Subgroups — C₃×#10602648 #14303041

Order 1920.

680 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×C₅×#535	C₂	1	1	yes	maximal, characteristic
2	C₃×#66153	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65526	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#3965	C₂	1	2	yes	maximal, complemented
5	C₃×#64753	C₂	1	2	yes	maximal, complemented
6	#10602648	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
9	C₃×C₅×#127	—	2	1	no	—
10	C₃×C₅×#124	—	2	1	no	—
11	C₃×C₅×#121	—	2	2	no	—
12	C₃×C₅×#120	C₄	1	4	yes	—
13	C₃×C₅×#117	C₂²	1	1	yes	characteristic
14	C₃×#4094	—	2	4	no	—
15	C₂×C₃×#681	—	2	2	no	—
16	C₂×C₃×#681	C₂²	1	2	yes	—
17	C₂×C₃×#681	C₂²	1	2	yes	—
18	C₃×#3968	—	2	1	no	—
19	C₃×#3965	—	2	1	no	—
20	C₃×#3965	—	2	4	no	—
21	C₃×#1914	—	5	1	no	maximal, Hall
22	C₅×#535	C₂×C₃	1	1	yes	characteristic
23	#66153	C₂×C₃	1	1	yes	characteristic, complemented
24	#65526	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂×#3965	C₂×C₃	1	2	yes	complemented
26	#64753	C₂×C₃	1	2	yes	complemented
27	C₂⁴×C₃×C₅	C₂×C₄	1	2	yes	—
28	C₃×C₅×#41	—	2	2	no	—
29	C₃×C₅×#41	—	2	2	no	—
30	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
31	C₂×C₃×C₅×D₄	—	2	4	no	—
32	C₂×C₃×C₅×D₄	—	2	2	no	—
33	C₂×C₃×C₅×D₄	—	2	1	no	—
34	C₂×C₃×C₅×D₄	—	2	1	no	—
35	C₂×C₃×C₅×D₄	—	2	2	no	—
36	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
37	C₂×C₃×C₅×D₄	D₄	1	1	yes	characteristic
38	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
39	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₃×C₅×C₄⋊C₄	—	2	1	no	—
41	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
42	C₃×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
43	C₃×C₄²×C₅	—	2	1	no	—
44	C₃×#681	—	2	4	no	—
45	C₃×#681	—	2	2	no	—
46	C₃×#681	—	2	2	no	—
47	C₃×#681	—	2	2	no	—
48	C₃×#681	D₄	1	4	yes	complemented
49	C₂×C₃×#168	—	2	4	no	—
50	C₂×C₃×#168	—	2	2	no	—
51	C₂×C₃×#168	—	2	2	no	—
52	C₃×#568	—	5	1	no	—
53	C₃×#535	D₅	1	1	yes	characteristic
54	C₃×#413	—	5	1	no	—
55	C₂×C₃×#100	—	5	2	no	—
56	C₃×#344	—	5	2	no	—
57	C₂×C₅×#41	C₂²×C₃	1	1	yes	characteristic
58	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
59	C₅×#127	—	2	1	no	—
60	C₅×#124	—	2	1	no	—
61	C₅×#121	—	2	2	no	—
62	C₅×#120	C₃×C₄	1	4	yes	—
63	C₅×#117	C₂²×C₃	1	1	yes	characteristic
64	#4094	—	2	4	no	—
65	C₂×#681	—	2	2	no	—
66	C₂×#681	C₂²×C₃	1	2	yes	—
67	C₂×#681	C₂²×C₃	1	2	yes	—
68	#3968	—	2	1	no	—
69	#3965	—	2	1	no	—
70	#3965	—	2	4	no	—
71	#1914	—	5	1	no	Sylow
72	C₂³×C₃×C₅	—	2	2	no	—
73	C₂³×C₃×C₅	—	2	4	no	—
74	C₂³×C₃×C₅	—	8	2	no	—
75	C₂³×C₃×C₅	C₂²⋊C₄	1	2	yes	—
76	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₃×C₅×Q₈	—	2	2	no	—
78	C₃×C₅×D₄	—	2	4	no	—
79	C₃×C₅×D₄	—	2	2	no	—
80	C₃×C₅×D₄	—	4	1	no	—
81	C₃×C₅×D₄	—	4	2	no	—
82	C₃×C₅×D₄	—	4	1	no	—
83	C₃×C₅×D₄	—	8	4	no	—
84	C₂×C₃×C₄×C₅	—	2	2	no	—
85	C₂×C₃×C₄×C₅	—	2	2	no	—
86	C₂×C₃×C₄×C₅	—	2	2	no	—
