Subgroups — C₃×#10601226 #14285019

Order 1920.

504 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#65811	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65215	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64705	C₂	1	2	yes	maximal, complemented
5	C₃×#64701	C₂	1	2	yes	maximal, complemented
6	#10601226	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×#709	C₄	1	2	yes	—
8	C₂²×C₃×C₅×C₈	—	2	1	no	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₃×#4076	—	2	1	no	—
11	C₃×#4076	—	2	1	no	—
12	C₃×#4021	C₄	1	2	yes	—
13	C₃×#4020	C₂²	1	1	yes	characteristic
14	C₃×C₄×D₂₀	—	2	1	no	—
15	C₃×C₄×D₂₀	—	2	1	no	—
16	C₃×#4015	C₂²	1	1	yes	characteristic
17	C₃×C₅×C₄⋊C₈	—	2	1	no	—
18	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
19	C₃×C₂²⋊#168	—	2	1	no	—
20	C₃×C₄⋊#168	C₂²	1	2	yes	—
21	C₃×#3935	—	2	1	no	—
22	C₃×#1551	—	5	1	no	maximal, Hall
23	#65811	C₂×C₃	1	1	yes	characteristic
24	C₂×C₅×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
25	#65215	C₂×C₃	1	1	yes	characteristic, complemented
26	#64705	C₂×C₃	1	2	yes	complemented
27	#64701	C₂×C₃	1	2	yes	complemented
28	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
31	C₂×C₃×C₂²⋊D₅	—	2	2	no	—
32	C₃×#709	—	4	4	no	—
33	C₂×C₃×D₂₀	—	2	1	no	—
34	C₂×C₃×D₂₀	—	2	1	no	—
35	C₂×C₃×D₂₀	C₂×C₄	1	2	yes	—
36	C₂×C₃×C₄×D₅	—	2	2	no	—
37	C₂×C₃×Dic₁₀	C₂×C₄	1	2	yes	—
38	C₂×C₃×C₅×C₈	—	2	2	no	—
39	C₂×C₃×C₅×C₈	—	4	2	no	—
40	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
43	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₃×#685	—	2	2	no	—
45	C₃×#685	—	2	2	no	—
46	C₃×C₄⋊Dic₅	—	2	1	no	—
47	C₃×C₄⋊Dic₅	—	2	1	no	—
48	C₂×C₃×#168	—	2	2	no	—
49	C₃×#532	—	5	1	no	—
50	C₃×#432	—	5	1	no	—
51	C₂×C₃×C₄⋊C₈	D₅	1	1	yes	characteristic, complemented
52	C₃×#333	—	5	2	no	—
53	C₃×#331	—	5	2	no	—
54	C₂×#709	C₃×C₄	1	2	yes	—
55	C₂²×C₅×C₈	—	2	1	no	—
56	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
57	#4076	—	2	1	no	—
58	#4076	—	2	1	no	—
59	#4021	C₃×C₄	1	2	yes	—
60	#4020	C₂²×C₃	1	1	yes	characteristic
61	C₄×D₂₀	—	2	1	no	—
62	C₄×D₂₀	—	2	1	no	—
63	#4015	C₂²×C₃	1	1	yes	characteristic
64	C₅×C₄⋊C₈	—	2	1	no	—
65	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
66	C₂²⋊#168	—	2	1	no	—
67	C₄⋊#168	C₂²×C₃	1	2	yes	—
68	#3935	—	2	1	no	—
69	#1551	—	5	1	no	Sylow
70	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₃×D₅	—	2	2	no	—
72	C₂×C₃×C₄×C₅	—	2	2	no	—
73	C₂×C₃×C₄×C₅	—	2	2	no	—
74	C₂×C₃×C₄×C₅	—	2	1	no	—
75	C₂×C₃×C₄×C₅	—	2	1	no	—
76	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₃×C₂²⋊D₅	—	4	4	no	—
83	C₂×C₃×Dic₅	—	2	2	no	—
84	C₃×D₂₀	—	4	2	no	—
85	C₃×D₂₀	—	4	2	no	—
86	C₃×D₂₀	—	4	1	no	—
87	C₃×D₂₀	—	4	1	no	—
88	C₃×C₄×D₅	—	4	4	no	—
89	C₃×Dic₁₀	—	4	2	no	—
90	C₃×C₅×C₈	—	4	2	no	—
91	C₃×#168	—	4	2	no	—
