Subgroups — C₃×#10593141 #13164948

Order 1728.

2596 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#63716	C₂	1	6	yes	maximal, complemented
3	C₃×#41×D₉	C₂	1	6	yes	maximal, complemented
4	C₃×#63710	C₂	1	9	yes	maximal, complemented
5	C₂×C₃×#3699	C₂	1	9	yes	maximal, complemented
6	C₃×#5966	—	3	1	no	maximal
7	#10593141	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₃×C₉×#41	C₂²	1	6	yes	complemented
9	C₂×C₃×D₄×C₉	C₂²	1	9	yes	complemented
10	C₂×C₃×#611	C₂²	1	18	yes	complemented
11	C₂²×C₃×Dic₉	C₂²	1	6	yes	complemented
12	C₃×#3701	C₂²	1	2	yes	complemented
13	C₃×Q₈×D₉	C₂²	1	6	yes	complemented
14	C₃×#3699	C₂²	1	18	yes	complemented
15	C₃×#3699	C₂²	1	36	yes	complemented
16	C₃×D₄×D₉	C₂²	1	18	yes	complemented
17	C₃×#3697	C₂²	1	18	yes	complemented
18	C₂×C₃×C₄×D₉	C₂²	1	9	yes	complemented
19	C₂×C₃×Dic₁₈	C₂²	1	9	yes	complemented
20	C₃²×H(5,𝔽₂)	D₃	1	1	yes	characteristic
21	C₃×#1010	—	3	6	no	—
22	C₃×D₃×#41	—	3	6	no	—
23	C₃×#1004	—	3	9	no	—
24	C₂×C₃×#237	—	3	9	no	—
25	C₉×H(5,𝔽₂)	—	2	1	no	—
26	C₉×H(5,𝔽₂)	C₂×C₃	1	1	yes	characteristic, complemented
27	#63716	C₂×C₃	1	6	yes	complemented
28	#41×D₉	C₂×C₃	1	6	yes	complemented
29	#63710	C₂×C₃	1	9	yes	complemented
30	C₂×#3699	C₂×C₃	1	9	yes	complemented
31	C₂³×C₃×C₉	C₂³	1	6	yes	—
32	C₂²×C₃×D₉	C₂³	1	9	yes	—
33	C₃×Q₈×C₉	C₂³	1	2	yes	—
34	C₃×D₄×C₉	C₂³	1	18	yes	—
35	C₂×C₃×C₄×C₉	C₂³	1	9	yes	—
36	C₃×#611	C₂³	1	36	yes	—
37	C₂×C₃×Dic₉	C₂³	1	18	yes	—
38	C₂×C₃×Dic₉	C₂³	1	9	yes	—
39	C₃×D₃₆	C₂³	1	6	yes	—
40	C₃×C₄×D₉	C₂³	1	6	yes	—
41	C₃×C₄×D₉	C₂³	1	18	yes	—
42	C₃×Dic₁₈	C₂³	1	18	yes	—
43	C₃×#585	—	9	1	no	—
44	#5966	—	3	1	no	—
45	C₃²×#41	C₂×D₃	1	6	yes	—
46	C₂×C₃²×D₄	C₂×D₃	1	9	yes	—
47	C₂×C₃×#67	—	3	18	no	—
48	C₂²×C₃×Dic₃	—	3	6	no	—
49	C₃×#239	—	3	2	no	—
50	C₃×D₃×Q₈	—	3	6	no	—
51	C₃×#237	—	3	18	no	—
52	C₃×#237	—	3	36	no	—
53	C₃×D₃×D₄	—	3	18	no	—
54	C₃×#235	—	3	18	no	—
55	C₂×C₃×C₄×D₃	—	3	9	no	—
56	C₂×C₃×Dic₆	—	3	9	no	—
