Subgroups — D₃×#64984 #14438783

Order 1920.

1520 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃×D₄₀	C₂	1	4	yes	maximal, complemented
2	D₃×#4020	C₂	1	2	yes	maximal, complemented
3	#11750796	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₅×D₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64984	C₂	1	1	yes	maximal, characteristic, complemented
6	#11748005	C₂	1	4	yes	maximal, complemented
7	#11747035	C₂	1	2	yes	maximal, complemented
8	C₂×#64984	—	3	1	no	maximal, Hall
9	C₂×D₃×D₂₀	—	2	4	no	—
10	C₂×D₃×D₂₀	C₂²	1	4	yes	—
11	C₂×D₁₂₀	C₂²	1	4	yes	complemented
12	#91763	C₂²	1	2	yes	complemented
13	C₂×C₅×D₃×C₈	C₂²	1	2	yes	—
14	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
15	C₂×C₃×D₄₀	C₂²	1	4	yes	complemented
16	C₃×#4020	C₂²	1	2	yes	complemented
17	#91458	—	2	4	no	—
18	#91458	C₂²	1	4	yes	complemented
19	C₂×#6653	C₂²	1	8	yes	complemented
20	D₃×D₄₀	—	2	16	no	—
21	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic, complemented
22	C₅×C₈×Dic₃	C₂²	1	2	yes	complemented
23	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
24	D₃×#493	—	5	1	no	maximal, Hall
25	C₂²×D₄₀	—	3	4	no	—
26	C₂×#4020	—	3	2	no	—
27	C₂×C₄×C₅×C₈	—	3	1	no	—
28	#64984	—	3	4	no	—
29	#64984	—	3	2	no	—
30	#64984	—	3	1	no	—
31	#64984	D₃	1	1	yes	characteristic, direct product factor
32	C₂²×D₃×D₅	—	2	4	no	—
33	C₂×D₆₀	—	2	4	no	—
34	C₂×D₆₀	C₂³	1	4	yes	—
35	C₂×C₄×C₅×D₃	D₄	1	2	yes	—
36	C₂×C₄×C₅×D₃	C₂³	1	1	yes	characteristic
37	C₂×C₃×D₂₀	—	2	4	no	—
38	C₂×C₃×D₂₀	C₂³	1	4	yes	—
39	C₂×#1202	—	2	4	no	—
40	C₂×#1202	—	2	8	no	—
41	D₃×D₂₀	—	2	16	no	—
42	D₃×D₂₀	—	4	16	no	—
43	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
44	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
45	D₁₂₀	—	2	8	no	—
46	C₄×C₅×Dic₃	D₄	1	2	yes	—
47	C₄×C₅×Dic₃	C₂³	1	1	yes	characteristic
48	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
49	C₅×D₃×C₈	D₄	1	8	yes	complemented
50	C₃×D₄₀	—	2	8	no	—
51	#6653	—	2	16	no	—
52	C₂×D₃×D₈	—	5	4	no	—
53	D₃×#127	—	5	2	no	—
54	C₃×#493	—	5	1	no	—
55	#5151	—	5	4	no	—
56	#5073	—	5	2	no	—
57	#4695	—	5	1	no	—
58	C₄×D₃×C₈	D₅	1	1	yes	characteristic, complemented
59	C₂²×D₂₀	—	3	4	no	—
60	C₂²×D₂₀	—	6	4	no	—
61	C₂²×C₅×C₈	—	3	2	no	—
62	C₂×C₄²×C₅	—	3	1	no	—
63	C₂×D₄₀	—	3	8	no	—
64	C₂×D₄₀	—	3	4	no	—
65	C₂×D₄₀	—	6	16	no	—
