Subgroups — C₃×#10602517 #14301577

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	C₂×C₃×C₅×#124	C₂	1	1	yes	maximal, characteristic
2	C₃×#66146	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3968	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65504	C₂	1	2	yes	maximal
5	C₃×#64743	C₂	1	2	yes	maximal, complemented
6	#10602517	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
9	C₃×C₅×#124	—	2	2	no	—
10	C₃×C₅×#124	C₄	1	4	yes	complemented
11	C₂×C₃×C₅×C₂²⋊C₄	—	2	2	no	—
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₃×Q₈×Dic₅	—	2	4	no	—
14	C₃×#4072	—	2	2	no	—
15	C₃×#4072	C₂²	1	2	yes	—
16	C₂×C₃×#681	C₂²	1	2	yes	—
17	C₃×#3968	—	2	2	no	—
18	C₃×#3964	—	2	4	no	—
19	C₃×#1888	—	5	1	no	maximal, Hall
20	C₂×C₅×#124	C₂×C₃	1	1	yes	characteristic
21	#66146	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂×#3968	C₂×C₃	1	1	yes	characteristic, complemented
23	#65504	C₂×C₃	1	2	yes	—
24	#64743	C₂×C₃	1	2	yes	complemented
25	C₂⁴×C₃×C₅	—	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	C₂×C₃×C₅×D₄	—	2	2	no	—
31	C₂×C₃×C₅×D₄	—	2	1	no	—
32	C₂×C₃×C₅×D₄	—	2	2	no	—
33	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
34	C₂²×C₃×C₄×C₅	—	2	1	no	—
35	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
36	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
37	C₃×C₅×C₂²⋊C₄	—	2	8	no	—
38	C₃×C₄²×C₅	—	2	1	no	—
39	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
40	C₃×C₂²⋊Dic₅	—	2	2	no	—
41	C₃×C₄⋊Dic₅	—	2	4	no	—
42	C₃×C₄⋊Dic₅	D₄	1	4	yes	—
43	C₃×C₄×Dic₅	—	2	4	no	—
44	C₃×C₄×Dic₅	—	2	2	no	—
45	C₃×#681	—	2	2	no	—
46	C₃×#681	—	2	2	no	—
47	C₂×C₃×#168	—	2	2	no	—
48	C₂×C₃×#124	D₅	1	1	yes	characteristic
49	C₃×#519	—	5	1	no	—
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×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
55	C₅×#124	—	2	2	no	—
56	C₅×#124	C₃×C₄	1	4	yes	complemented
57	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
58	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
59	Q₈×Dic₅	—	2	4	no	—
60	#4072	—	2	2	no	—
61	#4072	C₂²×C₃	1	2	yes	—
62	C₂×#681	C₂²×C₃	1	2	yes	—
63	#3968	—	2	2	no	—
64	#3964	—	2	4	no	—
65	#1888	—	5	1	no	Sylow
66	C₂³×C₃×C₅	—	2	2	no	—
67	C₂³×C₃×C₅	—	4	4	no	—
68	C₂³×C₃×C₅	—	4	1	no	—
69	C₂³×C₃×C₅	—	4	1	no	—
70	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
71	C₃×C₅×Q₈	—	2	4	no	—
72	C₃×C₅×Q₈	—	2	4	no	—
73	C₃×C₅×D₄	—	2	4	no	—
74	C₃×C₅×D₄	—	4	2	no	—
75	C₂×C₃×C₄×C₅	—	2	2	no	—
76	C₂×C₃×C₄×C₅	—	2	2	no	—
77	C₂×C₃×C₄×C₅	—	2	2	no	—
78	C₂×C₃×C₄×C₅	—	4	2	no	—
79	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
80	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
82	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	#41	1	2	yes	—
84	C₂×C₃×Dic₅	—	2	4	no	—
85	C₃×#168	—	4	2	no	—
86	C₂²×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
87	C₂²×C₃×D₄	C₂×D₅	1	1	yes	characteristic
88	C₂×C₃×M₁₆	—	5	2	no	—
89	C₃×#124	—	2	2	no	—
90	C₃×#124	Dic₅	1	4	yes	complemented
91	C₃×C₄×Q₈	—	10	4	no	—
92	C₃×#117	—	5	2	no	—
93	C₃×#117	—	10	2	no	—
94	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
95	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
96	C₃×#102	—	10	4	no	—
97	C₃×#101	—	10	2	no	—
98	C₂⁴×C₅	—	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	C₂×C₅×D₄	—	2	2	no	—
104	C₂×C₅×D₄	—	2	1	no	—
105	C₂×C₅×D₄	—	2	2	no	—
106	C₂×C₅×D₄	C₂×C₃×C₄	1	2	yes	—
107	C₂²×C₄×C₅	—	2	1	no	—
108	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
