Subgroups — #36379772

Order 22599.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21419391	C₃	1	1	yes	maximal, characteristic
2	C₃²×C₃₁⋊C₂₇	C₃	1	1	yes	maximal, characteristic, complemented
3	#21419232	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#6869	C₃	1	1	yes	maximal, characteristic, complemented
5	#14578912	—	3	1	no	—
6	#14578902	C₉	1	2	yes	—
7	#14578900	C₉	1	1	yes	characteristic
8	#14578897	—	3	1	no	—
9	#14578894	—	3	1	no	—
10	#14578892	—	3	1	no	—
11	C₃×C₃₁⋊C₂₇	—	3	1	no	—
12	C₃×C₃₁⋊C₂₇	—	3	3	no	—
13	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
14	C₃₁×C₂₇⋊C₃	—	3	1	no	—
15	C₃³×C₃₁	3₋¹⁺²	1	1	yes	characteristic
16	C₃×C₉×C₃₁	—	3	1	no	—
17	C₃×C₉×C₃₁	—	3	1	no	—
18	C₃×C₉×C₃₁	—	3	1	no	—
19	C₃×C₉×C₃₁	C₃×C₉	1	1	yes	characteristic
20	C₃×C₉×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
21	C₃×C₉×C₃₁	3₋¹⁺²	1	1	yes	characteristic
22	C₃²⋊(C₃₁⋊C₃)	—	3	1	no	—
23	#11719689	—	3	2	no	—
24	#11719687	—	9	1	no	—
25	#11719686	—	9	1	no	—
26	C₉×C₃₁⋊C₃	—	9	1	no	—
27	C₂₇×C₃₁	—	9	1	no	—
28	C₃₁⋊C₂₇	—	3	3	no	—
29	C₃₁⋊C₂₇	—	9	1	no	—
30	#10624526	—	31	1	no	maximal, Sylow
31	C₃²×C₃₁	—	3	3	no	—
32	C₃²×C₃₁	—	3	1	no	—
33	C₃²×C₃₁	C₃²⋊C₉	1	1	yes	characteristic
34	C₃×C₃₁⋊C₃	—	9	1	no	—
35	C₉×C₃₁	—	3	1	no	—
36	C₉×C₃₁	—	3	1	no	—
37	C₉×C₃₁	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
38	C₉×C₃₁	#731	1	1	yes	characteristic
39	C₉×C₃₁	#732	1	1	yes	characteristic
40	C₃₁⋊C₉	—	9	2	no	—
41	#6908	—	31	1	no	—
42	C₃²×C₂₇	—	31	1	no	—
43	#6869	—	31	1	no	—
44	#6869	C₃₁⋊C₃	1	1	yes	characteristic, complemented
45	C₃×C₃₁	—	3	3	no	—
46	C₃×C₃₁	—	3	1	no	—
47	C₃×C₃₁	H(3,𝔽₃)⋊C₉	1	1	yes	characteristic
48	C₃₁⋊C₃	—	27	1	no	—
49	#737	—	93	1	no	—
50	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
51	#733	—	31	2	no	—
52	#732	—	31	1	no	—
53	#731	—	93	1	no	—
54	AΣL(1,𝔽₂₇)	—	93	1	no	—
55	C₂₇⋊C₃	—	3	1	no	—
56	C₂₇⋊C₃	—	93	1	no	—
57	C₃×C₂₇	—	93	1	no	—
58	C₃×C₂₇	—	93	3	no	—
59	C₃₁	#10624526	1	1	yes	characteristic, complemented, Sylow
60	C₃³	#11719687	1	1	yes	characteristic
61	3₋¹⁺²	—	279	1	no	—
62	3₋¹⁺²	—	279	1	no	—
63	3₋¹⁺²	—	93	2	no	—
64	H(3,𝔽₃)	—	93	1	no	—
65	C₃×C₉	—	3	1	no	—
66	C₃×C₉	—	3	1	no	—
67	C₃×C₉	—	3	1	no	—
68	C₃×C₉	—	279	1	no	—
69	C₃×C₉	C₉×C₃₁⋊C₃	1	1	yes	characteristic
70	C₃×C₉	#11719686	1	1	yes	characteristic
71	C₃×C₉	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
72	C₂₇	—	9	1	no	—
73	C₂₇	—	279	1	no	—
74	C₂₇	—	93	3	no	—
75	C₃²	—	3	3	no	—
76	C₃²	—	3	1	no	—
77	C₃²	—	279	1	no	—
78	C₃²	(C₃×C₃₁⋊C₃)⋊C₉	1	1	yes	characteristic
79	C₉	—	3	1	no	—
80	C₉	—	3	1	no	—
81	C₉	—	279	2	no	—
82	C₉	#14578895	1	1	yes	characteristic
83	C₉	#14578898	1	1	yes	characteristic
84	C₉	#14578899	1	1	yes	characteristic
85	C₃	—	3	1	no	—
86	C₃	—	3	3	no	—
87	C₃	—	837	1	no	—
88	C₃	#21419233	1	1	yes	characteristic

	subgroup	G/H
center	C₉	#14578899
commutator	C₃×C₉×C₃₁	C₃×C₉
Frattini	C₃²×C₉	C₃×C₃₁⋊C₃
Fitting	C₃₁×#6869	C₃
socle	C₃×C₃₁	H(3,𝔽₃)⋊C₉
prime radical	#21419391	C₃
cosocle	C₃²×C₉×C₃₁	C₃²
hypercenter	#6869	C₃₁⋊C₃
hypocenter	C₃₁	#10624526
LCS	G ▷ C₃×C₉×C₃₁ ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₉ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ #6869

Complements

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

N	H ≅ G/N	type
2. C₃²×C₃₁⋊C₂₇	C₃	semidirect
3. #21419232	C₃	semidirect
4. C₃₁×#6869	C₃	semidirect
44. #6869	C₃₁⋊C₃	semidirect
59. C₃₁	#10624526	semidirect

Maximal subgroups

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

subgroup	index
#21419391	3
C₃²×C₃₁⋊C₂₇	3
#21419232	3
C₃₁×#6869	3
#10624526	31