Subgroups — C₃×#10602454 #14301574

Order 1920.

688 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#519	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#4093	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#3968	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65290	C₂	1	2	yes	maximal
5	C₃×#64752	C₂	1	2	yes	maximal, complemented
6	#10602454	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
8	C₂²×C₃×D₂₀	C₂²	1	1	yes	characteristic
9	C₃×C₄×C₅×Q₈	—	2	4	no	—
10	C₃×C₅×#117	—	2	2	no	—
11	C₃×C₅×#117	C₂²	1	2	yes	—
12	C₃×#4093	—	2	2	no	—
13	C₃×#4093	C₄	1	4	yes	complemented
14	C₂×C₃×#685	—	2	2	no	—
15	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
16	C₂×C₃×#681	C₂²	1	2	yes	—
17	C₃×#3968	—	2	2	no	—
18	C₃×#3941	—	2	4	no	—
19	C₃×#1888	—	5	1	no	maximal, Hall
20	C₅×#519	C₂×C₃	1	1	yes	characteristic, complemented
21	C₂×#4093	C₂×C₃	1	1	yes	characteristic
22	C₂×#3968	C₂×C₃	1	1	yes	characteristic, complemented
23	#65290	C₂×C₃	1	2	yes	—
24	#64752	C₂×C₃	1	2	yes	complemented
25	C₂³×C₃×D₅	—	2	1	no	—
26	C₂×C₃×C₅×Q₈	—	2	2	no	—
27	C₂×C₃×C₅×Q₈	—	2	2	no	—
28	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₅×Q₈	D₄	1	2	yes	—
30	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
31	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂²×C₃×Dic₅	—	2	1	no	—
33	C₂×C₃×D₂₀	—	2	2	no	—
34	C₂×C₃×D₂₀	—	2	2	no	—
35	C₂×C₃×D₂₀	—	2	1	no	—
36	C₂×C₃×D₂₀	C₂×C₄	1	2	yes	—
37	C₃×C₅×C₄⋊C₄	—	2	4	no	—
38	C₃×C₅×C₄⋊C₄	D₄	1	4	yes	—
39	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
40	C₃×C₄²×C₅	—	2	4	no	—
41	C₃×C₄²×C₅	—	2	2	no	—
42	C₃×#685	—	2	8	no	—
43	C₃×C₄×Dic₅	—	2	1	no	—
44	C₃×C₄×Dic₅	C₂×C₄	1	2	yes	—
45	C₃×#681	—	2	2	no	—
46	C₃×#681	—	2	2	no	—
47	C₂×C₃×#168	—	2	2	no	—
48	C₂×C₃×#124	—	5	1	no	—
49	C₃×#519	D₅	1	1	yes	characteristic, complemented
50	C₃×#418	—	5	2	no	—
51	C₂×C₃×#101	—	5	1	no	—
52	C₃×#339	—	5	2	no	—
53	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
54	C₂²×D₂₀	C₂²×C₃	1	1	yes	characteristic
55	C₄×C₅×Q₈	—	2	4	no	—
56	C₅×#117	—	2	2	no	—
57	C₅×#117	C₂²×C₃	1	2	yes	—
58	#4093	—	2	2	no	—
59	#4093	C₃×C₄	1	4	yes	complemented
60	C₂×#685	—	2	2	no	—
61	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
62	C₂×#681	C₂²×C₃	1	2	yes	—
63	#3968	—	2	2	no	—
64	#3941	—	2	4	no	—
65	#1888	—	5	1	no	Sylow
66	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₃×D₅	—	2	2	no	—
68	C₂²×C₃×D₅	—	4	4	no	—
69	C₂²×C₃×D₅	—	4	1	no	—
70	C₂²×C₃×D₅	—	4	1	no	—
71	C₃×C₅×Q₈	—	2	4	no	—
72	C₃×C₅×Q₈	—	2	4	no	—
73	C₂×C₃×C₄×C₅	—	2	2	no	—
74	C₂×C₃×C₄×C₅	—	2	2	no	—
75	C₂×C₃×C₄×C₅	—	2	4	no	—
76	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
