Subgroups — C₃×C₅×#1680 #14274714

Order 1920.

300 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#487	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#475	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#462	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×#334	C₂	1	1	yes	maximal, characteristic
6	C₃×C₅×#332	C₂	1	1	yes	maximal, characteristic
7	C₃×C₅×#332	C₂	1	1	yes	maximal, characteristic
8	C₅×#1680	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₅×Q₁₆	—	2	1	no	—
10	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
13	C₃×C₅×#106	—	2	1	no	—
14	C₃×C₅×C₄⋊C₈	—	2	1	no	—
15	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₃×C₅×#103	—	2	1	no	—
19	C₃×C₅×#103	—	2	1	no	—
20	C₃×C₅×#103	—	2	1	no	—
21	C₃×C₅×#103	—	2	1	no	—
22	C₃×C₄×C₅×C₈	—	2	1	no	—
23	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
24	C₃×#1680	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₅×#487	C₂×C₃	1	1	yes	characteristic
26	C₅×#475	C₂×C₃	1	1	yes	characteristic
27	C₅×#462	C₂×C₃	1	1	yes	characteristic
28	C₅×C₈×Q₈	C₂×C₃	1	1	yes	characteristic
29	C₅×#334	C₂×C₃	1	1	yes	characteristic
30	C₅×#332	C₂×C₃	1	1	yes	characteristic
31	C₅×#332	C₂×C₃	1	1	yes	characteristic
32	C₂×C₃×C₅×Q₈	—	2	1	no	—
33	C₂×C₃×C₅×Q₈	—	2	1	no	—
34	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
35	C₃×C₅×Q₁₆	—	4	2	no	—
36	C₂×C₃×C₅×C₈	—	2	1	no	—
37	C₂×C₃×C₅×C₈	—	2	1	no	—
38	C₂×C₃×C₅×C₈	—	2	1	no	—
39	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
40	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
41	C₃×C₅×C₄⋊C₄	—	2	1	no	—
42	C₃×C₅×C₄⋊C₄	—	2	1	no	—
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	D₄	1	1	yes	characteristic
47	C₃×C₄²×C₅	—	2	1	no	—
48	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
49	C₃×#487	C₂×C₅	1	1	yes	characteristic
50	C₃×#475	C₂×C₅	1	1	yes	characteristic
51	C₃×#462	C₂×C₅	1	1	yes	characteristic
52	C₃×C₈×Q₈	C₂×C₅	1	1	yes	characteristic
53	C₃×#334	C₂×C₅	1	1	yes	characteristic
54	C₃×#332	C₂×C₅	1	1	yes	characteristic
55	C₃×#332	C₂×C₅	1	1	yes	characteristic
56	C₂×C₅×Q₁₆	—	2	1	no	—
57	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
58	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
59	C₄×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
60	C₅×#106	—	2	1	no	—
61	C₅×C₄⋊C₈	—	2	1	no	—
62	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
63	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
64	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
65	C₅×#103	—	2	1	no	—
66	C₅×#103	—	2	1	no	—
67	C₅×#103	—	2	1	no	—
68	C₅×#103	—	2	1	no	—
69	C₄×C₅×C₈	—	2	1	no	—
70	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
71	#1680	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
72	C₃×C₅×Q₈	—	2	1	no	—
73	C₃×C₅×Q₈	—	4	1	no	—
74	C₃×C₅×Q₈	—	4	1	no	—
75	C₃×C₅×Q₈	—	4	1	no	—
76	C₃×C₅×Q₈	—	4	1	no	—
77	C₃×C₅×Q₈	SD₁₆	1	2	yes	—
78	C₂×C₃×C₄×C₅	—	2	1	no	—
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	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
85	C₃×C₅×C₈	—	2	1	no	—
86	C₃×C₅×C₈	—	2	1	no	—
87	C₃×C₅×C₈	—	4	1	no	—
88	C₃×C₅×C₈	—	4	1	no	—
89	C₃×C₅×C₈	—	4	1	no	—
90	C₂×C₃×Q₁₆	—	2	1	no	—
91	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
92	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
93	C₃×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
94	C₃×#106	—	2	1	no	—
95	C₃×C₄⋊C₈	—	2	1	no	—
