Subgroups — #490×C₂₅₇ #30656359

Order 16448.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#127×C₂₅₇	C₂	1	1	yes	maximal, characteristic
2	#125×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#102×C₂₅₇	C₂	1	4	yes	maximal, complemented
4	#97×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₄×C₂₅₇	—	2	2	no	—
6	C₂×D₄×C₂₅₇	C₂²	1	2	yes	—
7	C₂×C₈×C₂₅₇	C₂²	1	2	yes	—
8	C₄⋊C₄×C₂₅₇	—	2	2	no	—
9	C₄⋊C₄×C₂₅₇	C₂²	1	2	yes	—
10	C₄²×C₂₅₇	C₂²	1	1	yes	characteristic
11	C₂³×C₂₅₇	—	2	2	no	—
12	D₄×C₂₅₇	—	2	4	no	—
13	D₄×C₂₅₇	—	4	4	no	—
14	C₂×C₄×C₂₅₇	—	2	2	no	—
15	C₂×C₄×C₂₅₇	D₄	1	2	yes	—
16	C₂×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
17	C₈×C₂₅₇	—	2	2	no	—
18	C₂²×C₂₅₇	—	4	2	no	—
19	C₂²×C₂₅₇	—	4	4	no	—
20	C₂²×C₂₅₇	C₂×D₄	1	1	yes	characteristic
21	C₄×C₂₅₇	—	2	2	no	—
22	C₄×C₂₅₇	—	4	2	no	—
23	C₄×C₂₅₇	#41	1	2	yes	—
24	C₂×C₂₅₇	—	8	2	no	—
25	C₂×C₂₅₇	#124	1	1	yes	characteristic
26	C₂×C₂₅₇	AGL(1,ℤ/8)	1	2	yes	—
27	C₂₅₇	#490	1	1	yes	characteristic, direct product factor, Sylow
28	#490	C₂₅₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	#127	C₂×C₂₅₇	1	1	yes	characteristic
30	#125	C₂×C₂₅₇	1	1	yes	characteristic, complemented
31	#102	C₂×C₂₅₇	1	4	yes	complemented
32	#97	C₂×C₂₅₇	1	1	yes	characteristic, complemented
33	C₂×D₄	—	2	2	no	—
34	C₂×D₄	C₂²×C₂₅₇	1	2	yes	—
35	C₂×C₈	C₂²×C₂₅₇	1	2	yes	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	C₂²×C₂₅₇	1	2	yes	—
38	C₄²	C₂²×C₂₅₇	1	1	yes	characteristic
39	C₂³	—	2	2	no	—
40	D₄	—	2	4	no	—
41	D₄	—	4	4	no	—
42	C₂×C₄	—	2	2	no	—
43	C₂×C₄	D₄×C₂₅₇	1	2	yes	—
44	C₂×C₄	C₂³×C₂₅₇	1	1	yes	characteristic
45	C₈	—	2	2	no	—
46	C₂²	—	4	2	no	—
47	C₂²	—	4	4	no	—
48	C₂²	C₂×D₄×C₂₅₇	1	1	yes	characteristic
49	C₄	—	2	2	no	—
50	C₄	—	4	2	no	—
51	C₄	#41×C₂₅₇	1	2	yes	—
52	C₂	—	8	2	no	—
53	C₂	#124×C₂₅₇	1	1	yes	characteristic
54	C₂	AGL(1,ℤ/8)×C₂₅₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₅₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₂₅₇
prime radical	#127×C₂₅₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₅₇ ◁ C₄²×C₂₅₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #125×C₂₅₇	C₂	semidirect
3. #102×C₂₅₇	C₂	semidirect
4. #97×C₂₅₇	C₂	semidirect
27. C₂₅₇	#490	direct (N × H)
28. #490	C₂₅₇	direct (N × H)
30. #125	C₂×C₂₅₇	semidirect
31. #102	C₂×C₂₅₇	semidirect
32. #97	C₂×C₂₅₇	semidirect

Maximal subgroups

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

subgroup	index
#127×C₂₅₇	2
#125×C₂₅₇	2
#102×C₂₅₇	2
#97×C₂₅₇	2
#490	257

Subgroups — #490×C257 #30656359

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #490×C₂₅₇ #30656359