57	C₉×#41	—	2	6	no	—
58	C₉×#41	C₂²×C₃	1	6	yes	complemented
59	C₂×D₄×C₉	—	2	9	no	—
60	C₂×D₄×C₉	C₂²×C₃	1	9	yes	complemented
61	C₂×#611	C₂²×C₃	1	18	yes	complemented
62	C₂²×Dic₉	C₂²×C₃	1	6	yes	complemented
63	#3701	C₂²×C₃	1	2	yes	complemented
64	Q₈×D₉	C₂²×C₃	1	6	yes	complemented
65	#3699	C₂²×C₃	1	18	yes	complemented
66	#3699	C₂²×C₃	1	36	yes	complemented
67	D₄×D₉	C₂²×C₃	1	18	yes	complemented
68	#3697	C₂²×C₃	1	18	yes	complemented
69	C₂×C₄×D₉	C₂²×C₃	1	9	yes	complemented
70	C₂×Dic₁₈	C₂²×C₃	1	9	yes	complemented
71	C₂²×C₃×C₉	—	4	6	no	—
72	C₂²×C₃×C₉	C₂⁴	1	9	yes	—
73	C₂×C₃×D₉	—	4	9	no	—
74	C₂×C₃×D₉	C₂⁴	1	6	yes	—
75	C₃×C₄×C₉	C₂⁴	1	6	yes	—
76	C₃×Dic₉	C₂⁴	1	9	yes	—
77	C₃×Dic₉	C₂⁴	1	1	yes	characteristic
78	C₃×2₋¹⁺⁴	—	9	6	no	—
79	C₃×H(5,𝔽₂)	—	2	1	no	—
80	C₃×H(5,𝔽₂)	—	9	9	no	—
81	C₃×H(5,𝔽₂)	D₉	1	1	yes	characteristic, complemented
82	C₃×H(5,𝔽₂)	C₃×D₃	1	1	yes	characteristic
83	C₂×C₃×#41	—	9	6	no	—
84	C₂×C₃×#41	—	9	9	no	—
85	#1010	—	3	6	no	—
86	D₃×#41	—	3	6	no	—
87	#1004	—	3	9	no	—
88	C₂×#237	—	3	9	no	—
89	C₂³×C₃²	C₂²×D₃	1	6	yes	—
90	C₂²×C₃×D₃	—	3	9	no	—
91	C₃²×Q₈	C₂²×D₃	1	2	yes	—
92	C₃²×D₄	C₂²×D₃	1	18	yes	—
93	C₂×C₃²×C₄	C₂²×D₃	1	9	yes	—
94	C₃×#67	—	3	36	no	—
95	C₂×C₃×Dic₃	—	3	18	no	—
96	C₂×C₃×Dic₃	—	3	9	no	—
97	C₃×D₁₂	—	3	6	no	—
98	C₃×C₄×D₃	—	3	6	no	—
99	C₃×C₄×D₃	—	3	18	no	—
100	C₃×Dic₆	—	3	18	no	—
101	C₂³×C₉	—	2	6	no	—
102	C₂³×C₉	C₂³×C₃	1	6	yes	—
103	C₂²×D₉	C₂³×C₃	1	9	yes	—
104	Q₈×C₉	—	2	2	no	—
105	Q₈×C₉	C₂³×C₃	1	2	yes	—
106	D₄×C₉	—	2	18	no	—
107	D₄×C₉	C₂³×C₃	1	18	yes	—
108	C₂×C₄×C₉	—	2	9	no	—
109	C₂×C₄×C₉	C₂³×C₃	1	9	yes	—
110	#611	C₂³×C₃	1	36	yes	—
111	C₂×Dic₉	C₂³×C₃	1	18	yes	—
112	C₂×Dic₉	C₂³×C₃	1	9	yes	—
113	D₃₆	C₂³×C₃	1	6	yes	—
114	C₄×D₉	C₂³×C₃	1	6	yes	—
115	C₄×D₉	C₂³×C₃	1	18	yes	—
116	Dic₁₈	C₂³×C₃	1	18	yes	—
117	#585	—	9	1	no	Sylow
118	C₂×C₃×C₉	—	2	9	no	—
119	C₂×C₃×C₉	C₂⁵	1	1	yes	characteristic
120	C₃×D₉	—	2	6	no	—
121	C₃×#41	—	2	6	no	—
