Subgroups — D₅×#11925893 #27130910

Order 12100.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅×#10597245	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅⋊#11925893	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#11925893	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#10597245	C₂²	1	1	yes	characteristic, complemented, Hall
5	D₅×C₁₁⋊D₁₁	C₅	1	1	yes	maximal, characteristic, complemented
6	C₂×#11925893	—	5	1	no	maximal
7	C₅⋊(C₁₁⋊D₁₁)	C₂×C₅	1	1	yes	characteristic, complemented
8	D₅×C₁₁²	C₂×C₅	1	1	yes	characteristic, complemented
9	C₅×C₁₁⋊D₁₁	C₂×C₅	1	1	yes	characteristic, complemented
10	C₂×#10597245	—	5	1	no	—
11	#11925893	—	2	2	no	—
12	#11925893	—	5	1	no	—
13	#11925893	D₅	1	1	yes	characteristic, direct product factor
14	D₅×AGL(1,𝔽₁₁)	—	11	2	no	maximal
15	C₅×C₁₁²	C₂²×C₅	1	1	yes	characteristic, complemented
16	#10597245	—	2	2	no	—
17	#10597245	C₂×D₅	1	1	yes	characteristic, complemented
18	C₅⋊AGL(1,𝔽₁₁)	—	11	2	no	—
19	D₅×C₁₁⋊C₅	—	11	2	no	—
20	C₅×AGL(1,𝔽₁₁)	—	11	2	no	—
21	C₂×C₁₁⋊D₁₁	—	5	1	no	Hall
22	C₅×C₁₁⋊C₅	—	11	2	no	—
23	C₂×C₁₁²	—	5	1	no	—
24	C₁₁⋊D₁₁	—	5	1	no	—
25	C₁₁⋊D₁₁	C₅×D₅	1	1	yes	characteristic, complemented
26	D₅×D₁₁	—	11	2	no	—
27	D₅×D₁₁	—	55	2	no	—
28	C₂×AGL(1,𝔽₁₁)	—	55	2	no	—
29	C₁₁²	C₂×C₅×D₅	1	1	yes	characteristic, complemented, Sylow
30	D₅₅	—	11	2	no	—
31	D₅₅	—	55	2	no	—
32	C₅×D₁₁	—	11	2	no	—
33	C₅×D₁₁	—	55	2	no	—
34	D₅×C₁₁	—	5	2	no	—
35	D₅×C₁₁	AGL(1,𝔽₁₁)	1	2	yes	complemented
36	C₂×C₁₁⋊C₅	—	55	2	no	—
37	AGL(1,𝔽₁₁)	—	11	2	no	—
38	AGL(1,𝔽₁₁)	—	22	4	no	—
39	AGL(1,𝔽₁₁)	—	55	2	no	—
40	C₂×C₅×D₅	—	121	1	no	Hall
41	C₅×C₁₁	—	5	2	no	—
42	C₅×C₁₁	C₂×AGL(1,𝔽₁₁)	1	2	yes	complemented
43	C₁₁⋊C₅	—	11	2	no	—
44	C₁₁⋊C₅	—	22	4	no	—
45	C₂×C₅²	—	121	1	no	—
46	C₅×D₅	—	121	1	no	—
47	C₅×D₅	—	121	1	no	—
48	C₂×D₁₁	—	275	2	no	—
49	C₂×D₁₁	—	55	2	no	—
50	C₅²	—	121	1	no	Sylow
51	C₂×C₁₁	—	5	2	no	—
52	C₂×C₁₁	—	25	2	no	—
53	D₁₁	—	11	2	no	—
54	D₁₁	—	275	2	no	—
55	D₁₁	—	55	2	no	—
56	D₁₁	—	55	2	no	—
57	C₂²×C₅	—	605	1	no	—
58	C₂×D₅	—	121	1	no	—
59	C₁₁	—	5	2	no	—
60	C₁₁	D₅×AGL(1,𝔽₁₁)	1	2	yes	complemented
61	C₂×C₅	—	605	1	no	—
62	C₂×C₅	—	605	1	no	—
63	C₂×C₅	—	121	1	no	—
64	C₂×C₅	—	121	1	no	—
65	C₂×C₅	—	242	2	no	—
66	D₅	—	121	1	no	—
67	D₅	#11925893	1	1	yes	characteristic, direct product factor
68	C₅	—	121	1	no	—
69	C₅	—	242	2	no	—
70	C₅	C₂×#11925893	1	1	yes	characteristic, complemented
71	C₂²	—	605	1	no	Sylow
72	C₂	—	5	1	no	—
73	C₂	—	605	1	no	—
74	C₂	—	121	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₁₁²	C₂²×C₅
LCS	G ▷ C₅×C₁₁²
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. D₅×#10597245	C₂	semidirect
2. C₅⋊#11925893	C₂	semidirect
3. C₅×#11925893	C₂	semidirect
4. C₅×#10597245	C₂²	semidirect
5. D₅×C₁₁⋊D₁₁	C₅	semidirect
7. C₅⋊(C₁₁⋊D₁₁)	C₂×C₅	semidirect
8. D₅×C₁₁²	C₂×C₅	semidirect
9. C₅×C₁₁⋊D₁₁	C₂×C₅	semidirect
13. #11925893	D₅	direct (N × H)
15. C₅×C₁₁²	C₂²×C₅	semidirect
17. #10597245	C₂×D₅	semidirect
25. C₁₁⋊D₁₁	C₅×D₅	semidirect
29. C₁₁²	C₂×C₅×D₅	semidirect
35. D₅×C₁₁	AGL(1,𝔽₁₁)	semidirect
42. C₅×C₁₁	C₂×AGL(1,𝔽₁₁)	semidirect
60. C₁₁	D₅×AGL(1,𝔽₁₁)	semidirect
67. D₅	#11925893	direct (N × H)
70. C₅	C₂×#11925893	semidirect

Maximal subgroups

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

subgroup	index
D₅×#10597245	2
C₅⋊#11925893	2
C₅×#11925893	2
D₅×C₁₁⋊D₁₁	5
C₂×#11925893	5
D₅×AGL(1,𝔽₁₁)	11

Subgroups — D5×#11925893 #27130910

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×#11925893 #27130910