Q₈×C₁₇	1	1	yes	characteristic
92	C₂	C₁₇×#585	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₁₇
socle	C₂²×C₁₇	C₂⁵
prime radical	C₂⁴×C₁₇	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₇×#544	C₂	semidirect
3. C₁₇×#533	C₂	semidirect
6. C₁₇×C₄⋊Q₈	C₂²	semidirect
7. C₁₇×#125	C₂²	semidirect
9. C₄×Q₈×C₁₇	C₂²	semidirect
17. C₂²×C₄×C₁₇	Q₈	semidirect
36. #3458	C₁₇	direct (N × H)
44. #544	C₂×C₁₇	semidirect
46. #533	C₂×C₁₇	semidirect
52. C₄⋊Q₈	C₂²×C₁₇	semidirect
53. #125	C₂²×C₁₇	semidirect
55. C₄×Q₈	C₂²×C₁₇	semidirect
61. C₁₇	#3458	direct (N × H)
64. C₂²×C₄	Q₈×C₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₇×#544	2
C₁₇×#543	2
C₁₇×#533	2
C₂×C₁₇×#117	2
#3458	17

Subgroups — C17×#3458 #14526464

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#3458 #14526464