Subgroups — D₃×#65371 #14437985

Order 1920.

768 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×D₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	D₃×#4016	C₂	1	1	yes	maximal, characteristic
3	#11751183	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65371	C₂	1	1	yes	maximal, characteristic, complemented
5	D₃×#3967	C₂	1	4	yes	maximal
6	#11749192	C₂	1	4	yes	maximal, complemented
7	D₃×#3935	C₂	1	1	yes	maximal, characteristic
8	#11746931	C₂	1	1	yes	maximal, characteristic, complemented
9	#11746928	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#65371	—	3	1	no	maximal, Hall
11	C₂×C₅×D₃×Q₈	C₂²	1	2	yes	—
12	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	#91759	C₂²	1	1	yes	characteristic
14	C₅×#944	C₂²	1	2	yes	—
15	C₅×D₃×C₄⋊C₄	—	2	2	no	—
16	C₅×#888	—	2	2	no	—
17	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
18	C₅×#869	C₂²	1	1	yes	characteristic
19	C₃×#4016	C₂²	1	1	yes	characteristic
20	D₃×C₄⋊Dic₅	C₂²	1	2	yes	—
21	D₃×#683	—	2	2	no	—
22	#91381	C₂²	1	2	yes	—
23	#91379	—	2	2	no	—
24	C₂×D₃×#168	C₂²	1	2	yes	—
25	#91119	C₂²	1	4	yes	—
26	#91087	C₂²	1	1	yes	characteristic
27	C₃×#3967	C₂²	1	4	yes	—
28	C₃×#3935	C₂²	1	1	yes	characteristic
29	#90973	C₂²	1	4	yes	—
30	#90972	C₂²	1	4	yes	—
31	#90953	C₂²	1	2	yes	—
32	D₃×#491	—	5	1	no	maximal, Hall
33	C₂×C₅×C₄⋊Q₈	—	3	1	no	—
34	C₂×#4016	—	3	1	no	—
35	C₂×#3967	—	3	4	no	—
36	#65371	—	3	1	no	—
37	#65371	—	3	4	no	—
38	#65371	—	3	1	no	—
39	#65371	—	3	1	no	—
40	#65371	D₃	1	1	yes	characteristic, direct product factor
41	C₂×#3935	—	3	1	no	—
42	C₂×C₃×C₅×Q₈	C₂³	1	2	yes	—
43	C₅×D₃×Q₈	—	2	8	no	—
44	C₂×C₄×C₅×D₃	—	2	2	no	—
45	C₂×C₄×C₅×D₃	D₄	1	2	yes	—
46	C₂×C₄×C₅×D₃	C₂³	1	1	yes	characteristic
47	C₂×C₅×Dic₆	—	2	2	no	—
48	C₂×C₅×Dic₆	C₂³	1	2	yes	—
49	C₂×D₃×Dic₅	—	2	2	no	—
50	C₃×C₅×C₄⋊C₄	—	2	2	no	—
51	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
52	C₄⋊Dic₁₅	C₂³	1	2	yes	—
53	#6712	—	2	2	no	—
54	C₂×C₃⋊#168	C₂³	1	2	yes	—
55	C₅×C₄⋊Dic₃	—	2	2	no	—
56	C₅×Dic₃⋊C₄	—	2	4	no	—
57	C₄×C₅×Dic₃	D₄	1	2	yes	—
58	C₄×C₅×Dic₃	C₂³	1	1	yes	characteristic
59	C₃×C₄⋊Dic₅	C₂³	1	2	yes	—
60	C₃×#683	—	2	2	no	—
61	C₂×C₃×#168	C₂³	1	2	yes	—
62	Dic₃⋊Dic₅	—	2	2	no	—