122	C₃×#41	—	9	2	no	—
123	C₃×#41	—	9	18	no	—
124	C₃×#41	—	9	36	no	—
125	C₃×#41	—	9	18	no	—
126	C₃×#41	C₂×D₉	1	6	yes	complemented
127	C₃×#41	C₂×C₃×D₃	1	6	yes	—
128	C₂×C₃×Q₈	—	9	6	no	—
129	C₂×C₃×Q₈	—	9	9	no	—
130	C₂×C₃×D₄	—	2	9	no	—
131	C₂×C₃×D₄	—	9	18	no	—
132	C₂×C₃×D₄	—	9	18	no	—
133	C₂×C₃×D₄	C₂×D₉	1	9	yes	complemented
134	C₂×C₃×D₄	C₂×C₃×D₃	1	9	yes	—
135	C₂²×C₃×C₄	—	9	6	no	—
136	C₂²×C₃×C₄	—	9	9	no	—
137	C₂×#67	—	3	18	no	—
138	C₂²×Dic₃	—	3	6	no	—
139	#239	—	3	2	no	—
140	D₃×Q₈	—	3	6	no	—
141	#237	—	3	18	no	—
142	#237	—	3	36	no	—
143	D₃×D₄	—	3	18	no	—
144	#235	—	3	18	no	—
145	C₂×C₄×D₃	—	3	9	no	—
146	C₂×Dic₆	—	3	9	no	—
147	C₂²×C₃²	—	4	6	no	—
148	C₂²×C₃²	C₂³×D₃	1	9	yes	—
149	C₂×C₃×D₃	—	3	6	no	—
150	C₂×C₃×D₃	—	12	9	no	—
151	C₃²×C₄	C₂³×D₃	1	6	yes	—
152	C₃×Dic₃	—	3	9	no	—
153	C₃×Dic₃	—	3	1	no	—
154	C₂²×C₉	—	2	9	no	—
155	C₂²×C₉	—	4	6	no	—
156	C₂²×C₉	—	8	6	no	—
157	C₂²×C₉	C₂⁴×C₃	1	9	yes	—
158	C₂×D₉	—	4	9	no	—
159	C₂×D₉	C₂⁴×C₃	1	6	yes	—
160	C₄×C₉	—	2	6	no	—
161	C₄×C₉	C₂⁴×C₃	1	6	yes	—
162	Dic₉	C₂⁴×C₃	1	9	yes	—
163	Dic₉	C₂⁴×C₃	1	1	yes	characteristic
164	2₋¹⁺⁴	—	9	6	no	—
165	H(5,𝔽₂)	—	9	9	no	—
166	H(5,𝔽₂)	C₃×D₉	1	1	yes	characteristic, complemented
167	C₂×#41	—	9	6	no	—
168	C₂×#41	—	9	9	no	—
169	C₃×C₉	#585	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₃	—	2	6	no	—
171	C₂³×C₃	—	9	9	no	—
172	C₂³×C₃	C₂²×D₉	1	6	yes	—
173	C₂³×C₃	C₂²×C₃×D₃	1	6	yes	—
174	C₂²×D₃	—	3	9	no	—
175	C₃×Q₈	—	2	2	no	—
176	C₃×Q₈	—	9	18	no	—
177	C₃×Q₈	C₂²×D₉	1	2	yes	—
178	C₃×Q₈	C₂²×C₃×D₃	1	2	yes	—
179	C₃×D₄	—	2	18	no	—
180	C₃×D₄	—	9	36	no	—
181	C₃×D₄	—	9	6	no	—
182	C₃×D₄	C₂²×D₉	1	18	yes	—
183	C₃×D₄	C₂²×C₃×D₃	1	18	yes	—
184	C₂×C₃×C₄	—	2	9	no	—
185	C₂×C₃×C₄	—	9	9	no	—
186	C₂×C₃×C₄	—	9	18	no	—
187	C₂×C₃×C₄	—	9	6	no	—
188	C₂×C₃×C₄	—	9	18	no	—
189	C₂×C₃×C₄	C₂²×D₉	1	9	yes	—
190	C₂×C₃×C₄	C₂²×C₃×D₃	1	9	yes	—
191	#67	—	3	36	no	—
192	C₂×Dic₃	—	3	18	no	—
193	C₂×Dic₃	—	3	9	no	—
