Subgroups — C₃×#10601611 #14285223

Order 1920.

664 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×#4020	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#432	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65215	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64705	C₂	1	4	yes	maximal, complemented
5	#10601611	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×D₂₀	—	2	1	no	—
7	C₂²×C₃×D₂₀	—	2	1	no	—
8	C₂²×C₃×D₂₀	C₄	1	2	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₃×#4020	—	2	1	no	—
11	C₃×#4020	—	2	1	no	—
12	C₃×#4020	C₄	1	2	yes	—
13	C₃×#4020	C₂²	1	2	yes	—
14	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	complemented
15	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
16	C₃×C₅×#97	—	2	1	no	—
17	C₃×C₂²⋊#168	—	2	1	no	—
18	C₃×C₄⋊#168	C₂²	1	2	yes	complemented
19	C₃×#3935	—	2	1	no	—
20	C₃×#1555	—	5	1	no	maximal, Hall
21	C₂×#4020	C₂×C₃	1	1	yes	characteristic
22	C₅×#432	C₂×C₃	1	1	yes	characteristic, complemented
23	#65215	C₂×C₃	1	1	yes	characteristic, complemented
24	#64705	C₂×C₃	1	4	yes	complemented
25	C₂³×C₃×D₅	—	2	2	no	—
26	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
29	C₂×C₃×D₂₀	—	2	2	no	—
30	C₂×C₃×D₂₀	—	2	2	no	—
31	C₂×C₃×D₂₀	—	2	2	no	—
32	C₂×C₃×D₂₀	—	2	2	no	—
33	C₂×C₃×D₂₀	—	2	2	no	—
34	C₂×C₃×D₂₀	—	4	4	no	—
35	C₂×C₃×D₂₀	—	4	4	no	—
36	C₂×C₃×D₂₀	—	4	4	no	—
37	C₂×C₃×D₂₀	C₂×C₄	1	4	yes	—
38	C₂×C₃×C₅×C₈	—	2	2	no	—
39	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
40	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₂×C₃×#168	—	2	2	no	—
44	C₂×C₃×#127	—	5	1	no	—
45	C₃×#432	—	5	1	no	—
46	C₃×#432	D₅	1	1	yes	characteristic, complemented
47	C₃×#331	—	5	4	no	—
48	C₂²×D₂₀	—	2	1	no	—
49	C₂²×D₂₀	—	2	1	no	—
50	C₂²×D₂₀	C₃×C₄	1	2	yes	—
51	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
52	#4020	—	2	1	no	—
53	#4020	—	2	1	no	—
54	#4020	C₃×C₄	1	2	yes	—
55	#4020	C₂²×C₃	1	2	yes	—
56	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	complemented
57	C₅×C₂²⋊C₈	—	2	1	no	—
58	C₅×#97	—	2	1	no	—
59	C₂²⋊#168	—	2	1	no	—
60	C₄⋊#168	C₂²×C₃	1	2	yes	complemented
61	#3935	—	2	1	no	—
62	#1555	—	5	1	no	Sylow
63	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₃×D₅	—	2	4	no	—
65	C₂²×C₃×D₅	—	4	4	no	—
66	C₂²×C₃×D₅	—	4	4	no	—
67	C₂²×C₃×D₅	—	4	4	no	—
68	C₂×C₃×C₄×C₅	—	2	2	no	—
69	C₂×C₃×C₄×C₅	—	2	2	no	—
70	C₂×C₃×C₄×C₅	—	2	2	no	—
71	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₃×D₂₀	—	4	4	no	—
78	C₃×D₂₀	—	4	4	no	—
79	C₃×D₂₀	—	4	4	no	—
80	C₃×D₂₀	—	4	4	no	—
81	C₃×D₂₀	—	4	4	no	—
82	C₃×D₂₀	—	4	4	no	—
83	C₃×C₅×C₈	—	4	2	no	—
84	C₃×#168	—	4	2	no	—
85	C₂²×C₃×D₄	—	5	2	no	—
86	C₂²×C₃×D₄	—	10	1	no	—
87	C₂²×C₃×D₄	—	10	1	no	—
88	C₃×#127	—	5	2	no	—
