Subgroups — #13100401

Order 1600.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11717264	C₂	1	1	yes	maximal, characteristic
2	#11716897	C₂	1	2	yes	maximal
3	C₂×C₅²×Q₈	C₄	1	1	yes	characteristic
4	C₂×#6045	C₄	1	1	yes	characteristic
5	C₂×#6034	—	2	2	no	—
6	#87806	—	2	2	no	—
7	C₄×#1039	C₂²	1	1	yes	characteristic
8	#64931	—	5	2	no	maximal
9	C₅²×Q₈	—	4	1	no	—
10	C₂×C₄×C₅²	—	2	1	no	—
11	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
12	C₂×#1039	—	2	1	no	—
13	C₂×#1039	D₄	1	2	yes	—
14	#6045	—	4	1	no	—
15	#6034	—	4	2	no	—
16	#4090	—	5	2	no	—
17	#4090	—	10	2	no	—
18	#4004	—	5	4	no	—
19	C₂²×C₅²	C₂²⋊C₄	1	1	yes	characteristic
20	C₄×C₅²	—	2	1	no	—
21	C₄×C₅²	—	4	1	no	—
22	#1039	—	4	1	no	—
23	#1039	SD₁₆	1	2	yes	—
24	#1039	Q₁₆	1	2	yes	—
25	C₂×C₅×Q₈	—	2	2	no	—
26	C₂×C₅×Q₈	AGL(1,𝔽₅)	1	2	yes	—
27	C₂×Dic₁₀	—	5	2	no	—
28	C₂×Dic₁₀	—	10	2	no	—
29	C₂×#170	—	10	4	no	—
30	#683	—	10	4	no	—
31	#683	—	10	8	no	—
32	C₄×Dic₅	—	5	2	no	—
33	C₄×Dic₅	—	10	2	no	—
34	#333	—	25	1	no	Sylow
35	C₂×C₅²	#100	1	1	yes	characteristic
36	C₂×C₅²	#103	1	2	yes	—
37	C₅×Q₈	—	4	2	no	—
38	C₅×Q₈	—	4	4	no	—
39	C₂×C₄×C₅	—	2	4	no	—
40	C₂×C₄×C₅	—	2	2	no	—
41	C₂×C₄×C₅	—	2	2	no	—
42	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
43	C₂×Dic₅	—	5	4	no	—
44	C₂×Dic₅	—	10	4	no	—
45	C₂×Dic₅	—	10	4	no	—
46	C₂×Dic₅	—	10	2	no	—
47	Dic₁₀	—	20	4	no	—
48	Dic₁₀	—	20	2	no	—
49	#170	—	20	4	no	—
50	C₄⋊Q₈	—	25	1	no	—
51	C₄⋊C₈	—	25	2	no	—
52	C₅²	#333	1	1	yes	characteristic, complemented, Sylow
53	C₂²×C₅	—	2	2	no	—
54	C₂²×C₅	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
55	C₄×C₅	—	2	4	no	—
56	C₄×C₅	—	2	2	no	—
57	C₄×C₅	—	4	4	no	—
58	C₄×C₅	—	4	2	no	—
59	Dic₅	—	5	4	no	—
60	Dic₅	—	5	4	no	—
61	Dic₅	—	10	4	no	—
62	Dic₅	—	10	4	no	—
63	Dic₅	—	20	4	no	—
64	Dic₅	—	20	2	no	—
65	C₂×Q₈	—	25	1	no	—
66	C₂×Q₈	#1044	1	1	yes	characteristic
67	C₂×C₈	—	50	2	no	—
68	C₄⋊C₄	—	50	2	no	—
69	C₄²	—	25	1	no	—
70	C₂×C₅	—	2	4	no	—
71	C₂×C₅	—	2	2	no	—
72	C₂×C₅	#4009	1	4	yes	—
73	C₂×C₅	#4013	1	2	yes	—
74	Q₈	—	4	1	no	—
75	Q₈	—	100	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	25	2	no	—
78	C₂×C₄	—	50	1	no	—
79	C₂×C₄	C₂×#1044	1	1	yes	characteristic
80	C₈	—	100	2	no	—
81	C₅	—	2	2	no	—
82	C₅	#64931	1	2	yes	complemented
83	C₂²	C₂²⋊#1044	1	1	yes	characteristic
84	C₄	—	2	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	25	2	no	—
87	C₄	—	25	2	no	—
88	C₄	—	100	1	no	—
89	C₂	#11716902	1	2	yes	—
90	C₂	#11716906	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#1044
commutator	C₂×C₄×C₅²	C₂×C₄
Frattini	C₂×C₄	C₂×#1044
Fitting	C₂×C₅²×Q₈	C₄
socle, prime radical	C₂²×C₅²	C₂²⋊C₄
cosocle	C₄×#1039	C₂²
hypercenter	C₂×Q₈	#1044
hypocenter	C₅²	#333
LCS	G ▷ C₂×C₄×C₅² ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
52. C₅²	#333	semidirect
82. C₅	#64931	semidirect

Maximal subgroups

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

subgroup	index
#11717264	2
#11716897	2
#64931	5