Subgroups — Q₈×D₁₀₂₈ #30657404

Order 16448.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₂₅₇	C₂	1	2	yes	maximal, complemented
2	#22551592	C₂	1	3	yes	maximal, complemented
3	#22551572	C₂	1	6	yes	maximal, complemented
4	C₄×D₁₀₂₈	C₂	1	3	yes	maximal
5	C₄×Q₈×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₅₁₄	C₂²	1	6	yes	—
7	Q₈×D₂₅₇	—	2	8	no	—
8	C₂×D₁₀₂₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₂₅₇	C₂²	1	6	yes	—
10	C₄⋊Dic₂₅₇	C₂²	1	3	yes	—
11	C₄⋊Dic₂₅₇	C₂²	1	6	yes	—
12	#16861595	C₂²	1	6	yes	—
13	C₂×Q₈×C₂₅₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₂₅₇	C₂²	1	3	yes	—
15	C₄²×C₂₅₇	C₂²	1	3	yes	—
16	C₂²×D₂₅₇	C₂³	1	2	yes	—
17	Dic₅₁₄	—	2	12	no	—
18	D₁₀₂₈	Q₈	1	4	yes	direct product factor
19	C₂×Dic₂₅₇	C₂³	1	6	yes	—
20	C₄×D₂₅₇	—	2	12	no	—
21	Q₈×C₂₅₇	D₄	1	4	yes	complemented
22	C₂×C₄×C₂₅₇	C₂³	1	3	yes	—
23	C₂×C₄×C₂₅₇	C₂³	1	3	yes	—
24	C₂×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
25	C₂²×C₂₅₇	C₂⁴	1	1	yes	characteristic
26	C₂×D₂₅₇	—	2	2	no	—
27	C₂×D₂₅₇	—	2	2	no	—
28	C₂×D₂₅₇	C₂×Q₈	1	4	yes	complemented
29	C₄×C₂₅₇	—	2	3	no	—
30	C₄×C₂₅₇	C₂×D₄	1	6	yes	—
31	C₄×C₂₅₇	C₂×Q₈	1	2	yes	complemented
32	Dic₂₅₇	—	2	6	no	—
33	C₂×C₂₅₇	C₂²×D₄	1	1	yes	characteristic
34	C₂×C₂₅₇	C₂²×Q₈	1	1	yes	characteristic
35	C₂×C₂₅₇	2₋¹⁺⁴	1	1	yes	characteristic
36	D₂₅₇	—	2	4	no	—
37	C₂₅₇	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
38	D₄×Q₈	—	257	1	no	maximal, Sylow
39	C₂²×Q₈	—	257	2	no	—
40	C₄⋊Q₈	—	257	3	no	—
41	C₂²⋊Q₈	—	257	6	no	—
42	C₄×Q₈	D₂₅₇	1	1	yes	characteristic, complemented
43	C₄×D₄	—	257	3	no	—
44	C₂×Q₈	—	257	6	no	—
45	C₂×Q₈	—	514	8	no	—
46	C₂×Q₈	C₂×D₂₅₇	1	1	yes	characteristic
47	C₂×D₄	—	257	1	no	—
48	C₂²×C₄	—	257	6	no	—
49	C₄⋊C₄	—	257	3	no	—
50	C₄⋊C₄	—	257	6	no	—
51	C₄⋊C₄	C₂×D₂₅₇	1	3	yes	—
52	C₂²⋊C₄	—	257	6	no	—
53	C₄²	C₂×D₂₅₇	1	3	yes	—
54	C₂³	—	257	2	no	—
55	Q₈	—	514	12	no	—
56	Q₈	D₁₀₂₈	1	4	yes	direct product factor
57	D₄	—	257	4	no	—
58	C₂×C₄	—	257	6	no	—
59	C₂×C₄	—	514	12	no	—
60	C₂×C₄	C₂²×D₂₅₇	1	3	yes	—
61	C₂×C₄	C₂²×D₂₅₇	1	3	yes	—
62	C₂×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
63	C₂²	—	257	4	no	—
64	C₂²	—	514	2	no	—
65	C₂²	—	514	2	no	—
66	C₂²	C₂³×D₂₅₇	1	1	yes	characteristic
67	C₄	—	2	3	no	—
68	C₄	—	514	6	no	—
69	C₄	C₂×D₁₀₂₈	1	6	yes	—
70	C₄	Q₈×D₂₅₇	1	2	yes	complemented
71	C₂	—	514	4	no	—
72	C₂	C₂²×D₁₀₂₈	1	1	yes	characteristic
73	C₂	C₂×Q₈×D₂₅₇	1	1	yes	characteristic
74	C₂	#22551639	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₅₇
commutator, socle, cosocle	C₂²×C₂₅₇	C₂⁴
Fitting	C₄×Q₈×C₂₅₇	C₂
prime radical	C₂×D₁₀₂₈	C₂²
hypercenter	C₄×Q₈	D₂₅₇
hypocenter	C₂₅₇	D₄×Q₈
LCS	G ▷ C₂²×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×D₂₅₇	C₂	semidirect
2. #22551592	C₂	semidirect
3. #22551572	C₂	semidirect
5. C₄×Q₈×C₂₅₇	C₂	semidirect
18. D₁₀₂₈	Q₈	direct (N × H)
21. Q₈×C₂₅₇	D₄	semidirect
28. C₂×D₂₅₇	C₂×Q₈	semidirect
31. C₄×C₂₅₇	C₂×Q₈	semidirect
37. C₂₅₇	D₄×Q₈	semidirect
42. C₄×Q₈	D₂₅₇	semidirect
56. Q₈	D₁₀₂₈	direct (N × H)
70. C₄	Q₈×D₂₅₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×D₂₅₇	2
#22551592	2
#22551572	2
C₄×D₁₀₂₈	2
C₄×Q₈×C₂₅₇	2
D₄×Q₈	257

Subgroups — Q8×D1028 #30657404

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×D₁₀₂₈ #30657404