Subgroups — D₃×#64701 #14284637

Order 1920.

512 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃×#4015	C₂	1	1	yes	maximal, characteristic
2	C₅×D₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	D₃×C₄⋊#168	C₂	1	1	yes	maximal, characteristic
4	#11746302	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64701	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745863	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745861	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#64701	—	3	1	no	maximal, Hall
9	C₂×D₃×Dic₁₀	C₄	1	2	yes	—
10	#91758	C₂²	1	1	yes	characteristic
11	C₂×C₅×D₃×C₈	—	2	1	no	—
12	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
13	C₃×#4015	C₂²	1	1	yes	characteristic
14	#91469	—	2	1	no	—
15	#91469	—	2	1	no	—
16	D₃×C₄⋊Dic₅	—	2	1	no	—
17	D₃×C₄⋊Dic₅	—	2	1	no	—
18	#91389	C₄	1	2	yes	—
19	C₂×D₃×#168	—	2	1	no	—
20	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
22	C₅×Dic₃⋊C₈	—	2	1	no	—
23	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
24	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
25	Dic₃⋊#168	—	2	1	no	—
26	D₃×#333	—	5	1	no	maximal, Hall
27	C₂×#4015	—	3	1	no	—
28	C₂×C₅×C₄⋊C₈	—	3	1	no	—
29	C₂×C₄⋊#168	—	3	1	no	—
30	#64701	—	3	1	no	—
31	#64701	—	3	1	no	—
32	#64701	—	3	1	no	—
33	#64701	D₃	1	1	yes	characteristic, direct product factor
34	C₂×Dic₃₀	C₂×C₄	1	2	yes	—
35	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
37	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
38	C₂×C₃×Dic₁₀	C₂×C₄	1	2	yes	—
39	C₂×#1204	—	2	2	no	—
40	C₂×D₃×Dic₅	—	2	2	no	—
41	D₃×Dic₁₀	—	4	4	no	—
42	C₂×C₃×C₅×C₈	—	2	1	no	—
43	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₄⋊Dic₁₅	—	2	1	no	—
45	C₄⋊Dic₁₅	—	2	1	no	—
46	C₂×C₃⋊#168	—	2	1	no	—
47	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
48	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
49	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
50	C₂×C₅×C₃⋊C₈	—	2	1	no	—
51	C₅×D₃×C₈	—	4	2	no	—
52	C₃×C₄⋊Dic₅	—	2	1	no	—
53	C₃×C₄⋊Dic₅	—	2	1	no	—
54	C₂×C₃×#168	—	2	1	no	—
55	Dic₃⋊Dic₅	—	2	2	no	—
56	Dic₃⋊Dic₅	—	2	2	no	—
57	D₃×#168	—	4	2	no	—
58	D₃×C₄⋊Q₈	—	5	1	no	—
59	D₃×C₄⋊C₈	—	5	1	no	—
60	D₃×C₄⋊C₈	D₅	1	1	yes	characteristic
61	C₃×#333	—	5	1	no	—
62	#4546	—	5	1	no	—
63	#4481	—	5	1	no	—
64	#4481	—	5	1	no	—
65	C₂²×Dic₁₀	—	3	2	no	—
