Subgroups — C₃×#10615065 #14437792

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×#65811	C₂	1	1	yes	maximal, characteristic
3	C₃×#65382	C₂	1	2	yes	maximal, complemented
4	C₃×#65377	C₂	1	2	yes	maximal, complemented
5	C₃×#65357	C₂	1	4	yes	maximal, complemented
6	C₃×#65291	C₂	1	4	yes	maximal, complemented
7	C₂×C₃×C₄×#168	C₂	1	1	yes	maximal, characteristic, complemented
8	#10615065	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×#709	C₂²	1	2	yes	—
10	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₃×C₅×#125	C₂²	1	2	yes	—
12	C₃×C₅×C₂²⋊Q₈	—	2	2	no	—
13	C₃×C₄×C₅×Q₈	—	2	2	no	—
14	C₃×C₅×#117	C₂²	1	2	yes	—
15	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
16	C₃×#4076	—	2	2	no	—
17	C₂²×C₃×#168	C₂²	1	2	yes	—
18	C₃×#4021	C₂²	1	2	yes	—
19	C₃×#4020	C₂²	1	1	yes	characteristic
20	C₃×C₄×D₂₀	—	2	2	no	—
21	C₃×#4015	C₂²	1	1	yes	characteristic
22	C₃×#3942	C₂²	1	8	yes	—
23	C₃×#3941	C₂²	1	8	yes	—
24	C₃×C₄×#168	C₂²	1	2	yes	—
25	C₃×C₄×#168	C₂²	1	2	yes	—
26	C₃×#3131	—	5	1	no	maximal, Hall
27	C₅×#533	C₂×C₃	1	1	yes	characteristic, complemented
28	#65811	C₂×C₃	1	1	yes	characteristic
29	#65382	C₂×C₃	1	2	yes	complemented
30	#65377	C₂×C₃	1	2	yes	complemented
31	#65357	C₂×C₃	1	4	yes	complemented
32	#65291	C₂×C₃	1	4	yes	complemented
33	C₂×C₄×#168	C₂×C₃	1	1	yes	characteristic, complemented
34	C₂×C₃×C₅×Q₈	—	2	2	no	—
35	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
36	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
37	C₂×C₃×C₂²⋊D₅	—	2	2	no	—
38	C₃×#709	—	2	8	no	—
39	C₂×C₃×D₂₀	—	2	2	no	—
40	C₂×C₃×D₂₀	C₂³	1	2	yes	—
41	C₂×C₃×C₄×D₅	—	2	2	no	—
42	C₂×C₃×Dic₁₀	C₂³	1	2	yes	—
43	C₃×C₅×C₄⋊C₄	—	2	4	no	—
44	C₃×C₅×C₄⋊C₄	—	2	2	no	—
45	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
46	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
47	C₃×C₄²×C₅	—	2	2	no	—
48	C₃×C₄²×C₅	D₄	1	2	yes	—
49	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
50	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
51	C₃×#685	—	2	4	no	—
52	C₃×C₄⋊Dic₅	—	2	2	no	—
53	C₂×C₃×#168	—	2	4	no	—
54	C₂×C₃×#168	C₂³	1	4	yes	—
55	C₃×#533	D₅	1	1	yes	characteristic, complemented
56	C₃×#532	—	5	1	no	—
57	C₃×#488	—	5	4	no	—
58	C₃×#487	—	5	2	no	—
59	C₃×#486	—	5	2	no	—
60	C₃×#416	—	5	4	no	—
61	C₂×C₃×C₄×C₈	—	5	1	no	—
62	C₂×#709	C₂²×C₃	1	2	yes	—
63	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
64	C₅×#125	C₂²×C₃	1	2	yes	—
65	C₅×C₂²⋊Q₈	—	2	2	no	—
66	C₄×C₅×Q₈	—	2	2	no	—
67	C₅×#117	C₂²×C₃	1	2	yes	—
68	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
69	#4076	—	2	2	no	—
70	C₂²×#168	C₂²×C₃	1	2	yes	—
71	#4021	C₂²×C₃	1	2	yes	—
72	#4020	C₂²×C₃	1	1	yes	characteristic
73	C₄×D₂₀	—	2	2	no	—
74	#4015	C₂²×C₃	1	1	yes	characteristic
75	#3942	C₂²×C₃	1	8	yes	—
76	#3941	C₂²×C₃	1	8	yes	—
77	C₄×#168	C₂²×C₃	1	2	yes	—
