Subgroups — C₂×C₄×C₁₁²×C₂₅ #72201503

Order 24200.

336 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×C₁₁²	C₅	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₁×C₂₅	C₁₁	1	12	yes	maximal, direct product factor
3	C₂²×C₁₁²×C₂₅	C₂	1	1	yes	maximal, characteristic
4	C₄×C₁₁²×C₂₅	C₂	1	2	yes	maximal, direct product factor
5	C₂×C₁₁²×C₂₅	C₄	1	2	yes	direct product factor
6	C₂×C₁₁²×C₂₅	C₂²	1	1	yes	characteristic
7	C₁₁²×C₂₅	C₂×C₄	1	1	yes	characteristic, direct product factor, Hall
8	C₂²×C₅×C₁₁²	C₂×C₅	1	1	yes	characteristic
9	C₄×C₅×C₁₁²	C₂×C₅	1	2	yes	—
10	C₂×C₅×C₁₁²	C₄×C₅	1	2	yes	—
11	C₂×C₅×C₁₁²	C₂²×C₅	1	1	yes	characteristic
12	C₂²×C₁₁×C₂₅	C₂×C₁₁	1	12	yes	—
13	C₄×C₁₁×C₂₅	C₂×C₁₁	1	24	yes	direct product factor
14	C₂×C₄×C₁₁²	C₂₅	1	1	yes	characteristic, direct product factor, Hall
15	C₅×C₁₁²	C₂×C₄×C₅	1	1	yes	characteristic
16	C₂×C₁₁×C₂₅	C₄×C₁₁	1	24	yes	direct product factor
17	C₂×C₁₁×C₂₅	C₂²×C₁₁	1	12	yes	—
18	C₂²×C₁₁²	C₂×C₂₅	1	1	yes	characteristic
19	C₄×C₁₁²	C₂×C₂₅	1	2	yes	direct product factor
20	C₂×C₄×C₅×C₁₁	C₅×C₁₁	1	12	yes	—
21	C₁₁×C₂₅	C₂×C₄×C₁₁	1	12	yes	direct product factor
22	C₂×C₁₁²	C₄×C₂₅	1	2	yes	direct product factor
23	C₂×C₁₁²	C₂²×C₂₅	1	1	yes	characteristic
24	C₂²×C₅×C₁₁	C₂×C₅×C₁₁	1	12	yes	—
25	C₄×C₅×C₁₁	C₂×C₅×C₁₁	1	24	yes	—
26	C₂×C₄×C₂₅	C₁₁²	1	1	yes	characteristic, direct product factor, Hall
27	C₁₁²	C₂×C₄×C₂₅	1	1	yes	characteristic, direct product factor, Sylow
28	C₂×C₅×C₁₁	C₄×C₅×C₁₁	1	24	yes	—
29	C₂×C₅×C₁₁	C₂²×C₅×C₁₁	1	12	yes	—
30	C₂²×C₂₅	C₂×C₁₁²	1	1	yes	characteristic
31	C₄×C₂₅	C₂×C₁₁²	1	2	yes	direct product factor
32	C₂×C₄×C₁₁	C₁₁×C₂₅	1	12	yes	direct product factor
33	C₅×C₁₁	C₂×C₄×C₅×C₁₁	1	12	yes	—
34	C₂×C₂₅	C₄×C₁₁²	1	2	yes	direct product factor
35	C₂×C₂₅	C₂²×C₁₁²	1	1	yes	characteristic
36	C₂²×C₁₁	C₂×C₁₁×C₂₅	1	12	yes	—
37	C₄×C₁₁	C₂×C₁₁×C₂₅	1	24	yes	direct product factor
38	C₂×C₄×C₅	C₅×C₁₁²	1	1	yes	characteristic
39	C₂₅	C₂×C₄×C₁₁²	1	1	yes	characteristic, direct product factor, Sylow
40	C₂×C₁₁	C₄×C₁₁×C₂₅	1	24	yes	direct product factor
41	C₂×C₁₁	C₂²×C₁₁×C₂₅	1	12	yes	—
42	C₂²×C₅	C₂×C₅×C₁₁²	1	1	yes	characteristic
43	C₄×C₅	C₂×C₅×C₁₁²	1	2	yes	—
44	C₁₁	C₂×C₄×C₁₁×C₂₅	1	12	yes	direct product factor
45	C₂×C₅	C₄×C₅×C₁₁²	1	2	yes	—
46	C₂×C₅	C₂²×C₅×C₁₁²	1	1	yes	characteristic
47	C₂×C₄	C₁₁²×C₂₅	1	1	yes	characteristic, direct product factor, Sylow
48	C₅	C₂×C₄×C₅×C₁₁²	1	1	yes	characteristic
49	C₂²	C₂×C₁₁²×C₂₅	1	1	yes	characteristic
50	C₄	C₂×C₁₁²×C₂₅	1	2	yes	direct product factor
51	C₂	C₄×C₁₁²×C₂₅	1	2	yes	direct product factor
52	C₂	C₂²×C₁₁²×C₂₅	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₅	C₂²×C₅×C₁₁²
socle, prime radical	C₂²×C₅×C₁₁²	C₂×C₅
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₁₁×C₂₅	C₁₁	direct (N × H)
4. C₄×C₁₁²×C₂₅	C₂	direct (N × H)
5. C₂×C₁₁²×C₂₅	C₄	direct (N × H)
7. C₁₁²×C₂₅	C₂×C₄	direct (N × H)
13. C₄×C₁₁×C₂₅	C₂×C₁₁	direct (N × H)
14. C₂×C₄×C₁₁²	C₂₅	direct (N × H)
16. C₂×C₁₁×C₂₅	C₄×C₁₁	direct (N × H)
19. C₄×C₁₁²	C₂×C₂₅	direct (N × H)
21. C₁₁×C₂₅	C₂×C₄×C₁₁	direct (N × H)
22. C₂×C₁₁²	C₄×C₂₅	direct (N × H)
26. C₂×C₄×C₂₅	C₁₁²	direct (N × H)
27. C₁₁²	C₂×C₄×C₂₅	direct (N × H)
31. C₄×C₂₅	C₂×C₁₁²	direct (N × H)
32. C₂×C₄×C₁₁	C₁₁×C₂₅	direct (N × H)
34. C₂×C₂₅	C₄×C₁₁²	direct (N × H)
37. C₄×C₁₁	C₂×C₁₁×C₂₅	direct (N × H)
39. C₂₅	C₂×C₄×C₁₁²	direct (N × H)
40. C₂×C₁₁	C₄×C₁₁×C₂₅	direct (N × H)
44. C₁₁	C₂×C₄×C₁₁×C₂₅	direct (N × H)
47. C₂×C₄	C₁₁²×C₂₅	direct (N × H)
50. C₄	C₂×C₁₁²×C₂₅	direct (N × H)
51. C₂	C₄×C₁₁²×C₂₅	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅×C₁₁²	5
C₂×C₄×C₁₁×C₂₅	11
C₂²×C₁₁²×C₂₅	2
C₄×C₁₁²×C₂₅	2

Subgroups — C2×C4×C112×C25 #72201503

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₁₁²×C₂₅ #72201503