66	C₂²×C₅×C₈	—	6	1	no	—
67	C₂×C₄²×C₅	—	3	1	no	—
68	C₂×C₄⋊Dic₅	—	6	1	no	—
69	C₂×C₄⋊Dic₅	—	6	1	no	—
70	C₂²×#168	—	6	1	no	—
71	#4015	—	3	2	no	—
72	#4015	—	3	1	no	—
73	#4015	—	6	1	no	—
74	#4015	—	6	1	no	—
75	#4015	C₂×D₃	1	1	yes	characteristic
76	C₅×C₄⋊C₈	—	3	1	no	—
77	C₅×C₄⋊C₈	—	6	1	no	—
78	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
79	C₄⋊#168	—	3	1	no	—
80	C₄⋊#168	—	6	1	no	—
81	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
82	C₂×#333	—	15	1	no	Sylow
83	C₂²×C₅×D₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂×Dic₁₅	—	2	2	no	—
88	Dic₃₀	—	4	2	no	—
89	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
92	C₄×C₅×D₃	—	2	2	no	—
93	C₄×C₅×D₃	SD₁₆	1	2	yes	—
94	C₄×C₅×D₃	SD₁₆	1	2	yes	—
95	C₄×C₅×D₃	Q₁₆	1	2	yes	—
96	C₄×C₅×D₃	Q₁₆	1	2	yes	—
97	C₂×C₃×Dic₅	—	2	2	no	—
98	C₃×Dic₁₀	—	4	2	no	—
99	#1204	—	4	4	no	—
100	D₃×Dic₅	—	4	4	no	—
101	C₃×C₅×C₈	—	4	1	no	—
102	C₃⋊#168	—	4	1	no	—
103	C₃×#168	—	4	1	no	—
104	C₅×C₃⋊C₈	—	4	1	no	—
105	C₂×D₃×Q₈	—	5	2	no	—
106	C₃×C₄⋊Q₈	—	5	1	no	—
107	#944	—	5	2	no	—
108	C₂×D₃×C₈	—	2	1	no	—
109	C₂×D₃×C₈	—	10	1	no	—
110	D₃×C₄⋊C₄	—	10	1	no	—
111	D₃×C₄⋊C₄	—	10	1	no	—
112	#888	—	10	1	no	—
113	#888	—	10	1	no	—
114	C₄²×D₃	C₂×D₅	1	1	yes	characteristic
115	#869	—	5	1	no	—
116	C₃×C₄⋊C₈	—	5	1	no	—
117	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
118	Dic₃⋊C₈	—	2	1	no	—
119	Dic₃⋊C₈	—	10	1	no	—
120	C₄⋊(C₃⋊C₈)	—	5	1	no	—
121	C₄⋊(C₃⋊C₈)	C₂×D₅	1	1	yes	characteristic
122	C₂²×C₄×C₅	—	3	1	no	—
123	C₂²×C₄×C₅	—	3	1	no	—
124	C₂²×C₄×C₅	—	3	1	no	—
125	C₂²×Dic₅	—	6	2	no	—
126	C₂×Dic₁₀	—	3	2	no	—
127	C₂×Dic₁₀	—	6	2	no	—
128	C₂×Dic₁₀	—	12	4	no	—
129	C₂×Dic₁₀	C₄×D₃	1	2	yes	—
130	C₂×C₅×C₈	—	2	1	no	—
131	C₂×C₅×C₈	—	6	1	no	—
132	C₂×C₅×C₈	—	12	2	no	—
133	C₄²×C₅	—	3	1	no	—
134	C₄²×C₅	—	3	1	no	—
135	C₄²×C₅	—	3	1	no	—
136	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
137	C₄⋊Dic₅	—	2	1	no	—
138	C₄⋊Dic₅	—	2	1	no	—
139	C₄⋊Dic₅	—	6	2	no	—
140	C₄⋊Dic₅	—	6	2	no	—
141	C₄⋊Dic₅	—	6	1	no	—
142	C₄⋊Dic₅	—	6	1	no	—
143	C₂×#168	—	2	1	no	—
144	C₂×#168	—	6	1	no	—
145	C₂×#168	—	12	2	no	—
146	C₂×C₄⋊Q₈	—	15	1	no	—
147	C₂×C₄⋊C₈	—	3	1	no	—
148	C₂×C₄⋊C₈	—	15	1	no	—
149	#333	—	5	1	no	—
150	#333	—	15	1	no	—
151	#333	—	15	1	no	—
152	#333	—	15	1	no	—
153	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	C₂×C₅×D₃	#100	1	2	yes	—
