Subgroups — C₃×C₅×#2081 #14275115

Order 1920.

408 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#430	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×#418	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#418	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×C₅×#100	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×#344	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₅×#344	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₅×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2081	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
14	C₃×C₅×#117	C₂²	1	1	yes	characteristic
15	C₃×C₅×#117	C₂²	1	1	yes	characteristic
16	C₃×C₅×#108	—	2	2	no	—
17	C₃×C₅×#102	—	2	2	no	—
18	C₃×C₅×#102	—	2	2	no	—
19	C₃×C₅×#100	—	2	2	no	—
20	C₃×#2081	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₅×#430	C₂×C₃	1	1	yes	characteristic, complemented
22	C₅×#418	C₂×C₃	1	1	yes	characteristic
23	C₅×#418	C₂×C₃	1	1	yes	characteristic
24	C₂×C₅×#100	C₂×C₃	1	1	yes	characteristic
25	C₅×#344	C₂×C₃	1	1	yes	characteristic, complemented
26	C₅×#344	C₂×C₃	1	1	yes	characteristic, complemented
27	C₅×H(3,ℤ/4)	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂⁴×C₃×C₅	—	2	1	no	—
29	C₂×C₃×C₅×D₄	—	2	1	no	—
30	C₂×C₃×C₅×D₄	—	2	1	no	—
31	C₂×C₃×C₅×D₄	—	2	1	no	—
32	C₂×C₃×C₅×D₄	—	2	1	no	—
33	C₂×C₃×C₅×D₄	—	2	1	no	—
34	C₂×C₃×C₅×D₄	D₄	1	2	yes	—
35	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₃×C₅×M₁₆	—	2	1	no	—
37	C₃×C₅×M₁₆	—	2	1	no	—
38	C₃×C₅×M₁₆	—	2	1	no	—
39	C₃×C₅×M₁₆	—	2	1	no	—
40	C₃×C₅×M₁₆	—	2	1	no	—
41	C₃×C₅×M₁₆	—	2	1	no	—
42	C₃×C₅×M₁₆	—	2	1	no	—
43	C₃×C₅×M₁₆	—	2	1	no	—
44	C₂×C₃×C₅×C₈	—	2	1	no	—
45	C₂×C₃×C₅×C₈	—	2	1	no	—
46	C₂×C₃×C₅×C₈	—	2	1	no	—
47	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
48	C₃×C₅×C₄⋊C₄	—	2	1	no	—
49	C₃×C₅×C₄⋊C₄	—	2	1	no	—
50	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
51	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
52	C₃×C₄²×C₅	—	2	1	no	—
53	C₃×C₄²×C₅	—	2	1	no	—
54	C₃×#430	C₂×C₅	1	1	yes	characteristic, complemented
55	C₃×#418	C₂×C₅	1	1	yes	characteristic
56	C₃×#418	C₂×C₅	1	1	yes	characteristic
57	C₂×C₃×#100	C₂×C₅	1	1	yes	characteristic
58	C₃×#344	C₂×C₅	1	1	yes	characteristic, complemented
59	C₃×#344	C₂×C₅	1	1	yes	characteristic, complemented
60	C₃×H(3,ℤ/4)	C₂×C₅	1	1	yes	characteristic, complemented
61	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
62	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
64	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
65	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
66	C₅×#117	C₂²×C₃	1	1	yes	characteristic
67	C₅×#117	C₂²×C₃	1	1	yes	characteristic
68	C₅×#108	—	2	2	no	—
69	C₅×#102	—	2	2	no	—
70	C₅×#102	—	2	2	no	—
71	C₅×#100	—	2	2	no	—
72	#2081	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
73	C₂³×C₃×C₅	—	2	2	no	—
74	C₂³×C₃×C₅	—	4	1	no	—
75	C₂³×C₃×C₅	—	4	1	no	—
76	C₂³×C₃×C₅	—	8	2	no	—
77	C₂³×C₃×C₅	#41	1	1	yes	characteristic
78	C₃×C₅×D₄	—	2	2	no	—
79	C₃×C₅×D₄	—	2	2	no	—
80	C₃×C₅×D₄	—	4	1	no	—
81	C₃×C₅×D₄	—	4	1	no	—
82	C₂×C₃×C₄×C₅	—	4	1	no	—
