Subgroups — C₃×C₅×#898 #13087909

Order 1440.

266 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×(C₂×D₃)⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×(C₂×D₃)⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×(C₂×D₃)⋊C₄	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₅×Dic₃⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₄×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₅×#126	—	3	1	no	maximal
9	C₅×#898	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂²×C₃×C₅×D₃	C₂²	1	1	yes	characteristic
11	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
17	C₃×#898	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₃×C₅×C₄⋊C₄	—	2	1	no	—
19	C₃×C₅×C₄⋊C₄	—	3	1	no	—
20	C₃×C₅×C₄⋊C₄	—	3	1	no	—
21	C₃×C₅×C₄⋊C₄	D₃	1	1	yes	characteristic, complemented
22	C₃×C₅×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
23	C₃×C₅×C₂²⋊C₄	—	3	1	no	—
24	C₃×C₅×C₂²⋊C₄	—	3	1	no	—
25	C₃×C₅×C₂²⋊C₄	—	3	1	no	—
26	C₃×C₄²×C₅	—	3	1	no	—
27	C₅×(C₂×D₃)⋊C₄	C₂×C₃	1	1	yes	characteristic
28	C₅×(C₂×D₃)⋊C₄	C₂×C₃	1	1	yes	characteristic
29	C₅×(C₂×D₃)⋊C₄	C₂×C₃	1	1	yes	characteristic
30	C₅×C₄⋊Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
31	C₅×Dic₃⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
32	C₄×C₅×Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
33	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
34	C₂×C₃×C₅×D₃	—	2	1	no	—
35	C₂×C₃×C₅×D₃	—	2	1	no	—
36	C₂×C₃×C₅×D₃	—	2	1	no	—
37	C₃²×C₄×C₅	—	2	1	no	—
38	C₃²×C₄×C₅	—	2	1	no	—
39	C₃²×C₄×C₅	—	2	1	no	—
40	C₃×C₅×Dic₃	—	2	1	no	—
41	C₃×C₅×Dic₃	—	2	1	no	—
42	C₃×C₅×Dic₃	—	2	1	no	—
43	C₅×#126	—	3	1	no	Hall
44	C₃²×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
45	C₃×(C₂×D₃)⋊C₄	C₂×C₅	1	1	yes	characteristic
46	C₃×(C₂×D₃)⋊C₄	C₂×C₅	1	1	yes	characteristic
47	C₃×(C₂×D₃)⋊C₄	C₂×C₅	1	1	yes	characteristic
48	C₃×C₄⋊Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
49	C₃×Dic₃⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
50	C₃×C₄×Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
51	C₂³×C₃×C₅	—	3	1	no	—
52	C₂²×C₅×D₃	C₂²×C₃	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	—	2	1	no	—
54	C₂×C₃×C₄×C₅	—	2	1	no	—
55	C₂×C₃×C₄×C₅	—	2	1	no	—
56	C₂×C₃×C₄×C₅	—	3	1	no	—
57	C₂×C₃×C₄×C₅	—	3	1	no	—
58	C₂×C₃×C₄×C₅	—	3	1	no	—
59	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
64	C₂×C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
65	C₂×C₅×Dic₃	C₂²×C₃	1	1	yes	characteristic
66	C₂×C₅×Dic₃	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₅×Dic₃	C₂²×C₃	1	1	yes	characteristic
68	C₃×#126	—	3	1	no	—
