Subgroups — C₃×#10601225 #14284843

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#65811	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65213	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64706	C₂	1	2	yes	maximal, complemented
5	C₃×#64705	C₂	1	2	yes	maximal, complemented
6	#10601225	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×#709	—	2	1	no	—
8	C₂²×C₃×C₅×C₈	—	2	1	no	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₃×#4076	—	2	1	no	—
11	C₃×#4076	C₄	1	2	yes	—
12	C₂×C₃×#681	—	2	1	no	—
13	C₃×#4021	—	2	1	no	—
14	C₃×#4020	C₂²	1	1	yes	characteristic
15	C₃×C₄×D₂₀	—	2	1	no	—
16	C₃×C₄×D₂₀	C₄	1	2	yes	—
17	C₃×#4015	C₂²	1	1	yes	characteristic
18	C₃×C₅×C₄⋊C₈	—	2	1	no	—
19	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
20	C₃×C₄⋊#168	—	2	1	no	—
21	C₃×C₄⋊#168	C₂²	1	2	yes	—
22	C₃×#1547	—	5	1	no	maximal, Hall
23	#65811	C₂×C₃	1	1	yes	characteristic
24	C₂×C₅×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
25	#65213	C₂×C₃	1	1	yes	characteristic, complemented
26	#64706	C₂×C₃	1	2	yes	complemented
27	#64705	C₂×C₃	1	2	yes	complemented
28	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
31	C₂×C₃×C₂²⋊D₅	—	2	2	no	—
32	C₃×#709	—	4	2	no	—
33	C₃×#709	—	4	2	no	—
34	C₂×C₃×D₂₀	—	2	1	no	—
35	C₂×C₃×D₂₀	—	2	1	no	—
36	C₂×C₃×D₂₀	C₂×C₄	1	2	yes	—
37	C₂×C₃×C₄×D₅	—	2	2	no	—
38	C₂×C₃×Dic₁₀	—	2	1	no	—
39	C₂×C₃×C₅×C₈	—	2	2	no	—
40	C₂×C₃×C₅×C₈	—	4	2	no	—
41	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
43	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
44	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
45	C₃×#685	—	2	2	no	—
46	C₃×#685	—	2	2	no	—
47	C₃×C₄⋊Dic₅	—	2	1	no	—
48	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
49	C₃×#681	—	4	2	no	—
50	C₂×C₃×#168	—	2	2	no	—
51	C₃×#532	—	5	1	no	—
52	C₃×#423	—	5	1	no	—
53	C₂×C₃×C₄⋊C₈	D₅	1	1	yes	characteristic, complemented
54	C₃×#332	—	5	2	no	—
55	C₃×#331	—	5	2	no	—
56	C₂×#709	—	2	1	no	—
57	C₂²×C₅×C₈	—	2	1	no	—
58	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
59	#4076	—	2	1	no	—
60	#4076	C₃×C₄	1	2	yes	—
61	C₂×#681	—	2	1	no	—
62	#4021	—	2	1	no	—
63	#4020	C₂²×C₃	1	1	yes	characteristic
64	C₄×D₂₀	—	2	1	no	—
65	C₄×D₂₀	C₃×C₄	1	2	yes	—
66	#4015	C₂²×C₃	1	1	yes	characteristic
67	C₅×C₄⋊C₈	—	2	1	no	—
68	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
69	C₄⋊#168	—	2	1	no	—
70	C₄⋊#168	C₂²×C₃	1	2	yes	—
71	#1547	—	5	1	no	Sylow
72	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₃×D₅	—	2	2	no	—
74	C₂×C₃×C₄×C₅	—	2	1	no	—
75	C₂×C₃×C₄×C₅	—	2	1	no	—
76	C₂×C₃×C₄×C₅	—	4	1	no	—
77	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	D₈	1	2	yes	—
81	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
82	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₃×C₂²⋊D₅	—	4	4	no	—
86	C₂×C₃×Dic₅	—	2	2	no	—
87	C₃×D₂₀	—	4	2	no	—
88	C₃×D₂₀	—	4	1	no	—
89	C₃×D₂₀	—	4	1	no	—
90	C₃×D₂₀	—	4	1	no	—
91	C₃×D₂₀	—	4	1	no	—
92	C₃×C₄×D₅	—	4	4	no	—
93	C₃×Dic₁₀	—	4	1	no	—
94	C₃×Dic₁₀	—	4	1	no	—
95	C₃×C₅×C₈	—	4	2	no	—
96	C₃×#168	—	4	2	no	—
97	C₂×C₃×#41	—	10	1	no	—
