Subgroups — #28356726

Order 13851.

164 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	(C₁₉⋊C₃)⋊(C₂₇⋊C₃)	C₃	1	2	yes	maximal
2	#17987589	C₃	1	1	yes	maximal, characteristic
3	C₃×C₂₇×C₁₉⋊C₃	C₃	1	1	yes	maximal, characteristic
4	#13095344	—	3	2	no	—
5	#13095343	—	3	2	no	—
6	#13095342	C₉	1	6	yes	complemented
7	(C₁₉⋊C₃)⋊C₂₇	—	3	2	no	—
8	#13095339	—	3	1	no	—
9	#13095338	—	3	1	no	—
10	C₃²⋊#4296	C₉	1	3	yes	—
11	C₉×#4296	—	3	1	no	—
12	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
13	C₃×#10587018	C₉	1	2	yes	complemented
14	C₂₇×C₁₉⋊C₃	—	3	1	no	—
15	C₂₇×C₁₉⋊C₃	—	3	2	no	—
16	C₃×C₁₉×C₂₇	C₉	1	1	yes	characteristic, complemented
17	#10624468	—	19	1	no	maximal, Sylow
18	C₃²×C₁₉⋊C₃	C₃×C₉	1	1	yes	characteristic
19	C₃×#4296	—	3	3	no	—
20	C₃×C₉×C₁₉	C₃×C₉	1	1	yes	characteristic
21	C₃×#4294	C₃×C₉	1	2	yes	—
22	C₉×C₁₉⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
23	C₉×C₁₉⋊C₃	3₋¹⁺²	1	2	yes	—
24	C₉×C₁₉⋊C₃	3₋¹⁺²	1	2	yes	—
25	C₉×C₁₉⋊C₃	3₋¹⁺²	1	2	yes	—
26	C₉×C₁₉⋊C₃	3₋¹⁺²	1	1	yes	characteristic
27	C₉×C₁₉⋊C₃	3₋¹⁺²	1	1	yes	characteristic
28	#10587020	—	3	6	no	—
29	C₁₉×C₂₇	—	3	1	no	—
30	#10587018	—	3	2	no	—
31	C₃×C₂₇⋊C₃	—	19	2	no	—
32	C₃²×C₂₇	—	19	1	no	—
33	#6887	—	19	1	no	—
34	C₃²×C₁₉	C₉²	1	1	yes	characteristic
35	C₃×C₁₉⋊C₃	—	3	3	no	—
36	C₃×C₁₉⋊C₃	C₃²⋊C₉	1	1	yes	characteristic
37	C₃×C₁₉⋊C₃	C₉⋊C₉	1	2	yes	—
38	#4296	—	9	3	no	—
39	C₉×C₁₉	C₃²⋊C₉	1	1	yes	characteristic
40	C₉×C₁₉	C₉⋊C₉	1	1	yes	characteristic
41	C₉×C₁₉	C₉⋊C₉	1	1	yes	characteristic
42	#4294	C₃²⋊C₉	1	2	yes	—
43	#4294	C₉⋊C₉	1	2	yes	—
44	#4294	C₉⋊C₉	1	2	yes	—
45	C₃²×C₉	—	19	1	no	—
46	C₂₇⋊C₃	—	19	6	no	—
47	C₂₇⋊C₃	—	57	2	no	—
48	C₂₇⋊C₃	—	57	2	no	—
49	C₃×C₂₇	—	19	2	no	—
50	C₃×C₂₇	—	57	1	no	—
51	C₃×C₂₇	—	57	2	no	—
52	C₃×C₂₇	—	57	2	no	—
53	C₃×C₂₇	#4296	1	1	yes	characteristic, complemented
54	C₉⋊C₉	—	57	1	no	—
55	C₉⋊C₉	—	57	1	no	—
56	C₃²⋊C₉	—	19	3	no	—
57	C₉²	—	57	1	no	—
58	C₃×C₁₉	—	3	1	no	—
59	C₃×C₁₉	#6855	1	1	yes	characteristic
60	C₁₉⋊C₃	—	3	2	no	—
61	C₁₉⋊C₃	#6873	1	3	yes	—
62	C₃³	—	19	1	no	—
63	C₃×C₉	—	19	2	no	—
64	C₃×C₉	—	19	1	no	—
65	C₃×C₉	—	19	2	no	—
66	C₃×C₉	—	19	1	no	—
67	C₃×C₉	—	19	2	no	—
68	C₃×C₉	—	19	1	no	—
69	C₃×C₉	—	19	2	no	—
70	C₃×C₉	—	57	3	no	—
71	C₃×C₉	C₃×#4296	1	1	yes	characteristic
72	C₂₇	—	3	1	no	—
73	C₂₇	—	57	6	no	—
74	C₂₇	—	57	2	no	—
75	C₁₉	#10624468	1	1	yes	characteristic, complemented, Sylow
76	C₃²	—	19	1	no	—
77	C₃²	—	19	2	no	—
78	C₃²	—	57	3	no	—
79	C₃²	C₉×#4296	1	1	yes	characteristic
80	C₉	—	19	2	no	—
81	C₉	—	19	2	no	—
82	C₉	—	19	2	no	—
83	C₉	—	171	3	no	—
84	C₉	C₃²⋊#4296	1	1	yes	characteristic
85	C₉	#13095338	1	1	yes	characteristic
86	C₉	#13095339	1	1	yes	characteristic
87	C₃	—	3	1	no	—
88	C₃	—	19	3	no	—
89	C₃	—	57	2	no	—
90	C₃	(C₃×#4296)⋊C₉	1	1	yes	characteristic

	subgroup	G/H
center	C₉	C₃²⋊#4296
commutator	C₃²×C₁₉	C₉²
Frattini	C₃×C₉	C₃×#4296
Fitting	C₃×C₁₉×C₂₇	C₉
socle	C₃×C₁₉	#6855
prime radical	C₃²×C₁₉⋊C₃	C₃×C₉
cosocle	C₃×C₉×C₁₉⋊C₃	C₃²
hypercenter	C₃×C₂₇	#4296
hypocenter	C₁₉	#10624468
LCS	G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₉ ◁ C₃×C₉ ◁ C₃×C₂₇

Complements

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

N	H ≅ G/N	type
6. #13095342	C₉	semidirect
13. C₃×#10587018	C₉	semidirect
16. C₃×C₁₉×C₂₇	C₉	semidirect
53. C₃×C₂₇	#4296	semidirect
75. C₁₉	#10624468	semidirect

Maximal subgroups

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

subgroup	index
(C₁₉⋊C₃)⋊(C₂₇⋊C₃)	3
#17987589	3
C₃×C₂₇×C₁₉⋊C₃	3
#10624468	19