Subgroups — Q₈×#6045 #13107094

Order 1600.

848 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅²×Q₈	C₂	1	1	yes	maximal, characteristic
2	Q₈×#1039	C₂	1	2	yes	maximal
3	#11717208	C₂	1	6	yes	maximal
4	#11717189	C₂	1	3	yes	maximal
5	C₄×#6045	C₂	1	3	yes	maximal
6	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
7	C₂×#6045	C₂²	1	6	yes	—
8	C₂×#6045	C₂²	1	1	yes	characteristic
9	C₅²×C₄⋊C₄	C₂²	1	3	yes	—
10	C₄²×C₅²	C₂²	1	3	yes	—
11	C₄⋊#1039	C₂²	1	3	yes	—
12	C₄⋊#1039	C₂²	1	6	yes	—
13	#87806	C₂²	1	6	yes	—
14	C₄×#1039	C₂²	1	6	yes	—
15	Q₈×Dic₁₀	—	5	6	no	maximal
16	C₅²×Q₈	Q₈	1	4	yes	complemented
17	C₂×C₄×C₅²	C₂³	1	3	yes	—
18	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
19	C₂×C₄×C₅²	C₂³	1	3	yes	—
20	C₂×#1039	C₂³	1	6	yes	—
21	C₂×#1039	C₂³	1	2	yes	—
22	#6045	—	4	6	no	—
23	#6045	Q₈	1	4	yes	direct product factor
24	C₄×C₅×Q₈	D₅	1	6	yes	—
25	Q₈×Dic₅	—	5	12	no	—
26	#4034	—	5	36	no	—
27	#4015	—	5	18	no	—
28	C₄×Dic₁₀	—	5	18	no	—
29	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
30	C₄×C₅²	—	2	3	no	—
31	C₄×C₅²	C₂×Q₈	1	6	yes	—
32	C₄×C₅²	C₂×Q₈	1	2	yes	—
33	#1039	—	2	6	no	—
34	#1039	C₂×Q₈	1	4	yes	—
35	C₂×C₅×Q₈	C₂×D₅	1	6	yes	—
36	C₂×Dic₁₀	—	5	6	no	—
37	C₂×Dic₁₀	—	5	36	no	—
38	C₅×C₄⋊C₄	C₂×D₅	1	18	yes	—
39	C₄²×C₅	C₂×D₅	1	18	yes	—
40	C₄⋊Dic₅	—	5	18	no	—
41	C₄⋊Dic₅	—	5	36	no	—
42	#683	—	5	36	no	—
43	C₄×Dic₅	—	5	36	no	—
44	Q₈²	—	25	1	no	Sylow
45	C₂×C₅²	C₂²×Q₈	1	1	yes	characteristic
46	C₂×C₅²	C₂²×Q₈	1	1	yes	characteristic
47	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
48	C₅×Q₈	Dic₁₀	1	24	yes	complemented
49	C₂×C₄×C₅	C₂²×D₅	1	18	yes	—
50	C₂×C₄×C₅	C₂²×D₅	1	18	yes	—
51	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
52	C₂×Dic₅	—	5	12	no	—
53	C₂×Dic₅	—	5	36	no	—
54	Dic₁₀	—	5	24	no	—
55	Dic₁₀	—	20	36	no	—
56	C₄⋊Q₈	—	25	6	no	—
57	C₄⋊Q₈	—	25	3	no	—
58	C₄×Q₈	—	25	2	no	—
59	C₄×Q₈	—	25	3	no	—
60	C₄×Q₈	C₅⋊D₅	1	1	yes	characteristic
61	C₅²	Q₈²	1	1	yes	characteristic, complemented, Sylow
62	C₂²×C₅	C₂³×D₅	1	6	yes	—
63	C₄×C₅	—	2	18	no	—
64	C₄×C₅	C₂×Dic₁₀	1	36	yes	—
65	C₄×C₅	Q₈×D₅	1	12	yes	—
66	Dic₅	—	5	24	no	—
67	Dic₅	—	10	36	no	—
68	C₂×Q₈	—	25	1	no	—
69	C₂×Q₈	—	25	6	no	—
70	C₂×Q₈	C₂×C₅⋊D₅	1	1	yes	characteristic
71	C₄⋊C₄	—	25	6	no	—
72	C₄⋊C₄	—	25	6	no	—
73	C₄⋊C₄	—	25	3	no	—
74	C₄⋊C₄	C₂×C₅⋊D₅	1	3	yes	—
75	C₄²	—	25	6	no	—
76	C₄²	C₂×C₅⋊D₅	1	3	yes	—
77	C₂×C₅	C₂²×Dic₁₀	1	6	yes	—
78	C₂×C₅	C₂×Q₈×D₅	1	6	yes	—
79	C₂×C₅	#4149	1	6	yes	—
80	Q₈	—	25	4	no	—
81	Q₈	—	100	6	no	—
82	Q₈	#6045	1	4	yes	direct product factor
83	C₂×C₄	—	25	2	no	—
84	C₂×C₄	—	25	6	no	—
85	C₂×C₄	C₂²×C₅⋊D₅	1	3	yes	—
86	C₂×C₄	C₂²×C₅⋊D₅	1	3	yes	—
87	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
88	C₅	Q₈×Dic₁₀	1	6	yes	complemented
89	C₂²	C₂³×C₅⋊D₅	1	1	yes	characteristic
90	C₄	—	2	3	no	—
91	C₄	—	25	4	no	—
92	C₄	—	50	6	no	—
93	C₄	C₂×#6045	1	6	yes	—
94	C₄	Q₈×C₅⋊D₅	1	2	yes	—
95	C₂	C₂²×#6045	1	1	yes	characteristic
96	C₂	C₂×Q₈×C₅⋊D₅	1	1	yes	characteristic
97	C₂	#11717522	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₅⋊D₅
commutator, socle, prime radical, cosocle	C₂²×C₅²	C₂⁴
Fitting	C₄×C₅²×Q₈	C₂
hypercenter	C₄×Q₈	C₅⋊D₅
hypocenter	C₅²	Q₈²
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
16. C₅²×Q₈	Q₈	semidirect
23. #6045	Q₈	direct (N × H)
48. C₅×Q₈	Dic₁₀	semidirect
61. C₅²	Q₈²	semidirect
82. Q₈	#6045	direct (N × H)
88. C₅	Q₈×Dic₁₀	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅²×Q₈	2
Q₈×#1039	2
#11717208	2
#11717189	2
C₄×#6045	2
Q₈×Dic₁₀	5

Subgroups — Q8×#6045 #13107094

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#6045 #13107094