Subgroups — C₃×#10593143 #13164956

Order 1728.

2576 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#63715	C₂	1	10	yes	maximal, complemented
3	C₃×#41×D₉	C₂	1	10	yes	maximal, complemented
4	C₃×#63713	C₂	1	5	yes	maximal, complemented
5	C₂×C₃×#3701	C₂	1	5	yes	maximal, complemented
6	C₃×#5968	—	3	1	no	maximal
7	#10593143	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₃×C₉×#41	C₂²	1	10	yes	complemented
9	C₂×C₃×Q₈×C₉	C₂²	1	5	yes	complemented
10	C₃×#3701	C₂²	1	10	yes	complemented
11	C₃×#3701	C₂²	1	20	yes	complemented
12	C₃×Q₈×D₉	C₂²	1	10	yes	complemented
13	C₃×#3699	C₂²	1	10	yes	complemented
14	C₃×D₄×D₉	C₂²	1	30	yes	complemented
15	C₃×#3697	C₂²	1	30	yes	complemented
16	C₂×C₃×D₃₆	C₂²	1	15	yes	complemented
17	C₂×C₃×C₄×D₉	C₂²	1	15	yes	complemented
18	C₃²×2₋¹⁺⁴	D₃	1	1	yes	characteristic
19	C₃×#1009	—	3	10	no	—
20	C₃×D₃×#41	—	3	10	no	—
21	C₃×#1007	—	3	5	no	—
22	C₂×C₃×#239	—	3	5	no	—
23	C₉×2₋¹⁺⁴	—	2	1	no	—
24	C₉×2₋¹⁺⁴	C₂×C₃	1	1	yes	characteristic, complemented
25	#63715	C₂×C₃	1	10	yes	complemented
26	#41×D₉	C₂×C₃	1	10	yes	complemented
27	#63713	C₂×C₃	1	5	yes	complemented
28	C₂×#3701	C₂×C₃	1	5	yes	complemented
29	C₂²×C₃×D₉	C₂³	1	15	yes	—
30	C₃×Q₈×C₉	C₂³	1	10	yes	—
31	C₃×D₄×C₉	C₂³	1	10	yes	—
32	C₂×C₃×C₄×C₉	C₂³	1	15	yes	—
33	C₃×#611	C₂³	1	20	yes	—
34	C₂×C₃×Dic₉	C₂³	1	5	yes	—
35	C₃×D₃₆	C₂³	1	30	yes	—
36	C₃×C₄×D₉	C₂³	1	10	yes	—
37	C₃×C₄×D₉	C₂³	1	30	yes	—
38	C₃×Dic₁₈	C₂³	1	10	yes	—
39	C₃×#585	—	9	1	no	—
40	#5968	—	3	1	no	—
41	C₃²×#41	C₂×D₃	1	10	yes	—
42	C₂×C₃²×Q₈	C₂×D₃	1	5	yes	—
43	C₃×#239	—	3	10	no	—
44	C₃×#239	—	3	20	no	—
45	C₃×D₃×Q₈	—	3	10	no	—
46	C₃×#237	—	3	10	no	—
47	C₃×D₃×D₄	—	3	30	no	—
48	C₃×#235	—	3	30	no	—
49	C₂×C₃×D₁₂	—	3	15	no	—
50	C₂×C₃×C₄×D₃	—	3	15	no	—
51	C₉×#41	—	2	10	no	—
52	C₉×#41	C₂²×C₃	1	10	yes	complemented
53	C₂×Q₈×C₉	—	2	5	no	—
54	C₂×Q₈×C₉	C₂²×C₃	1	5	yes	complemented
55	#3701	C₂²×C₃	1	10	yes	complemented
56	#3701	C₂²×C₃	1	20	yes	complemented
57	Q₈×D₉	C₂²×C₃	1	10	yes	complemented
58	#3699	C₂²×C₃	1	10	yes	complemented
59	D₄×D₉	C₂²×C₃	1	30	yes	complemented
60	#3697	C₂²×C₃	1	30	yes	complemented