92	C₂×C₃×#41	—	5	2	no	—
93	C₂²×C₃×C₈	—	2	1	no	—
94	C₃×C₄⋊Q₈	—	5	1	no	—
95	C₃×#127	—	5	1	no	—
96	C₃×#124	—	5	2	no	—
97	C₃×#121	—	10	1	no	—
98	C₃×#121	—	10	1	no	—
99	C₃×C₄×D₄	—	10	1	no	—
100	C₃×C₄×D₄	—	10	1	no	—
101	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
102	C₃×C₄⋊C₈	—	2	1	no	—
103	C₃×C₄⋊C₈	—	5	2	no	—
104	C₃×C₄⋊C₈	C₂×D₅	1	2	yes	—
105	C₃×C₂²⋊C₈	—	10	1	no	—
106	C₃×#97	—	10	1	no	—
107	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
110	C₂×C₂²⋊D₅	—	2	2	no	—
111	#709	—	4	4	no	—
112	C₂×D₂₀	—	2	1	no	—
113	C₂×D₂₀	—	2	1	no	—
114	C₂×D₂₀	C₂×C₃×C₄	1	2	yes	—
115	C₂×C₄×D₅	—	2	2	no	—
116	C₂×Dic₁₀	C₂×C₃×C₄	1	2	yes	—
117	C₂×C₅×C₈	—	2	2	no	—
118	C₂×C₅×C₈	—	4	2	no	—
119	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
120	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
121	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
122	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
123	#685	—	2	2	no	—
124	#685	—	2	2	no	—
125	C₄⋊Dic₅	—	2	1	no	—
126	C₄⋊Dic₅	—	2	1	no	—
127	C₂×#168	—	2	2	no	—
128	#532	—	5	1	no	—
129	#432	—	5	1	no	—
130	C₂×C₄⋊C₈	C₃×D₅	1	1	yes	characteristic, complemented
131	#333	—	5	2	no	—
132	#331	—	5	2	no	—
133	C₂²×C₃×C₅	—	2	1	no	—
134	C₂²×C₃×C₅	—	2	1	no	—
135	C₂²×C₃×C₅	#100	1	2	yes	—
136	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₃×D₅	—	4	2	no	—
138	C₂×C₃×D₅	—	4	2	no	—
139	C₂×C₃×D₅	—	4	2	no	—
140	C₃×C₄×C₅	—	2	1	no	—
141	C₃×C₄×C₅	—	2	1	no	—
142	C₃×C₄×C₅	—	2	1	no	—
143	C₃×C₄×C₅	—	2	1	no	—
144	C₃×C₄×C₅	#135	1	2	yes	—
145	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₃×C₄×C₅	#137	1	1	yes	characteristic
147	C₃×Dic₅	—	4	2	no	—
148	C₃×#41	—	20	4	no	—
149	C₂×C₃×Q₈	—	5	2	no	—
150	C₂×C₃×D₄	—	5	2	no	—
151	C₂×C₃×D₄	—	10	2	no	—
152	C₂×C₃×D₄	—	10	1	no	—
153	C₂×C₃×D₄	—	10	1	no	—
154	C₂²×C₃×C₄	—	10	2	no	—
155	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
156	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
157	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
158	C₂×C₃×C₈	—	2	2	no	—
159	C₂×C₃×C₈	—	4	2	no	—
160	C₂×C₃×C₈	—	10	2	no	—
161	C₃×C₄⋊C₄	—	10	1	no	—
162	C₃×C₄⋊C₄	—	10	1	no	—
163	C₃×C₂²⋊C₄	—	10	2	no	—
164	C₃×C₂²⋊C₄	—	10	2	no	—
165	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
166	C₃×C₄²	D₂₀	1	1	yes	characteristic
167	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
168	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
169	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
170	C₂²×D₅	—	2	2	no	—
171	C₂×C₄×C₅	—	2	2	no	—
172	C₂×C₄×C₅	—	2	2	no	—
173	C₂×C₄×C₅	—	2	1	no	—
174	C₂×C₄×C₅	—	2	1	no	—
175	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
176	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
177	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
