Subgroups — C₃×#10602437 #14298585

Order 1920.

524 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×C₄×C₅×#41	C₂	1	1	yes	maximal, characteristic
2	C₃×#65529	C₂	1	1	yes	maximal, characteristic
3	C₃×#65526	C₂	1	1	yes	maximal, characteristic
4	C₃×#64751	C₂	1	2	yes	maximal, complemented
5	C₃×#64743	C₂	1	2	yes	maximal, complemented
6	#10602437	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
8	C₃×C₄×C₅×Q₈	—	2	1	no	—
9	C₃×C₄×C₅×D₄	—	2	2	no	—
10	C₃×C₄×C₅×D₄	—	2	1	no	—
11	C₃×C₅×#117	C₄	1	2	yes	complemented
12	C₃×C₅×#117	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄²×C₅	C₄	1	2	yes	complemented
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₃×#4072	C₂²	1	2	yes	—
16	C₂×C₃×#681	C₂²	1	2	yes	—
17	C₃×#3968	—	2	1	no	—
18	C₃×#3966	—	2	2	no	—
19	C₃×#3965	—	2	1	no	—
20	C₃×#1827	—	5	1	no	maximal, Hall
21	C₄×C₅×#41	C₂×C₃	1	1	yes	characteristic
22	#65529	C₂×C₃	1	1	yes	characteristic
23	#65526	C₂×C₃	1	1	yes	characteristic
24	#64751	C₂×C₃	1	2	yes	complemented
25	#64743	C₂×C₃	1	2	yes	complemented
26	C₃×C₅×#41	—	2	2	no	—
27	C₃×C₅×#41	—	2	2	no	—
28	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
29	C₂×C₃×C₅×D₄	D₄	1	2	yes	—
30	C₂×C₃×C₅×D₄	D₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	—	2	2	no	—
32	C₂²×C₃×C₄×C₅	—	2	1	no	—
33	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
34	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₃×C₅×C₄⋊C₄	—	2	2	no	—
36	C₃×C₅×C₄⋊C₄	—	2	1	no	—
37	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
38	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
39	C₃×C₄²×C₅	—	2	2	no	—
40	C₃×C₄²×C₅	—	2	1	no	—
41	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
43	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
44	C₃×C₂²⋊Dic₅	—	2	2	no	—
45	C₃×C₄⋊Dic₅	—	2	2	no	—
46	C₃×C₄×Dic₅	—	2	2	no	—
47	C₃×#681	—	2	2	no	—
48	C₃×#681	—	2	2	no	—
49	C₂×C₃×#168	—	2	2	no	—
50	C₃×C₄×#41	D₅	1	1	yes	characteristic
51	C₃×#413	—	5	1	no	—
52	C₃×#410	—	5	1	no	—
53	C₃×#339	—	5	2	no	—
54	C₃×H(3,ℤ/4)	—	5	2	no	—
55	C₂×C₅×#41	C₂²×C₃	1	1	yes	characteristic
56	C₄×C₅×Q₈	—	2	1	no	—
57	C₄×C₅×D₄	—	2	2	no	—
58	C₄×C₅×D₄	—	2	1	no	—
59	C₅×#117	C₃×C₄	1	2	yes	complemented
60	C₅×#117	C₂²×C₃	1	1	yes	characteristic
61	C₂×C₄²×C₅	C₃×C₄	1	2	yes	complemented
62	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
63	#4072	C₂²×C₃	1	2	yes	—
64	C₂×#681	C₂²×C₃	1	2	yes	—
65	#3968	—	2	1	no	—
66	#3966	—	2	2	no	—
67	#3965	—	2	1	no	—
68	#1827	—	5	1	no	Sylow
69	C₂³×C₃×C₅	C₂²⋊C₄	1	2	yes	—
70	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
71	C₃×C₅×Q₈	—	2	2	no	—
72	C₃×C₅×D₄	—	2	4	no	—
73	C₃×C₅×D₄	—	2	2	no	—
74	C₂×C₃×C₄×C₅	—	2	1	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	—	2	1	no	—
