Subgroups — C₃×C₅×#1671 #14274705

Order 1920.

332 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#498	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×#476	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#461	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×C₈×D₄	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×#334	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₅×#333	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₅×#332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1671	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₅×SD₁₆	—	2	1	no	—
10	C₂²×C₃×C₅×C₈	—	2	1	no	—
11	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
14	C₃×C₅×#106	—	2	1	no	—
15	C₃×C₅×#106	—	2	1	no	—
16	C₃×C₅×#106	—	2	1	no	—
17	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₃×C₅×#103	—	2	1	no	—
21	C₃×C₅×#102	—	2	1	no	—
22	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
23	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
24	C₃×#1671	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₅×#498	C₂×C₃	1	1	yes	characteristic, complemented
26	C₅×#476	C₂×C₃	1	1	yes	characteristic
27	C₅×#461	C₂×C₃	1	1	yes	characteristic
28	C₅×C₈×D₄	C₂×C₃	1	1	yes	characteristic
29	C₅×#334	C₂×C₃	1	1	yes	characteristic, complemented
30	C₅×#333	C₂×C₃	1	1	yes	characteristic, complemented
31	C₅×#332	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂×C₃×C₅×Q₈	—	2	1	no	—
33	C₂×C₃×C₅×Q₈	—	2	1	no	—
34	C₂×C₃×C₅×D₄	D₄	1	1	yes	characteristic
35	C₂²×C₃×C₄×C₅	—	2	1	no	—
36	C₃×C₅×SD₁₆	—	4	2	no	—
37	C₂×C₃×C₅×C₈	—	2	1	no	—
38	C₂×C₃×C₅×C₈	—	2	1	no	—
39	C₂×C₃×C₅×C₈	—	2	1	no	—
40	C₂×C₃×C₅×C₈	—	4	2	no	—
41	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
42	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
43	C₃×C₅×C₄⋊C₄	—	2	1	no	—
44	C₃×C₅×C₄⋊C₄	—	2	1	no	—
45	C₃×C₅×C₄⋊C₄	—	2	1	no	—
46	C₃×C₅×C₄⋊C₄	—	2	1	no	—
47	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
48	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
49	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
50	C₃×#498	C₂×C₅	1	1	yes	characteristic, complemented
51	C₃×#476	C₂×C₅	1	1	yes	characteristic
52	C₃×#461	C₂×C₅	1	1	yes	characteristic
53	C₃×C₈×D₄	C₂×C₅	1	1	yes	characteristic
54	C₃×#334	C₂×C₅	1	1	yes	characteristic, complemented
55	C₃×#333	C₂×C₅	1	1	yes	characteristic, complemented
56	C₃×#332	C₂×C₅	1	1	yes	characteristic, complemented
57	C₂×C₅×SD₁₆	—	2	1	no	—
58	C₂²×C₅×C₈	—	2	1	no	—
59	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
60	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
61	C₄×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
62	C₅×#106	—	2	1	no	—
63	C₅×#106	—	2	1	no	—
64	C₅×#106	—	2	1	no	—
65	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
66	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
67	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
68	C₅×#103	—	2	1	no	—