61	C₂×D₃₆	C₂²×C₃	1	15	yes	complemented
62	C₂×C₄×D₉	C₂²×C₃	1	15	yes	complemented
63	C₂²×C₃×C₉	C₂⁴	1	5	yes	—
64	C₂×C₃×D₉	—	4	15	no	—
65	C₂×C₃×D₉	C₂⁴	1	10	yes	—
66	C₃×C₄×C₉	C₂⁴	1	10	yes	—
67	C₃×Dic₉	C₂⁴	1	5	yes	—
68	C₃×Dic₉	C₂⁴	1	1	yes	characteristic
69	C₃×2₋¹⁺⁴	—	2	1	no	—
70	C₃×2₋¹⁺⁴	—	9	5	no	—
71	C₃×2₋¹⁺⁴	D₉	1	1	yes	characteristic, complemented
72	C₃×2₋¹⁺⁴	C₃×D₃	1	1	yes	characteristic
73	C₃×H(5,𝔽₂)	—	9	10	no	—
74	C₂×C₃×#41	—	9	10	no	—
75	C₂×C₃×#41	—	9	5	no	—
76	#1009	—	3	10	no	—
77	D₃×#41	—	3	10	no	—
78	#1007	—	3	5	no	—
79	C₂×#239	—	3	5	no	—
80	C₂²×C₃×D₃	—	3	15	no	—
81	C₃²×Q₈	C₂²×D₃	1	10	yes	—
82	C₃²×D₄	C₂²×D₃	1	10	yes	—
83	C₂×C₃²×C₄	C₂²×D₃	1	15	yes	—
84	C₃×#67	—	3	20	no	—
85	C₂×C₃×Dic₃	—	3	5	no	—
86	C₃×D₁₂	—	3	30	no	—
87	C₃×C₄×D₃	—	3	10	no	—
88	C₃×C₄×D₃	—	3	30	no	—
89	C₃×Dic₆	—	3	10	no	—
90	C₂²×D₉	C₂³×C₃	1	15	yes	—
91	Q₈×C₉	—	2	10	no	—
92	Q₈×C₉	C₂³×C₃	1	10	yes	—
93	D₄×C₉	—	2	10	no	—
94	D₄×C₉	C₂³×C₃	1	10	yes	—
95	C₂×C₄×C₉	—	2	15	no	—
96	C₂×C₄×C₉	C₂³×C₃	1	15	yes	—
97	#611	C₂³×C₃	1	20	yes	—
98	C₂×Dic₉	C₂³×C₃	1	5	yes	—
99	D₃₆	C₂³×C₃	1	30	yes	—
100	C₄×D₉	C₂³×C₃	1	10	yes	—
101	C₄×D₉	C₂³×C₃	1	30	yes	—
102	Dic₁₈	C₂³×C₃	1	10	yes	—
103	#585	—	9	1	no	Sylow
104	C₂×C₃×C₉	—	2	5	no	—
105	C₂×C₃×C₉	C₂⁵	1	1	yes	characteristic
106	C₃×D₉	—	2	10	no	—
107	C₃×#41	—	2	10	no	—
108	C₃×#41	—	9	10	no	—
109	C₃×#41	—	9	20	no	—
110	C₃×#41	—	9	10	no	—
111	C₃×#41	—	9	30	no	—
112	C₃×#41	C₂×D₉	1	10	yes	complemented
113	C₃×#41	C₂×C₃×D₃	1	10	yes	—
114	C₂×C₃×Q₈	—	2	5	no	—
115	C₂×C₃×Q₈	—	9	10	no	—
116	C₂×C₃×Q₈	C₂×D₉	1	5	yes	complemented
117	C₂×C₃×Q₈	C₂×C₃×D₃	1	5	yes	—
118	C₂×C₃×D₄	—	9	30	no	—
119	C₂×C₃×D₄	—	9	15	no	—
120	C₂²×C₃×C₄	—	9	15	no	—
121	#239	—	3	10	no	—
122	#239	—	3	20	no	—
123	D₃×Q₈	—	3	10	no	—
124	#237	—	3	10	no	—
125	D₃×D₄	—	3	30	no	—
126	#235	—	3	30	no	—
127	C₂×D₁₂	—	3	15	no	—
128	C₂×C₄×D₃	—	3	15	no	—
129	C₂²×C₃²	C₂³×D₃	1	5	yes	—
130	C₂×C₃×D₃	—	3	10	no	—
131	C₂×C₃×D₃	—	12	15	no	—
132	C₃²×C₄	C₂³×D₃	1	10	yes	—
133	C₃×Dic₃	—	3	5	no	—
134	C₃×Dic₃	—	3	1	no	—
135	C₂²×C₉	—	2	5	no	—
136	C₂²×C₉	C₂⁴×C₃	1	5	yes	—
137	C₂×D₉	—	4	15	no	—
138	C₂×D₉	C₂⁴×C₃	1	10	yes	—
139	C₄×C₉	—	2	10	no	—
