Subgroups — C₃×#10600864 #14302879

Order 1920.

724 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#66139	C₂	1	1	yes	maximal, characteristic
2	C₃×#66132	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₅×#100	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65428	C₂	1	2	yes	maximal, complemented
5	C₃×#64777	C₂	1	2	yes	maximal, complemented
6	#10600864	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
8	C₂×C₃×#709	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
10	C₃×#4099	—	2	2	no	—
11	C₂×C₃×C₂²⋊Dic₅	—	2	2	no	—
12	C₃×#4086	C₄	1	2	yes	complemented
13	C₃×#4084	—	2	2	no	—
14	C₃×#4082	C₄	1	2	yes	complemented
15	C₃×D₄×Dic₅	—	2	4	no	—
16	C₃×#4072	C₂²	1	1	yes	characteristic
17	C₃×#4072	C₂²	1	2	yes	—
18	C₃×C₅×#100	—	2	2	no	—
19	C₃×#3957	—	2	4	no	—
20	C₃×#1911	—	5	1	no	maximal, Hall
21	#66139	C₂×C₃	1	1	yes	characteristic
22	#66132	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂×C₅×#100	C₂×C₃	1	1	yes	characteristic, complemented
24	#65428	C₂×C₃	1	2	yes	complemented
25	#64777	C₂×C₃	1	2	yes	complemented
26	C₂⁴×C₃×C₅	D₄	1	2	yes	—
27	C₂×C₃×C₅×D₄	—	2	2	no	—
28	C₂×C₃×C₅×D₄	—	2	2	no	—
29	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₅×D₄	D₄	1	2	yes	—
31	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂×C₃×C₂²⋊D₅	—	2	1	no	—
33	C₂×C₃×C₂²⋊D₅	—	2	4	no	—
34	C₂²×C₃×Dic₅	—	2	4	no	—
35	C₃×#709	—	2	2	no	—
36	C₃×#709	—	2	2	no	—
37	C₂×C₃×D₂₀	C₂×C₄	1	1	yes	characteristic, complemented
38	C₂×C₃×C₄×D₅	—	2	1	no	—
39	C₂×C₃×Dic₁₀	C₂×C₄	1	1	yes	characteristic, complemented
40	C₃×C₅×M₁₆	—	2	2	no	—
41	C₃×C₅×M₁₆	—	2	2	no	—
42	C₂×C₃×C₅×C₈	—	2	2	no	—
43	C₃×C₂²⋊Dic₅	—	2	4	no	—
44	C₃×C₂²⋊Dic₅	—	2	4	no	—
45	C₃×C₂²⋊Dic₅	—	2	2	no	—
46	C₃×C₂²⋊Dic₅	—	2	1	no	—
47	C₃×C₄⋊Dic₅	—	2	2	no	—
48	C₃×C₄⋊Dic₅	—	2	1	no	—
49	C₃×C₄×Dic₅	C₂×C₄	1	2	yes	complemented
50	C₃×C₄×Dic₅	D₄	1	4	yes	complemented
51	C₃×#535	—	5	1	no	—
52	C₃×#518	—	5	1	no	—
53	C₃×#421	—	5	2	no	—
54	C₂×C₃×#100	D₅	1	1	yes	characteristic, complemented
55	C₃×H(3,ℤ/4)	—	5	2	no	—
56	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
57	C₂×#709	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
59	#4099	—	2	2	no	—
60	C₂×C₂²⋊Dic₅	—	2	2	no	—
61	#4086	C₃×C₄	1	2	yes	complemented
62	#4084	—	2	2	no	—
63	#4082	C₃×C₄	1	2	yes	complemented
64	D₄×Dic₅	—	2	4	no	—
65	#4072	C₂²×C₃	1	1	yes	characteristic
