Subgroups — #13103094

Order 1600.

808 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, complemented
2	C₂×#87808	C₂	1	3	yes	maximal
3	#11716698	C₂	1	3	yes	maximal, complemented
4	C₂³×C₅⋊D₅	C₂²	1	1	yes	characteristic
5	C₂×C₅²×Q₈	—	2	6	no	—
6	C₂²×C₄×C₅²	C₂²	1	3	yes	—
7	C₂²×#1039	C₂²	1	3	yes	—
8	#87808	—	2	6	no	—
9	#65520	—	5	6	no	maximal
10	C₂³×C₅²	C₂³	1	1	yes	characteristic
11	C₂²×C₅⋊D₅	—	2	3	no	—
12	C₂²×C₅⋊D₅	—	2	3	no	—
13	C₂²×C₅⋊D₅	Q₈	1	2	yes	complemented
14	C₅²×Q₈	—	2	8	no	—
15	C₂×C₄×C₅²	—	2	6	no	—
16	C₂×C₄×C₅²	D₄	1	6	yes	—
17	C₂×#1039	—	2	3	no	—
18	C₂×#1039	—	2	6	no	—
19	C₂²×C₅×Q₈	D₅	1	6	yes	complemented
20	C₂×#685	—	5	18	no	—
21	#3963	—	5	18	no	—
22	C₂²×C₅²	C₂×D₄	1	3	yes	—
23	C₂²×C₅²	C₂×Q₈	1	1	yes	characteristic
24	C₂²×C₅²	#41	1	3	yes	—
25	C₂×C₅⋊D₅	—	2	6	no	—
26	C₂×C₅⋊D₅	—	4	1	no	—
27	C₂×C₅⋊D₅	—	4	3	no	—
28	C₄×C₅²	—	2	6	no	—
29	#1039	—	4	3	no	—
30	C₂³×D₅	—	5	6	no	—
31	C₂×C₅×Q₈	—	2	36	no	—
32	C₂²×C₄×C₅	C₂×D₅	1	18	yes	—
33	C₂²×Dic₅	—	5	18	no	—
34	#685	—	10	36	no	—
35	#393	—	25	1	no	Sylow
36	C₂×C₅²	#120	1	1	yes	characteristic
37	C₂×C₅²	C₂²⋊Q₈	1	3	yes	—
38	C₂×C₅²	#124	1	3	yes	—
39	C₅⋊D₅	—	4	2	no	—
40	C₂³×C₅	C₂²×D₅	1	6	yes	—
41	C₂²×D₅	—	5	12	no	—
42	C₂²×D₅	—	10	18	no	—
43	C₂²×D₅	—	10	18	no	—
44	C₅×Q₈	—	2	48	no	—
45	C₂×C₄×C₅	—	2	36	no	—
46	C₂×C₄×C₅	C₂²⋊D₅	1	36	yes	—
47	C₂×Dic₅	—	10	36	no	—
48	C₂×Dic₅	—	10	18	no	—
49	C₂²×Q₈	C₅⋊D₅	1	1	yes	characteristic, complemented
50	C₂×C₂²⋊C₄	—	25	3	no	—
51	#95	—	25	3	no	—
52	C₅²	#393	1	1	yes	characteristic, complemented, Sylow
53	C₂²×C₅	Q₈×D₅	1	6	yes	—
54	C₂²×C₅	#713	1	18	yes	—
55	C₂²×C₅	C₂×C₂²⋊D₅	1	18	yes	—
56	C₂×D₅	—	10	36	no	—
57	C₂×D₅	—	20	6	no	—
58	C₂×D₅	—	20	18	no	—
59	C₄×C₅	—	2	36	no	—
60	Dic₅	—	20	18	no	—
61	C₂⁴	—	25	1	no	—
62	C₂×Q₈	—	2	6	no	—
63	C₂²×C₄	—	25	3	no	—
64	C₂²×C₄	C₂×C₅⋊D₅	1	3	yes	—
65	C₂²⋊C₄	—	50	6	no	—
66	C₂×C₅	#4092	1	18	yes	—
67	C₂×C₅	#4093	1	18	yes	—
68	C₂×C₅	#4099	1	6	yes	—
69	D₅	—	20	12	no	—
70	C₂³	—	25	2	no	—
71	C₂³	—	50	3	no	—
72	C₂³	—	50	3	no	—
73	C₂³	C₂²×C₅⋊D₅	1	1	yes	characteristic
74	Q₈	—	2	8	no	—
75	C₂×C₄	—	2	6	no	—
76	C₂×C₄	—	50	6	no	—
77	C₂×C₄	—	50	3	no	—
78	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	6	yes	—
79	C₅	#65520	1	6	yes	complemented
80	C₂²	—	50	6	no	—
81	C₂²	—	100	3	no	—
82	C₂²	—	100	1	no	—
83	C₂²	Q₈×C₅⋊D₅	1	1	yes	characteristic
84	C₂²	#87904	1	3	yes	—
85	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	3	yes	—
86	C₄	—	2	6	no	—
87	C₄	—	100	3	no	—
88	C₂	—	100	2	no	—
89	C₂	#11717266	1	3	yes	—
90	C₂	#11717267	1	3	yes	—
91	C₂	#11717273	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×C₅⋊D₅
commutator, socle, cosocle	C₂³×C₅²	C₂³
Fitting	C₂²×C₅²×Q₈	C₂
prime radical	C₂³×C₅⋊D₅	C₂²
hypercenter	C₂²×Q₈	C₅⋊D₅
hypocenter	C₅²	#393
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
1. C₂²×C₅²×Q₈	C₂	semidirect
3. #11716698	C₂	semidirect
13. C₂²×C₅⋊D₅	Q₈	semidirect
19. C₂²×C₅×Q₈	D₅	semidirect
49. C₂²×Q₈	C₅⋊D₅	semidirect
52. C₅²	#393	semidirect
79. C₅	#65520	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×#87808	2
#11716698	2
#65520	5