98	C₂×C₃×M₁₆	—	10	1	no	—
99	C₂²×C₃×C₈	—	2	1	no	—
100	C₃×C₄⋊Q₈	—	5	1	no	—
101	C₃×#127	—	5	1	no	—
102	C₃×#124	—	10	1	no	—
103	C₃×#121	—	5	2	no	—
104	C₃×#121	—	10	1	no	—
105	C₃×C₄×D₄	—	5	2	no	—
106	C₃×C₄×D₄	—	10	1	no	—
107	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
108	C₃×C₄⋊C₈	—	2	1	no	—
109	C₃×C₄⋊C₈	—	5	2	no	—
110	C₃×C₄⋊C₈	—	10	1	no	—
111	C₃×C₄⋊C₈	C₂×D₅	1	2	yes	—
112	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
115	C₂×C₂²⋊D₅	—	2	2	no	—
116	#709	—	4	2	no	—
117	#709	—	4	2	no	—
118	C₂×D₂₀	—	2	1	no	—
119	C₂×D₂₀	—	2	1	no	—
120	C₂×D₂₀	C₂×C₃×C₄	1	2	yes	—
121	C₂×C₄×D₅	—	2	2	no	—
122	C₂×Dic₁₀	—	2	1	no	—
123	C₂×C₅×C₈	—	2	2	no	—
124	C₂×C₅×C₈	—	4	2	no	—
125	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
126	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
127	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
128	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
129	#685	—	2	2	no	—
130	#685	—	2	2	no	—
131	C₄⋊Dic₅	—	2	1	no	—
132	C₄⋊Dic₅	C₂×C₃×C₄	1	2	yes	—
133	#681	—	4	2	no	—
134	C₂×#168	—	2	2	no	—
135	#532	—	5	1	no	—
136	#423	—	5	1	no	—
137	C₂×C₄⋊C₈	C₃×D₅	1	1	yes	characteristic, complemented
138	#332	—	5	2	no	—
139	#331	—	5	2	no	—
140	C₂²×C₃×C₅	—	2	1	no	—
141	C₂²×C₃×C₅	—	2	1	no	—
142	C₂²×C₃×C₅	#102	1	2	yes	—
143	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₃×D₅	—	4	2	no	—
145	C₂×C₃×D₅	—	4	2	no	—
146	C₂×C₃×D₅	—	4	2	no	—
147	C₃×C₄×C₅	—	2	1	no	—
148	C₃×C₄×C₅	—	2	1	no	—
149	C₃×C₄×C₅	—	2	1	no	—
150	C₃×C₄×C₅	#102	1	2	yes	—
151	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
152	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
153	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
154	C₃×C₄×C₅	#137	1	1	yes	characteristic
155	C₃×Dic₅	—	4	2	no	—
156	C₃×#41	—	20	2	no	—
157	C₃×#41	—	20	2	no	—
158	C₂×C₃×Q₈	—	10	1	no	—
159	C₂×C₃×D₄	—	5	2	no	—
160	C₂×C₃×D₄	—	10	2	no	—
161	C₂×C₃×D₄	—	10	1	no	—
162	C₂×C₃×D₄	—	10	1	no	—
163	C₂²×C₃×C₄	—	10	2	no	—
164	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
165	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
166	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
167	C₃×M₁₆	—	20	2	no	—
168	C₂×C₃×C₈	—	2	2	no	—
169	C₂×C₃×C₈	—	4	2	no	—
170	C₂×C₃×C₈	—	10	2	no	—
171	C₃×C₄⋊C₄	—	5	2	no	—
172	C₃×C₄⋊C₄	—	10	1	no	—
173	C₃×C₂²⋊C₄	—	10	2	no	—
174	C₃×C₂²⋊C₄	—	10	2	no	—
175	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
176	C₃×C₄²	D₂₀	1	1	yes	characteristic
177	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
178	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
179	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
180	C₂²×D₅	—	2	2	no	—
181	C₂×C₄×C₅	—	2	1	no	—
182	C₂×C₄×C₅	—	2	1	no	—
183	C₂×C₄×C₅	—	4	1	no	—
184	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
185	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
186	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
187	C₂×C₄×C₅	C₃×D₈	1	2	yes	—
188	C₂×C₄×C₅	C₃×SD₁₆	1	2	yes	—
189	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
190	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
