Subgroups — C₃×#10600223 #14295634

Order 1920.

640 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×#709	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#376	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3937	C₂	1	2	yes	maximal
4	C₃×C₄×#681	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64743	C₂	1	2	yes	maximal, complemented
6	#10600223	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×#709	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₃×C₄×C₂²⋊D₅	—	2	2	no	—
11	C₃×#4072	C₂²	1	1	yes	characteristic
12	C₂×C₃×#681	C₂²	1	2	yes	—
13	C₃×C₄×D₂₀	C₄	1	2	yes	complemented
14	C₃×#4018	—	2	1	no	—
15	C₃×C₄²×D₅	—	2	1	no	—
16	C₃×C₄×Dic₁₀	C₄	1	2	yes	complemented
17	C₃×C₅×#95	—	2	2	no	—
18	C₃×#3937	C₄	1	8	yes	complemented
19	C₃×#3935	—	2	1	no	—
20	C₃×C₄×#168	—	2	1	no	—
21	C₃×#1759	—	5	1	no	maximal, Hall
22	C₄×#709	C₂×C₃	1	1	yes	characteristic
23	C₅×#376	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×#3937	C₂×C₃	1	2	yes	—
25	C₄×#681	C₂×C₃	1	1	yes	characteristic, complemented
26	#64743	C₂×C₃	1	2	yes	complemented
27	C₂²×C₃×C₄×C₅	—	2	2	no	—
28	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
29	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
30	C₂×C₃×C₂²⋊D₅	—	2	1	no	—
31	C₃×#709	—	2	2	no	—
32	C₃×#709	C₂×C₄	1	4	yes	—
33	C₂×C₃×D₂₀	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃×C₄×D₅	—	2	1	no	—
35	C₂×C₃×C₄×D₅	—	2	2	no	—
36	C₂×C₃×Dic₁₀	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄²×C₅	—	2	2	no	—
38	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
39	C₃×C₄²×C₅	C₂×C₄	1	4	yes	complemented
40	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₃×C₂²⋊Dic₅	—	2	1	no	—
43	C₃×#685	—	2	2	no	—
44	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
45	C₃×#683	—	2	2	no	—
46	C₃×C₄×Dic₅	—	2	2	no	—
47	C₃×C₄×Dic₅	—	2	1	no	—
48	C₃×#681	—	2	2	no	—
49	C₃×#681	C₂×C₄	1	4	yes	complemented
50	C₂×C₃×#168	—	2	2	no	—
51	C₃×C₄×#41	—	5	1	no	—
52	C₂×C₃×#104	—	5	2	no	—
53	C₃×C₄×M₁₆	—	5	1	no	—
54	C₃×#376	D₅	1	1	yes	characteristic, complemented
55	C₃×#339	—	5	2	no	—
56	C₂×#709	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₄²×C₅	C₂²×C₃	1	2	yes	—
58	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
59	C₄×C₂²⋊D₅	—	2	2	no	—
60	#4072	C₂²×C₃	1	1	yes	characteristic
61	C₂×#681	C₂²×C₃	1	2	yes	—
62	C₄×D₂₀	C₃×C₄	1	2	yes	complemented
63	#4018	—	2	1	no	—
64	C₄²×D₅	—	2	1	no	—
65	C₄×Dic₁₀	C₃×C₄	1	2	yes	complemented
66	C₅×#95	—	2	2	no	—
67	#3937	C₃×C₄	1	8	yes	complemented
68	#3935	—	2	1	no	—
69	C₄×#168	—	2	1	no	—
70	#1759	—	5	1	no	Sylow
71	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₃×D₅	—	2	1	no	—
73	C₂×C₃×C₄×C₅	—	2	4	no	—
74	C₂×C₃×C₄×C₅	—	4	2	no	—
75	C₂×C₃×C₄×C₅	—	4	4	no	—
76	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
77	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
78	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂²×C₄	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	—	2	4	no	—