83	C₂×C₃×C₄×C₅	—	4	1	no	—
84	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
90	C₃×C₅×C₈	—	2	2	no	—
91	C₃×C₅×C₈	—	4	1	no	—
92	C₃×C₅×C₈	—	4	1	no	—
93	C₃×C₅×C₈	—	4	1	no	—
94	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
95	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
96	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
97	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
98	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
99	C₃×#117	C₂²×C₅	1	1	yes	characteristic
100	C₃×#117	C₂²×C₅	1	1	yes	characteristic
101	C₃×#108	—	2	2	no	—
102	C₃×#102	—	2	2	no	—
103	C₃×#102	—	2	2	no	—
104	C₃×#100	—	2	2	no	—
105	C₂⁴×C₅	—	2	1	no	—
106	C₂×C₅×D₄	—	2	1	no	—
107	C₂×C₅×D₄	—	2	1	no	—
108	C₂×C₅×D₄	—	2	1	no	—
109	C₂×C₅×D₄	—	2	1	no	—
110	C₂×C₅×D₄	—	2	1	no	—
111	C₂×C₅×D₄	C₃×D₄	1	2	yes	—
112	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
113	C₅×M₁₆	—	2	1	no	—
114	C₅×M₁₆	—	2	1	no	—
115	C₅×M₁₆	—	2	1	no	—
116	C₅×M₁₆	—	2	1	no	—
117	C₅×M₁₆	—	2	1	no	—
118	C₅×M₁₆	—	2	1	no	—
119	C₅×M₁₆	—	2	1	no	—
120	C₅×M₁₆	—	2	1	no	—
121	C₂×C₅×C₈	—	2	1	no	—
122	C₂×C₅×C₈	—	2	1	no	—
123	C₂×C₅×C₈	—	2	1	no	—
124	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
125	C₅×C₄⋊C₄	—	2	1	no	—
126	C₅×C₄⋊C₄	—	2	1	no	—
127	C₅×C₂²⋊C₄	—	2	1	no	—
128	C₅×C₂²⋊C₄	—	2	1	no	—
129	C₄²×C₅	—	2	1	no	—
130	C₄²×C₅	—	2	1	no	—
131	#430	C₂×C₃×C₅	1	1	yes	characteristic, complemented
132	#418	C₂×C₃×C₅	1	1	yes	characteristic
133	#418	C₂×C₃×C₅	1	1	yes	characteristic
134	C₂×#100	C₂×C₃×C₅	1	1	yes	characteristic
135	#344	C₂×C₃×C₅	1	1	yes	characteristic, complemented
136	#344	C₂×C₃×C₅	1	1	yes	characteristic, complemented
137	H(3,ℤ/4)	C₂×C₃×C₅	1	1	yes	characteristic, complemented
138	C₂²×C₃×C₅	—	4	1	no	—
139	C₂²×C₃×C₅	—	4	2	no	—
140	C₂²×C₃×C₅	—	8	2	no	—
141	C₂²×C₃×C₅	—	8	2	no	—
142	C₂²×C₃×C₅	—	8	2	no	—
143	C₂²×C₃×C₅	#123	1	1	yes	characteristic
144	C₂²×C₃×C₅	#124	1	1	yes	characteristic
145	C₂²×C₃×C₅	#126	1	1	yes	characteristic
146	C₃×C₄×C₅	—	8	1	no	—
147	C₃×C₄×C₅	—	8	1	no	—
148	C₃×C₄×C₅	#121	1	1	yes	characteristic
149	C₃×C₄×C₅	#123	1	1	yes	characteristic
150	C₃×C₄×C₅	#123	1	1	yes	characteristic
151	C₃×C₄×C₅	#126	1	1	yes	characteristic
152	C₂⁴×C₃	—	2	1	no	—
153	C₂×C₃×D₄	—	2	1	no	—
154	C₂×C₃×D₄	—	2	1	no	—
155	C₂×C₃×D₄	—	2	1	no	—
156	C₂×C₃×D₄	—	2	1	no	—
157	C₂×C₃×D₄	—	2	1	no	—
158	C₂×C₃×D₄	C₅×D₄	1	2	yes	—
159	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
160	C₃×M₁₆	—	2	1	no	—
161	C₃×M₁₆	—	2	1	no	—
162	C₃×M₁₆	—	2	1	no	—
163	C₃×M₁₆	—	2	1	no	—
164	C₃×M₁₆	—	2	1	no	—
165	C₃×M₁₆	—	2	1	no	—
166	C₃×M₁₆	—	2	1	no	—
167	C₃×M₁₆	—	2	1	no	—
168	C₂×C₃×C₈	—	2	1	no	—
169	C₂×C₃×C₈	—	2	1	no	—
170	C₂×C₃×C₈	—	2	1	no	—
171	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
172	C₃×C₄⋊C₄	—	2	1	no	—
173	C₃×C₄⋊C₄	—	2	1	no	—
174	C₃×C₂²⋊C₄	—	2	1	no	—
175	C₃×C₂²⋊C₄	—	2	1	no	—
176	C₃×C₄²	—	2	1	no	—
177	C₃×C₄²	—	2	1	no	—
178	C₂³×C₅	—	2	2	no	—
179	C₂³×C₅	—	4	1	no	—
180	C₂³×C₅	—	4	1	no	—
181	C₂³×C₅	—	8	2	no	—
182	C₂³×C₅	C₃×#41	1	1	yes	characteristic
183	C₅×D₄	—	2	2	no	—
184	C₅×D₄	—	2	2	no	—
185	C₅×D₄	—	4	1	no	—
186	C₅×D₄	—	4	1	no	—
