Subgroups — #28356981

Order 13851.

292 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×(C₁₉⋊C₃)⋊C₂₇	C₃	1	3	yes	maximal, complemented
2	C₃²×#10587020	C₃	1	1	yes	maximal, characteristic, complemented
3	#17987557	C₃	1	6	yes	maximal, complemented
4	C₃²⋊#10587020	C₃	1	2	yes	maximal, complemented
5	#17987406	C₃	1	1	yes	maximal, characteristic
6	(C₁₉⋊C₃)⋊C₂₇	—	3	9	no	—
7	C₃×#10587020	—	3	3	no	—
8	C₃×#10587020	C₃²	1	2	yes	—
9	C₃×#10587020	C₃²	1	6	yes	—
10	C₃×#10587020	C₃²	1	1	yes	characteristic
11	C₃×C₃²⋊(C₁₉⋊C₃)	C₉	1	1	yes	characteristic
12	C₃²×#4294	C₃²	1	1	yes	characteristic
13	C₃×C₉×C₁₉⋊C₃	C₃²	1	3	yes	—
14	C₃²⋊#4294	C₉	1	1	yes	characteristic
15	#13095303	C₉	1	6	yes	—
16	C₁₉×C₃²⋊C₉	C₉	1	1	yes	characteristic
17	C₂₇×H(3,𝔽₃)	—	19	1	no	maximal, Sylow
18	C₃³×C₁₉	C₃×C₉	1	1	yes	characteristic
19	C₃²×C₁₉⋊C₃	C₃×C₉	1	3	yes	—
20	C₃×C₉×C₁₉	C₃×C₉	1	3	yes	—
21	C₃²⋊(C₁₉⋊C₃)	C₂₇	1	9	yes	complemented
22	C₃×#4294	—	3	1	no	—
23	C₃×#4294	—	3	2	no	—
24	C₃×#4294	C₃×C₉	1	2	yes	—
25	C₃×#4294	C₃×C₉	1	3	yes	—
26	C₃×#4294	C₃³	1	1	yes	characteristic
27	C₉×C₁₉⋊C₃	—	3	3	no	—
28	C₉×C₁₉⋊C₃	—	3	6	no	—
29	#10587020	—	3	2	no	—
30	#10587020	—	3	6	no	—
31	#10587020	H(3,𝔽₃)	1	3	yes	complemented
32	C₃²×C₂₇	—	19	1	no	—
33	C₃²×C₂₇	—	19	3	no	—
34	C₉×H(3,𝔽₃)	—	19	1	no	—
35	C₃²⋊C₂₇	—	19	2	no	—
36	C₃²⋊C₂₇	—	19	6	no	—
37	C₃²×C₁₉	—	3	1	no	—
38	C₃²×C₁₉	—	3	2	no	—
39	C₃²×C₁₉	C₃×C₂₇	1	3	yes	complemented
40	C₃²×C₁₉	C₃²×C₉	1	1	yes	characteristic
41	C₃×C₁₉⋊C₃	—	3	3	no	—
42	C₃×C₁₉⋊C₃	—	3	6	no	—
43	C₃×C₁₉⋊C₃	C₃×C₂₇	1	9	yes	complemented
44	C₉×C₁₉	—	3	3	no	—
45	#4294	—	3	3	no	—
46	#4294	—	3	2	no	—
47	#4294	C₃×H(3,𝔽₃)	1	1	yes	characteristic
48	#4294	#737	1	2	yes	—
49	C₃×H(3,𝔽₃)	—	19	1	no	—
50	C₃²×C₉	—	19	1	no	—
51	C₃²×C₉	—	19	3	no	—
52	C₃×C₂₇	—	19	1	no	—
53	C₃×C₂₇	—	19	2	no	—
54	C₃×C₂₇	—	19	6	no	—
55	C₃×C₂₇	—	57	3	no	—
56	C₃×C₂₇	—	57	9	no	—
57	C₃²⋊C₉	—	19	1	no	—
58	C₃²⋊C₉	—	19	6	no	—
59	C₃²⋊C₉	#4296	1	1	yes	characteristic
60	C₃×C₁₉	—	3	3	no	—
61	C₃×C₁₉	C₉×H(3,𝔽₃)	1	1	yes	characteristic
62	C₃×C₁₉	C₃²×C₂₇	1	1	yes	characteristic
63	C₃×C₁₉	#6903	1	2	yes	—
64	C₁₉⋊C₃	—	3	9	no	—
65	C₃³	—	19	3	no	—
66	C₃³	C₃×#4296	1	1	yes	characteristic
67	H(3,𝔽₃)	—	19	9	no	—
68	C₃×C₉	—	19	1	no	—
69	C₃×C₉	—	19	2	no	—
70	C₃×C₉	—	19	3	no	—
71	C₃×C₉	—	57	3	no	—
72	C₃×C₉	—	57	6	no	—
73	C₃×C₉	—	57	1	no	—
74	C₃×C₉	—	57	2	no	—
75	C₃×C₉	C₃×#4296	1	3	yes	—
76	C₂₇	—	19	3	no	—
77	C₂₇	—	57	2	no	—
78	C₂₇	—	57	6	no	—
79	C₁₉	C₂₇×H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
80	C₃²	—	3	1	no	—
81	C₃²	—	3	2	no	—
82	C₃²	—	19	9	no	—
83	C₃²	—	57	3	no	—
84	C₃²	—	57	6	no	—
85	C₃²	(C₁₉⋊C₃)⋊C₂₇	1	3	yes	complemented
86	C₃²	C₃²×#4296	1	1	yes	characteristic
87	C₉	—	3	3	no	—
88	C₉	—	19	1	no	—
89	C₉	—	19	2	no	—
90	C₉	—	57	3	no	—
91	C₉	—	57	2	no	—
92	C₃	—	3	3	no	—
93	C₃	—	57	9	no	—
94	C₃	H(3,𝔽₃)×#4296	1	1	yes	characteristic
95	C₃	C₃×(C₁₉⋊C₃)⋊C₂₇	1	1	yes	characteristic
96	C₃	#17987599	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×#4296
commutator	C₃×C₁₉	C₃²×C₂₇
Fitting	C₁₉×C₃²⋊C₉	C₉
socle	C₃²×C₁₉	C₃²×C₉
prime radical	C₃×C₃²⋊(C₁₉⋊C₃)	C₉
cosocle	C₃×#4294	C₃³
hypercenter	C₃²⋊C₉	#4296
hypocenter	C₁₉	C₂₇×H(3,𝔽₃)
LCS	G ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃²⋊C₉

Complements

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

N	H ≅ G/N	type
1. C₃×(C₁₉⋊C₃)⋊C₂₇	C₃	semidirect
2. C₃²×#10587020	C₃	semidirect
3. #17987557	C₃	semidirect
4. C₃²⋊#10587020	C₃	semidirect
21. C₃²⋊(C₁₉⋊C₃)	C₂₇	semidirect
31. #10587020	H(3,𝔽₃)	semidirect
39. C₃²×C₁₉	C₃×C₂₇	semidirect
43. C₃×C₁₉⋊C₃	C₃×C₂₇	semidirect
79. C₁₉	C₂₇×H(3,𝔽₃)	semidirect
85. C₃²	(C₁₉⋊C₃)⋊C₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₃×(C₁₉⋊C₃)⋊C₂₇	3
C₃²×#10587020	3
#17987557	3
C₃²⋊#10587020	3
#17987406	3
C₂₇×H(3,𝔽₃)	19