155	C₂×C₅×D₃	#103	1	2	yes	—
156	C₂×C₅×D₃	#103	1	2	yes	—
157	C₃×C₄×C₅	—	2	1	no	—
158	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
159	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
160	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
161	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
162	Dic₁₅	—	4	2	no	—
163	C₃×Dic₅	—	4	2	no	—
164	C₅×Dic₃	—	2	1	no	—
165	C₅×Dic₃	#103	1	2	yes	—
166	C₅×Dic₃	#103	1	2	yes	—
167	C₂×C₃×Q₈	—	5	2	no	—
168	D₃×Q₈	—	20	4	no	—
169	C₂×C₄×D₃	—	10	2	no	—
170	C₂×C₄×D₃	C₄×D₅	1	1	yes	characteristic
171	C₂×C₄×D₃	D₂₀	1	1	yes	characteristic
172	C₂×C₄×D₃	C₂²⋊D₅	1	1	yes	characteristic
173	C₂×Dic₆	—	5	2	no	—
174	C₂×Dic₆	—	10	2	no	—
175	C₂×C₃×C₈	—	2	1	no	—
176	C₂×C₃×C₈	—	10	1	no	—
177	C₃×C₄⋊C₄	—	10	1	no	—
178	C₃×C₄⋊C₄	—	10	1	no	—
179	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
180	C₄⋊Dic₃	—	10	1	no	—
181	C₄⋊Dic₃	—	10	1	no	—
182	Dic₃⋊C₄	—	10	2	no	—
183	Dic₃⋊C₄	—	10	2	no	—
184	C₄×Dic₃	C₄×D₅	1	1	yes	characteristic
185	C₄×Dic₃	D₂₀	1	1	yes	characteristic
186	C₄×Dic₃	C₂²⋊D₅	1	1	yes	characteristic
187	C₂×C₃⋊C₈	—	2	1	no	—
188	C₂×C₃⋊C₈	—	10	1	no	—
189	D₃×C₈	—	4	2	no	—
190	D₃×C₈	—	20	2	no	—
191	C₂³×C₅	—	3	1	no	—
192	C₂×C₄×C₅	—	3	1	no	—
193	C₂×C₄×C₅	—	3	1	no	—
194	C₂×C₄×C₅	—	3	2	no	—
195	C₂×C₄×C₅	—	3	2	no	—
196	C₂×C₄×C₅	—	3	2	no	—
197	C₂×C₄×C₅	—	3	2	no	—
198	C₂×C₄×C₅	—	3	1	no	—
199	C₂×C₄×C₅	—	6	2	no	—
200	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
201	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
202	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
203	C₂×Dic₅	—	2	2	no	—
204	C₂×Dic₅	—	6	2	no	—
205	C₂×Dic₅	—	12	4	no	—
206	Dic₁₀	—	4	2	no	—
207	Dic₁₀	—	12	4	no	—
208	Dic₁₀	—	12	2	no	—
209	C₅×C₈	—	4	1	no	—
210	C₅×C₈	—	12	1	no	—
211	#168	—	4	1	no	—
212	#168	—	12	1	no	—
213	C₂²×Q₈	—	15	2	no	—
214	C₂²×C₈	—	6	1	no	—
215	C₂²×C₈	—	30	1	no	—
216	C₄⋊Q₈	—	5	1	no	—
217	C₄⋊Q₈	—	15	2	no	—
218	C₄⋊Q₈	—	15	1	no	—
219	C₄⋊Q₈	—	30	1	no	—
220	C₄⋊Q₈	—	30	1	no	—
221	C₂×C₄⋊C₄	—	30	1	no	—
222	C₂×C₄⋊C₄	—	30	1	no	—
223	C₂×C₄²	—	3	1	no	—
224	C₄⋊C₈	—	3	1	no	—
225	C₄⋊C₈	—	5	1	no	—
226	C₄⋊C₈	—	6	1	no	—
227	C₄⋊C₈	—	15	1	no	—
228	C₄⋊C₈	—	30	1	no	—
229	C₄⋊C₈	D₃×D₅	1	1	yes	characteristic
230	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
231	C₂×C₃×C₅	C₂×#103	1	1	yes	characteristic
