Subgroups — C₃×C₅×#878 #13087889

Order 1440.

330 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×C₅×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#217	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×C₅×Dic₃⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₅×C₂²⋊Q₈	—	3	1	no	maximal
9	C₅×#878	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	—	2	2	no	—
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	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
18	C₃×C₅×Dic₆	—	2	2	no	—
19	C₃×#878	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₂×C₃×C₅×Q₈	—	3	1	no	—
21	C₂²×C₃×C₄×C₅	—	3	1	no	—
22	C₂²×C₅×Dic₃	C₂×C₃	1	1	yes	characteristic
23	C₂×C₅×Dic₆	C₂×C₃	1	1	yes	characteristic, complemented
24	C₃×C₅×C₄⋊C₄	—	3	1	no	—
25	C₃×C₅×C₄⋊C₄	—	3	1	no	—
26	C₃×C₅×C₄⋊C₄	—	3	1	no	—
27	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
28	C₃×C₅×C₂²⋊C₄	—	3	1	no	—
29	C₃×C₅×C₂²⋊C₄	D₃	1	1	yes	characteristic
30	C₃×C₅×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
31	C₅×#217	C₂×C₃	1	1	yes	characteristic
32	C₅×C₄⋊Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
33	C₅×Dic₃⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
34	C₅×Dic₃⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
35	C₂²×C₃²×C₅	—	2	1	no	—
36	C₂²×C₃²×C₅	—	2	1	no	—
37	C₂²×C₃²×C₅	Q₈	1	2	yes	complemented
38	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
39	C₃²×C₄×C₅	—	2	1	no	—
40	C₃²×C₄×C₅	—	2	1	no	—
41	C₃×C₅×Dic₃	—	2	1	no	—
42	C₃×C₅×Dic₃	—	2	1	no	—
43	C₃×C₅×Dic₃	—	2	1	no	—
44	C₃×C₅×Dic₃	D₄	1	2	yes	—
45	C₅×C₂²⋊Q₈	—	3	1	no	Hall
46	C₂²×C₃×Dic₃	C₂×C₅	1	1	yes	characteristic
47	C₂×C₃×Dic₆	C₂×C₅	1	1	yes	characteristic, complemented
48	C₃²×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic
49	C₃×#217	C₂×C₅	1	1	yes	characteristic
50	C₃×C₄⋊Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
51	C₃×Dic₃⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
52	C₃×Dic₃⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
53	C₂³×C₃×C₅	—	2	1	no	—
54	C₂³×C₃×C₅	C₂×D₃	1	1	yes	characteristic
55	C₂³×C₃×C₅	C₂²×C₃	1	1	yes	characteristic
56	C₃×C₅×Q₈	—	6	2	no	—
57	C₂×C₃×C₄×C₅	—	2	1	no	—
58	C₂×C₃×C₄×C₅	—	2	1	no	—
59	C₂×C₃×C₄×C₅	—	3	1	no	—
60	C₂×C₃×C₄×C₅	—	3	1	no	—
61	C₂×C₃×C₄×C₅	—	3	1	no	—
62	C₂×C₃×C₄×C₅	—	3	1	no	—
63	C₂×C₃×C₄×C₅	—	6	2	no	—
64	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₅×Dic₃	—	2	2	no	—
69	C₂×C₅×Dic₃	C₂²×C₃	1	1	yes	characteristic
70	C₂×C₅×Dic₃	C₂²×C₃	1	1	yes	characteristic
71	C₂×C₅×Dic₃	C₂²×C₃	1	1	yes	characteristic
72	C₂×C₅×Dic₃	C₂²×C₃	1	1	yes	characteristic
73	C₅×Dic₆	—	2	2	no	—
74	C₃×C₂²⋊Q₈	—	3	1	no	—
75	#878	C₃×C₅	1	1	yes	characteristic, direct product factor
76	C₂×C₃²×C₅	—	2	2	no	—
