Subgroups — #36380949

Order 22599.

320 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#14578890	C₃	1	3	yes	maximal
2	C₉²×C₃₁⋊C₃	C₃	1	1	yes	maximal, characteristic
3	#21419222	C₃	1	2	yes	maximal, complemented
4	#21419221	C₃	1	3	yes	maximal, complemented
5	C₂₇×C₃₁⋊C₉	C₃	1	3	yes	maximal, complemented
6	C₃₁×#6864	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃×C₉×C₃₁⋊C₃	C₉	1	3	yes	—
8	C₃×C₉×C₃₁⋊C₃	C₃²	1	1	yes	characteristic
9	#14578890	C₉	1	27	yes	complemented
10	C₃×C₃₁⋊C₂₇	C₃²	1	6	yes	—
11	C₉×C₃₁⋊C₉	C₉	1	6	yes	—
12	C₉×C₃₁⋊C₉	C₃²	1	1	yes	characteristic
13	C₉×C₃₁⋊C₉	C₃²	1	1	yes	characteristic
14	C₃×C₂₇×C₃₁	C₃²	1	3	yes	—
15	C₉²×C₃₁	C₃²	1	1	yes	characteristic
16	C₃²×C₃₁⋊C₃	C₃×C₉	1	1	yes	characteristic
17	C₃×C₉×C₃₁	C₃×C₉	1	3	yes	—
18	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
19	C₃×C₃₁⋊C₉	C₃×C₉	1	3	yes	—
20	C₃×C₃₁⋊C₉	C₃×C₉	1	2	yes	—
21	C₃×C₃₁⋊C₉	C₃×C₉	1	3	yes	—
22	C₉×C₃₁⋊C₃	—	3	9	no	—
23	C₉×C₃₁⋊C₃	C₃×C₉	1	9	yes	—
24	C₂₇×C₃₁	C₃×C₉	1	9	yes	complemented
25	C₃₁⋊C₂₇	C₃×C₉	1	18	yes	complemented
26	C₉×C₂₇⋊C₃	—	31	1	no	maximal, Sylow
27	C₃²×C₃₁	C₃²×C₉	1	1	yes	characteristic
28	C₃×C₃₁⋊C₃	—	3	3	no	—
29	C₃×C₃₁⋊C₃	C₉²	1	3	yes	—
30	C₉×C₃₁	—	3	3	no	—
31	C₉×C₃₁	C₃²×C₉	1	3	yes	—
32	C₃₁⋊C₉	—	3	3	no	—
33	C₃₁⋊C₉	C₉²	1	6	yes	—
34	C₃₁⋊C₉	C₂₇⋊C₃	1	9	yes	complemented
35	C₃×C₂₇⋊C₃	—	31	3	no	—
36	C₃×C₉²	—	31	1	no	—
37	#6864	—	31	3	no	—
38	#6864	—	31	2	no	—
39	#6864	C₃₁⋊C₃	1	1	yes	characteristic, complemented
40	C₉×C₂₇	—	31	3	no	—
41	C₃×C₃₁	C₃×C₉²	1	1	yes	characteristic
42	C₃×C₃₁	C₃×C₂₇⋊C₃	1	3	yes	—
43	C₃₁⋊C₃	—	3	3	no	—
44	C₃²×C₉	—	31	1	no	—
45	C₃²×C₉	—	31	3	no	—
46	C₂₇⋊C₃	—	31	27	no	—
47	C₃×C₂₇	—	31	6	no	—
48	C₃×C₂₇	C₃×C₃₁⋊C₃	1	3	yes	—
49	C₉²	—	31	6	no	—
50	C₉²	—	31	1	no	—
51	C₉²	—	31	1	no	—
52	C₉²	C₃×C₃₁⋊C₃	1	1	yes	characteristic
53	C₃₁	C₉×C₂₇⋊C₃	1	1	yes	characteristic, complemented, Sylow
54	C₃³	—	31	1	no	—
55	C₃×C₉	—	31	2	no	—
56	C₃×C₉	—	31	3	no	—
57	C₃×C₉	—	31	9	no	—
58	C₃×C₉	—	31	3	no	—
59	C₃×C₉	—	93	9	no	—
60	C₃×C₉	C₉×C₃₁⋊C₃	1	3	yes	—
61	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
62	C₂₇	—	31	18	no	—
63	C₂₇	C₉×C₃₁⋊C₃	1	9	yes	complemented
64	C₃²	—	31	3	no	—
65	C₃²	—	93	3	no	—
66	C₃²	C₃×C₉×C₃₁⋊C₃	1	1	yes	characteristic
67	C₉	—	3	3	no	—
68	C₉	—	31	6	no	—
69	C₉	—	31	9	no	—
70	C₉	—	93	3	no	—
71	C₉	C₃×C₉×C₃₁⋊C₃	1	3	yes	—
72	C₃	—	93	3	no	—
73	C₃	C₉²×C₃₁⋊C₃	1	1	yes	characteristic
74	C₃	C₂₇⋊C₃×C₃₁⋊C₃	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃×C₉	C₃²×C₃₁⋊C₃
commutator	C₃×C₃₁	C₃×C₉²
Fitting	C₃₁×#6864	C₃
socle	C₃²×C₃₁	C₃²×C₉
prime radical	C₃²×C₃₁⋊C₃	C₃×C₉
cosocle	C₃×C₉×C₃₁	C₃³
hypercenter	#6864	C₃₁⋊C₃
hypocenter	C₃₁	C₉×C₂₇⋊C₃
LCS	G ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃×C₉ ◁ #6864

Complements

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

N	H ≅ G/N	type
3. #21419222	C₃	semidirect
4. #21419221	C₃	semidirect
5. C₂₇×C₃₁⋊C₉	C₃	semidirect
6. C₃₁×#6864	C₃	semidirect
9. #14578890	C₉	semidirect
24. C₂₇×C₃₁	C₃×C₉	semidirect
25. C₃₁⋊C₂₇	C₃×C₉	semidirect
34. C₃₁⋊C₉	C₂₇⋊C₃	semidirect
39. #6864	C₃₁⋊C₃	semidirect
53. C₃₁	C₉×C₂₇⋊C₃	semidirect
63. C₂₇	C₉×C₃₁⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×#14578890	3
C₉²×C₃₁⋊C₃	3
#21419222	3
#21419221	3
C₂₇×C₃₁⋊C₉	3
C₃₁×#6864	3
C₉×C₂₇⋊C₃	31