Subgroups — #13102553

Order 1600.

384 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11717192	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅²×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#11716687	C₂	1	2	yes	maximal
4	#11716681	C₂	1	2	yes	maximal, complemented
5	#11716670	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
7	C₂×C₅²×C₈	C₂²	1	2	yes	—
8	C₄²×C₅²	C₂²	1	1	yes	characteristic
9	#87808	C₄	1	2	yes	—
10	#87806	C₄	1	2	yes	—
11	C₄×#1039	C₂²	1	1	yes	characteristic
12	C₂×#6029	C₂²	1	2	yes	—
13	#64979	—	5	6	no	maximal
14	C₂²×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
16	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
17	C₂×#1039	C₂×C₄	1	2	yes	—
18	C₂×#1039	C₂×C₄	1	1	yes	characteristic
19	C₄×C₅⋊D₅	—	2	2	no	—
20	C₅²×C₈	—	2	2	no	—
21	#6029	—	2	2	no	—
22	#4018	—	5	6	no	—
23	C₄×C₅×C₈	D₅	1	6	yes	complemented
24	#3952	—	5	12	no	—
25	#3946	—	5	12	no	—
26	#3935	—	5	6	no	—
27	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₅⋊D₅	—	2	1	no	—
29	C₂×C₅⋊D₅	—	2	2	no	—
30	C₄×C₅²	—	2	2	no	—
31	C₄×C₅²	#41	1	2	yes	—
32	#1039	—	2	1	no	—
33	#1039	—	2	2	no	—
34	C₂×C₄×D₅	—	5	6	no	—
35	C₂×C₅×C₈	C₂×D₅	1	12	yes	—
36	C₄²×C₅	C₂×D₅	1	6	yes	—
37	#685	—	5	12	no	—
38	#683	—	5	12	no	—
39	C₄×Dic₅	—	5	6	no	—
40	C₂×#168	—	5	12	no	—
41	#433	—	25	1	no	Sylow
42	C₂×C₅²	#117	1	1	yes	characteristic
43	C₂×C₅²	#131	1	2	yes	—
44	C₅⋊D₅	—	4	1	no	—
45	C₂²×D₅	—	5	6	no	—
46	C₂×C₄×C₅	C₄×D₅	1	12	yes	—
47	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
48	C₂×Dic₅	—	5	12	no	—
49	C₂×Dic₅	—	5	6	no	—
50	C₄×D₅	—	10	12	no	—
51	C₅×C₈	—	2	12	no	—
52	#168	—	10	12	no	—
53	#117	—	25	1	no	—
54	C₄⋊C₈	—	25	2	no	—
55	C₂²⋊C₈	—	25	2	no	—
56	#97	—	25	1	no	—
57	C₄×C₈	C₅⋊D₅	1	1	yes	characteristic, complemented
58	C₅²	#433	1	1	yes	characteristic, complemented, Sylow
59	C₂²×C₅	C₂×C₄×D₅	1	6	yes	—
60	C₂×D₅	—	10	6	no	—
61	C₂×D₅	—	10	12	no	—
62	C₄×C₅	—	2	12	no	—
63	C₄×C₅	#709	1	12	yes	—
64	Dic₅	—	10	6	no	—
65	Dic₅	—	10	12	no	—
66	C₂²×C₄	—	25	1	no	—
67	C₂×C₈	—	25	2	no	—
68	C₂×C₈	C₂×C₅⋊D₅	1	2	yes	—
69	C₄⋊C₄	—	25	2	no	—
70	C₂²⋊C₄	—	25	2	no	—
71	C₄²	—	25	1	no	—
72	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
73	C₂×C₅	#4018	1	6	yes	—
74	C₂×C₅	#4047	1	12	yes	—
75	D₅	—	20	6	no	—
76	C₂³	—	25	1	no	—
77	C₂×C₄	—	25	2	no	—
78	C₂×C₄	—	25	1	no	—
79	C₂×C₄	—	50	2	no	—
80	C₂×C₄	C₄×C₅⋊D₅	1	2	yes	—
81	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
82	C₈	—	2	2	no	—
83	C₈	—	50	2	no	—
84	C₅	#64979	1	6	yes	complemented
85	C₂²	—	50	1	no	—
86	C₂²	—	50	2	no	—
87	C₂²	C₂×C₄×C₅⋊D₅	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	50	1	no	—
90	C₄	—	50	2	no	—
91	C₄	#87900	1	2	yes	—
92	C₂	—	100	1	no	—
93	C₂	#11717192	1	1	yes	characteristic
94	C₂	#11717221	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×C₅⋊D₅
commutator, socle	C₂²×C₅²	C₂²×C₄
Fitting	C₄×C₅²×C₈	C₂
prime radical	C₂²×C₅⋊D₅	C₂×C₄
cosocle	C₂×C₄×C₅²	C₂³
hypercenter	C₄×C₈	C₅⋊D₅
hypocenter	C₅²	#433
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₄×C₅²×C₈	C₂	semidirect
4. #11716681	C₂	semidirect
5. #11716670	C₂	semidirect
23. C₄×C₅×C₈	D₅	semidirect
57. C₄×C₈	C₅⋊D₅	semidirect
58. C₅²	#433	semidirect
84. C₅	#64979	semidirect

Maximal subgroups

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

subgroup	index
#11717192	2
C₄×C₅²×C₈	2
#11716687	2
#11716681	2
#11716670	2
#64979	5