Subgroups — C₃×C₅×#1915 #14274949

Order 1920.

616 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#568	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#536	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#413	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×C₅×#101	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#344	C₂	1	2	yes	maximal, complemented
6	C₅×#1915	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
9	C₃×C₅×#131	—	2	4	no	—
10	C₂×C₃×C₅×M₁₆	—	2	2	no	—
11	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
12	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
13	C₃×C₅×C₄⋊Q₈	—	2	1	no	—
14	C₃×C₅×#124	—	2	1	no	—
15	C₃×C₅×#123	C₄	1	4	yes	—
16	C₃×C₅×C₂²⋊Q₈	—	2	2	no	—
17	C₃×C₅×#117	C₂²	1	1	yes	characteristic
18	C₃×C₅×#101	—	2	1	no	—
19	C₃×C₅×#101	—	2	4	no	—
20	C₃×C₅×#100	—	2	1	no	—
21	C₃×#1915	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₅×#568	C₂×C₃	1	1	yes	characteristic
23	C₅×#536	C₂×C₃	1	1	yes	characteristic
24	C₅×#413	C₂×C₃	1	1	yes	characteristic
25	C₂×C₅×#101	C₂×C₃	1	2	yes	complemented
26	C₅×#344	C₂×C₃	1	2	yes	complemented
27	C₃×C₅×#41	—	2	2	no	—
28	C₃×C₅×#41	—	2	2	no	—
29	C₂×C₃×C₅×Q₈	—	2	1	no	—
30	C₂×C₃×C₅×Q₈	—	2	1	no	—
31	C₂×C₃×C₅×Q₈	—	2	2	no	—
32	C₂×C₃×C₅×Q₈	—	2	2	no	—
33	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
34	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
35	C₂×C₃×C₅×D₄	D₄	1	1	yes	characteristic
36	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
37	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
38	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
39	C₃×C₅×M₁₆	—	2	4	no	—
40	C₃×C₅×M₁₆	—	2	2	no	—
41	C₃×C₅×M₁₆	—	2	2	no	—
42	C₃×C₅×M₁₆	—	2	2	no	—
43	C₃×C₅×M₁₆	D₄	1	4	yes	—
44	C₂×C₃×C₅×C₈	—	2	4	no	—
45	C₂×C₃×C₅×C₈	—	2	2	no	—
46	C₂×C₃×C₅×C₈	—	2	2	no	—
47	C₃×C₅×C₄⋊C₄	—	2	4	no	—
48	C₃×C₅×C₄⋊C₄	—	2	1	no	—
49	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
50	C₃×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
51	C₃×C₄²×C₅	—	2	1	no	—
52	C₃×#568	C₂×C₅	1	1	yes	characteristic
53	C₃×#536	C₂×C₅	1	1	yes	characteristic
54	C₃×#413	C₂×C₅	1	1	yes	characteristic
55	C₂×C₃×#101	C₂×C₅	1	2	yes	complemented
56	C₃×#344	C₂×C₅	1	2	yes	complemented
57	C₂×C₅×#41	C₂²×C₃	1	1	yes	characteristic
58	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
59	C₅×#131	—	2	4	no	—
60	C₂×C₅×M₁₆	—	2	2	no	—
61	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
62	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
63	C₅×C₄⋊Q₈	—	2	1	no	—
64	C₅×#124	—	2	1	no	—
65	C₅×#123	C₃×C₄	1	4	yes	—
66	C₅×C₂²⋊Q₈	—	2	2	no	—
67	C₅×#117	C₂²×C₃	1	1	yes	characteristic
68	C₅×#101	—	2	1	no	—
69	C₅×#101	—	2	4	no	—
70	C₅×#100	—	2	1	no	—
71	#1915	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
72	C₂³×C₃×C₅	C₂²⋊C₄	1	2	yes	—
73	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₃×C₅×Q₈	—	2	2	no	—