69	C₅×#102	—	2	1	no	—
70	C₅×C₂²⋊C₈	—	2	1	no	—
71	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
72	#1671	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
73	C₂³×C₃×C₅	—	2	1	no	—
74	C₃×C₅×Q₈	—	4	1	no	—
75	C₃×C₅×Q₈	—	4	1	no	—
76	C₃×C₅×Q₈	—	8	1	no	—
77	C₃×C₅×D₄	—	2	1	no	—
78	C₃×C₅×D₄	SD₁₆	1	2	yes	—
79	C₂×C₃×C₄×C₅	—	2	1	no	—
80	C₂×C₃×C₄×C₅	—	4	1	no	—
81	C₂×C₃×C₄×C₅	—	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	C₃×C₅×C₈	—	2	1	no	—
88	C₃×C₅×C₈	—	4	1	no	—
89	C₃×C₅×C₈	—	4	1	no	—
90	C₃×C₅×C₈	—	4	1	no	—
91	C₃×C₅×C₈	SD₁₆	1	2	yes	complemented
92	C₂×C₃×SD₁₆	—	2	1	no	—
93	C₂²×C₃×C₈	—	2	1	no	—
94	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
95	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
96	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
97	C₃×#106	—	2	1	no	—
98	C₃×#106	—	2	1	no	—
99	C₃×#106	—	2	1	no	—
100	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
101	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
102	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
103	C₃×#103	—	2	1	no	—
104	C₃×#102	—	2	1	no	—
105	C₃×C₂²⋊C₈	—	2	1	no	—
106	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
107	C₂×C₅×Q₈	—	2	1	no	—
108	C₂×C₅×Q₈	—	2	1	no	—
109	C₂×C₅×D₄	C₃×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₅	—	2	1	no	—
111	C₅×SD₁₆	—	4	2	no	—
112	C₂×C₅×C₈	—	2	1	no	—
113	C₂×C₅×C₈	—	2	1	no	—
114	C₂×C₅×C₈	—	2	1	no	—
115	C₂×C₅×C₈	—	4	2	no	—
116	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
117	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
118	C₅×C₄⋊C₄	—	2	1	no	—
119	C₅×C₄⋊C₄	—	2	1	no	—
120	C₅×C₄⋊C₄	—	2	1	no	—
121	C₅×C₄⋊C₄	—	2	1	no	—
122	C₅×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
123	C₅×C₂²⋊C₄	—	2	1	no	—
124	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
125	#498	C₂×C₃×C₅	1	1	yes	characteristic, complemented
126	#476	C₂×C₃×C₅	1	1	yes	characteristic
127	#461	C₂×C₃×C₅	1	1	yes	characteristic
128	C₈×D₄	C₂×C₃×C₅	1	1	yes	characteristic
129	#334	C₂×C₃×C₅	1	1	yes	characteristic, complemented
130	#333	C₂×C₃×C₅	1	1	yes	characteristic, complemented
131	#332	C₂×C₃×C₅	1	1	yes	characteristic, complemented
132	C₂²×C₃×C₅	—	2	2	no	—
133	C₂²×C₃×C₅	—	4	1	no	—
134	C₂²×C₃×C₅	—	4	1	no	—
135	C₂²×C₃×C₅	#121	1	1	yes	characteristic
136	C₃×C₄×C₅	—	2	1	no	—
137	C₃×C₄×C₅	—	4	1	no	—
138	C₃×C₄×C₅	—	8	1	no	—
139	C₃×C₄×C₅	—	8	1	no	—
140	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
141	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
142	C₃×C₄×C₅	#135	1	1	yes	characteristic
143	C₃×C₄×C₅	#137	1	1	yes	characteristic
144	C₂×C₃×Q₈	—	2	1	no	—
145	C₂×C₃×Q₈	—	2	1	no	—
146	C₂×C₃×D₄	C₅×D₄	1	1	yes	characteristic
147	C₂²×C₃×C₄	—	2	1	no	—
148	C₃×SD₁₆	—	4	2	no	—
149	C₂×C₃×C₈	—	2	1	no	—
150	C₂×C₃×C₈	—	2	1	no	—
151	C₂×C₃×C₈	—	2	1	no	—
152	C₂×C₃×C₈	—	4	2	no	—
153	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