96	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
97	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
98	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
99	C₃×#103	—	2	1	no	—
100	C₃×#103	—	2	1	no	—
101	C₃×#103	—	2	1	no	—
102	C₃×#103	—	2	1	no	—
103	C₃×C₄×C₈	—	2	1	no	—
104	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
105	C₂×C₅×Q₈	—	2	1	no	—
106	C₂×C₅×Q₈	—	2	1	no	—
107	C₂×C₅×Q₈	C₃×D₄	1	1	yes	characteristic
108	C₅×Q₁₆	—	4	2	no	—
109	C₂×C₅×C₈	—	2	1	no	—
110	C₂×C₅×C₈	—	2	1	no	—
111	C₂×C₅×C₈	—	2	1	no	—
112	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
113	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
114	C₅×C₄⋊C₄	—	2	1	no	—
115	C₅×C₄⋊C₄	—	2	1	no	—
116	C₅×C₄⋊C₄	—	2	1	no	—
117	C₅×C₄⋊C₄	—	2	1	no	—
118	C₅×C₄⋊C₄	—	2	1	no	—
119	C₅×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
120	C₄²×C₅	—	2	1	no	—
121	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
122	#487	C₂×C₃×C₅	1	1	yes	characteristic
123	#475	C₂×C₃×C₅	1	1	yes	characteristic
124	#462	C₂×C₃×C₅	1	1	yes	characteristic
125	C₈×Q₈	C₂×C₃×C₅	1	1	yes	characteristic
126	#334	C₂×C₃×C₅	1	1	yes	characteristic
127	#332	C₂×C₃×C₅	1	1	yes	characteristic
128	#332	C₂×C₃×C₅	1	1	yes	characteristic
129	C₂²×C₃×C₅	#121	1	1	yes	characteristic
130	C₃×C₄×C₅	—	2	2	no	—
131	C₃×C₄×C₅	—	2	1	no	—
132	C₃×C₄×C₅	—	4	1	no	—
133	C₃×C₄×C₅	—	8	1	no	—
134	C₃×C₄×C₅	—	8	1	no	—
135	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
136	C₃×C₄×C₅	#135	1	1	yes	characteristic
137	C₃×C₄×C₅	#135	1	1	yes	characteristic
138	C₃×C₄×C₅	#137	1	1	yes	characteristic
139	C₂×C₃×Q₈	—	2	1	no	—
140	C₂×C₃×Q₈	—	2	1	no	—
141	C₂×C₃×Q₈	C₅×D₄	1	1	yes	characteristic
142	C₃×Q₁₆	—	4	2	no	—
143	C₂×C₃×C₈	—	2	1	no	—
144	C₂×C₃×C₈	—	2	1	no	—
145	C₂×C₃×C₈	—	2	1	no	—
146	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
147	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
148	C₃×C₄⋊C₄	—	2	1	no	—
149	C₃×C₄⋊C₄	—	2	1	no	—
150	C₃×C₄⋊C₄	—	2	1	no	—
151	C₃×C₄⋊C₄	—	2	1	no	—
152	C₃×C₄⋊C₄	—	2	1	no	—
153	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
154	C₃×C₄²	—	2	1	no	—
155	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
156	C₅×Q₈	—	2	1	no	—
157	C₅×Q₈	—	4	1	no	—
158	C₅×Q₈	—	4	1	no	—
159	C₅×Q₈	—	4	1	no	—
160	C₅×Q₈	—	4	1	no	—
161	C₅×Q₈	C₃×SD₁₆	1	2	yes	—
162	C₂×C₄×C₅	—	2	1	no	—
163	C₂×C₄×C₅	—	2	1	no	—
164	C₂×C₄×C₅	—	4	1	no	—
165	C₂×C₄×C₅	—	4	1	no	—
166	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
167	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
168	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
169	C₅×C₈	—	2	1	no	—
170	C₅×C₈	—	2	1	no	—
171	C₅×C₈	—	4	1	no	—
172	C₅×C₈	—	4	1	no	—
173	C₅×C₈	—	4	1	no	—
174	C₂×Q₁₆	—	2	1	no	—
175	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
176	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
177	C₄×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
178	#106	—	2	1	no	—
179	C₄⋊C₈	—	2	1	no	—
180	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
181	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
182	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
183	#103	—	2	1	no	—
184	#103	—	2	1	no	—
185	#103	—	2	1	no	—
186	#103	—	2	1	no	—
187	C₄×C₈	—	2	1	no	—
188	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
189	C₂×C₃×C₅	#464	1	1	yes	characteristic
190	C₂×C₃×C₅	#465	1	1	yes	characteristic
191	C₂×C₃×C₅	#473	1	1	yes	characteristic
192	C₃×Q₈	—	2	1	no	—