178	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
179	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
181	C₂²⋊D₅	—	4	4	no	—
182	C₂×Dic₅	—	2	2	no	—
183	D₂₀	—	4	2	no	—
184	D₂₀	—	4	2	no	—
185	D₂₀	—	4	1	no	—
186	D₂₀	—	4	1	no	—
187	C₄×D₅	—	4	4	no	—
188	Dic₁₀	—	4	2	no	—
189	C₅×C₈	—	4	2	no	—
190	#168	—	4	2	no	—
191	C₂×#41	—	5	2	no	—
192	C₂²×C₈	—	2	1	no	—
193	C₄⋊Q₈	—	5	1	no	—
194	#127	—	5	1	no	—
195	#124	—	5	2	no	—
196	#121	—	10	1	no	—
197	#121	—	10	1	no	—
198	C₄×D₄	—	10	1	no	—
199	C₄×D₄	—	10	1	no	—
200	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
201	C₄⋊C₈	—	2	1	no	—
202	C₄⋊C₈	—	5	2	no	—
203	C₄⋊C₈	C₂×C₃×D₅	1	2	yes	—
204	C₂²⋊C₈	—	10	1	no	—
205	#97	—	10	1	no	—
206	C₂×C₃×C₅	—	2	2	no	—
207	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
208	C₂×C₃×C₅	#416	1	1	yes	characteristic
209	C₂×C₃×C₅	#417	1	1	yes	characteristic
210	C₃×D₅	—	8	2	no	—
211	C₂³×C₃	—	10	2	no	—
212	C₂³×C₃	#685	1	1	yes	characteristic
213	C₃×Q₈	—	20	2	no	—
214	C₃×D₄	—	20	2	no	—
215	C₃×D₄	—	20	4	no	—
216	C₃×D₄	—	20	2	no	—
217	C₃×D₄	—	20	1	no	—
218	C₃×D₄	—	20	1	no	—
219	C₂×C₃×C₄	—	2	2	no	—
220	C₂×C₃×C₄	—	2	2	no	—
221	C₂×C₃×C₄	—	2	1	no	—
222	C₂×C₃×C₄	—	2	1	no	—
223	C₂×C₃×C₄	—	10	2	no	—
224	C₂×C₃×C₄	—	20	4	no	—
225	C₂×C₃×C₄	#685	1	1	yes	characteristic
226	C₂×C₃×C₄	#685	1	1	yes	characteristic
227	C₂×C₃×C₄	#685	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
230	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
231	C₃×C₈	—	4	2	no	—
232	C₃×C₈	—	20	2	no	—
233	C₂²×C₅	—	2	1	no	—
234	C₂²×C₅	—	2	1	no	—
235	C₂²×C₅	C₃×#100	1	2	yes	—
236	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
237	C₂×D₅	—	4	2	no	—
238	C₂×D₅	—	4	2	no	—
239	C₂×D₅	—	4	2	no	—
240	C₄×C₅	—	2	1	no	—
241	C₄×C₅	—	2	1	no	—
242	C₄×C₅	—	2	1	no	—
243	C₄×C₅	—	2	1	no	—
244	C₄×C₅	C₃×#135	1	2	yes	—
245	C₄×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
246	C₄×C₅	C₃×#137	1	1	yes	characteristic
247	Dic₅	—	4	2	no	—
248	#41	—	20	4	no	—
249	C₂×Q₈	—	5	2	no	—
250	C₂×D₄	—	5	2	no	—
251	C₂×D₄	—	10	2	no	—
252	C₂×D₄	—	10	1	no	—
253	C₂×D₄	—	10	1	no	—
254	C₂²×C₄	—	10	2	no	—
255	C₂²×C₄	C₃×C₄×D₅	1	1	yes	characteristic
256	C₂²×C₄	C₃×D₂₀	1	1	yes	characteristic
257	C₂²×C₄	C₃×C₂²⋊D₅	1	1	yes	characteristic
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	4	2	no	—
260	C₂×C₈	—	10	2	no	—
261	C₄⋊C₄	—	10	1	no	—
262	C₄⋊C₄	—	10	1	no	—
263	C₂²⋊C₄	—	10	2	no	—
264	C₂²⋊C₄	—	10	2	no	—
265	C₄²	C₃×C₄×D₅	1	1	yes	characteristic
266	C₄²	C₃×D₂₀	1	1	yes	characteristic
267	C₄²	C₃×C₂²⋊D₅	1	1	yes	characteristic
268	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
269	C₃×C₅	#1551	1	1	yes	characteristic, complemented, Hall
270	C₂²×C₃	—	2	1	no	—
271	C₂²×C₃	—	2	1	no	—
272	C₂²×C₃	—	20	2	no	—
273	C₂²×C₃	—	20	2	no	—
274	C₂²×C₃	—	20	2	no	—
