Subgroups — C₃×C₅×#1497 #14274531

Order 1920.

404 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×#533	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#432	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#431	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×Q₈⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₅×#1497	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₃×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
7	C₃×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
8	C₃×C₅×#125	—	2	1	no	—
9	C₃×C₅×C₂²⋊Q₈	—	2	1	no	—
10	C₃×C₅×C₂²⋊Q₈	C₄	1	2	yes	—
11	C₃×C₄×C₅×Q₈	—	2	1	no	—
12	C₃×C₄×C₅×Q₈	C₂²	1	2	yes	—
13	C₃×C₅×#117	—	2	1	no	—
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₃×C₅×C₄⋊C₈	—	2	1	no	—
16	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
17	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
18	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
19	C₃×C₅×#97	—	2	1	no	—
20	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
21	C₃×#1497	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₅×#533	C₂×C₃	1	1	yes	characteristic
23	C₅×#432	C₂×C₃	1	1	yes	characteristic
24	C₅×#431	C₂×C₃	1	1	yes	characteristic
25	C₅×Q₈⋊C₈	C₂×C₃	1	4	yes	complemented
26	C₂×C₃×C₅×Q₈	—	2	1	no	—
27	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
31	C₂×C₃×C₅×C₈	—	2	2	no	—
32	C₂×C₃×C₅×C₈	—	2	2	no	—
33	C₃×C₅×C₄⋊C₄	—	2	1	no	—
34	C₃×C₅×C₄⋊C₄	—	2	1	no	—
35	C₃×C₅×C₄⋊C₄	—	2	1	no	—
36	C₃×C₅×C₄⋊C₄	—	2	2	no	—
37	C₃×C₅×C₄⋊C₄	—	2	2	no	—
38	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
39	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
40	C₃×C₄²×C₅	—	2	2	no	—
41	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
43	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
44	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
45	C₃×#533	C₂×C₅	1	1	yes	characteristic
46	C₃×#432	C₂×C₅	1	1	yes	characteristic
47	C₃×#431	C₂×C₅	1	1	yes	characteristic
48	C₃×Q₈⋊C₈	C₂×C₅	1	4	yes	complemented
49	C₅×C₄⋊Q₈	C₃×C₄	1	1	yes	characteristic
50	C₅×C₄⋊Q₈	C₃×C₄	1	1	yes	characteristic
51	C₅×#125	—	2	1	no	—
52	C₅×C₂²⋊Q₈	—	2	1	no	—
53	C₅×C₂²⋊Q₈	C₃×C₄	1	2	yes	—
54	C₄×C₅×Q₈	—	2	1	no	—
55	C₄×C₅×Q₈	C₂²×C₃	1	2	yes	—
56	C₅×#117	—	2	1	no	—
57	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
58	C₅×C₄⋊C₈	—	2	1	no	—
59	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
60	C₅×C₂²⋊C₈	—	2	1	no	—
61	C₅×C₂²⋊C₈	—	2	1	no	—
62	C₅×#97	—	2	1	no	—
63	C₄×C₅×C₈	C₂²×C₃	1	2	yes	—
64	#1497	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
65	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
66	C₃×C₅×Q₈	—	2	4	no	—
67	C₃×C₅×Q₈	—	4	2	no	—
68	C₂×C₃×C₄×C₅	—	2	2	no	—
69	C₂×C₃×C₄×C₅	—	2	2	no	—
70	C₂×C₃×C₄×C₅	—	2	2	no	—
71	C₂×C₃×C₄×C₅	—	2	2	no	—
72	C₂×C₃×C₄×C₅	—	4	1	no	—
73	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₃×C₅×C₈	—	4	2	no	—
80	C₃×C₅×C₈	—	4	2	no	—
81	C₃×C₄⋊Q₈	C₄×C₅	1	1	yes	characteristic
82	C₃×C₄⋊Q₈	C₄×C₅	1	1	yes	characteristic
83	C₃×#125	—	2	1	no	—
84	C₃×C₂²⋊Q₈	—	2	1	no	—
85	C₃×C₂²⋊Q₈	C₄×C₅	1	2	yes	—
86	C₃×C₄×Q₈	—	2	1	no	—
87	C₃×C₄×Q₈	C₂²×C₅	1	2	yes	—
88	C₃×#117	—	2	1	no	—
89	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
90	C₃×C₄⋊C₈	—	2	1	no	—
91	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
92	C₃×C₂²⋊C₈	—	2	1	no	—
93	C₃×C₂²⋊C₈	—	2	1	no	—
94	C₃×#97	—	2	1	no	—