232	C₂×C₃×C₅	C₂×#103	1	1	yes	characteristic
233	C₅×D₃	#333	1	4	yes	complemented
234	C₂²×D₃	#685	1	1	yes	characteristic
235	C₃×Q₈	—	20	2	no	—
236	C₂×C₃×C₄	—	10	2	no	—
237	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
238	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
239	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
240	C₂×Dic₃	—	10	2	no	—
241	C₂×Dic₃	#685	1	1	yes	characteristic
242	C₂×Dic₃	#685	1	1	yes	characteristic
243	C₂×Dic₃	#685	1	1	yes	characteristic
244	C₄×D₃	—	2	2	no	—
245	C₄×D₃	—	20	4	no	—
246	C₄×D₃	#677	1	2	yes	—
247	C₄×D₃	Dic₂₀	1	2	yes	—
248	C₄×D₃	#687	1	2	yes	—
249	C₄×D₃	#689	1	2	yes	—
250	Dic₆	—	20	4	no	—
251	Dic₆	—	20	2	no	—
252	C₃×C₈	—	4	1	no	—
253	C₃×C₈	—	20	1	no	—
254	C₃⋊C₈	—	4	1	no	—
255	C₃⋊C₈	—	20	1	no	—
256	C₂²×C₅	—	3	2	no	—
257	C₂²×C₅	—	3	2	no	—
258	C₂²×C₅	—	3	2	no	—
259	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
260	C₄×C₅	—	2	1	no	—
261	C₄×C₅	—	3	2	no	—
262	C₄×C₅	—	3	2	no	—
263	C₄×C₅	—	6	1	no	—
264	C₄×C₅	D₃×SD₁₆	1	1	yes	characteristic
265	C₄×C₅	D₃×SD₁₆	1	1	yes	characteristic
266	C₄×C₅	D₃×Q₁₆	1	1	yes	characteristic
267	C₄×C₅	D₃×Q₁₆	1	1	yes	characteristic
268	Dic₅	—	4	2	no	—
269	Dic₅	—	12	2	no	—
270	C₂×Q₈	—	5	2	no	—
271	C₂×Q₈	—	15	2	no	—
272	C₂×Q₈	—	30	2	no	—
273	C₂×Q₈	—	60	4	no	—
274	C₂²×C₄	—	3	1	no	—
275	C₂²×C₄	—	3	1	no	—
276	C₂²×C₄	—	3	1	no	—
277	C₂²×C₄	—	30	2	no	—
278	C₂×C₈	—	2	1	no	—
279	C₂×C₈	—	6	1	no	—
280	C₂×C₈	—	10	1	no	—
281	C₂×C₈	—	12	2	no	—
282	C₂×C₈	—	30	1	no	—
283	C₂×C₈	—	60	2	no	—
284	C₄⋊C₄	—	10	1	no	—
285	C₄⋊C₄	—	10	1	no	—
286	C₄⋊C₄	—	30	2	no	—
287	C₄⋊C₄	—	30	2	no	—
288	C₄⋊C₄	—	30	1	no	—
289	C₄⋊C₄	—	30	1	no	—
290	C₄²	—	3	1	no	—
291	C₄²	—	3	1	no	—
292	C₄²	—	3	1	no	—
293	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
294	C₃×C₅	C₂×#333	1	1	yes	characteristic, complemented, Hall
295	C₂²×C₃	C₂×#685	1	1	yes	characteristic
296	C₂×D₃	#3942	1	2	yes	—
297	C₂×D₃	#3948	1	2	yes	—
298	C₂×D₃	#3955	1	2	yes	—
299	C₃×C₄	—	2	1	no	—
300	C₃×C₄	—	20	2	no	—
301	C₃×C₄	C₂×#677	1	1	yes	characteristic
302	C₃×C₄	C₂×Dic₂₀	1	1	yes	characteristic
303	C₃×C₄	C₂×#687	1	1	yes	characteristic
304	C₃×C₄	C₂×#689	1	1	yes	characteristic
305	Dic₃	—	2	1	no	—
306	Dic₃	—	20	2	no	—
307	Dic₃	#3942	1	2	yes	—
308	Dic₃	#3948	1	2	yes	—
309	C₂×C₅	—	3	4	no	—
310	C₂×C₅	D₃×#100	1	1	yes	characteristic
