Subgroups — #13102945

Order 1600.

360 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	#11717190	C₂	1	1	yes	maximal, characteristic
3	#11716702	C₂	1	4	yes	maximal
4	#11716670	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅²×Q₈	C₂²	1	2	yes	—
6	C₅²×C₄⋊C₄	—	2	2	no	—
7	C₄²×C₅²	C₂²	1	1	yes	characteristic
8	C₄⋊#1039	C₂²	1	2	yes	—
9	#87806	—	2	2	no	—
10	C₂×#6029	C₂²	1	2	yes	—
11	#65371	—	5	6	no	maximal
12	C₅²×Q₈	—	2	4	no	—
13	C₂×C₄×C₅²	—	2	2	no	—
14	C₂×C₄×C₅²	D₄	1	2	yes	—
15	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
16	C₂×#1039	—	2	2	no	—
17	#6029	—	2	2	no	—
18	C₅×C₄⋊Q₈	D₅	1	6	yes	—
19	#4016	—	5	6	no	—
20	#3967	—	5	24	no	—
21	#3935	—	5	6	no	—
22	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
23	C₄×C₅²	—	2	2	no	—
24	C₄×C₅²	—	4	2	no	—
25	C₄×C₅²	#41	1	2	yes	—
26	#1039	—	4	2	no	—
27	C₂×C₅×Q₈	C₂×D₅	1	12	yes	—
28	C₅×C₄⋊C₄	—	2	12	no	—
29	C₄²×C₅	C₂×D₅	1	6	yes	—
30	C₄⋊Dic₅	—	5	12	no	—
31	#683	—	10	12	no	—
32	C₂×#168	—	5	12	no	—
33	#491	—	25	1	no	Sylow
34	C₂×C₅²	#124	1	1	yes	characteristic
35	C₂×C₅²	#137	1	2	yes	—
36	C₅×Q₈	—	2	24	no	—
37	C₂×C₄×C₅	—	2	12	no	—
38	C₂×C₄×C₅	C₂²⋊D₅	1	12	yes	—
39	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
40	C₂×Dic₅	—	10	12	no	—
41	#168	—	10	12	no	—
42	C₄⋊Q₈	C₅⋊D₅	1	1	yes	characteristic
43	#125	—	25	1	no	—
44	#103	—	25	4	no	—
45	#97	—	25	1	no	—
46	C₅²	#491	1	1	yes	characteristic, complemented, Sylow
47	C₂²×C₅	C₂×C₂²⋊D₅	1	6	yes	—
48	C₄×C₅	—	2	12	no	—
49	C₄×C₅	—	4	12	no	—
50	C₄×C₅	#711	1	12	yes	—
51	Dic₅	—	20	12	no	—
52	C₂×Q₈	C₂×C₅⋊D₅	1	2	yes	—
53	C₂×C₈	—	25	2	no	—
54	C₄⋊C₄	—	2	2	no	—
55	C₄⋊C₄	—	25	2	no	—
56	C₄⋊C₄	—	50	2	no	—
57	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
58	C₂×C₅	#4082	1	6	yes	—
59	C₂×C₅	#4088	1	12	yes	—
60	Q₈	—	2	4	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	50	2	no	—
63	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	2	yes	—
64	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
65	C₈	—	50	2	no	—
66	C₅	#65371	1	6	yes	complemented
67	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
68	C₄	—	2	2	no	—
69	C₄	—	4	2	no	—
70	C₄	—	100	2	no	—
71	C₄	#87902	1	2	yes	—
72	C₂	#11717256	1	1	yes	characteristic
73	C₂	#11717262	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊(C₅⋊D₅)
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₅²×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₅²	C₂×D₄
hypercenter	C₄⋊Q₈	C₅⋊D₅
hypocenter	C₅²	#491
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
46. C₅²	#491	semidirect
66. C₅	#65371	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
#11717190	2
#11716702	2
#11716670	2
#65371	5