66	C₂×D₄₀	C₂×D₃	1	4	yes	complemented
67	#4020	—	3	4	no	—
68	#4020	—	3	2	no	—
69	#4020	—	6	4	no	—
70	#4020	C₂×D₃	1	2	yes	complemented
71	C₄×C₅×C₈	—	3	2	no	—
72	C₄×C₅×C₈	—	3	1	no	—
73	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic, complemented
74	C₂×#493	—	15	1	no	Sylow
75	C₂²×D₁₅	—	2	4	no	—
76	C₂²×C₅×D₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₃×D₅	—	2	4	no	—
78	C₂×D₃×D₅	—	4	8	no	—
79	C₂×D₃×D₅	—	4	16	no	—
80	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
81	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
82	D₆₀	—	2	8	no	—
83	D₆₀	—	4	8	no	—
84	C₂×C₅×Dic₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₅×Dic₃	C₂×D₄	1	2	yes	—
86	C₄×C₅×D₃	D₈	1	8	yes	complemented
87	C₄×C₅×D₃	C₂×D₄	1	4	yes	—
88	C₃×D₂₀	—	2	8	no	—
89	C₃×D₂₀	—	4	8	no	—
90	#1202	—	4	8	no	—
91	#1202	—	4	16	no	—
92	C₃×C₅×C₈	C₂×D₄	1	4	yes	complemented
93	C₅×C₃⋊C₈	C₂×D₄	1	4	yes	complemented
94	C₂×D₃×D₄	—	5	4	no	—
95	C₂×D₃×D₄	—	10	4	no	—
96	C₂×C₃×D₈	—	5	4	no	—
97	C₃×#127	—	5	2	no	—
98	#940	—	5	4	no	—
99	#940	—	10	4	no	—
100	C₂×#213	—	5	8	no	—
101	D₃×D₈	—	10	16	no	—
102	C₂×D₂₄	—	5	4	no	—
103	C₂×D₃×C₈	C₂×D₅	1	2	yes	—
104	#874	—	5	2	no	—
105	C₄²×D₃	C₂×D₅	1	1	yes	characteristic
106	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic, complemented
107	C₈×Dic₃	C₂×D₅	1	2	yes	complemented
108	C₄×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic, complemented
109	C₂³×D₅	—	6	4	no	—
110	C₂²×C₄×C₅	—	3	2	no	—
111	C₂²×C₄×C₅	—	3	1	no	—
112	C₂×D₂₀	—	2	4	no	—
113	C₂×D₂₀	—	3	4	no	—
114	C₂×D₂₀	—	6	16	no	—
115	C₂×D₂₀	—	6	4	no	—
116	C₂×D₂₀	—	6	8	no	—
117	C₂×D₂₀	—	6	4	no	—
118	C₂×D₂₀	—	12	16	no	—
119	C₂×D₂₀	C₂²×D₃	1	4	yes	—
120	C₂×C₅×C₈	—	3	2	no	—
121	C₂×C₅×C₈	—	3	8	no	—
122	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
123	C₄²×C₅	—	3	2	no	—
124	C₄²×C₅	—	3	1	no	—
125	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
126	D₄₀	—	2	8	no	—
127	D₄₀	—	6	16	no	—
128	D₄₀	—	6	8	no	—
129	C₂²×D₈	—	15	4	no	—
130	C₂×#127	—	15	2	no	—
131	#493	—	5	1	no	—
132	#493	—	15	2	no	—
133	#493	—	15	4	no	—
134	#493	—	15	1	no	—
135	C₂×C₄×C₈	—	3	1	no	—
136	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
137	C₂×D₁₅	—	4	4	no	—
138	C₂×D₁₅	—	4	8	no	—
139	C₂×C₅×D₃	#127	1	2	yes	—
140	C₂×C₅×D₃	C₂×D₈	1	4	yes	—
141	C₂×C₃×D₅	—	4	4	no	—
