Subgroups — C₅²×#458 #13103195

Order 1600.

640 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
2	C₅²×#123	C₂	1	3	yes	maximal, complemented
3	C₅²×#99	C₂	1	3	yes	maximal, complemented
4	C₅²×#41	—	2	3	no	—
5	C₂×C₅²×D₄	—	2	3	no	—
6	C₂×C₅²×D₄	C₂²	1	3	yes	—
7	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic, complemented
8	C₅²×C₄⋊C₄	—	2	3	no	—
9	C₅²×C₂²⋊C₄	—	2	3	no	—
10	C₅²×C₂²⋊C₄	C₂²	1	3	yes	complemented
11	C₅×#458	C₅	1	6	yes	maximal, direct product factor
12	C₂³×C₅²	—	2	1	no	—
13	C₂³×C₅²	D₄	1	3	yes	—
14	C₂³×C₅²	C₂³	1	1	yes	characteristic
15	C₅²×Q₈	—	2	1	no	—
16	C₅²×D₄	—	2	3	no	—
17	C₅²×D₄	—	2	6	no	—
18	C₂×C₄×C₅²	—	2	3	no	—
19	C₂×C₄×C₅²	—	2	3	no	—
20	C₂×C₄×C₅²	—	2	3	no	—
21	C₂×C₄×C₅²	D₄	1	3	yes	—
22	C₅×H(5,𝔽₂)	C₂×C₅	1	6	yes	—
23	C₅×#123	C₂×C₅	1	18	yes	complemented
24	C₅×#99	C₂×C₅	1	18	yes	complemented
25	C₂²×C₅²	—	2	3	no	—
26	C₂²×C₅²	—	4	1	no	—
27	C₂²×C₅²	—	4	3	no	—
28	C₂²×C₅²	—	8	1	no	—
29	C₂²×C₅²	C₂×D₄	1	3	yes	—
30	C₄×C₅²	—	2	3	no	—
31	C₄×C₅²	—	4	3	no	—
32	C₄×C₅²	—	4	1	no	—
33	C₅×#41	—	2	18	no	—
34	C₂×C₅×D₄	—	2	18	no	—
35	C₂×C₅×D₄	C₂²×C₅	1	18	yes	—
36	C₂²×C₄×C₅	C₂²×C₅	1	6	yes	complemented
37	C₅×C₄⋊C₄	—	2	18	no	—
38	C₅×C₂²⋊C₄	—	2	18	no	—
39	C₅×C₂²⋊C₄	C₂²×C₅	1	18	yes	complemented
40	#458	C₅²	1	1	yes	characteristic, direct product factor, Sylow
41	C₂×C₅²	—	2	3	no	—
42	C₂×C₅²	—	4	3	no	—
43	C₂×C₅²	#120	1	1	yes	characteristic
44	C₂³×C₅	—	2	6	no	—
45	C₂³×C₅	C₅×D₄	1	18	yes	—
46	C₂³×C₅	C₂³×C₅	1	6	yes	—
47	C₅×Q₈	—	2	6	no	—
48	C₅×D₄	—	2	18	no	—
49	C₅×D₄	—	2	36	no	—
50	C₂×C₄×C₅	—	2	18	no	—
51	C₂×C₄×C₅	—	2	18	no	—
52	C₂×C₄×C₅	—	2	18	no	—
53	C₂×C₄×C₅	C₅×D₄	1	18	yes	—
54	H(5,𝔽₂)	C₂×C₅²	1	1	yes	characteristic
55	#123	C₂×C₅²	1	3	yes	complemented
56	#99	C₂×C₅²	1	3	yes	complemented
57	C₅²	#458	1	1	yes	characteristic, direct product factor, Sylow
58	C₂²×C₅	—	2	18	no	—
59	C₂²×C₅	—	4	6	no	—
60	C₂²×C₅	—	4	18	no	—
61	C₂²×C₅	—	8	6	no	—
62	C₂²×C₅	C₂×C₅×D₄	1	18	yes	—
63	C₄×C₅	—	2	18	no	—
64	C₄×C₅	—	4	18	no	—
65	C₄×C₅	—	4	6	no	—
66	#41	—	2	3	no	—
67	C₂×D₄	—	2	3	no	—
68	C₂×D₄	C₂²×C₅²	1	3	yes	—
69	C₂²×C₄	C₂²×C₅²	1	1	yes	characteristic, complemented
70	C₄⋊C₄	—	2	3	no	—
71	C₂²⋊C₄	—	2	3	no	—
72	C₂²⋊C₄	C₂²×C₅²	1	3	yes	complemented
73	C₂×C₅	—	2	18	no	—
74	C₂×C₅	—	4	18	no	—
75	C₂×C₅	C₅×#120	1	6	yes	—
76	C₂³	—	2	1	no	—
77	C₂³	C₅²×D₄	1	3	yes	—
78	C₂³	C₂³×C₅²	1	1	yes	characteristic
79	Q₈	—	2	1	no	—
80	D₄	—	2	3	no	—
81	D₄	—	2	6	no	—
82	C₂×C₄	—	2	3	no	—
83	C₂×C₄	—	2	3	no	—
84	C₂×C₄	—	2	3	no	—
85	C₂×C₄	C₅²×D₄	1	3	yes	—
86	C₅	C₅×#458	1	6	yes	direct product factor
87	C₂²	—	2	3	no	—
88	C₂²	—	4	1	no	—
89	C₂²	—	4	3	no	—
90	C₂²	—	8	1	no	—
91	C₂²	C₂×C₅²×D₄	1	3	yes	—
92	C₄	—	2	3	no	—
93	C₄	—	4	3	no	—
94	C₄	—	4	1	no	—
95	C₂	—	2	3	no	—
96	C₂	—	4	3	no	—
97	C₂	C₅²×#120	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅²	#120
commutator, Frattini, cosocle	C₂³	C₂³×C₅²
prime radical	C₅²×H(5,𝔽₂)	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅² ◁ C₂³×C₅² ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅²×#123	C₂	semidirect
3. C₅²×#99	C₂	semidirect
7. C₂²×C₄×C₅²	C₂²	semidirect
10. C₅²×C₂²⋊C₄	C₂²	semidirect
11. C₅×#458	C₅	direct (N × H)
23. C₅×#123	C₂×C₅	semidirect
24. C₅×#99	C₂×C₅	semidirect
36. C₂²×C₄×C₅	C₂²×C₅	semidirect
39. C₅×C₂²⋊C₄	C₂²×C₅	semidirect
40. #458	C₅²	direct (N × H)
55. #123	C₂×C₅²	semidirect
56. #99	C₂×C₅²	semidirect
57. C₅²	#458	direct (N × H)
69. C₂²×C₄	C₂²×C₅²	semidirect
72. C₂²⋊C₄	C₂²×C₅²	semidirect
86. C₅	C₅×#458	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅²×H(5,𝔽₂)	2
C₅²×#123	2
C₅²×#99	2
C₅×#458	5

Subgroups — C52×#458 #13103195

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×#458 #13103195