Subgroups — #36379573

Order 22599.

238 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21419308	C₃	1	2	yes	maximal, complemented
2	#21419226	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#6866	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₃²⋊(C₃₁⋊C₃)	—	3	2	no	—
5	C₃×C₃²⋊(C₃₁⋊C₃)	—	3	6	no	—
6	C₃²⋊(C₃₁⋊C₉)	—	3	1	no	—
7	C₃⁴×C₃₁	C₃²	1	1	yes	characteristic
8	C₃₁×C₃²⋊C₉	—	3	1	no	—
9	C₃³×C₃₁	—	3	2	no	—
10	C₃³×C₃₁	—	3	1	no	—
11	C₃³×C₃₁	—	3	1	no	—
12	C₃³×C₃₁	—	9	3	no	—
13	C₃³×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
14	C₃²×C₃₁⋊C₃	—	3	6	no	—
15	C₃×C₉×C₃₁	—	9	1	no	—
16	C₃²⋊(C₃₁⋊C₃)	—	9	18	no	—
17	C₃²⋊(C₃₁⋊C₃)	—	9	2	no	—
18	C₃²⋊(C₃₁⋊C₃)	—	9	2	no	—
19	C₃²⋊(C₃₁⋊C₃)	—	9	2	no	—
20	C₃×C₃₁⋊C₉	—	9	1	no	—
21	#10624474	—	31	1	no	maximal, Sylow
22	C₃²×C₃₁	—	3	6	no	—
23	C₃²×C₃₁	—	9	1	no	—
24	C₃²×C₃₁	—	9	3	no	—
25	C₃²×C₃₁	—	9	3	no	—
26	C₃²×C₃₁	—	9	4	no	—
27	C₃²×C₃₁	—	9	1	no	—
28	C₃²×C₃₁	AΣL(1,𝔽₂₇)	1	2	yes	—
29	C₃²×C₃₁	#732	1	1	yes	characteristic
30	C₃²×C₃₁	#732	1	1	yes	characteristic
31	C₃×C₃₁⋊C₃	—	9	18	no	—
32	C₃×C₃₁⋊C₃	—	9	6	no	—
33	C₉×C₃₁	—	27	1	no	—
34	C₃₁⋊C₉	—	27	1	no	—
35	C₃²⋊H(3,𝔽₃)	—	31	2	no	—
36	#6866	—	31	1	no	—
37	#6866	C₃₁⋊C₃	1	1	yes	characteristic, complemented
38	C₃×C₃₁	—	3	1	no	—
39	C₃×C₃₁	—	3	1	no	—
40	C₃×C₃₁	—	3	2	no	—
41	C₃×C₃₁	—	9	3	no	—
42	C₃×C₃₁	#6856	1	1	yes	characteristic
43	C₃₁⋊C₃	—	27	6	no	—
44	C₃⁴	C₃×C₃₁⋊C₃	1	1	yes	characteristic
45	C₃×H(3,𝔽₃)	—	93	6	no	—
46	C₃×H(3,𝔽₃)	—	93	2	no	—
47	C₃²⋊C₉	—	3	1	no	—
48	C₃²⋊C₉	—	93	1	no	—
49	C₃₁	#10624474	1	1	yes	characteristic, complemented, Sylow
50	C₃³	—	3	2	no	—
51	C₃³	—	3	1	no	—
52	C₃³	—	3	1	no	—
53	C₃³	—	9	3	no	—
54	C₃³	—	93	6	no	—
55	C₃³	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
56	H(3,𝔽₃)	—	279	18	no	—
57	H(3,𝔽₃)	—	279	2	no	—
58	H(3,𝔽₃)	—	279	2	no	—
59	H(3,𝔽₃)	—	279	2	no	—
60	C₃×C₉	—	9	1	no	—
61	C₃×C₉	—	279	1	no	—
62	C₃²	—	3	6	no	—
63	C₃²	—	9	1	no	—
64	C₃²	—	9	3	no	—
65	C₃²	—	9	3	no	—
66	C₃²	—	9	4	no	—
67	C₃²	—	9	1	no	—
68	C₃²	—	279	18	no	—
69	C₃²	—	279	6	no	—
70	C₃²	#14578895	1	2	yes	—
71	C₃²	#14578899	1	1	yes	characteristic
72	C₃²	#14578900	1	1	yes	characteristic
73	C₉	—	27	1	no	—
74	C₉	—	837	1	no	—
75	C₃	—	3	1	no	—
76	C₃	—	3	1	no	—
77	C₃	—	3	2	no	—
78	C₃	—	9	3	no	—
79	C₃	—	837	6	no	—
80	C₃	#21419201	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#21419201
commutator, cosocle	C₃⁴×C₃₁	C₃²
Frattini	C₃⁴	C₃×C₃₁⋊C₃
Fitting	C₃₁×#6866	C₃
socle	C₃×C₃₁	#6856
hypercenter	#6866	C₃₁⋊C₃
hypocenter	C₃₁	#10624474
LCS	G ▷ C₃⁴×C₃₁ ▷ C₃³×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃⁴ ◁ #6866

Complements

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

N	H ≅ G/N	type
1. #21419308	C₃	semidirect
2. #21419226	C₃	semidirect
3. C₃₁×#6866	C₃	semidirect
37. #6866	C₃₁⋊C₃	semidirect
49. C₃₁	#10624474	semidirect

Maximal subgroups

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

subgroup	index
#21419308	3
#21419226	3
C₃₁×#6866	3
#10624474	31