Subgroups — C₃×#10600840 #14302828

Order 1920.

688 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×#66164	C₂	1	1	yes	maximal, characteristic
2	C₃×#66153	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#410	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65290	C₂	1	2	yes	maximal, complemented
5	C₃×#64751	C₂	1	2	yes	maximal, complemented
6	#10600840	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
8	C₂²×C₃×D₂₀	C₂²	1	1	yes	characteristic
9	C₃×C₅×#117	C₂²	1	2	yes	—
10	C₃×#4094	—	2	4	no	—
11	C₃×#4093	C₄	1	2	yes	complemented
12	C₃×#4086	C₄	1	2	yes	complemented
13	C₃×#4083	—	2	2	no	—
14	C₃×#4076	—	2	2	no	—
15	C₃×#4072	C₂²	1	1	yes	characteristic
16	C₂×C₃×#681	—	2	2	no	—
17	C₂×C₃×#681	C₂²	1	2	yes	—
18	C₃×C₅×#99	—	2	2	no	—
19	C₃×#3941	—	2	4	no	—
20	C₃×#1910	—	5	1	no	maximal, Hall
21	#66164	C₂×C₃	1	1	yes	characteristic
22	#66153	C₂×C₃	1	1	yes	characteristic, complemented
23	C₅×#410	C₂×C₃	1	1	yes	characteristic, complemented
24	#65290	C₂×C₃	1	2	yes	complemented
25	#64751	C₂×C₃	1	2	yes	complemented
26	C₂³×C₃×D₅	—	2	1	no	—
27	C₃×C₅×#41	—	2	2	no	—
28	C₃×C₅×#41	—	2	2	no	—
29	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₅×D₄	D₄	1	2	yes	—
32	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
33	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
34	C₂×C₃×C₂²⋊D₅	—	2	4	no	—
35	C₂×C₃×D₂₀	—	2	2	no	—
36	C₂×C₃×D₂₀	—	2	2	no	—
37	C₂×C₃×D₂₀	—	2	1	no	—
38	C₂×C₃×D₂₀	C₂×C₄	1	2	yes	—
39	C₃×C₅×C₄⋊C₄	—	2	2	no	—
40	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
41	C₃×C₄²×C₅	—	2	2	no	—
42	C₃×C₂²⋊Dic₅	—	2	1	no	—
43	C₃×#685	—	2	4	no	—
44	C₃×C₄⋊Dic₅	—	2	1	no	—
45	C₃×C₄×Dic₅	C₂×C₄	1	2	yes	—
46	C₃×#681	—	2	4	no	—
47	C₃×#681	—	2	2	no	—
48	C₃×#681	—	2	2	no	—
49	C₂×C₃×#168	—	2	4	no	—
50	C₂×C₃×#168	D₄	1	4	yes	complemented
51	C₃×#568	—	5	1	no	—
52	C₃×#535	—	5	1	no	—
53	C₃×#418	—	5	2	no	—
54	C₃×#410	D₅	1	1	yes	characteristic, complemented
55	C₃×H(3,ℤ/4)	—	5	2	no	—
56	C₂×C₅×#41	C₂²×C₃	1	1	yes	characteristic
57	C₂²×D₂₀	C₂²×C₃	1	1	yes	characteristic
58	C₅×#117	C₂²×C₃	1	2	yes	—
59	#4094	—	2	4	no	—
60	#4093	C₃×C₄	1	2	yes	complemented
61	#4086	C₃×C₄	1	2	yes	complemented
62	#4083	—	2	2	no	—
63	#4076	—	2	2	no	—
64	#4072	C₂²×C₃	1	1	yes	characteristic
65	C₂×#681	—	2	2	no	—
66	C₂×#681	C₂²×C₃	1	2	yes	—
67	C₅×#99	—	2	2	no	—
68	#3941	—	2	4	no	—
69	#1910	—	5	1	no	Sylow
70	C₂³×C₃×C₅	C₂²⋊C₄	1	2	yes	—
71	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₃×D₅	—	2	2	no	—
73	C₂²×C₃×D₅	—	4	2	no	—
74	C₂²×C₃×D₅	—	8	2	no	—
75	C₃×C₅×Q₈	—	2	2	no	—
76	C₃×C₅×D₄	—	2	4	no	—
77	C₃×C₅×D₄	—	2	2	no	—
78	C₂×C₃×C₄×C₅	—	2	2	no	—
79	C₂×C₃×C₄×C₅	—	2	2	no	—
80	C₂×C₃×C₄×C₅	—	4	2	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	2	yes	—
84	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	#41	1	2	yes	—
86	C₃×C₂²⋊D₅	—	8	4	no	—
87	C₂×C₃×Dic₅	—	2	2	no	—