83	C₂×C₃×Dic₅	—	2	1	no	—
84	C₂×C₃×Dic₅	—	2	2	no	—
85	C₃×D₂₀	—	2	1	no	—
86	C₃×D₂₀	C₄²	1	2	yes	—
87	C₃×C₄×D₅	—	2	4	no	—
88	C₃×C₄×D₅	—	4	4	no	—
89	C₃×Dic₁₀	—	2	1	no	—
90	C₃×Dic₁₀	C₄²	1	2	yes	—
91	C₃×#168	—	2	4	no	—
92	C₂×C₃×#41	—	5	1	no	—
93	C₂×C₃×M₁₆	—	5	2	no	—
94	C₃×C₄×Q₈	—	5	2	no	—
95	C₃×C₄×D₄	—	5	2	no	—
96	C₃×C₄×D₄	—	10	2	no	—
97	C₃×#117	—	5	1	no	—
98	C₃×#117	—	10	1	no	—
99	C₂×C₃×C₄²	—	10	1	no	—
100	C₂×C₃×C₄²	C₂×D₅	1	2	yes	—
101	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
102	C₃×#104	—	5	8	no	—
103	C₃×#97	—	10	1	no	—
104	C₃×C₄×C₈	—	10	1	no	—
105	C₃×#95	—	2	2	no	—
106	C₂²×C₄×C₅	—	2	2	no	—
107	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
108	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
109	C₂×C₂²⋊D₅	—	2	1	no	—
110	#709	—	2	2	no	—
111	#709	C₂×C₃×C₄	1	4	yes	—
112	C₂×D₂₀	C₂×C₃×C₄	1	1	yes	characteristic
113	C₂×C₄×D₅	—	2	1	no	—
114	C₂×C₄×D₅	—	2	2	no	—
115	C₂×Dic₁₀	C₂×C₃×C₄	1	1	yes	characteristic
116	C₄²×C₅	—	2	2	no	—
117	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
118	C₄²×C₅	C₂×C₃×C₄	1	4	yes	complemented
119	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
120	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
121	C₂²⋊Dic₅	—	2	1	no	—
122	#685	—	2	2	no	—
123	C₄⋊Dic₅	C₂×C₃×C₄	1	2	yes	—
124	#683	—	2	2	no	—
125	C₄×Dic₅	—	2	2	no	—
126	C₄×Dic₅	—	2	1	no	—
127	#681	—	2	2	no	—
128	#681	C₂×C₃×C₄	1	4	yes	complemented
129	C₂×#168	—	2	2	no	—
130	C₄×#41	—	5	1	no	—
131	C₂×#104	—	5	2	no	—
132	C₄×M₁₆	—	5	1	no	—
133	#376	C₃×D₅	1	1	yes	characteristic, complemented
134	#339	—	5	2	no	—
135	C₂²×C₃×C₅	—	2	2	no	—
136	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
138	C₂×C₃×D₅	—	2	2	no	—
139	C₂×C₃×D₅	—	4	2	no	—
140	C₃×C₄×C₅	—	2	2	no	—
141	C₃×C₄×C₅	—	2	2	no	—
142	C₃×C₄×C₅	—	4	4	no	—
143	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
144	C₃×C₄×C₅	#117	1	1	yes	characteristic
145	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
146	C₃×Dic₅	—	2	2	no	—
147	C₃×Dic₅	—	4	2	no	—
148	C₃×#41	—	5	4	no	—
149	C₃×#41	—	10	2	no	—
150	C₂×C₃×Q₈	—	5	1	no	—
151	C₂×C₃×D₄	—	5	1	no	—
152	C₂×C₃×D₄	—	10	1	no	—
153	C₂²×C₃×C₄	—	2	2	no	—
154	C₂²×C₃×C₄	—	10	1	no	—
155	C₂²×C₃×C₄	—	10	2	no	—
156	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
157	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
158	C₃×M₁₆	—	5	4	no	—
159	C₃×M₁₆	—	10	2	no	—
160	C₂×C₃×C₈	—	10	2	no	—
161	C₃×C₄⋊C₄	—	5	2	no	—
162	C₃×C₄⋊C₄	—	10	2	no	—
163	C₃×C₂²⋊C₄	—	10	1	no	—
164	C₃×C₂²⋊C₄	—	10	2	no	—
165	C₃×C₄²	—	2	2	no	—
166	C₃×C₄²	—	10	2	no	—
167	C₃×C₄²	—	10	1	no	—
168	C₃×C₄²	C₄×D₅	1	2	yes	—
169	C₃×C₄²	C₄×D₅	1	4	yes	complemented
170	C₃×C₄²	D₂₀	1	1	yes	characteristic
171	C₃×C₄²	D₂₀	1	1	yes	characteristic
172	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
173	C₂²×D₅	—	2	1	no	—
174	C₂×C₄×C₅	—	2	4	no	—
175	C₂×C₄×C₅	—	4	2	no	—
176	C₂×C₄×C₅	—	4	4	no	—