63	#6669	—	2	2	no	—
64	#6668	—	2	2	no	—
65	D₃×#168	—	2	4	no	—
66	D₃×C₄⋊Q₈	D₅	1	1	yes	characteristic
67	D₃×#125	—	5	1	no	—
68	C₃×#491	—	5	1	no	—
69	#5082	—	5	1	no	—
70	#5069	—	5	1	no	—
71	D₃×#103	—	5	4	no	—
72	#4797	—	5	4	no	—
73	D₃×#97	—	5	1	no	—
74	#4703	—	5	1	no	—
75	C₂²×C₅×Q₈	—	3	2	no	—
76	C₅×C₄⋊Q₈	—	3	2	no	—
77	C₅×C₄⋊Q₈	—	3	1	no	—
78	C₅×C₄⋊Q₈	—	6	2	no	—
79	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
80	C₂×C₅×C₄⋊C₄	—	6	2	no	—
81	C₂×C₄²×C₅	—	3	1	no	—
82	C₂×C₄⋊Dic₅	—	3	2	no	—
83	C₂×#683	—	6	2	no	—
84	C₂²×#168	—	3	2	no	—
85	#4016	—	3	1	no	—
86	#4016	—	3	2	no	—
87	#4016	—	6	2	no	—
88	#4016	C₂×D₃	1	1	yes	characteristic
89	#3967	—	3	4	no	—
90	#3967	—	3	4	no	—
91	#3967	—	3	4	no	—
92	#3967	C₂×D₃	1	4	yes	—
93	#3935	—	3	1	no	—
94	#3935	—	3	2	no	—
95	#3935	C₂×D₃	1	1	yes	characteristic
96	C₂×#491	—	15	1	no	Sylow
97	C₂²×C₅×D₃	C₂×D₄	1	1	yes	characteristic
98	C₃×C₅×Q₈	—	2	4	no	—
99	C₂×C₃×C₄×C₅	—	2	2	no	—
100	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
101	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
102	C₂×Dic₁₅	—	2	2	no	—
103	C₂×C₅×Dic₃	—	2	2	no	—
104	C₂×C₅×Dic₃	C₂×D₄	1	2	yes	—
105	C₂×C₅×Dic₃	C₂×D₄	1	1	yes	characteristic
106	C₄×C₅×D₃	—	2	4	no	—
107	C₄×C₅×D₃	—	4	4	no	—
108	C₄×C₅×D₃	#41	1	4	yes	—
109	C₅×Dic₆	—	2	4	no	—
110	C₅×Dic₆	—	4	4	no	—
111	C₂×C₃×Dic₅	—	2	2	no	—
112	D₃×Dic₅	—	4	4	no	—
113	C₃⋊#168	—	2	2	no	—
114	C₃×#168	—	2	2	no	—
115	C₂×D₃×Q₈	C₂×D₅	1	2	yes	—
116	C₃×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
117	C₃×#125	—	5	1	no	—
118	#944	C₂×D₅	1	2	yes	—
119	C₂×D₃×C₈	—	5	2	no	—
120	D₃×C₄⋊C₄	—	2	2	no	—
121	D₃×C₄⋊C₄	—	5	2	no	—
122	D₃×C₄⋊C₄	—	10	2	no	—
123	#890	—	5	2	no	—
124	#889	—	10	2	no	—
125	#888	—	2	2	no	—
126	C₄²×D₃	C₂×D₅	1	1	yes	characteristic
127	#870	—	5	1	no	—
128	#869	C₂×D₅	1	1	yes	characteristic
129	C₃×#103	—	5	4	no	—
130	C₃×#97	—	5	1	no	—
131	#835	—	5	4	no	—
132	#816	—	5	4	no	—
133	#815	—	5	2	no	—
134	#810	—	5	4	no	—
135	#803	—	5	1	no	—
136	C₂×C₅×Q₈	—	3	2	no	—
137	C₂×C₅×Q₈	—	6	2	no	—
138	C₂×C₅×Q₈	—	6	8	no	—
139	C₂×C₅×Q₈	C₂²×D₃	1	2	yes	—
140	C₂²×C₄×C₅	—	3	1	no	—
141	C₂²×C₄×C₅	—	3	2	no	—
142	C₂²×C₄×C₅	—	6	2	no	—
143	C₂²×Dic₅	—	6	2	no	—
144	C₅×C₄⋊C₄	—	2	2	no	—
