Subgroups — #518×C₂₅₇ #30656387

Order 16448.

338 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₄×C₂₅₇	C₂	1	1	yes	maximal, characteristic
2	C₄×D₄×C₂₅₇	C₂	1	8	yes	maximal, complemented
3	#117×C₂₅₇	C₂	1	2	yes	maximal, complemented
4	C₂×C₂²⋊C₄×C₂₅₇	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₂₅₇	C₂²	1	2	yes	—
6	C₂×D₄×C₂₅₇	C₄	1	8	yes	complemented
7	C₂×D₄×C₂₅₇	C₂²	1	4	yes	—
8	C₂²×C₄×C₂₅₇	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₂₅₇	C₂²	1	8	yes	complemented
10	C₄⋊C₄×C₂₅₇	C₂²	1	4	yes	complemented
11	C₂²⋊C₄×C₂₅₇	C₂²	1	4	yes	—
12	C₂²⋊C₄×C₂₅₇	C₂²	1	8	yes	complemented
13	C₄²×C₂₅₇	C₂²	1	4	yes	complemented
14	C₂³×C₂₅₇	—	4	4	no	—
15	C₂³×C₂₅₇	C₂×C₄	1	8	yes	complemented
16	C₂³×C₂₅₇	C₂³	1	1	yes	characteristic
17	C₂³×C₂₅₇	C₂³	1	4	yes	—
18	D₄×C₂₅₇	C₂×C₄	1	16	yes	complemented
19	C₂×C₄×C₂₅₇	—	4	8	no	—
20	C₂×C₄×C₂₅₇	C₂×C₄	1	4	yes	complemented
21	C₂×C₄×C₂₅₇	C₂³	1	8	yes	—
22	C₂×C₄×C₂₅₇	C₂³	1	2	yes	—
23	C₂²×C₂₅₇	—	2	8	no	—
24	C₂²×C₂₅₇	—	2	2	no	—
25	C₂²×C₂₅₇	—	4	8	no	—
26	C₂²×C₂₅₇	C₂²×C₄	1	2	yes	—
27	C₂²×C₂₅₇	C₂²×C₄	1	8	yes	—
28	C₂²×C₂₅₇	C₂⁴	1	1	yes	characteristic
29	C₄×C₂₅₇	—	2	8	no	—
30	C₄×C₂₅₇	C₂²×C₄	1	4	yes	—
31	C₂×C₂₅₇	—	2	2	no	—
32	C₂×C₂₅₇	—	2	8	no	—
33	C₂×C₂₅₇	C₂³×C₄	1	1	yes	characteristic
34	C₂×C₂₅₇	H(5,𝔽₂)	1	2	yes	—
35	C₂₅₇	#518	1	1	yes	characteristic, direct product factor, Sylow
36	#518	C₂₅₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂²×D₄	C₂×C₂₅₇	1	1	yes	characteristic
38	C₄×D₄	C₂×C₂₅₇	1	8	yes	complemented
39	#117	C₂×C₂₅₇	1	2	yes	complemented
40	C₂×C₂²⋊C₄	C₂×C₂₅₇	1	4	yes	complemented
41	C₂⁴	C₂²×C₂₅₇	1	2	yes	—
42	C₂×D₄	C₄×C₂₅₇	1	8	yes	complemented
43	C₂×D₄	C₂²×C₂₅₇	1	4	yes	—
44	C₂²×C₄	C₂²×C₂₅₇	1	1	yes	characteristic
45	C₂²×C₄	C₂²×C₂₅₇	1	8	yes	complemented
46	C₄⋊C₄	C₂²×C₂₅₇	1	4	yes	complemented
47	C₂²⋊C₄	C₂²×C₂₅₇	1	4	yes	—
48	C₂²⋊C₄	C₂²×C₂₅₇	1	8	yes	complemented
49	C₄²	C₂²×C₂₅₇	1	4	yes	complemented
50	C₂³	—	4	4	no	—
51	C₂³	C₂×C₄×C₂₅₇	1	8	yes	complemented
52	C₂³	C₂³×C₂₅₇	1	1	yes	characteristic
53	C₂³	C₂³×C₂₅₇	1	4	yes	—
54	D₄	C₂×C₄×C₂₅₇	1	16	yes	complemented
55	C₂×C₄	—	4	8	no	—
56	C₂×C₄	C₂×C₄×C₂₅₇	1	4	yes	complemented
57	C₂×C₄	C₂³×C₂₅₇	1	8	yes	—
58	C₂×C₄	C₂³×C₂₅₇	1	2	yes	—
59	C₂²	—	2	8	no	—
60	C₂²	—	2	2	no	—
61	C₂²	—	4	8	no	—
62	C₂²	C₂²×C₄×C₂₅₇	1	2	yes	—
63	C₂²	C₂²×C₄×C₂₅₇	1	8	yes	—
64	C₂²	C₂⁴×C₂₅₇	1	1	yes	characteristic
65	C₄	—	2	8	no	—
66	C₄	C₂²×C₄×C₂₅₇	1	4	yes	—
67	C₂	—	2	2	no	—
68	C₂	—	2	8	no	—
69	C₂	C₂³×C₄×C₂₅₇	1	1	yes	characteristic
70	C₂	H(5,𝔽₂)×C₂₅₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₅₇	C₂⁴
commutator	C₂	C₂³×C₄×C₂₅₇
Frattini, cosocle	C₂²	C₂⁴×C₂₅₇
prime radical	C₂²×D₄×C₂₅₇	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₅₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×D₄×C₂₅₇	C₂	semidirect
3. #117×C₂₅₇	C₂	semidirect
4. C₂×C₂²⋊C₄×C₂₅₇	C₂	semidirect
6. C₂×D₄×C₂₅₇	C₄	semidirect
9. C₂²×C₄×C₂₅₇	C₂²	semidirect
10. C₄⋊C₄×C₂₅₇	C₂²	semidirect
12. C₂²⋊C₄×C₂₅₇	C₂²	semidirect
13. C₄²×C₂₅₇	C₂²	semidirect
15. C₂³×C₂₅₇	C₂×C₄	semidirect
18. D₄×C₂₅₇	C₂×C₄	semidirect
20. C₂×C₄×C₂₅₇	C₂×C₄	semidirect
35. C₂₅₇	#518	direct (N × H)
36. #518	C₂₅₇	direct (N × H)
38. C₄×D₄	C₂×C₂₅₇	semidirect
39. #117	C₂×C₂₅₇	semidirect
40. C₂×C₂²⋊C₄	C₂×C₂₅₇	semidirect
42. C₂×D₄	C₄×C₂₅₇	semidirect
45. C₂²×C₄	C₂²×C₂₅₇	semidirect
46. C₄⋊C₄	C₂²×C₂₅₇	semidirect
48. C₂²⋊C₄	C₂²×C₂₅₇	semidirect
49. C₄²	C₂²×C₂₅₇	semidirect
51. C₂³	C₂×C₄×C₂₅₇	semidirect
54. D₄	C₂×C₄×C₂₅₇	semidirect
56. C₂×C₄	C₂×C₄×C₂₅₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₄×C₂₅₇	2
C₄×D₄×C₂₅₇	2
#117×C₂₅₇	2
C₂×C₂²⋊C₄×C₂₅₇	2
#518	257

Subgroups — #518×C257 #30656387

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #518×C₂₅₇ #30656387