69	#898	C₃×C₅	1	1	yes	characteristic, direct product factor
70	C₂×C₃²×C₅	#41	1	1	yes	characteristic
71	C₂×C₃²×C₅	#41	1	1	yes	characteristic
72	C₂×C₃²×C₅	#41	1	1	yes	characteristic
73	C₃×C₅×D₃	—	4	1	no	—
74	C₅×C₄⋊C₄	—	3	1	no	—
75	C₅×C₄⋊C₄	—	3	1	no	—
76	C₅×C₄⋊C₄	C₃×D₃	1	1	yes	characteristic, complemented
77	C₅×C₂²⋊C₄	—	3	1	no	—
78	C₅×C₂²⋊C₄	—	3	1	no	—
79	C₅×C₂²⋊C₄	—	3	1	no	—
80	C₄²×C₅	—	3	1	no	—
81	C₂²×C₃×D₃	C₂²×C₅	1	1	yes	characteristic
82	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
83	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
84	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
85	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic
86	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic
87	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic
88	C₂²×C₃×C₅	—	2	1	no	—
89	C₂²×C₃×C₅	—	6	1	no	—
90	C₂²×C₃×C₅	—	6	1	no	—
91	C₂²×C₃×C₅	—	6	1	no	—
92	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
93	C₂²×C₃×C₅	C₂³×C₃	1	1	yes	characteristic
94	C₂×C₅×D₃	—	2	1	no	—
95	C₂×C₅×D₃	—	2	1	no	—
96	C₂×C₅×D₃	—	2	1	no	—
97	C₃×C₄×C₅	—	2	2	no	—
98	C₃×C₄×C₅	—	2	1	no	—
99	C₃×C₄×C₅	—	2	1	no	—
100	C₃×C₄×C₅	—	2	1	no	—
101	C₃×C₄×C₅	—	2	1	no	—
102	C₃×C₄×C₅	—	2	1	no	—
103	C₃×C₄×C₅	—	2	1	no	—
104	C₃×C₄×C₅	—	4	1	no	—
105	C₃×C₄×C₅	—	4	1	no	—
106	C₃×C₄×C₅	—	6	1	no	—
107	C₃×C₄×C₅	—	6	1	no	—
108	C₃×C₄×C₅	—	6	1	no	—
109	C₅×Dic₃	—	2	1	no	—
110	C₅×Dic₃	—	2	1	no	—
111	C₅×Dic₃	—	2	1	no	—
112	C₃×C₄⋊C₄	—	2	1	no	—
113	C₃×C₄⋊C₄	—	3	1	no	—
114	C₃×C₄⋊C₄	—	3	1	no	—
115	C₃×C₄⋊C₄	C₅×D₃	1	1	yes	characteristic, complemented
116	C₃×C₄⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic, complemented
117	C₃×C₂²⋊C₄	—	3	1	no	—
118	C₃×C₂²⋊C₄	—	3	1	no	—
119	C₃×C₂²⋊C₄	—	3	1	no	—
120	C₃×C₄²	—	3	1	no	—
121	(C₂×D₃)⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic
122	(C₂×D₃)⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic
123	(C₂×D₃)⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic
124	C₄⋊Dic₃	C₂×C₃×C₅	1	1	yes	characteristic, complemented
125	Dic₃⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic, complemented
126	C₄×Dic₃	C₂×C₃×C₅	1	1	yes	characteristic, complemented
127	C₃²×C₅	#126	1	1	yes	characteristic, complemented, Hall
128	C₂³×C₅	—	3	1	no	—
129	C₂×C₄×C₅	—	3	1	no	—
130	C₂×C₄×C₅	—	3	1	no	—
131	C₂×C₄×C₅	—	3	1	no	—
132	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
135	C₂²×C₃²	C₂³×C₅	1	1	yes	characteristic
136	C₂×C₃×D₃	—	2	1	no	—
137	C₂×C₃×D₃	—	2	1	no	—
138	C₂×C₃×D₃	—	2	1	no	—
139	C₃²×C₄	—	2	1	no	—
140	C₃²×C₄	—	2	1	no	—