145	C₅×C₄⋊C₄	—	6	4	no	—
146	C₅×C₄⋊C₄	—	6	2	no	—
147	C₄²×C₅	—	3	2	no	—
148	C₄²×C₅	—	3	1	no	—
149	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
150	C₄⋊Dic₅	—	3	2	no	—
151	C₄⋊Dic₅	—	6	2	no	—
152	C₄⋊Dic₅	C₂²×D₃	1	2	yes	—
153	#683	—	2	2	no	—
154	#683	—	6	2	no	—
155	#683	—	6	2	no	—
156	#683	—	6	2	no	—
157	C₂×#168	—	3	2	no	—
158	C₂×#168	—	6	4	no	—
159	C₂×#168	C₂²×D₃	1	2	yes	—
160	C₂×C₄⋊Q₈	—	3	1	no	—
161	C₂×#125	—	15	1	no	—
162	#491	—	5	1	no	—
163	#491	—	15	1	no	—
164	#491	—	15	4	no	—
165	#491	—	15	1	no	—
166	#491	—	15	1	no	—
167	C₂×#103	—	15	4	no	—
168	C₂×#97	—	15	1	no	—
169	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
170	C₂×C₅×D₃	#124	1	2	yes	—
171	C₂×C₅×D₃	#137	1	4	yes	—
172	C₃×C₄×C₅	—	2	2	no	—
173	C₃×C₄×C₅	—	4	2	no	—
174	C₃×C₄×C₅	C₂×#41	1	2	yes	—
175	Dic₁₅	—	4	2	no	—
176	C₃×Dic₅	—	4	2	no	—
177	C₅×Dic₃	—	2	2	no	—
178	C₅×Dic₃	—	4	2	no	—
179	C₅×Dic₃	#124	1	2	yes	—
180	C₂×C₃×Q₈	C₂²×D₅	1	2	yes	—
181	D₃×Q₈	—	2	8	no	—
182	C₂×C₄×D₃	—	2	2	no	—
183	C₂×C₄×D₃	—	10	2	no	—
184	C₂×C₄×D₃	C₂²⋊D₅	1	2	yes	—
185	C₂×C₄×D₃	C₂²×D₅	1	1	yes	characteristic
186	C₂×Dic₆	—	2	2	no	—
187	C₂×Dic₆	C₂²×D₅	1	2	yes	—
188	C₂×C₃×C₈	—	5	2	no	—
189	C₃×C₄⋊C₄	—	2	2	no	—
190	C₃×C₄⋊C₄	—	5	2	no	—
191	C₃×C₄⋊C₄	—	10	2	no	—
192	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
193	C₄⋊Dic₃	—	2	2	no	—
194	C₄⋊Dic₃	—	5	2	no	—
195	C₄⋊Dic₃	—	10	2	no	—
196	Dic₃⋊C₄	—	2	4	no	—
197	Dic₃⋊C₄	—	10	2	no	—
198	Dic₃⋊C₄	—	10	2	no	—
199	Dic₃⋊C₄	—	10	2	no	—
200	C₄×Dic₃	C₂²⋊D₅	1	2	yes	—
201	C₄×Dic₃	C₂²×D₅	1	1	yes	characteristic
202	C₂×C₃⋊C₈	—	5	2	no	—
203	D₃×C₈	—	10	4	no	—
204	C₂³×C₅	—	3	1	no	—
205	C₅×Q₈	—	2	4	no	—
206	C₅×Q₈	—	6	4	no	—
207	C₅×Q₈	—	12	4	no	—
208	C₂×C₄×C₅	—	2	2	no	—
209	C₂×C₄×C₅	—	3	1	no	—
210	C₂×C₄×C₅	—	3	4	no	—
211	C₂×C₄×C₅	—	3	2	no	—
212	C₂×C₄×C₅	—	6	4	no	—
213	C₂×C₄×C₅	—	6	2	no	—
214	C₂×C₄×C₅	—	12	4	no	—
215	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
216	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
217	C₂×Dic₅	—	2	2	no	—
218	C₂×Dic₅	—	6	2	no	—
219	C₂×Dic₅	—	12	4	no	—
220	#168	—	2	2	no	—
221	#168	—	6	2	no	—
222	C₂²×Q₈	—	3	2	no	—
223	C₂²×C₈	—	15	2	no	—
224	C₄⋊Q₈	—	3	2	no	—
225	C₄⋊Q₈	—	3	1	no	—
