Subgroups — C₃×C₅×#1912 #14274946

Order 1920.

628 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#536	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#519	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#421	C₂	1	2	yes	maximal
4	C₂×C₃×C₅×#101	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×H(3,ℤ/4)	C₂	1	2	yes	maximal, complemented
6	C₅×#1912	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	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
10	C₃×C₅×C₄⋊Q₈	C₄	1	2	yes	complemented
11	C₃×C₅×#124	C₄	1	2	yes	complemented
12	C₃×C₅×#123	—	2	2	no	—
13	C₃×C₅×C₂²⋊Q₈	—	2	2	no	—
14	C₃×C₄×C₅×Q₈	—	2	4	no	—
15	C₃×C₅×#117	—	2	2	no	—
16	C₃×C₅×#117	C₂²	1	1	yes	characteristic
17	C₃×C₅×#117	C₂²	1	2	yes	—
18	C₃×C₅×#104	—	2	4	no	—
19	C₃×C₅×#101	—	2	2	no	—
20	C₃×#1912	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₅×#536	C₂×C₃	1	1	yes	characteristic
22	C₅×#519	C₂×C₃	1	1	yes	characteristic, complemented
23	C₅×#421	C₂×C₃	1	2	yes	—
24	C₂×C₅×#101	C₂×C₃	1	1	yes	characteristic, complemented
25	C₅×H(3,ℤ/4)	C₂×C₃	1	2	yes	complemented
26	C₃×C₅×#41	—	2	2	no	—
27	C₃×C₅×#41	—	2	2	no	—
28	C₂×C₃×C₅×Q₈	—	2	2	no	—
29	C₂×C₃×C₅×Q₈	—	2	2	no	—
30	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
32	C₂×C₃×C₅×Q₈	D₄	1	2	yes	—
33	C₂×C₃×C₅×D₄	—	2	1	no	—
34	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₃×C₄×C₅	—	2	1	no	—
36	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
37	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
38	C₃×C₅×M₁₆	—	2	2	no	—
39	C₃×C₅×M₁₆	—	2	2	no	—
40	C₂×C₃×C₅×C₈	—	2	2	no	—
41	C₃×C₅×C₄⋊C₄	—	2	4	no	—
42	C₃×C₅×C₄⋊C₄	—	2	4	no	—
43	C₃×C₅×C₄⋊C₄	—	2	2	no	—
44	C₃×C₅×C₄⋊C₄	—	2	1	no	—
45	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
46	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
47	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
48	C₃×C₄²×C₅	—	2	4	no	—
49	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
50	C₃×C₄²×C₅	D₄	1	4	yes	—
51	C₃×#536	C₂×C₅	1	1	yes	characteristic
52	C₃×#519	C₂×C₅	1	1	yes	characteristic, complemented
53	C₃×#421	C₂×C₅	1	2	yes	—
54	C₂×C₃×#101	C₂×C₅	1	1	yes	characteristic, complemented
55	C₃×H(3,ℤ/4)	C₂×C₅	1	2	yes	complemented
56	C₂×C₅×#41	C₂²×C₃	1	1	yes	characteristic
57	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
59	C₅×C₄⋊Q₈	C₃×C₄	1	2	yes	complemented
60	C₅×#124	C₃×C₄	1	2	yes	complemented
61	C₅×#123	—	2	2	no	—
62	C₅×C₂²⋊Q₈	—	2	2	no	—
63	C₄×C₅×Q₈	—	2	4	no	—
64	C₅×#117	—	2	2	no	—
65	C₅×#117	C₂²×C₃	1	1	yes	characteristic
66	C₅×#117	C₂²×C₃	1	2	yes	—
67	C₅×#104	—	2	4	no	—
68	C₅×#101	—	2	2	no	—
69	#1912	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
70	C₂³×C₃×C₅	—	2	1	no	—
71	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
72	C₃×C₅×Q₈	—	2	4	no	—
73	C₃×C₅×Q₈	—	2	4	no	—
74	C₃×C₅×Q₈	—	2	2	no	—
75	C₃×C₅×D₄	—	2	2	no	—
76	C₃×C₅×D₄	—	4	1	no	—
77	C₃×C₅×D₄	—	4	1	no	—
78	C₂×C₃×C₄×C₅	—	2	2	no	—
79	C₂×C₃×C₄×C₅	—	2	2	no	—
