Subgroups — #28357047

Order 13851.

1178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(3,𝔽₃)×#4296	C₃	1	27	yes	maximal, complemented
2	C₃×C₃²⋊#4296	C₃	1	12	yes	maximal, complemented
3	C₃×H(3,𝔽₃)×C₁₉⋊C₃	C₃	1	1	yes	maximal, characteristic
4	C₃²×#4296	C₃²	1	36	yes	complemented
5	C₃²⋊#4296	C₃²	1	9	yes	—
6	C₃²⋊#4296	C₃²	1	72	yes	complemented
7	C₃³×C₁₉⋊C₃	C₃²	1	4	yes	—
8	C₃×C₃²⋊(C₁₉⋊C₃)	C₉	1	8	yes	complemented
9	H(3,𝔽₃)×C₁₉⋊C₃	C₉	1	18	yes	complemented
10	H(3,𝔽₃)×C₁₉⋊C₃	C₃²	1	9	yes	—
11	C₃×C₁₉×H(3,𝔽₃)	C₉	1	1	yes	characteristic, complemented
12	#10624862	—	19	1	no	maximal, Sylow
13	C₃³×C₁₉	C₃×C₉	1	4	yes	complemented
14	C₃²×C₁₉⋊C₃	—	9	4	no	—
15	C₃²×C₁₉⋊C₃	—	9	8	no	—
16	C₃²×C₁₉⋊C₃	C₃×C₉	1	24	yes	complemented
17	C₃²×C₁₉⋊C₃	C₃×C₉	1	8	yes	complemented
18	C₃²×C₁₉⋊C₃	C₃³	1	12	yes	—
19	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
20	C₃×#4296	—	9	36	no	—
21	C₃×#4296	C₃³	1	27	yes	—
22	C₁₉×H(3,𝔽₃)	C₃×C₉	1	9	yes	complemented
23	C₃²⋊(C₁₉⋊C₃)	C₃×C₉	1	72	yes	complemented
24	C₃²×H(3,𝔽₃)	—	19	1	no	—
25	C₉×H(3,𝔽₃)	—	19	27	no	—
26	C₃×C₃²⋊C₉	—	19	12	no	—
27	C₃²×C₁₉	—	9	4	no	—
28	C₃²×C₁₉	C₃²×C₉	1	12	yes	—
29	C₃²×C₁₉	C₃²×C₉	1	1	yes	characteristic
30	C₃×C₁₉⋊C₃	—	3	12	no	—
31	C₃×C₁₉⋊C₃	—	3	1	no	—
32	C₃×C₁₉⋊C₃	—	3	2	no	—
33	C₃×C₁₉⋊C₃	—	3	24	no	—
34	C₃×C₁₉⋊C₃	—	9	24	no	—
35	C₃×C₁₉⋊C₃	—	9	8	no	—
36	C₃×C₁₉⋊C₃	C₃²×C₉	1	24	yes	—
37	C₃×C₁₉⋊C₃	C₃²×C₉	1	2	yes	—
38	C₃×C₁₉⋊C₃	C₃⁴	1	1	yes	characteristic
39	#4296	—	3	27	no	—
40	C₃⁴	—	19	4	no	—
41	C₃×H(3,𝔽₃)	—	19	8	no	—
42	C₃×H(3,𝔽₃)	—	19	18	no	—
43	C₃×H(3,𝔽₃)	—	19	9	no	—
44	C₃×H(3,𝔽₃)	#4296	1	1	yes	characteristic, complemented
45	C₃²×C₉	—	19	36	no	—
46	C₃²⋊C₉	—	19	9	no	—
47	C₃²⋊C₉	—	19	72	no	—
48	C₃×C₁₉	—	3	1	no	—
49	C₃×C₁₉	—	3	12	no	—
50	C₃×C₁₉	C₃³×C₉	1	1	yes	characteristic
51	C₁₉⋊C₃	—	3	2	no	—
52	C₁₉⋊C₃	—	3	24	no	—
53	C₁₉⋊C₃	H(5,𝔽₃)	1	1	yes	characteristic
54	C₁₉⋊C₃	3₋¹⁺⁴	1	2	yes	—
55	C₃³	—	19	1	no	—
56	C₃³	—	19	12	no	—
57	C₃³	—	19	8	no	—
58	C₃³	—	19	24	no	—
59	C₃³	—	171	4	no	—
60	C₃³	—	171	8	no	—
61	C₃³	C₃×#4296	1	4	yes	complemented
62	H(3,𝔽₃)	—	19	72	no	—
63	H(3,𝔽₃)	C₃×#4296	1	9	yes	complemented
64	C₃×C₉	—	19	27	no	—
65	C₃×C₉	—	171	36	no	—
66	C₁₉	#10624862	1	1	yes	characteristic, complemented, Sylow
67	C₃²	—	9	4	no	—
68	C₃²	—	19	1	no	—
69	C₃²	—	19	24	no	—
70	C₃²	—	19	2	no	—
71	C₃²	—	57	12	no	—
72	C₃²	—	57	1	no	—
73	C₃²	—	57	2	no	—
74	C₃²	—	57	24	no	—
75	C₃²	—	171	8	no	—
76	C₃²	—	171	24	no	—
77	C₃²	C₃²×#4296	1	12	yes	—
78	C₃²	C₃²×#4296	1	1	yes	characteristic
79	C₉	—	57	27	no	—
80	C₃	—	3	1	no	—
81	C₃	—	3	12	no	—
82	C₃	—	19	1	no	—
83	C₃	—	19	2	no	—
84	C₃	—	57	2	no	—
85	C₃	—	57	24	no	—
86	C₃	C₃³×#4296	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃	C₃³×#4296
commutator, socle	C₃×C₁₉	C₃³×C₉
Fitting	C₃×C₁₉×H(3,𝔽₃)	C₉
prime radical	C₃×H(3,𝔽₃)×C₁₉⋊C₃	C₃
cosocle	C₃×C₁₉⋊C₃	C₃⁴
hypercenter	C₃×H(3,𝔽₃)	#4296
hypocenter	C₁₉	#10624862
LCS	G ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. H(3,𝔽₃)×#4296	C₃	semidirect
2. C₃×C₃²⋊#4296	C₃	semidirect
4. C₃²×#4296	C₃²	semidirect
6. C₃²⋊#4296	C₃²	semidirect
8. C₃×C₃²⋊(C₁₉⋊C₃)	C₉	semidirect
9. H(3,𝔽₃)×C₁₉⋊C₃	C₉	semidirect
11. C₃×C₁₉×H(3,𝔽₃)	C₉	semidirect
13. C₃³×C₁₉	C₃×C₉	semidirect
16. C₃²×C₁₉⋊C₃	C₃×C₉	semidirect
17. C₃²×C₁₉⋊C₃	C₃×C₉	semidirect
22. C₁₉×H(3,𝔽₃)	C₃×C₉	semidirect
23. C₃²⋊(C₁₉⋊C₃)	C₃×C₉	semidirect
44. C₃×H(3,𝔽₃)	#4296	semidirect
61. C₃³	C₃×#4296	semidirect
63. H(3,𝔽₃)	C₃×#4296	semidirect
66. C₁₉	#10624862	semidirect

Maximal subgroups

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

subgroup	index
H(3,𝔽₃)×#4296	3
C₃×C₃²⋊#4296	3
C₃×H(3,𝔽₃)×C₁₉⋊C₃	3
#10624862	19