Subgroups — #36345163

Order 21780.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⋊(C₃⋊#11925895)	C₂	1	2	yes	maximal, complemented
2	C₃²×#11925892	C₂	1	1	yes	maximal, characteristic, complemented
3	#20300008	—	3	4	no	maximal
4	C₃²×#10597243	C₂²	1	1	yes	characteristic, complemented, Hall
5	#17983775	C₅	1	1	yes	maximal, characteristic, complemented, Hall
6	C₃⋊#11925895	—	3	8	no	—
7	C₃×#11925892	D₃	1	4	yes	complemented
8	#11925892⋊C₂	—	9	1	no	Hall
9	C₁₁×C₃⋊D₃₃	C₂×C₅	1	2	yes	complemented
10	C₃²×C₁₁⋊D₁₁	C₂×C₅	1	1	yes	characteristic, complemented
11	C₃⋊D₃×AGL(1,𝔽₁₁)	—	11	2	no	maximal
12	C₃×#10597243	C₂×D₃	1	4	yes	complemented
13	D₃₃⋊D₁₁	—	3	4	no	—
14	#11925895	—	9	2	no	—
15	#11925892	C₃⋊D₃	1	1	yes	characteristic, complemented
16	C₃²×C₁₁²	C₂²×C₅	1	1	yes	characteristic, complemented, Hall
17	C₃⋊(C₃⋊AGL(1,𝔽₁₁))	—	11	2	no	—
18	C₃⋊D₃×C₁₁⋊C₅	—	11	2	no	—
19	C₃²×AGL(1,𝔽₁₁)	—	11	2	no	—
20	C₃²×AGL(1,𝔽₁₁)	—	22	5	no	—
21	C₁₁×D₃₃	—	3	8	no	—
22	C₃×C₁₁⋊D₁₁	C₅×D₃	1	4	yes	complemented
23	D₃×AGL(1,𝔽₁₁)	—	33	8	no	—
24	#10597243	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented, Hall
25	C₃²×C₁₁⋊C₅	—	11	2	no	—
26	C₃²×C₁₁⋊C₅	—	22	5	no	—
27	D₁₁²	—	9	1	no	Hall
28	C₃⋊D₃×D₁₁	—	11	2	no	—
29	C₃×C₁₁²	C₂×C₅×D₃	1	4	yes	complemented
30	C₃⋊AGL(1,𝔽₁₁)	—	33	8	no	—
31	D₃×C₁₁⋊C₅	—	33	8	no	—
32	C₃×AGL(1,𝔽₁₁)	—	11	8	no	—
33	C₃×AGL(1,𝔽₁₁)	—	22	20	no	—
34	C₁₁⋊D₁₁	C₅×C₃⋊D₃	1	1	yes	characteristic, complemented
35	C₁₁×D₁₁	—	9	2	no	—
36	C₂×AGL(1,𝔽₁₁)	—	99	2	no	—
37	C₃⋊D₃₃	AGL(1,𝔽₁₁)	1	2	yes	complemented
38	C₁₁×C₃⋊D₃	—	11	2	no	—
39	C₃²×D₁₁	—	11	2	no	—
40	C₃²×D₁₁	—	22	5	no	—
41	C₂×C₅×C₃⋊D₃	—	121	1	no	Hall
42	C₃×C₁₁⋊C₅	—	11	8	no	—
43	C₃×C₁₁⋊C₅	—	22	20	no	—
44	D₃×D₁₁	—	33	8	no	—
45	C₁₁²	C₂×C₅×C₃⋊D₃	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₁₁⋊C₅	—	99	2	no	—
47	AGL(1,𝔽₁₁)	—	11	2	no	—
48	AGL(1,𝔽₁₁)	—	22	5	no	—
49	AGL(1,𝔽₁₁)	—	99	2	no	—
50	C₃²×C₁₁	—	2	5	no	—
51	C₃²×C₁₁	C₂×AGL(1,𝔽₁₁)	1	2	yes	complemented