66	#4072	C₂²×C₃	1	2	yes	—
67	C₅×#100	—	2	2	no	—
68	#3957	—	2	4	no	—
69	#1911	—	5	1	no	Sylow
70	C₂³×C₃×C₅	—	2	2	no	—
71	C₂³×C₃×C₅	—	2	2	no	—
72	C₂³×C₃×C₅	—	8	2	no	—
73	C₂³×C₃×C₅	C₂²⋊C₄	1	4	yes	—
74	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₃×D₅	—	2	1	no	—
76	C₃×C₅×D₄	—	2	4	no	—
77	C₃×C₅×D₄	—	2	4	no	—
78	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
80	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	#41	1	2	yes	—
82	C₃×C₂²⋊D₅	—	4	1	no	—
83	C₃×C₂²⋊D₅	—	4	1	no	—
84	C₃×C₂²⋊D₅	—	8	4	no	—
85	C₂×C₃×Dic₅	—	2	2	no	—
86	C₂×C₃×Dic₅	—	2	1	no	—
87	C₂×C₃×Dic₅	—	2	4	no	—
88	C₂×C₃×Dic₅	—	8	4	no	—
89	C₃×D₂₀	—	2	2	no	—
90	C₃×C₄×D₅	—	4	2	no	—
91	C₃×Dic₁₀	—	2	2	no	—
92	C₃×C₅×C₈	—	4	2	no	—
93	C₂×C₃×#41	—	5	1	no	—
94	C₂²×C₃×D₄	C₂×D₅	1	1	yes	characteristic
95	C₂×C₃×M₁₆	C₂×D₅	1	2	yes	—
96	C₃×#127	—	5	2	no	—
97	C₃×#124	—	5	2	no	—
98	C₃×#121	—	10	2	no	—
99	C₃×#120	—	10	2	no	—
100	C₃×C₄×D₄	—	10	4	no	—
101	C₃×#117	—	5	2	no	—
102	C₃×#117	—	5	1	no	—
103	C₂×C₃×C₂²⋊C₄	—	10	2	no	—
104	C₃×#104	—	10	4	no	—
105	C₃×#100	—	2	2	no	—
106	C₂⁴×C₅	C₃×D₄	1	2	yes	—
107	C₂×C₅×D₄	—	2	2	no	—
108	C₂×C₅×D₄	—	2	2	no	—
109	C₂×C₅×D₄	C₂×C₃×C₄	1	2	yes	—
110	C₂×C₅×D₄	C₃×D₄	1	2	yes	—
111	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
112	C₂×C₂²⋊D₅	—	2	1	no	—
113	C₂×C₂²⋊D₅	—	2	4	no	—
114	C₂²×Dic₅	—	2	4	no	—
115	#709	—	2	2	no	—
116	#709	—	2	2	no	—
117	C₂×D₂₀	C₂×C₃×C₄	1	1	yes	characteristic, complemented
118	C₂×C₄×D₅	—	2	1	no	—
119	C₂×Dic₁₀	C₂×C₃×C₄	1	1	yes	characteristic, complemented
120	C₅×M₁₆	—	2	2	no	—
121	C₅×M₁₆	—	2	2	no	—
122	C₂×C₅×C₈	—	2	2	no	—
123	C₂²⋊Dic₅	—	2	4	no	—
124	C₂²⋊Dic₅	—	2	4	no	—
125	C₂²⋊Dic₅	—	2	2	no	—
126	C₂²⋊Dic₅	—	2	1	no	—
127	C₄⋊Dic₅	—	2	2	no	—
128	C₄⋊Dic₅	—	2	1	no	—
129	C₄×Dic₅	C₂×C₃×C₄	1	2	yes	complemented
130	C₄×Dic₅	C₃×D₄	1	4	yes	complemented
131	#535	—	5	1	no	—
132	#518	—	5	1	no	—
133	#421	—	5	2	no	—
134	C₂×#100	C₃×D₅	1	1	yes	characteristic, complemented
135	H(3,ℤ/4)	—	5	2	no	—
136	C₂²×C₃×C₅	—	2	4	no	—
137	C₂²×C₃×C₅	—	4	1	no	—
138	C₂²×C₃×C₅	—	4	4	no	—
139	C₂²×C₃×C₅	—	4	4	no	—
140	C₂²×C₃×C₅	—	8	2	no	—
141	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×C₃×C₅	#120	1	1	yes	characteristic
143	C₂²×C₃×C₅	#123	1	1	yes	characteristic
144	C₂×C₃×D₅	—	4	2	no	—
145	C₂×C₃×D₅	—	4	1	no	—
146	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
147	C₃×C₄×C₅	#121	1	2	yes	—