80	C₂×C₃×C₄×C₅	—	2	2	no	—
81	C₂×C₃×C₄×C₅	—	2	1	no	—
82	C₂×C₃×C₄×C₅	—	2	4	no	—
83	C₂×C₃×C₄×C₅	—	4	2	no	—
84	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
85	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
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	#41	1	2	yes	—
89	C₃×C₅×C₈	—	4	2	no	—
90	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
91	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
92	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
93	C₃×C₄⋊Q₈	C₄×C₅	1	2	yes	complemented
94	C₃×#124	C₄×C₅	1	2	yes	complemented
95	C₃×#123	—	2	2	no	—
96	C₃×C₂²⋊Q₈	—	2	2	no	—
97	C₃×C₄×Q₈	—	2	4	no	—
98	C₃×#117	—	2	2	no	—
99	C₃×#117	C₂²×C₅	1	1	yes	characteristic
100	C₃×#117	C₂²×C₅	1	2	yes	—
101	C₃×#104	—	2	4	no	—
102	C₃×#101	—	2	2	no	—
103	C₅×#41	—	2	2	no	—
104	C₅×#41	—	2	2	no	—
105	C₂×C₅×Q₈	—	2	2	no	—
106	C₂×C₅×Q₈	—	2	2	no	—
107	C₂×C₅×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
108	C₂×C₅×Q₈	C₂×C₃×C₄	1	2	yes	—
109	C₂×C₅×Q₈	C₃×D₄	1	2	yes	—
110	C₂×C₅×D₄	—	2	1	no	—
111	C₂×C₅×D₄	C₂×C₃×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₅	—	2	1	no	—
113	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
114	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
115	C₅×M₁₆	—	2	2	no	—
116	C₅×M₁₆	—	2	2	no	—
117	C₂×C₅×C₈	—	2	2	no	—
118	C₅×C₄⋊C₄	—	2	4	no	—
119	C₅×C₄⋊C₄	—	2	4	no	—
120	C₅×C₄⋊C₄	—	2	2	no	—
121	C₅×C₄⋊C₄	—	2	1	no	—
122	C₅×C₂²⋊C₄	—	2	2	no	—
123	C₅×C₂²⋊C₄	—	2	1	no	—
124	C₅×C₂²⋊C₄	—	2	4	no	—
125	C₄²×C₅	—	2	4	no	—
126	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
127	C₄²×C₅	C₃×D₄	1	4	yes	—
128	#536	C₂×C₃×C₅	1	1	yes	characteristic
129	#519	C₂×C₃×C₅	1	1	yes	characteristic, complemented
130	#421	C₂×C₃×C₅	1	2	yes	—
131	C₂×#101	C₂×C₃×C₅	1	1	yes	characteristic, complemented
132	H(3,ℤ/4)	C₂×C₃×C₅	1	2	yes	complemented
133	C₂²×C₃×C₅	—	4	1	no	—
134	C₂²×C₃×C₅	—	4	2	no	—
135	C₂²×C₃×C₅	—	4	1	no	—
136	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂²×C₃×C₅	#120	1	1	yes	characteristic
138	C₂²×C₃×C₅	#123	1	1	yes	characteristic
139	C₃×C₄×C₅	—	2	4	no	—
140	C₃×C₄×C₅	—	4	4	no	—
141	C₃×C₄×C₅	—	4	2	no	—
142	C₃×C₄×C₅	—	4	1	no	—
143	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
144	C₃×C₄×C₅	#121	1	2	yes	—
145	C₃×#41	—	2	2	no	—
146	C₃×#41	—	2	2	no	—
147	C₂×C₃×Q₈	—	2	2	no	—
148	C₂×C₃×Q₈	—	2	2	no	—
149	C₂×C₃×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
150	C₂×C₃×Q₈	C₂×C₄×C₅	1	2	yes	—
151	C₂×C₃×Q₈	C₅×D₄	1	2	yes	—
152	C₂×C₃×D₄	—	2	1	no	—
153	C₂×C₃×D₄	C₂×C₄×C₅	1	1	yes	characteristic
154	C₂²×C₃×C₄	—	2	1	no	—
155	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
156	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
157	C₃×M₁₆	—	2	2	no	—
158	C₃×M₁₆	—	2	2	no	—
159	C₂×C₃×C₈	—	2	2	no	—
160	C₃×C₄⋊C₄	—	2	4	no	—
161	C₃×C₄⋊C₄	—	2	4	no	—
162	C₃×C₄⋊C₄	—	2	2	no	—
163	C₃×C₄⋊C₄	—	2	1	no	—
164	C₃×C₂²⋊C₄	—	2	2	no	—
165	C₃×C₂²⋊C₄	—	2	1	no	—
166	C₃×C₂²⋊C₄	—	2	4	no	—