75	C₃×C₅×Q₈	—	4	1	no	—
76	C₃×C₅×Q₈	—	4	2	no	—
77	C₃×C₅×Q₈	—	4	1	no	—
78	C₃×C₅×D₄	—	2	4	no	—
79	C₃×C₅×D₄	—	2	2	no	—
80	C₂×C₃×C₄×C₅	—	2	2	no	—
81	C₂×C₃×C₄×C₅	—	2	2	no	—
82	C₂×C₃×C₄×C₅	—	2	2	no	—
83	C₂×C₃×C₄×C₅	—	2	2	no	—
84	C₂×C₃×C₄×C₅	—	2	4	no	—
85	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
86	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
87	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	#41	1	2	yes	—
90	C₃×C₅×C₈	—	2	4	no	—
91	C₃×C₅×C₈	—	4	2	no	—
92	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
93	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
94	C₃×#131	—	2	4	no	—
95	C₂×C₃×M₁₆	—	2	2	no	—
96	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
97	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
98	C₃×C₄⋊Q₈	—	2	1	no	—
99	C₃×#124	—	2	1	no	—
100	C₃×#123	C₄×C₅	1	4	yes	—
101	C₃×C₂²⋊Q₈	—	2	2	no	—
102	C₃×#117	C₂²×C₅	1	1	yes	characteristic
103	C₃×#101	—	2	1	no	—
104	C₃×#101	—	2	4	no	—
105	C₃×#100	—	2	1	no	—
106	C₅×#41	—	2	2	no	—
107	C₅×#41	—	2	2	no	—
108	C₂×C₅×Q₈	—	2	1	no	—
109	C₂×C₅×Q₈	—	2	1	no	—
110	C₂×C₅×Q₈	—	2	2	no	—
111	C₂×C₅×Q₈	—	2	2	no	—
112	C₂×C₅×Q₈	C₃×D₄	1	1	yes	characteristic
113	C₂×C₅×D₄	C₂×C₃×C₄	1	2	yes	—
114	C₂×C₅×D₄	C₃×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
116	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
117	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
118	C₅×M₁₆	—	2	4	no	—
119	C₅×M₁₆	—	2	2	no	—
120	C₅×M₁₆	—	2	2	no	—
121	C₅×M₁₆	—	2	2	no	—
122	C₅×M₁₆	C₃×D₄	1	4	yes	—
123	C₂×C₅×C₈	—	2	4	no	—
124	C₂×C₅×C₈	—	2	2	no	—
125	C₂×C₅×C₈	—	2	2	no	—
126	C₅×C₄⋊C₄	—	2	4	no	—
127	C₅×C₄⋊C₄	—	2	1	no	—
128	C₅×C₂²⋊C₄	—	2	4	no	—
129	C₅×C₂²⋊C₄	C₂×C₃×C₄	1	2	yes	—
130	C₄²×C₅	—	2	1	no	—
131	#568	C₂×C₃×C₅	1	1	yes	characteristic
132	#536	C₂×C₃×C₅	1	1	yes	characteristic
133	#413	C₂×C₃×C₅	1	1	yes	characteristic
134	C₂×#101	C₂×C₃×C₅	1	2	yes	complemented
135	#344	C₂×C₃×C₅	1	2	yes	complemented
136	C₂²×C₃×C₅	—	2	2	no	—
137	C₂²×C₃×C₅	—	4	1	no	—
138	C₂²×C₃×C₅	—	4	2	no	—
139	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
141	C₃×C₄×C₅	—	2	2	no	—
142	C₃×C₄×C₅	—	4	2	no	—
143	C₃×C₄×C₅	—	4	2	no	—
144	C₃×C₄×C₅	#120	1	1	yes	characteristic
145	C₃×C₄×C₅	#121	1	2	yes	—
146	C₃×C₄×C₅	#123	1	1	yes	characteristic
147	C₃×#41	—	2	2	no	—
148	C₃×#41	—	2	2	no	—
149	C₂×C₃×Q₈	—	2	2	no	—
150	C₂×C₃×Q₈	—	2	1	no	—
151	C₂×C₃×Q₈	—	2	1	no	—
152	C₂×C₃×Q₈	—	2	2	no	—
153	C₂×C₃×Q₈	C₅×D₄	1	1	yes	characteristic
154	C₂×C₃×D₄	C₂×C₄×C₅	1	2	yes	—
155	C₂×C₃×D₄	C₅×D₄	1	1	yes	characteristic
156	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
157	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
158	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
159	C₃×M₁₆	—	2	4	no	—
160	C₃×M₁₆	—	2	2	no	—
161	C₃×M₁₆	—	2	2	no	—
162	C₃×M₁₆	—	2	2	no	—
163	C₃×M₁₆	C₅×D₄	1	4	yes	—
