Subgroups — C₃×C₅×#3131 #14276165

Order 1920.

800 conjugacy classes of subgroups.

The subgroup lattice has 316 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, complemented
2	C₃×C₅×#532	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#488	C₂	1	4	yes	maximal, complemented
4	C₃×C₅×#487	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#486	C₂	1	2	yes	maximal, complemented
6	C₃×C₅×#416	C₂	1	4	yes	maximal, complemented
7	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#3131	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₅×#41	C₂²	1	2	yes	—
10	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
11	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₃×C₅×#127	C₂²	1	1	yes	characteristic
14	C₃×C₅×#125	C₂²	1	2	yes	—
15	C₃×C₅×#124	C₂²	1	2	yes	—
16	C₃×C₅×C₂²⋊Q₈	—	2	2	no	—
17	C₃×C₅×#121	—	2	2	no	—
18	C₃×C₄×C₅×Q₈	—	2	2	no	—
19	C₃×C₄×C₅×D₄	—	2	2	no	—
20	C₃×C₅×#117	C₂²	1	2	yes	—
21	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
22	C₃×C₅×#103	C₂²	1	8	yes	—
23	C₃×C₅×#102	C₂²	1	8	yes	—
24	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
25	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
26	C₃×#3131	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
27	C₅×#533	C₂×C₃	1	1	yes	characteristic, complemented
28	C₅×#532	C₂×C₃	1	1	yes	characteristic
29	C₅×#488	C₂×C₃	1	4	yes	complemented
30	C₅×#487	C₂×C₃	1	2	yes	complemented
31	C₅×#486	C₂×C₃	1	2	yes	complemented
32	C₅×#416	C₂×C₃	1	4	yes	complemented
33	C₂×C₄×C₅×C₈	C₂×C₃	1	1	yes	characteristic, complemented
34	C₃×C₅×#41	—	2	8	no	—
35	C₂×C₃×C₅×Q₈	—	2	2	no	—
36	C₂×C₃×C₅×Q₈	C₂³	1	2	yes	—
37	C₂×C₃×C₅×D₄	—	2	2	no	—
38	C₂×C₃×C₅×D₄	—	2	2	no	—
39	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
40	C₂²×C₃×C₄×C₅	—	2	2	no	—
41	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
42	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
43	C₂×C₃×C₅×C₈	—	2	4	no	—
44	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
45	C₃×C₅×C₄⋊C₄	—	2	2	no	—
46	C₃×C₅×C₄⋊C₄	—	2	4	no	—
47	C₃×C₅×C₄⋊C₄	—	2	2	no	—
48	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
49	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
50	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
51	C₃×C₄²×C₅	—	2	2	no	—
52	C₃×C₄²×C₅	D₄	1	2	yes	—
53	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
54	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
55	C₃×#533	C₂×C₅	1	1	yes	characteristic, complemented
56	C₃×#532	C₂×C₅	1	1	yes	characteristic
57	C₃×#488	C₂×C₅	1	4	yes	complemented
58	C₃×#487	C₂×C₅	1	2	yes	complemented
59	C₃×#486	C₂×C₅	1	2	yes	complemented
60	C₃×#416	C₂×C₅	1	4	yes	complemented
61	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic, complemented
62	C₂×C₅×#41	C₂²×C₃	1	2	yes	—
63	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
64	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
65	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
66	C₅×#127	C₂²×C₃	1	1	yes	characteristic
67	C₅×#125	C₂²×C₃	1	2	yes	—
68	C₅×#124	C₂²×C₃	1	2	yes	—
69	C₅×C₂²⋊Q₈	—	2	2	no	—
70	C₅×#121	—	2	2	no	—
71	C₄×C₅×Q₈	—	2	2	no	—
