Subgroups — C₃×C₅×#879 #13087891

Order 1440.

274 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#217	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#217	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×Dic₃⋊C₄	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×#879	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂³×C₃²×C₅	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×C₃×C₅×Dic₃	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×#879	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₃×C₅×C₄⋊C₄	—	3	1	no	—
19	C₃×C₅×C₄⋊C₄	—	3	1	no	—
20	C₃×C₅×C₄⋊C₄	—	3	1	no	—
21	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
22	C₃×C₅×C₂²⋊C₄	—	3	1	no	—
23	C₃×C₅×C₂²⋊C₄	—	3	1	no	—
24	C₃×C₅×C₂²⋊C₄	D₃	1	1	yes	characteristic
25	C₃×C₅×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
26	C₃×C₄²×C₅	—	3	1	no	—
27	C₅×#217	C₂×C₃	1	1	yes	characteristic
28	C₅×#217	C₂×C₃	1	1	yes	characteristic
29	C₅×C₄⋊Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
30	C₅×Dic₃⋊C₄	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	—	2	1	no	—
34	C₂²×C₃²×C₅	—	2	1	no	—
35	C₂²×C₃²×C₅	—	2	1	no	—
36	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
37	C₃²×C₄×C₅	—	2	1	no	—
38	C₃²×C₄×C₅	—	2	1	no	—
39	C₃×C₅×Dic₃	—	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×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic
45	C₃×#217	C₂×C₅	1	1	yes	characteristic
46	C₃×#217	C₂×C₅	1	1	yes	characteristic
47	C₃×C₄⋊Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
48	C₃×Dic₃⋊C₄	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	—	2	1	no	—
52	C₂³×C₃×C₅	C₂×D₃	1	1	yes	characteristic
53	C₂³×C₃×C₅	C₂²×C₃	1	1	yes	characteristic
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	—	3	1	no	—
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×C₅×Dic₃	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	#879	C₃×C₅	1	1	yes	characteristic, direct product factor
70	C₂×C₃²×C₅	—	4	1	no	—
71	C₂×C₃²×C₅	#41	1	1	yes	characteristic
72	C₂×C₃²×C₅	#41	1	1	yes	characteristic
73	C₂×C₃²×C₅	#41	1	1	yes	characteristic
74	C₅×C₄⋊C₄	—	3	1	no	—
75	C₅×C₄⋊C₄	—	3	1	no	—
76	C₅×C₄⋊C₄	—	3	1	no	—
77	C₅×C₂²⋊C₄	—	3	1	no	—
78	C₅×C₂²⋊C₄	—	3	1	no	—
79	C₅×C₂²⋊C₄	C₃×D₃	1	1	yes	characteristic
80	C₄²×C₅	—	3	1	no	—
81	C₂³×C₃²	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×C₃×Dic₃	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	2	no	—
89	C₂²×C₃×C₅	—	2	1	no	—
90	C₂²×C₃×C₅	—	2	1	no	—
91	C₂²×C₃×C₅	—	2	1	no	—
92	C₂²×C₃×C₅	—	2	1	no	—
93	C₂²×C₃×C₅	—	2	1	no	—
94	C₂²×C₃×C₅	—	2	1	no	—
95	C₂²×C₃×C₅	—	2	1	no	—
96	C₂²×C₃×C₅	—	4	1	no	—
97	C₂²×C₃×C₅	—	4	1	no	—
98	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
99	C₂²×C₃×C₅	C₂³×C₃	1	1	yes	characteristic
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	C₃×C₄×C₅	—	6	1	no	—
110	C₅×Dic₃	—	2	1	no	—
111	C₅×Dic₃	—	2	1	no	—
112	C₅×Dic₃	—	2	1	no	—
113	C₅×Dic₃	—	2	1	no	—
114	C₃×C₄⋊C₄	—	3	1	no	—
115	C₃×C₄⋊C₄	—	3	1	no	—
116	C₃×C₄⋊C₄	—	3	1	no	—
117	C₃×C₂²⋊C₄	—	2	1	no	—
118	C₃×C₂²⋊C₄	—	3	1	no	—
