Subgroups — Q₈×D₃₅₆ #19152265

Order 5696.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₈₉	C₂	1	2	yes	maximal, complemented
2	#15668012	C₂	1	3	yes	maximal, complemented
3	#15667992	C₂	1	6	yes	maximal, complemented
4	C₄×D₃₅₆	C₂	1	3	yes	maximal
5	C₄×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₁₇₈	C₂²	1	6	yes	—
7	Q₈×D₈₉	—	2	8	no	—
8	C₂×D₃₅₆	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₈₉	C₂²	1	6	yes	—
10	C₄⋊Dic₈₉	C₂²	1	3	yes	—
11	C₄⋊Dic₈₉	C₂²	1	6	yes	—
12	#13083551	C₂²	1	6	yes	—
13	C₂×Q₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₈₉	C₂²	1	3	yes	—
15	C₄²×C₈₉	C₂²	1	3	yes	—
16	C₂²×D₈₉	C₂³	1	2	yes	—
17	Q₈×C₈₉	D₄	1	4	yes	complemented
18	C₂×C₄×C₈₉	C₂³	1	3	yes	—
19	C₂×C₄×C₈₉	C₂³	1	3	yes	—
20	C₂×C₄×C₈₉	C₂³	1	1	yes	characteristic
21	C₂×Dic₈₉	C₂³	1	6	yes	—
22	D₃₅₆	Q₈	1	4	yes	direct product factor
23	C₄×D₈₉	—	2	12	no	—
24	Dic₁₇₈	—	2	12	no	—
25	C₂²×C₈₉	C₂⁴	1	1	yes	characteristic
26	C₂×D₈₉	—	2	2	no	—
27	C₂×D₈₉	—	2	2	no	—
28	C₂×D₈₉	C₂×Q₈	1	4	yes	complemented
29	C₄×C₈₉	—	2	3	no	—
30	C₄×C₈₉	C₂×D₄	1	6	yes	—
31	C₄×C₈₉	C₂×Q₈	1	2	yes	complemented
32	Dic₈₉	—	2	6	no	—
33	C₂×C₈₉	C₂²×D₄	1	1	yes	characteristic
34	C₂×C₈₉	C₂²×Q₈	1	1	yes	characteristic
35	C₂×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
36	D₈₉	—	2	4	no	—
37	C₈₉	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
38	D₄×Q₈	—	89	1	no	maximal, Sylow
39	C₂²×Q₈	—	89	2	no	—
40	C₄⋊Q₈	—	89	3	no	—
41	C₂²⋊Q₈	—	89	6	no	—
42	C₄×Q₈	D₈₉	1	1	yes	characteristic, complemented
43	C₄×D₄	—	89	3	no	—
44	C₂×Q₈	—	89	6	no	—
45	C₂×Q₈	—	178	8	no	—
46	C₂×Q₈	C₂×D₈₉	1	1	yes	characteristic
47	C₂×D₄	—	89	1	no	—
48	C₂²×C₄	—	89	6	no	—
49	C₄⋊C₄	—	89	3	no	—
50	C₄⋊C₄	—	89	6	no	—
51	C₄⋊C₄	C₂×D₈₉	1	3	yes	—
52	C₂²⋊C₄	—	89	6	no	—
53	C₄²	C₂×D₈₉	1	3	yes	—
54	C₂³	—	89	2	no	—
55	Q₈	—	178	12	no	—
56	Q₈	D₃₅₆	1	4	yes	direct product factor
57	D₄	—	89	4	no	—
58	C₂×C₄	—	89	6	no	—
59	C₂×C₄	—	178	12	no	—
60	C₂×C₄	C₂²×D₈₉	1	3	yes	—
61	C₂×C₄	C₂²×D₈₉	1	3	yes	—
62	C₂×C₄	C₂²×D₈₉	1	1	yes	characteristic
63	C₂²	—	89	4	no	—
64	C₂²	—	178	2	no	—
65	C₂²	—	178	2	no	—
66	C₂²	C₂³×D₈₉	1	1	yes	characteristic
67	C₄	—	2	3	no	—
68	C₄	—	178	6	no	—
69	C₄	C₂×D₃₅₆	1	6	yes	—
70	C₄	Q₈×D₈₉	1	2	yes	complemented
71	C₂	—	178	4	no	—
72	C₂	C₂²×D₃₅₆	1	1	yes	characteristic
73	C₂	C₂×Q₈×D₈₉	1	1	yes	characteristic
74	C₂	#15668059	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂⁴
Fitting	C₄×Q₈×C₈₉	C₂
hypercenter	C₄×Q₈	D₈₉
hypocenter	C₈₉	D₄×Q₈
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×D₈₉	C₂	semidirect
2. #15668012	C₂	semidirect
3. #15667992	C₂	semidirect
5. C₄×Q₈×C₈₉	C₂	semidirect
17. Q₈×C₈₉	D₄	semidirect
22. D₃₅₆	Q₈	direct (N × H)
28. C₂×D₈₉	C₂×Q₈	semidirect
31. C₄×C₈₉	C₂×Q₈	semidirect
37. C₈₉	D₄×Q₈	semidirect
42. C₄×Q₈	D₈₉	semidirect
56. Q₈	D₃₅₆	direct (N × H)
70. C₄	Q₈×D₈₉	semidirect

Subgroups — Q8×D356 #19152265

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×D₃₅₆ #19152265