Subgroups — #21419411

Order 7533.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14578913	C₃	1	2	yes	maximal, complemented
2	#14578911	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×#11719689	C₃	1	1	yes	maximal, characteristic, complemented
4	#14578901	C₃	1	2	yes	maximal, complemented
5	#14578899	C₃	1	1	yes	maximal, characteristic, complemented
6	#14578898	C₃	1	2	yes	maximal, complemented
7	#14578896	C₃	1	1	yes	maximal, characteristic, complemented
8	#14578894	C₃	1	2	yes	maximal, complemented
9	C₃₁×AΣL(1,𝔽₂₇)	C₃	1	1	yes	maximal, characteristic, complemented
10	C₃³×C₃₁	C₃²	1	1	yes	characteristic
11	3₋¹⁺²×C₃₁	C₃²	1	2	yes	—
12	H(3,𝔽₃)×C₃₁	C₃²	1	1	yes	characteristic
13	C₃²⋊(C₃₁⋊C₃)	C₃²	1	2	yes	—
14	#11719689	—	3	2	no	—
15	#11719689	—	3	1	no	—
16	#11719689	—	3	2	no	—
17	#11719689	C₃²	1	2	yes	—
18	#11719689	C₃²	1	2	yes	—
19	C₃×C₃₁⋊C₉	—	3	1	no	—
20	C₃×C₃₁⋊C₉	C₃²	1	1	yes	characteristic
21	C₃×C₃₁⋊C₉	C₃²	1	1	yes	characteristic
22	C₃×C₃₁⋊C₉	C₃²	1	1	yes	characteristic
23	#11719687	—	3	2	no	—
24	#11719686	—	3	2	no	—
25	C₉×C₃₁⋊C₃	—	3	2	no	—
26	C₃²×C₃₁	—	3	1	no	—
27	C₃²×C₃₁	—	3	1	no	—
28	C₃²×C₃₁	—	9	1	no	—
29	C₃²×C₃₁	C₃³	1	1	yes	characteristic
30	C₃×C₃₁⋊C₃	—	3	2	no	—
31	C₉×C₃₁	—	3	2	no	—
32	C₃₁⋊C₉	—	3	2	no	—
33	C₃₁⋊C₉	—	3	2	no	—
34	C₃₁⋊C₉	—	3	1	no	—
35	C₃₁⋊C₉	—	3	1	no	—
36	C₃₁⋊C₉	H(3,𝔽₃)	1	2	yes	—
37	C₃₁⋊C₉	H(3,𝔽₃)	1	1	yes	characteristic
38	#6913	—	31	1	no	maximal, Sylow
39	C₃×C₃₁	—	3	1	no	—
40	C₃×C₃₁	—	9	1	no	—
41	C₃×C₃₁	—	9	1	no	—
42	C₃×C₃₁	C₃×H(3,𝔽₃)	1	1	yes	characteristic
43	C₃₁⋊C₃	—	9	2	no	—
44	#737	—	31	2	no	—
45	#737	—	31	1	no	—
46	C₃×3₋¹⁺²	—	31	1	no	—
47	#733	—	31	2	no	—
48	#732	—	31	1	no	—
49	#731	—	31	2	no	—
50	#731	—	31	1	no	—
51	AΣL(1,𝔽₂₇)	—	31	2	no	—
52	AΣL(1,𝔽₂₇)	C₃₁⋊C₃	1	1	yes	characteristic, complemented
53	C₃₁	#6913	1	1	yes	characteristic, complemented, Sylow
54	C₃³	C₃×C₃₁⋊C₃	1	1	yes	characteristic
55	3₋¹⁺²	—	31	2	no	—
56	3₋¹⁺²	—	31	2	no	—
57	3₋¹⁺²	—	93	2	no	—
58	3₋¹⁺²	—	93	2	no	—
59	3₋¹⁺²	—	93	2	no	—
60	3₋¹⁺²	—	93	1	no	—
61	3₋¹⁺²	—	93	2	no	—
62	3₋¹⁺²	C₃×C₃₁⋊C₃	1	2	yes	—
63	H(3,𝔽₃)	—	31	2	no	—
64	H(3,𝔽₃)	C₃×C₃₁⋊C₃	1	1	yes	characteristic
65	C₃×C₉	—	31	1	no	—
66	C₃×C₉	—	31	1	no	—
67	C₃×C₉	—	31	1	no	—
68	C₃×C₉	—	93	2	no	—
69	C₃×C₉	—	93	1	no	—
70	C₃²	—	3	1	no	—
71	C₃²	—	3	1	no	—
72	C₃²	—	9	1	no	—
73	C₃²	—	93	2	no	—
74	C₃²	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
75	C₉	—	3	2	no	—
76	C₉	—	31	2	no	—
77	C₉	—	31	1	no	—
78	C₉	—	93	1	no	—
79	C₉	—	93	1	no	—
80	C₉	—	93	2	no	—
81	C₉	—	93	2	no	—
82	C₃	—	3	1	no	—
83	C₃	—	9	1	no	—
84	C₃	—	9	1	no	—
85	C₃	—	279	2	no	—
86	C₃	H(3,𝔽₃)×C₃₁⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃	H(3,𝔽₃)×C₃₁⋊C₃
commutator	C₃²×C₃₁	C₃³
Frattini	C₃²	C₃²×C₃₁⋊C₃
Fitting	C₃₁×AΣL(1,𝔽₂₇)	C₃
socle	C₃×C₃₁	C₃×H(3,𝔽₃)
hypercenter	AΣL(1,𝔽₂₇)	C₃₁⋊C₃
hypocenter	C₃₁	#6913
LCS	G ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃ ◁ C₃² ◁ AΣL(1,𝔽₂₇)

Complements

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

N	H ≅ G/N	type
1. #14578913	C₃	semidirect
2. #14578911	C₃	semidirect
3. C₃×#11719689	C₃	semidirect
4. #14578901	C₃	semidirect
5. #14578899	C₃	semidirect
6. #14578898	C₃	semidirect
7. #14578896	C₃	semidirect
8. #14578894	C₃	semidirect
9. C₃₁×AΣL(1,𝔽₂₇)	C₃	semidirect
52. AΣL(1,𝔽₂₇)	C₃₁⋊C₃	semidirect
53. C₃₁	#6913	semidirect