79	C₂×C₃×C₄×C₅	—	4	2	no	—
80	C₂×C₃×C₄×C₅	—	4	4	no	—
81	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
82	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	#41	1	2	yes	—
85	C₂×C₃×C₄×C₅	#41	1	2	yes	—
86	C₂×C₃×Dic₅	—	4	2	no	—
87	C₃×#168	—	4	2	no	—
88	C₂×C₃×#41	C₂×D₅	1	1	yes	characteristic
89	C₂×C₃×M₁₆	—	5	2	no	—
90	C₃×C₄×Q₈	—	2	1	no	—
91	C₃×C₄×D₄	—	2	2	no	—
92	C₃×C₄×D₄	—	2	1	no	—
93	C₃×#117	—	5	2	no	—
94	C₃×#117	Dic₅	1	2	yes	complemented
95	C₃×#117	C₂×D₅	1	1	yes	characteristic
96	C₂×C₃×C₄²	Dic₅	1	2	yes	complemented
97	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
98	C₃×#101	—	10	1	no	—
99	C₃×#100	—	10	1	no	—
100	C₃×#99	—	10	2	no	—
101	C₅×#41	—	2	2	no	—
102	C₅×#41	—	2	2	no	—
103	C₂×C₅×Q₈	C₃×D₄	1	1	yes	characteristic
104	C₂×C₅×D₄	C₃×D₄	1	2	yes	—
105	C₂×C₅×D₄	C₃×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₅	—	2	2	no	—
107	C₂²×C₄×C₅	—	2	1	no	—
108	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
109	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
110	C₅×C₄⋊C₄	—	2	2	no	—
111	C₅×C₄⋊C₄	—	2	1	no	—
112	C₅×C₂²⋊C₄	—	2	2	no	—
113	C₅×C₂²⋊C₄	—	2	1	no	—
114	C₄²×C₅	—	2	2	no	—
115	C₄²×C₅	—	2	1	no	—
116	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
117	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
118	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
119	C₂²⋊Dic₅	—	2	2	no	—
120	C₄⋊Dic₅	—	2	2	no	—
121	C₄×Dic₅	—	2	2	no	—
122	#681	—	2	2	no	—
123	#681	—	2	2	no	—
124	C₂×#168	—	2	2	no	—
125	C₄×#41	C₃×D₅	1	1	yes	characteristic
126	#413	—	5	1	no	—
127	#410	—	5	1	no	—
128	#339	—	5	2	no	—
129	H(3,ℤ/4)	—	5	2	no	—
130	C₂²×C₃×C₅	—	2	2	no	—
131	C₂²×C₃×C₅	—	4	1	no	—
132	C₂²×C₃×C₅	—	4	2	no	—
133	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂²×C₃×C₅	#123	1	2	yes	—
135	C₃×C₄×C₅	—	2	2	no	—
136	C₃×C₄×C₅	—	2	2	no	—
137	C₃×C₄×C₅	—	4	2	no	—
138	C₃×C₄×C₅	—	4	1	no	—
139	C₃×C₄×C₅	#117	1	1	yes	characteristic
140	C₃×C₄×C₅	#117	1	1	yes	characteristic
141	C₃×C₄×C₅	#123	1	2	yes	—
142	C₃×Dic₅	—	8	2	no	—
143	C₃×#41	—	2	2	no	—
144	C₃×#41	—	2	2	no	—
145	C₂×C₃×Q₈	C₂²⋊D₅	1	1	yes	characteristic
146	C₂×C₃×D₄	C₂²⋊D₅	1	2	yes	—
147	C₂×C₃×D₄	C₂²⋊D₅	1	1	yes	characteristic
148	C₂²×C₃×C₄	—	2	2	no	—
149	C₂²×C₃×C₄	—	2	1	no	—
150	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
151	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
152	C₃×M₁₆	—	10	2	no	—
153	C₃×M₁₆	—	10	2	no	—
154	C₂×C₃×C₈	—	10	2	no	—
155	C₃×C₄⋊C₄	—	2	2	no	—
156	C₃×C₄⋊C₄	—	2	1	no	—
157	C₃×C₄⋊C₄	—	10	2	no	—
158	C₃×C₂²⋊C₄	—	2	2	no	—
159	C₃×C₂²⋊C₄	—	2	1	no	—
160	C₃×C₂²⋊C₄	—	10	2	no	—
161	C₃×C₄²	—	2	2	no	—
162	C₃×C₄²	—	2	1	no	—
163	C₃×C₄²	—	10	2	no	—
164	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
165	C₃×C₄²	C₂×Dic₅	1	2	yes	—
