Subgroups — #21419394

Order 7533.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	3₋¹⁺²×C₃₁⋊C₃	C₃	1	2	yes	maximal, complemented
2	C₃×C₃²⋊(C₃₁⋊C₃)	C₃	1	1	yes	maximal, characteristic, complemented
3	H(3,𝔽₃)×C₃₁⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
4	#14578895	C₃	1	4	yes	maximal, complemented
5	#14578894	C₃	1	2	yes	maximal, complemented
6	#14578893	C₃	1	2	yes	maximal, complemented
7	C₃₁×AΣL(1,𝔽₂₇)	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃³×C₃₁	C₃²	1	1	yes	characteristic, complemented
9	C₃²×C₃₁⋊C₃	—	3	1	no	—
10	C₃²×C₃₁⋊C₃	C₃²	1	1	yes	characteristic
11	C₃²×C₃₁⋊C₃	C₃²	1	2	yes	complemented
12	3₋¹⁺²×C₃₁	C₃²	1	2	yes	complemented
13	H(3,𝔽₃)×C₃₁	C₃²	1	1	yes	characteristic
14	C₃²⋊(C₃₁⋊C₃)	—	3	2	no	—
15	C₃²⋊(C₃₁⋊C₃)	—	3	2	no	—
16	C₃²⋊(C₃₁⋊C₃)	—	3	1	no	—
17	C₃²⋊(C₃₁⋊C₃)	C₃²	1	2	yes	—
18	#11719689	C₃²	1	4	yes	complemented
19	#11719687	—	3	2	no	—
20	#11719686	—	3	2	no	—
21	C₉×C₃₁⋊C₃	—	3	2	no	—
22	C₃²×C₃₁	—	3	1	no	—
23	C₃²×C₃₁	—	3	1	no	—
24	C₃²×C₃₁	—	9	1	no	—
25	C₃²×C₃₁	C₃³	1	1	yes	characteristic
26	C₃×C₃₁⋊C₃	—	3	2	no	—
27	C₃×C₃₁⋊C₃	—	3	2	no	—
28	C₃×C₃₁⋊C₃	—	9	1	no	—
29	C₃×C₃₁⋊C₃	—	9	3	no	—
30	C₃×C₃₁⋊C₃	—	9	2	no	—
31	C₃×C₃₁⋊C₃	H(3,𝔽₃)	1	2	yes	—
32	C₃×C₃₁⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
33	C₉×C₃₁	—	3	2	no	—
34	C₃₁⋊C₉	—	3	4	no	—
35	#6909	—	31	1	no	maximal, Sylow
36	C₃×C₃₁	—	3	1	no	—
37	C₃×C₃₁	—	9	1	no	—
38	C₃×C₃₁	—	9	1	no	—
39	C₃×C₃₁	C₃×H(3,𝔽₃)	1	1	yes	characteristic
40	C₃₁⋊C₃	—	3	1	no	—
41	C₃₁⋊C₃	—	3	2	no	—
42	C₃₁⋊C₃	—	9	2	no	—
43	C₃₁⋊C₃	—	9	2	no	—
44	C₃×3₋¹⁺²	—	31	2	no	—
45	C₃×H(3,𝔽₃)	—	31	1	no	—
46	C₃×H(3,𝔽₃)	—	31	1	no	—
47	AΣL(1,𝔽₂₇)	—	31	2	no	—
48	AΣL(1,𝔽₂₇)	—	31	4	no	—
49	AΣL(1,𝔽₂₇)	—	31	2	no	—
50	AΣL(1,𝔽₂₇)	C₃₁⋊C₃	1	1	yes	characteristic, complemented
51	C₃₁	#6909	1	1	yes	characteristic, complemented, Sylow
52	C₃³	—	31	1	no	—
53	C₃³	—	31	2	no	—
54	C₃³	—	93	1	no	—
55	C₃³	C₃×C₃₁⋊C₃	1	1	yes	characteristic, complemented
56	3₋¹⁺²	—	31	4	no	—
57	3₋¹⁺²	—	93	2	no	—
58	3₋¹⁺²	—	93	2	no	—
59	3₋¹⁺²	C₃×C₃₁⋊C₃	1	2	yes	complemented
60	H(3,𝔽₃)	—	31	2	no	—
61	H(3,𝔽₃)	—	93	2	no	—
62	H(3,𝔽₃)	—	93	1	no	—
63	H(3,𝔽₃)	—	93	2	no	—
64	H(3,𝔽₃)	C₃×C₃₁⋊C₃	1	1	yes	characteristic
65	C₃×C₉	—	93	2	no	—
66	C₃²	—	3	1	no	—
67	C₃²	—	3	1	no	—
68	C₃²	—	9	1	no	—
69	C₃²	—	279	2	no	—
70	C₃²	—	279	1	no	—
71	C₃²	—	279	3	no	—
72	C₃²	—	31	2	no	—
73	C₃²	—	31	1	no	—
74	C₃²	—	93	2	no	—
75	C₃²	—	93	2	no	—
76	C₃²	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
77	C₉	—	3	2	no	—
78	C₉	—	93	4	no	—
79	C₃	—	3	1	no	—
80	C₃	—	9	1	no	—
81	C₃	—	9	1	no	—
82	C₃	—	279	2	no	—
83	C₃	—	279	2	no	—
84	C₃	—	93	1	no	—
85	C₃	—	93	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	#6909
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. 3₋¹⁺²×C₃₁⋊C₃	C₃	semidirect
2. C₃×C₃²⋊(C₃₁⋊C₃)	C₃	semidirect
3. H(3,𝔽₃)×C₃₁⋊C₃	C₃	semidirect
4. #14578895	C₃	semidirect
5. #14578894	C₃	semidirect
6. #14578893	C₃	semidirect
7. C₃₁×AΣL(1,𝔽₂₇)	C₃	semidirect
8. C₃³×C₃₁	C₃²	semidirect
11. C₃²×C₃₁⋊C₃	C₃²	semidirect
12. 3₋¹⁺²×C₃₁	C₃²	semidirect
18. #11719689	C₃²	semidirect
50. AΣL(1,𝔽₂₇)	C₃₁⋊C₃	semidirect
51. C₃₁	#6909	semidirect
55. C₃³	C₃×C₃₁⋊C₃	semidirect
59. 3₋¹⁺²	C₃×C₃₁⋊C₃	semidirect