95	C₃×C₄×C₈	C₂²×C₅	1	2	yes	—
96	C₂×C₅×Q₈	—	2	1	no	—
97	C₂×C₅×Q₈	C₂×C₃×C₄	1	2	yes	—
98	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
101	C₂×C₅×C₈	—	2	2	no	—
102	C₂×C₅×C₈	—	2	2	no	—
103	C₅×C₄⋊C₄	—	2	1	no	—
104	C₅×C₄⋊C₄	—	2	1	no	—
105	C₅×C₄⋊C₄	—	2	1	no	—
106	C₅×C₄⋊C₄	—	2	2	no	—
107	C₅×C₄⋊C₄	—	2	2	no	—
108	C₅×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
109	C₅×C₂²⋊C₄	—	2	2	no	—
110	C₄²×C₅	—	2	2	no	—
111	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
112	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
113	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
114	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
115	#533	C₂×C₃×C₅	1	1	yes	characteristic
116	#432	C₂×C₃×C₅	1	1	yes	characteristic
117	#431	C₂×C₃×C₅	1	1	yes	characteristic
118	Q₈⋊C₈	C₂×C₃×C₅	1	4	yes	complemented
119	C₂²×C₃×C₅	—	2	1	no	—
120	C₂²×C₃×C₅	—	2	1	no	—
121	C₂²×C₃×C₅	—	2	1	no	—
122	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₃×C₄×C₅	—	2	1	no	—
124	C₃×C₄×C₅	—	2	1	no	—
125	C₃×C₄×C₅	—	2	1	no	—
126	C₃×C₄×C₅	—	2	1	no	—
127	C₃×C₄×C₅	—	4	2	no	—
128	C₃×C₄×C₅	—	4	2	no	—
129	C₃×C₄×C₅	#131	1	2	yes	—
130	C₃×C₄×C₅	#137	1	2	yes	—
131	C₂×C₃×Q₈	—	2	1	no	—
132	C₂×C₃×Q₈	C₂×C₄×C₅	1	2	yes	—
133	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
134	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
135	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
136	C₂×C₃×C₈	—	2	2	no	—
137	C₂×C₃×C₈	—	2	2	no	—
138	C₃×C₄⋊C₄	—	2	1	no	—
139	C₃×C₄⋊C₄	—	2	1	no	—
140	C₃×C₄⋊C₄	—	2	1	no	—
141	C₃×C₄⋊C₄	—	2	2	no	—
142	C₃×C₄⋊C₄	—	2	2	no	—
143	C₃×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
144	C₃×C₂²⋊C₄	—	2	2	no	—
145	C₃×C₄²	—	2	2	no	—
146	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
147	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
148	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
149	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
150	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
151	C₅×Q₈	—	2	4	no	—
152	C₅×Q₈	—	4	2	no	—
153	C₂×C₄×C₅	—	2	2	no	—
154	C₂×C₄×C₅	—	2	2	no	—
155	C₂×C₄×C₅	—	2	2	no	—
156	C₂×C₄×C₅	—	2	2	no	—
157	C₂×C₄×C₅	—	4	1	no	—
158	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
159	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
160	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
161	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
164	C₅×C₈	—	4	2	no	—
165	C₅×C₈	—	4	2	no	—
166	C₄⋊Q₈	C₃×C₄×C₅	1	1	yes	characteristic
167	C₄⋊Q₈	C₃×C₄×C₅	1	1	yes	characteristic
168	#125	—	2	1	no	—
169	C₂²⋊Q₈	—	2	1	no	—
170	C₂²⋊Q₈	C₃×C₄×C₅	1	2	yes	—
171	C₄×Q₈	—	2	1	no	—
172	C₄×Q₈	C₂²×C₃×C₅	1	2	yes	—
173	#117	—	2	1	no	—
174	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
175	C₄⋊C₈	—	2	1	no	—
176	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
177	C₂²⋊C₈	—	2	1	no	—
178	C₂²⋊C₈	—	2	1	no	—
179	#97	—	2	1	no	—
180	C₄×C₈	C₂²×C₃×C₅	1	2	yes	—
181	C₂×C₃×C₅	—	4	1	no	—
182	C₂×C₃×C₅	#408	1	1	yes	characteristic
183	C₂×C₃×C₅	#419	1	1	yes	characteristic
184	C₂×C₃×C₅	#421	1	1	yes	characteristic
185	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
186	C₃×Q₈	—	2	4	no	—
187	C₃×Q₈	—	4	2	no	—
188	C₂×C₃×C₄	—	2	2	no	—
189	C₂×C₃×C₄	—	2	2	no	—
190	C₂×C₃×C₄	—	2	2	no	—
191	C₂×C₃×C₄	—	2	2	no	—
192	C₂×C₃×C₄	—	4	1	no	—
193	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