89	C₃×#127	—	5	2	no	—
90	C₃×#127	—	10	1	no	—
91	C₃×#127	—	10	1	no	—
92	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
93	C₃×C₄⋊C₈	—	5	2	no	—
94	C₃×C₄⋊C₈	C₂×D₅	1	2	yes	complemented
95	C₃×C₂²⋊C₈	—	2	1	no	—
96	C₃×C₂²⋊C₈	—	10	1	no	—
97	C₃×#97	—	2	1	no	—
98	C₃×#97	—	10	1	no	—
99	C₂³×D₅	—	2	2	no	—
100	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
103	C₂×D₂₀	—	2	2	no	—
104	C₂×D₂₀	—	2	2	no	—
105	C₂×D₂₀	—	2	2	no	—
106	C₂×D₂₀	—	2	2	no	—
107	C₂×D₂₀	—	2	2	no	—
108	C₂×D₂₀	—	4	4	no	—
109	C₂×D₂₀	—	4	4	no	—
110	C₂×D₂₀	—	4	4	no	—
111	C₂×D₂₀	C₂×C₃×C₄	1	4	yes	—
112	C₂×C₅×C₈	—	2	2	no	—
113	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
114	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
115	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
116	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
117	C₂×#168	—	2	2	no	—
118	C₂×#127	—	5	1	no	—
119	#432	—	5	1	no	—
120	#432	C₃×D₅	1	1	yes	characteristic, complemented
121	#331	—	5	4	no	—
122	C₂²×C₃×C₅	—	2	1	no	—
123	C₂²×C₃×C₅	—	2	1	no	—
124	C₂²×C₃×C₅	#100	1	2	yes	—
125	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₃×D₅	—	4	4	no	—
127	C₂×C₃×D₅	—	4	4	no	—
128	C₂×C₃×D₅	—	4	4	no	—
129	C₂×C₃×D₅	—	8	8	no	—
130	C₃×C₄×C₅	—	2	1	no	—
131	C₃×C₄×C₅	—	2	1	no	—
132	C₃×C₄×C₅	—	2	1	no	—
133	C₃×C₄×C₅	—	2	1	no	—
134	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
135	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
136	C₂⁴×C₃	—	10	2	no	—
137	C₂×C₃×D₄	—	5	4	no	—
138	C₂×C₃×D₄	—	10	2	no	—
139	C₂×C₃×D₄	—	10	2	no	—
140	C₂×C₃×D₄	—	10	2	no	—
141	C₂×C₃×D₄	—	10	2	no	—
142	C₂×C₃×D₄	—	10	2	no	—
143	C₂×C₃×D₄	—	20	4	no	—
144	C₂×C₃×D₄	—	20	4	no	—
145	C₂×C₃×D₄	—	20	4	no	—
146	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
147	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
148	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
149	C₂×C₃×C₈	—	2	2	no	—
150	C₂×C₃×C₈	—	10	2	no	—
151	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
152	C₃×C₄²	D₂₀	1	1	yes	characteristic
153	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
154	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
155	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
156	C₂²×D₅	—	2	4	no	—
157	C₂²×D₅	—	4	4	no	—
158	C₂²×D₅	—	4	4	no	—
159	C₂²×D₅	—	4	4	no	—
160	C₂×C₄×C₅	—	2	2	no	—
161	C₂×C₄×C₅	—	2	2	no	—
162	C₂×C₄×C₅	—	2	2	no	—
163	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
164	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
165	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
166	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
167	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
169	D₂₀	—	4	4	no	—