193	C₃×Q₈	—	4	1	no	—
194	C₃×Q₈	—	4	1	no	—
195	C₃×Q₈	—	4	1	no	—
196	C₃×Q₈	—	4	1	no	—
197	C₃×Q₈	C₅×SD₁₆	1	2	yes	—
198	C₂×C₃×C₄	—	2	1	no	—
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	—	4	1	no	—
201	C₂×C₃×C₄	—	4	1	no	—
202	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
203	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
204	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
205	C₃×C₈	—	2	1	no	—
206	C₃×C₈	—	2	1	no	—
207	C₃×C₈	—	4	1	no	—
208	C₃×C₈	—	4	1	no	—
209	C₃×C₈	—	4	1	no	—
210	C₂²×C₅	C₃×#121	1	1	yes	characteristic
211	C₄×C₅	—	2	2	no	—
212	C₄×C₅	—	2	1	no	—
213	C₄×C₅	—	4	1	no	—
214	C₄×C₅	—	8	1	no	—
215	C₄×C₅	—	8	1	no	—
216	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
217	C₄×C₅	C₃×#135	1	1	yes	characteristic
218	C₄×C₅	C₃×#135	1	1	yes	characteristic
219	C₄×C₅	C₃×#137	1	1	yes	characteristic
220	C₂×Q₈	—	2	1	no	—
221	C₂×Q₈	—	2	1	no	—
222	C₂×Q₈	C₃×C₅×D₄	1	1	yes	characteristic
223	Q₁₆	—	4	2	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
228	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	C₃×C₅×D₄	1	1	yes	characteristic
235	C₄²	—	2	1	no	—
236	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
237	C₃×C₅	#1680	1	1	yes	characteristic, direct product factor, Hall
238	C₂²×C₃	C₅×#121	1	1	yes	characteristic
239	C₃×C₄	—	2	2	no	—
240	C₃×C₄	—	2	1	no	—
241	C₃×C₄	—	4	1	no	—
242	C₃×C₄	—	8	1	no	—
243	C₃×C₄	—	8	1	no	—
244	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
245	C₃×C₄	C₅×#135	1	1	yes	characteristic
246	C₃×C₄	C₅×#135	1	1	yes	characteristic
247	C₃×C₄	C₅×#137	1	1	yes	characteristic
248	C₂×C₅	C₃×#464	1	1	yes	characteristic
249	C₂×C₅	C₃×#465	1	1	yes	characteristic
250	C₂×C₅	C₃×#473	1	1	yes	characteristic
251	Q₈	—	2	1	no	—
252	Q₈	—	4	1	no	—
253	Q₈	—	4	1	no	—
254	Q₈	—	4	1	no	—
255	Q₈	—	4	1	no	—
256	Q₈	C₃×C₅×SD₁₆	1	2	yes	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
262	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
263	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
264	C₈	—	2	1	no	—
265	C₈	—	2	1	no	—
266	C₈	—	4	1	no	—
267	C₈	—	4	1	no	—
268	C₈	—	4	1	no	—
269	C₂×C₃	C₅×#464	1	1	yes	characteristic
270	C₂×C₃	C₅×#465	1	1	yes	characteristic
271	C₂×C₃	C₅×#473	1	1	yes	characteristic
272	C₅	C₃×#1680	1	1	yes	characteristic, direct product factor, Sylow
273	C₂²	C₃×C₅×#121	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	1	no	—
277	C₄	—	8	1	no	—
278	C₄	—	8	1	no	—
279	C₄	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
280	C₄	C₃×C₅×#135	1	1	yes	characteristic
281	C₄	C₃×C₅×#135	1	1	yes	characteristic
282	C₄	C₃×C₅×#137	1	1	yes	characteristic
283	C₃	C₅×#1680	1	1	yes	characteristic, direct product factor, Sylow
284	C₂	C₃×C₅×#464	1	1	yes	characteristic
285	C₂	C₃×C₅×#465	1	1	yes	characteristic
286	C₂	C₃×C₅×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₃×C₅	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₃×C₅
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
8. C₅×#1680	C₃	direct (N × H)
24. C₃×#1680	C₅	direct (N × H)
71. #1680	C₃×C₅	direct (N × H)
237. C₃×C₅	#1680	direct (N × H)
272. C₅	C₃×#1680	direct (N × H)
283. C₃	C₅×#1680	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×#487	2
C₃×C₅×#475	2
C₃×C₅×#462	2
C₃×C₅×C₈×Q₈	2
C₃×C₅×#334	2
C₃×C₅×#332	2
C₅×#1680	3
C₃×#1680	5

Subgroups — C3×C5×#1680 #14274714

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1680 #14274714