88	C₃×D₂₀	—	2	4	no	—
89	C₃×D₂₀	—	4	1	no	—
90	C₃×D₂₀	—	4	1	no	—
91	C₃×#168	—	2	4	no	—
92	C₂×C₃×#41	C₂×D₅	1	1	yes	characteristic
93	C₂²×C₃×D₄	—	5	1	no	—
94	C₃×#131	—	10	4	no	—
95	C₂×C₃×M₁₆	—	5	2	no	—
96	C₂×C₃×M₁₆	—	10	2	no	—
97	C₃×#127	—	5	2	no	—
98	C₃×#124	—	5	2	no	—
99	C₃×#121	—	10	2	no	—
100	C₃×#120	—	10	2	no	—
101	C₃×#117	—	5	1	no	—
102	C₃×#117	C₂×D₅	1	2	yes	—
103	C₃×#102	—	10	4	no	—
104	C₃×#99	—	2	2	no	—
105	C₂³×D₅	—	2	1	no	—
106	C₅×#41	—	2	2	no	—
107	C₅×#41	—	2	2	no	—
108	C₂×C₅×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
109	C₂×C₅×D₄	C₂×C₃×C₄	1	1	yes	characteristic
110	C₂×C₅×D₄	C₃×D₄	1	2	yes	—
111	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
112	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
113	C₂×C₂²⋊D₅	—	2	4	no	—
114	C₂×D₂₀	—	2	2	no	—
115	C₂×D₂₀	—	2	2	no	—
116	C₂×D₂₀	—	2	1	no	—
117	C₂×D₂₀	C₂×C₃×C₄	1	2	yes	—
118	C₅×C₄⋊C₄	—	2	2	no	—
119	C₅×C₂²⋊C₄	—	2	2	no	—
120	C₄²×C₅	—	2	2	no	—
121	C₂²⋊Dic₅	—	2	1	no	—
122	#685	—	2	4	no	—
123	C₄⋊Dic₅	—	2	1	no	—
124	C₄×Dic₅	C₂×C₃×C₄	1	2	yes	—
125	#681	—	2	4	no	—
126	#681	—	2	2	no	—
127	#681	—	2	2	no	—
128	C₂×#168	—	2	4	no	—
129	C₂×#168	C₃×D₄	1	4	yes	complemented
130	#568	—	5	1	no	—
131	#535	—	5	1	no	—
132	#418	—	5	2	no	—
133	#410	C₃×D₅	1	1	yes	characteristic, complemented
134	H(3,ℤ/4)	—	5	2	no	—
135	C₂²×C₃×C₅	—	2	2	no	—
136	C₂²×C₃×C₅	—	4	1	no	—
137	C₂²×C₃×C₅	—	4	2	no	—
138	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₃×C₅	#121	1	2	yes	—
140	C₂×C₃×D₅	—	4	4	no	—
141	C₂×C₃×D₅	—	4	2	no	—
142	C₂×C₃×D₅	—	8	4	no	—
143	C₃×C₄×C₅	—	2	2	no	—
144	C₃×C₄×C₅	—	8	2	no	—
145	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
146	C₃×C₄×C₅	#120	1	1	yes	characteristic
147	C₃×C₄×C₅	#123	1	1	yes	characteristic
148	C₃×Dic₅	—	4	2	no	—
149	C₂⁴×C₃	—	10	1	no	—
150	C₃×#41	—	2	2	no	—
151	C₃×#41	—	2	2	no	—
152	C₂×C₃×Q₈	C₄×D₅	1	1	yes	characteristic
153	C₂×C₃×D₄	—	5	2	no	—
154	C₂×C₃×D₄	—	10	4	no	—
155	C₂×C₃×D₄	—	10	1	no	—
156	C₂×C₃×D₄	—	10	2	no	—
157	C₂×C₃×D₄	—	10	2	no	—
158	C₂×C₃×D₄	C₄×D₅	1	1	yes	characteristic
159	C₂×C₃×D₄	C₂²⋊D₅	1	2	yes	—
160	C₂²×C₃×C₄	D₂₀	1	2	yes	—
161	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
162	C₃×M₁₆	—	10	4	no	—
163	C₃×M₁₆	—	10	2	no	—
164	C₃×M₁₆	—	10	2	no	—
165	C₂×C₃×C₈	—	5	4	no	—
166	C₂×C₃×C₈	—	10	4	no	—
167	C₃×C₄⋊C₄	—	2	2	no	—
168	C₃×C₄⋊C₄	—	10	1	no	—
169	C₃×C₂²⋊C₄	—	2	2	no	—
170	C₃×C₂²⋊C₄	—	10	1	no	—
171	C₃×C₂²⋊C₄	—	10	4	no	—
172	C₃×C₄²	—	2	2	no	—
173	C₃×C₄²	—	5	2	no	—
174	C₂³×C₅	C₃×C₂²⋊C₄	1	2	yes	—
175	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
176	C₂²×D₅	—	2	2	no	—
177	C₂²×D₅	—	4	2	no	—
178	C₂²×D₅	—	8	2	no	—
179	C₅×Q₈	—	2	2	no	—
180	C₅×D₄	—	2	4	no	—
181	C₅×D₄	—	2	2	no	—
182	C₂×C₄×C₅	—	2	2	no	—
183	C₂×C₄×C₅	—	2	2	no	—
184	C₂×C₄×C₅	—	4	2	no	—
185	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