187	C₂×C₄×C₅	—	4	1	no	—
188	C₂×C₄×C₅	—	4	1	no	—
189	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
190	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
191	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
192	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
193	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
194	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
195	C₅×C₈	—	2	2	no	—
196	C₅×C₈	—	4	1	no	—
197	C₅×C₈	—	4	1	no	—
198	C₅×C₈	—	4	1	no	—
199	C₂²×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
200	C₂×M₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
201	C₂×M₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
202	C₂×M₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
203	C₂×M₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
204	#117	C₂²×C₃×C₅	1	1	yes	characteristic
205	#117	C₂²×C₃×C₅	1	1	yes	characteristic
206	#108	—	2	2	no	—
207	#102	—	2	2	no	—
208	#102	—	2	2	no	—
209	#100	—	2	2	no	—
210	C₂×C₃×C₅	—	2	1	no	—
211	C₂×C₃×C₅	—	2	1	no	—
212	C₂×C₃×C₅	—	2	1	no	—
213	C₂×C₃×C₅	—	8	2	no	—
214	C₂×C₃×C₅	#397	1	1	yes	characteristic
215	C₂³×C₃	—	2	2	no	—
216	C₂³×C₃	—	4	1	no	—
217	C₂³×C₃	—	4	1	no	—
218	C₂³×C₃	—	8	2	no	—
219	C₂³×C₃	C₅×#41	1	1	yes	characteristic
220	C₃×D₄	—	2	2	no	—
221	C₃×D₄	—	2	2	no	—
222	C₃×D₄	—	4	1	no	—
223	C₃×D₄	—	4	1	no	—
224	C₂×C₃×C₄	—	4	1	no	—
225	C₂×C₃×C₄	—	4	1	no	—
226	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
227	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
228	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
229	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
230	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
231	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
232	C₃×C₈	—	2	2	no	—
233	C₃×C₈	—	4	1	no	—
234	C₃×C₈	—	4	1	no	—
235	C₃×C₈	—	4	1	no	—
236	C₂²×C₅	—	4	1	no	—
237	C₂²×C₅	—	4	2	no	—
238	C₂²×C₅	—	8	2	no	—
239	C₂²×C₅	—	8	2	no	—
240	C₂²×C₅	—	8	2	no	—
241	C₂²×C₅	C₃×#123	1	1	yes	characteristic
242	C₂²×C₅	C₃×#124	1	1	yes	characteristic
243	C₂²×C₅	C₃×#126	1	1	yes	characteristic
244	C₄×C₅	—	8	1	no	—
245	C₄×C₅	—	8	1	no	—
246	C₄×C₅	C₃×#121	1	1	yes	characteristic
247	C₄×C₅	C₃×#123	1	1	yes	characteristic
248	C₄×C₅	C₃×#123	1	1	yes	characteristic
249	C₄×C₅	C₃×#126	1	1	yes	characteristic
250	C₂⁴	—	2	1	no	—
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	—	2	1	no	—
253	C₂×D₄	—	2	1	no	—
254	C₂×D₄	—	2	1	no	—
255	C₂×D₄	—	2	1	no	—
256	C₂×D₄	C₃×C₅×D₄	1	2	yes	—
257	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
258	M₁₆	—	2	1	no	—
259	M₁₆	—	2	1	no	—
260	M₁₆	—	2	1	no	—
261	M₁₆	—	2	1	no	—
262	M₁₆	—	2	1	no	—
263	M₁₆	—	2	1	no	—
264	M₁₆	—	2	1	no	—
265	M₁₆	—	2	1	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	—	2	1	no	—
269	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
270	C₄⋊C₄	—	2	1	no	—
271	C₄⋊C₄	—	2	1	no	—
272	C₂²⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	2	1	no	—
274	C₄²	—	2	1	no	—
275	C₄²	—	2	1	no	—
276	C₃×C₅	#2081	1	1	yes	characteristic, direct product factor, Hall
277	C₂²×C₃	—	4	1	no	—
278	C₂²×C₃	—	4	2	no	—