226	C₄⋊Q₈	—	6	2	no	—
227	C₄⋊Q₈	D₃×D₅	1	1	yes	characteristic
228	#125	—	5	1	no	—
229	#125	—	15	2	no	—
230	#125	—	15	1	no	—
231	#125	—	30	2	no	—
232	C₂×C₄⋊C₄	—	6	2	no	—
233	C₂×C₄⋊C₄	—	15	2	no	—
234	C₂×C₄⋊C₄	—	30	2	no	—
235	C₂×C₄²	—	3	1	no	—
236	#103	—	5	4	no	—
237	#103	—	15	4	no	—
238	#103	—	15	4	no	—
239	#103	—	15	4	no	—
240	#97	—	5	1	no	—
241	#97	—	15	2	no	—
242	#97	—	15	1	no	—
243	C₂×C₃×C₅	C₂×#124	1	1	yes	characteristic
244	C₂×C₃×C₅	C₂×#137	1	2	yes	—
245	C₅×D₃	#491	1	4	yes	complemented
246	C₂²×D₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
247	C₃×Q₈	—	2	4	no	—
248	C₂×C₃×C₄	—	2	2	no	—
249	C₂×C₃×C₄	—	10	2	no	—
250	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
251	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
252	C₂×Dic₃	—	2	2	no	—
253	C₂×Dic₃	—	10	2	no	—
254	C₂×Dic₃	C₂×C₂²⋊D₅	1	2	yes	—
255	C₂×Dic₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
256	C₄×D₃	—	2	4	no	—
257	C₄×D₃	—	4	4	no	—
258	C₄×D₃	—	20	4	no	—
259	C₄×D₃	#711	1	4	yes	—
260	Dic₆	—	2	4	no	—
261	Dic₆	—	4	4	no	—
262	C₃×C₈	—	10	2	no	—
263	C₃⋊C₈	—	10	2	no	—
264	C₂²×C₅	—	3	2	no	—
265	C₂²×C₅	—	3	4	no	—
266	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
267	C₄×C₅	—	2	2	no	—
268	C₄×C₅	—	3	2	no	—
269	C₄×C₅	—	4	2	no	—
270	C₄×C₅	—	6	2	no	—
271	C₄×C₅	—	12	2	no	—
272	C₄×C₅	D₃×#41	1	2	yes	—
273	Dic₅	—	4	2	no	—
274	Dic₅	—	12	2	no	—
275	C₂×Q₈	—	3	2	no	—
276	C₂×Q₈	—	6	2	no	—
277	C₂×Q₈	—	6	8	no	—
278	C₂×Q₈	C₂×D₃×D₅	1	2	yes	—
279	C₂²×C₄	—	3	1	no	—
280	C₂²×C₄	—	3	2	no	—
281	C₂²×C₄	—	6	2	no	—
282	C₂²×C₄	—	30	2	no	—
283	C₂×C₈	—	5	2	no	—
284	C₂×C₈	—	15	2	no	—
285	C₂×C₈	—	30	4	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	5	2	no	—
288	C₄⋊C₄	—	6	4	no	—
289	C₄⋊C₄	—	6	2	no	—
290	C₄⋊C₄	—	10	2	no	—
291	C₄⋊C₄	—	15	2	no	—
292	C₄⋊C₄	—	30	2	no	—
293	C₄⋊C₄	—	30	2	no	—
294	C₄⋊C₄	—	30	2	no	—
295	C₄⋊C₄	—	30	2	no	—
296	C₄²	—	3	2	no	—
297	C₄²	—	3	1	no	—
298	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
299	C₃×C₅	C₂×#491	1	1	yes	characteristic, complemented, Hall
300	C₂²×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
301	C₂×D₃	#4082	1	2	yes	—
302	C₂×D₃	#4088	1	4	yes	—
303	C₃×C₄	—	2	2	no	—
304	C₃×C₄	—	4	2	no	—