311	C₂×C₅	D₃×#103	1	1	yes	characteristic
312	C₂×C₅	D₃×#103	1	1	yes	characteristic
313	C₂³	—	3	1	no	—
314	Q₈	—	20	2	no	—
315	Q₈	—	60	2	no	—
316	Q₈	—	60	4	no	—
317	C₂×C₄	—	3	1	no	—
318	C₂×C₄	—	3	1	no	—
319	C₂×C₄	—	3	2	no	—
320	C₂×C₄	—	3	2	no	—
321	C₂×C₄	—	3	2	no	—
322	C₂×C₄	—	3	2	no	—
323	C₂×C₄	—	3	1	no	—
324	C₂×C₄	—	6	2	no	—
325	C₂×C₄	—	10	2	no	—
326	C₂×C₄	—	30	2	no	—
327	C₂×C₄	—	60	4	no	—
328	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
329	C₂×C₄	D₃×D₂₀	1	1	yes	characteristic
330	C₂×C₄	D₃×C₂²⋊D₅	1	1	yes	characteristic
331	C₈	—	4	1	no	—
332	C₈	—	12	1	no	—
333	C₈	—	20	1	no	—
334	C₈	—	60	1	no	—
335	C₂×C₃	C₂×#3942	1	1	yes	characteristic
336	C₂×C₃	C₂×#3948	1	1	yes	characteristic
337	C₂×C₃	C₂×#3955	1	1	yes	characteristic
338	D₃	#64701	1	4	yes	direct product factor
339	C₅	D₃×#333	1	1	yes	characteristic, complemented, Sylow
340	C₂²	—	3	2	no	—
341	C₂²	—	3	2	no	—
342	C₂²	—	3	2	no	—
343	C₂²	D₃×#685	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	3	2	no	—
346	C₄	—	3	2	no	—
347	C₄	—	6	1	no	—
348	C₄	—	20	2	no	—
349	C₄	—	60	2	no	—
350	C₄	D₃×#677	1	1	yes	characteristic
351	C₄	D₃×Dic₂₀	1	1	yes	characteristic
352	C₄	D₃×#687	1	1	yes	characteristic
353	C₄	D₃×#689	1	1	yes	characteristic
354	C₃	C₂×#64701	1	1	yes	characteristic, complemented, Sylow
355	C₂	—	3	4	no	—
356	C₂	D₃×#3942	1	1	yes	characteristic
357	C₂	D₃×#3948	1	1	yes	characteristic
358	C₂	D₃×#3955	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×#685
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₅×C₄⋊C₈	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×C₅×D₃	C₂²⋊C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄⋊C₈	D₃×D₅
hypocenter	C₃×C₅	C₂×#333
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
4. #11746302	C₂	semidirect
5. C₃×#64701	C₂	semidirect
6. #11745863	C₂	semidirect
7. #11745861	C₂	semidirect
33. #64701	D₃	direct (N × H)
233. C₅×D₃	#333	semidirect
294. C₃×C₅	C₂×#333	semidirect
338. D₃	#64701	direct (N × H)
339. C₅	D₃×#333	semidirect
354. C₃	C₂×#64701	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
D₃×#4015	2
C₅×D₃×C₄⋊C₈	2
D₃×C₄⋊#168	2
#11746302	2
C₃×#64701	2
#11745863	2
#11745861	2
C₂×#64701	3
D₃×#333	5

Subgroups — D3×#64701 #14284637

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×#64701 #14284637