186	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
187	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
188	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
189	C₂×C₄×C₅	C₃×#41	1	2	yes	—
190	C₂²⋊D₅	—	8	4	no	—
191	C₂×Dic₅	—	2	2	no	—
192	D₂₀	—	2	4	no	—
193	D₂₀	—	4	1	no	—
194	D₂₀	—	4	1	no	—
195	#168	—	2	4	no	—
196	C₂×#41	C₂×C₃×D₅	1	1	yes	characteristic
197	C₂²×D₄	—	5	1	no	—
198	#131	—	10	4	no	—
199	C₂×M₁₆	—	5	2	no	—
200	C₂×M₁₆	—	10	2	no	—
201	#127	—	5	2	no	—
202	#124	—	5	2	no	—
203	#121	—	10	2	no	—
204	#120	—	10	2	no	—
205	#117	—	5	1	no	—
206	#117	C₂×C₃×D₅	1	2	yes	—
207	#102	—	10	4	no	—
208	#99	—	2	2	no	—
209	C₂×C₃×C₅	—	2	1	no	—
210	C₂×C₃×C₅	—	2	2	no	—
211	C₂×C₃×C₅	—	4	2	no	—
212	C₂×C₃×C₅	#386	1	1	yes	characteristic
213	C₃×D₅	—	8	2	no	—
214	C₂³×C₃	—	10	2	no	—
215	C₂³×C₃	—	20	2	no	—
216	C₂³×C₃	—	40	2	no	—
217	C₂³×C₃	#685	1	2	yes	—
218	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
219	C₃×Q₈	—	2	2	no	—
220	C₃×D₄	—	2	4	no	—
221	C₃×D₄	—	2	2	no	—
222	C₃×D₄	—	10	4	no	—
223	C₃×D₄	—	20	1	no	—
224	C₃×D₄	—	20	1	no	—
225	C₃×D₄	—	40	4	no	—
226	C₂×C₃×C₄	—	2	2	no	—
227	C₂×C₃×C₄	—	2	2	no	—
228	C₂×C₃×C₄	—	4	2	no	—
229	C₂×C₃×C₄	—	10	2	no	—
230	C₂×C₃×C₄	#685	1	2	yes	—
231	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
232	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
233	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
234	C₂×C₃×C₄	#711	1	2	yes	—
235	C₃×C₈	—	10	4	no	—
236	C₂²×C₅	—	2	2	no	—
237	C₂²×C₅	—	4	1	no	—
238	C₂²×C₅	—	4	2	no	—
239	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
240	C₂²×C₅	C₃×#121	1	2	yes	—
241	C₂×D₅	—	4	4	no	—
242	C₂×D₅	—	4	2	no	—
243	C₂×D₅	—	8	4	no	—
244	C₄×C₅	—	2	2	no	—
245	C₄×C₅	—	8	2	no	—
246	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
247	C₄×C₅	C₃×#120	1	1	yes	characteristic
248	C₄×C₅	C₃×#123	1	1	yes	characteristic
249	Dic₅	—	4	2	no	—
250	C₂⁴	—	10	1	no	—
251	#41	—	2	2	no	—
252	#41	—	2	2	no	—
253	C₂×Q₈	C₃×C₄×D₅	1	1	yes	characteristic
254	C₂×D₄	—	5	2	no	—
255	C₂×D₄	—	10	4	no	—
256	C₂×D₄	—	10	1	no	—
257	C₂×D₄	—	10	2	no	—
258	C₂×D₄	—	10	2	no	—
259	C₂×D₄	C₃×C₄×D₅	1	1	yes	characteristic
260	C₂×D₄	C₃×C₂²⋊D₅	1	2	yes	—
261	C₂²×C₄	C₃×D₂₀	1	2	yes	—
262	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
263	M₁₆	—	10	4	no	—
264	M₁₆	—	10	2	no	—
265	M₁₆	—	10	2	no	—
266	C₂×C₈	—	5	4	no	—
267	C₂×C₈	—	10	4	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	10	1	no	—
270	C₂²⋊C₄	—	2	2	no	—
271	C₂²⋊C₄	—	10	1	no	—
272	C₂²⋊C₄	—	10	4	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	5	2	no	—
275	C₃×C₅	#1910	1	1	yes	characteristic, complemented, Hall
276	C₂²×C₃	—	2	2	no	—
277	C₂²×C₃	—	4	1	no	—
278	C₂²×C₃	—	4	2	no	—
279	C₂²×C₃	—	20	4	no	—
280	C₂²×C₃	—	20	2	no	—
281	C₂²×C₃	—	40	4	no	—
282	C₂²×C₃	C₂×#685	1	1	yes	characteristic
283	C₂²×C₃	#4085	1	2	yes	—
284	C₃×C₄	—	2	2	no	—