164	C₂×C₃×C₈	—	2	4	no	—
165	C₂×C₃×C₈	—	2	2	no	—
166	C₂×C₃×C₈	—	2	2	no	—
167	C₃×C₄⋊C₄	—	2	4	no	—
168	C₃×C₄⋊C₄	—	2	1	no	—
169	C₃×C₂²⋊C₄	—	2	4	no	—
170	C₃×C₂²⋊C₄	C₂×C₄×C₅	1	2	yes	—
171	C₃×C₄²	—	2	1	no	—
172	C₂³×C₅	C₃×C₂²⋊C₄	1	2	yes	—
173	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
174	C₅×Q₈	—	2	2	no	—
175	C₅×Q₈	—	4	1	no	—
176	C₅×Q₈	—	4	2	no	—
177	C₅×Q₈	—	4	1	no	—
178	C₅×D₄	—	2	4	no	—
179	C₅×D₄	—	2	2	no	—
180	C₂×C₄×C₅	—	2	2	no	—
181	C₂×C₄×C₅	—	2	2	no	—
182	C₂×C₄×C₅	—	2	2	no	—
183	C₂×C₄×C₅	—	2	2	no	—
184	C₂×C₄×C₅	—	2	4	no	—
185	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
186	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
187	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
188	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
189	C₂×C₄×C₅	C₃×#41	1	2	yes	—
190	C₅×C₈	—	2	4	no	—
191	C₅×C₈	—	4	2	no	—
192	C₂×#41	C₂²×C₃×C₅	1	1	yes	characteristic
193	C₂²×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
194	#131	—	2	4	no	—
195	C₂×M₁₆	—	2	2	no	—
196	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
197	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
198	C₄⋊Q₈	—	2	1	no	—
199	#124	—	2	1	no	—
200	#123	C₃×C₄×C₅	1	4	yes	—
201	C₂²⋊Q₈	—	2	2	no	—
202	#117	C₂²×C₃×C₅	1	1	yes	characteristic
203	#101	—	2	1	no	—
204	#101	—	2	4	no	—
205	#100	—	2	1	no	—
206	C₂×C₃×C₅	—	2	1	no	—
207	C₂×C₃×C₅	—	2	2	no	—
208	C₂×C₃×C₅	—	4	2	no	—
209	C₂×C₃×C₅	#386	1	1	yes	characteristic
210	C₂³×C₃	C₅×C₂²⋊C₄	1	2	yes	—
211	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
212	C₃×Q₈	—	2	2	no	—
213	C₃×Q₈	—	4	1	no	—
214	C₃×Q₈	—	4	2	no	—
215	C₃×Q₈	—	4	1	no	—
216	C₃×D₄	—	2	4	no	—
217	C₃×D₄	—	2	2	no	—
218	C₂×C₃×C₄	—	2	2	no	—
219	C₂×C₃×C₄	—	2	2	no	—
220	C₂×C₃×C₄	—	2	2	no	—
221	C₂×C₃×C₄	—	2	2	no	—
222	C₂×C₃×C₄	—	2	4	no	—
223	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
224	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
225	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
226	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
227	C₂×C₃×C₄	C₅×#41	1	2	yes	—
228	C₃×C₈	—	2	4	no	—
229	C₃×C₈	—	4	2	no	—
230	C₂²×C₅	—	2	2	no	—
231	C₂²×C₅	—	4	1	no	—
232	C₂²×C₅	—	4	2	no	—
233	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
234	C₂²×C₅	C₃×C₄×D₄	1	2	yes	—
235	C₄×C₅	—	2	2	no	—
236	C₄×C₅	—	4	2	no	—
237	C₄×C₅	—	4	2	no	—
238	C₄×C₅	C₃×#120	1	1	yes	characteristic
239	C₄×C₅	C₃×#121	1	2	yes	—
240	C₄×C₅	C₃×#123	1	1	yes	characteristic
241	#41	—	2	2	no	—
242	#41	—	2	2	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	C₃×C₅×D₄	1	1	yes	characteristic
248	C₂×D₄	C₂×C₃×C₄×C₅	1	2	yes	—
249	C₂×D₄	C₃×C₅×D₄	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₃×C₄×C₅	1	2	yes	—
251	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
252	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
253	M₁₆	—	2	4	no	—