77	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
79	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	#41	1	2	yes	—
81	C₂×C₃×Dic₅	—	2	2	no	—
82	C₂×C₃×Dic₅	—	4	2	no	—
83	C₃×D₂₀	—	2	4	no	—
84	C₃×D₂₀	—	4	2	no	—
85	C₃×#168	—	4	2	no	—
86	C₂²×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
87	C₂²×C₃×D₄	—	5	1	no	—
88	C₂×C₃×M₁₆	—	5	2	no	—
89	C₃×#124	—	5	4	no	—
90	C₃×#124	—	10	2	no	—
91	C₃×C₄×Q₈	—	2	4	no	—
92	C₃×#117	—	2	2	no	—
93	C₃×#117	C₂×D₅	1	2	yes	—
94	C₂×C₃×C₂²⋊C₄	—	10	2	no	—
95	C₂×C₃×C₄²	—	5	1	no	—
96	C₃×#102	—	10	4	no	—
97	C₃×#101	—	10	2	no	—
98	C₂³×D₅	—	2	1	no	—
99	C₂×C₅×Q₈	—	2	2	no	—
100	C₂×C₅×Q₈	—	2	2	no	—
101	C₂×C₅×Q₈	C₂×C₃×C₄	1	2	yes	—
102	C₂×C₅×Q₈	C₃×D₄	1	2	yes	—
103	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
104	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
105	C₂²×Dic₅	—	2	1	no	—
106	C₂×D₂₀	—	2	2	no	—
107	C₂×D₂₀	—	2	2	no	—
108	C₂×D₂₀	—	2	1	no	—
109	C₂×D₂₀	C₂×C₃×C₄	1	2	yes	—
110	C₅×C₄⋊C₄	—	2	4	no	—
111	C₅×C₄⋊C₄	C₃×D₄	1	4	yes	—
112	C₅×C₂²⋊C₄	—	2	2	no	—
113	C₄²×C₅	—	2	4	no	—
114	C₄²×C₅	—	2	2	no	—
115	#685	—	2	8	no	—
116	C₄×Dic₅	—	2	1	no	—
117	C₄×Dic₅	C₂×C₃×C₄	1	2	yes	—
118	#681	—	2	2	no	—
119	#681	—	2	2	no	—
120	C₂×#168	—	2	2	no	—
121	C₂×#124	—	5	1	no	—
122	#519	C₃×D₅	1	1	yes	characteristic, complemented
123	#418	—	5	2	no	—
124	C₂×#101	—	5	1	no	—
125	#339	—	5	2	no	—
126	C₂²×C₃×C₅	—	2	2	no	—
127	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂²×C₃×C₅	#120	1	1	yes	characteristic
129	C₂²×C₃×C₅	#123	1	1	yes	characteristic
130	C₂×C₃×D₅	—	4	4	no	—
131	C₂×C₃×D₅	—	4	2	no	—
132	C₂×C₃×D₅	—	8	2	no	—
133	C₂×C₃×D₅	—	8	2	no	—
134	C₃×C₄×C₅	—	2	4	no	—
135	C₃×C₄×C₅	—	4	4	no	—
136	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
137	C₃×C₄×C₅	#121	1	2	yes	—
138	C₃×Dic₅	—	4	2	no	—
139	C₂⁴×C₃	—	10	1	no	—
140	C₂×C₃×Q₈	—	2	2	no	—
141	C₂×C₃×Q₈	—	2	2	no	—
142	C₂×C₃×Q₈	C₄×D₅	1	2	yes	—
143	C₂×C₃×Q₈	D₂₀	1	2	yes	—
144	C₂×C₃×D₄	—	5	2	no	—
145	C₂×C₃×D₄	—	10	1	no	—
146	C₂×C₃×D₄	—	10	2	no	—
147	C₂×C₃×D₄	—	10	2	no	—
148	C₂²×C₃×C₄	—	10	1	no	—
149	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
150	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
151	C₃×M₁₆	—	10	2	no	—
152	C₃×M₁₆	—	10	2	no	—
153	C₂×C₃×C₈	—	10	2	no	—
154	C₃×C₄⋊C₄	—	2	4	no	—
155	C₃×C₄⋊C₄	C₂²⋊D₅	1	4	yes	—
156	C₃×C₂²⋊C₄	—	2	2	no	—
157	C₃×C₂²⋊C₄	—	10	8	no	—
158	C₃×C₄²	—	2	4	no	—
159	C₃×C₄²	—	2	2	no	—
160	C₃×C₄²	—	5	2	no	—
161	C₃×C₄²	—	10	1	no	—
162	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
163	C₂²×D₅	—	2	2	no	—
164	C₂²×D₅	—	4	4	no	—
165	C₂²×D₅	—	4	1	no	—
166	C₂²×D₅	—	4	1	no	—
167	C₅×Q₈	—	2	4	no	—
168	C₅×Q₈	—	2	4	no	—
169	C₂×C₄×C₅	—	2	2	no	—
