Subgroups — C₂×C₁₇×#124 #11762467

Order 1088.

330 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₄×C₁₇	C₂	1	1	yes	maximal, characteristic
3	C₁₇×#124	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂²⋊C₄×C₁₇	C₂	1	4	yes	maximal, complemented
5	C₂×C₄²×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁴×C₁₇	C₂²	1	2	yes	—
7	C₂×Q₈×C₁₇	—	2	4	no	—
8	C₂×Q₈×C₁₇	C₂²	1	4	yes	complemented
9	C₂×D₄×C₁₇	—	2	4	no	—
10	C₂×D₄×C₁₇	C₂²	1	4	yes	—
11	C₂²×C₄×C₁₇	C₂²	1	2	yes	—
12	C₂²×C₄×C₁₇	C₂²	1	2	yes	—
13	C₂²×C₄×C₁₇	C₂²	1	1	yes	characteristic
14	C₂²⋊C₄×C₁₇	C₂²	1	16	yes	complemented
15	C₄²×C₁₇	C₂²	1	4	yes	complemented
16	C₂³×C₁₇	—	2	4	no	—
17	C₂³×C₁₇	—	2	8	no	—
18	C₂³×C₁₇	C₂³	1	4	yes	—
19	C₂³×C₁₇	C₂³	1	1	yes	characteristic
20	Q₈×C₁₇	—	2	8	no	—
21	D₄×C₁₇	—	2	8	no	—
22	C₂×C₄×C₁₇	—	2	4	no	—
23	C₂×C₄×C₁₇	—	2	4	no	—
24	C₂×C₄×C₁₇	D₄	1	4	yes	—
25	C₂×C₄×C₁₇	C₂³	1	4	yes	—
26	C₂×C₄×C₁₇	C₂³	1	4	yes	—
27	C₂×C₄×C₁₇	C₂³	1	2	yes	—
28	C₂²×C₁₇	—	2	4	no	—
29	C₂²×C₁₇	—	2	8	no	—
30	C₂²×C₁₇	—	4	8	no	—
31	C₂²×C₁₇	C₂×D₄	1	2	yes	—
32	C₂²×C₁₇	#41	1	4	yes	—
33	C₂²×C₁₇	C₂⁴	1	1	yes	characteristic
34	C₄×C₁₇	—	2	4	no	—
35	C₄×C₁₇	—	2	4	no	—
36	C₄×C₁₇	C₂×D₄	1	4	yes	—
37	C₂×#124	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₂×C₁₇	—	4	4	no	—
39	C₂×C₁₇	#124	1	4	yes	direct product factor
40	C₂×C₁₇	C₂²×D₄	1	1	yes	characteristic
41	C₂×C₁₇	C₂×#41	1	2	yes	—
42	C₂²×Q₈	C₂×C₁₇	1	1	yes	characteristic, complemented
43	C₂²×D₄	C₂×C₁₇	1	1	yes	characteristic
44	#124	C₂×C₁₇	1	8	yes	direct product factor
45	C₂×C₂²⋊C₄	C₂×C₁₇	1	4	yes	complemented
46	C₂×C₄²	C₂×C₁₇	1	1	yes	characteristic, complemented
47	C₁₇	C₂×#124	1	1	yes	characteristic, direct product factor, Sylow
48	C₂⁴	C₂²×C₁₇	1	2	yes	—
49	C₂×Q₈	—	2	4	no	—
50	C₂×Q₈	C₂²×C₁₇	1	4	yes	complemented
51	C₂×D₄	—	2	4	no	—
52	C₂×D₄	C₂²×C₁₇	1	4	yes	—
53	C₂²×C₄	C₂²×C₁₇	1	2	yes	—
54	C₂²×C₄	C₂²×C₁₇	1	2	yes	—
55	C₂²×C₄	C₂²×C₁₇	1	1	yes	characteristic
56	C₂²⋊C₄	C₂²×C₁₇	1	16	yes	complemented
57	C₄²	C₂²×C₁₇	1	4	yes	complemented
58	C₂³	—	2	4	no	—
59	C₂³	—	2	8	no	—
60	C₂³	C₂³×C₁₇	1	4	yes	—
61	C₂³	C₂³×C₁₇	1	1	yes	characteristic
62	Q₈	—	2	8	no	—
63	D₄	—	2	8	no	—
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	D₄×C₁₇	1	4	yes	—
67	C₂×C₄	C₂³×C₁₇	1	4	yes	—
68	C₂×C₄	C₂³×C₁₇	1	4	yes	—
69	C₂×C₄	C₂³×C₁₇	1	2	yes	—
70	C₂²	—	2	4	no	—
71	C₂²	—	2	8	no	—
72	C₂²	—	4	8	no	—
73	C₂²	C₂×D₄×C₁₇	1	2	yes	—
74	C₂²	#41×C₁₇	1	4	yes	—
75	C₂²	C₂⁴×C₁₇	1	1	yes	characteristic
76	C₄	—	2	4	no	—
77	C₄	—	2	4	no	—
78	C₄	C₂×D₄×C₁₇	1	4	yes	—
79	C₂	—	4	4	no	—
80	C₂	C₁₇×#124	1	4	yes	direct product factor
81	C₂	C₂²×D₄×C₁₇	1	1	yes	characteristic
82	C₂	C₂×#41×C₁₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₇	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₁₇
prime radical	C₂²×D₄×C₁₇	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×Q₈×C₁₇	C₂	semidirect
3. C₁₇×#124	C₂	direct (N × H)
4. C₂×C₂²⋊C₄×C₁₇	C₂	semidirect
5. C₂×C₄²×C₁₇	C₂	semidirect
8. C₂×Q₈×C₁₇	C₂²	semidirect
14. C₂²⋊C₄×C₁₇	C₂²	semidirect
15. C₄²×C₁₇	C₂²	semidirect
37. C₂×#124	C₁₇	direct (N × H)
39. C₂×C₁₇	#124	direct (N × H)
42. C₂²×Q₈	C₂×C₁₇	semidirect
44. #124	C₂×C₁₇	direct (N × H)
45. C₂×C₂²⋊C₄	C₂×C₁₇	semidirect
46. C₂×C₄²	C₂×C₁₇	semidirect
47. C₁₇	C₂×#124	direct (N × H)
50. C₂×Q₈	C₂²×C₁₇	semidirect
56. C₂²⋊C₄	C₂²×C₁₇	semidirect
57. C₄²	C₂²×C₁₇	semidirect
80. C₂	C₁₇×#124	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Q₈×C₁₇	2
C₂²×D₄×C₁₇	2
C₁₇×#124	2
C₂×C₂²⋊C₄×C₁₇	2
C₂×C₄²×C₁₇	2
C₂×#124	17

Subgroups — C2×C17×#124 #11762467

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₇×#124 #11762467