141	C₃²×C₄	—	2	1	no	—
142	C₃×Dic₃	—	2	1	no	—
143	C₃×Dic₃	—	2	1	no	—
144	C₃×Dic₃	—	2	1	no	—
145	#126	—	3	1	no	Sylow
146	C₂×C₃×C₅	—	2	1	no	—
147	C₂×C₃×C₅	—	2	1	no	—
148	C₂×C₃×C₅	—	2	1	no	—
149	C₂×C₃×C₅	—	12	1	no	—
150	C₂×C₃×C₅	#235	1	1	yes	characteristic
151	C₂×C₃×C₅	#237	1	1	yes	characteristic
152	C₂×C₃×C₅	#239	1	1	yes	characteristic
153	C₂×C₃×C₅	C₃×#41	1	1	yes	characteristic
154	C₂×C₃×C₅	C₃×#41	1	1	yes	characteristic
155	C₂×C₃×C₅	C₃×#41	1	1	yes	characteristic
156	C₅×D₃	—	4	1	no	—
157	C₂³×C₃	—	3	1	no	—
158	C₂²×D₃	C₂²×C₃×C₅	1	1	yes	characteristic
159	C₂×C₃×C₄	—	2	1	no	—
160	C₂×C₃×C₄	—	2	1	no	—
161	C₂×C₃×C₄	—	2	1	no	—
162	C₂×C₃×C₄	—	3	1	no	—
163	C₂×C₃×C₄	—	3	1	no	—
164	C₂×C₃×C₄	—	3	1	no	—
165	C₂×C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
171	C₂×Dic₃	C₂²×C₃×C₅	1	1	yes	characteristic
172	C₂×Dic₃	C₂²×C₃×C₅	1	1	yes	characteristic
173	C₂×Dic₃	C₂²×C₃×C₅	1	1	yes	characteristic
174	C₂²×C₅	—	6	1	no	—
175	C₂²×C₅	—	6	1	no	—
176	C₂²×C₅	—	6	1	no	—
177	C₂²×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
178	C₄×C₅	—	2	1	no	—
179	C₄×C₅	—	2	1	no	—
180	C₄×C₅	—	2	1	no	—
181	C₄×C₅	—	6	1	no	—
182	C₄×C₅	—	6	1	no	—
183	C₄×C₅	—	6	1	no	—
184	C₂×C₃²	C₅×#41	1	1	yes	characteristic
185	C₂×C₃²	C₅×#41	1	1	yes	characteristic
186	C₂×C₃²	C₅×#41	1	1	yes	characteristic
187	C₃×D₃	—	4	1	no	—
188	C₄⋊C₄	—	3	1	no	—
189	C₄⋊C₄	—	3	1	no	—
190	C₄⋊C₄	C₃×C₅×D₃	1	1	yes	characteristic, complemented
191	C₂²⋊C₄	—	3	1	no	—
192	C₂²⋊C₄	—	3	1	no	—
193	C₂²⋊C₄	—	3	1	no	—
194	C₄²	—	3	1	no	—
195	C₃×C₅	—	2	1	no	—
196	C₃×C₅	#898	1	1	yes	characteristic, direct product factor
197	C₃×C₅	C₃×#126	1	1	yes	characteristic, complemented
198	C₂²×C₃	—	2	1	no	—
199	C₂²×C₃	—	6	1	no	—
200	C₂²×C₃	—	6	1	no	—
201	C₂²×C₃	—	6	1	no	—
202	C₂²×C₃	C₂²×C₅×D₃	1	1	yes	characteristic
203	C₂²×C₃	C₂³×C₃×C₅	1	1	yes	characteristic
204	C₂×D₃	—	2	1	no	—
205	C₂×D₃	—	2	1	no	—
206	C₂×D₃	—	2	1	no	—
207	C₃×C₄	—	2	2	no	—
208	C₃×C₄	—	2	1	no	—
209	C₃×C₄	—	2	1	no	—
210	C₃×C₄	—	2	1	no	—
211	C₃×C₄	—	2	1	no	—
212	C₃×C₄	—	2	1	no	—
213	C₃×C₄	—	2	1	no	—
214	C₃×C₄	—	4	1	no	—
215	C₃×C₄	—	4	1	no	—
216	C₃×C₄	—	6	1	no	—
217	C₃×C₄	—	6	1	no	—
218	C₃×C₄	—	6	1	no	—
219	Dic₃	—	2	1	no	—
220	Dic₃	—	2	1	no	—
221	Dic₃	—	2	1	no	—
222	C₂×C₅	—	12	1	no	—
223	C₂×C₅	C₃×#235	1	1	yes	characteristic
224	C₂×C₅	C₃×#237	1	1	yes	characteristic
225	C₂×C₅	C₃×#239	1	1	yes	characteristic
226	C₃²	C₅×#126	1	1	yes	characteristic, complemented, Sylow
227	C₂³	—	3	1	no	—
228	C₂×C₄	—	3	1	no	—
229	C₂×C₄	—	3	1	no	—
230	C₂×C₄	—	3	1	no	—
231	C₂×C₄	C₂×C₃×C₅×D₃	1	1	yes	characteristic