78	C₄×#168	C₂²×C₃	1	2	yes	—
79	#3131	—	5	1	no	Sylow
80	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₃×D₅	—	2	2	no	—
82	C₃×C₅×Q₈	—	4	4	no	—
83	C₂×C₃×C₄×C₅	—	2	4	no	—
84	C₂×C₃×C₄×C₅	—	2	4	no	—
85	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
86	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
88	C₂×C₃×C₄×C₅	#41	1	4	yes	—
89	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
90	C₃×C₂²⋊D₅	—	4	4	no	—
91	C₂×C₃×Dic₅	—	2	2	no	—
92	C₃×D₂₀	—	2	4	no	—
93	C₃×D₂₀	—	4	4	no	—
94	C₃×C₄×D₅	—	4	4	no	—
95	C₃×Dic₁₀	—	2	4	no	—
96	C₃×#168	—	2	4	no	—
97	C₂×C₃×#41	—	5	2	no	—
98	C₂²×C₃×C₈	—	5	2	no	—
99	C₃×C₄⋊Q₈	—	5	1	no	—
100	C₃×C₄⋊Q₈	C₂×D₅	1	2	yes	—
101	C₃×#127	—	5	1	no	—
102	C₃×#125	C₂×D₅	1	2	yes	—
103	C₃×#124	—	5	2	no	—
104	C₃×C₂²⋊Q₈	—	2	2	no	—
105	C₃×#121	—	10	2	no	—
106	C₃×C₄×Q₈	—	2	2	no	—
107	C₃×C₄×D₄	—	10	2	no	—
108	C₃×#117	C₂×D₅	1	2	yes	—
109	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
110	C₃×#103	—	5	8	no	—
111	C₃×#102	—	5	8	no	—
112	C₃×C₄×C₈	—	5	2	no	—
113	C₃×C₄×C₈	—	5	2	no	—
114	C₂×C₅×Q₈	—	2	2	no	—
115	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
116	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
117	C₂×C₂²⋊D₅	—	2	2	no	—
118	#709	—	2	8	no	—
119	C₂×D₂₀	—	2	2	no	—
120	C₂×D₂₀	C₂³×C₃	1	2	yes	—
121	C₂×C₄×D₅	—	2	2	no	—
122	C₂×Dic₁₀	C₂³×C₃	1	2	yes	—
123	C₅×C₄⋊C₄	—	2	4	no	—
124	C₅×C₄⋊C₄	—	2	2	no	—
125	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
126	C₅×C₂²⋊C₄	—	2	2	no	—
127	C₄²×C₅	—	2	2	no	—
128	C₄²×C₅	C₃×D₄	1	2	yes	—
129	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
130	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
131	#685	—	2	4	no	—
132	C₄⋊Dic₅	—	2	2	no	—
133	C₂×#168	—	2	4	no	—
134	C₂×#168	C₂³×C₃	1	4	yes	—
135	#533	C₃×D₅	1	1	yes	characteristic, complemented
136	#532	—	5	1	no	—
137	#488	—	5	4	no	—
138	#487	—	5	2	no	—
139	#486	—	5	2	no	—
140	#416	—	5	4	no	—
141	C₂×C₄×C₈	—	5	1	no	—
142	C₂²×C₃×C₅	—	2	2	no	—
143	C₂²×C₃×C₅	#124	1	2	yes	—
144	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₃×D₅	—	4	2	no	—
146	C₂×C₃×D₅	—	4	4	no	—
147	C₃×C₄×C₅	—	2	2	no	—
148	C₃×C₄×C₅	—	4	4	no	—
149	C₃×C₄×C₅	#124	1	2	yes	—
150	C₃×C₄×C₅	#135	1	4	yes	—
151	C₃×C₄×C₅	C₂×#41	1	2	yes	—
152	C₃×Dic₅	—	4	2	no	—
153	C₃×#41	—	10	8	no	—
154	C₂×C₃×Q₈	—	2	2	no	—
155	C₂×C₃×Q₈	—	5	2	no	—
156	C₂×C₃×D₄	—	5	2	no	—
157	C₂×C₃×D₄	—	10	2	no	—
158	C₂×C₃×D₄	—	10	2	no	—
159	C₂²×C₃×C₄	—	10	2	no	—
160	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
161	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
162	C₂×C₃×C₈	—	5	4	no	—
163	C₂×C₃×C₈	—	10	4	no	—
164	C₃×C₄⋊C₄	—	2	4	no	—
165	C₃×C₄⋊C₄	—	2	2	no	—
166	C₃×C₄⋊C₄	—	10	2	no	—
167	C₃×C₄⋊C₄	C₂²×D₅	1	4	yes	—
168	C₃×C₂²⋊C₄	—	2	2	no	—
169	C₃×C₂²⋊C₄	—	10	4	no	—
170	C₃×C₄²	—	2	2	no	—