170	D₂₀	—	4	4	no	—
171	D₂₀	—	4	4	no	—
172	D₂₀	—	4	4	no	—
173	D₂₀	—	4	4	no	—
174	D₂₀	—	4	4	no	—
175	C₅×C₈	—	4	2	no	—
176	#168	—	4	2	no	—
177	C₂²×D₄	—	5	2	no	—
178	C₂²×D₄	—	10	1	no	—
179	C₂²×D₄	—	10	1	no	—
180	#127	—	5	2	no	—
181	#127	—	5	2	no	—
182	#127	—	10	1	no	—
183	#127	—	10	1	no	—
184	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
185	C₄⋊C₈	—	5	2	no	—
186	C₄⋊C₈	C₂×C₃×D₅	1	2	yes	complemented
187	C₂²⋊C₈	—	2	1	no	—
188	C₂²⋊C₈	—	10	1	no	—
189	#97	—	2	1	no	—
190	#97	—	10	1	no	—
191	C₂×C₃×C₅	—	2	2	no	—
192	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
193	C₂×C₃×C₅	#418	1	1	yes	characteristic
194	C₂×C₃×C₅	#418	1	1	yes	characteristic
195	C₃×D₅	—	8	4	no	—
196	C₂³×C₃	—	10	4	no	—
197	C₂³×C₃	—	20	4	no	—
198	C₂³×C₃	—	20	4	no	—
199	C₂³×C₃	—	20	4	no	—
200	C₂³×C₃	#685	1	1	yes	characteristic
201	C₃×D₄	—	20	4	no	—
202	C₃×D₄	—	20	4	no	—
203	C₃×D₄	—	20	4	no	—
204	C₃×D₄	—	20	4	no	—
205	C₃×D₄	—	20	4	no	—
206	C₃×D₄	—	20	4	no	—
207	C₂×C₃×C₄	—	2	2	no	—
208	C₂×C₃×C₄	—	2	2	no	—
209	C₂×C₃×C₄	—	2	2	no	—
210	C₂×C₃×C₄	#685	1	1	yes	characteristic
211	C₂×C₃×C₄	#685	1	1	yes	characteristic
212	C₂×C₃×C₄	#685	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
216	C₃×C₈	—	4	2	no	—
217	C₃×C₈	—	20	2	no	—
218	C₂²×C₅	—	2	1	no	—
219	C₂²×C₅	—	2	1	no	—
220	C₂²×C₅	C₃×#100	1	2	yes	—
221	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
222	C₂×D₅	—	4	4	no	—
223	C₂×D₅	—	4	4	no	—
224	C₂×D₅	—	4	4	no	—
225	C₂×D₅	—	8	8	no	—
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	—	2	1	no	—
228	C₄×C₅	—	2	1	no	—
229	C₄×C₅	—	2	1	no	—
230	C₄×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
231	C₄×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
232	C₂⁴	—	10	2	no	—
233	C₂×D₄	—	5	4	no	—
234	C₂×D₄	—	10	2	no	—
235	C₂×D₄	—	10	2	no	—
236	C₂×D₄	—	10	2	no	—
237	C₂×D₄	—	10	2	no	—
238	C₂×D₄	—	10	2	no	—
239	C₂×D₄	—	20	4	no	—
240	C₂×D₄	—	20	4	no	—
241	C₂×D₄	—	20	4	no	—
242	C₂²×C₄	C₃×C₄×D₅	1	1	yes	characteristic
243	C₂²×C₄	C₃×D₂₀	1	1	yes	characteristic
244	C₂²×C₄	C₃×C₂²⋊D₅	1	1	yes	characteristic
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	10	2	no	—
247	C₄²	C₃×C₄×D₅	1	1	yes	characteristic
248	C₄²	C₃×D₂₀	1	1	yes	characteristic
249	C₄²	C₃×C₂²⋊D₅	1	1	yes	characteristic
250	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
251	C₃×C₅	#1555	1	1	yes	characteristic, complemented, Hall
252	C₂²×C₃	—	2	1	no	—
253	C₂²×C₃	—	2	1	no	—
254	C₂²×C₃	—	20	4	no	—
255	C₂²×C₃	—	20	4	no	—
256	C₂²×C₃	—	20	4	no	—
257	C₂²×C₃	—	40	8	no	—
258	C₂²×C₃	#3955	1	2	yes	—
259	C₂²×C₃	C₂×#685	1	1	yes	characteristic
260	C₃×C₄	—	2	1	no	—
261	C₃×C₄	—	2	1	no	—
262	C₃×C₄	—	2	1	no	—
263	C₃×C₄	—	2	1	no	—
264	C₃×C₄	#4054	1	2	yes	—