170	C₂×C₄×C₅	—	2	2	no	—
171	C₂×C₄×C₅	—	2	4	no	—
172	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	4	yes	—
173	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
175	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
176	C₂×C₄×C₅	C₃×#41	1	2	yes	—
177	C₂×Dic₅	—	2	2	no	—
178	C₂×Dic₅	—	4	2	no	—
179	D₂₀	—	2	4	no	—
180	D₂₀	—	4	2	no	—
181	#168	—	4	2	no	—
182	C₂²×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
183	C₂²×D₄	—	5	1	no	—
184	C₂×M₁₆	—	5	2	no	—
185	#124	—	5	4	no	—
186	#124	—	10	2	no	—
187	C₄×Q₈	—	2	4	no	—
188	#117	—	2	2	no	—
189	#117	C₂×C₃×D₅	1	2	yes	—
190	C₂×C₂²⋊C₄	—	10	2	no	—
191	C₂×C₄²	—	5	1	no	—
192	#102	—	10	4	no	—
193	#101	—	10	2	no	—
194	C₂×C₃×C₅	—	2	1	no	—
195	C₂×C₃×C₅	—	2	1	no	—
196	C₂×C₃×C₅	#386	1	1	yes	characteristic
197	C₂×C₃×C₅	#454	1	2	yes	—
198	C₃×D₅	—	8	2	no	—
199	C₂³×C₃	—	10	2	no	—
200	C₂³×C₃	—	20	4	no	—
201	C₂³×C₃	—	20	1	no	—
202	C₂³×C₃	—	20	1	no	—
203	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
204	C₃×Q₈	—	2	4	no	—
205	C₃×Q₈	—	2	4	no	—
206	C₃×D₄	—	10	4	no	—
207	C₃×D₄	—	20	2	no	—
208	C₂×C₃×C₄	—	2	2	no	—
209	C₂×C₃×C₄	—	2	2	no	—
210	C₂×C₃×C₄	—	2	4	no	—
211	C₂×C₃×C₄	—	10	2	no	—
212	C₂×C₃×C₄	—	20	2	no	—
213	C₂×C₃×C₄	#685	1	4	yes	—
214	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
216	C₂×C₃×C₄	#709	1	2	yes	—
217	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
218	C₃×C₈	—	20	2	no	—
219	C₂²×C₅	—	2	2	no	—
220	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
221	C₂²×C₅	C₃×#120	1	1	yes	characteristic
222	C₂²×C₅	C₃×#123	1	1	yes	characteristic
223	C₂×D₅	—	4	4	no	—
224	C₂×D₅	—	4	2	no	—
225	C₂×D₅	—	8	2	no	—
226	C₂×D₅	—	8	2	no	—
227	C₄×C₅	—	2	4	no	—
228	C₄×C₅	—	4	4	no	—
229	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
230	C₄×C₅	C₃×#121	1	2	yes	—
231	Dic₅	—	4	2	no	—
232	C₂⁴	—	10	1	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	C₃×C₄×D₅	1	2	yes	—
236	C₂×Q₈	C₃×D₂₀	1	2	yes	—
237	C₂×D₄	—	5	2	no	—
238	C₂×D₄	—	10	1	no	—
239	C₂×D₄	—	10	2	no	—
240	C₂×D₄	—	10	2	no	—
241	C₂²×C₄	—	10	1	no	—
242	C₂²×C₄	C₃×C₂²⋊D₅	1	2	yes	—
243	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
244	M₁₆	—	10	2	no	—
245	M₁₆	—	10	2	no	—
246	C₂×C₈	—	10	2	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	C₃×C₂²⋊D₅	1	4	yes	—
249	C₂²⋊C₄	—	2	2	no	—
250	C₂²⋊C₄	—	10	8	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	5	2	no	—
254	C₄²	—	10	1	no	—
255	C₃×C₅	#1888	1	1	yes	characteristic, complemented, Hall
256	C₂²×C₃	—	2	2	no	—
257	C₂²×C₃	—	20	4	no	—
258	C₂²×C₃	—	20	2	no	—
259	C₂²×C₃	—	40	2	no	—
260	C₂²×C₃	—	40	2	no	—
261	C₂²×C₃	C₂×#685	1	1	yes	characteristic
262	C₂²×C₃	#4075	1	1	yes	characteristic