191	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
192	C₂²⋊D₅	—	4	4	no	—
193	C₂×Dic₅	—	2	2	no	—
194	D₂₀	—	4	2	no	—
195	D₂₀	—	4	1	no	—
196	D₂₀	—	4	1	no	—
197	D₂₀	—	4	1	no	—
198	D₂₀	—	4	1	no	—
199	C₄×D₅	—	4	4	no	—
200	Dic₁₀	—	4	1	no	—
201	Dic₁₀	—	4	1	no	—
202	C₅×C₈	—	4	2	no	—
203	#168	—	4	2	no	—
204	C₂×#41	—	10	1	no	—
205	C₂×M₁₆	—	10	1	no	—
206	C₂²×C₈	—	2	1	no	—
207	C₄⋊Q₈	—	5	1	no	—
208	#127	—	5	1	no	—
209	#124	—	10	1	no	—
210	#121	—	5	2	no	—
211	#121	—	10	1	no	—
212	C₄×D₄	—	5	2	no	—
213	C₄×D₄	—	10	1	no	—
214	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	—	5	2	no	—
217	C₄⋊C₈	—	10	1	no	—
218	C₄⋊C₈	C₂×C₃×D₅	1	2	yes	—
219	C₂×C₃×C₅	—	2	2	no	—
220	C₂×C₃×C₅	#413	1	1	yes	characteristic
221	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
222	C₂×C₃×C₅	#417	1	1	yes	characteristic
223	C₃×D₅	—	8	2	no	—
224	C₂³×C₃	—	10	2	no	—
225	C₂³×C₃	#685	1	1	yes	characteristic
226	C₃×Q₈	—	20	1	no	—
227	C₃×Q₈	—	20	1	no	—
228	C₃×D₄	—	20	2	no	—
229	C₃×D₄	—	20	4	no	—
230	C₃×D₄	—	20	1	no	—
231	C₃×D₄	—	20	1	no	—
232	C₃×D₄	—	20	1	no	—
233	C₃×D₄	—	20	1	no	—
234	C₂×C₃×C₄	—	2	1	no	—
235	C₂×C₃×C₄	—	2	1	no	—
236	C₂×C₃×C₄	—	4	1	no	—
237	C₂×C₃×C₄	—	10	2	no	—
238	C₂×C₃×C₄	—	20	4	no	—
239	C₂×C₃×C₄	#677	1	2	yes	—
240	C₂×C₃×C₄	D₄₀	1	2	yes	—
241	C₂×C₃×C₄	#685	1	1	yes	characteristic
242	C₂×C₃×C₄	#685	1	1	yes	characteristic
243	C₂×C₃×C₄	#685	1	1	yes	characteristic
244	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
245	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
246	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
247	C₃×C₈	—	4	2	no	—
248	C₃×C₈	—	20	2	no	—
249	C₂²×C₅	—	2	1	no	—
250	C₂²×C₅	—	2	1	no	—
251	C₂²×C₅	C₃×#102	1	2	yes	—
252	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
253	C₂×D₅	—	4	2	no	—
254	C₂×D₅	—	4	2	no	—
255	C₂×D₅	—	4	2	no	—
256	C₄×C₅	—	2	1	no	—
257	C₄×C₅	—	2	1	no	—
258	C₄×C₅	—	2	1	no	—
259	C₄×C₅	C₃×#102	1	2	yes	—
260	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
261	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
262	C₄×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
263	C₄×C₅	C₃×#137	1	1	yes	characteristic
264	Dic₅	—	4	2	no	—
265	#41	—	20	2	no	—
266	#41	—	20	2	no	—
267	C₂×Q₈	—	10	1	no	—
268	C₂×D₄	—	5	2	no	—
269	C₂×D₄	—	10	2	no	—
270	C₂×D₄	—	10	1	no	—
271	C₂×D₄	—	10	1	no	—
272	C₂²×C₄	—	10	2	no	—
273	C₂²×C₄	C₃×C₄×D₅	1	1	yes	characteristic
274	C₂²×C₄	C₃×D₂₀	1	1	yes	characteristic
275	C₂²×C₄	C₃×C₂²⋊D₅	1	1	yes	characteristic
276	M₁₆	—	20	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	4	2	no	—
279	C₂×C₈	—	10	2	no	—
280	C₄⋊C₄	—	5	2	no	—
281	C₄⋊C₄	—	10	1	no	—
282	C₂²⋊C₄	—	10	2	no	—
283	C₂²⋊C₄	—	10	2	no	—
284	C₄²	C₃×C₄×D₅	1	1	yes	characteristic
285	C₄²	C₃×D₂₀	1	1	yes	characteristic
286	C₄²	C₃×C₂²⋊D₅	1	1	yes	characteristic
287	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
288	C₃×C₅	#1547	1	1	yes	characteristic, complemented, Hall
289	C₂²×C₃	—	2	1	no	—
290	C₂²×C₃	—	2	1	no	—
291	C₂²×C₃	—	20	2	no	—
292	C₂²×C₃	—	20	2	no	—
293	C₂²×C₃	—	20	2	no	—
294	C₂²×C₃	#3953	1	2	yes	—