119	C₃×C₂²⋊C₄	—	3	1	no	—
120	C₃×C₂²⋊C₄	C₅×D₃	1	1	yes	characteristic
121	C₃×C₂²⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic
122	C₃×C₄²	—	3	1	no	—
123	#217	C₂×C₃×C₅	1	1	yes	characteristic
124	#217	C₂×C₃×C₅	1	1	yes	characteristic
125	C₄⋊Dic₃	C₂×C₃×C₅	1	1	yes	characteristic, complemented
126	Dic₃⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic, complemented
127	Dic₃⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic, complemented
128	C₄×Dic₃	C₂×C₃×C₅	1	1	yes	characteristic, complemented
129	C₃²×C₅	#126	1	1	yes	characteristic, complemented, Hall
130	C₂³×C₅	C₂×C₃×D₃	1	1	yes	characteristic
131	C₂×C₄×C₅	—	3	1	no	—
132	C₂×C₄×C₅	—	3	1	no	—
133	C₂×C₄×C₅	—	3	1	no	—
134	C₂×C₄×C₅	—	3	1	no	—
135	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
137	C₂²×C₃²	—	2	1	no	—
138	C₂²×C₃²	—	2	1	no	—
139	C₂²×C₃²	—	2	1	no	—
140	C₂²×C₃²	C₂³×C₅	1	1	yes	characteristic
141	C₃²×C₄	—	2	1	no	—
142	C₃²×C₄	—	2	1	no	—
143	C₃×Dic₃	—	2	1	no	—
144	C₃×Dic₃	—	2	1	no	—
145	C₃×Dic₃	—	2	1	no	—
146	C₃×Dic₃	—	2	1	no	—
147	#126	—	3	1	no	Sylow
148	C₂×C₃×C₅	—	2	1	no	—
149	C₂×C₃×C₅	—	2	1	no	—
150	C₂×C₃×C₅	—	2	1	no	—
151	C₂×C₃×C₅	—	4	2	no	—
152	C₂×C₃×C₅	—	4	1	no	—
153	C₂×C₃×C₅	—	4	1	no	—
154	C₂×C₃×C₅	#235	1	1	yes	characteristic
155	C₂×C₃×C₅	#237	1	1	yes	characteristic
156	C₂×C₃×C₅	#237	1	1	yes	characteristic
157	C₂×C₃×C₅	C₃×#41	1	1	yes	characteristic
158	C₂×C₃×C₅	C₃×#41	1	1	yes	characteristic
159	C₂×C₃×C₅	C₃×#41	1	1	yes	characteristic
160	C₂³×C₃	—	2	1	no	—
161	C₂³×C₃	C₂×C₅×D₃	1	1	yes	characteristic
162	C₂³×C₃	C₂²×C₃×C₅	1	1	yes	characteristic
163	C₂×C₃×C₄	—	2	1	no	—
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	3	1	no	—
166	C₂×C₃×C₄	—	3	1	no	—
167	C₂×C₃×C₄	—	3	1	no	—
168	C₂×C₃×C₄	—	3	1	no	—
169	C₂×C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
173	C₂×Dic₃	C₂²×C₃×C₅	1	1	yes	characteristic
174	C₂×Dic₃	C₂²×C₃×C₅	1	1	yes	characteristic
175	C₂×Dic₃	C₂²×C₃×C₅	1	1	yes	characteristic
176	C₂×Dic₃	C₂²×C₃×C₅	1	1	yes	characteristic
177	C₂²×C₅	—	2	1	no	—
178	C₂²×C₅	—	2	1	no	—
179	C₂²×C₅	—	2	1	no	—
180	C₂²×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
181	C₄×C₅	—	2	1	no	—
182	C₄×C₅	—	2	1	no	—
183	C₄×C₅	—	6	1	no	—
184	C₄×C₅	—	6	1	no	—
185	C₄×C₅	—	6	1	no	—
186	C₄×C₅	—	6	1	no	—
187	C₂×C₃²	—	4	1	no	—
188	C₂×C₃²	C₅×#41	1	1	yes	characteristic
189	C₂×C₃²	C₅×#41	1	1	yes	characteristic
190	C₂×C₃²	C₅×#41	1	1	yes	characteristic
191	C₄⋊C₄	—	3	1	no	—
192	C₄⋊C₄	—	3	1	no	—
193	C₄⋊C₄	—	3	1	no	—
194	C₂²⋊C₄	—	3	1	no	—
195	C₂²⋊C₄	—	3	1	no	—
196	C₂²⋊C₄	C₃×C₅×D₃	1	1	yes	characteristic
197	C₄²	—	3	1	no	—
198	C₃×C₅	—	2	1	no	—
199	C₃×C₅	#879	1	1	yes	characteristic, direct product factor
200	C₃×C₅	C₃×#126	1	1	yes	characteristic, complemented
201	C₂²×C₃	—	2	2	no	—
202	C₂²×C₃	—	2	1	no	—
203	C₂²×C₃	—	2	1	no	—
204	C₂²×C₃	—	2	1	no	—
205	C₂²×C₃	—	2	1	no	—
206	C₂²×C₃	—	2	1	no	—
207	C₂²×C₃	—	2	1	no	—
208	C₂²×C₃	—	2	1	no	—
209	C₂²×C₃	—	4	1	no	—
210	C₂²×C₃	—	4	1	no	—
211	C₂²×C₃	C₂²×C₅×D₃	1	1	yes	characteristic
212	C₂²×C₃	C₂³×C₃×C₅	1	1	yes	characteristic