194	D₁₂	—	3	6	no	—
195	C₄×D₃	—	3	6	no	—
196	C₄×D₃	—	3	18	no	—
197	Dic₆	—	3	18	no	—
198	C₂×C₃²	—	2	9	no	—
199	C₂×C₃²	C₂⁴×D₃	1	1	yes	characteristic
200	C₃×D₃	—	6	6	no	—
201	C₂×C₉	—	2	9	no	—
202	C₂×C₉	—	2	1	no	—
203	C₂×C₉	—	4	9	no	—
204	C₂×C₉	C₂⁵×C₃	1	1	yes	characteristic
205	D₉	—	2	6	no	—
206	#41	—	9	2	no	—
207	#41	—	9	18	no	—
208	#41	—	9	36	no	—
209	#41	—	9	18	no	—
210	#41	C₂×C₃×D₉	1	6	yes	complemented
211	C₂×Q₈	—	9	6	no	—
212	C₂×Q₈	—	9	9	no	—
213	C₂×D₄	—	9	18	no	—
214	C₂×D₄	—	9	18	no	—
215	C₂×D₄	C₂×C₃×D₉	1	9	yes	complemented
216	C₂²×C₄	—	9	6	no	—
217	C₂²×C₄	—	9	9	no	—
218	C₂²×C₃	—	2	9	no	—
219	C₂²×C₃	—	4	6	no	—
220	C₂²×C₃	—	4	6	no	—
221	C₂²×C₃	—	8	6	no	—
222	C₂²×C₃	—	9	6	no	—
223	C₂²×C₃	—	36	9	no	—
224	C₂²×C₃	C₂³×D₉	1	9	yes	—
225	C₂²×C₃	C₂³×C₃×D₃	1	9	yes	—
226	C₂×D₃	—	3	6	no	—
227	C₂×D₃	—	12	9	no	—
228	C₃×C₄	—	2	6	no	—
229	C₃×C₄	—	9	1	no	—
230	C₃×C₄	—	9	9	no	—
231	C₃×C₄	C₂³×D₉	1	6	yes	—
232	C₃×C₄	C₂³×C₃×D₃	1	6	yes	—
233	Dic₃	—	3	9	no	—
234	Dic₃	—	3	1	no	—
235	C₃²	#5966	1	1	yes	characteristic
236	C₉	—	2	1	no	—
237	C₉	C₃×#585	1	1	yes	characteristic, complemented
238	C₂³	—	9	9	no	—
239	C₂³	C₂²×C₃×D₉	1	6	yes	—
240	Q₈	—	9	18	no	—
241	Q₈	C₂²×C₃×D₉	1	2	yes	—
242	D₄	—	9	36	no	—
243	D₄	—	9	6	no	—
244	D₄	C₂²×C₃×D₉	1	18	yes	—
245	C₂×C₄	—	9	9	no	—
246	C₂×C₄	—	9	18	no	—
247	C₂×C₄	—	9	6	no	—
248	C₂×C₄	—	9	18	no	—
249	C₂×C₄	C₂²×C₃×D₉	1	9	yes	—
250	C₂×C₃	—	2	9	no	—
251	C₂×C₃	—	2	9	no	—
252	C₂×C₃	—	2	1	no	—
253	C₂×C₃	—	4	9	no	—
254	C₂×C₃	—	18	6	no	—
255	C₂×C₃	C₂⁴×D₉	1	1	yes	characteristic
256	C₂×C₃	C₂⁴×C₃×D₃	1	1	yes	characteristic
257	D₃	—	6	6	no	—
258	C₂²	—	4	6	no	—
259	C₂²	—	9	6	no	—
260	C₂²	—	36	9	no	—
261	C₂²	C₂³×C₃×D₉	1	9	yes	—
262	C₄	—	9	1	no	—
263	C₄	—	9	9	no	—
264	C₄	C₂³×C₃×D₉	1	6	yes	—
265	C₃	—	2	1	no	—
266	C₃	#10593141	1	1	yes	characteristic, direct product factor
267	C₃	C₃×#5966	1	1	yes	characteristic
268	C₂	—	2	9	no	—
269	C₂	—	18	6	no	—
