Subgroups — C₂₅×#458 #13098607

Order 1600.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₅×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
2	C₂₅×#123	C₂	1	3	yes	maximal, complemented
3	C₂₅×#99	C₂	1	3	yes	maximal, complemented
4	#41×C₂₅	—	2	3	no	—
5	C₂×D₄×C₂₅	—	2	3	no	—
6	C₂×D₄×C₂₅	C₂²	1	3	yes	—
7	C₂²×C₄×C₂₅	C₂²	1	1	yes	characteristic, complemented
8	C₄⋊C₄×C₂₅	—	2	3	no	—
9	C₂²⋊C₄×C₂₅	—	2	3	no	—
10	C₂²⋊C₄×C₂₅	C₂²	1	3	yes	complemented
11	C₅×#458	C₅	1	1	yes	maximal, characteristic
12	C₂³×C₂₅	—	2	1	no	—
13	C₂³×C₂₅	D₄	1	3	yes	—
14	C₂³×C₂₅	C₂³	1	1	yes	characteristic
15	Q₈×C₂₅	—	2	1	no	—
16	D₄×C₂₅	—	2	3	no	—
17	D₄×C₂₅	—	2	6	no	—
18	C₂×C₄×C₂₅	—	2	3	no	—
19	C₂×C₄×C₂₅	—	2	3	no	—
20	C₂×C₄×C₂₅	—	2	3	no	—
21	C₂×C₄×C₂₅	D₄	1	3	yes	—
22	C₅×H(5,𝔽₂)	C₂×C₅	1	1	yes	characteristic
23	C₅×#123	C₂×C₅	1	3	yes	—
24	C₅×#99	C₂×C₅	1	3	yes	—
25	C₂²×C₂₅	—	2	3	no	—
26	C₂²×C₂₅	—	4	1	no	—
27	C₂²×C₂₅	—	4	3	no	—
28	C₂²×C₂₅	—	8	1	no	—
29	C₂²×C₂₅	C₂×D₄	1	3	yes	—
30	C₄×C₂₅	—	2	3	no	—
31	C₄×C₂₅	—	4	3	no	—
32	C₄×C₂₅	—	4	1	no	—
33	C₅×#41	—	2	3	no	—
34	C₂×C₅×D₄	—	2	3	no	—
35	C₂×C₅×D₄	C₂²×C₅	1	3	yes	—
36	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
37	C₅×C₄⋊C₄	—	2	3	no	—
38	C₅×C₂²⋊C₄	—	2	3	no	—
39	C₅×C₂²⋊C₄	C₂²×C₅	1	3	yes	—
40	#458	C₂₅	1	1	yes	characteristic, direct product factor, Sylow
41	C₂×C₂₅	—	2	3	no	—
42	C₂×C₂₅	—	4	3	no	—
43	C₂×C₂₅	#120	1	1	yes	characteristic
44	C₂³×C₅	—	2	1	no	—
45	C₂³×C₅	C₅×D₄	1	3	yes	—
46	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
47	C₅×Q₈	—	2	1	no	—
48	C₅×D₄	—	2	3	no	—
49	C₅×D₄	—	2	6	no	—
50	C₂×C₄×C₅	—	2	3	no	—
51	C₂×C₄×C₅	—	2	3	no	—
52	C₂×C₄×C₅	—	2	3	no	—
53	C₂×C₄×C₅	C₅×D₄	1	3	yes	—
54	H(5,𝔽₂)	C₂×C₂₅	1	1	yes	characteristic
55	#123	C₂×C₂₅	1	3	yes	complemented
56	#99	C₂×C₂₅	1	3	yes	complemented
57	C₂₅	#458	1	1	yes	characteristic, direct product factor, Sylow
58	C₂²×C₅	—	2	3	no	—
59	C₂²×C₅	—	4	1	no	—
60	C₂²×C₅	—	4	3	no	—
61	C₂²×C₅	—	8	1	no	—
62	C₂²×C₅	C₂×C₅×D₄	1	3	yes	—
63	C₄×C₅	—	2	3	no	—
64	C₄×C₅	—	4	3	no	—
65	C₄×C₅	—	4	1	no	—
66	#41	—	2	3	no	—
67	C₂×D₄	—	2	3	no	—
68	C₂×D₄	C₂²×C₂₅	1	3	yes	—
69	C₂²×C₄	C₂²×C₂₅	1	1	yes	characteristic, complemented
70	C₄⋊C₄	—	2	3	no	—
71	C₂²⋊C₄	—	2	3	no	—
72	C₂²⋊C₄	C₂²×C₂₅	1	3	yes	complemented
73	C₂×C₅	—	2	3	no	—
74	C₂×C₅	—	4	3	no	—
75	C₂×C₅	C₅×#120	1	1	yes	characteristic
76	C₂³	—	2	1	no	—
77	C₂³	D₄×C₂₅	1	3	yes	—
78	C₂³	C₂³×C₂₅	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	D₄×C₂₅	1	3	yes	—
86	C₅	C₅×#458	1	1	yes	characteristic
87	C₂²	—	2	3	no	—
88	C₂²	—	4	1	no	—
89	C₂²	—	4	3	no	—
90	C₂²	—	8	1	no	—
91	C₂²	C₂×D₄×C₂₅	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	C₂₅×#120	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₂₅	#120
commutator	C₂³	C₂³×C₂₅
Frattini, cosocle	C₂³×C₅	C₂³×C₅
socle	C₂×C₅	C₅×#120
prime radical	C₅×H(5,𝔽₂)	C₂×C₅
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₅ ◁ C₂³×C₂₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₅×#123	C₂	semidirect
3. C₂₅×#99	C₂	semidirect
7. C₂²×C₄×C₂₅	C₂²	semidirect
10. C₂²⋊C₄×C₂₅	C₂²	semidirect
40. #458	C₂₅	direct (N × H)
55. #123	C₂×C₂₅	semidirect
56. #99	C₂×C₂₅	semidirect
57. C₂₅	#458	direct (N × H)
69. C₂²×C₄	C₂²×C₂₅	semidirect
72. C₂²⋊C₄	C₂²×C₂₅	semidirect

Maximal subgroups

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

subgroup	index
C₂₅×H(5,𝔽₂)	2
C₂₅×#123	2
C₂₅×#99	2
C₅×#458	5

Subgroups — C25×#458 #13098607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₅×#458 #13098607