109	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
110	C₅×C₂²⋊C₄	—	2	8	no	—
111	C₄²×C₅	—	2	1	no	—
112	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
113	C₂²⋊Dic₅	—	2	2	no	—
114	C₄⋊Dic₅	—	2	4	no	—
115	C₄⋊Dic₅	C₃×D₄	1	4	yes	—
116	C₄×Dic₅	—	2	4	no	—
117	C₄×Dic₅	—	2	2	no	—
118	#681	—	2	2	no	—
119	#681	—	2	2	no	—
120	C₂×#168	—	2	2	no	—
121	C₂×#124	C₃×D₅	1	1	yes	characteristic
122	#519	—	5	1	no	—
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	—	4	4	no	—
128	C₂²×C₃×C₅	—	4	2	no	—
129	C₂²×C₃×C₅	—	8	2	no	—
130	C₂²×C₃×C₅	—	8	2	no	—
131	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂²×C₃×C₅	#120	1	1	yes	characteristic
133	C₂²×C₃×C₅	#123	1	1	yes	characteristic
134	C₃×C₄×C₅	—	2	4	no	—
135	C₃×C₄×C₅	—	4	2	no	—
136	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
137	C₃×C₄×C₅	#121	1	2	yes	—
138	C₃×Dic₅	—	4	4	no	—
139	C₂⁴×C₃	—	2	1	no	—
140	C₂×C₃×Q₈	—	2	2	no	—
141	C₂×C₃×Q₈	—	2	2	no	—
142	C₂×C₃×Q₈	C₂×Dic₅	1	2	yes	—
143	C₂×C₃×Q₈	C₂²⋊D₅	1	2	yes	—
144	C₂×C₃×D₄	—	2	2	no	—
145	C₂×C₃×D₄	—	2	2	no	—
146	C₂×C₃×D₄	—	2	1	no	—
147	C₂×C₃×D₄	C₂×Dic₅	1	2	yes	—
148	C₂²×C₃×C₄	—	2	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	—	5	4	no	—
155	C₃×C₄⋊C₄	—	10	4	no	—
156	C₃×C₂²⋊C₄	—	2	8	no	—
157	C₃×C₂²⋊C₄	—	10	2	no	—
158	C₃×C₄²	—	2	1	no	—
159	C₃×C₄²	—	10	4	no	—
160	C₃×C₄²	—	10	2	no	—
161	C₃×C₄²	C₂×Dic₅	1	2	yes	—
162	C₂³×C₅	—	2	2	no	—
163	C₂³×C₅	—	4	4	no	—
164	C₂³×C₅	—	4	1	no	—
165	C₂³×C₅	—	4	1	no	—
166	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
167	C₅×Q₈	—	2	4	no	—
168	C₅×Q₈	—	2	4	no	—
169	C₅×D₄	—	2	4	no	—
170	C₅×D₄	—	4	2	no	—
171	C₂×C₄×C₅	—	2	2	no	—
172	C₂×C₄×C₅	—	2	2	no	—
173	C₂×C₄×C₅	—	2	2	no	—
174	C₂×C₄×C₅	—	4	2	no	—
175	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	4	yes	—
176	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
178	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
179	C₂×C₄×C₅	C₃×#41	1	2	yes	—
180	C₂×Dic₅	—	2	4	no	—
181	#168	—	4	2	no	—
182	C₂²×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
183	C₂²×D₄	C₂×C₃×D₅	1	1	yes	characteristic
184	C₂×M₁₆	—	5	2	no	—
185	#124	—	2	2	no	—
186	#124	C₃×Dic₅	1	4	yes	complemented
187	C₄×Q₈	—	10	4	no	—
188	#117	—	5	2	no	—
189	#117	—	10	2	no	—
190	C₂×C₂²⋊C₄	—	2	2	no	—
191	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
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	—	8	2	no	—
197	C₂×C₃×C₅	#386	1	1	yes	characteristic
198	C₂×C₃×C₅	#454	1	2	yes	—
199	C₂³×C₃	—	2	2	no	—
200	C₂³×C₃	—	4	4	no	—
201	C₂³×C₃	—	4	1	no	—
202	C₂³×C₃	—	4	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	—	2	4	no	—
207	C₃×D₄	—	4	2	no	—
208	C₂×C₃×C₄	—	2	2	no	—
209	C₂×C₃×C₄	—	2	2	no	—
210	C₂×C₃×C₄	—	2	2	no	—
211	C₂×C₃×C₄	—	4	2	no	—
212	C₂×C₃×C₄	—	10	4	no	—
213	C₂×C₃×C₄	C₂²⋊Dic₅	1	4	yes	—
214	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
215	C₂×C₃×C₄	#711	1	2	yes	—
216	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
217	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
218	C₃×C₈	—	20	2	no	—
219	C₂²×C₅	—	2	2	no	—
220	C₂²×C₅	—	4	4	no	—
221	C₂²×C₅	—	4	2	no	—
222	C₂²×C₅	—	8	2	no	—
223	C₂²×C₅	—	8	2	no	—
224	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
225	C₂²×C₅	C₃×#120	1	1	yes	characteristic
226	C₂²×C₅	C₃×#123	1	1	yes	characteristic
227	C₄×C₅	—	2	4	no	—
228	C₄×C₅	—	4	2	no	—
229	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