194	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
195	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
199	C₃×C₈	—	4	2	no	—
200	C₃×C₈	—	4	2	no	—
201	C₂²×C₅	—	2	1	no	—
202	C₂²×C₅	—	2	1	no	—
203	C₂²×C₅	—	2	1	no	—
204	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
205	C₄×C₅	—	2	1	no	—
206	C₄×C₅	—	2	1	no	—
207	C₄×C₅	—	2	1	no	—
208	C₄×C₅	—	2	1	no	—
209	C₄×C₅	—	4	2	no	—
210	C₄×C₅	—	4	2	no	—
211	C₄×C₅	C₃×#131	1	2	yes	—
212	C₄×C₅	C₃×#137	1	2	yes	—
213	C₂×Q₈	—	2	1	no	—
214	C₂×Q₈	C₂×C₃×C₄×C₅	1	2	yes	—
215	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
216	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
217	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₄⋊C₄	—	2	1	no	—
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	C₂×C₃×C₄×C₅	1	2	yes	—
226	C₂²⋊C₄	—	2	2	no	—
227	C₄²	—	2	2	no	—
228	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
229	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
230	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
231	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
232	C₃×C₅	#1497	1	1	yes	characteristic, direct product factor, Hall
233	C₂²×C₃	—	2	1	no	—
234	C₂²×C₃	—	2	1	no	—
235	C₂²×C₃	—	2	1	no	—
236	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
237	C₃×C₄	—	2	1	no	—
238	C₃×C₄	—	2	1	no	—
239	C₃×C₄	—	2	1	no	—
240	C₃×C₄	—	2	1	no	—
241	C₃×C₄	—	4	2	no	—
242	C₃×C₄	—	4	2	no	—
243	C₃×C₄	C₅×#131	1	2	yes	—
244	C₃×C₄	C₅×#137	1	2	yes	—
245	C₂×C₅	—	4	1	no	—
246	C₂×C₅	C₃×#408	1	1	yes	characteristic
247	C₂×C₅	C₃×#419	1	1	yes	characteristic
248	C₂×C₅	C₃×#421	1	1	yes	characteristic
249	C₂³	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
250	Q₈	—	2	4	no	—
251	Q₈	—	4	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
258	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
259	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
260	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
261	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
262	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
263	C₈	—	4	2	no	—
264	C₈	—	4	2	no	—
265	C₂×C₃	—	4	1	no	—
266	C₂×C₃	C₅×#408	1	1	yes	characteristic
267	C₂×C₃	C₅×#419	1	1	yes	characteristic
268	C₂×C₃	C₅×#421	1	1	yes	characteristic
269	C₅	C₃×#1497	1	1	yes	characteristic, direct product factor, Sylow
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	4	2	no	—
279	C₄	—	4	2	no	—
280	C₄	C₃×C₅×#131	1	2	yes	—
281	C₄	C₃×C₅×#137	1	2	yes	—
282	C₃	C₅×#1497	1	1	yes	characteristic, direct product factor, Sylow
283	C₂	—	4	1	no	—
284	C₂	C₃×C₅×#408	1	1	yes	characteristic
285	C₂	C₃×C₅×#419	1	1	yes	characteristic
286	C₂	C₃×C₅×#421	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₄²	C₂³×C₃×C₅
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂³×C₃×C₅	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×Q₈⋊C₈	C₂	semidirect
5. C₅×#1497	C₃	direct (N × H)
21. C₃×#1497	C₅	direct (N × H)
25. C₅×Q₈⋊C₈	C₂×C₃	semidirect
48. C₃×Q₈⋊C₈	C₂×C₅	semidirect
64. #1497	C₃×C₅	direct (N × H)
118. Q₈⋊C₈	C₂×C₃×C₅	semidirect
232. C₃×C₅	#1497	direct (N × H)
269. C₅	C₃×#1497	direct (N × H)
282. C₃	C₅×#1497	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×#533	2
C₃×C₅×#432	2
C₃×C₅×#431	2
C₃×C₅×Q₈⋊C₈	2
C₅×#1497	3
C₃×#1497	5

Subgroups — C3×C5×#1497 #14274531

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1497 #14274531