263	C₂²×C₃	#4099	1	1	yes	characteristic
264	C₃×C₄	—	2	4	no	—
265	C₃×C₄	—	4	4	no	—
266	C₃×C₄	—	20	2	no	—
267	C₃×C₄	C₄×C₂²⋊D₅	1	2	yes	—
268	C₃×C₄	#4076	1	2	yes	—
269	C₂×C₅	—	2	1	no	—
270	C₂×C₅	—	2	1	no	—
271	C₂×C₅	C₃×#386	1	1	yes	characteristic
272	C₂×C₅	C₃×#454	1	2	yes	—
273	D₅	—	8	2	no	—
274	C₂³	—	10	2	no	—
275	C₂³	—	20	4	no	—
276	C₂³	—	20	1	no	—
277	C₂³	—	20	1	no	—
278	C₂³	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
279	Q₈	—	2	4	no	—
280	Q₈	—	2	4	no	—
281	D₄	—	10	4	no	—
282	D₄	—	20	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	10	2	no	—
287	C₂×C₄	—	20	2	no	—
288	C₂×C₄	C₃×#685	1	4	yes	—
289	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
290	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
291	C₂×C₄	C₃×#709	1	2	yes	—
292	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	2	yes	—
293	C₈	—	20	2	no	—
294	C₂×C₃	—	2	1	no	—
295	C₂×C₃	—	2	1	no	—
296	C₂×C₃	—	40	2	no	—
297	C₂×C₃	#65502	1	1	yes	characteristic
298	C₂×C₃	#65534	1	2	yes	—
299	C₅	C₃×#1888	1	1	yes	characteristic, complemented, Sylow
300	C₂²	—	2	2	no	—
301	C₂²	—	20	4	no	—
302	C₂²	—	20	2	no	—
303	C₂²	—	40	2	no	—
304	C₂²	—	40	2	no	—
305	C₂²	C₂×C₃×#685	1	1	yes	characteristic
306	C₂²	C₃×#4075	1	1	yes	characteristic
307	C₂²	C₃×#4099	1	1	yes	characteristic
308	C₄	—	2	4	no	—
309	C₄	—	4	4	no	—
310	C₄	—	20	2	no	—
311	C₄	C₃×C₄×C₂²⋊D₅	1	2	yes	—
312	C₄	C₃×#4076	1	2	yes	—
313	C₃	#10602454	1	1	yes	characteristic, direct product factor, Sylow
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	—	40	2	no	—
317	C₂	C₃×#65502	1	1	yes	characteristic
318	C₂	C₃×#65534	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	#4099
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₃×C₅×#519	C₂
socle	C₂²×C₃×C₅	#120
prime radical	C₂²×C₃×D₂₀	C₂²
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₃×#519	D₅
hypocenter	C₅	C₃×#1888
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×#519

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#519	C₂	semidirect
3. C₂×C₃×#3968	C₂	semidirect
5. C₃×#64752	C₂	semidirect
6. #10602454	C₃	direct (N × H)
13. C₃×#4093	C₄	semidirect
20. C₅×#519	C₂×C₃	semidirect
22. C₂×#3968	C₂×C₃	semidirect
24. #64752	C₂×C₃	semidirect
49. C₃×#519	D₅	semidirect
59. #4093	C₃×C₄	semidirect
122. #519	C₃×D₅	semidirect
255. C₃×C₅	#1888	semidirect
299. C₅	C₃×#1888	semidirect
313. C₃	#10602454	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×#519	2
C₂×C₃×#4093	2
C₂×C₃×#3968	2
C₃×#65290	2
C₃×#64752	2
#10602454	3
C₃×#1888	5

Subgroups — C3×#10602454 #14301574

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10602454 #14301574