140	C₄×C₉	C₂⁴×C₃	1	10	yes	—
141	Dic₉	C₂⁴×C₃	1	5	yes	—
142	Dic₉	C₂⁴×C₃	1	1	yes	characteristic
143	2₋¹⁺⁴	—	9	5	no	—
144	2₋¹⁺⁴	C₃×D₉	1	1	yes	characteristic, complemented
145	H(5,𝔽₂)	—	9	10	no	—
146	C₂×#41	—	9	10	no	—
147	C₂×#41	—	9	5	no	—
148	C₃×C₉	#585	1	1	yes	characteristic, complemented, Sylow
149	C₂³×C₃	—	9	15	no	—
150	C₂²×D₃	—	3	15	no	—
151	C₃×Q₈	—	2	10	no	—
152	C₃×Q₈	—	9	10	no	—
153	C₃×Q₈	C₂²×D₉	1	10	yes	—
154	C₃×Q₈	C₂²×C₃×D₃	1	10	yes	—
155	C₃×D₄	—	2	10	no	—
156	C₃×D₄	—	9	20	no	—
157	C₃×D₄	—	9	30	no	—
158	C₃×D₄	C₂²×D₉	1	10	yes	—
159	C₃×D₄	C₂²×C₃×D₃	1	10	yes	—
160	C₂×C₃×C₄	—	2	15	no	—
161	C₂×C₃×C₄	—	9	5	no	—
162	C₂×C₃×C₄	—	9	10	no	—
163	C₂×C₃×C₄	—	9	30	no	—
164	C₂×C₃×C₄	C₂²×D₉	1	15	yes	—
165	C₂×C₃×C₄	C₂²×C₃×D₃	1	15	yes	—
166	#67	—	3	20	no	—
167	C₂×Dic₃	—	3	5	no	—
168	D₁₂	—	3	30	no	—
169	C₄×D₃	—	3	10	no	—
170	C₄×D₃	—	3	30	no	—
171	Dic₆	—	3	10	no	—
172	C₂×C₃²	—	2	5	no	—
173	C₂×C₃²	C₂⁴×D₃	1	1	yes	characteristic
174	C₃×D₃	—	6	10	no	—
175	C₂×C₉	—	2	5	no	—
176	C₂×C₉	—	2	1	no	—
177	C₂×C₉	—	4	5	no	—
178	C₂×C₉	C₂⁵×C₃	1	1	yes	characteristic
179	D₉	—	2	10	no	—
180	#41	—	9	10	no	—
181	#41	—	9	20	no	—
182	#41	—	9	10	no	—
183	#41	—	9	30	no	—
184	#41	C₂×C₃×D₉	1	10	yes	complemented
185	C₂×Q₈	—	9	10	no	—
186	C₂×Q₈	C₂×C₃×D₉	1	5	yes	complemented
187	C₂×D₄	—	9	30	no	—
188	C₂×D₄	—	9	15	no	—
189	C₂²×C₄	—	9	15	no	—
190	C₂²×C₃	—	2	5	no	—
191	C₂²×C₃	—	9	10	no	—
192	C₂²×C₃	—	36	15	no	—
193	C₂²×C₃	C₂³×D₉	1	5	yes	—
194	C₂²×C₃	C₂³×C₃×D₃	1	5	yes	—
195	C₂×D₃	—	3	10	no	—
196	C₂×D₃	—	12	15	no	—
197	C₃×C₄	—	2	10	no	—
198	C₃×C₄	—	9	1	no	—
199	C₃×C₄	—	9	5	no	—
200	C₃×C₄	C₂³×D₉	1	10	yes	—
201	C₃×C₄	C₂³×C₃×D₃	1	10	yes	—
202	Dic₃	—	3	5	no	—
203	Dic₃	—	3	1	no	—
204	C₃²	#5968	1	1	yes	characteristic
205	C₉	—	2	1	no	—
206	C₉	C₃×#585	1	1	yes	characteristic, complemented
207	C₂³	—	9	15	no	—
208	Q₈	—	9	10	no	—
209	Q₈	C₂²×C₃×D₉	1	10	yes	—
210	D₄	—	9	20	no	—
211	D₄	—	9	30	no	—
212	D₄	C₂²×C₃×D₉	1	10	yes	—
213	C₂×C₄	—	9	5	no	—
214	C₂×C₄	—	9	10	no	—
215	C₂×C₄	—	9	30	no	—
216	C₂×C₄	C₂²×C₃×D₉	1	15	yes	—
217	C₂×C₃	—	2	5	no	—
218	C₂×C₃	—	2	5	no	—
219	C₂×C₃	—	2	1	no	—