166	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
167	C₂³×C₅	C₃×C₂²⋊C₄	1	2	yes	—
168	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
169	C₅×Q₈	—	2	2	no	—
170	C₅×D₄	—	2	4	no	—
171	C₅×D₄	—	2	2	no	—
172	C₂×C₄×C₅	—	2	1	no	—
173	C₂×C₄×C₅	—	2	2	no	—
174	C₂×C₄×C₅	—	2	2	no	—
175	C₂×C₄×C₅	—	2	2	no	—
176	C₂×C₄×C₅	—	2	1	no	—
177	C₂×C₄×C₅	—	4	2	no	—
178	C₂×C₄×C₅	—	4	4	no	—
179	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
180	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
182	C₂×C₄×C₅	C₃×#41	1	2	yes	—
183	C₂×C₄×C₅	C₃×#41	1	2	yes	—
184	C₂×Dic₅	—	4	2	no	—
185	#168	—	4	2	no	—
186	C₂×#41	C₂×C₃×D₅	1	1	yes	characteristic
187	C₂×M₁₆	—	5	2	no	—
188	C₄×Q₈	—	2	1	no	—
189	C₄×D₄	—	2	2	no	—
190	C₄×D₄	—	2	1	no	—
191	#117	—	5	2	no	—
192	#117	C₃×Dic₅	1	2	yes	complemented
193	#117	C₂×C₃×D₅	1	1	yes	characteristic
194	C₂×C₄²	C₃×Dic₅	1	2	yes	complemented
195	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
196	#101	—	10	1	no	—
197	#100	—	10	1	no	—
198	#99	—	10	2	no	—
199	C₂×C₃×C₅	—	2	2	no	—
200	C₂×C₃×C₅	—	2	1	no	—
201	C₂×C₃×C₅	—	4	2	no	—
202	C₂×C₃×C₅	#381	1	1	yes	characteristic
203	C₂³×C₃	C₂²⋊Dic₅	1	2	yes	—
204	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
205	C₃×Q₈	—	2	2	no	—
206	C₃×D₄	—	2	4	no	—
207	C₃×D₄	—	2	2	no	—
208	C₂×C₃×C₄	—	2	1	no	—
209	C₂×C₃×C₄	—	2	2	no	—
210	C₂×C₃×C₄	—	2	2	no	—
211	C₂×C₃×C₄	—	2	2	no	—
212	C₂×C₃×C₄	—	2	1	no	—
213	C₂×C₃×C₄	—	4	2	no	—
214	C₂×C₃×C₄	—	4	4	no	—
215	C₂×C₃×C₄	—	20	2	no	—
216	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
217	C₂×C₃×C₄	#709	1	2	yes	—
218	C₂×C₃×C₄	#709	1	2	yes	—
219	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
221	C₃×C₈	—	20	2	no	—
222	C₂²×C₅	—	2	2	no	—
223	C₂²×C₅	—	4	1	no	—
224	C₂²×C₅	—	4	2	no	—
225	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
226	C₂²×C₅	C₃×#123	1	2	yes	—
227	C₄×C₅	—	2	2	no	—
228	C₄×C₅	—	2	2	no	—
229	C₄×C₅	—	4	2	no	—
230	C₄×C₅	—	4	1	no	—
231	C₄×C₅	C₃×#117	1	1	yes	characteristic
232	C₄×C₅	C₃×#117	1	1	yes	characteristic
233	C₄×C₅	C₃×#123	1	2	yes	—
234	Dic₅	—	8	2	no	—
235	#41	—	2	2	no	—
236	#41	—	2	2	no	—
237	C₂×Q₈	C₃×C₂²⋊D₅	1	1	yes	characteristic
238	C₂×D₄	C₃×C₂²⋊D₅	1	2	yes	—
239	C₂×D₄	C₃×C₂²⋊D₅	1	1	yes	characteristic
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	C₂×C₃×Dic₅	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	2	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	10	2	no	—
250	C₂²⋊C₄	—	2	2	no	—
251	C₂²⋊C₄	—	2	1	no	—
252	C₂²⋊C₄	—	10	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	10	2	no	—
256	C₄²	C₂×C₃×Dic₅	1	1	yes	characteristic
257	C₄²	C₂×C₃×Dic₅	1	2	yes	—
258	C₄²	C₂×C₃×Dic₅	1	1	yes	characteristic
259	C₃×C₅	#1827	1	1	yes	characteristic, complemented, Hall
260	C₂²×C₃	—	2	2	no	—