275	C₂²×C₃	#3955	1	2	yes	—
276	C₂²×C₃	C₂×#685	1	1	yes	characteristic
277	C₃×C₄	—	2	1	no	—
278	C₃×C₄	—	2	1	no	—
279	C₃×C₄	—	2	1	no	—
280	C₃×C₄	—	2	1	no	—
281	C₃×C₄	—	20	2	no	—
282	C₃×C₄	#4050	1	2	yes	—
283	C₃×C₄	#4078	1	1	yes	characteristic
284	C₃×C₄	#4088	1	1	yes	characteristic
285	C₂×C₅	—	2	2	no	—
286	C₂×C₅	C₂×C₃×#100	1	1	yes	characteristic
287	C₂×C₅	C₃×#416	1	1	yes	characteristic
288	C₂×C₅	C₃×#417	1	1	yes	characteristic
289	D₅	—	8	2	no	—
290	C₂³	—	10	2	no	—
291	C₂³	C₃×#685	1	1	yes	characteristic
292	Q₈	—	20	2	no	—
293	D₄	—	20	2	no	—
294	D₄	—	20	4	no	—
295	D₄	—	20	1	no	—
296	D₄	—	20	1	no	—
297	D₄	—	20	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	10	2	no	—
303	C₂×C₄	—	20	4	no	—
304	C₂×C₄	C₃×#685	1	1	yes	characteristic
305	C₂×C₄	C₃×#685	1	1	yes	characteristic
306	C₂×C₄	C₃×#685	1	1	yes	characteristic
307	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
308	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
309	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
310	C₈	—	4	2	no	—
311	C₈	—	20	2	no	—
312	C₂×C₃	—	2	2	no	—
313	C₂×C₃	—	40	2	no	—
314	C₂×C₃	#65255	1	1	yes	characteristic
315	C₂×C₃	#65402	1	1	yes	characteristic
316	C₂×C₃	C₂×#3955	1	1	yes	characteristic
317	C₅	C₃×#1551	1	1	yes	characteristic, complemented, Sylow
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	20	2	no	—
321	C₂²	—	20	2	no	—
322	C₂²	—	20	2	no	—
323	C₂²	C₃×#3955	1	2	yes	—
324	C₂²	C₂×C₃×#685	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	20	2	no	—
330	C₄	C₃×#4050	1	2	yes	—
331	C₄	C₃×#4078	1	1	yes	characteristic
332	C₄	C₃×#4088	1	1	yes	characteristic
333	C₃	#10601226	1	1	yes	characteristic, direct product factor, Sylow
334	C₂	—	2	2	no	—
335	C₂	—	40	2	no	—
336	C₂	C₃×#65255	1	1	yes	characteristic
337	C₂	C₃×#65402	1	1	yes	characteristic
338	C₂	C₂×C₃×#3955	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	C₂×C₃×C₅×C₄⋊C₈	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₃×#65811	C₂
cosocle	C₄²×C₅	C₂³×C₃
hypercenter	C₂×C₃×C₄⋊C₈	D₅
hypocenter	C₅	C₃×#1551
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×C₄⋊C₈	C₂	semidirect
3. C₃×#65215	C₂	semidirect
4. C₃×#64705	C₂	semidirect
5. C₃×#64701	C₂	semidirect
6. #10601226	C₃	direct (N × H)
24. C₂×C₅×C₄⋊C₈	C₂×C₃	semidirect
25. #65215	C₂×C₃	semidirect
26. #64705	C₂×C₃	semidirect
27. #64701	C₂×C₃	semidirect
51. C₂×C₃×C₄⋊C₈	D₅	semidirect
130. C₂×C₄⋊C₈	C₃×D₅	semidirect
269. C₃×C₅	#1551	semidirect
317. C₅	C₃×#1551	semidirect
333. C₃	#10601226	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×#65811	2
C₂×C₃×C₅×C₄⋊C₈	2
C₃×#65215	2
C₃×#64705	2
C₃×#64701	2
#10601226	3
C₃×#1551	5

Subgroups — C3×#10601226 #14285019

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10601226 #14285019