Subgroups — #36379661

Order 22599.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21419230	C₃	1	1	yes	maximal, characteristic
2	#21419230	C₃	1	1	yes	maximal, characteristic
3	C₃₁⋊#6855	C₃	1	1	yes	maximal, characteristic
4	C₃₁×#6868	C₃	1	1	yes	maximal, characteristic
5	C₃²×C₃₁⋊C₉	—	3	1	no	—
6	C₃²⋊(C₃₁⋊C₉)	—	3	1	no	—
7	C₃²⋊(C₃₁⋊C₉)	—	3	1	no	—
8	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
9	C₃₁×C₃²⋊C₉	—	3	1	no	—
10	C₃³×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
11	C₃×C₉×C₃₁	—	3	1	no	—
12	C₃×C₉×C₃₁	—	3	1	no	—
13	C₃×C₉×C₃₁	—	3	2	no	—
14	C₃×C₉×C₃₁	—	9	1	no	—
15	C₃×C₃₁⋊C₉	—	3	3	no	—
16	C₃×C₃₁⋊C₉	—	9	1	no	—
17	C₃×C₃₁⋊C₉	—	9	1	no	—
18	C₃×C₃₁⋊C₉	—	9	1	no	—
19	C₃×C₃₁⋊C₉	—	9	1	no	—
20	C₃×C₃₁⋊C₉	—	9	1	no	—
21	#10624492	—	31	1	no	maximal, Sylow
22	C₃²×C₃₁	—	3	1	no	—
23	C₃²×C₃₁	—	3	1	no	—
24	C₃²×C₃₁	—	3	1	no	—
25	C₃²×C₃₁	#731	1	1	yes	characteristic
26	C₃²×C₃₁	#732	1	1	yes	characteristic
27	C₃²×C₃₁	#733	1	2	yes	—
28	C₉×C₃₁	—	9	1	no	—
29	C₉×C₃₁	—	27	1	no	—
30	C₃₁⋊C₉	—	9	3	no	—
31	C₃₁⋊C₉	—	27	1	no	—
32	C₃₁⋊C₉	—	27	1	no	—
33	#6868	—	31	1	no	—
34	#6868	—	31	1	no	—
35	#6868	C₃₁⋊C₃	1	1	yes	characteristic
36	#6855	—	31	1	no	—
37	C₃×C₃₁	—	3	1	no	—
38	C₃×C₃₁	—	3	2	no	—
39	C₃×C₃₁	#6861	1	1	yes	characteristic
40	C₃×C₃₁	#6878	1	1	yes	characteristic
41	C₃×C₃₁	#6883	1	1	yes	characteristic
42	C₃×C₃₁	#6883	1	1	yes	characteristic
43	C₃²×C₉	—	93	1	no	—
44	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
45	C₃²⋊C₉	—	3	1	no	—
46	C₃²⋊C₉	—	93	1	no	—
47	C₃²⋊C₉	—	93	1	no	—
48	C₃₁	#10624492	1	1	yes	characteristic, complemented, Sylow
49	C₃³	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
50	C₃×C₉	—	3	1	no	—
51	C₃×C₉	—	3	1	no	—
52	C₃×C₉	—	3	2	no	—
53	C₃×C₉	—	9	1	no	—
54	C₃×C₉	—	279	1	no	—
55	C₃×C₉	—	279	1	no	—
56	C₃×C₉	—	279	1	no	—
57	C₃×C₉	—	279	1	no	—
58	C₃×C₉	—	279	1	no	—
59	C₃×C₉	—	93	3	no	—
60	C₃²	—	3	1	no	—
61	C₃²	—	3	1	no	—
62	C₃²	—	3	1	no	—
63	C₃²	#14578897	1	1	yes	characteristic
64	C₃²	#14578899	1	1	yes	characteristic
65	C₃²	#14578901	1	2	yes	—
66	C₉	—	9	1	no	—
67	C₉	—	279	3	no	—
68	C₉	—	27	1	no	—
69	C₉	—	837	1	no	—
70	C₉	—	837	1	no	—
71	C₃	—	3	1	no	—
72	C₃	—	3	2	no	—
73	C₃	#21419215	1	1	yes	characteristic
74	C₃	#21419259	1	1	yes	characteristic
75	C₃	#21419279	1	1	yes	characteristic
76	C₃	#21419280	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	#14578899
commutator, cosocle	C₃²×C₉×C₃₁	C₃²
Frattini	C₃²×C₉	C₃×C₃₁⋊C₃
Fitting	C₃₁×#6868	C₃
socle	C₃²×C₃₁	#732
prime radical	C₃³×C₃₁	H(3,𝔽₃)
hypercenter	#6868	C₃₁⋊C₃
hypocenter	C₃₁	#10624492
LCS	G ▷ C₃²×C₉×C₃₁ ▷ C₃³×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃²×C₉ ◁ #6868

Complements

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

N	H ≅ G/N	type
48. C₃₁	#10624492	semidirect

Maximal subgroups

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

subgroup	index
#21419230	3
C₃₁⋊#6855	3
C₃₁×#6868	3
#10624492	31