77	C₂×C₃²×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃²×C₅	C₂×Q₈	1	1	yes	characteristic
79	C₂×C₃²×C₅	#41	1	1	yes	characteristic
80	C₂×C₅×Q₈	—	3	1	no	—
81	C₂²×C₄×C₅	—	3	1	no	—
82	C₅×C₄⋊C₄	—	3	1	no	—
83	C₅×C₄⋊C₄	—	3	1	no	—
84	C₅×C₄⋊C₄	—	3	1	no	—
85	C₅×C₂²⋊C₄	—	3	1	no	—
86	C₅×C₂²⋊C₄	C₃×D₃	1	1	yes	characteristic
87	C₂³×C₃²	C₂²×C₅	1	1	yes	characteristic
88	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
89	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
90	C₂×C₃×Dic₃	—	2	2	no	—
91	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic
92	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic
93	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic
94	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic
95	C₃×Dic₆	—	2	2	no	—
96	C₂²×C₃×C₅	—	2	1	no	—
97	C₂²×C₃×C₅	—	2	1	no	—
98	C₂²×C₃×C₅	—	2	1	no	—
99	C₂²×C₃×C₅	—	2	1	no	—
100	C₂²×C₃×C₅	—	2	2	no	—
101	C₂²×C₃×C₅	—	2	1	no	—
102	C₂²×C₃×C₅	—	4	1	no	—
103	C₂²×C₃×C₅	—	4	1	no	—
104	C₂²×C₃×C₅	Dic₆	1	2	yes	complemented
105	C₂²×C₃×C₅	C₃×Q₈	1	2	yes	complemented
106	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
107	C₂²×C₃×C₅	C₂³×C₃	1	1	yes	characteristic
108	C₃×C₄×C₅	—	2	1	no	—
109	C₃×C₄×C₅	—	2	1	no	—
110	C₃×C₄×C₅	—	2	1	no	—
111	C₃×C₄×C₅	—	2	1	no	—
112	C₃×C₄×C₅	—	3	2	no	—
113	C₃×C₄×C₅	—	4	1	no	—
114	C₃×C₄×C₅	—	4	1	no	—
115	C₃×C₄×C₅	—	6	1	no	—
116	C₃×C₄×C₅	—	6	1	no	—
117	C₃×C₄×C₅	—	6	1	no	—
118	C₅×Dic₃	—	2	1	no	—
119	C₅×Dic₃	—	2	1	no	—
120	C₅×Dic₃	—	2	1	no	—
121	C₅×Dic₃	C₃×D₄	1	2	yes	—
122	C₂×C₃×Q₈	—	3	1	no	—
123	C₂²×C₃×C₄	—	3	1	no	—
124	C₂²×Dic₃	C₂×C₃×C₅	1	1	yes	characteristic
125	C₂×Dic₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
126	C₃×C₄⋊C₄	—	3	1	no	—
127	C₃×C₄⋊C₄	—	3	1	no	—
128	C₃×C₄⋊C₄	—	3	1	no	—
129	C₃×C₂²⋊C₄	—	2	1	no	—
130	C₃×C₂²⋊C₄	—	3	1	no	—
131	C₃×C₂²⋊C₄	C₅×D₃	1	1	yes	characteristic
132	C₃×C₂²⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic
133	#217	C₂×C₃×C₅	1	1	yes	characteristic
134	C₄⋊Dic₃	C₂×C₃×C₅	1	1	yes	characteristic, complemented
135	Dic₃⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic, complemented
136	Dic₃⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic, complemented
137	C₃²×C₅	C₂²⋊Q₈	1	1	yes	characteristic, complemented, Hall
138	C₂³×C₅	C₂×C₃×D₃	1	1	yes	characteristic
139	C₅×Q₈	—	6	2	no	—
140	C₂×C₄×C₅	—	3	1	no	—
141	C₂×C₄×C₅	—	3	1	no	—
142	C₂×C₄×C₅	—	3	1	no	—
143	C₂×C₄×C₅	—	3	1	no	—
144	C₂×C₄×C₅	—	6	2	no	—
145	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
147	C₂²×C₃²	—	2	1	no	—
148	C₂²×C₃²	—	2	1	no	—
149	C₂²×C₃²	C₅×Q₈	1	2	yes	complemented
150	C₂²×C₃²	C₂³×C₅	1	1	yes	characteristic
151	C₃²×C₄	—	2	1	no	—
152	C₃²×C₄	—	2	1	no	—
153	C₃×Dic₃	—	2	1	no	—
154	C₃×Dic₃	—	2	1	no	—
155	C₃×Dic₃	—	2	1	no	—
