Subgroups — C₂²×#63218 #11763568

Order 1088.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₁₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#3622	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#63218	C₂	1	12	yes	maximal, direct product factor
5	C₂³×D₁₇	C₄	1	1	yes	characteristic
6	C₂²×C₄×C₁₇	C₂²	1	1	yes	characteristic
7	C₂²×Dic₁₇	C₄	1	1	yes	characteristic
8	C₂×C₄×D₁₇	C₄	1	6	yes	—
9	C₂×C₄×D₁₇	C₂²	1	6	yes	—
10	C₂×C₈×C₁₇	C₂²	1	6	yes	complemented
11	C₂×#3622	C₂²	1	6	yes	complemented
12	#63218	C₂²	1	16	yes	direct product factor
13	C₂³×C₁₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×D₁₇	—	2	4	no	—
15	C₂²×D₁₇	C₂×C₄	1	6	yes	—
16	C₂×C₄×C₁₇	C₂×C₄	1	3	yes	—
17	C₂×C₄×C₁₇	C₂³	1	3	yes	—
18	C₂×Dic₁₇	C₂×C₄	1	6	yes	—
19	C₄×D₁₇	C₂×C₄	1	12	yes	—
20	C₄×D₁₇	C₂³	1	4	yes	—
21	C₈×C₁₇	C₂³	1	4	yes	complemented
22	#3622	C₂³	1	4	yes	complemented
23	C₂²×C₁₇	M₁₆	1	4	yes	complemented
24	C₂²×C₁₇	C₂²×C₄	1	3	yes	—
25	C₂×D₁₇	—	2	12	no	—
26	C₂×D₁₇	C₂²×C₄	1	4	yes	—
27	C₄×C₁₇	C₂²×C₄	1	3	yes	—
28	C₄×C₁₇	C₂⁴	1	1	yes	characteristic
29	Dic₁₇	C₂²×C₄	1	4	yes	—
30	C₂²×M₁₆	—	17	1	no	maximal, Sylow
31	C₂×C₁₇	C₂×M₁₆	1	6	yes	complemented
32	C₂×C₁₇	C₂³×C₄	1	1	yes	characteristic
33	D₁₇	—	2	4	no	—
34	C₂³×C₄	—	17	1	no	—
35	C₂×M₁₆	—	17	12	no	—
36	C₂²×C₈	—	17	1	no	—
37	C₂²×C₈	D₁₇	1	1	yes	characteristic, complemented
38	C₁₇	C₂²×M₁₆	1	1	yes	characteristic, complemented, Sylow
39	C₂⁴	—	17	1	no	—
40	C₂²×C₄	—	17	1	no	—
41	C₂²×C₄	—	17	6	no	—
42	C₂²×C₄	—	17	6	no	—
43	C₂²×C₄	C₂×D₁₇	1	1	yes	characteristic
44	M₁₆	—	17	16	no	—
45	C₂×C₈	—	17	6	no	—
46	C₂×C₈	C₂×D₁₇	1	6	yes	complemented
47	C₂³	—	17	6	no	—
48	C₂³	—	34	4	no	—
49	C₂³	C₄×D₁₇	1	1	yes	characteristic
50	C₂×C₄	—	17	6	no	—
51	C₂×C₄	—	17	12	no	—
52	C₂×C₄	—	17	4	no	—
53	C₂×C₄	C₄×D₁₇	1	3	yes	—
54	C₂×C₄	C₂²×D₁₇	1	3	yes	—
55	C₈	—	17	4	no	—
56	C₈	C₂²×D₁₇	1	4	yes	complemented
57	C₂²	—	17	4	no	—
58	C₂²	—	34	12	no	—
59	C₂²	#63218	1	4	yes	direct product factor
60	C₂²	C₂×C₄×D₁₇	1	3	yes	—
61	C₄	—	17	4	no	—
62	C₄	C₂×C₄×D₁₇	1	3	yes	—
63	C₄	C₂³×D₁₇	1	1	yes	characteristic
64	C₂	—	34	4	no	—
65	C₂	C₂×#63218	1	6	yes	direct product factor
66	C₂	C₂²×C₄×D₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₇
commutator	C₂×C₁₇	C₂³×C₄
Frattini	C₄	C₂³×D₁₇
Fitting	C₂²×C₈×C₁₇	C₂
socle	C₂³×C₁₇	C₂×C₄
prime radical	C₂³×D₁₇	C₄
cosocle	C₄×C₁₇	C₂⁴
hypercenter	C₂²×C₈	D₁₇
hypocenter	C₁₇	C₂²×M₁₆
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×C₈×C₁₇	C₂	semidirect
3. C₂²×#3622	C₂	semidirect
4. C₂×#63218	C₂	direct (N × H)
10. C₂×C₈×C₁₇	C₂²	semidirect
11. C₂×#3622	C₂²	semidirect
12. #63218	C₂²	direct (N × H)
21. C₈×C₁₇	C₂³	semidirect
22. #3622	C₂³	semidirect
23. C₂²×C₁₇	M₁₆	semidirect
31. C₂×C₁₇	C₂×M₁₆	semidirect
37. C₂²×C₈	D₁₇	semidirect
38. C₁₇	C₂²×M₁₆	semidirect
46. C₂×C₈	C₂×D₁₇	semidirect
56. C₈	C₂²×D₁₇	semidirect
59. C₂²	#63218	direct (N × H)
65. C₂	C₂×#63218	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×C₄×D₁₇	2
C₂²×C₈×C₁₇	2
C₂²×#3622	2
C₂×#63218	2
C₂²×M₁₆	17

Subgroups — C22×#63218 #11763568

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#63218 #11763568