Subgroups — #13097754

Order 1600.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#11716383	C₂	1	2	yes	maximal
3	C₂×Q₈×C₂₅	C₄	1	1	yes	characteristic
4	C₂×Q₈×C₂₅	C₂²	1	1	yes	characteristic
5	C₄⋊C₄×C₂₅	—	2	2	no	—
6	C₄²×C₂₅	C₄	1	1	yes	characteristic
7	#87716	—	2	2	no	—
8	#64768	—	5	1	no	maximal
9	Q₈×C₂₅	—	4	1	no	—
10	Q₈×C₂₅	—	4	1	no	—
11	C₂×C₄×C₂₅	—	2	1	no	—
12	C₂×C₄×C₂₅	—	2	1	no	—
13	C₂×C₄×C₂₅	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₂₅	D₄	1	2	yes	—
15	#6014	—	4	2	no	—
16	C₅×C₄⋊Q₈	D₅	1	1	yes	characteristic
17	#3968	—	5	2	no	—
18	C₂²×C₂₅	C₂²⋊C₄	1	1	yes	characteristic
19	C₄×C₂₅	—	2	2	no	—
20	C₄×C₂₅	—	2	2	no	—
21	C₄×C₂₅	—	2	1	no	—
22	C₄×C₂₅	—	4	1	no	—
23	C₂×C₅×Q₈	Dic₅	1	1	yes	characteristic
24	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic
25	C₅×C₄⋊C₄	—	2	2	no	—
26	C₄²×C₅	Dic₅	1	1	yes	characteristic
27	#681	—	10	2	no	—
28	#356	—	25	1	no	Sylow
29	C₂×C₂₅	—	2	1	no	—
30	C₂×C₂₅	#99	1	1	yes	characteristic
31	C₅×Q₈	—	4	1	no	—
32	C₅×Q₈	—	4	1	no	—
33	C₂×C₄×C₅	—	2	1	no	—
34	C₂×C₄×C₅	—	2	1	no	—
35	C₂×C₄×C₅	C₂×Dic₅	1	1	yes	characteristic
36	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
37	#168	—	20	2	no	—
38	C₄⋊Q₈	D₂₅	1	1	yes	characteristic
39	#101	—	25	2	no	—
40	C₂₅	#356	1	1	yes	characteristic, complemented, Sylow
41	C₂²×C₅	C₂²⋊Dic₅	1	1	yes	characteristic
42	C₄×C₅	—	2	2	no	—
43	C₄×C₅	—	2	2	no	—
44	C₄×C₅	—	2	1	no	—
45	C₄×C₅	—	4	1	no	—
46	C₂×Q₈	Dic₂₅	1	1	yes	characteristic
47	C₂×Q₈	C₂×D₂₅	1	1	yes	characteristic
48	M₁₆	—	50	2	no	—
49	C₄⋊C₄	—	2	2	no	—
50	C₄²	Dic₂₅	1	1	yes	characteristic
51	C₂×C₅	—	2	1	no	—
52	C₂×C₅	#3966	1	1	yes	characteristic
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×Dic₂₅	1	1	yes	characteristic
58	C₂×C₄	C₂²⋊D₂₅	1	2	yes	—
59	C₈	—	100	2	no	—
60	C₅	#64768	1	1	yes	characteristic
61	C₂²	C₂²⋊Dic₂₅	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	#11716382	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11716382
commutator	C₂×C₄×C₂₅	C₂×C₄
Frattini	C₂×C₅×Q₈	C₂×D₅
Fitting	C₂₅×C₄⋊Q₈	C₂
socle	C₂×C₅	#3966
prime radical	C₂²×C₅	C₂²⋊Dic₅
cosocle	C₂×Q₈×C₂₅	C₂²
hypercenter	C₄⋊Q₈	D₂₅
hypocenter	C₂₅	#356
LCS	G ▷ C₂×C₄×C₂₅ ▷ C₂²×C₂₅ ▷ C₂×C₂₅ ▷ C₂₅
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. C₂₅	#356	semidirect

Maximal subgroups

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

subgroup	index
C₂₅×C₄⋊Q₈	2
#11716383	2
#64768	5