72	C₄×C₅×D₄	—	2	2	no	—
73	C₅×#117	C₂²×C₃	1	2	yes	—
74	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
75	C₅×#103	C₂²×C₃	1	8	yes	—
76	C₅×#102	C₂²×C₃	1	8	yes	—
77	C₄×C₅×C₈	C₂²×C₃	1	2	yes	—
78	C₄×C₅×C₈	C₂²×C₃	1	2	yes	—
79	#3131	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
80	C₂³×C₃×C₅	—	2	2	no	—
81	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
82	C₃×C₅×Q₈	—	2	4	no	—
83	C₃×C₅×Q₈	—	4	4	no	—
84	C₃×C₅×D₄	—	2	4	no	—
85	C₃×C₅×D₄	—	4	4	no	—
86	C₃×C₅×D₄	—	4	4	no	—
87	C₂×C₃×C₄×C₅	—	2	4	no	—
88	C₂×C₃×C₄×C₅	—	2	4	no	—
89	C₂×C₃×C₄×C₅	—	2	2	no	—
90	C₂×C₃×C₄×C₅	—	4	4	no	—
91	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
92	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
93	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
94	C₂×C₃×C₄×C₅	#41	1	4	yes	—
95	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
96	C₃×C₅×C₈	—	2	4	no	—
97	C₂×C₃×#41	C₂²×C₅	1	2	yes	—
98	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
99	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
100	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
101	C₃×#127	C₂²×C₅	1	1	yes	characteristic
102	C₃×#125	C₂²×C₅	1	2	yes	—
103	C₃×#124	C₂²×C₅	1	2	yes	—
104	C₃×C₂²⋊Q₈	—	2	2	no	—
105	C₃×#121	—	2	2	no	—
106	C₃×C₄×Q₈	—	2	2	no	—
107	C₃×C₄×D₄	—	2	2	no	—
108	C₃×#117	C₂²×C₅	1	2	yes	—
109	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
110	C₃×#103	C₂²×C₅	1	8	yes	—
111	C₃×#102	C₂²×C₅	1	8	yes	—
112	C₃×C₄×C₈	C₂²×C₅	1	2	yes	—
113	C₃×C₄×C₈	C₂²×C₅	1	2	yes	—
114	C₅×#41	—	2	8	no	—
115	C₂×C₅×Q₈	—	2	2	no	—
116	C₂×C₅×Q₈	C₂³×C₃	1	2	yes	—
117	C₂×C₅×D₄	—	2	2	no	—
118	C₂×C₅×D₄	—	2	2	no	—
119	C₂×C₅×D₄	C₂³×C₃	1	2	yes	—
120	C₂²×C₄×C₅	—	2	2	no	—
121	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
122	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
123	C₂×C₅×C₈	—	2	4	no	—
124	C₂×C₅×C₈	C₂³×C₃	1	4	yes	—
125	C₅×C₄⋊C₄	—	2	2	no	—
126	C₅×C₄⋊C₄	—	2	4	no	—
127	C₅×C₄⋊C₄	—	2	2	no	—
128	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
129	C₅×C₂²⋊C₄	—	2	2	no	—
130	C₅×C₂²⋊C₄	—	2	4	no	—
131	C₄²×C₅	—	2	2	no	—
132	C₄²×C₅	C₃×D₄	1	2	yes	—
133	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
134	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
135	#533	C₂×C₃×C₅	1	1	yes	characteristic, complemented
136	#532	C₂×C₃×C₅	1	1	yes	characteristic
137	#488	C₂×C₃×C₅	1	4	yes	complemented
138	#487	C₂×C₃×C₅	1	2	yes	complemented
139	#486	C₂×C₃×C₅	1	2	yes	complemented
140	#416	C₂×C₃×C₅	1	4	yes	complemented
141	C₂×C₄×C₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
142	C₂²×C₃×C₅	—	2	2	no	—
143	C₂²×C₃×C₅	—	4	2	no	—
144	C₂²×C₃×C₅	—	4	4	no	—
145	C₂²×C₃×C₅	#124	1	2	yes	—
146	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
147	C₃×C₄×C₅	—	2	2	no	—
148	C₃×C₄×C₅	—	4	2	no	—
149	C₃×C₄×C₅	—	4	4	no	—
150	C₃×C₄×C₅	#124	1	2	yes	—
151	C₃×C₄×C₅	#135	1	4	yes	—
152	C₃×C₄×C₅	C₂×#41	1	2	yes	—
153	C₃×#41	—	2	8	no	—
154	C₂×C₃×Q₈	—	2	2	no	—
155	C₂×C₃×Q₈	C₂³×C₅	1	2	yes	—
156	C₂×C₃×D₄	—	2	2	no	—
157	C₂×C₃×D₄	—	2	2	no	—
158	C₂×C₃×D₄	C₂³×C₅	1	2	yes	—
159	C₂²×C₃×C₄	—	2	2	no	—