261	C₂²×C₃	—	4	1	no	—
262	C₂²×C₃	—	4	2	no	—
263	C₂²×C₃	#4075	1	2	yes	—
264	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
265	C₃×C₄	—	2	2	no	—
266	C₃×C₄	—	2	2	no	—
267	C₃×C₄	—	4	2	no	—
268	C₃×C₄	—	4	1	no	—
269	C₃×C₄	—	40	2	no	—
270	C₃×C₄	#4072	1	1	yes	characteristic
271	C₃×C₄	#4072	1	1	yes	characteristic
272	C₃×C₄	#4075	1	2	yes	—
273	C₂×C₅	—	2	2	no	—
274	C₂×C₅	—	2	1	no	—
275	C₂×C₅	—	4	2	no	—
276	C₂×C₅	C₃×#381	1	1	yes	characteristic
277	C₂³	C₃×C₂²⋊Dic₅	1	2	yes	—
278	C₂³	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
279	Q₈	—	2	2	no	—
280	D₄	—	2	4	no	—
281	D₄	—	2	2	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	20	2	no	—
290	C₂×C₄	C₃×C₂²⋊Dic₅	1	2	yes	—
291	C₂×C₄	C₃×#709	1	2	yes	—
292	C₂×C₄	C₃×#709	1	2	yes	—
293	C₂×C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
294	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
295	C₈	—	20	2	no	—
296	C₂×C₃	—	2	1	no	—
297	C₂×C₃	—	2	2	no	—
298	C₂×C₃	—	4	2	no	—
299	C₂×C₃	#65500	1	1	yes	characteristic
300	C₅	C₃×#1827	1	1	yes	characteristic, complemented, Sylow
301	C₂²	—	2	2	no	—
302	C₂²	—	4	1	no	—
303	C₂²	—	4	2	no	—
304	C₂²	C₃×#4075	1	2	yes	—
305	C₂²	C₂×C₃×C₂²⋊Dic₅	1	1	yes	characteristic
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	4	2	no	—
309	C₄	—	4	1	no	—
310	C₄	—	40	2	no	—
311	C₄	C₃×#4072	1	1	yes	characteristic
312	C₄	C₃×#4072	1	1	yes	characteristic
313	C₄	C₃×#4075	1	2	yes	—
314	C₃	#10602437	1	1	yes	characteristic, direct product factor, Sylow
315	C₂	—	2	1	no	—
316	C₂	—	2	2	no	—
317	C₂	—	4	2	no	—
318	C₂	C₃×#65500	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	#4072
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₃×C₄×C₅×#41	C₂
socle	C₂×C₃×C₅	#381
prime radical	C₂×C₃×C₅×#41	C₂²
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₃×C₄×#41	D₅
hypocenter	C₅	C₃×#1827
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₄ ◁ C₃×#117 ◁ C₃×C₄×#41

Complements

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

N	H ≅ G/N	type
4. C₃×#64751	C₂	semidirect
5. C₃×#64743	C₂	semidirect
6. #10602437	C₃	direct (N × H)
11. C₃×C₅×#117	C₄	semidirect
13. C₂×C₃×C₄²×C₅	C₄	semidirect
24. #64751	C₂×C₃	semidirect
25. #64743	C₂×C₃	semidirect
59. C₅×#117	C₃×C₄	semidirect
61. C₂×C₄²×C₅	C₃×C₄	semidirect
94. C₃×#117	Dic₅	semidirect
96. C₂×C₃×C₄²	Dic₅	semidirect
192. #117	C₃×Dic₅	semidirect
194. C₂×C₄²	C₃×Dic₅	semidirect
259. C₃×C₅	#1827	semidirect
300. C₅	C₃×#1827	semidirect
314. C₃	#10602437	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×C₄×C₅×#41	2
C₃×#65529	2
C₃×#65526	2
C₃×#64751	2
C₃×#64743	2
#10602437	3
C₃×#1827	5

Subgroups — C3×#10602437 #14298585

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10602437 #14298585