142	C₂×C₃×D₅	—	4	8	no	—
143	D₃×D₅	—	8	16	no	—
144	C₃×C₄×C₅	C₂×D₈	1	4	yes	complemented
145	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
146	C₅×Dic₃	#127	1	2	yes	—
147	C₅×Dic₃	C₂×D₈	1	4	yes	complemented
148	C₂³×D₃	—	10	4	no	—
149	C₂×C₃×D₄	—	5	4	no	—
150	C₂×C₃×D₄	—	10	4	no	—
151	C₂×#67	—	10	8	no	—
152	C₂×#67	—	10	4	no	—
153	D₃×D₄	—	10	16	no	—
154	D₃×D₄	—	20	16	no	—
155	C₂×D₁₂	—	5	4	no	—
156	C₂×D₁₂	—	10	4	no	—
157	C₂×C₄×D₃	D₂₀	1	2	yes	—
158	C₂×C₄×D₃	C₂²×D₅	1	1	yes	characteristic
159	C₃×D₈	—	10	8	no	—
160	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
161	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
162	#213	—	10	16	no	—
163	C₄×Dic₃	D₂₀	1	2	yes	—
164	C₄×Dic₃	C₂²×D₅	1	1	yes	characteristic
165	C₂×C₃⋊C₈	C₂²×D₅	1	2	yes	—
166	D₂₄	—	10	8	no	—
167	D₃×C₈	D₂₀	1	8	yes	complemented
168	C₂³×C₅	—	3	1	no	—
169	C₂²×D₅	—	2	4	no	—
170	C₂²×D₅	—	6	4	no	—
171	C₂²×D₅	—	12	8	no	—
172	C₂²×D₅	—	12	16	no	—
173	C₂×C₄×C₅	—	3	1	no	—
174	C₂×C₄×C₅	—	3	2	no	—
175	C₂×C₄×C₅	—	3	8	no	—
176	C₂×C₄×C₅	—	3	4	no	—
177	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
178	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
179	D₂₀	—	2	8	no	—
180	D₂₀	—	4	8	no	—
181	D₂₀	—	6	8	no	—
182	D₂₀	—	12	8	no	—
183	D₂₀	—	12	16	no	—
184	D₂₀	—	12	8	no	—
185	C₅×C₈	—	3	4	no	—
186	C₅×C₈	D₃×D₄	1	4	yes	complemented
187	C₂²×D₄	—	15	4	no	—
188	C₂²×D₄	—	30	4	no	—
189	C₂×D₈	—	5	4	no	—
190	C₂×D₈	—	15	8	no	—
191	C₂×D₈	—	15	4	no	—
192	C₂×D₈	—	30	16	no	—
193	C₂²×C₈	—	3	2	no	—
194	#127	—	5	2	no	—
195	#127	—	15	4	no	—
196	#127	—	15	2	no	—
197	#127	—	30	4	no	—
198	C₂×C₄²	—	3	1	no	—
199	C₄×C₈	—	3	2	no	—
200	C₄×C₈	—	3	1	no	—
201	C₄×C₈	D₃×D₅	1	1	yes	characteristic, complemented
202	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
203	C₂×C₃×C₅	C₂²×D₈	1	2	yes	—
204	D₁₅	—	8	4	no	—
205	C₃×D₅	—	8	4	no	—
206	C₅×D₃	#493	1	4	yes	complemented
207	C₂³×C₃	—	10	4	no	—
208	C₂²×D₃	—	10	4	no	—
209	C₂²×D₃	—	20	8	no	—
210	C₂²×D₃	—	20	16	no	—
211	C₂²×D₃	C₂×D₂₀	1	1	yes	characteristic
212	C₃×D₄	—	10	8	no	—
213	C₃×D₄	—	20	8	no	—
214	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
215	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
216	#67	—	20	8	no	—
217	#67	—	20	16	no	—
218	C₂×Dic₃	C₂×D₂₀	1	1	yes	characteristic
219	C₂×Dic₃	C₂×D₂₀	1	2	yes	—
220	D₁₂	—	10	8	no	—