156	C₃×Dic₃	C₅×D₄	1	2	yes	—
157	C₂²⋊Q₈	—	3	1	no	Sylow
158	C₂×C₃×C₅	—	2	2	no	—
159	C₂×C₃×C₅	—	2	2	no	—
160	C₂×C₃×C₅	—	2	1	no	—
161	C₂×C₃×C₅	—	2	1	no	—
162	C₂×C₃×C₅	—	2	1	no	—
163	C₂×C₃×C₅	—	4	2	no	—
164	C₂×C₃×C₅	C₂×Dic₆	1	1	yes	characteristic
165	C₂×C₃×C₅	D₃×D₄	1	1	yes	characteristic
166	C₂×C₃×C₅	#237	1	1	yes	characteristic
167	C₂×C₃×C₅	C₂×C₃×D₄	1	1	yes	characteristic
168	C₂×C₃×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
169	C₂×C₃×C₅	C₃×#41	1	1	yes	characteristic
170	C₂³×C₃	—	2	1	no	—
171	C₂³×C₃	C₂×C₅×D₃	1	1	yes	characteristic
172	C₂³×C₃	C₂²×C₃×C₅	1	1	yes	characteristic
173	C₃×Q₈	—	6	2	no	—
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	—	2	1	no	—
176	C₂×C₃×C₄	—	3	1	no	—
177	C₂×C₃×C₄	—	3	1	no	—
178	C₂×C₃×C₄	—	3	1	no	—
179	C₂×C₃×C₄	—	3	1	no	—
180	C₂×C₃×C₄	—	6	2	no	—
181	C₂×C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
185	C₂×Dic₃	—	2	2	no	—
186	C₂×Dic₃	C₂²×C₃×C₅	1	1	yes	characteristic
187	C₂×Dic₃	C₂²×C₃×C₅	1	1	yes	characteristic
188	C₂×Dic₃	C₂²×C₃×C₅	1	1	yes	characteristic
189	C₂×Dic₃	C₂²×C₃×C₅	1	1	yes	characteristic
190	Dic₆	—	2	2	no	—
191	C₂²×C₅	—	2	1	no	—
192	C₂²×C₅	—	2	1	no	—
193	C₂²×C₅	C₃×Dic₆	1	2	yes	complemented
194	C₂²×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
195	C₄×C₅	—	2	1	no	—
196	C₄×C₅	—	2	1	no	—
197	C₄×C₅	—	3	2	no	—
198	C₄×C₅	—	6	1	no	—
199	C₄×C₅	—	6	1	no	—
200	C₄×C₅	—	6	1	no	—
201	C₂×C₃²	—	2	2	no	—
202	C₂×C₃²	C₂×C₅×D₄	1	1	yes	characteristic
203	C₂×C₃²	C₂×C₅×Q₈	1	1	yes	characteristic
204	C₂×C₃²	C₅×#41	1	1	yes	characteristic
205	C₂×Q₈	—	3	1	no	—
206	C₂²×C₄	—	3	1	no	—
207	C₄⋊C₄	—	3	1	no	—
208	C₄⋊C₄	—	3	1	no	—
209	C₄⋊C₄	—	3	1	no	—
210	C₂²⋊C₄	—	3	1	no	—
211	C₂²⋊C₄	C₃×C₅×D₃	1	1	yes	characteristic
212	C₃×C₅	—	2	1	no	—
213	C₃×C₅	#878	1	1	yes	characteristic, direct product factor
214	C₃×C₅	C₃×C₂²⋊Q₈	1	1	yes	characteristic, complemented
215	C₂²×C₃	—	2	1	no	—
216	C₂²×C₃	—	2	1	no	—
217	C₂²×C₃	—	2	1	no	—
218	C₂²×C₃	—	2	1	no	—
219	C₂²×C₃	—	2	2	no	—
220	C₂²×C₃	—	2	1	no	—
221	C₂²×C₃	—	4	1	no	—
222	C₂²×C₃	—	4	1	no	—
223	C₂²×C₃	C₅×Dic₆	1	2	yes	complemented
224	C₂²×C₃	C₃×C₅×Q₈	1	2	yes	complemented
225	C₂²×C₃	C₂²×C₅×D₃	1	1	yes	characteristic
226	C₂²×C₃	C₂³×C₃×C₅	1	1	yes	characteristic
227	C₃×C₄	—	2	1	no	—
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	2	1	no	—
230	C₃×C₄	—	2	1	no	—
231	C₃×C₄	—	3	2	no	—
232	C₃×C₄	—	4	1	no	—
233	C₃×C₄	—	4	1	no	—
234	C₃×C₄	—	6	1	no	—
235	C₃×C₄	—	6	1	no	—
236	C₃×C₄	—	6	1	no	—
237	Dic₃	—	2	1	no	—
238	Dic₃	—	2	1	no	—
239	Dic₃	—	2	1	no	—
240	Dic₃	C₃×C₅×D₄	1	2	yes	—
241	C₂×C₅	—	2	2	no	—
242	C₂×C₅	C₂×C₃×Dic₆	1	1	yes	characteristic
243	C₂×C₅	C₃×D₃×D₄	1	1	yes	characteristic
244	C₂×C₅	C₃×#237	1	1	yes	characteristic
245	C₃²	C₅×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