254	M₁₆	—	2	2	no	—
255	M₁₆	—	2	2	no	—
256	M₁₆	—	2	2	no	—
257	M₁₆	C₃×C₅×D₄	1	4	yes	—
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	2	4	no	—
264	C₂²⋊C₄	C₂×C₃×C₄×C₅	1	2	yes	—
265	C₄²	—	2	1	no	—
266	C₃×C₅	#1915	1	1	yes	characteristic, direct product factor, Hall
267	C₂²×C₃	—	2	2	no	—
268	C₂²×C₃	—	4	1	no	—
269	C₂²×C₃	—	4	2	no	—
270	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
271	C₂²×C₃	C₄×C₅×D₄	1	2	yes	—
272	C₃×C₄	—	2	2	no	—
273	C₃×C₄	—	4	2	no	—
274	C₃×C₄	—	4	2	no	—
275	C₃×C₄	C₅×#120	1	1	yes	characteristic
276	C₃×C₄	C₅×#121	1	2	yes	—
277	C₃×C₄	C₅×#123	1	1	yes	characteristic
278	C₂×C₅	—	2	1	no	—
279	C₂×C₅	—	2	2	no	—
280	C₂×C₅	—	4	2	no	—
281	C₂×C₅	C₃×#386	1	1	yes	characteristic
282	C₂³	C₃×C₅×C₂²⋊C₄	1	2	yes	—
283	C₂³	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
284	Q₈	—	2	2	no	—
285	Q₈	—	4	1	no	—
286	Q₈	—	4	2	no	—
287	Q₈	—	4	1	no	—
288	D₄	—	2	4	no	—
289	D₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
296	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
297	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
298	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
299	C₂×C₄	C₃×C₅×#41	1	2	yes	—
300	C₈	—	2	4	no	—
301	C₈	—	4	2	no	—
302	C₂×C₃	—	2	1	no	—
303	C₂×C₃	—	2	2	no	—
304	C₂×C₃	—	4	2	no	—
305	C₂×C₃	C₅×#386	1	1	yes	characteristic
306	C₅	C₃×#1915	1	1	yes	characteristic, direct product factor, Sylow
307	C₂²	—	2	2	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	4	2	no	—
310	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
311	C₂²	C₃×C₄×C₅×D₄	1	2	yes	—
312	C₄	—	2	2	no	—
313	C₄	—	4	2	no	—
314	C₄	—	4	2	no	—
315	C₄	C₃×C₅×#120	1	1	yes	characteristic
316	C₄	C₃×C₅×#121	1	2	yes	—
317	C₄	C₃×C₅×#123	1	1	yes	characteristic
318	C₃	C₅×#1915	1	1	yes	characteristic, direct product factor, Sylow
319	C₂	—	2	1	no	—
320	C₂	—	2	2	no	—
321	C₂	—	4	2	no	—
322	C₂	C₃×C₅×#386	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃×C₅	#386
commutator	C₂³	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
prime radical	C₂×C₃×C₅×#41	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₅ ◁ C₂²×C₃×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₃×C₅×#101	C₂	semidirect
5. C₃×C₅×#344	C₂	semidirect
6. C₅×#1915	C₃	direct (N × H)
21. C₃×#1915	C₅	direct (N × H)
25. C₂×C₅×#101	C₂×C₃	semidirect
26. C₅×#344	C₂×C₃	semidirect
55. C₂×C₃×#101	C₂×C₅	semidirect
56. C₃×#344	C₂×C₅	semidirect
71. #1915	C₃×C₅	direct (N × H)
134. C₂×#101	C₂×C₃×C₅	semidirect
135. #344	C₂×C₃×C₅	semidirect
266. C₃×C₅	#1915	direct (N × H)
306. C₅	C₃×#1915	direct (N × H)
318. C₃	C₅×#1915	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×C₅×#568	2
C₃×C₅×#536	2
C₃×C₅×#413	2
C₂×C₃×C₅×#101	2
C₃×C₅×#344	2
C₅×#1915	3
C₃×#1915	5

Subgroups — C3×C5×#1915 #14274949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1915 #14274949