270	C₂	C₂⁴×C₃×D₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂⁴×D₉
commutator, cosocle	C₂×C₉	C₂⁵×C₃
Frattini	C₂×C₃	C₂⁴×C₃×D₃
Fitting	C₃×C₉×H(5,𝔽₂)	C₂
socle	C₂×C₃²	C₂⁴×D₃
hypercenter	C₃×H(5,𝔽₂)	D₉
hypocenter	C₉	C₃×#585
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₃ ◁ C₃×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₃×C₉×H(5,𝔽₂)	C₂	semidirect
2. C₃×#63716	C₂	semidirect
3. C₃×#41×D₉	C₂	semidirect
4. C₃×#63710	C₂	semidirect
5. C₂×C₃×#3699	C₂	semidirect
7. #10593141	C₃	direct (N × H)
8. C₃×C₉×#41	C₂²	semidirect
9. C₂×C₃×D₄×C₉	C₂²	semidirect
10. C₂×C₃×#611	C₂²	semidirect
11. C₂²×C₃×Dic₉	C₂²	semidirect
12. C₃×#3701	C₂²	semidirect
13. C₃×Q₈×D₉	C₂²	semidirect
14. C₃×#3699	C₂²	semidirect
15. C₃×#3699	C₂²	semidirect
16. C₃×D₄×D₉	C₂²	semidirect
17. C₃×#3697	C₂²	semidirect
18. C₂×C₃×C₄×D₉	C₂²	semidirect
19. C₂×C₃×Dic₁₈	C₂²	semidirect
26. C₉×H(5,𝔽₂)	C₂×C₃	semidirect
27. #63716	C₂×C₃	semidirect
28. #41×D₉	C₂×C₃	semidirect
29. #63710	C₂×C₃	semidirect
30. C₂×#3699	C₂×C₃	semidirect
58. C₉×#41	C₂²×C₃	semidirect
60. C₂×D₄×C₉	C₂²×C₃	semidirect
61. C₂×#611	C₂²×C₃	semidirect
62. C₂²×Dic₉	C₂²×C₃	semidirect
63. #3701	C₂²×C₃	semidirect
64. Q₈×D₉	C₂²×C₃	semidirect
65. #3699	C₂²×C₃	semidirect
66. #3699	C₂²×C₃	semidirect
67. D₄×D₉	C₂²×C₃	semidirect
68. #3697	C₂²×C₃	semidirect
69. C₂×C₄×D₉	C₂²×C₃	semidirect
70. C₂×Dic₁₈	C₂²×C₃	semidirect
81. C₃×H(5,𝔽₂)	D₉	semidirect
126. C₃×#41	C₂×D₉	semidirect
133. C₂×C₃×D₄	C₂×D₉	semidirect
166. H(5,𝔽₂)	C₃×D₉	semidirect
169. C₃×C₉	#585	semidirect
210. #41	C₂×C₃×D₉	semidirect
215. C₂×D₄	C₂×C₃×D₉	semidirect
237. C₉	C₃×#585	semidirect
266. C₃	#10593141	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×C₉×H(5,𝔽₂)	2
C₃×#63716	2
C₃×#41×D₉	2
C₃×#63710	2
C₂×C₃×#3699	2
C₃×#5966	3
#10593141	3

Subgroups — C3×#10593141 #13164948

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10593141 #13164948