160	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
161	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
162	C₂×C₃×C₈	—	2	4	no	—
163	C₂×C₃×C₈	C₂³×C₅	1	4	yes	—
164	C₃×C₄⋊C₄	—	2	2	no	—
165	C₃×C₄⋊C₄	—	2	4	no	—
166	C₃×C₄⋊C₄	—	2	2	no	—
167	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
168	C₃×C₂²⋊C₄	—	2	2	no	—
169	C₃×C₂²⋊C₄	—	2	4	no	—
170	C₃×C₄²	—	2	2	no	—
171	C₃×C₄²	C₅×D₄	1	2	yes	—
172	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
173	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
174	C₂³×C₅	—	2	2	no	—
175	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
176	C₅×Q₈	—	2	4	no	—
177	C₅×Q₈	—	4	4	no	—
178	C₅×D₄	—	2	4	no	—
179	C₅×D₄	—	4	4	no	—
180	C₅×D₄	—	4	4	no	—
181	C₂×C₄×C₅	—	2	4	no	—
182	C₂×C₄×C₅	—	2	4	no	—
183	C₂×C₄×C₅	—	2	2	no	—
184	C₂×C₄×C₅	—	4	4	no	—
185	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
186	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
187	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
188	C₂×C₄×C₅	C₃×#41	1	4	yes	—
189	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
190	C₅×C₈	—	2	4	no	—
191	C₂×#41	C₂²×C₃×C₅	1	2	yes	—
192	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
193	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
194	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
195	#127	C₂²×C₃×C₅	1	1	yes	characteristic
196	#125	C₂²×C₃×C₅	1	2	yes	—
197	#124	C₂²×C₃×C₅	1	2	yes	—
198	C₂²⋊Q₈	—	2	2	no	—
199	#121	—	2	2	no	—
200	C₄×Q₈	—	2	2	no	—
201	C₄×D₄	—	2	2	no	—
202	#117	C₂²×C₃×C₅	1	2	yes	—
203	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
204	#103	C₂²×C₃×C₅	1	8	yes	—
205	#102	C₂²×C₃×C₅	1	8	yes	—
206	C₄×C₈	C₂²×C₃×C₅	1	2	yes	—
207	C₄×C₈	C₂²×C₃×C₅	1	2	yes	—
208	C₂×C₃×C₅	—	2	2	no	—
209	C₂×C₃×C₅	—	8	2	no	—
210	C₂×C₃×C₅	C₂×#124	1	1	yes	characteristic
211	C₂×C₃×C₅	C₂×#135	1	2	yes	—
212	C₂³×C₃	—	2	2	no	—
213	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
214	C₃×Q₈	—	2	4	no	—
215	C₃×Q₈	—	4	4	no	—
216	C₃×D₄	—	2	4	no	—
217	C₃×D₄	—	4	4	no	—
218	C₃×D₄	—	4	4	no	—
219	C₂×C₃×C₄	—	2	4	no	—
220	C₂×C₃×C₄	—	2	4	no	—
221	C₂×C₃×C₄	—	2	2	no	—
222	C₂×C₃×C₄	—	4	4	no	—
223	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
224	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
225	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
226	C₂×C₃×C₄	C₅×#41	1	4	yes	—
227	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
228	C₃×C₈	—	2	4	no	—
229	C₂²×C₅	—	2	2	no	—
230	C₂²×C₅	—	4	4	no	—
231	C₂²×C₅	—	4	2	no	—
232	C₂²×C₅	C₃×#124	1	2	yes	—
233	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
234	C₄×C₅	—	2	2	no	—
235	C₄×C₅	—	4	4	no	—
236	C₄×C₅	—	4	2	no	—
237	C₄×C₅	C₃×#124	1	2	yes	—
238	C₄×C₅	C₃×#135	1	4	yes	—
239	C₄×C₅	C₂×C₃×#41	1	2	yes	—
240	#41	—	2	8	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	C₂³×C₃×C₅	1	2	yes	—
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	C₂³×C₃×C₅	1	2	yes	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
248	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	C₂³×C₃×C₅	1	4	yes	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	C₂³×C₃×C₅	1	4	yes	—