167	C₃×C₄²	—	2	4	no	—
168	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
169	C₃×C₄²	C₅×D₄	1	4	yes	—
170	C₂³×C₅	—	2	1	no	—
171	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
172	C₅×Q₈	—	2	4	no	—
173	C₅×Q₈	—	2	4	no	—
174	C₅×Q₈	—	2	2	no	—
175	C₅×D₄	—	2	2	no	—
176	C₅×D₄	—	4	1	no	—
177	C₅×D₄	—	4	1	no	—
178	C₂×C₄×C₅	—	2	2	no	—
179	C₂×C₄×C₅	—	2	2	no	—
180	C₂×C₄×C₅	—	2	2	no	—
181	C₂×C₄×C₅	—	2	1	no	—
182	C₂×C₄×C₅	—	2	4	no	—
183	C₂×C₄×C₅	—	4	2	no	—
184	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	4	yes	—
185	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
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	C₃×#41	1	2	yes	—
189	C₅×C₈	—	4	2	no	—
190	C₂×#41	C₂²×C₃×C₅	1	1	yes	characteristic
191	C₂²×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
192	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
193	C₄⋊Q₈	C₃×C₄×C₅	1	2	yes	complemented
194	#124	C₃×C₄×C₅	1	2	yes	complemented
195	#123	—	2	2	no	—
196	C₂²⋊Q₈	—	2	2	no	—
197	C₄×Q₈	—	2	4	no	—
198	#117	—	2	2	no	—
199	#117	C₂²×C₃×C₅	1	1	yes	characteristic
200	#117	C₂²×C₃×C₅	1	2	yes	—
201	#104	—	2	4	no	—
202	#101	—	2	2	no	—
203	C₂×C₃×C₅	—	2	1	no	—
204	C₂×C₃×C₅	—	2	1	no	—
205	C₂×C₃×C₅	—	2	1	no	—
206	C₂×C₃×C₅	—	8	1	no	—
207	C₂×C₃×C₅	#386	1	1	yes	characteristic
208	C₂³×C₃	—	2	1	no	—
209	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
210	C₃×Q₈	—	2	4	no	—
211	C₃×Q₈	—	2	4	no	—
212	C₃×Q₈	—	2	2	no	—
213	C₃×D₄	—	2	2	no	—
214	C₃×D₄	—	4	1	no	—
215	C₃×D₄	—	4	1	no	—
216	C₂×C₃×C₄	—	2	2	no	—
217	C₂×C₃×C₄	—	2	2	no	—
218	C₂×C₃×C₄	—	2	2	no	—
219	C₂×C₃×C₄	—	2	1	no	—
220	C₂×C₃×C₄	—	2	4	no	—
221	C₂×C₃×C₄	—	4	2	no	—
222	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	4	yes	—
223	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
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	C₅×#41	1	2	yes	—
227	C₃×C₈	—	4	2	no	—
228	C₂²×C₅	—	4	1	no	—
229	C₂²×C₅	—	4	2	no	—
230	C₂²×C₅	—	4	1	no	—
231	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
232	C₂²×C₅	C₃×#120	1	1	yes	characteristic
233	C₂²×C₅	C₃×#123	1	1	yes	characteristic
234	C₄×C₅	—	2	4	no	—
235	C₄×C₅	—	4	4	no	—
236	C₄×C₅	—	4	2	no	—
237	C₄×C₅	—	4	1	no	—
238	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
239	C₄×C₅	C₃×#121	1	2	yes	—
240	#41	—	2	2	no	—
241	#41	—	2	2	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
245	C₂×Q₈	C₂×C₃×C₄×C₅	1	2	yes	—
246	C₂×Q₈	C₃×C₅×D₄	1	2	yes	—
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
251	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
252	M₁₆	—	2	2	no	—
253	M₁₆	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	2	2	no	—
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	2	4	no	—
262	C₄²	—	2	4	no	—
263	C₄²	C₂×C₃×C₄×C₅	1	2	yes	—
264	C₄²	C₃×C₅×D₄	1	4	yes	—
265	C₃×C₅	#1912	1	1	yes	characteristic, direct product factor, Hall
266	C₂²×C₃	—	4	1	no	—
267	C₂²×C₃	—	4	2	no	—
268	C₂²×C₃	—	4	1	no	—