305	C₃×C₄	—	20	2	no	—
306	C₃×C₄	C₂×#711	1	2	yes	—
307	Dic₃	—	2	2	no	—
308	Dic₃	—	4	2	no	—
309	Dic₃	—	20	2	no	—
310	Dic₃	#4082	1	2	yes	—
311	C₂×C₅	—	3	4	no	—
312	C₂×C₅	D₃×#124	1	1	yes	characteristic
313	C₂×C₅	D₃×#137	1	2	yes	—
314	C₂³	—	3	1	no	—
315	Q₈	—	2	4	no	—
316	Q₈	—	6	4	no	—
317	Q₈	—	12	4	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	3	1	no	—
320	C₂×C₄	—	3	4	no	—
321	C₂×C₄	—	3	2	no	—
322	C₂×C₄	—	6	4	no	—
323	C₂×C₄	—	6	2	no	—
324	C₂×C₄	—	10	2	no	—
325	C₂×C₄	—	12	4	no	—
326	C₂×C₄	—	30	2	no	—
327	C₂×C₄	—	60	4	no	—
328	C₂×C₄	D₃×C₂²⋊D₅	1	2	yes	—
329	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
330	C₈	—	10	2	no	—
331	C₈	—	30	2	no	—
332	C₂×C₃	C₂×#4082	1	1	yes	characteristic
333	C₂×C₃	C₂×#4088	1	2	yes	—
334	D₃	#65371	1	4	yes	direct product factor
335	C₅	D₃×#491	1	1	yes	characteristic, complemented, Sylow
336	C₂²	—	3	2	no	—
337	C₂²	—	3	4	no	—
338	C₂²	C₂×D₃×C₂²⋊D₅	1	1	yes	characteristic
339	C₄	—	2	2	no	—
340	C₄	—	3	2	no	—
341	C₄	—	4	2	no	—
342	C₄	—	6	2	no	—
343	C₄	—	12	2	no	—
344	C₄	—	20	2	no	—
345	C₄	—	60	2	no	—
346	C₄	D₃×#711	1	2	yes	—
347	C₃	C₂×#65371	1	1	yes	characteristic, complemented, Sylow
348	C₂	—	3	4	no	—
349	C₂	D₃×#4082	1	1	yes	characteristic
350	C₂	D₃×#4088	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₃×C₂²⋊D₅
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₃×C₅×C₄⋊Q₈	C₂²
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂²×C₅×D₃	C₂×D₄
hypercenter	C₄⋊Q₈	D₃×D₅
hypocenter	C₃×C₅	C₂×#491
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
3. #11751183	C₂	semidirect
4. C₃×#65371	C₂	semidirect
6. #11749192	C₂	semidirect
8. #11746931	C₂	semidirect
9. #11746928	C₂	semidirect
40. #65371	D₃	direct (N × H)
245. C₅×D₃	#491	semidirect
299. C₃×C₅	C₂×#491	semidirect
334. D₃	#65371	direct (N × H)
335. C₅	D₃×#491	semidirect
347. C₃	C₂×#65371	semidirect

Maximal subgroups

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

subgroup	index
C₅×D₃×C₄⋊Q₈	2
D₃×#4016	2
#11751183	2
C₃×#65371	2
D₃×#3967	2
#11749192	2
D₃×#3935	2
#11746931	2
#11746928	2
C₂×#65371	3
D₃×#491	5

Subgroups — D3×#65371 #14437985

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×#65371 #14437985