Subgroups — C₃×#10604650 #14282378

Order 1920.

552 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#66059	C₂	1	1	yes	maximal, characteristic
2	C₃×#65681	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×D₅⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64871	C₂	1	4	yes	maximal, complemented
5	#10604650	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×#713	—	2	1	no	—
7	C₂×C₃×#700	—	2	1	no	—
8	C₂×C₃×D₅⋊C₈	—	2	1	no	—
9	C₃×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
10	C₃×#4093	—	2	1	no	—
11	C₃×#4041	—	2	1	no	—
12	C₃×#4041	C₄	1	2	yes	—
13	C₃×#4039	—	2	1	no	—
14	C₃×#4039	C₂²	1	2	yes	—
15	C₃×#4020	C₄	1	1	yes	characteristic
16	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
17	C₃×C₄⋊#170	—	2	1	no	—
18	C₃×C₄⋊#170	C₂²	1	2	yes	—
19	C₃×C₄×#170	—	2	1	no	—
20	C₃×C₄×#170	C₂²	1	2	yes	—
21	C₃×#1492	—	5	1	no	maximal, Hall
22	#66059	C₂×C₃	1	1	yes	characteristic
23	#65681	C₂×C₃	1	1	yes	characteristic, complemented
24	C₄×D₅⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
25	#64871	C₂×C₃	1	4	yes	complemented
26	C₂×C₃×C₅×Q₈	—	2	1	no	—
27	C₃×#713	—	4	2	no	—
28	C₃×#713	—	4	2	no	—
29	C₂×C₃×D₂₀	—	2	2	no	—
30	C₂×C₃×D₂₀	—	2	1	no	—
31	C₂×C₃×D₂₀	—	2	1	no	—
32	C₂×C₃×D₂₀	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₄×D₅	—	2	2	no	—
34	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
36	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
37	C₂×C₃×#170	—	2	2	no	—
38	C₂×C₃×#170	—	2	2	no	—
39	C₃×#700	—	4	2	no	—
40	C₃×D₅⋊C₈	—	2	4	no	—
41	C₃×C₅×C₄⋊C₄	—	2	1	no	—
42	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
43	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
44	C₃×#685	—	2	2	no	—
45	C₃×#685	—	2	2	no	—
46	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
47	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
48	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
49	C₃×#532	—	5	1	no	—
50	C₃×#423	—	5	1	no	—
51	C₂×C₃×C₄×C₈	—	5	1	no	—
52	C₃×D₄⋊C₈	—	5	4	no	—
53	C₂×#713	—	2	1	no	—
54	C₂×#700	—	2	1	no	—
55	C₂×D₅⋊C₈	—	2	1	no	—
56	C₅×C₄⋊Q₈	C₃×C₄	1	1	yes	characteristic
57	#4093	—	2	1	no	—
58	#4041	—	2	1	no	—
59	#4041	C₃×C₄	1	2	yes	—
60	#4039	—	2	1	no	—
61	#4039	C₂²×C₃	1	2	yes	—
62	#4020	C₃×C₄	1	1	yes	characteristic
63	C₄²×D₅	C₂²×C₃	1	1	yes	characteristic
64	C₄⋊#170	—	2	1	no	—
65	C₄⋊#170	C₂²×C₃	1	2	yes	—
66	C₄×#170	—	2	1	no	—
67	C₄×#170	C₂²×C₃	1	2	yes	—
68	#1492	—	5	1	no	Sylow
69	C₂²×C₃×D₅	—	2	2	no	—
70	C₂²×C₃×D₅	C₂²⋊C₄	1	1	yes	characteristic
71	C₃×C₅×Q₈	—	4	2	no	—
72	C₂×C₃×C₄×C₅	—	2	2	no	—
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	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
79	C₃×D₂₀	—	2	4	no	—
80	C₃×D₂₀	—	4	4	no	—
81	C₃×D₂₀	—	4	2	no	—
82	C₃×D₂₀	—	4	2	no	—
83	C₃×C₄×D₅	—	2	2	no	—
