Subgroups — C₉×#13095333 #28355921

Order 13851.

986 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#13095333	C₃	1	1	yes	maximal, characteristic
2	#17987415	C₃	1	2	yes	maximal, complemented
3	3₋¹⁺²×#4294	C₃	1	2	yes	maximal, complemented
4	C₉×#10587023	C₃	1	3	yes	maximal, complemented
5	C₉×#10587022	C₃	1	3	yes	maximal, complemented
6	C₉×#10587025	C₃	1	1	yes	maximal, characteristic, complemented
7	#17987410	C₃	1	6	yes	maximal, complemented
8	#17987409	C₃	1	6	yes	maximal, complemented
9	C₉×C₃²⋊(C₁₉⋊C₃)	C₃	1	1	yes	maximal, characteristic
10	#17987406	C₃	1	2	yes	maximal
11	#17987404	C₃	1	6	yes	maximal, complemented
12	#17987402	C₃	1	2	yes	maximal, complemented
13	C₃×C₉×#4294	C₃	1	1	yes	maximal, characteristic, complemented
14	C₉²×C₁₉⋊C₃	C₃	1	3	yes	maximal, complemented
15	C₉×C₁₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
16	#13095333	C₉	1	27	yes	direct product factor
17	C₃×#10587025	C₃²	1	1	yes	characteristic
18	C₃×#10587022	C₃²	1	3	yes	—
19	C₃×#10587023	C₃²	1	3	yes	—
20	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	1	yes	characteristic
21	C₃²×#4294	C₃²	1	2	yes	—
22	C₃²×#4294	C₃²	1	1	yes	characteristic
23	C₃×C₉×C₁₉⋊C₃	C₃²	1	6	yes	—
24	C₃×C₉×C₁₉⋊C₃	C₃²	1	3	yes	—
25	C₃×C₉×C₁₉⋊C₃	C₃²	1	3	yes	—
26	#13095307	C₃²	1	6	yes	—
27	#13095306	C₃²	1	6	yes	—
28	#13095305	C₃²	1	6	yes	—
29	#13095305	C₃²	1	6	yes	—
30	#13095305	C₃²	1	6	yes	—
31	#13095305	C₃²	1	6	yes	—
32	#13095305	C₃²	1	6	yes	—
33	C₃²⋊#4294	C₃²	1	2	yes	—
34	C₃²⋊#4294	C₃²	1	2	yes	—
35	#13095303	C₃²	1	6	yes	—
36	#13095303	C₃²	1	6	yes	—
37	#13095303	C₃²	1	6	yes	—
38	C₉×#4294	C₃²	1	1	yes	characteristic
39	C₉×#4294	C₃²	1	6	yes	—
40	C₉×#4294	C₃²	1	6	yes	—
41	C₉×#4294	C₃²	1	6	yes	—
42	C₉×#4294	C₃²	1	3	yes	—
43	C₉×#4294	C₃²	1	2	yes	—
44	C₉×#4294	C₃²	1	2	yes	—
45	C₉×#4294	C₃²	1	2	yes	—
46	C₉×#4294	C₃²	1	1	yes	characteristic
47	C₉×#4294	C₃²	1	1	yes	characteristic
48	C₃×C₁₉×3₋¹⁺²	C₃²	1	1	yes	characteristic
49	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
50	C₁₉×C₉⋊C₉	C₃²	1	6	yes	—
51	C₁₉×C₃²⋊C₉	C₃²	1	2	yes	—
52	C₉²×C₁₉	C₃²	1	3	yes	—
53	C₉×#737	—	19	1	no	maximal, Sylow
54	C₃³×C₁₉	C₃³	1	1	yes	characteristic
55	C₃²×C₁₉⋊C₃	C₃³	1	3	yes	—
56	C₁₉×3₋¹⁺²	C₃×C₉	1	9	yes	complemented
57	C₃×C₉×C₁₉	—	3	3	no	—
58	C₃×C₉×C₁₉	C₃³	1	6	yes	—
59	C₃×C₉×C₁₉	C₃³	1	2	yes	—
60	C₃×C₉×C₁₉	C₃³	1	3	yes	—
61	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
62	C₃²⋊(C₁₉⋊C₃)	C₃×C₉	1	9	yes	—
63	#10587025	C₃×C₉	1	9	yes	complemented
64	C₃×#4294	—	3	6	no	—
65	C₃×#4294	C₃×C₉	1	9	yes	complemented
66	C₃×#4294	C₃³	1	2	yes	—
67	C₃×#4294	C₃³	1	2	yes	—
68	C₃×#4294	C₃³	1	6	yes	—
69	C₃×#4294	C₃³	1	6	yes	—
70	C₃×#4294	C₃³	1	1	yes	characteristic