220	C₂×C₃	—	4	5	no	—
221	C₂×C₃	—	18	10	no	—
222	C₂×C₃	C₂⁴×D₉	1	1	yes	characteristic
223	C₂×C₃	C₂⁴×C₃×D₃	1	1	yes	characteristic
224	D₃	—	6	10	no	—
225	C₂²	—	9	10	no	—
226	C₂²	—	36	15	no	—
227	C₂²	C₂³×C₃×D₉	1	5	yes	—
228	C₄	—	9	1	no	—
229	C₄	—	9	5	no	—
230	C₄	C₂³×C₃×D₉	1	10	yes	—
231	C₃	—	2	1	no	—
232	C₃	#10593143	1	1	yes	characteristic, direct product factor
233	C₃	C₃×#5968	1	1	yes	characteristic
234	C₂	—	2	5	no	—
235	C₂	—	18	10	no	—
236	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×C₉×2₋¹⁺⁴	C₂
socle	C₂×C₃²	C₂⁴×D₃
hypercenter	C₃×2₋¹⁺⁴	D₉
hypocenter	C₉	C₃×#585
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₃ ◁ C₃×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. C₃×C₉×2₋¹⁺⁴	C₂	semidirect
2. C₃×#63715	C₂	semidirect
3. C₃×#41×D₉	C₂	semidirect
4. C₃×#63713	C₂	semidirect
5. C₂×C₃×#3701	C₂	semidirect
7. #10593143	C₃	direct (N × H)
8. C₃×C₉×#41	C₂²	semidirect
9. C₂×C₃×Q₈×C₉	C₂²	semidirect
10. C₃×#3701	C₂²	semidirect
11. C₃×#3701	C₂²	semidirect
12. C₃×Q₈×D₉	C₂²	semidirect
13. C₃×#3699	C₂²	semidirect
14. C₃×D₄×D₉	C₂²	semidirect
15. C₃×#3697	C₂²	semidirect
16. C₂×C₃×D₃₆	C₂²	semidirect
17. C₂×C₃×C₄×D₉	C₂²	semidirect
24. C₉×2₋¹⁺⁴	C₂×C₃	semidirect
25. #63715	C₂×C₃	semidirect
26. #41×D₉	C₂×C₃	semidirect
27. #63713	C₂×C₃	semidirect
28. C₂×#3701	C₂×C₃	semidirect
52. C₉×#41	C₂²×C₃	semidirect
54. C₂×Q₈×C₉	C₂²×C₃	semidirect
55. #3701	C₂²×C₃	semidirect
56. #3701	C₂²×C₃	semidirect
57. Q₈×D₉	C₂²×C₃	semidirect
58. #3699	C₂²×C₃	semidirect
59. D₄×D₉	C₂²×C₃	semidirect
60. #3697	C₂²×C₃	semidirect
61. C₂×D₃₆	C₂²×C₃	semidirect
62. C₂×C₄×D₉	C₂²×C₃	semidirect
71. C₃×2₋¹⁺⁴	D₉	semidirect
112. C₃×#41	C₂×D₉	semidirect
116. C₂×C₃×Q₈	C₂×D₉	semidirect
144. 2₋¹⁺⁴	C₃×D₉	semidirect
148. C₃×C₉	#585	semidirect
184. #41	C₂×C₃×D₉	semidirect
186. C₂×Q₈	C₂×C₃×D₉	semidirect
206. C₉	C₃×#585	semidirect
232. C₃	#10593143	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×C₉×2₋¹⁺⁴	2
C₃×#63715	2
C₃×#41×D₉	2
C₃×#63713	2
C₂×C₃×#3701	2
C₃×#5968	3
#10593143	3

Subgroups — C3×#10593143 #13164956

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10593143 #13164956