148	C₃×Dic₅	—	4	4	no	—
149	C₃×Dic₅	—	4	2	no	—
150	C₃×Dic₅	—	4	1	no	—
151	C₂⁴×C₃	D₂₀	1	2	yes	—
152	C₃×#41	—	10	2	no	—
153	C₃×#41	—	10	2	no	—
154	C₂×C₃×Q₈	—	5	1	no	—
155	C₂×C₃×D₄	—	2	2	no	—
156	C₂×C₃×D₄	—	2	2	no	—
157	C₂×C₃×D₄	—	5	1	no	—
158	C₂×C₃×D₄	—	10	1	no	—
159	C₂×C₃×D₄	—	10	4	no	—
160	C₂×C₃×D₄	C₄×D₅	1	2	yes	—
161	C₂×C₃×D₄	C₂²⋊D₅	1	2	yes	—
162	C₂²×C₃×C₄	—	10	4	no	—
163	C₂²×C₃×C₄	—	10	1	no	—
164	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
165	C₃×M₁₆	—	2	2	no	—
166	C₃×M₁₆	—	2	2	no	—
167	C₂×C₃×C₈	—	2	2	no	—
168	C₃×C₄⋊C₄	—	10	2	no	—
169	C₃×C₄⋊C₄	—	10	1	no	—
170	C₃×C₂²⋊C₄	—	10	4	no	—
171	C₃×C₂²⋊C₄	—	10	4	no	—
172	C₃×C₂²⋊C₄	—	10	2	no	—
173	C₃×C₂²⋊C₄	—	10	1	no	—
174	C₃×C₄²	—	5	2	no	—
175	C₃×C₄²	—	5	4	no	—
176	C₂³×C₅	—	2	2	no	—
177	C₂³×C₅	—	2	2	no	—
178	C₂³×C₅	—	8	2	no	—
179	C₂³×C₅	C₃×C₂²⋊C₄	1	4	yes	—
180	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
181	C₂²×D₅	—	2	1	no	—
182	C₅×D₄	—	2	4	no	—
183	C₅×D₄	—	2	4	no	—
184	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
185	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
186	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
187	C₂×C₄×C₅	C₃×#41	1	2	yes	—
188	C₂²⋊D₅	—	4	1	no	—
189	C₂²⋊D₅	—	4	1	no	—
190	C₂²⋊D₅	—	8	4	no	—
191	C₂×Dic₅	—	2	2	no	—
192	C₂×Dic₅	—	2	1	no	—
193	C₂×Dic₅	—	2	4	no	—
194	C₂×Dic₅	—	8	4	no	—
195	D₂₀	—	2	2	no	—
196	C₄×D₅	—	4	2	no	—
197	Dic₁₀	—	2	2	no	—
198	C₅×C₈	—	4	2	no	—
199	C₂×#41	—	5	1	no	—
200	C₂²×D₄	C₂×C₃×D₅	1	1	yes	characteristic
201	C₂×M₁₆	C₂×C₃×D₅	1	2	yes	—
202	#127	—	5	2	no	—
203	#124	—	5	2	no	—
204	#121	—	10	2	no	—
205	#120	—	10	2	no	—
206	C₄×D₄	—	10	4	no	—
207	#117	—	5	2	no	—
208	#117	—	5	1	no	—
209	C₂×C₂²⋊C₄	—	10	2	no	—
210	#104	—	10	4	no	—
211	#100	—	2	2	no	—
212	C₂×C₃×C₅	—	2	1	no	—
213	C₂×C₃×C₅	—	2	1	no	—
214	C₂×C₃×C₅	—	2	1	no	—
215	C₂×C₃×C₅	—	4	4	no	—
216	C₂×C₃×C₅	#386	1	1	yes	characteristic
217	C₃×D₅	—	8	1	no	—
218	C₂³×C₃	—	2	2	no	—
219	C₂³×C₃	—	2	2	no	—
220	C₂³×C₃	—	8	2	no	—
221	C₂³×C₃	—	10	1	no	—
222	C₂³×C₃	#685	1	4	yes	—
223	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
224	C₃×Q₈	—	10	2	no	—
225	C₃×D₄	—	2	4	no	—
226	C₃×D₄	—	2	4	no	—
227	C₃×D₄	—	10	2	no	—
228	C₃×D₄	—	20	1	no	—
229	C₃×D₄	—	20	1	no	—
230	C₃×D₄	—	40	4	no	—
231	C₂×C₃×C₄	—	10	4	no	—
232	C₂×C₃×C₄	—	10	2	no	—
233	C₂×C₃×C₄	—	10	1	no	—
234	C₂×C₃×C₄	—	20	2	no	—