177	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
178	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
179	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
181	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
182	C₂×C₄×C₅	C₃×#41	1	2	yes	—
183	C₂²⋊D₅	—	2	4	no	—
184	C₂×Dic₅	—	2	1	no	—
185	C₂×Dic₅	—	2	2	no	—
186	D₂₀	—	2	1	no	—
187	D₂₀	C₃×C₄²	1	2	yes	—
188	C₄×D₅	—	2	4	no	—
189	C₄×D₅	—	4	4	no	—
190	Dic₁₀	—	2	1	no	—
191	Dic₁₀	C₃×C₄²	1	2	yes	—
192	#168	—	2	4	no	—
193	C₂×#41	—	5	1	no	—
194	C₂×M₁₆	—	5	2	no	—
195	C₄×Q₈	—	5	2	no	—
196	C₄×D₄	—	5	2	no	—
197	C₄×D₄	—	10	2	no	—
198	#117	—	5	1	no	—
199	#117	—	10	1	no	—
200	C₂×C₄²	—	10	1	no	—
201	C₂×C₄²	C₂×C₃×D₅	1	2	yes	—
202	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
203	#104	—	5	8	no	—
204	#97	—	10	1	no	—
205	C₄×C₈	—	10	1	no	—
206	#95	—	2	2	no	—
207	C₂×C₃×C₅	—	2	2	no	—
208	C₂×C₃×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
209	C₂×C₃×C₅	#421	1	2	yes	—
210	C₃×D₅	—	4	2	no	—
211	C₂³×C₃	—	10	1	no	—
212	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
213	C₃×Q₈	—	5	2	no	—
214	C₃×Q₈	—	10	1	no	—
215	C₃×D₄	—	5	2	no	—
216	C₃×D₄	—	10	4	no	—
217	C₃×D₄	—	10	1	no	—
218	C₂×C₃×C₄	—	2	4	no	—
219	C₂×C₃×C₄	—	4	2	no	—
220	C₂×C₃×C₄	—	4	4	no	—
221	C₂×C₃×C₄	—	10	1	no	—
222	C₂×C₃×C₄	—	10	4	no	—
223	C₂×C₃×C₄	—	10	2	no	—
224	C₂×C₃×C₄	—	20	4	no	—
225	C₂×C₃×C₄	#685	1	2	yes	—
226	C₂×C₃×C₄	#685	1	2	yes	—
227	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
229	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
230	C₂×C₃×C₄	#709	1	2	yes	—
231	C₃×C₈	—	10	4	no	—
232	C₂²×C₅	—	2	2	no	—
233	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
234	C₂²×C₅	C₃×C₄×D₄	1	2	yes	—
235	C₂×D₅	—	2	2	no	—
236	C₂×D₅	—	4	2	no	—
237	C₄×C₅	—	2	2	no	—
238	C₄×C₅	—	2	2	no	—
239	C₄×C₅	—	4	4	no	—
240	C₄×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
241	C₄×C₅	C₃×#117	1	1	yes	characteristic
242	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
243	Dic₅	—	2	2	no	—
244	Dic₅	—	4	2	no	—
245	#41	—	5	4	no	—
246	#41	—	10	2	no	—
247	C₂×Q₈	—	5	1	no	—
248	C₂×D₄	—	5	1	no	—
249	C₂×D₄	—	10	1	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	10	1	no	—
252	C₂²×C₄	—	10	2	no	—
253	C₂²×C₄	C₃×C₂²⋊D₅	1	2	yes	—
254	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
255	M₁₆	—	5	4	no	—
256	M₁₆	—	10	2	no	—
257	C₂×C₈	—	10	2	no	—
258	C₄⋊C₄	—	5	2	no	—
259	C₄⋊C₄	—	10	2	no	—
260	C₂²⋊C₄	—	10	1	no	—
261	C₂²⋊C₄	—	10	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	10	2	no	—
264	C₄²	—	10	1	no	—
265	C₄²	C₃×C₄×D₅	1	2	yes	—
266	C₄²	C₃×C₄×D₅	1	4	yes	complemented
267	C₄²	C₃×D₂₀	1	1	yes	characteristic
268	C₄²	C₃×D₂₀	1	1	yes	characteristic
269	C₃×C₅	#1759	1	1	yes	characteristic, complemented, Hall
270	C₂²×C₃	—	2	2	no	—
271	C₂²×C₃	—	10	2	no	—
272	C₂²×C₃	—	20	2	no	—
273	C₂²×C₃	C₂×#685	1	1	yes	characteristic
274	C₂²×C₃	C₄×C₂²⋊D₅	1	2	yes	—
275	C₃×C₄	—	2	2	no	—
276	C₃×C₄	—	2	2	no	—
