Subgroups — #13102862

Order 1600.

456 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×#87804	C₂	1	1	yes	maximal, characteristic
3	C₂²×#6029	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#6029	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
6	C₅²×C₄⋊C₄	C₄	1	2	yes	—
7	C₅²×C₂²⋊C₄	C₄	1	2	yes	—
8	C₄²×C₅²	C₂²	1	2	yes	—
9	#87804	—	2	2	no	—
10	C₂×#6029	—	2	2	no	—
11	C₂×#6029	C₂²	1	2	yes	—
12	C₂×#6029	C₂²	1	2	yes	—
13	#65288	—	5	6	no	maximal
14	C₂³×C₅²	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₅²	C₂×C₄	1	4	yes	—
16	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₅²	D₄	1	2	yes	—
18	C₂×C₄×C₅²	Q₈	1	2	yes	—
19	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
20	#6029	—	2	2	no	—
21	#6029	—	2	2	no	—
22	C₅×#117	D₅	1	6	yes	—
23	C₂×#681	—	5	6	no	—
24	C₂²×#168	—	5	6	no	—
25	C₄⋊#168	—	5	24	no	—
26	C₂²×C₅²	—	2	2	no	—
27	C₂²×C₅²	C₄⋊C₄	1	2	yes	—
28	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₅²	—	2	4	no	—
30	C₄×C₅²	C₄⋊C₄	1	2	yes	—
31	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
32	C₄×C₅²	C₂×Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₅	C₂×D₅	1	6	yes	—
34	C₅×C₄⋊C₄	Dic₅	1	12	yes	—
35	C₅×C₂²⋊C₄	Dic₅	1	12	yes	—
36	C₄²×C₅	C₂×D₅	1	12	yes	—
37	#681	—	10	12	no	—
38	C₂×#168	—	5	12	no	—
39	C₂×#168	—	5	12	no	—
40	C₂×#168	—	10	12	no	—
41	#424	—	25	1	no	Sylow
42	C₂×C₅²	—	2	2	no	—
43	C₂×C₅²	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₅²	#131	1	2	yes	—
45	C₂³×C₅	C₂×Dic₅	1	6	yes	—
46	C₂×C₄×C₅	Dic₁₀	1	12	yes	—
47	C₂×C₄×C₅	D₂₀	1	12	yes	—
48	C₂×C₄×C₅	C₂×Dic₅	1	24	yes	—
49	C₂×C₄×C₅	C₂×Dic₅	1	6	yes	—
50	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
51	#168	—	10	12	no	—
52	#168	—	10	12	no	—
53	C₂×M₁₆	—	25	1	no	—
54	C₂²×C₈	—	25	1	no	—
55	#117	C₅⋊D₅	1	1	yes	characteristic
56	C₄⋊C₈	—	25	4	no	—
57	C₅²	#424	1	1	yes	characteristic, complemented, Sylow
58	C₂²×C₅	—	2	12	no	—
59	C₂²×C₅	C₄⋊Dic₅	1	12	yes	—
60	C₂²×C₅	C₂²×Dic₅	1	6	yes	—
61	C₄×C₅	—	2	24	no	—
62	C₄×C₅	C₄⋊Dic₅	1	12	yes	—
63	C₄×C₅	C₂×Dic₁₀	1	6	yes	—
64	C₄×C₅	C₂×D₂₀	1	6	yes	—
65	C₂²×C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
66	M₁₆	—	50	2	no	—
67	C₂×C₈	—	25	2	no	—
68	C₂×C₈	—	25	2	no	—
69	C₂×C₈	—	50	2	no	—
70	C₄⋊C₄	#1039	1	2	yes	—
71	C₂²⋊C₄	#1039	1	2	yes	—
72	C₄²	C₂×C₅⋊D₅	1	2	yes	—
73	C₂×C₅	—	2	12	no	—
74	C₂×C₅	C₂×C₄⋊Dic₅	1	6	yes	—
75	C₂×C₅	#4094	1	12	yes	—
76	C₂³	C₂×#1039	1	1	yes	characteristic
77	C₂×C₄	#6045	1	2	yes	—
78	C₂×C₄	C₄⋊(C₅⋊D₅)	1	2	yes	—
79	C₂×C₄	C₂×#1039	1	4	yes	—
80	C₂×C₄	C₂×#1039	1	1	yes	characteristic
81	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
82	C₈	—	50	2	no	—
83	C₈	—	50	2	no	—
84	C₅	#65288	1	6	yes	complemented
85	C₂²	—	2	2	no	—
86	C₂²	C₄⋊#1039	1	2	yes	—
87	C₂²	C₂²×#1039	1	1	yes	characteristic
88	C₄	—	2	4	no	—
89	C₄	C₄⋊#1039	1	2	yes	—
90	C₄	C₂×#6045	1	1	yes	characteristic
91	C₄	C₂×C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
92	C₂	—	2	2	no	—
93	C₂	C₂×C₄⋊#1039	1	1	yes	characteristic
94	C₂	#11717268	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×C₅⋊D₅
commutator, socle	C₂²×C₅²	C₂²×C₄
Fitting	C₅²×#117	C₂
prime radical	C₂³×C₅²	C₂×C₄
cosocle	C₂×C₄×C₅²	C₂³
hypercenter	#117	C₅⋊D₅
hypocenter	C₅²	#424
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
4. C₄⋊#6029	C₂	semidirect
57. C₅²	#424	semidirect
84. C₅	#65288	semidirect

Maximal subgroups

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

subgroup	index
C₅²×#117	2
C₂×#87804	2
C₂²×#6029	2
C₄⋊#6029	2
#65288	5