246	C₂³	C₂×C₃×C₅×D₃	1	1	yes	characteristic
247	Q₈	—	6	2	no	—
248	C₂×C₄	—	3	1	no	—
249	C₂×C₄	—	3	1	no	—
250	C₂×C₄	—	3	1	no	—
251	C₂×C₄	—	3	1	no	—
252	C₂×C₄	—	6	2	no	—
253	C₂×C₄	C₂×C₃×C₅×D₃	1	1	yes	characteristic
254	C₂×C₄	C₂×C₃×C₅×D₃	1	1	yes	characteristic
255	C₂×C₃	—	2	2	no	—
256	C₂×C₃	—	2	2	no	—
257	C₂×C₃	—	2	1	no	—
258	C₂×C₃	—	2	1	no	—
259	C₂×C₃	—	2	1	no	—
260	C₂×C₃	—	4	2	no	—
261	C₂×C₃	C₂×C₅×Dic₆	1	1	yes	characteristic
262	C₂×C₃	C₅×D₃×D₄	1	1	yes	characteristic
263	C₂×C₃	C₅×#237	1	1	yes	characteristic
264	C₂×C₃	C₂×C₃×C₅×D₄	1	1	yes	characteristic
265	C₂×C₃	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
266	C₂×C₃	C₃×C₅×#41	1	1	yes	characteristic
267	C₅	C₃×#878	1	1	yes	characteristic, direct product factor, Sylow
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	C₃×C₅×Dic₆	1	2	yes	complemented
271	C₂²	C₂²×C₃×C₅×D₃	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	3	2	no	—
275	C₄	—	6	1	no	—
276	C₄	—	6	1	no	—
277	C₄	—	6	1	no	—
278	C₃	—	2	1	no	—
279	C₃	C₅×#878	1	1	yes	characteristic, direct product factor
280	C₃	C₃×C₅×C₂²⋊Q₈	1	1	yes	characteristic, complemented
281	C₂	—	2	2	no	—
282	C₂	C₂×C₃×C₅×Dic₆	1	1	yes	characteristic
283	C₂	C₃×C₅×D₃×D₄	1	1	yes	characteristic
284	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×C₂²⋊Q₈
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
2. C₂×C₃×C₅×Dic₆	C₂	semidirect
5. C₃×C₅×C₄⋊Dic₃	C₂	semidirect
6. C₃×C₅×Dic₃⋊C₄	C₂	semidirect
7. C₃×C₅×Dic₃⋊C₄	C₂	semidirect
9. C₅×#878	C₃	direct (N × H)
19. C₃×#878	C₅	direct (N × H)
23. C₂×C₅×Dic₆	C₂×C₃	semidirect
32. C₅×C₄⋊Dic₃	C₂×C₃	semidirect
33. C₅×Dic₃⋊C₄	C₂×C₃	semidirect
34. C₅×Dic₃⋊C₄	C₂×C₃	semidirect
37. C₂²×C₃²×C₅	Q₈	semidirect
47. C₂×C₃×Dic₆	C₂×C₅	semidirect
50. C₃×C₄⋊Dic₃	C₂×C₅	semidirect
51. C₃×Dic₃⋊C₄	C₂×C₅	semidirect
52. C₃×Dic₃⋊C₄	C₂×C₅	semidirect
75. #878	C₃×C₅	direct (N × H)
104. C₂²×C₃×C₅	Dic₆	semidirect
105. C₂²×C₃×C₅	C₃×Q₈	semidirect
125. C₂×Dic₆	C₂×C₃×C₅	semidirect
134. C₄⋊Dic₃	C₂×C₃×C₅	semidirect
135. Dic₃⋊C₄	C₂×C₃×C₅	semidirect
136. Dic₃⋊C₄	C₂×C₃×C₅	semidirect
137. C₃²×C₅	C₂²⋊Q₈	semidirect
149. C₂²×C₃²	C₅×Q₈	semidirect
193. C₂²×C₅	C₃×Dic₆	semidirect
213. C₃×C₅	#878	direct (N × H)
214. C₃×C₅	C₃×C₂²⋊Q₈	semidirect
223. C₂²×C₃	C₅×Dic₆	semidirect
224. C₂²×C₃	C₃×C₅×Q₈	semidirect
245. C₃²	C₅×C₂²⋊Q₈	semidirect
267. C₅	C₃×#878	direct (N × H)
270. C₂²	C₃×C₅×Dic₆	semidirect
279. C₃	C₅×#878	direct (N × H)
280. C₃	C₃×C₅×C₂²⋊Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₅×Dic₃	2
C₂×C₃×C₅×Dic₆	2
C₃²×C₅×C₂²⋊C₄	2
C₃×C₅×#217	2
C₃×C₅×C₄⋊Dic₃	2
C₃×C₅×Dic₃⋊C₄	2
C₃×C₅×C₂²⋊Q₈	3
C₅×#878	3
C₃×#878	5

Subgroups — C3×C5×#878 #13087889

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#878 #13087889