84	C₃×C₄×D₅	—	2	2	no	—
85	C₃×C₄×D₅	—	4	2	no	—
86	C₃×C₄×D₅	—	4	2	no	—
87	C₃×C₄×D₅	D₈	1	2	yes	—
88	C₃×C₄×D₅	SD₁₆	1	2	yes	—
89	C₃×#170	—	2	4	no	—
90	C₃×#170	—	4	2	no	—
91	C₂×C₃×#41	—	10	1	no	—
92	C₂×C₃×M₁₆	—	10	1	no	—
93	C₂²×C₃×C₈	—	10	1	no	—
94	C₃×C₄⋊Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic
95	C₃×#127	—	5	1	no	—
96	C₃×#124	—	10	1	no	—
97	C₃×#121	—	5	2	no	—
98	C₃×#121	—	10	1	no	—
99	C₃×C₄×D₄	—	5	2	no	—
100	C₃×C₄×D₄	—	10	1	no	—
101	C₂×C₃×C₄²	—	5	1	no	—
102	C₃×C₄⋊C₈	—	5	2	no	—
103	C₃×C₄⋊C₈	—	10	1	no	—
104	C₃×C₄×C₈	—	5	2	no	—
105	C₃×C₄×C₈	—	10	1	no	—
106	C₂×C₅×Q₈	—	2	1	no	—
107	#713	—	4	2	no	—
108	#713	—	4	2	no	—
109	C₂×D₂₀	—	2	2	no	—
110	C₂×D₂₀	—	2	1	no	—
111	C₂×D₂₀	—	2	1	no	—
112	C₂×D₂₀	C₂×C₃×C₄	1	2	yes	—
113	C₂×C₄×D₅	—	2	2	no	—
114	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
115	C₂×C₄×D₅	C₃×D₄	1	1	yes	characteristic
116	C₂×C₄×D₅	C₃×D₄	1	1	yes	characteristic
117	C₂×#170	—	2	2	no	—
118	C₂×#170	—	2	2	no	—
119	#700	—	4	2	no	—
120	D₅⋊C₈	—	2	4	no	—
121	C₅×C₄⋊C₄	—	2	1	no	—
122	C₅×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
123	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
124	#685	—	2	2	no	—
125	#685	—	2	2	no	—
126	C₄×Dic₅	C₃×D₄	1	1	yes	characteristic
127	C₄×Dic₅	C₃×D₄	1	1	yes	characteristic
128	C₄×Dic₅	C₂³×C₃	1	1	yes	characteristic
129	#532	—	5	1	no	—
130	#423	—	5	1	no	—
131	C₂×C₄×C₈	—	5	1	no	—
132	D₄⋊C₈	—	5	4	no	—
133	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₃×D₅	—	2	1	no	—
135	C₂×C₃×D₅	—	2	1	no	—
136	C₂×C₃×D₅	—	4	2	no	—
137	C₂×C₃×D₅	—	4	2	no	—
138	C₂×C₃×D₅	—	4	2	no	—
139	C₂×C₃×D₅	#102	1	2	yes	—
140	C₃×C₄×C₅	—	4	2	no	—
141	C₃×C₄×C₅	#102	1	2	yes	—
142	C₃×C₄×C₅	#104	1	2	yes	—
143	C₃×C₄×C₅	#131	1	2	yes	—
144	C₃×Dic₅	—	2	1	no	—
145	C₃×Dic₅	—	2	1	no	—
146	C₃×Dic₅	C₂×D₈	1	1	yes	characteristic
147	C₃×Dic₅	C₂×SD₁₆	1	1	yes	characteristic
148	C₃×#41	—	20	2	no	—
149	C₃×#41	—	20	2	no	—
150	C₂×C₃×Q₈	—	2	1	no	—
151	C₂×C₃×D₄	—	5	2	no	—
152	C₂×C₃×D₄	—	10	2	no	—
153	C₂×C₃×D₄	—	10	1	no	—
154	C₂×C₃×D₄	—	10	1	no	—
155	C₂²×C₃×C₄	—	5	1	no	—
156	C₂²×C₃×C₄	—	5	1	no	—
157	C₂²×C₃×C₄	—	5	1	no	—
158	C₂²×C₃×C₄	—	10	2	no	—
159	C₃×M₁₆	—	20	2	no	—
160	C₂×C₃×C₈	—	10	2	no	—
161	C₂×C₃×C₈	—	10	4	no	—
162	C₂×C₃×C₈	—	10	2	no	—
163	C₃×C₄⋊C₄	—	2	1	no	—
164	C₃×C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
165	C₃×C₂²⋊C₄	—	10	2	no	—
166	C₃×C₂²⋊C₄	—	10	2	no	—
167	C₃×C₄²	—	5	1	no	—
168	C₃×C₄²	—	5	1	no	—
169	C₃×C₄²	—	5	1	no	—
170	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
171	C₂²×D₅	—	2	2	no	—
172	C₂²×D₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
173	C₅×Q₈	—	4	2	no	—
174	C₂×C₄×C₅	—	2	2	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	C₂²×C₃×C₄	1	1	yes	characteristic
178	C₂×Dic₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
