Subgroups — #13102764

Order 1600.

336 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11717221	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅²×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#11716679	C₂	1	1	yes	maximal, characteristic, complemented
4	#11716665	C₂	1	2	yes	maximal, complemented
5	C₁₆×C₅⋊D₅	C₂	1	2	yes	maximal, complemented
6	#87900	C₄	1	1	yes	characteristic
7	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
8	#87804	C₄	1	1	yes	characteristic
9	#87799	C₄	1	2	yes	—
10	C₈×C₅⋊D₅	C₂²	1	2	yes	—
11	C₅²×C₁₆	C₂²	1	1	yes	characteristic
12	C₅²×C₁₆	C₂²	1	1	yes	characteristic
13	#87756	C₂²	1	2	yes	—
14	#65190	—	5	6	no	maximal
15	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
16	C₂²⋊(C₅⋊D₅)	C₈	1	2	yes	—
17	C₄⋊(C₅⋊D₅)	C₈	1	1	yes	characteristic
18	C₄×C₅⋊D₅	C₂×C₄	1	2	yes	—
19	#6045	C₈	1	1	yes	characteristic
20	C₅²×C₈	C₂×C₄	1	1	yes	characteristic
21	C₅²×C₈	C₂³	1	1	yes	characteristic
22	#6029	C₂×C₄	1	2	yes	—
23	#4047	—	5	6	no	—
24	C₂×C₅×C₁₆	D₅	1	6	yes	complemented
25	#3944	—	5	6	no	—
26	#3930	—	5	12	no	—
27	D₅×C₁₆	—	5	12	no	—
28	C₂²×C₅²	C₂×C₈	1	1	yes	characteristic
29	C₂×C₅⋊D₅	C₂×C₈	1	2	yes	—
30	C₄×C₅²	C₂×C₈	1	1	yes	characteristic
31	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
32	#1039	C₂×C₈	1	2	yes	—
33	#709	—	5	6	no	—
34	C₂×C₅×C₈	C₂×D₅	1	6	yes	—
35	#681	—	5	6	no	—
36	#676	—	5	12	no	—
37	C₈×D₅	—	5	12	no	—
38	C₅×C₁₆	C₂×D₅	1	6	yes	—
39	C₅×C₁₆	C₂×D₅	1	6	yes	—
40	#672	—	5	12	no	—
41	#504	—	25	1	no	Sylow
42	C₂×C₅²	—	2	1	no	—
43	C₂×C₅²	C₂²×C₈	1	1	yes	characteristic
44	C₅⋊D₅	—	2	2	no	—
45	C₂×C₄×C₅	C₄×D₅	1	6	yes	—
46	C₂²⋊D₅	—	5	12	no	—
47	D₂₀	—	5	6	no	—
48	C₄×D₅	—	5	12	no	—
49	Dic₁₀	—	5	6	no	—
50	C₅×C₈	C₄×D₅	1	6	yes	—
51	C₅×C₈	C₂²×D₅	1	6	yes	—
52	#168	—	5	12	no	—
53	#131	—	25	1	no	—
54	M₃₂	—	25	1	no	—
55	M₃₂	—	25	2	no	—
56	C₂×C₁₆	—	25	2	no	—
57	C₂×C₁₆	C₅⋊D₅	1	1	yes	characteristic, complemented
58	C₅²	#504	1	1	yes	characteristic, complemented, Sylow
59	C₂²×C₅	C₈×D₅	1	6	yes	—
60	C₂×D₅	—	5	12	no	—
61	C₄×C₅	C₈×D₅	1	6	yes	—
62	C₄×C₅	C₂×C₄×D₅	1	6	yes	—
63	Dic₅	—	5	12	no	—
64	#41	—	25	1	no	—
65	M₁₆	—	25	1	no	—
66	M₁₆	—	25	2	no	—
67	C₂×C₈	—	25	2	no	—
68	C₂×C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
69	C₁₆	—	25	2	no	—
70	C₁₆	C₂×C₅⋊D₅	1	1	yes	characteristic
71	C₁₆	C₂×C₅⋊D₅	1	1	yes	characteristic
72	C₂×C₅	—	2	6	no	—
73	C₂×C₅	C₂×C₈×D₅	1	6	yes	—
74	D₅	—	10	12	no	—
75	Q₈	—	25	1	no	—
76	D₄	—	25	2	no	—
77	D₄	—	25	1	no	—
78	C₂×C₄	—	25	2	no	—
79	C₂×C₄	C₄×C₅⋊D₅	1	1	yes	characteristic
80	C₈	—	25	2	no	—
81	C₈	C₄×C₅⋊D₅	1	1	yes	characteristic
82	C₈	C₂²×C₅⋊D₅	1	1	yes	characteristic
83	C₅	#65190	1	6	yes	complemented
84	C₂²	—	25	2	no	—
85	C₂²	C₈×C₅⋊D₅	1	1	yes	characteristic
86	C₄	—	25	2	no	—
87	C₄	C₈×C₅⋊D₅	1	1	yes	characteristic
88	C₄	C₂×C₄×C₅⋊D₅	1	1	yes	characteristic
89	C₂	—	2	1	no	—
90	C₂	—	50	2	no	—
91	C₂	C₂×C₈×C₅⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂×C₅⋊D₅
commutator, socle	C₂×C₅²	C₂²×C₈
Frattini	C₈	C₂²×C₅⋊D₅
Fitting	C₂×C₅²×C₁₆	C₂
prime radical	#87900	C₄
cosocle	C₅²×C₈	C₂³
hypercenter	C₂×C₁₆	C₅⋊D₅
hypocenter	C₅²	#504
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₅²×C₁₆	C₂	semidirect
3. #11716679	C₂	semidirect
4. #11716665	C₂	semidirect
5. C₁₆×C₅⋊D₅	C₂	semidirect
24. C₂×C₅×C₁₆	D₅	semidirect
57. C₂×C₁₆	C₅⋊D₅	semidirect
58. C₅²	#504	semidirect
83. C₅	#65190	semidirect

Maximal subgroups

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

subgroup	index
#11717221	2
C₂×C₅²×C₁₆	2
#11716679	2
#11716665	2
C₁₆×C₅⋊D₅	2
#65190	5