230	C₄×C₅	C₃×#121	1	2	yes	—
231	Dic₅	—	4	4	no	—
232	C₂⁴	—	2	1	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	C₂×C₃×Dic₅	1	2	yes	—
236	C₂×Q₈	C₃×C₂²⋊D₅	1	2	yes	—
237	C₂×D₄	—	2	2	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	C₂×C₃×Dic₅	1	2	yes	—
241	C₂²×C₄	—	2	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	—	5	4	no	—
248	C₄⋊C₄	—	10	4	no	—
249	C₂²⋊C₄	—	2	8	no	—
250	C₂²⋊C₄	—	10	2	no	—
251	C₄²	—	2	1	no	—
252	C₄²	—	10	4	no	—
253	C₄²	—	10	2	no	—
254	C₄²	C₂×C₃×Dic₅	1	2	yes	—
255	C₃×C₅	#1888	1	1	yes	characteristic, complemented, Hall
256	C₂²×C₃	—	2	2	no	—
257	C₂²×C₃	—	4	4	no	—
258	C₂²×C₃	—	4	2	no	—
259	C₂²×C₃	—	8	2	no	—
260	C₂²×C₃	—	8	2	no	—
261	C₂²×C₃	#4081	1	1	yes	characteristic
262	C₂²×C₃	#4083	1	1	yes	characteristic
263	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
264	C₃×C₄	—	2	4	no	—
265	C₃×C₄	—	4	2	no	—
266	C₃×C₄	—	20	4	no	—
267	C₃×C₄	D₄×Dic₅	1	2	yes	—
268	C₃×C₄	#4085	1	2	yes	—
269	C₂×C₅	—	2	1	no	—
270	C₂×C₅	—	2	1	no	—
271	C₂×C₅	—	8	2	no	—
272	C₂×C₅	C₃×#386	1	1	yes	characteristic
273	C₂×C₅	C₃×#454	1	2	yes	—
274	C₂³	—	2	2	no	—
275	C₂³	—	4	4	no	—
276	C₂³	—	4	1	no	—
277	C₂³	—	4	1	no	—
278	C₂³	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
279	Q₈	—	2	4	no	—
280	Q₈	—	2	4	no	—
281	D₄	—	2	4	no	—
282	D₄	—	4	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	10	4	no	—
288	C₂×C₄	C₃×C₂²⋊Dic₅	1	4	yes	—
289	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
290	C₂×C₄	C₃×#711	1	2	yes	—
291	C₂×C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
292	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
293	C₈	—	20	2	no	—
294	C₂×C₃	—	2	1	no	—
295	C₂×C₃	—	2	1	no	—
296	C₂×C₃	—	8	2	no	—
297	C₂×C₃	#65351	1	2	yes	—
298	C₂×C₃	#65509	1	1	yes	characteristic
299	C₅	C₃×#1888	1	1	yes	characteristic, complemented, Sylow
300	C₂²	—	2	2	no	—
301	C₂²	—	4	4	no	—
302	C₂²	—	4	2	no	—
303	C₂²	—	8	2	no	—
304	C₂²	—	8	2	no	—
305	C₂²	C₃×#4081	1	1	yes	characteristic
306	C₂²	C₃×#4083	1	1	yes	characteristic
307	C₂²	C₂×C₃×C₂²⋊Dic₅	1	1	yes	characteristic
308	C₄	—	2	4	no	—
309	C₄	—	4	2	no	—
310	C₄	—	20	4	no	—
311	C₄	C₃×D₄×Dic₅	1	2	yes	—
312	C₄	C₃×#4085	1	2	yes	—
313	C₃	#10602517	1	1	yes	characteristic, direct product factor, Sylow
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	—	8	2	no	—
317	C₂	C₃×#65351	1	2	yes	—
318	C₂	C₃×#65509	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	#4083
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₂×C₃×C₅×#124	C₂
socle	C₂²×C₃×C₅	#120
prime radical	C₂²×C₃×C₅×D₄	C₂²
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₂×C₃×#124	D₅
hypocenter	C₅	C₃×#1888
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#124

Complements

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

N	H ≅ G/N	type
2. C₃×#66146	C₂	semidirect
3. C₂×C₃×#3968	C₂	semidirect
5. C₃×#64743	C₂	semidirect
6. #10602517	C₃	direct (N × H)
10. C₃×C₅×#124	C₄	semidirect
21. #66146	C₂×C₃	semidirect
22. C₂×#3968	C₂×C₃	semidirect
24. #64743	C₂×C₃	semidirect
56. C₅×#124	C₃×C₄	semidirect
90. C₃×#124	Dic₅	semidirect
186. #124	C₃×Dic₅	semidirect
255. C₃×C₅	#1888	semidirect
299. C₅	C₃×#1888	semidirect
313. C₃	#10602517	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×C₅×#124	2
C₃×#66146	2
C₂×C₃×#3968	2
C₃×#65504	2
C₃×#64743	2
#10602517	3
C₃×#1888	5

Subgroups — C3×#10602517 #14301577

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10602517 #14301577