179	C₂×Dic₅	C₂×C₃×D₄	1	1	yes	characteristic
180	C₂×Dic₅	C₂×C₃×D₄	1	1	yes	characteristic
181	D₂₀	—	2	4	no	—
182	D₂₀	—	4	4	no	—
183	D₂₀	—	4	2	no	—
184	D₂₀	—	4	2	no	—
185	C₄×D₅	—	2	2	no	—
186	C₄×D₅	—	2	2	no	—
187	C₄×D₅	—	4	2	no	—
188	C₄×D₅	—	4	2	no	—
189	C₄×D₅	C₃×D₈	1	2	yes	—
190	C₄×D₅	C₃×SD₁₆	1	2	yes	—
191	#170	—	2	4	no	—
192	#170	—	4	2	no	—
193	C₂×#41	—	10	1	no	—
194	C₂×M₁₆	—	10	1	no	—
195	C₂²×C₈	—	10	1	no	—
196	C₄⋊Q₈	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
197	#127	—	5	1	no	—
198	#124	—	10	1	no	—
199	#121	—	5	2	no	—
200	#121	—	10	1	no	—
201	C₄×D₄	—	5	2	no	—
202	C₄×D₄	—	10	1	no	—
203	C₂×C₄²	—	5	1	no	—
204	C₄⋊C₈	—	5	2	no	—
205	C₄⋊C₈	—	10	1	no	—
206	C₄×C₈	—	5	2	no	—
207	C₄×C₈	—	10	1	no	—
208	C₂×C₃×C₅	#408	1	1	yes	characteristic
209	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
210	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
211	C₃×D₅	—	2	2	no	—
212	C₃×D₅	—	8	2	no	—
213	C₂³×C₃	—	5	1	no	—
214	C₂³×C₃	—	10	2	no	—
215	C₃×Q₈	—	4	2	no	—
216	C₃×D₄	—	10	4	no	—
217	C₃×D₄	—	20	4	no	—
218	C₃×D₄	—	20	2	no	—
219	C₃×D₄	—	20	2	no	—
220	C₂×C₃×C₄	—	2	2	no	—
221	C₂×C₃×C₄	—	5	1	no	—
222	C₂×C₃×C₄	—	5	2	no	—
223	C₂×C₃×C₄	—	5	2	no	—
224	C₂×C₃×C₄	—	5	1	no	—
225	C₂×C₃×C₄	—	5	1	no	—
226	C₂×C₃×C₄	—	10	2	no	—
227	C₂×C₃×C₄	—	10	2	no	—
228	C₂×C₃×C₄	—	20	2	no	—
229	C₂×C₃×C₄	—	20	2	no	—
230	C₂×C₃×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
231	C₂×C₃×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
232	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
233	C₃×C₈	—	10	4	no	—
234	C₃×C₈	—	20	2	no	—
235	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
236	C₂×D₅	—	2	1	no	—
237	C₂×D₅	—	2	1	no	—
238	C₂×D₅	—	4	2	no	—
239	C₂×D₅	—	4	2	no	—
240	C₂×D₅	—	4	2	no	—
241	C₂×D₅	C₃×#102	1	2	yes	—
242	C₄×C₅	—	4	2	no	—
243	C₄×C₅	C₃×#102	1	2	yes	—
244	C₄×C₅	C₃×#104	1	2	yes	—
245	C₄×C₅	C₃×#131	1	2	yes	—
246	Dic₅	—	2	1	no	—
247	Dic₅	—	2	1	no	—
248	Dic₅	C₂×C₃×D₈	1	1	yes	characteristic
249	Dic₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
250	#41	—	20	2	no	—
251	#41	—	20	2	no	—
252	C₂×Q₈	—	2	1	no	—
253	C₂×D₄	—	5	2	no	—
254	C₂×D₄	—	10	2	no	—
255	C₂×D₄	—	10	1	no	—
256	C₂×D₄	—	10	1	no	—
257	C₂²×C₄	—	5	1	no	—
258	C₂²×C₄	—	5	1	no	—
259	C₂²×C₄	—	5	1	no	—
260	C₂²×C₄	—	10	2	no	—
261	M₁₆	—	20	2	no	—
262	C₂×C₈	—	10	2	no	—
263	C₂×C₈	—	10	4	no	—
264	C₂×C₈	—	10	2	no	—
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
267	C₂²⋊C₄	—	10	2	no	—
268	C₂²⋊C₄	—	10	2	no	—
269	C₄²	—	5	1	no	—
270	C₄²	—	5	1	no	—
271	C₄²	—	5	1	no	—
272	C₄²	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
273	C₃×C₅	#1492	1	1	yes	characteristic, complemented, Hall
274	C₂²×C₃	—	5	2	no	—
275	C₂²×C₃	—	10	1	no	—
276	C₂²×C₃	—	10	1	no	—
277	C₂²×C₃	—	20	2	no	—