52	C₂×C₃²×C₅	—	121	1	no	—
53	C₅×C₃⋊D₃	—	121	2	no	—
54	D₃₃	—	3	8	no	—
55	C₃×D₁₁	—	11	8	no	—
56	C₃×D₁₁	—	22	20	no	—
57	D₃×C₁₁	—	33	8	no	—
58	C₂×C₅×D₃	—	363	4	no	—
59	C₁₁⋊C₅	—	11	2	no	—
60	C₁₁⋊C₅	—	22	5	no	—
61	C₃²×C₅	—	121	1	no	Hall
62	C₂×D₁₁	—	99	2	no	—
63	C₂×C₃⋊D₃	—	121	1	no	Hall
64	C₃×C₁₁	—	2	20	no	—
65	C₃×C₁₁	D₃×AGL(1,𝔽₁₁)	1	8	yes	complemented
66	C₂×C₃×C₅	—	121	4	no	—
67	C₅×D₃	—	363	8	no	—
68	C₂×C₁₁	—	99	2	no	—
69	D₁₁	—	9	2	no	—
70	D₁₁	—	11	2	no	—
71	D₁₁	—	22	5	no	—
72	C₂²×C₅	—	1089	1	no	Hall
73	C₂×C₃²	—	121	1	no	—
74	C₃⋊D₃	—	11	2	no	—
75	C₃×C₅	—	121	4	no	—
76	C₂×D₃	—	363	4	no	—
77	C₁₁	—	2	5	no	—
78	C₁₁	C₃⋊D₃×AGL(1,𝔽₁₁)	1	2	yes	complemented
79	C₂×C₅	—	1089	2	no	—
80	C₂×C₅	—	121	1	no	—
81	C₃²	#11925892⋊C₂	1	1	yes	characteristic, complemented, Sylow
82	C₂×C₃	—	121	4	no	—
83	D₃	—	33	8	no	—
84	C₅	—	121	1	no	Sylow
85	C₂²	—	1089	1	no	Sylow
86	C₃	#20300008	1	4	yes	complemented
87	C₂	—	99	2	no	—
88	C₂	—	121	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃²×C₁₁²	C₂²×C₅
LCS	G ▷ C₃²×C₁₁²
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₃⋊(C₃⋊#11925895)	C₂	semidirect
2. C₃²×#11925892	C₂	semidirect
4. C₃²×#10597243	C₂²	semidirect
5. #17983775	C₅	semidirect
7. C₃×#11925892	D₃	semidirect
9. C₁₁×C₃⋊D₃₃	C₂×C₅	semidirect
10. C₃²×C₁₁⋊D₁₁	C₂×C₅	semidirect
12. C₃×#10597243	C₂×D₃	semidirect
15. #11925892	C₃⋊D₃	semidirect
16. C₃²×C₁₁²	C₂²×C₅	semidirect
22. C₃×C₁₁⋊D₁₁	C₅×D₃	semidirect
24. #10597243	C₂×C₃⋊D₃	semidirect
29. C₃×C₁₁²	C₂×C₅×D₃	semidirect
34. C₁₁⋊D₁₁	C₅×C₃⋊D₃	semidirect
37. C₃⋊D₃₃	AGL(1,𝔽₁₁)	semidirect
45. C₁₁²	C₂×C₅×C₃⋊D₃	semidirect
51. C₃²×C₁₁	C₂×AGL(1,𝔽₁₁)	semidirect
65. C₃×C₁₁	D₃×AGL(1,𝔽₁₁)	semidirect
78. C₁₁	C₃⋊D₃×AGL(1,𝔽₁₁)	semidirect
81. C₃²	#11925892⋊C₂	semidirect
86. C₃	#20300008	semidirect

Maximal subgroups

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

subgroup	index
C₃⋊(C₃⋊#11925895)	2
C₃²×#11925892	2
#20300008	3
#17983775	5
C₃⋊D₃×AGL(1,𝔽₁₁)	11