154	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
155	C₃×C₄⋊C₄	—	2	1	no	—
156	C₃×C₄⋊C₄	—	2	1	no	—
157	C₃×C₄⋊C₄	—	2	1	no	—
158	C₃×C₄⋊C₄	—	2	1	no	—
159	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
160	C₃×C₂²⋊C₄	—	2	1	no	—
161	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
162	C₂³×C₅	—	2	1	no	—
163	C₅×Q₈	—	4	1	no	—
164	C₅×Q₈	—	4	1	no	—
165	C₅×Q₈	—	8	1	no	—
166	C₅×D₄	—	2	1	no	—
167	C₅×D₄	C₃×SD₁₆	1	2	yes	—
168	C₂×C₄×C₅	—	2	1	no	—
169	C₂×C₄×C₅	—	4	1	no	—
170	C₂×C₄×C₅	—	4	1	no	—
171	C₂×C₄×C₅	—	4	1	no	—
172	C₂×C₄×C₅	—	4	1	no	—
173	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
175	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
176	C₅×C₈	—	2	1	no	—
177	C₅×C₈	—	4	1	no	—
178	C₅×C₈	—	4	1	no	—
179	C₅×C₈	—	4	1	no	—
180	C₅×C₈	C₃×SD₁₆	1	2	yes	complemented
181	C₂×SD₁₆	—	2	1	no	—
182	C₂²×C₈	—	2	1	no	—
183	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
184	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
185	C₄×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
186	#106	—	2	1	no	—
187	#106	—	2	1	no	—
188	#106	—	2	1	no	—
189	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
190	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
191	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
192	#103	—	2	1	no	—
193	#102	—	2	1	no	—
194	C₂²⋊C₈	—	2	1	no	—
195	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
196	C₂×C₃×C₅	—	4	2	no	—
197	C₂×C₃×C₅	#461	1	1	yes	characteristic
198	C₂×C₃×C₅	#465	1	1	yes	characteristic
199	C₂×C₃×C₅	#471	1	1	yes	characteristic
200	C₂³×C₃	—	2	1	no	—
201	C₃×Q₈	—	4	1	no	—
202	C₃×Q₈	—	4	1	no	—
203	C₃×Q₈	—	8	1	no	—
204	C₃×D₄	—	2	1	no	—
205	C₃×D₄	C₅×SD₁₆	1	2	yes	—
206	C₂×C₃×C₄	—	2	1	no	—
207	C₂×C₃×C₄	—	4	1	no	—
208	C₂×C₃×C₄	—	4	1	no	—
209	C₂×C₃×C₄	—	4	1	no	—
210	C₂×C₃×C₄	—	4	1	no	—
211	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
213	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
214	C₃×C₈	—	2	1	no	—
215	C₃×C₈	—	4	1	no	—
216	C₃×C₈	—	4	1	no	—
217	C₃×C₈	—	4	1	no	—
218	C₃×C₈	C₅×SD₁₆	1	2	yes	complemented
219	C₂²×C₅	—	2	2	no	—
220	C₂²×C₅	—	4	1	no	—
221	C₂²×C₅	—	4	1	no	—
222	C₂²×C₅	C₃×#121	1	1	yes	characteristic
223	C₄×C₅	—	2	1	no	—
224	C₄×C₅	—	4	1	no	—
225	C₄×C₅	—	8	1	no	—
226	C₄×C₅	—	8	1	no	—
227	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
228	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
229	C₄×C₅	C₃×#135	1	1	yes	characteristic
230	C₄×C₅	C₃×#137	1	1	yes	characteristic
231	C₂×Q₈	—	2	1	no	—
232	C₂×Q₈	—	2	1	no	—
233	C₂×D₄	C₃×C₅×D₄	1	1	yes	characteristic
234	C₂²×C₄	—	2	1	no	—
235	SD₁₆	—	4	2	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
241	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	C₃×C₅×D₄	1	1	yes	characteristic
247	C₂²⋊C₄	—	2	1	no	—
248	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
249	C₃×C₅	#1671	1	1	yes	characteristic, direct product factor, Hall
250	C₂²×C₃	—	2	2	no	—