87	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
88	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
89	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
90	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
91	C₂×C₃×C₄×C₅	#41	1	2	yes	—
92	C₃×#168	—	2	4	no	—
93	C₃×#168	—	4	2	no	—
94	C₂×C₃×#41	C₂×D₅	1	1	yes	characteristic
95	C₂²×C₃×D₄	C₂×D₅	1	1	yes	characteristic
96	C₃×#131	—	10	4	no	—
97	C₂×C₃×M₁₆	—	5	2	no	—
98	C₂×C₃×M₁₆	—	5	2	no	—
99	C₂×C₃×M₁₆	—	10	2	no	—
100	C₃×#127	—	2	1	no	—
101	C₃×#124	—	2	1	no	—
102	C₃×#121	—	2	2	no	—
103	C₃×#120	Dic₅	1	4	yes	—
104	C₃×#117	C₂×D₅	1	1	yes	characteristic
105	C₃×#101	—	10	1	no	—
106	C₃×#100	—	10	4	no	—
107	C₃×#100	—	10	1	no	—
108	C₂⁴×C₅	C₂×C₃×C₄	1	2	yes	—
109	C₅×#41	—	2	2	no	—
110	C₅×#41	—	2	2	no	—
111	C₂×C₅×Q₈	C₃×D₄	1	1	yes	characteristic
112	C₂×C₅×D₄	—	2	4	no	—
113	C₂×C₅×D₄	—	2	1	no	—
114	C₂×C₅×D₄	—	2	1	no	—
115	C₂×C₅×D₄	—	2	2	no	—
116	C₂×C₅×D₄	—	2	2	no	—
117	C₂×C₅×D₄	C₂×C₃×C₄	1	2	yes	—
118	C₂×C₅×D₄	C₃×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
120	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
121	C₅×C₄⋊C₄	—	2	1	no	—
122	C₅×C₂²⋊C₄	—	2	4	no	—
123	C₅×C₂²⋊C₄	C₂×C₃×C₄	1	2	yes	—
124	C₄²×C₅	—	2	1	no	—
125	#681	—	2	4	no	—
126	#681	—	2	2	no	—
127	#681	—	2	2	no	—
128	#681	—	2	2	no	—
129	#681	C₃×D₄	1	4	yes	complemented
130	C₂×#168	—	2	4	no	—
131	C₂×#168	—	2	2	no	—
132	C₂×#168	—	2	2	no	—
133	#568	—	5	1	no	—
134	#535	C₃×D₅	1	1	yes	characteristic
135	#413	—	5	1	no	—
136	C₂×#100	—	5	2	no	—
137	#344	—	5	2	no	—
138	C₂²×C₃×C₅	—	2	2	no	—
139	C₂²×C₃×C₅	—	4	1	no	—
140	C₂²×C₃×C₅	—	4	2	no	—
141	C₂²×C₃×C₅	—	4	4	no	—
142	C₂²×C₃×C₅	—	4	2	no	—
143	C₂²×C₃×C₅	—	8	4	no	—
144	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
146	C₃×C₄×C₅	—	2	2	no	—
147	C₃×C₄×C₅	—	4	2	no	—
148	C₃×C₄×C₅	#120	1	1	yes	characteristic
149	C₃×C₄×C₅	#121	1	2	yes	—
150	C₃×C₄×C₅	#123	1	1	yes	characteristic
151	C₂⁴×C₃	C₂×Dic₅	1	2	yes	—
152	C₃×#41	—	2	2	no	—
153	C₃×#41	—	2	2	no	—
154	C₂×C₃×Q₈	C₂²⋊D₅	1	1	yes	characteristic
155	C₂×C₃×D₄	—	2	4	no	—
156	C₂×C₃×D₄	—	2	2	no	—
157	C₂×C₃×D₄	—	2	2	no	—
158	C₂×C₃×D₄	—	2	1	no	—
159	C₂×C₃×D₄	—	2	1	no	—
160	C₂×C₃×D₄	C₂×Dic₅	1	2	yes	—
161	C₂×C₃×D₄	C₂²⋊D₅	1	1	yes	characteristic
162	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
163	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
164	C₃×M₁₆	—	5	4	no	—
165	C₃×M₁₆	—	10	4	no	—
166	C₃×M₁₆	—	10	2	no	—
167	C₃×M₁₆	—	10	2	no	—
168	C₃×M₁₆	—	10	2	no	—
169	C₂×C₃×C₈	—	10	4	no	—
170	C₂×C₃×C₈	—	10	2	no	—
171	C₂×C₃×C₈	—	10	2	no	—
172	C₃×C₄⋊C₄	—	2	1	no	—
173	C₃×C₂²⋊C₄	—	2	4	no	—
174	C₃×C₂²⋊C₄	C₂×Dic₅	1	2	yes	—
175	C₃×C₄²	—	2	1	no	—
176	C₂³×C₅	—	2	2	no	—
177	C₂³×C₅	—	2	4	no	—
178	C₂³×C₅	—	8	2	no	—
179	C₂³×C₅	C₃×C₂²⋊C₄	1	2	yes	—
180	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
181	C₅×Q₈	—	2	2	no	—
182	C₅×D₄	—	2	4	no	—