278	C₂²×C₃	—	20	2	no	—
279	C₂²×C₃	—	20	2	no	—
280	C₂²×C₃	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
281	C₃×C₄	—	4	2	no	—
282	C₃×C₄	—	5	1	no	—
283	C₃×C₄	—	5	1	no	—
284	C₃×C₄	—	10	1	no	—
285	C₃×C₄	—	10	1	no	—
286	C₃×C₄	#4007	1	2	yes	—
287	C₃×C₄	#4010	1	2	yes	—
288	C₃×C₄	#4133	1	2	yes	—
289	C₂×C₅	C₃×#408	1	1	yes	characteristic
290	C₂×C₅	C₂×C₃×#102	1	1	yes	characteristic
291	C₂×C₅	C₂×C₃×#104	1	1	yes	characteristic
292	D₅	—	2	2	no	—
293	D₅	—	8	2	no	—
294	C₂³	—	5	1	no	—
295	C₂³	—	10	2	no	—
296	Q₈	—	4	2	no	—
297	D₄	—	10	4	no	—
298	D₄	—	20	4	no	—
299	D₄	—	20	2	no	—
300	D₄	—	20	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	5	1	no	—
303	C₂×C₄	—	5	2	no	—
304	C₂×C₄	—	5	2	no	—
305	C₂×C₄	—	5	1	no	—
306	C₂×C₄	—	5	1	no	—
307	C₂×C₄	—	10	2	no	—
308	C₂×C₄	—	10	2	no	—
309	C₂×C₄	—	20	2	no	—
310	C₂×C₄	—	20	2	no	—
311	C₂×C₄	C₃×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
312	C₂×C₄	C₃×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
313	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
314	C₈	—	10	4	no	—
315	C₈	—	20	2	no	—
316	C₂×C₃	—	10	2	no	—
317	C₂×C₃	—	40	2	no	—
318	C₂×C₃	C₂×#4007	1	1	yes	characteristic
319	C₂×C₃	C₂×#4010	1	1	yes	characteristic
320	C₂×C₃	#65787	1	1	yes	characteristic
321	C₅	C₃×#1492	1	1	yes	characteristic, complemented, Sylow
322	C₂²	—	5	2	no	—
323	C₂²	—	10	1	no	—
324	C₂²	—	10	1	no	—
325	C₂²	—	20	2	no	—
326	C₂²	—	20	2	no	—
327	C₂²	—	20	2	no	—
328	C₂²	C₂×C₃×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
329	C₄	—	4	2	no	—
330	C₄	—	5	1	no	—
331	C₄	—	5	1	no	—
332	C₄	—	10	1	no	—
333	C₄	—	10	1	no	—
334	C₄	C₃×#4007	1	2	yes	—
335	C₄	C₃×#4010	1	2	yes	—
336	C₄	C₃×#4133	1	2	yes	—
337	C₃	#10604650	1	1	yes	characteristic, direct product factor, Sylow
338	C₂	—	10	2	no	—
339	C₂	—	40	2	no	—
340	C₂	C₂×C₃×#4007	1	1	yes	characteristic
341	C₂	C₂×C₃×#4010	1	1	yes	characteristic
342	C₂	C₃×#65787	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×C₂²⋊AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₅×C₄⋊Q₈	C₄
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₃×#66059	C₂
cosocle	C₄×Dic₅	C₂³×C₃
hypercenter	C₃×C₄⋊Q₈	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#1492
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. C₃×#65681	C₂	semidirect
3. C₃×C₄×D₅⋊C₈	C₂	semidirect
4. C₃×#64871	C₂	semidirect
5. #10604650	C₃	direct (N × H)
23. #65681	C₂×C₃	semidirect
24. C₄×D₅⋊C₈	C₂×C₃	semidirect
25. #64871	C₂×C₃	semidirect
273. C₃×C₅	#1492	semidirect
321. C₅	C₃×#1492	semidirect
337. C₃	#10604650	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×#66059	2
C₃×#65681	2
C₃×C₄×D₅⋊C₈	2
C₃×#64871	2
#10604650	3
C₃×#1492	5

Subgroups — C3×#10604650 #14282378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10604650 #14282378