Subgroups — #510×C₂₅₇ #30656379

Order 16448.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#135×C₂₅₇	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₁₆×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₃₂×C₂₅₇	C₂	1	2	yes	maximal, complemented
4	SD₃₂×C₂₅₇	C₂	1	2	yes	maximal, complemented
5	M₃₂×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#41×C₂₅₇	—	2	1	no	—
7	C₂×Q₈×C₂₅₇	—	2	1	no	—
8	Q₁₆×C₂₅₇	—	2	1	no	—
9	Q₁₆×C₂₅₇	C₂²	1	2	yes	—
10	Q₁₆×C₂₅₇	C₂²	1	1	yes	characteristic
11	SD₁₆×C₂₅₇	—	2	1	no	—
12	D₈×C₂₅₇	C₂²	1	1	yes	characteristic
13	C₂×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
14	C₁₆×C₂₅₇	C₂²	1	2	yes	—
15	Q₈×C₂₅₇	—	2	1	no	—
16	Q₈×C₂₅₇	—	2	2	no	—
17	Q₈×C₂₅₇	—	4	1	no	—
18	D₄×C₂₅₇	—	2	1	no	—
19	D₄×C₂₅₇	—	4	1	no	—
20	C₂×C₄×C₂₅₇	—	4	1	no	—
21	C₂×C₄×C₂₅₇	—	4	1	no	—
22	C₂×C₄×C₂₅₇	D₄	1	1	yes	characteristic
23	C₈×C₂₅₇	D₄	1	1	yes	characteristic
24	C₈×C₂₅₇	C₂³	1	1	yes	characteristic
25	C₂²×C₂₅₇	—	4	1	no	—
26	C₂²×C₂₅₇	D₈	1	1	yes	characteristic
27	C₄×C₂₅₇	—	4	1	no	—
28	C₄×C₂₅₇	—	4	2	no	—
29	C₄×C₂₅₇	D₈	1	1	yes	characteristic
30	C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
31	C₂×C₂₅₇	—	2	1	no	—
32	C₂×C₂₅₇	—	8	1	no	—
33	C₂×C₂₅₇	C₂×D₈	1	1	yes	characteristic
34	C₂₅₇	#510	1	1	yes	characteristic, direct product factor, Sylow
35	#510	C₂₅₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	#135	C₂×C₂₅₇	1	1	yes	characteristic
37	C₂×Q₁₆	C₂×C₂₅₇	1	1	yes	characteristic, complemented
38	Q₃₂	C₂×C₂₅₇	1	2	yes	complemented
39	SD₃₂	C₂×C₂₅₇	1	2	yes	complemented
40	M₃₂	C₂×C₂₅₇	1	1	yes	characteristic, complemented
41	#41	—	2	1	no	—
42	C₂×Q₈	—	2	1	no	—
43	Q₁₆	—	2	1	no	—
44	Q₁₆	C₂²×C₂₅₇	1	2	yes	—
45	Q₁₆	C₂²×C₂₅₇	1	1	yes	characteristic
46	SD₁₆	—	2	1	no	—
47	D₈	C₂²×C₂₅₇	1	1	yes	characteristic
48	C₂×C₈	C₂²×C₂₅₇	1	1	yes	characteristic
49	C₁₆	C₂²×C₂₅₇	1	2	yes	—
50	Q₈	—	2	1	no	—
51	Q₈	—	2	2	no	—
52	Q₈	—	4	1	no	—
53	D₄	—	2	1	no	—
54	D₄	—	4	1	no	—
55	C₂×C₄	—	4	1	no	—
56	C₂×C₄	—	4	1	no	—
57	C₂×C₄	D₄×C₂₅₇	1	1	yes	characteristic
58	C₈	D₄×C₂₅₇	1	1	yes	characteristic
59	C₈	C₂³×C₂₅₇	1	1	yes	characteristic
60	C₂²	—	4	1	no	—
61	C₂²	D₈×C₂₅₇	1	1	yes	characteristic
62	C₄	—	4	1	no	—
63	C₄	—	4	2	no	—
64	C₄	D₈×C₂₅₇	1	1	yes	characteristic
65	C₄	C₂×D₄×C₂₅₇	1	1	yes	characteristic
66	C₂	—	2	1	no	—
67	C₂	—	8	1	no	—
68	C₂	C₂×D₈×C₂₅₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₅₇	C₂×D₈
commutator, Frattini, cosocle	C₈	C₂³×C₂₅₇
prime radical	#135×C₂₅₇	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₅₇ ◁ C₂×C₄×C₂₅₇ ◁ C₂×C₈×C₂₅₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×Q₁₆×C₂₅₇	C₂	semidirect
3. Q₃₂×C₂₅₇	C₂	semidirect
4. SD₃₂×C₂₅₇	C₂	semidirect
5. M₃₂×C₂₅₇	C₂	semidirect
34. C₂₅₇	#510	direct (N × H)
35. #510	C₂₅₇	direct (N × H)
37. C₂×Q₁₆	C₂×C₂₅₇	semidirect
38. Q₃₂	C₂×C₂₅₇	semidirect
39. SD₃₂	C₂×C₂₅₇	semidirect
40. M₃₂	C₂×C₂₅₇	semidirect

Maximal subgroups

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

subgroup	index
#135×C₂₅₇	2
C₂×Q₁₆×C₂₅₇	2
Q₃₂×C₂₅₇	2
SD₃₂×C₂₅₇	2
M₃₂×C₂₅₇	2
#510	257

Subgroups — #510×C257 #30656379

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #510×C₂₅₇ #30656379