Subgroups — Q₈×#6034 #13104925

Order 1600.

274 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#1039	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#6034	C₂	1	3	yes	maximal
3	C₄×#6034	C₂	1	3	yes	maximal
4	C₂×C₅²×Q₈	C₄	1	1	yes	characteristic
5	C₂×#6034	C₂²	1	3	yes	—
6	C₂×#6034	C₂²	1	1	yes	characteristic
7	C₄⋊#1039	C₄	1	3	yes	—
8	C₄×#1039	C₂²	1	3	yes	—
9	Q₈×#170	—	5	2	no	maximal
10	C₅²×Q₈	C₈	1	4	yes	complemented
11	C₂×C₄×C₅²	C₂×C₄	1	3	yes	—
12	C₂×#1039	C₂×C₄	1	3	yes	—
13	C₂×#1039	C₂³	1	1	yes	characteristic
14	#6034	—	2	3	no	—
15	#6034	Q₈	1	2	yes	direct product factor
16	Q₈×Dic₅	—	5	2	no	—
17	Q₈×Dic₅	—	10	2	no	—
18	C₄⋊#170	—	5	6	no	—
19	C₄×#170	—	5	6	no	—
20	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
21	C₄×C₅²	C₂×C₈	1	6	yes	—
22	#1039	—	2	3	no	—
23	#1039	C₂×Q₈	1	1	yes	characteristic
24	#1039	#41	1	1	yes	characteristic
25	C₂×C₅×Q₈	—	2	2	no	—
26	C₂×C₅×Q₈	AGL(1,𝔽₅)	1	2	yes	—
27	C₂×#170	—	5	2	no	—
28	C₂×#170	—	5	6	no	—
29	C₄⋊Dic₅	—	5	6	no	—
30	C₄⋊Dic₅	—	10	6	no	—
31	C₄×Dic₅	—	5	6	no	—
32	C₄×Dic₅	—	10	6	no	—
33	C₈×Q₈	—	25	1	no	Sylow
34	C₂×C₅²	C₄×Q₈	1	1	yes	characteristic
35	C₂×C₅²	C₂²×C₈	1	1	yes	characteristic
36	C₂×C₅²	#131	1	1	yes	characteristic
37	C₅×Q₈	—	2	8	no	—
38	C₅×Q₈	#170	1	8	yes	complemented
39	C₂×C₄×C₅	—	2	6	no	—
40	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	6	yes	—
41	C₂×Dic₅	—	5	2	no	—
42	C₂×Dic₅	—	5	6	no	—
43	C₂×Dic₅	—	10	2	no	—
44	C₂×Dic₅	—	10	6	no	—
45	#170	—	5	4	no	—
46	#170	—	10	6	no	—
47	C₄×Q₈	—	25	1	no	—
48	C₄⋊C₈	—	25	3	no	—
49	C₄×C₈	—	25	3	no	—
50	C₅²	C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
51	C₂²×C₅	—	2	2	no	—
52	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	2	yes	—
53	C₄×C₅	—	2	12	no	—
54	C₄×C₅	C₂×#170	1	12	yes	—
55	Dic₅	—	5	2	no	—
56	Dic₅	—	5	2	no	—
57	Dic₅	—	10	2	no	—
58	Dic₅	—	10	2	no	—
59	Dic₅	—	10	6	no	—
60	Dic₅	—	20	6	no	—
61	C₂×Q₈	#1044	1	1	yes	characteristic
62	C₂×C₈	—	25	1	no	—
63	C₂×C₈	—	25	3	no	—
64	C₄⋊C₄	—	25	3	no	—
65	C₄²	—	25	3	no	—
66	C₂×C₅	—	2	2	no	—
67	C₂×C₅	—	2	2	no	—
68	C₂×C₅	—	2	2	no	—
69	C₂×C₅	#4133	1	2	yes	—
70	C₂×C₅	Q₈×AGL(1,𝔽₅)	1	2	yes	—
71	C₂×C₅	C₂²×#170	1	2	yes	—
72	Q₈	#6034	1	4	yes	direct product factor
73	C₂×C₄	—	25	1	no	—
74	C₂×C₄	—	25	3	no	—
75	C₂×C₄	C₂×#1044	1	3	yes	—
76	C₈	—	25	2	no	—
77	C₈	—	50	3	no	—
78	C₅	—	2	2	no	—
79	C₅	Q₈×#170	1	2	yes	complemented
80	C₂²	C₂²×#1044	1	1	yes	characteristic
81	C₄	—	25	1	no	—
82	C₄	—	25	1	no	—
83	C₄	—	50	3	no	—
84	C₄	C₂×#6034	1	6	yes	—
85	C₂	#11717454	1	1	yes	characteristic
86	C₂	Q₈×#1044	1	1	yes	characteristic
87	C₂	C₂²×#6034	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#1044
commutator	C₂×C₅²	C₂²×C₈
Fitting	C₂×C₅²×Q₈	C₄
socle, prime radical	C₂²×C₅²	C₂²×C₄
cosocle	C₂×#1039	C₂³
hypercenter	C₂×Q₈	#1044
hypocenter	C₅²	C₈×Q₈
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
10. C₅²×Q₈	C₈	semidirect
15. #6034	Q₈	direct (N × H)
38. C₅×Q₈	#170	semidirect
50. C₅²	C₈×Q₈	semidirect
72. Q₈	#6034	direct (N × H)
79. C₅	Q₈×#170	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#1039	2
C₄⋊#6034	2
C₄×#6034	2
Q₈×#170	5

Subgroups — Q8×#6034 #13104925

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#6034 #13104925