221	D₁₂	—	20	8	no	—
222	C₄×D₃	D₄₀	1	8	yes	complemented
223	C₄×D₃	C₂×D₂₀	1	4	yes	—
224	C₃×C₈	C₂×D₂₀	1	4	yes	complemented
225	C₃⋊C₈	C₂×D₂₀	1	4	yes	complemented
226	C₂²×C₅	—	3	2	no	—
227	C₂²×C₅	—	3	4	no	—
228	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
229	C₂×D₅	—	4	4	no	—
230	C₂×D₅	—	4	8	no	—
231	C₂×D₅	—	12	4	no	—
232	C₂×D₅	—	12	8	no	—
233	C₂×D₅	—	24	16	no	—
234	C₄×C₅	—	3	2	no	—
235	C₄×C₅	—	3	4	no	—
236	C₄×C₅	D₃×D₈	1	4	yes	complemented
237	C₄×C₅	C₂×D₃×D₄	1	2	yes	—
238	C₂⁴	—	30	4	no	—
239	C₂×D₄	—	5	4	no	—
240	C₂×D₄	—	10	4	no	—
241	C₂×D₄	—	15	4	no	—
242	C₂×D₄	—	30	16	no	—
243	C₂×D₄	—	30	8	no	—
244	C₂×D₄	—	30	4	no	—
245	C₂×D₄	—	30	4	no	—
246	C₂×D₄	—	60	16	no	—
247	C₂²×C₄	—	3	2	no	—
248	C₂²×C₄	—	3	1	no	—
249	D₈	—	10	8	no	—
250	D₈	—	30	16	no	—
251	D₈	—	30	8	no	—
252	C₂×C₈	—	3	2	no	—
253	C₂×C₈	—	3	8	no	—
254	C₂×C₈	C₂×D₃×D₅	1	2	yes	—
255	C₄²	—	3	2	no	—
256	C₄²	—	3	1	no	—
257	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
258	C₃×C₅	C₂×#493	1	1	yes	characteristic, complemented, Hall
259	C₂²×C₃	—	20	4	no	—
260	C₂²×C₃	—	20	8	no	—
261	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
262	C₂×D₃	—	20	4	no	—
263	C₂×D₃	—	20	8	no	—
264	C₂×D₃	—	40	16	no	—
265	C₂×D₃	#4020	1	2	yes	—
266	C₂×D₃	C₂×D₄₀	1	4	yes	—
267	C₃×C₄	C₂×D₄₀	1	4	yes	complemented
268	C₃×C₄	C₂²×D₂₀	1	2	yes	—
269	Dic₃	#4020	1	2	yes	—
270	Dic₃	C₂×D₄₀	1	4	yes	complemented
271	C₂×C₅	—	3	4	no	—
272	C₂×C₅	D₃×#127	1	1	yes	characteristic
273	C₂×C₅	C₂×D₃×D₈	1	2	yes	—
274	D₅	—	8	4	no	—
275	D₅	—	24	4	no	—
276	C₂³	—	3	1	no	—
277	C₂³	—	10	4	no	—
278	C₂³	—	30	4	no	—
279	C₂³	—	60	8	no	—
280	C₂³	—	60	16	no	—
281	D₄	—	10	8	no	—
282	D₄	—	20	8	no	—
283	D₄	—	30	8	no	—
284	D₄	—	60	8	no	—
285	D₄	—	60	16	no	—
286	D₄	—	60	8	no	—
287	C₂×C₄	—	3	1	no	—
288	C₂×C₄	—	3	2	no	—
289	C₂×C₄	—	3	4	no	—
290	C₂×C₄	—	3	8	no	—
291	C₂×C₄	D₃×D₂₀	1	2	yes	—
292	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
293	C₈	—	3	4	no	—
294	C₈	D₃×D₂₀	1	4	yes	complemented
295	C₂×C₃	—	40	4	no	—
296	C₂×C₃	C₂×#4020	1	1	yes	characteristic
297	C₂×C₃	C₂²×D₄₀	1	2	yes	—
298	D₃	—	40	4	no	—
299	D₃	#64984	1	4	yes	direct product factor
300	C₅	D₃×#493	1	1	yes	characteristic, complemented, Sylow
301	C₂²	—	3	2	no	—
302	C₂²	—	3	4	no	—
303	C₂²	—	20	4	no	—
304	C₂²	—	20	8	no	—
305	C₂²	—	60	4	no	—