279	C₂²×C₃	—	8	2	no	—
280	C₂²×C₃	—	8	2	no	—
281	C₂²×C₃	—	8	2	no	—
282	C₂²×C₃	C₅×#123	1	1	yes	characteristic
283	C₂²×C₃	C₅×#124	1	1	yes	characteristic
284	C₂²×C₃	C₅×#126	1	1	yes	characteristic
285	C₃×C₄	—	8	1	no	—
286	C₃×C₄	—	8	1	no	—
287	C₃×C₄	C₅×#121	1	1	yes	characteristic
288	C₃×C₄	C₅×#123	1	1	yes	characteristic
289	C₃×C₄	C₅×#123	1	1	yes	characteristic
290	C₃×C₄	C₅×#126	1	1	yes	characteristic
291	C₂×C₅	—	2	1	no	—
292	C₂×C₅	—	2	1	no	—
293	C₂×C₅	—	2	1	no	—
294	C₂×C₅	—	8	2	no	—
295	C₂×C₅	C₃×#397	1	1	yes	characteristic
296	C₂³	—	2	2	no	—
297	C₂³	—	4	1	no	—
298	C₂³	—	4	1	no	—
299	C₂³	—	8	2	no	—
300	C₂³	C₃×C₅×#41	1	1	yes	characteristic
301	D₄	—	2	2	no	—
302	D₄	—	2	2	no	—
303	D₄	—	4	1	no	—
304	D₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
308	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
309	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
310	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
311	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
312	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
313	C₈	—	2	2	no	—
314	C₈	—	4	1	no	—
315	C₈	—	4	1	no	—
316	C₈	—	4	1	no	—
317	C₂×C₃	—	2	1	no	—
318	C₂×C₃	—	2	1	no	—
319	C₂×C₃	—	2	1	no	—
320	C₂×C₃	—	8	2	no	—
321	C₂×C₃	C₅×#397	1	1	yes	characteristic
322	C₅	C₃×#2081	1	1	yes	characteristic, direct product factor, Sylow
323	C₂²	—	4	1	no	—
324	C₂²	—	4	2	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	8	2	no	—
328	C₂²	C₃×C₅×#123	1	1	yes	characteristic
329	C₂²	C₃×C₅×#124	1	1	yes	characteristic
330	C₂²	C₃×C₅×#126	1	1	yes	characteristic
331	C₄	—	8	1	no	—
332	C₄	—	8	1	no	—
333	C₄	C₃×C₅×#121	1	1	yes	characteristic
334	C₄	C₃×C₅×#123	1	1	yes	characteristic
335	C₄	C₃×C₅×#123	1	1	yes	characteristic
336	C₄	C₃×C₅×#126	1	1	yes	characteristic
337	C₃	C₅×#2081	1	1	yes	characteristic, direct product factor, Sylow
338	C₂	—	2	1	no	—
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	—	8	2	no	—
342	C₂	C₃×C₅×#397	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃×C₅	#397
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
prime radical	C₂²×C₃×C₅×D₄	C₂²
LCS	G ▷ C₂²×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
1. C₃×C₅×#430	C₂	semidirect
5. C₃×C₅×#344	C₂	semidirect
6. C₃×C₅×#344	C₂	semidirect
7. C₃×C₅×H(3,ℤ/4)	C₂	semidirect
8. C₅×#2081	C₃	direct (N × H)
20. C₃×#2081	C₅	direct (N × H)
21. C₅×#430	C₂×C₃	semidirect
25. C₅×#344	C₂×C₃	semidirect
26. C₅×#344	C₂×C₃	semidirect
27. C₅×H(3,ℤ/4)	C₂×C₃	semidirect
54. C₃×#430	C₂×C₅	semidirect
58. C₃×#344	C₂×C₅	semidirect
59. C₃×#344	C₂×C₅	semidirect
60. C₃×H(3,ℤ/4)	C₂×C₅	semidirect
72. #2081	C₃×C₅	direct (N × H)
131. #430	C₂×C₃×C₅	semidirect
135. #344	C₂×C₃×C₅	semidirect
136. #344	C₂×C₃×C₅	semidirect
137. H(3,ℤ/4)	C₂×C₃×C₅	semidirect
276. C₃×C₅	#2081	direct (N × H)
322. C₅	C₃×#2081	direct (N × H)
337. C₃	C₅×#2081	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×#430	2
C₃×C₅×#418	2
C₂×C₃×C₅×#100	2
C₃×C₅×#344	2
C₃×C₅×H(3,ℤ/4)	2
C₅×#2081	3
C₃×#2081	5

Subgroups — C3×C5×#2081 #14275115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2081 #14275115