Subgroups — #13103092

Order 1600.

728 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅²×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#87806	C₂	1	3	yes	maximal
3	#11716698	C₂	1	3	yes	maximal
4	C₂×C₅²×Q₈	—	2	6	no	—
5	C₂²×C₄×C₅²	C₂²	1	3	yes	—
6	C₂²×#1039	C₂²	1	1	yes	characteristic
7	C₂²×#1039	C₂²	1	3	yes	—
8	#87806	—	2	6	no	—
9	#65518	—	5	6	no	maximal
10	C₂³×C₅²	C₂³	1	1	yes	characteristic
11	C₅²×Q₈	—	2	8	no	—
12	C₂×C₄×C₅²	—	2	6	no	—
13	C₂×C₄×C₅²	D₄	1	6	yes	—
14	C₂×#1039	—	2	3	no	—
15	C₂×#1039	—	2	3	no	—
16	C₂×#1039	—	2	3	no	—
17	C₂×#1039	Q₈	1	6	yes	complemented
18	C₂²×C₅×Q₈	D₅	1	6	yes	—
19	C₂×#683	—	5	18	no	—
20	#3963	—	5	18	no	—
21	C₂²×C₅²	C₂×D₄	1	3	yes	—
22	C₂²×C₅²	C₂×Q₈	1	3	yes	—
23	C₂²×C₅²	#41	1	1	yes	characteristic
24	C₄×C₅²	—	2	6	no	—
25	#1039	—	2	6	no	—
26	#1039	—	4	1	no	—
27	C₂×C₅×Q₈	—	2	36	no	—
28	C₂²×C₄×C₅	C₂×D₅	1	18	yes	—
29	C₂²×Dic₅	—	5	18	no	—
30	C₂²×Dic₅	—	5	6	no	—
31	#683	—	10	36	no	—
32	#395	—	25	1	no	Sylow
33	C₂×C₅²	#120	1	1	yes	characteristic
34	C₂×C₅²	C₂²⋊Q₈	1	3	yes	—
35	C₂×C₅²	C₄⋊Q₈	1	3	yes	—
36	C₂³×C₅	C₂²×D₅	1	6	yes	—
37	C₅×Q₈	—	2	48	no	—
38	C₂×C₄×C₅	—	2	36	no	—
39	C₂×C₄×C₅	C₂²⋊D₅	1	36	yes	—
40	C₂×Dic₅	—	5	36	no	—
41	C₂×Dic₅	—	10	18	no	—
42	C₂×Dic₅	—	10	18	no	—
43	C₂×Dic₅	—	10	18	no	—
44	C₂²×Q₈	C₅⋊D₅	1	1	yes	characteristic
45	C₂×C₄⋊C₄	—	25	3	no	—
46	#95	—	25	3	no	—
47	C₅²	#395	1	1	yes	characteristic, complemented, Sylow
48	C₂²×C₅	Q₈×D₅	1	18	yes	—
49	C₂²×C₅	#713	1	6	yes	—
50	C₂²×C₅	C₂×C₂²⋊D₅	1	18	yes	—
51	C₄×C₅	—	2	36	no	—
52	Dic₅	—	10	36	no	—
53	Dic₅	—	20	6	no	—
54	C₂×Q₈	—	2	6	no	—
55	C₂²×C₄	—	25	3	no	—
56	C₂²×C₄	—	25	1	no	—
57	C₂²×C₄	C₂×C₅⋊D₅	1	3	yes	—
58	C₄⋊C₄	—	50	6	no	—
59	C₂×C₅	#4090	1	18	yes	—
60	C₂×C₅	#4092	1	18	yes	—
61	C₂×C₅	#4099	1	6	yes	—
62	C₂³	C₂²×C₅⋊D₅	1	1	yes	characteristic
63	Q₈	—	2	8	no	—
64	C₂×C₄	—	2	6	no	—
65	C₂×C₄	—	25	6	no	—
66	C₂×C₄	—	50	3	no	—
67	C₂×C₄	—	50	3	no	—
68	C₂×C₄	—	50	3	no	—
69	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	6	yes	—
70	C₅	#65518	1	6	yes	complemented
71	C₂²	Q₈×C₅⋊D₅	1	3	yes	—
72	C₂²	#87904	1	1	yes	characteristic
73	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	3	yes	—
74	C₄	—	2	6	no	—
75	C₄	—	50	6	no	—
76	C₄	—	100	1	no	—
77	C₂	#11717264	1	3	yes	—
78	C₂	#11717266	1	3	yes	—
79	C₂	#11717273	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×C₅⋊D₅
commutator, socle, prime radical, cosocle	C₂³×C₅²	C₂³
Fitting	C₂²×C₅²×Q₈	C₂
hypercenter	C₂²×Q₈	C₅⋊D₅
hypocenter	C₅²	#395
LCS	G ▷ C₂³×C₅² ▷ C₅²
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
17. C₂×#1039	Q₈	semidirect
47. C₅²	#395	semidirect
70. C₅	#65518	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅²×Q₈	2
C₂×#87806	2
#11716698	2
#65518	5