255	C₂²⋊C₄	—	2	2	no	—
256	C₂²⋊C₄	—	2	4	no	—
257	C₄²	—	2	2	no	—
258	C₄²	C₃×C₅×D₄	1	2	yes	—
259	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
260	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
261	C₃×C₅	#3131	1	1	yes	characteristic, direct product factor, Hall
262	C₂²×C₃	—	2	2	no	—
263	C₂²×C₃	—	4	2	no	—
264	C₂²×C₃	—	4	4	no	—
265	C₂²×C₃	C₅×#124	1	2	yes	—
266	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
267	C₃×C₄	—	2	2	no	—
268	C₃×C₄	—	4	2	no	—
269	C₃×C₄	—	4	4	no	—
270	C₃×C₄	C₅×#124	1	2	yes	—
271	C₃×C₄	C₅×#135	1	4	yes	—
272	C₃×C₄	C₂×C₅×#41	1	2	yes	—
273	C₂×C₅	—	2	2	no	—
274	C₂×C₅	—	8	2	no	—
275	C₂×C₅	C₂×C₃×#124	1	1	yes	characteristic
276	C₂×C₅	C₂×C₃×#135	1	2	yes	—
277	C₂³	—	2	2	no	—
278	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
279	Q₈	—	2	4	no	—
280	Q₈	—	4	4	no	—
281	D₄	—	2	4	no	—
282	D₄	—	4	4	no	—
283	D₄	—	4	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
289	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
290	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
291	C₂×C₄	C₃×C₅×#41	1	4	yes	—
292	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
293	C₈	—	2	4	no	—
294	C₂×C₃	—	2	2	no	—
295	C₂×C₃	—	8	2	no	—
296	C₂×C₃	C₂×C₅×#124	1	1	yes	characteristic
297	C₂×C₃	C₂×C₅×#135	1	2	yes	—
298	C₅	C₃×#3131	1	1	yes	characteristic, direct product factor, Sylow
299	C₂²	—	2	2	no	—
300	C₂²	—	4	2	no	—
301	C₂²	—	4	4	no	—
302	C₂²	C₃×C₅×#124	1	2	yes	—
303	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	4	2	no	—
306	C₄	—	4	4	no	—
307	C₄	C₃×C₅×#124	1	2	yes	—
308	C₄	C₃×C₅×#135	1	4	yes	—
309	C₄	C₂×C₃×C₅×#41	1	2	yes	—
310	C₃	C₅×#3131	1	1	yes	characteristic, direct product factor, Sylow
311	C₂	—	2	2	no	—
312	C₂	—	8	2	no	—
313	C₂	C₂×C₃×C₅×#124	1	1	yes	characteristic
314	C₂	C₂×C₃×C₅×#135	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₃×C₅×#532	C₂
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
1. C₃×C₅×#533	C₂	semidirect
3. C₃×C₅×#488	C₂	semidirect
4. C₃×C₅×#487	C₂	semidirect
5. C₃×C₅×#486	C₂	semidirect
6. C₃×C₅×#416	C₂	semidirect
7. C₂×C₃×C₄×C₅×C₈	C₂	semidirect
8. C₅×#3131	C₃	direct (N × H)
26. C₃×#3131	C₅	direct (N × H)
27. C₅×#533	C₂×C₃	semidirect
29. C₅×#488	C₂×C₃	semidirect
30. C₅×#487	C₂×C₃	semidirect
31. C₅×#486	C₂×C₃	semidirect
32. C₅×#416	C₂×C₃	semidirect
33. C₂×C₄×C₅×C₈	C₂×C₃	semidirect
55. C₃×#533	C₂×C₅	semidirect
57. C₃×#488	C₂×C₅	semidirect
58. C₃×#487	C₂×C₅	semidirect
59. C₃×#486	C₂×C₅	semidirect
60. C₃×#416	C₂×C₅	semidirect
61. C₂×C₃×C₄×C₈	C₂×C₅	semidirect
79. #3131	C₃×C₅	direct (N × H)
135. #533	C₂×C₃×C₅	semidirect
137. #488	C₂×C₃×C₅	semidirect
138. #487	C₂×C₃×C₅	semidirect
139. #486	C₂×C₃×C₅	semidirect
140. #416	C₂×C₃×C₅	semidirect
141. C₂×C₄×C₈	C₂×C₃×C₅	semidirect
261. C₃×C₅	#3131	direct (N × H)
298. C₅	C₃×#3131	direct (N × H)
310. C₃	C₅×#3131	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×#532	2
C₃×C₅×#488	2
C₃×C₅×#487	2
C₃×C₅×#486	2
C₃×C₅×#416	2
C₂×C₃×C₄×C₅×C₈	2
C₅×#3131	3
C₃×#3131	5

Subgroups — C3×C5×#3131 #14276165

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3131 #14276165