71	C₃×#4294	C₃³	1	3	yes	—
72	C₃×#4294	C₃³	1	1	yes	characteristic
73	C₃×#4294	C₃³	1	1	yes	characteristic
74	C₃×#4294	C₃³	1	2	yes	—
75	#10587023	C₃×C₉	1	27	yes	complemented
76	#10587022	C₃×C₉	1	27	yes	complemented
77	C₉×C₁₉⋊C₃	—	3	9	no	—
78	C₉×C₁₉⋊C₃	—	3	18	no	—
79	C₉×C₁₉⋊C₃	C₃×C₉	1	27	yes	complemented
80	C₃×#737	—	19	1	no	—
81	C₉×3₋¹⁺²	—	19	2	no	—
82	C₉×3₋¹⁺²	—	19	1	no	—
83	C₉×3₋¹⁺²	—	19	6	no	—
84	C₉×3₋¹⁺²	—	19	6	no	—
85	C₉×3₋¹⁺²	—	19	2	no	—
86	C₉×3₋¹⁺²	—	19	3	no	—
87	C₉×3₋¹⁺²	—	19	3	no	—
88	C₉×3₋¹⁺²	C₁₉⋊C₃	1	1	yes	characteristic, complemented
89	C₉×H(3,𝔽₃)	—	19	1	no	—
90	C₉×H(3,𝔽₃)	—	19	2	no	—
91	#6887	—	19	6	no	—
92	#6887	—	19	2	no	—
93	C₃×C₉²	—	19	1	no	—
94	C₃×C₉²	—	19	3	no	—
95	C₃²×C₁₉	—	3	2	no	—
96	C₃²×C₁₉	—	3	1	no	—
97	C₃²×C₁₉	C₃²×C₉	1	3	yes	—
98	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
99	C₃×C₁₉⋊C₃	—	3	6	no	—
100	C₃×C₁₉⋊C₃	—	3	3	no	—
101	C₃×C₁₉⋊C₃	C₃²×C₉	1	9	yes	—
102	C₉×C₁₉	—	3	6	no	—
103	C₉×C₁₉	—	3	2	no	—
104	C₉×C₁₉	C₃²×C₉	1	9	yes	—
105	C₉×C₁₉	#737	1	3	yes	complemented
106	#4294	—	3	6	no	—
107	#4294	—	3	6	no	—
108	#4294	—	3	2	no	—
109	#4294	—	3	2	no	—
110	#4294	C₃²×C₉	1	3	yes	—
111	#4294	C₃²×C₉	1	9	yes	—
112	#4294	C₃²×C₉	1	3	yes	—
113	#4294	C₃²×C₉	1	3	yes	—
114	#4294	#737	1	6	yes	complemented
115	#737	—	19	27	no	—
116	C₃×3₋¹⁺²	—	19	1	no	—
117	C₃×3₋¹⁺²	—	19	3	no	—
118	C₃×3₋¹⁺²	—	19	3	no	—
119	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	1	yes	characteristic
120	C₃×H(3,𝔽₃)	—	19	1	no	—
121	C₃²×C₉	—	19	1	no	—
122	C₃²×C₉	—	19	3	no	—
123	C₃²×C₉	—	19	2	no	—
124	C₃²×C₉	—	19	6	no	—
125	C₃²×C₉	—	19	3	no	—
126	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
127	C₉⋊C₉	—	19	6	no	—
128	C₉⋊C₉	—	19	6	no	—
129	C₉⋊C₉	—	19	6	no	—
130	C₉⋊C₉	—	19	6	no	—
131	C₉⋊C₉	—	19	6	no	—
132	C₉⋊C₉	—	19	6	no	—
133	C₉⋊C₉	—	19	6	no	—
134	C₉⋊C₉	C₃×C₁₉⋊C₃	1	6	yes	—
135	C₃²⋊C₉	—	19	2	no	—
136	C₃²⋊C₉	—	19	2	no	—
137	C₃²⋊C₉	—	19	6	no	—
138	C₃²⋊C₉	—	19	6	no	—
139	C₃²⋊C₉	—	19	6	no	—
140	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	2	yes	—
141	C₉²	—	19	2	no	—
142	C₉²	—	19	6	no	—
143	C₉²	—	19	2	no	—
144	C₉²	—	19	2	no	—
145	C₉²	—	19	6	no	—
146	C₉²	—	19	6	no	—
147	C₉²	—	19	1	no	—
148	C₉²	—	19	3	no	—
149	C₉²	—	19	1	no	—
150	C₉²	—	19	1	no	—
151	C₉²	C₃×C₁₉⋊C₃	1	3	yes	—
152	C₃×C₁₉	—	3	3	no	—
153	C₃×C₁₉	C₃³×C₉	1	1	yes	characteristic
154	C₃×C₁₉	C₃×#737	1	2	yes	—
155	C₃×C₁₉	C₃×#737	1	1	yes	characteristic
156	C₁₉⋊C₃	—	3	9	no	—
157	C₃³	—	19	3	no	—
158	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
159	3₋¹⁺²	—	19	9	no	—
160	3₋¹⁺²	—	19	27	no	—
161	3₋¹⁺²	—	19	27	no	—
162	3₋¹⁺²	C₉×C₁₉⋊C₃	1	9	yes	complemented
163	H(3,𝔽₃)	—	19	9	no	—
164	C₃×C₉	—	3	3	no	—
165	C₃×C₉	—	19	9	no	—