183	C₅×D₄	—	2	2	no	—
184	C₅×D₄	—	4	1	no	—
185	C₅×D₄	—	4	2	no	—
186	C₅×D₄	—	4	1	no	—
187	C₅×D₄	—	8	4	no	—
188	C₂×C₄×C₅	—	2	2	no	—
189	C₂×C₄×C₅	—	2	2	no	—
190	C₂×C₄×C₅	—	2	2	no	—
191	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
192	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
193	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
194	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
195	C₂×C₄×C₅	C₃×#41	1	2	yes	—
196	#168	—	2	4	no	—
197	#168	—	4	2	no	—
198	C₂×#41	C₂×C₃×D₅	1	1	yes	characteristic
199	C₂²×D₄	C₂×C₃×D₅	1	1	yes	characteristic
200	#131	—	10	4	no	—
201	C₂×M₁₆	—	5	2	no	—
202	C₂×M₁₆	—	5	2	no	—
203	C₂×M₁₆	—	10	2	no	—
204	#127	—	2	1	no	—
205	#124	—	2	1	no	—
206	#121	—	2	2	no	—
207	#120	C₃×Dic₅	1	4	yes	—
208	#117	C₂×C₃×D₅	1	1	yes	characteristic
209	#101	—	10	1	no	—
210	#100	—	10	4	no	—
211	#100	—	10	1	no	—
212	C₂×C₃×C₅	—	2	1	no	—
213	C₂×C₃×C₅	—	2	2	no	—
214	C₂×C₃×C₅	—	4	2	no	—
215	C₂×C₃×C₅	—	8	2	no	—
216	C₂×C₃×C₅	#386	1	1	yes	characteristic
217	C₂³×C₃	—	2	2	no	—
218	C₂³×C₃	—	2	4	no	—
219	C₂³×C₃	—	8	2	no	—
220	C₂³×C₃	C₂²⋊Dic₅	1	2	yes	—
221	C₂³×C₃	C₂²×Dic₅	1	1	yes	characteristic
222	C₃×Q₈	—	2	2	no	—
223	C₃×D₄	—	2	4	no	—
224	C₃×D₄	—	2	2	no	—
225	C₃×D₄	—	4	1	no	—
226	C₃×D₄	—	4	1	no	—
227	C₃×D₄	—	4	2	no	—
228	C₃×D₄	—	8	4	no	—
229	C₂×C₃×C₄	—	2	2	no	—
230	C₂×C₃×C₄	—	2	2	no	—
231	C₂×C₃×C₄	—	2	2	no	—
232	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
233	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
234	C₂×C₃×C₄	#711	1	2	yes	—
235	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
236	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
237	C₃×C₈	—	10	4	no	—
238	C₃×C₈	—	20	2	no	—
239	C₂²×C₅	—	2	2	no	—
240	C₂²×C₅	—	4	1	no	—
241	C₂²×C₅	—	4	2	no	—
242	C₂²×C₅	—	4	4	no	—
243	C₂²×C₅	—	4	2	no	—
244	C₂²×C₅	—	8	4	no	—
245	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
246	C₂²×C₅	C₃×C₄×D₄	1	2	yes	—
247	C₄×C₅	—	2	2	no	—
248	C₄×C₅	—	4	2	no	—
249	C₄×C₅	C₃×#120	1	1	yes	characteristic
250	C₄×C₅	C₃×#121	1	2	yes	—
251	C₄×C₅	C₃×#123	1	1	yes	characteristic
252	C₂⁴	C₂×C₃×Dic₅	1	2	yes	—
253	#41	—	2	2	no	—
254	#41	—	2	2	no	—
255	C₂×Q₈	C₃×C₂²⋊D₅	1	1	yes	characteristic
256	C₂×D₄	—	2	4	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	2	1	no	—
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	C₂×C₃×Dic₅	1	2	yes	—
262	C₂×D₄	C₃×C₂²⋊D₅	1	1	yes	characteristic
263	C₂²×C₄	C₃×C₂²⋊D₅	1	2	yes	—
264	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
265	M₁₆	—	5	4	no	—
266	M₁₆	—	10	4	no	—
267	M₁₆	—	10	2	no	—
268	M₁₆	—	10	2	no	—
269	M₁₆	—	10	2	no	—
270	C₂×C₈	—	10	4	no	—
271	C₂×C₈	—	10	2	no	—
272	C₂×C₈	—	10	2	no	—
273	C₄⋊C₄	—	2	1	no	—
274	C₂²⋊C₄	—	2	4	no	—
275	C₂²⋊C₄	C₂×C₃×Dic₅	1	2	yes	—
276	C₄²	—	2	1	no	—
277	C₃×C₅	#1914	1	1	yes	characteristic, complemented, Hall
278	C₂²×C₃	—	2	2	no	—
279	C₂²×C₃	—	4	1	no	—