235	C₂×C₃×C₄	—	40	4	no	—
236	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
237	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
238	C₂×C₃×C₄	#711	1	2	yes	—
239	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
240	C₃×C₈	—	4	2	no	—
241	C₂²×C₅	—	2	4	no	—
242	C₂²×C₅	—	4	1	no	—
243	C₂²×C₅	—	4	4	no	—
244	C₂²×C₅	—	4	4	no	—
245	C₂²×C₅	—	8	2	no	—
246	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
247	C₂²×C₅	C₃×#120	1	1	yes	characteristic
248	C₂²×C₅	C₃×#123	1	1	yes	characteristic
249	C₂×D₅	—	4	2	no	—
250	C₂×D₅	—	4	1	no	—
251	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
252	C₄×C₅	C₃×#121	1	2	yes	—
253	Dic₅	—	4	4	no	—
254	Dic₅	—	4	2	no	—
255	Dic₅	—	4	1	no	—
256	C₂⁴	C₃×D₂₀	1	2	yes	—
257	#41	—	10	2	no	—
258	#41	—	10	2	no	—
259	C₂×Q₈	—	5	1	no	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	5	1	no	—
263	C₂×D₄	—	10	1	no	—
264	C₂×D₄	—	10	4	no	—
265	C₂×D₄	C₃×C₄×D₅	1	2	yes	—
266	C₂×D₄	C₃×C₂²⋊D₅	1	2	yes	—
267	C₂²×C₄	—	10	4	no	—
268	C₂²×C₄	—	10	1	no	—
269	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
270	M₁₆	—	2	2	no	—
271	M₁₆	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₄⋊C₄	—	10	2	no	—
274	C₄⋊C₄	—	10	1	no	—
275	C₂²⋊C₄	—	10	4	no	—
276	C₂²⋊C₄	—	10	4	no	—
277	C₂²⋊C₄	—	10	2	no	—
278	C₂²⋊C₄	—	10	1	no	—
279	C₄²	—	5	2	no	—
280	C₄²	—	5	4	no	—
281	C₃×C₅	#1911	1	1	yes	characteristic, complemented, Hall
282	C₂²×C₃	—	2	4	no	—
283	C₂²×C₃	—	4	1	no	—
284	C₂²×C₃	—	4	4	no	—
285	C₂²×C₃	—	4	4	no	—
286	C₂²×C₃	—	8	2	no	—
287	C₂²×C₃	—	20	2	no	—
288	C₂²×C₃	—	20	1	no	—
289	C₂²×C₃	#4028	1	1	yes	characteristic
290	C₂²×C₃	#4032	1	1	yes	characteristic
291	C₂²×C₃	C₂×#685	1	1	yes	characteristic
292	C₃×C₄	—	20	4	no	—
293	C₃×C₄	—	20	2	no	—
294	C₃×C₄	—	20	1	no	—
295	C₃×C₄	#4027	1	2	yes	—
296	C₃×C₄	#4085	1	2	yes	—
297	C₂×C₅	—	2	1	no	—
298	C₂×C₅	—	2	1	no	—
299	C₂×C₅	—	2	1	no	—
300	C₂×C₅	—	4	4	no	—
301	C₂×C₅	C₃×#386	1	1	yes	characteristic
302	D₅	—	8	1	no	—
303	C₂³	—	2	2	no	—
304	C₂³	—	2	2	no	—
305	C₂³	—	8	2	no	—
306	C₂³	—	10	1	no	—
307	C₂³	C₃×#685	1	4	yes	—
308	C₂³	C₂×C₃×D₂₀	1	1	yes	characteristic
309	Q₈	—	10	2	no	—
310	D₄	—	2	4	no	—
311	D₄	—	2	4	no	—
312	D₄	—	10	2	no	—
313	D₄	—	20	1	no	—
314	D₄	—	20	1	no	—
315	D₄	—	40	4	no	—
316	C₂×C₄	—	10	4	no	—
317	C₂×C₄	—	10	2	no	—
318	C₂×C₄	—	10	1	no	—
319	C₂×C₄	—	20	2	no	—
320	C₂×C₄	—	40	4	no	—