166	C₃×C₉	—	19	1	no	—
167	C₃×C₉	—	19	2	no	—
168	C₃×C₉	—	19	3	no	—
169	C₃×C₉	—	19	1	no	—
170	C₃×C₉	—	19	1	no	—
171	C₃×C₉	—	19	27	no	—
172	C₃×C₉	—	19	2	no	—
173	C₃×C₉	—	19	2	no	—
174	C₃×C₉	—	19	6	no	—
175	C₃×C₉	—	19	6	no	—
176	C₃×C₉	—	57	18	no	—
177	C₃×C₉	—	57	6	no	—
178	C₃×C₉	—	57	9	no	—
179	C₃×C₉	C₃²×C₁₉⋊C₃	1	2	yes	—
180	C₃×C₉	C₃²×C₁₉⋊C₃	1	6	yes	—
181	C₃×C₉	C₃²×C₁₉⋊C₃	1	3	yes	—
182	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
183	C₁₉	C₉×#737	1	1	yes	characteristic, complemented, Sylow
184	C₃²	—	3	2	no	—
185	C₃²	—	3	1	no	—
186	C₃²	—	19	9	no	—
187	C₃²	—	57	6	no	—
188	C₃²	—	57	3	no	—
189	C₃²	C₃×C₉×C₁₉⋊C₃	1	3	yes	—
190	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
191	C₉	—	3	6	no	—
192	C₉	—	3	2	no	—
193	C₉	—	19	3	no	—
194	C₉	—	19	3	no	—
195	C₉	—	19	3	no	—
196	C₉	—	19	9	no	—
197	C₉	—	19	6	no	—
198	C₉	—	57	6	no	—
199	C₉	—	57	2	no	—
200	C₉	—	57	2	no	—
201	C₉	—	57	6	no	—
202	C₉	C₃×C₉×C₁₉⋊C₃	1	9	yes	—
203	C₉	#13095333	1	3	yes	direct product factor
204	C₃	—	3	3	no	—
205	C₃	—	57	9	no	—
206	C₃	C₃²×C₉×C₁₉⋊C₃	1	1	yes	characteristic
207	C₃	C₁₉⋊C₃×#737	1	2	yes	—
208	C₃	C₃×#13095333	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₉	C₃²×C₁₉⋊C₃
commutator	C₃×C₁₉	C₃³×C₉
Frattini	C₃²	C₃³×C₁₉⋊C₃
Fitting	C₉×C₁₉×3₋¹⁺²	C₃
socle, cosocle	C₃²×C₁₉	C₃⁴
prime radical	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²
hypercenter	C₉×3₋¹⁺²	C₁₉⋊C₃
hypocenter	C₁₉	C₉×#737
LCS	G ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃×C₉ ◁ C₉×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
2. #17987415	C₃	semidirect
3. 3₋¹⁺²×#4294	C₃	semidirect
4. C₉×#10587023	C₃	semidirect
5. C₉×#10587022	C₃	semidirect
6. C₉×#10587025	C₃	semidirect
7. #17987410	C₃	semidirect
8. #17987409	C₃	semidirect
11. #17987404	C₃	semidirect
12. #17987402	C₃	semidirect
13. C₃×C₉×#4294	C₃	semidirect
14. C₉²×C₁₉⋊C₃	C₃	semidirect
15. C₉×C₁₉×3₋¹⁺²	C₃	semidirect
16. #13095333	C₉	direct (N × H)
56. C₁₉×3₋¹⁺²	C₃×C₉	semidirect
63. #10587025	C₃×C₉	semidirect
65. C₃×#4294	C₃×C₉	semidirect
75. #10587023	C₃×C₉	semidirect
76. #10587022	C₃×C₉	semidirect
79. C₉×C₁₉⋊C₃	C₃×C₉	semidirect
88. C₉×3₋¹⁺²	C₁₉⋊C₃	semidirect
105. C₉×C₁₉	#737	semidirect
114. #4294	#737	semidirect
162. 3₋¹⁺²	C₉×C₁₉⋊C₃	semidirect
183. C₁₉	C₉×#737	semidirect
203. C₉	#13095333	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#13095333	3
#17987415	3
3₋¹⁺²×#4294	3
C₉×#10587023	3
C₉×#10587022	3
C₉×#10587025	3
#17987410	3
#17987409	3
C₉×C₃²⋊(C₁₉⋊C₃)	3
#17987406	3
#17987404	3
#17987402	3
C₃×C₉×#4294	3
C₉²×C₁₉⋊C₃	3
C₉×C₁₉×3₋¹⁺²	3
C₉×#737	19

Subgroups — C9×#13095333 #28355921

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#13095333 #28355921