Subgroups — #28356982

Order 13851.

256 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#13095342	C₃	1	3	yes	maximal, complemented
2	C₃²×#10587020	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃²⋊#10587020	C₃	1	2	yes	maximal, complemented
4	C₃²⋊#10587020	C₃	1	6	yes	maximal, complemented
5	H(3,𝔽₃)×#4294	C₃	1	1	yes	maximal, characteristic
6	#13095342	—	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	C₃²×#4294	C₃²	1	1	yes	characteristic
12	C₃²×#4294	C₃²	1	3	yes	—
13	C₃²⋊#4294	C₉	1	6	yes	—
14	C₃²⋊#4294	C₉	1	2	yes	—
15	C₃×C₁₉×H(3,𝔽₃)	C₉	1	1	yes	characteristic
16	C₂₇⋊H(3,𝔽₃)	—	19	1	no	maximal, Sylow
17	C₃³×C₁₉	C₃×C₉	1	3	yes	—
18	C₃³×C₁₉	C₃×C₉	1	1	yes	characteristic
19	C₁₉×H(3,𝔽₃)	—	3	3	no	—
20	C₃×#4294	—	3	1	no	—
21	C₃×#4294	—	3	3	no	—
22	C₃×#4294	—	3	2	no	—
23	C₃×#4294	—	3	6	no	—
24	C₃×#4294	C₃×C₉	1	2	yes	—
25	C₃×#4294	C₃×C₉	1	6	yes	—
26	C₃×#4294	C₃³	1	1	yes	characteristic
27	#10587020	—	3	2	no	—
28	#10587020	—	3	6	no	—
29	#10587020	H(3,𝔽₃)	1	3	yes	complemented
30	C₃×C₂₇⋊C₃	—	19	3	no	—
31	C₃²×C₂₇	—	19	1	no	—
32	C₉×H(3,𝔽₃)	—	19	1	no	—
33	C₃²⋊C₂₇	—	19	6	no	—
34	C₃²⋊C₂₇	—	19	2	no	—
35	C₃²×C₁₉	—	3	1	no	—
36	C₃²×C₁₉	—	3	2	no	—
37	C₃²×C₁₉	—	3	3	no	—
38	C₃²×C₁₉	—	3	3	no	—
39	C₃²×C₁₉	—	3	6	no	—
40	C₃²×C₁₉	C₂₇⋊C₃	1	3	yes	complemented
41	C₃²×C₁₉	C₃²×C₉	1	1	yes	characteristic
42	#4294	—	3	2	no	—
43	#4294	—	3	6	no	—
44	#4294	C₃×H(3,𝔽₃)	1	1	yes	characteristic
45	#4294	#737	1	2	yes	—
46	C₃×H(3,𝔽₃)	#4296	1	1	yes	characteristic
47	C₃²×C₉	—	19	1	no	—
48	C₃²×C₉	—	19	3	no	—
49	C₂₇⋊C₃	—	57	9	no	—
50	C₃×C₂₇	—	19	1	no	—
51	C₃×C₂₇	—	19	2	no	—
52	C₃×C₂₇	—	19	6	no	—
53	C₃×C₂₇	—	57	3	no	—
54	C₃²⋊C₉	—	19	6	no	—
55	C₃²⋊C₉	—	19	2	no	—
56	C₃×C₁₉	—	3	3	no	—
57	C₃×C₁₉	—	9	3	no	—
58	C₃×C₁₉	C₉×H(3,𝔽₃)	1	1	yes	characteristic
59	C₃×C₁₉	C₃×C₂₇⋊C₃	1	1	yes	characteristic
60	C₃×C₁₉	#6903	1	2	yes	—
61	C₃³	C₃×#4296	1	3	yes	—
62	C₃³	C₃×#4296	1	1	yes	characteristic
63	H(3,𝔽₃)	—	3	3	no	—
64	C₃×C₉	—	19	1	no	—
65	C₃×C₉	—	19	2	no	—
66	C₃×C₉	—	19	6	no	—
67	C₃×C₉	—	57	1	no	—
68	C₃×C₉	—	57	3	no	—
69	C₃×C₉	—	57	2	no	—
70	C₃×C₉	—	57	6	no	—
71	C₂₇	—	19	3	no	—
72	C₂₇	—	57	2	no	—
73	C₂₇	—	57	6	no	—
74	C₁₉	C₂₇⋊H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
75	C₃²	—	3	1	no	—
76	C₃²	—	3	2	no	—
77	C₃²	—	3	3	no	—
78	C₃²	—	3	3	no	—
79	C₃²	—	3	6	no	—
80	C₃²	#13095342	1	3	yes	complemented
81	C₃²	C₃²×#4296	1	1	yes	characteristic
82	C₉	—	19	1	no	—
83	C₉	—	19	2	no	—
84	C₉	—	57	2	no	—
85	C₉	—	57	6	no	—
86	C₃	—	3	3	no	—
87	C₃	—	9	3	no	—
88	C₃	H(3,𝔽₃)×#4296	1	1	yes	characteristic
89	C₃	C₃×#13095342	1	1	yes	characteristic
90	C₃	#17987598	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₃×#13095342	C₃	semidirect
2. C₃²×#10587020	C₃	semidirect
3. C₃²⋊#10587020	C₃	semidirect
4. C₃²⋊#10587020	C₃	semidirect
29. #10587020	H(3,𝔽₃)	semidirect
40. C₃²×C₁₉	C₂₇⋊C₃	semidirect
74. C₁₉	C₂₇⋊H(3,𝔽₃)	semidirect
80. C₃²	#13095342	semidirect

Maximal subgroups

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

subgroup	index
C₃×#13095342	3
C₃²×#10587020	3
C₃²⋊#10587020	3
H(3,𝔽₃)×#4294	3
C₂₇⋊H(3,𝔽₃)	19