171	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
172	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
173	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
174	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
175	C₂²×D₅	—	2	2	no	—
176	C₅×Q₈	—	4	4	no	—
177	C₂×C₄×C₅	—	2	4	no	—
178	C₂×C₄×C₅	—	2	4	no	—
179	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
180	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
182	C₂×C₄×C₅	C₃×#41	1	4	yes	—
183	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
184	C₂²⋊D₅	—	4	4	no	—
185	C₂×Dic₅	—	2	2	no	—
186	D₂₀	—	2	4	no	—
187	D₂₀	—	4	4	no	—
188	C₄×D₅	—	4	4	no	—
189	Dic₁₀	—	2	4	no	—
190	#168	—	2	4	no	—
191	C₂×#41	—	5	2	no	—
192	C₂²×C₈	—	5	2	no	—
193	C₄⋊Q₈	—	5	1	no	—
194	C₄⋊Q₈	C₂×C₃×D₅	1	2	yes	—
195	#127	—	5	1	no	—
196	#125	C₂×C₃×D₅	1	2	yes	—
197	#124	—	5	2	no	—
198	C₂²⋊Q₈	—	2	2	no	—
199	#121	—	10	2	no	—
200	C₄×Q₈	—	2	2	no	—
201	C₄×D₄	—	10	2	no	—
202	#117	C₂×C₃×D₅	1	2	yes	—
203	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
204	#103	—	5	8	no	—
205	#102	—	5	8	no	—
206	C₄×C₈	—	5	2	no	—
207	C₄×C₈	—	5	2	no	—
208	C₂×C₃×C₅	—	2	2	no	—
209	C₂×C₃×C₅	C₂×#124	1	1	yes	characteristic
210	C₂×C₃×C₅	C₂×#135	1	2	yes	—
211	C₃×D₅	—	8	2	no	—
212	C₂³×C₃	—	10	2	no	—
213	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
214	C₃×Q₈	—	4	4	no	—
215	C₃×Q₈	—	10	4	no	—
216	C₃×D₄	—	10	4	no	—
217	C₃×D₄	—	20	4	no	—
218	C₃×D₄	—	20	4	no	—
219	C₂×C₃×C₄	—	2	4	no	—
220	C₂×C₃×C₄	—	2	4	no	—
221	C₂×C₃×C₄	—	10	2	no	—
222	C₂×C₃×C₄	—	20	4	no	—
223	C₂×C₃×C₄	#713	1	4	yes	—
224	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
225	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
226	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
227	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
228	C₃×C₈	—	10	4	no	—
229	C₂²×C₅	—	2	2	no	—
230	C₂²×C₅	C₃×#124	1	2	yes	—
231	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
232	C₂×D₅	—	4	4	no	—
233	C₂×D₅	—	4	2	no	—
234	C₄×C₅	—	2	2	no	—
235	C₄×C₅	—	4	4	no	—
236	C₄×C₅	C₃×#124	1	2	yes	—
237	C₄×C₅	C₃×#135	1	4	yes	—
238	C₄×C₅	C₂×C₃×#41	1	2	yes	—
239	Dic₅	—	4	2	no	—
240	#41	—	10	8	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	5	2	no	—
243	C₂×D₄	—	5	2	no	—
244	C₂×D₄	—	10	2	no	—
245	C₂×D₄	—	10	2	no	—
246	C₂²×C₄	—	10	2	no	—
247	C₂²×C₄	C₃×C₂²⋊D₅	1	2	yes	—
248	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
249	C₂×C₈	—	5	4	no	—
250	C₂×C₈	—	10	4	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	10	2	no	—
254	C₄⋊C₄	C₂²×C₃×D₅	1	4	yes	—
255	C₂²⋊C₄	—	2	2	no	—
256	C₂²⋊C₄	—	10	4	no	—
257	C₄²	—	2	2	no	—
258	C₄²	C₃×C₂²⋊D₅	1	2	yes	—
259	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
260	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
261	C₃×C₅	#3131	1	1	yes	characteristic, complemented, Hall