285	C₃×C₄	—	8	2	no	—
286	C₃×C₄	—	20	2	no	—
287	C₃×C₄	#4027	1	2	yes	—
288	C₃×C₄	#4028	1	1	yes	characteristic
289	C₃×C₄	#4032	1	1	yes	characteristic
290	C₂×C₅	—	2	1	no	—
291	C₂×C₅	—	2	2	no	—
292	C₂×C₅	—	4	2	no	—
293	C₂×C₅	C₃×#386	1	1	yes	characteristic
294	D₅	—	8	2	no	—
295	C₂³	—	10	2	no	—
296	C₂³	—	20	2	no	—
297	C₂³	—	40	2	no	—
298	C₂³	C₃×#685	1	2	yes	—
299	C₂³	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
300	Q₈	—	2	2	no	—
301	D₄	—	2	4	no	—
302	D₄	—	2	2	no	—
303	D₄	—	10	4	no	—
304	D₄	—	20	1	no	—
305	D₄	—	20	1	no	—
306	D₄	—	40	4	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	10	2	no	—
311	C₂×C₄	C₃×#685	1	2	yes	—
312	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
313	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
314	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
315	C₂×C₄	C₃×#711	1	2	yes	—
316	C₈	—	10	4	no	—
317	C₂×C₃	—	2	2	no	—
318	C₂×C₃	—	2	1	no	—
319	C₂×C₃	—	4	2	no	—
320	C₂×C₃	—	40	2	no	—
321	C₂×C₃	#65247	1	1	yes	characteristic
322	C₅	C₃×#1910	1	1	yes	characteristic, complemented, Sylow
323	C₂²	—	2	2	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	4	2	no	—
326	C₂²	—	20	4	no	—
327	C₂²	—	20	2	no	—
328	C₂²	—	40	4	no	—
329	C₂²	C₂×C₃×#685	1	1	yes	characteristic
330	C₂²	C₃×#4085	1	2	yes	—
331	C₄	—	2	2	no	—
332	C₄	—	8	2	no	—
333	C₄	—	20	2	no	—
334	C₄	C₃×#4027	1	2	yes	—
335	C₄	C₃×#4028	1	1	yes	characteristic
336	C₄	C₃×#4032	1	1	yes	characteristic
337	C₃	#10600840	1	1	yes	characteristic, direct product factor, Sylow
338	C₂	—	2	2	no	—
339	C₂	—	2	1	no	—
340	C₂	—	4	2	no	—
341	C₂	—	40	2	no	—
342	C₂	C₃×#65247	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	#65247
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₃×C₅×#410	C₂
socle	C₂×C₃×C₅	#386
prime radical	C₃×#66164	C₂
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₃×#410	D₅
hypocenter	C₅	C₃×#1910
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×#410

Complements

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

N	H ≅ G/N	type
2. C₃×#66153	C₂	semidirect
3. C₃×C₅×#410	C₂	semidirect
4. C₃×#65290	C₂	semidirect
5. C₃×#64751	C₂	semidirect
6. #10600840	C₃	direct (N × H)
11. C₃×#4093	C₄	semidirect
12. C₃×#4086	C₄	semidirect
22. #66153	C₂×C₃	semidirect
23. C₅×#410	C₂×C₃	semidirect
24. #65290	C₂×C₃	semidirect
25. #64751	C₂×C₃	semidirect
50. C₂×C₃×#168	D₄	semidirect
54. C₃×#410	D₅	semidirect
60. #4093	C₃×C₄	semidirect
61. #4086	C₃×C₄	semidirect
129. C₂×#168	C₃×D₄	semidirect
133. #410	C₃×D₅	semidirect
275. C₃×C₅	#1910	semidirect
322. C₅	C₃×#1910	semidirect
337. C₃	#10600840	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×#66164	2
C₃×#66153	2
C₃×C₅×#410	2
C₃×#65290	2
C₃×#64751	2
#10600840	3
C₃×#1910	5

Subgroups — C3×#10600840 #14302828

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10600840 #14302828