251	C₂²×C₃	—	4	1	no	—
252	C₂²×C₃	—	4	1	no	—
253	C₂²×C₃	C₅×#121	1	1	yes	characteristic
254	C₃×C₄	—	2	1	no	—
255	C₃×C₄	—	4	1	no	—
256	C₃×C₄	—	8	1	no	—
257	C₃×C₄	—	8	1	no	—
258	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
259	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
260	C₃×C₄	C₅×#135	1	1	yes	characteristic
261	C₃×C₄	C₅×#137	1	1	yes	characteristic
262	C₂×C₅	—	4	2	no	—
263	C₂×C₅	C₃×#461	1	1	yes	characteristic
264	C₂×C₅	C₃×#465	1	1	yes	characteristic
265	C₂×C₅	C₃×#471	1	1	yes	characteristic
266	C₂³	—	2	1	no	—
267	Q₈	—	4	1	no	—
268	Q₈	—	4	1	no	—
269	Q₈	—	8	1	no	—
270	D₄	—	2	1	no	—
271	D₄	C₃×C₅×SD₁₆	1	2	yes	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
278	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
279	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
280	C₈	—	2	1	no	—
281	C₈	—	4	1	no	—
282	C₈	—	4	1	no	—
283	C₈	—	4	1	no	—
284	C₈	C₃×C₅×SD₁₆	1	2	yes	complemented
285	C₂×C₃	—	4	2	no	—
286	C₂×C₃	C₅×#461	1	1	yes	characteristic
287	C₂×C₃	C₅×#465	1	1	yes	characteristic
288	C₂×C₃	C₅×#471	1	1	yes	characteristic
289	C₅	C₃×#1671	1	1	yes	characteristic, direct product factor, Sylow
290	C₂²	—	2	2	no	—
291	C₂²	—	4	1	no	—
292	C₂²	—	4	1	no	—
293	C₂²	C₃×C₅×#121	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	4	1	no	—
296	C₄	—	8	1	no	—
297	C₄	—	8	1	no	—
298	C₄	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
299	C₄	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
300	C₄	C₃×C₅×#135	1	1	yes	characteristic
301	C₄	C₃×C₅×#137	1	1	yes	characteristic
302	C₃	C₅×#1671	1	1	yes	characteristic, direct product factor, Sylow
303	C₂	—	4	2	no	—
304	C₂	C₃×C₅×#461	1	1	yes	characteristic
305	C₂	C₃×C₅×#465	1	1	yes	characteristic
306	C₂	C₃×C₅×#471	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₃×C₅
prime radical	C₂×C₃×C₅×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#498	C₂	semidirect
5. C₃×C₅×#334	C₂	semidirect
6. C₃×C₅×#333	C₂	semidirect
7. C₃×C₅×#332	C₂	semidirect
8. C₅×#1671	C₃	direct (N × H)
24. C₃×#1671	C₅	direct (N × H)
25. C₅×#498	C₂×C₃	semidirect
29. C₅×#334	C₂×C₃	semidirect
30. C₅×#333	C₂×C₃	semidirect
31. C₅×#332	C₂×C₃	semidirect
50. C₃×#498	C₂×C₅	semidirect
54. C₃×#334	C₂×C₅	semidirect
55. C₃×#333	C₂×C₅	semidirect
56. C₃×#332	C₂×C₅	semidirect
72. #1671	C₃×C₅	direct (N × H)
91. C₃×C₅×C₈	SD₁₆	semidirect
125. #498	C₂×C₃×C₅	semidirect
129. #334	C₂×C₃×C₅	semidirect
130. #333	C₂×C₃×C₅	semidirect
131. #332	C₂×C₃×C₅	semidirect
180. C₅×C₈	C₃×SD₁₆	semidirect
218. C₃×C₈	C₅×SD₁₆	semidirect
249. C₃×C₅	#1671	direct (N × H)
284. C₈	C₃×C₅×SD₁₆	semidirect
289. C₅	C₃×#1671	direct (N × H)
302. C₃	C₅×#1671	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×#498	2
C₃×C₅×#476	2
C₃×C₅×#461	2
C₃×C₅×C₈×D₄	2
C₃×C₅×#334	2
C₃×C₅×#333	2
C₃×C₅×#332	2
C₅×#1671	3
C₃×#1671	5

Subgroups — C3×C5×#1671 #14274705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1671 #14274705