Subgroups — C₂×#22551584 #30657259

Order 16448.

226 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22551584	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊Dic₂₅₇	C₂	1	2	yes	maximal
3	C₂×#16861597	C₂	1	4	yes	maximal
4	C₂×C₄²×C₂₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₅₇	C₂²	1	4	yes	—
6	C₄⋊Dic₂₅₇	C₂²	1	8	yes	—
7	#16861597	C₂²	1	16	yes	—
8	C₂²×C₄×C₂₅₇	C₂²	1	2	yes	—
9	C₂²×C₄×C₂₅₇	C₂²	1	1	yes	characteristic
10	C₄²×C₂₅₇	C₂²	1	4	yes	—
11	C₂×Dic₂₅₇	—	2	8	no	—
12	C₂×Dic₂₅₇	C₂³	1	8	yes	—
13	C₂³×C₂₅₇	C₂³	1	1	yes	characteristic
14	C₂×C₄×C₂₅₇	—	2	4	no	—
15	C₂×C₄×C₂₅₇	Q₈	1	4	yes	—
16	C₂×C₄×C₂₅₇	C₂³	1	4	yes	—
17	C₂×C₄×C₂₅₇	C₂³	1	2	yes	—
18	C₂²×C₂₅₇	C₂×Q₈	1	2	yes	—
19	C₂²×C₂₅₇	#41	1	4	yes	—
20	C₂²×C₂₅₇	C₂⁴	1	1	yes	characteristic
21	C₄×C₂₅₇	—	2	4	no	—
22	C₄×C₂₅₇	C₂×Q₈	1	4	yes	—
23	Dic₂₅₇	—	2	8	no	—
24	C₂×C₂₅₇	#125	1	4	yes	complemented
25	C₂×C₂₅₇	C₂²×Q₈	1	1	yes	characteristic
26	C₂×C₂₅₇	C₂×#41	1	2	yes	—
27	C₂₅₇	C₂×#125	1	1	yes	characteristic, complemented, Sylow
28	C₂×#125	—	257	1	no	maximal, Sylow
29	#125	—	257	8	no	—
30	C₂×C₄⋊C₄	—	257	4	no	—
31	C₂×C₄⋊C₄	—	257	2	no	—
32	C₂×C₄²	D₂₅₇	1	1	yes	characteristic
33	C₂²×C₄	—	257	4	no	—
34	C₂²×C₄	C₂×D₂₅₇	1	2	yes	—
35	C₂²×C₄	C₂×D₂₅₇	1	1	yes	characteristic
36	C₄⋊C₄	—	257	16	no	—
37	C₄⋊C₄	—	257	8	no	—
38	C₄²	C₂×D₂₅₇	1	4	yes	—
39	C₂³	C₂²×D₂₅₇	1	1	yes	characteristic
40	C₂×C₄	—	257	8	no	—
41	C₂×C₄	—	2	4	no	—
42	C₂×C₄	—	514	8	no	—
43	C₂×C₄	Dic₅₁₄	1	4	yes	—
44	C₂×C₄	C₂²×D₂₅₇	1	4	yes	—
45	C₂×C₄	C₂²×D₂₅₇	1	2	yes	—
46	C₂²	C₂×Dic₅₁₄	1	2	yes	—
47	C₂²	#16861618	1	4	yes	—
48	C₂²	C₂³×D₂₅₇	1	1	yes	characteristic
49	C₄	—	2	4	no	—
50	C₄	—	514	8	no	—
51	C₄	C₂×Dic₅₁₄	1	4	yes	—
52	C₂	#22551584	1	4	yes	direct product factor
53	C₂	C₂²×Dic₅₁₄	1	1	yes	characteristic
54	C₂	C₂×#16861618	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₂₅₇
commutator, cosocle	C₂²×C₂₅₇	C₂⁴
Frattini	C₂²	C₂³×D₂₅₇
Fitting	C₂×C₄²×C₂₅₇	C₂
socle, prime radical	C₂³×C₂₅₇	C₂³
hypercenter	C₂×C₄²	D₂₅₇
hypocenter	C₂₅₇	C₂×#125
LCS	G ▷ C₂²×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
1. #22551584	C₂	direct (N × H)
24. C₂×C₂₅₇	#125	semidirect
27. C₂₅₇	C₂×#125	semidirect
52. C₂	#22551584	direct (N × H)

Maximal subgroups

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

subgroup	index
#22551584	2
C₂×C₄⋊Dic₂₅₇	2
C₂×#16861597	2
C₂×C₄²×C₂₅₇	2
C₂×#125	257

Subgroups — C2×#22551584 #30657259

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22551584 #30657259