321	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
322	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
323	C₂×C₄	C₃×#711	1	2	yes	—
324	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
325	C₈	—	4	2	no	—
326	C₂×C₃	—	2	1	no	—
327	C₂×C₃	—	2	1	no	—
328	C₂×C₃	—	2	1	no	—
329	C₂×C₃	—	4	4	no	—
330	C₂×C₃	—	40	1	no	—
331	C₂×C₃	#65247	1	1	yes	characteristic
332	C₅	C₃×#1911	1	1	yes	characteristic, complemented, Sylow
333	C₂²	—	2	4	no	—
334	C₂²	—	4	1	no	—
335	C₂²	—	4	4	no	—
336	C₂²	—	4	4	no	—
337	C₂²	—	8	2	no	—
338	C₂²	—	20	2	no	—
339	C₂²	—	20	1	no	—
340	C₂²	C₃×#4028	1	1	yes	characteristic
341	C₂²	C₃×#4032	1	1	yes	characteristic
342	C₂²	C₂×C₃×#685	1	1	yes	characteristic
343	C₄	—	20	4	no	—
344	C₄	—	20	2	no	—
345	C₄	—	20	1	no	—
346	C₄	C₃×#4027	1	2	yes	—
347	C₄	C₃×#4085	1	2	yes	—
348	C₃	#10600864	1	1	yes	characteristic, direct product factor, Sylow
349	C₂	—	2	1	no	—
350	C₂	—	2	1	no	—
351	C₂	—	2	1	no	—
352	C₂	—	4	4	no	—
353	C₂	—	40	1	no	—
354	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	C₂×C₃×C₅×#100	C₂
socle	C₂×C₃×C₅	#386
prime radical	C₃×#66139	C₂
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₂×C₃×#100	D₅
hypocenter	C₅	C₃×#1911
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#100

Complements

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

N	H ≅ G/N	type
2. C₃×#66132	C₂	semidirect
3. C₂×C₃×C₅×#100	C₂	semidirect
4. C₃×#65428	C₂	semidirect
5. C₃×#64777	C₂	semidirect
6. #10600864	C₃	direct (N × H)
12. C₃×#4086	C₄	semidirect
14. C₃×#4082	C₄	semidirect
22. #66132	C₂×C₃	semidirect
23. C₂×C₅×#100	C₂×C₃	semidirect
24. #65428	C₂×C₃	semidirect
25. #64777	C₂×C₃	semidirect
37. C₂×C₃×D₂₀	C₂×C₄	semidirect
39. C₂×C₃×Dic₁₀	C₂×C₄	semidirect
49. C₃×C₄×Dic₅	C₂×C₄	semidirect
50. C₃×C₄×Dic₅	D₄	semidirect
54. C₂×C₃×#100	D₅	semidirect
61. #4086	C₃×C₄	semidirect
63. #4082	C₃×C₄	semidirect
117. C₂×D₂₀	C₂×C₃×C₄	semidirect
119. C₂×Dic₁₀	C₂×C₃×C₄	semidirect
129. C₄×Dic₅	C₂×C₃×C₄	semidirect
130. C₄×Dic₅	C₃×D₄	semidirect
134. C₂×#100	C₃×D₅	semidirect
281. C₃×C₅	#1911	semidirect
332. C₅	C₃×#1911	semidirect
348. C₃	#10600864	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×#66139	2
C₃×#66132	2
C₂×C₃×C₅×#100	2
C₃×#65428	2
C₃×#64777	2
#10600864	3
C₃×#1911	5

Subgroups — C3×#10600864 #14302879

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10600864 #14302879