306	C₂²	—	60	8	no	—
307	C₂²	—	120	16	no	—
308	C₂²	C₂×D₃×D₂₀	1	1	yes	characteristic
309	C₄	—	3	2	no	—
310	C₄	—	3	4	no	—
311	C₄	D₃×D₄₀	1	4	yes	complemented
312	C₄	C₂×D₃×D₂₀	1	2	yes	—
313	C₃	C₂×#64984	1	1	yes	characteristic, complemented, Sylow
314	C₂	—	3	4	no	—
315	C₂	—	40	4	no	—
316	C₂	—	120	4	no	—
317	C₂	D₃×#4020	1	1	yes	characteristic
318	C₂	C₂×D₃×D₄₀	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₃×D₂₀
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₃×C₄×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	C₂²×D₄
hypercenter	C₄×C₈	D₃×D₅
hypocenter	C₃×C₅	C₂×#493
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×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
1. C₂×D₃×D₄₀	C₂	semidirect
2. D₃×#4020	C₂	semidirect
3. #11750796	C₂	semidirect
4. C₄×C₅×D₃×C₈	C₂	semidirect
5. C₃×#64984	C₂	semidirect
6. #11748005	C₂	semidirect
7. #11747035	C₂	semidirect
11. C₂×D₁₂₀	C₂²	semidirect
12. #91763	C₂²	semidirect
15. C₂×C₃×D₄₀	C₂²	semidirect
16. C₃×#4020	C₂²	semidirect
18. #91458	C₂²	semidirect
19. C₂×#6653	C₂²	semidirect
21. C₃×C₄×C₅×C₈	C₂²	semidirect
22. C₅×C₈×Dic₃	C₂²	semidirect
23. C₄×C₅×C₃⋊C₈	C₂²	semidirect
31. #64984	D₃	direct (N × H)
49. C₅×D₃×C₈	D₄	semidirect
58. C₄×D₃×C₈	D₅	semidirect
66. C₂×D₄₀	C₂×D₃	semidirect
70. #4020	C₂×D₃	semidirect
73. C₄×C₅×C₈	C₂×D₃	semidirect
86. C₄×C₅×D₃	D₈	semidirect
92. C₃×C₅×C₈	C₂×D₄	semidirect
93. C₅×C₃⋊C₈	C₂×D₄	semidirect
106. C₃×C₄×C₈	C₂×D₅	semidirect
107. C₈×Dic₃	C₂×D₅	semidirect
108. C₄×C₃⋊C₈	C₂×D₅	semidirect
144. C₃×C₄×C₅	C₂×D₈	semidirect
147. C₅×Dic₃	C₂×D₈	semidirect
167. D₃×C₈	D₂₀	semidirect
186. C₅×C₈	D₃×D₄	semidirect
201. C₄×C₈	D₃×D₅	semidirect
206. C₅×D₃	#493	semidirect
222. C₄×D₃	D₄₀	semidirect
224. C₃×C₈	C₂×D₂₀	semidirect
225. C₃⋊C₈	C₂×D₂₀	semidirect
236. C₄×C₅	D₃×D₈	semidirect
258. C₃×C₅	C₂×#493	semidirect
267. C₃×C₄	C₂×D₄₀	semidirect
270. Dic₃	C₂×D₄₀	semidirect
294. C₈	D₃×D₂₀	semidirect
299. D₃	#64984	direct (N × H)
300. C₅	D₃×#493	semidirect
311. C₄	D₃×D₄₀	semidirect
313. C₃	C₂×#64984	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃×D₄₀	2
D₃×#4020	2
#11750796	2
C₄×C₅×D₃×C₈	2
C₃×#64984	2
#11748005	2
#11747035	2
C₂×#64984	3
D₃×#493	5

Subgroups — D3×#64984 #14438783

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×#64984 #14438783