295	C₂²×C₃	C₂×#685	1	1	yes	characteristic
296	C₃×C₄	—	2	1	no	—
297	C₃×C₄	—	2	1	no	—
298	C₃×C₄	—	2	1	no	—
299	C₃×C₄	—	20	2	no	—
300	C₃×C₄	#3953	1	2	yes	—
301	C₃×C₄	C₂×#677	1	1	yes	characteristic
302	C₃×C₄	C₂×D₄₀	1	1	yes	characteristic
303	C₃×C₄	#4078	1	1	yes	characteristic
304	C₃×C₄	#4088	1	1	yes	characteristic
305	C₂×C₅	—	2	2	no	—
306	C₂×C₅	C₃×#413	1	1	yes	characteristic
307	C₂×C₅	C₂×C₃×#102	1	1	yes	characteristic
308	C₂×C₅	C₃×#417	1	1	yes	characteristic
309	D₅	—	8	2	no	—
310	C₂³	—	10	2	no	—
311	C₂³	C₃×#685	1	1	yes	characteristic
312	Q₈	—	20	1	no	—
313	Q₈	—	20	1	no	—
314	D₄	—	20	2	no	—
315	D₄	—	20	4	no	—
316	D₄	—	20	1	no	—
317	D₄	—	20	1	no	—
318	D₄	—	20	1	no	—
319	D₄	—	20	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	10	2	no	—
324	C₂×C₄	—	20	4	no	—
325	C₂×C₄	C₃×#677	1	2	yes	—
326	C₂×C₄	C₃×D₄₀	1	2	yes	—
327	C₂×C₄	C₃×#685	1	1	yes	characteristic
328	C₂×C₄	C₃×#685	1	1	yes	characteristic
329	C₂×C₄	C₃×#685	1	1	yes	characteristic
330	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
331	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
332	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
333	C₈	—	4	2	no	—
334	C₈	—	20	2	no	—
335	C₂×C₃	—	2	2	no	—
336	C₂×C₃	—	40	2	no	—
337	C₂×C₃	#65255	1	1	yes	characteristic
338	C₂×C₃	C₂×#3953	1	1	yes	characteristic
339	C₂×C₃	#65421	1	1	yes	characteristic
340	C₅	C₃×#1547	1	1	yes	characteristic, complemented, Sylow
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	20	2	no	—
344	C₂²	—	20	2	no	—
345	C₂²	—	20	2	no	—
346	C₂²	C₃×#3953	1	2	yes	—
347	C₂²	C₂×C₃×#685	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	20	2	no	—
352	C₄	C₃×#3953	1	2	yes	—
353	C₄	C₂×C₃×#677	1	1	yes	characteristic
354	C₄	C₂×C₃×D₄₀	1	1	yes	characteristic
355	C₄	C₃×#4078	1	1	yes	characteristic
356	C₄	C₃×#4088	1	1	yes	characteristic
357	C₃	#10601225	1	1	yes	characteristic, direct product factor, Sylow
358	C₂	—	2	2	no	—
359	C₂	—	40	2	no	—
360	C₂	C₃×#65255	1	1	yes	characteristic
361	C₂	C₂×C₃×#3953	1	1	yes	characteristic
362	C₂	C₃×#65421	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×#685
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₄²	C₂²×C₃×D₅
Fitting	C₂×C₃×C₅×C₄⋊C₈	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₃×#65811	C₂
cosocle	C₄²×C₅	C₂³×C₃
hypercenter	C₂×C₃×C₄⋊C₈	D₅
hypocenter	C₅	C₃×#1547
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×C₄⋊C₈	C₂	semidirect
3. C₃×#65213	C₂	semidirect
4. C₃×#64706	C₂	semidirect
5. C₃×#64705	C₂	semidirect
6. #10601225	C₃	direct (N × H)
24. C₂×C₅×C₄⋊C₈	C₂×C₃	semidirect
25. #65213	C₂×C₃	semidirect
26. #64706	C₂×C₃	semidirect
27. #64705	C₂×C₃	semidirect
53. C₂×C₃×C₄⋊C₈	D₅	semidirect
137. C₂×C₄⋊C₈	C₃×D₅	semidirect
288. C₃×C₅	#1547	semidirect
340. C₅	C₃×#1547	semidirect
357. C₃	#10601225	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×#65811	2
C₂×C₃×C₅×C₄⋊C₈	2
C₃×#65213	2
C₃×#64706	2
C₃×#64705	2
#10601225	3
C₃×#1547	5

Subgroups — C3×#10601225 #14284843

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10601225 #14284843