232	C₂×C₄	C₂×C₃×C₅×D₃	1	1	yes	characteristic
233	C₂×C₄	C₂×C₃×C₅×D₃	1	1	yes	characteristic
234	C₂×C₃	—	2	1	no	—
235	C₂×C₃	—	2	1	no	—
236	C₂×C₃	—	2	1	no	—
237	C₂×C₃	—	12	1	no	—
238	C₂×C₃	C₅×#235	1	1	yes	characteristic
239	C₂×C₃	C₅×#237	1	1	yes	characteristic
240	C₂×C₃	C₅×#239	1	1	yes	characteristic
241	C₂×C₃	C₃×C₅×#41	1	1	yes	characteristic
242	C₂×C₃	C₃×C₅×#41	1	1	yes	characteristic
243	C₂×C₃	C₃×C₅×#41	1	1	yes	characteristic
244	D₃	—	4	1	no	—
245	C₅	C₃×#898	1	1	yes	characteristic, direct product factor, Sylow
246	C₂²	—	6	1	no	—
247	C₂²	—	6	1	no	—
248	C₂²	—	6	1	no	—
249	C₂²	C₂²×C₃×C₅×D₃	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	6	1	no	—
254	C₄	—	6	1	no	—
255	C₄	—	6	1	no	—
256	C₃	—	2	1	no	—
257	C₃	C₅×#898	1	1	yes	characteristic, direct product factor
258	C₃	C₃×C₅×#126	1	1	yes	characteristic, complemented
259	C₂	—	12	1	no	—
260	C₂	C₃×C₅×#235	1	1	yes	characteristic
261	C₂	C₃×C₅×#237	1	1	yes	characteristic
262	C₂	C₃×C₅×#239	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₅	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂³×C₃×C₅
Frattini	C₂²	C₂²×C₃×C₅×D₃
Fitting	C₃²×C₅×C₄⋊C₄	C₂
socle	C₂²×C₃²×C₅	C₂³
prime radical	C₂²×C₃×C₅×D₃	C₂²
hypercenter	C₃×C₅×C₄⋊C₄	D₃
hypocenter	C₃	C₃×C₅×#126
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₅×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅×C₄⋊C₄	C₂	semidirect
5. C₃×C₅×C₄⋊Dic₃	C₂	semidirect
6. C₃×C₅×Dic₃⋊C₄	C₂	semidirect
7. C₃×C₄×C₅×Dic₃	C₂	semidirect
9. C₅×#898	C₃	direct (N × H)
17. C₃×#898	C₅	direct (N × H)
21. C₃×C₅×C₄⋊C₄	D₃	semidirect
22. C₃×C₅×C₄⋊C₄	C₂×C₃	semidirect
30. C₅×C₄⋊Dic₃	C₂×C₃	semidirect
31. C₅×Dic₃⋊C₄	C₂×C₃	semidirect
32. C₄×C₅×Dic₃	C₂×C₃	semidirect
44. C₃²×C₄⋊C₄	C₂×C₅	semidirect
48. C₃×C₄⋊Dic₃	C₂×C₅	semidirect
49. C₃×Dic₃⋊C₄	C₂×C₅	semidirect
50. C₃×C₄×Dic₃	C₂×C₅	semidirect
69. #898	C₃×C₅	direct (N × H)
76. C₅×C₄⋊C₄	C₃×D₃	semidirect
115. C₃×C₄⋊C₄	C₅×D₃	semidirect
116. C₃×C₄⋊C₄	C₂×C₃×C₅	semidirect
124. C₄⋊Dic₃	C₂×C₃×C₅	semidirect
125. Dic₃⋊C₄	C₂×C₃×C₅	semidirect
126. C₄×Dic₃	C₂×C₃×C₅	semidirect
127. C₃²×C₅	#126	semidirect
190. C₄⋊C₄	C₃×C₅×D₃	semidirect
196. C₃×C₅	#898	direct (N × H)
197. C₃×C₅	C₃×#126	semidirect
226. C₃²	C₅×#126	semidirect
245. C₅	C₃×#898	direct (N × H)
257. C₃	C₅×#898	direct (N × H)
258. C₃	C₃×C₅×#126	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₅×C₄⋊C₄	2
C₃×C₅×(C₂×D₃)⋊C₄	2
C₃×C₅×C₄⋊Dic₃	2
C₃×C₅×Dic₃⋊C₄	2
C₃×C₄×C₅×Dic₃	2
C₃×C₅×#126	3
C₅×#898	3
C₃×#898	5

Subgroups — C3×C5×#898 #13087909

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#898 #13087909