280	C₂²×C₃	—	4	2	no	—
281	C₂²×C₃	—	4	4	no	—
282	C₂²×C₃	—	4	2	no	—
283	C₂²×C₃	—	8	4	no	—
284	C₂²×C₃	D₄×Dic₅	1	2	yes	—
285	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
286	C₃×C₄	—	2	2	no	—
287	C₃×C₄	—	4	2	no	—
288	C₃×C₄	#4081	1	1	yes	characteristic
289	C₃×C₄	#4083	1	1	yes	characteristic
290	C₃×C₄	#4085	1	2	yes	—
291	C₂×C₅	—	2	1	no	—
292	C₂×C₅	—	2	2	no	—
293	C₂×C₅	—	4	2	no	—
294	C₂×C₅	—	8	2	no	—
295	C₂×C₅	C₃×#386	1	1	yes	characteristic
296	C₂³	—	2	2	no	—
297	C₂³	—	2	4	no	—
298	C₂³	—	8	2	no	—
299	C₂³	C₃×C₂²⋊Dic₅	1	2	yes	—
300	C₂³	C₂²×C₃×Dic₅	1	1	yes	characteristic
301	Q₈	—	2	2	no	—
302	D₄	—	2	4	no	—
303	D₄	—	2	2	no	—
304	D₄	—	4	1	no	—
305	D₄	—	4	1	no	—
306	D₄	—	4	2	no	—
307	D₄	—	8	4	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	C₃×C₂²⋊Dic₅	1	2	yes	—
312	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
313	C₂×C₄	C₃×#711	1	2	yes	—
314	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
315	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
316	C₈	—	10	4	no	—
317	C₈	—	20	2	no	—
318	C₂×C₃	—	2	1	no	—
319	C₂×C₃	—	2	2	no	—
320	C₂×C₃	—	4	2	no	—
321	C₂×C₃	—	8	2	no	—
322	C₂×C₃	#65509	1	1	yes	characteristic
323	C₅	C₃×#1914	1	1	yes	characteristic, complemented, Sylow
324	C₂²	—	2	2	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	4	2	no	—
327	C₂²	—	4	4	no	—
328	C₂²	—	4	2	no	—
329	C₂²	—	8	4	no	—
330	C₂²	C₃×D₄×Dic₅	1	2	yes	—
331	C₂²	C₂×C₃×C₂²⋊Dic₅	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	4	2	no	—
334	C₄	C₃×#4081	1	1	yes	characteristic
335	C₄	C₃×#4083	1	1	yes	characteristic
336	C₄	C₃×#4085	1	2	yes	—
337	C₃	#10602648	1	1	yes	characteristic, direct product factor, Sylow
338	C₂	—	2	1	no	—
339	C₂	—	2	2	no	—
340	C₂	—	4	2	no	—
341	C₂	—	8	2	no	—
342	C₂	C₃×#65509	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	#65509
commutator	C₂³×C₅	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting, prime radical	C₃×C₅×#535	C₂
socle	C₂×C₃×C₅	#386
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₃×#535	D₅
hypocenter	C₅	C₃×#1914
LCS	G ▷ C₂³×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×#535

Complements

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

N	H ≅ G/N	type
2. C₃×#66153	C₂	semidirect
3. C₃×#65526	C₂	semidirect
4. C₂×C₃×#3965	C₂	semidirect
5. C₃×#64753	C₂	semidirect
6. #10602648	C₃	direct (N × H)
23. #66153	C₂×C₃	semidirect
24. #65526	C₂×C₃	semidirect
25. C₂×#3965	C₂×C₃	semidirect
26. #64753	C₂×C₃	semidirect
48. C₃×#681	D₄	semidirect
129. #681	C₃×D₄	semidirect
277. C₃×C₅	#1914	semidirect
323. C₅	C₃×#1914	semidirect
337. C₃	#10602648	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×C₅×#535	2
C₃×#66153	2
C₃×#65526	2
C₂×C₃×#3965	2
C₃×#64753	2
#10602648	3
C₃×#1914	5

Subgroups — C3×#10602648 #14303041

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10602648 #14303041