262	C₂²×C₃	—	2	2	no	—
263	C₂²×C₃	—	20	2	no	—
264	C₂²×C₃	—	20	4	no	—
265	C₂²×C₃	#4093	1	2	yes	—
266	C₂²×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
267	C₃×C₄	—	2	2	no	—
268	C₃×C₄	—	4	4	no	—
269	C₃×C₄	—	20	2	no	—
270	C₃×C₄	#4093	1	2	yes	—
271	C₃×C₄	#4096	1	4	yes	—
272	C₃×C₄	C₂×#713	1	2	yes	—
273	C₂×C₅	—	2	2	no	—
274	C₂×C₅	C₂×C₃×#124	1	1	yes	characteristic
275	C₂×C₅	C₂×C₃×#135	1	2	yes	—
276	D₅	—	8	2	no	—
277	C₂³	—	10	2	no	—
278	C₂³	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
279	Q₈	—	4	4	no	—
280	Q₈	—	10	4	no	—
281	D₄	—	10	4	no	—
282	D₄	—	20	4	no	—
283	D₄	—	20	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	10	2	no	—
287	C₂×C₄	—	20	4	no	—
288	C₂×C₄	C₃×#713	1	4	yes	—
289	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	2	yes	—
290	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
291	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	2	yes	—
292	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
293	C₈	—	10	4	no	—
294	C₂×C₃	—	2	2	no	—
295	C₂×C₃	—	40	2	no	—
296	C₂×C₃	C₂×#4093	1	1	yes	characteristic
297	C₂×C₃	C₂×#4096	1	2	yes	—
298	C₅	C₃×#3131	1	1	yes	characteristic, complemented, Sylow
299	C₂²	—	2	2	no	—
300	C₂²	—	20	2	no	—
301	C₂²	—	20	4	no	—
302	C₂²	C₃×#4093	1	2	yes	—
303	C₂²	C₂²×C₃×C₂²⋊D₅	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	4	4	no	—
306	C₄	—	20	2	no	—
307	C₄	C₃×#4093	1	2	yes	—
308	C₄	C₃×#4096	1	4	yes	—
309	C₄	C₂×C₃×#713	1	2	yes	—
310	C₃	#10615065	1	1	yes	characteristic, direct product factor, Sylow
311	C₂	—	2	2	no	—
312	C₂	—	40	2	no	—
313	C₂	C₂×C₃×#4093	1	1	yes	characteristic
314	C₂	C₂×C₃×#4096	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂×C₂²⋊D₅
commutator, cosocle	C₂×C₄×C₅	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₃×C₅×#533	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₃×#65811	C₂
hypercenter	C₃×#533	D₅
hypocenter	C₅	C₃×#3131
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₃×#533

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#533	C₂	semidirect
3. C₃×#65382	C₂	semidirect
4. C₃×#65377	C₂	semidirect
5. C₃×#65357	C₂	semidirect
6. C₃×#65291	C₂	semidirect
7. C₂×C₃×C₄×#168	C₂	semidirect
8. #10615065	C₃	direct (N × H)
27. C₅×#533	C₂×C₃	semidirect
29. #65382	C₂×C₃	semidirect
30. #65377	C₂×C₃	semidirect
31. #65357	C₂×C₃	semidirect
32. #65291	C₂×C₃	semidirect
33. C₂×C₄×#168	C₂×C₃	semidirect
55. C₃×#533	D₅	semidirect
135. #533	C₃×D₅	semidirect
261. C₃×C₅	#3131	semidirect
298. C₅	C₃×#3131	semidirect
310. C₃	#10615065	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×#65811	2
C₃×#65382	2
C₃×#65377	2
C₃×#65357	2
C₃×#65291	2
C₂×C₃×C₄×#168	2
#10615065	3
C₃×#3131	5

Subgroups — C3×#10615065 #14437792

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10615065 #14437792