Subgroups — #36379524

Order 22599.

208 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#14578912	C₃	1	1	yes	maximal, characteristic
2	C₃²⋊(C₃₁⋊C₂₇)	C₃	1	2	yes	maximal, complemented
3	C₃₁×C₃²⋊C₂₇	C₃	1	1	yes	maximal, characteristic, complemented
4	#14578912	—	3	3	no	—
5	#14578912	—	3	6	no	—
6	C₃×#11719689	C₉	1	2	yes	—
7	C₃×#11719686	—	3	1	no	—
8	C₃×#11719687	—	3	1	no	—
9	C₃×C₃²⋊(C₃₁⋊C₃)	C₉	1	1	yes	characteristic
10	C₃×C₉×C₃₁⋊C₃	—	3	1	no	—
11	C₃×C₃₁⋊C₂₇	—	3	2	no	—
12	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
13	C₃×C₂₇×C₃₁	—	3	1	no	—
14	C₃³×C₃₁	C₃×C₉	1	1	yes	characteristic
15	C₃²×C₃₁⋊C₃	—	3	1	no	—
16	C₃×C₉×C₃₁	—	3	1	no	—
17	C₃×C₉×C₃₁	—	3	2	no	—
18	C₃×C₉×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
19	C₃×C₉×C₃₁	3₋¹⁺²	1	1	yes	characteristic
20	C₃×C₉×C₃₁	3₋¹⁺²	1	1	yes	characteristic
21	C₃²⋊(C₃₁⋊C₃)	—	3	3	no	—
22	#11719689	—	3	6	no	—
23	C₃×C₃₁⋊C₉	—	3	2	no	—
24	#11719687	—	3	9	no	—
25	#11719686	—	3	9	no	—
26	C₉×C₃₁⋊C₃	—	3	3	no	—
27	C₉×C₃₁⋊C₃	—	3	6	no	—
28	C₂₇×C₃₁	—	9	1	no	—
29	C₃₁⋊C₂₇	—	9	2	no	—
30	#10624456	—	31	1	no	maximal, Sylow
31	C₃²×C₃₁	—	3	1	no	—
32	C₃²×C₃₁	—	3	2	no	—
33	C₃²×C₃₁	—	3	1	no	—
34	C₃²×C₃₁	C₃²⋊C₉	1	1	yes	characteristic
35	C₃×C₃₁⋊C₃	—	3	3	no	—
36	C₃×C₃₁⋊C₃	—	9	2	no	—
37	C₃×C₃₁⋊C₃	—	9	1	no	—
38	C₉×C₃₁	—	3	1	no	—
39	C₉×C₃₁	—	3	1	no	—
40	C₉×C₃₁	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
41	C₉×C₃₁	AΣL(1,𝔽₂₇)	1	2	yes	—
42	C₃₁⋊C₉	—	3	6	no	—
43	C₃×#737	—	31	1	no	—
44	C₃²⋊C₂₇	—	31	2	no	—
45	C₃²⋊C₂₇	C₃₁⋊C₃	1	1	yes	characteristic, complemented
46	C₃×C₃₁	—	9	1	no	—
47	C₃×C₃₁	#6866	1	1	yes	characteristic
48	C₃×C₃₁	#6872	1	2	yes	—
49	C₃×C₃₁	#6872	1	1	yes	characteristic
50	C₃₁⋊C₃	—	9	3	no	—
51	#737	—	93	3	no	—
52	#737	—	93	6	no	—
53	C₃×3₋¹⁺²	—	31	2	no	—
54	C₃×3₋¹⁺²	—	93	1	no	—
55	C₃×3₋¹⁺²	—	93	1	no	—
56	C₃×H(3,𝔽₃)	—	31	1	no	—
57	C₃²×C₉	—	93	1	no	—
58	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
59	C₃×C₂₇	—	3	1	no	—
60	C₃×C₂₇	—	93	2	no	—
61	C₃₁	#10624456	1	1	yes	characteristic, complemented, Sylow
62	C₃³	—	93	1	no	—
63	C₃³	C₉×C₃₁⋊C₃	1	1	yes	characteristic
64	3₋¹⁺²	—	93	9	no	—
65	3₋¹⁺²	—	93	9	no	—
66	3₋¹⁺²	—	93	6	no	—
67	H(3,𝔽₃)	—	93	3	no	—
68	C₃×C₉	—	3	1	no	—
69	C₃×C₉	—	3	2	no	—
70	C₃×C₉	—	93	2	no	—
71	C₃×C₉	—	93	3	no	—
72	C₃×C₉	—	93	6	no	—
73	C₃×C₉	#11719686	1	1	yes	characteristic
74	C₃×C₉	#11719687	1	1	yes	characteristic
75	C₃×C₉	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
76	C₂₇	—	9	1	no	—
77	C₂₇	—	279	2	no	—
78	C₃²	—	3	1	no	—
79	C₃²	—	3	2	no	—
80	C₃²	—	3	1	no	—
81	C₃²	—	279	2	no	—
82	C₃²	—	279	1	no	—
83	C₃²	—	93	3	no	—
84	C₃²	(C₃×C₃₁⋊C₃)⋊C₉	1	1	yes	characteristic
85	C₉	—	3	1	no	—
86	C₉	—	3	1	no	—
87	C₉	—	93	6	no	—
88	C₉	#14578893	1	1	yes	characteristic
89	C₉	#14578895	1	2	yes	—
90	C₃	—	9	1	no	—
91	C₃	—	279	3	no	—
92	C₃	#21419225	1	1	yes	characteristic
93	C₃	#21419240	1	2	yes	—
94	C₃	#21419242	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₉	C₃²⋊(C₃₁⋊C₃)
commutator	C₃³×C₃₁	C₃×C₉
Frattini	C₃²×C₉	C₃×C₃₁⋊C₃
Fitting	C₃₁×C₃²⋊C₂₇	C₃
socle	C₃²×C₃₁	C₃²⋊C₉
prime radical	C₃×C₃²⋊(C₃₁⋊C₃)	C₉
cosocle	C₃²×C₉×C₃₁	C₃²
hypercenter	C₃²⋊C₂₇	C₃₁⋊C₃
hypocenter	C₃₁	#10624456
LCS	G ▷ C₃³×C₃₁ ▷ C₃²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ C₃²⋊C₂₇

Complements

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

N	H ≅ G/N	type
2. C₃²⋊(C₃₁⋊C₂₇)	C₃	semidirect
3. C₃₁×C₃²⋊C₂₇	C₃	semidirect
45. C₃²⋊C₂₇	C₃₁⋊C₃	semidirect
61. C₃₁	#10624456	semidirect

Maximal subgroups

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

subgroup	index
C₃×#14578912	3
C₃²⋊(C₃₁⋊C₂₇)	3
C₃₁×C₃²⋊C₂₇	3
#10624456	31