265	C₃×C₄	#4095	1	2	yes	—
266	C₂×C₅	—	2	2	no	—
267	C₂×C₅	C₂×C₃×#100	1	1	yes	characteristic
268	C₂×C₅	C₃×#418	1	1	yes	characteristic
269	C₂×C₅	C₃×#418	1	1	yes	characteristic
270	D₅	—	8	4	no	—
271	C₂³	—	10	4	no	—
272	C₂³	—	20	4	no	—
273	C₂³	—	20	4	no	—
274	C₂³	—	20	4	no	—
275	C₂³	C₃×#685	1	1	yes	characteristic
276	D₄	—	20	4	no	—
277	D₄	—	20	4	no	—
278	D₄	—	20	4	no	—
279	D₄	—	20	4	no	—
280	D₄	—	20	4	no	—
281	D₄	—	20	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	C₃×#685	1	1	yes	characteristic
286	C₂×C₄	C₃×#685	1	1	yes	characteristic
287	C₂×C₄	C₃×#685	1	1	yes	characteristic
288	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
289	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
290	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
291	C₈	—	4	2	no	—
292	C₈	—	20	2	no	—
293	C₂×C₃	—	2	2	no	—
294	C₂×C₃	—	40	4	no	—
295	C₂×C₃	#65290	1	1	yes	characteristic
296	C₂×C₃	C₂×#3955	1	1	yes	characteristic
297	C₂×C₃	#65420	1	1	yes	characteristic
298	C₅	C₃×#1555	1	1	yes	characteristic, complemented, Sylow
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	20	4	no	—
302	C₂²	—	20	4	no	—
303	C₂²	—	20	4	no	—
304	C₂²	—	40	8	no	—
305	C₂²	C₃×#3955	1	2	yes	—
306	C₂²	C₂×C₃×#685	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	C₃×#4054	1	2	yes	—
312	C₄	C₃×#4095	1	2	yes	—
313	C₃	#10601611	1	1	yes	characteristic, direct product factor, Sylow
314	C₂	—	2	2	no	—
315	C₂	—	40	4	no	—
316	C₂	C₃×#65290	1	1	yes	characteristic
317	C₂	C₂×C₃×#3955	1	1	yes	characteristic
318	C₂	C₃×#65420	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×#685
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₄²	C₂²×C₃×D₅
Fitting	C₃×C₅×#432	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂×C₃×#4020	C₂
cosocle	C₄²×C₅	C₂³×C₃
hypercenter	C₃×#432	D₅
hypocenter	C₅	C₃×#1555
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₃×#432

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#432	C₂	semidirect
3. C₃×#65215	C₂	semidirect
4. C₃×#64705	C₂	semidirect
5. #10601611	C₃	direct (N × H)
14. C₃×C₅×C₄⋊C₈	C₂²	semidirect
18. C₃×C₄⋊#168	C₂²	semidirect
22. C₅×#432	C₂×C₃	semidirect
23. #65215	C₂×C₃	semidirect
24. #64705	C₂×C₃	semidirect
46. C₃×#432	D₅	semidirect
56. C₅×C₄⋊C₈	C₂²×C₃	semidirect
60. C₄⋊#168	C₂²×C₃	semidirect
94. C₃×C₄⋊C₈	C₂×D₅	semidirect
120. #432	C₃×D₅	semidirect
186. C₄⋊C₈	C₂×C₃×D₅	semidirect
251. C₃×C₅	#1555	semidirect
298. C₅	C₃×#1555	semidirect
313. C₃	#10601611	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×#4020	2
C₃×C₅×#432	2
C₃×#65215	2
C₃×#64705	2
#10601611	3
C₃×#1555	5

Subgroups — C3×#10601611 #14285223

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10601611 #14285223