269	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
270	C₂²×C₃	C₅×#120	1	1	yes	characteristic
271	C₂²×C₃	C₅×#123	1	1	yes	characteristic
272	C₃×C₄	—	2	4	no	—
273	C₃×C₄	—	4	4	no	—
274	C₃×C₄	—	4	2	no	—
275	C₃×C₄	—	4	1	no	—
276	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
277	C₃×C₄	C₅×#121	1	2	yes	—
278	C₂×C₅	—	2	1	no	—
279	C₂×C₅	—	2	1	no	—
280	C₂×C₅	—	2	1	no	—
281	C₂×C₅	—	8	1	no	—
282	C₂×C₅	C₃×#386	1	1	yes	characteristic
283	C₂³	—	2	1	no	—
284	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
285	Q₈	—	2	4	no	—
286	Q₈	—	2	4	no	—
287	Q₈	—	2	2	no	—
288	D₄	—	2	2	no	—
289	D₄	—	4	1	no	—
290	D₄	—	4	1	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	4	yes	—
298	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
299	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
300	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
301	C₂×C₄	C₃×C₅×#41	1	2	yes	—
302	C₈	—	4	2	no	—
303	C₂×C₃	—	2	1	no	—
304	C₂×C₃	—	2	1	no	—
305	C₂×C₃	—	2	1	no	—
306	C₂×C₃	—	8	1	no	—
307	C₂×C₃	C₅×#386	1	1	yes	characteristic
308	C₅	C₃×#1912	1	1	yes	characteristic, direct product factor, Sylow
309	C₂²	—	4	1	no	—
310	C₂²	—	4	2	no	—
311	C₂²	—	4	1	no	—
312	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
313	C₂²	C₃×C₅×#120	1	1	yes	characteristic
314	C₂²	C₃×C₅×#123	1	1	yes	characteristic
315	C₄	—	2	4	no	—
316	C₄	—	4	4	no	—
317	C₄	—	4	2	no	—
318	C₄	—	4	1	no	—
319	C₄	C₃×C₄×C₅×D₄	1	2	yes	—
320	C₄	C₃×C₅×#121	1	2	yes	—
321	C₃	C₅×#1912	1	1	yes	characteristic, direct product factor, Sylow
322	C₂	—	2	1	no	—
323	C₂	—	2	1	no	—
324	C₂	—	2	1	no	—
325	C₂	—	8	1	no	—
326	C₂	C₃×C₅×#386	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃×C₅	#386
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
prime radical	C₂×C₃×C₅×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ 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
2. C₃×C₅×#519	C₂	semidirect
4. C₂×C₃×C₅×#101	C₂	semidirect
5. C₃×C₅×H(3,ℤ/4)	C₂	semidirect
6. C₅×#1912	C₃	direct (N × H)
10. C₃×C₅×C₄⋊Q₈	C₄	semidirect
11. C₃×C₅×#124	C₄	semidirect
20. C₃×#1912	C₅	direct (N × H)
22. C₅×#519	C₂×C₃	semidirect
24. C₂×C₅×#101	C₂×C₃	semidirect
25. C₅×H(3,ℤ/4)	C₂×C₃	semidirect
52. C₃×#519	C₂×C₅	semidirect
54. C₂×C₃×#101	C₂×C₅	semidirect
55. C₃×H(3,ℤ/4)	C₂×C₅	semidirect
59. C₅×C₄⋊Q₈	C₃×C₄	semidirect
60. C₅×#124	C₃×C₄	semidirect
69. #1912	C₃×C₅	direct (N × H)
93. C₃×C₄⋊Q₈	C₄×C₅	semidirect
94. C₃×#124	C₄×C₅	semidirect
129. #519	C₂×C₃×C₅	semidirect
131. C₂×#101	C₂×C₃×C₅	semidirect
132. H(3,ℤ/4)	C₂×C₃×C₅	semidirect
193. C₄⋊Q₈	C₃×C₄×C₅	semidirect
194. #124	C₃×C₄×C₅	semidirect
265. C₃×C₅	#1912	direct (N × H)
308. C₅	C₃×#1912	direct (N × H)
321. C₃	C₅×#1912	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×#536	2
C₃×C₅×#519	2
C₃×C₅×#421	2
C₂×C₃×C₅×#101	2
C₃×C₅×H(3,ℤ/4)	2
C₅×#1912	3
C₃×#1912	5

Subgroups — C3×C5×#1912 #14274946

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1912 #14274946