213	C₃×C₄	—	2	1	no	—
214	C₃×C₄	—	2	1	no	—
215	C₃×C₄	—	2	1	no	—
216	C₃×C₄	—	2	1	no	—
217	C₃×C₄	—	4	1	no	—
218	C₃×C₄	—	4	1	no	—
219	C₃×C₄	—	6	1	no	—
220	C₃×C₄	—	6	1	no	—
221	C₃×C₄	—	6	1	no	—
222	C₃×C₄	—	6	1	no	—
223	Dic₃	—	2	1	no	—
224	Dic₃	—	2	1	no	—
225	Dic₃	—	2	1	no	—
226	Dic₃	—	2	1	no	—
227	C₂×C₅	—	4	1	no	—
228	C₂×C₅	C₃×#235	1	1	yes	characteristic
229	C₂×C₅	C₃×#237	1	1	yes	characteristic
230	C₂×C₅	C₃×#237	1	1	yes	characteristic
231	C₃²	C₅×#126	1	1	yes	characteristic, complemented, Sylow
232	C₂³	C₂×C₃×C₅×D₃	1	1	yes	characteristic
233	C₂×C₄	—	3	1	no	—
234	C₂×C₄	—	3	1	no	—
235	C₂×C₄	—	3	1	no	—
236	C₂×C₄	—	3	1	no	—
237	C₂×C₄	C₂×C₃×C₅×D₃	1	1	yes	characteristic
238	C₂×C₄	C₂×C₃×C₅×D₃	1	1	yes	characteristic
239	C₂×C₃	—	2	1	no	—
240	C₂×C₃	—	2	1	no	—
241	C₂×C₃	—	2	1	no	—
242	C₂×C₃	—	4	2	no	—
243	C₂×C₃	—	4	1	no	—
244	C₂×C₃	—	4	1	no	—
245	C₂×C₃	C₅×#235	1	1	yes	characteristic
246	C₂×C₃	C₅×#237	1	1	yes	characteristic
247	C₂×C₃	C₅×#237	1	1	yes	characteristic
248	C₂×C₃	C₃×C₅×#41	1	1	yes	characteristic
249	C₂×C₃	C₃×C₅×#41	1	1	yes	characteristic
250	C₂×C₃	C₃×C₅×#41	1	1	yes	characteristic
251	C₅	C₃×#879	1	1	yes	characteristic, direct product factor, Sylow
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	C₂²×C₃×C₅×D₃	1	1	yes	characteristic
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	6	1	no	—
259	C₄	—	6	1	no	—
260	C₄	—	6	1	no	—
261	C₄	—	6	1	no	—
262	C₃	—	2	1	no	—
263	C₃	C₅×#879	1	1	yes	characteristic, direct product factor
264	C₃	C₃×C₅×#126	1	1	yes	characteristic, complemented
265	C₂	—	4	1	no	—
266	C₂	C₃×C₅×#235	1	1	yes	characteristic
267	C₂	C₃×C₅×#237	1	1	yes	characteristic
268	C₂	C₃×C₅×#237	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×C₂²⋊C₄	C₂
socle	C₂²×C₃²×C₅	C₂³
prime radical	C₂³×C₃²×C₅	C₂²
hypercenter	C₃×C₅×C₂²⋊C₄	D₃
hypocenter	C₃	C₃×C₅×#126
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₅×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×C₄⋊Dic₃	C₂	semidirect
5. C₃×C₅×Dic₃⋊C₄	C₂	semidirect
6. C₃×C₅×Dic₃⋊C₄	C₂	semidirect
7. C₃×C₄×C₅×Dic₃	C₂	semidirect
9. C₅×#879	C₃	direct (N × H)
17. C₃×#879	C₅	direct (N × H)
29. C₅×C₄⋊Dic₃	C₂×C₃	semidirect
30. C₅×Dic₃⋊C₄	C₂×C₃	semidirect
31. C₅×Dic₃⋊C₄	C₂×C₃	semidirect
32. C₄×C₅×Dic₃	C₂×C₃	semidirect
47. C₃×C₄⋊Dic₃	C₂×C₅	semidirect
48. C₃×Dic₃⋊C₄	C₂×C₅	semidirect
49. C₃×Dic₃⋊C₄	C₂×C₅	semidirect
50. C₃×C₄×Dic₃	C₂×C₅	semidirect
69. #879	C₃×C₅	direct (N × H)
125. C₄⋊Dic₃	C₂×C₃×C₅	semidirect
126. Dic₃⋊C₄	C₂×C₃×C₅	semidirect
127. Dic₃⋊C₄	C₂×C₃×C₅	semidirect
128. C₄×Dic₃	C₂×C₃×C₅	semidirect
129. C₃²×C₅	#126	semidirect
199. C₃×C₅	#879	direct (N × H)
200. C₃×C₅	C₃×#126	semidirect
231. C₃²	C₅×#126	semidirect
251. C₅	C₃×#879	direct (N × H)
263. C₃	C₅×#879	direct (N × H)
264. 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×C₂²⋊C₄	2
C₃×C₅×#217	2
C₃×C₅×C₄⋊Dic₃	2
C₃×C₅×Dic₃⋊C₄	2
C₃×C₄×C₅×Dic₃	2
C₃×C₅×#126	3
C₅×#879	3
C₃×#879	5

Subgroups — C3×C5×#879 #13087891

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#879 #13087891