277	C₃×C₄	—	4	4	no	—
278	C₃×C₄	—	10	2	no	—
279	C₃×C₄	—	20	2	no	—
280	C₃×C₄	C₄²×D₅	1	1	yes	characteristic
281	C₃×C₄	#4018	1	1	yes	characteristic
282	C₃×C₄	C₄×D₂₀	1	2	yes	—
283	C₂×C₅	—	2	2	no	—
284	C₂×C₅	C₃×C₄×C₂²⋊C₄	1	1	yes	characteristic
285	C₂×C₅	C₃×#421	1	2	yes	—
286	D₅	—	4	2	no	—
287	C₂³	—	10	1	no	—
288	C₂³	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
289	Q₈	—	5	2	no	—
290	Q₈	—	10	1	no	—
291	D₄	—	5	2	no	—
292	D₄	—	10	4	no	—
293	D₄	—	10	1	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	10	1	no	—
298	C₂×C₄	—	10	4	no	—
299	C₂×C₄	—	10	2	no	—
300	C₂×C₄	—	20	4	no	—
301	C₂×C₄	C₃×#685	1	2	yes	—
302	C₂×C₄	C₃×#685	1	2	yes	—
303	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
304	C₂×C₄	C₂×C₃×C₄×D₅	1	2	yes	—
305	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
306	C₂×C₄	C₃×#709	1	2	yes	—
307	C₈	—	10	4	no	—
308	C₂×C₃	—	2	2	no	—
309	C₂×C₃	—	20	2	no	—
310	C₂×C₃	C₄×#685	1	1	yes	characteristic
311	C₂×C₃	#65294	1	2	yes	—
312	C₅	C₃×#1759	1	1	yes	characteristic, complemented, Sylow
313	C₂²	—	2	2	no	—
314	C₂²	—	10	2	no	—
315	C₂²	—	20	2	no	—
316	C₂²	C₂×C₃×#685	1	1	yes	characteristic
317	C₂²	C₃×C₄×C₂²⋊D₅	1	2	yes	—
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	4	4	no	—
321	C₄	—	10	2	no	—
322	C₄	—	20	2	no	—
323	C₄	C₃×C₄²×D₅	1	1	yes	characteristic
324	C₄	C₃×#4018	1	1	yes	characteristic
325	C₄	C₃×C₄×D₂₀	1	2	yes	—
326	C₃	#10600223	1	1	yes	characteristic, direct product factor, Sylow
327	C₂	—	2	2	no	—
328	C₂	—	20	2	no	—
329	C₂	C₃×C₄×#685	1	1	yes	characteristic
330	C₂	C₃×#65294	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₂₀
commutator	C₄×C₅	C₂×C₃×C₄²
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₃×C₅×#376	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂×C₃×#709	C₂²
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₃×#376	D₅
hypocenter	C₅	C₃×#1759
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₃×#376

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#376	C₂	semidirect
4. C₃×C₄×#681	C₂	semidirect
5. C₃×#64743	C₂	semidirect
6. #10600223	C₃	direct (N × H)
13. C₃×C₄×D₂₀	C₄	semidirect
16. C₃×C₄×Dic₁₀	C₄	semidirect
18. C₃×#3937	C₄	semidirect
23. C₅×#376	C₂×C₃	semidirect
25. C₄×#681	C₂×C₃	semidirect
26. #64743	C₂×C₃	semidirect
39. C₃×C₄²×C₅	C₂×C₄	semidirect
49. C₃×#681	C₂×C₄	semidirect
54. C₃×#376	D₅	semidirect
62. C₄×D₂₀	C₃×C₄	semidirect
65. C₄×Dic₁₀	C₃×C₄	semidirect
67. #3937	C₃×C₄	semidirect
118. C₄²×C₅	C₂×C₃×C₄	semidirect
128. #681	C₂×C₃×C₄	semidirect
133. #376	C₃×D₅	semidirect
169. C₃×C₄²	C₄×D₅	semidirect
266. C₄²	C₃×C₄×D₅	semidirect
269. C₃×C₅	#1759	semidirect
312. C₅	C₃×#1759	semidirect
326. C₃	#10600223	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×C₄×#709	2
C₃×C₅×#376	2
C₂×C₃×#3937	2
C₃×C₄×#681	2
C₃×#64743	2
#10600223	3
C₃×#1759	5

Subgroups — C3×#10600223 #14295634

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10600223 #14295634