Subgroups — #13099580

Order 1600.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#11716703	C₂	1	2	yes	maximal
3	C₂×C₅²×Q₈	C₄	1	1	yes	characteristic
4	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
5	C₅²×C₄⋊C₄	—	2	2	no	—
6	C₄²×C₅²	C₄	1	1	yes	characteristic
7	#87804	—	2	2	no	—
8	#64768	—	5	6	no	maximal
9	C₅²×Q₈	—	4	1	no	—
10	C₅²×Q₈	—	4	1	no	—
11	C₂×C₄×C₅²	—	2	1	no	—
12	C₂×C₄×C₅²	—	2	1	no	—
13	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₅²	D₄	1	2	yes	—
15	#6029	—	4	2	no	—
16	C₅×C₄⋊Q₈	D₅	1	6	yes	—
17	#3968	—	5	12	no	—
18	C₂²×C₅²	C₂²⋊C₄	1	1	yes	characteristic
19	C₄×C₅²	—	2	2	no	—
20	C₄×C₅²	—	2	2	no	—
21	C₄×C₅²	—	2	1	no	—
22	C₄×C₅²	—	4	1	no	—
23	C₂×C₅×Q₈	Dic₅	1	6	yes	—
24	C₂×C₅×Q₈	C₂×D₅	1	6	yes	—
25	C₅×C₄⋊C₄	—	2	12	no	—
26	C₄²×C₅	Dic₅	1	6	yes	—
27	#681	—	10	12	no	—
28	#356	—	25	1	no	Sylow
29	C₂×C₅²	—	2	1	no	—
30	C₂×C₅²	#99	1	1	yes	characteristic
31	C₅×Q₈	—	4	6	no	—
32	C₅×Q₈	—	4	6	no	—
33	C₂×C₄×C₅	—	2	6	no	—
34	C₂×C₄×C₅	—	2	6	no	—
35	C₂×C₄×C₅	C₂×Dic₅	1	6	yes	—
36	C₂×C₄×C₅	C₂²⋊D₅	1	12	yes	—
37	#168	—	20	12	no	—
38	C₄⋊Q₈	C₅⋊D₅	1	1	yes	characteristic
39	#101	—	25	2	no	—
40	C₅²	#356	1	1	yes	characteristic, complemented, Sylow
41	C₂²×C₅	C₂²⋊Dic₅	1	6	yes	—
42	C₄×C₅	—	2	12	no	—
43	C₄×C₅	—	2	12	no	—
44	C₄×C₅	—	2	6	no	—
45	C₄×C₅	—	4	6	no	—
46	C₂×Q₈	#1039	1	1	yes	characteristic
47	C₂×Q₈	C₂×C₅⋊D₅	1	1	yes	characteristic
48	M₁₆	—	50	2	no	—
49	C₄⋊C₄	—	2	2	no	—
50	C₄²	#1039	1	1	yes	characteristic
51	C₂×C₅	—	2	6	no	—
52	C₂×C₅	#3966	1	6	yes	—
53	Q₈	—	4	1	no	—
54	Q₈	—	4	1	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	C₂×#1039	1	1	yes	characteristic
58	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	2	yes	—
59	C₈	—	100	2	no	—
60	C₅	#64768	1	6	yes	complemented
61	C₂²	C₂²⋊#1039	1	1	yes	characteristic
62	C₄	—	2	2	no	—
63	C₄	—	2	2	no	—
64	C₄	—	2	1	no	—
65	C₄	—	4	1	no	—
66	C₂	—	2	1	no	—
67	C₂	#11716701	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11716701
commutator	C₂×C₄×C₅²	C₂×C₄
Frattini	C₂×Q₈	C₂×C₅⋊D₅
Fitting	C₅²×C₄⋊Q₈	C₂
socle	C₂×C₅²	#99
prime radical	C₂²×C₅²	C₂²⋊C₄
cosocle	C₂×C₅²×Q₈	C₂²
hypercenter	C₄⋊Q₈	C₅⋊D₅
hypocenter	C₅²	#356
LCS	G ▷ C₂×C₄×C₅² ▷ C₂²×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
40. C₅²	#356	semidirect
60. C₅	#64768	semidirect

Maximal subgroups

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

subgroup	index
C₅²×C₄⋊Q₈	2
#11716703	2
#64768	5