Subgroups — C₂×#22551592 #30657281

Order 16448.

290 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₅₁₄	C₂	1	2	yes	maximal
2	#22551592	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊Dic₂₅₇	C₂	1	4	yes	maximal
4	C₂×C₄²×C₂₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₅₁₄	—	2	8	no	—
6	C₂×Dic₅₁₄	C₂²	1	8	yes	—
7	C₂²×Dic₂₅₇	C₂²	1	4	yes	—
8	C₄⋊Dic₂₅₇	C₂²	1	16	yes	—
9	C₂²×C₄×C₂₅₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₂₅₇	C₂²	1	2	yes	—
11	C₄²×C₂₅₇	C₂²	1	4	yes	—
12	Dic₅₁₄	—	2	16	no	—
13	C₂×Dic₂₅₇	—	2	8	no	—
14	C₂×Dic₂₅₇	C₂³	1	8	yes	—
15	C₂³×C₂₅₇	C₂³	1	1	yes	characteristic
16	C₂×C₄×C₂₅₇	D₄	1	4	yes	—
17	C₂×C₄×C₂₅₇	Q₈	1	8	yes	complemented
18	C₂×C₄×C₂₅₇	C₂³	1	2	yes	—
19	C₂×C₄×C₂₅₇	C₂³	1	4	yes	—
20	C₂²×C₂₅₇	C₂×D₄	1	2	yes	—
21	C₂²×C₂₅₇	C₂×Q₈	1	4	yes	—
22	C₂²×C₂₅₇	C₂⁴	1	1	yes	characteristic
23	C₄×C₂₅₇	C₂×D₄	1	4	yes	—
24	C₄×C₂₅₇	C₂×Q₈	1	8	yes	complemented
25	Dic₂₅₇	—	2	8	no	—
26	C₂×C₂₅₇	C₄⋊Q₈	1	4	yes	complemented
27	C₂×C₂₅₇	C₂²×D₄	1	1	yes	characteristic
28	C₂×C₂₅₇	C₂²×Q₈	1	2	yes	—
29	C₂₅₇	C₂×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
30	C₂×C₄⋊Q₈	—	257	1	no	maximal, Sylow
31	C₂²×Q₈	—	257	2	no	—
32	C₄⋊Q₈	—	257	8	no	—
33	C₂×C₄⋊C₄	—	257	4	no	—
34	C₂×C₄²	D₂₅₇	1	1	yes	characteristic
35	C₂×Q₈	—	257	8	no	—
36	C₂×Q₈	—	514	8	no	—
37	C₂²×C₄	—	257	4	no	—
38	C₂²×C₄	C₂×D₂₅₇	1	1	yes	characteristic
39	C₂²×C₄	C₂×D₂₅₇	1	2	yes	—
40	C₄⋊C₄	—	257	16	no	—
41	C₄²	C₂×D₂₅₇	1	4	yes	—
42	C₂³	C₂²×D₂₅₇	1	1	yes	characteristic
43	Q₈	—	514	16	no	—
44	C₂×C₄	—	257	8	no	—
45	C₂×C₄	—	514	8	no	—
46	C₂×C₄	D₁₀₂₈	1	4	yes	—
47	C₂×C₄	Dic₅₁₄	1	8	yes	complemented
48	C₂×C₄	C₂²×D₂₅₇	1	2	yes	—
49	C₂×C₄	C₂²×D₂₅₇	1	4	yes	—
50	C₂²	C₂×D₁₀₂₈	1	2	yes	—
51	C₂²	C₂×Dic₅₁₄	1	4	yes	—
52	C₂²	C₂³×D₂₅₇	1	1	yes	characteristic
53	C₄	—	514	8	no	—
54	C₄	C₂×D₁₀₂₈	1	4	yes	—
55	C₄	C₂×Dic₅₁₄	1	8	yes	complemented
56	C₂	#22551592	1	4	yes	direct product factor
57	C₂	C₂²×D₁₀₂₈	1	1	yes	characteristic
58	C₂	C₂²×Dic₅₁₄	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×C₄⋊Q₈
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
2. #22551592	C₂	direct (N × H)
17. C₂×C₄×C₂₅₇	Q₈	semidirect
24. C₄×C₂₅₇	C₂×Q₈	semidirect
26. C₂×C₂₅₇	C₄⋊Q₈	semidirect
29. C₂₅₇	C₂×C₄⋊Q₈	semidirect
47. C₂×C₄	Dic₅₁₄	semidirect
55. C₄	C₂×Dic₅₁₄	semidirect
56. C₂	#22551592	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×Dic₅₁₄	2
#22551592	2
C₂×C₄⋊Dic₂₅₇	2
C₂×C₄²×C₂₅₇	2
C₂×C₄⋊Q₈	257

Subgroups — C2×#22551592 #30657281

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22551592 #30657281