Subgroups — #13102988

Order 1600.

408 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#87804	C₂	1	2	yes	maximal
3	#11716686	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
5	C₅²×M₁₆	C₄	1	4	yes	—
6	C₂×C₅²×C₈	C₂²	1	2	yes	—
7	#87804	—	2	2	no	—
8	#87804	C₂²	1	4	yes	—
9	C₂×#6029	—	2	2	no	—
10	#65414	—	5	6	no	maximal
11	C₂³×C₅²	Q₈	1	1	yes	characteristic
12	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
13	C₂×C₄×C₅²	D₄	1	2	yes	—
14	C₂×C₄×C₅²	Q₈	1	1	yes	characteristic
15	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
16	C₅²×C₈	C₂×C₄	1	4	yes	—
17	#6029	—	2	4	no	—
18	C₂×C₅×M₁₆	D₅	1	6	yes	—
19	C₂×#681	—	5	12	no	—
20	#3951	—	5	24	no	—
21	C₂²×C₅²	—	4	1	no	—
22	C₂²×C₅²	C₄⋊C₄	1	2	yes	—
23	C₂²×C₅²	C₂×Q₈	1	1	yes	characteristic
24	C₄×C₅²	C₄⋊C₄	1	2	yes	—
25	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
26	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₄×C₅	C₂×D₅	1	6	yes	—
28	C₅×M₁₆	Dic₅	1	24	yes	—
29	C₂×C₅×C₈	C₂×D₅	1	12	yes	—
30	#681	—	5	24	no	—
31	#681	—	10	12	no	—
32	C₂×#168	—	10	12	no	—
33	#430	—	25	1	no	Sylow
34	C₂×C₅²	—	2	2	no	—
35	C₂×C₅²	—	2	1	no	—
36	C₂×C₅²	C₂×C₄⋊C₄	1	1	yes	characteristic
37	C₂³×C₅	Dic₁₀	1	6	yes	—
38	C₂×C₄×C₅	Dic₁₀	1	6	yes	—
39	C₂×C₄×C₅	D₂₀	1	12	yes	—
40	C₂×C₄×C₅	C₂×Dic₅	1	12	yes	—
41	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
42	C₅×C₈	C₂×Dic₅	1	24	yes	—
43	#168	—	10	24	no	—
44	C₂×M₁₆	—	25	2	no	—
45	C₂×M₁₆	C₅⋊D₅	1	1	yes	characteristic
46	#108	—	25	4	no	—
47	C₅²	#430	1	1	yes	characteristic, complemented, Sylow
48	C₂²×C₅	—	4	6	no	—
49	C₂²×C₅	C₄⋊Dic₅	1	12	yes	—
50	C₂²×C₅	C₂×Dic₁₀	1	6	yes	—
51	C₄×C₅	C₄⋊Dic₅	1	12	yes	—
52	C₄×C₅	C₂×D₂₀	1	6	yes	—
53	C₄×C₅	C₂²×Dic₅	1	6	yes	—
54	C₂²×C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
55	M₁₆	—	25	4	no	—
56	M₁₆	—	50	2	no	—
57	M₁₆	#1039	1	4	yes	—
58	C₂×C₈	—	50	2	no	—
59	C₂×C₈	C₂×C₅⋊D₅	1	2	yes	—
60	C₂×C₅	—	2	12	no	—
61	C₂×C₅	—	2	6	no	—
62	C₂×C₅	C₂×C₄⋊Dic₅	1	6	yes	—
63	C₂³	#6045	1	1	yes	characteristic
64	C₂×C₄	#6045	1	1	yes	characteristic
65	C₂×C₄	C₄⋊(C₅⋊D₅)	1	2	yes	—
66	C₂×C₄	C₂×#1039	1	2	yes	—
67	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
68	C₈	—	50	4	no	—
69	C₈	C₂×#1039	1	4	yes	—
70	C₅	#65414	1	6	yes	complemented
71	C₂²	—	4	1	no	—
72	C₂²	C₄⋊#1039	1	2	yes	—
73	C₂²	C₂×#6045	1	1	yes	characteristic
74	C₄	C₄⋊#1039	1	2	yes	—
75	C₄	C₂×C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
76	C₄	C₂²×#1039	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	—	2	2	no	—
79	C₂	C₂×C₄⋊#1039	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×C₄⋊(C₅⋊D₅)
commutator	C₄×C₅²	C₂²×C₄
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₂×C₅²×M₁₆	C₂
socle	C₂×C₅²	C₂×C₄⋊C₄
prime radical	C₂³×C₅²	Q₈
cosocle	C₂×C₄×C₅²	C₂³
hypercenter	C₂×M₁₆	C₅⋊D₅
hypocenter	C₅²	#430
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #11716686	C₂	semidirect
47. C₅²	#430	semidirect
70. C₅	#65414	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅²×M₁₆	2
C₂×#87804	2
#11716686	2
#65414	5