Subgroups — #14478517

Order 1920.

740 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11753971	C₂	1	2	yes	maximal, complemented
2	#11751005	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4901	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×D₈₀	C₂	1	1	yes	maximal, characteristic, complemented
5	#11747543	C₂	1	2	yes	maximal, complemented
6	#11746835	C₂	1	2	yes	maximal, complemented
7	#11746820	C₂	1	4	yes	maximal, complemented
8	D₃×D₈₀	C₂	1	2	yes	maximal, complemented
9	#10613744	—	3	1	no	maximal, Hall
10	#92017	—	2	2	no	—
11	#91793	C₂²	1	2	yes	complemented
12	C₅×#901	C₂²	1	1	yes	characteristic
13	C₂×C₃×D₄₀	C₂²	1	2	yes	complemented
14	#91312	—	2	2	no	—
15	#91273	C₂²	1	4	yes	complemented
16	#91254	—	2	4	no	—
17	D₃×D₄₀	C₂²	1	4	yes	complemented
18	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic, complemented
19	Dic₁₂₀	C₂²	1	1	yes	characteristic, complemented
20	#91084	C₂²	1	2	yes	complemented
21	D₂₄₀	C₂²	1	1	yes	characteristic, complemented
22	C₅×#812	C₂²	1	1	yes	characteristic, complemented
23	C₅×#798	C₂²	1	2	yes	complemented
24	C₅×D₃×C₁₆	C₂²	1	2	yes	complemented
25	C₃×D₈₀	C₂²	1	2	yes	complemented
26	C₃×D₈₀	C₂²	1	2	yes	complemented
27	#90942	C₂²	1	4	yes	complemented
28	#90938	C₂²	1	4	yes	complemented
29	#82019	—	5	1	no	maximal, Hall
30	#66086	—	3	2	no	—
31	C₅×#504	—	3	1	no	—
32	#65197	—	3	2	no	—
33	#65197	—	3	4	no	—
34	#65193	—	3	1	no	—
35	#65193	—	3	2	no	—
36	C₂×D₈₀	—	3	2	no	—
37	C₂×D₈₀	D₃	1	1	yes	characteristic, complemented
38	#6817	—	2	2	no	—
39	C₅×#235	D₄	1	1	yes	characteristic
40	C₂×C₃×D₂₀	—	2	2	no	—
41	D₅×#67	—	4	4	no	—
42	#6777	—	4	2	no	—
43	D₃×D₂₀	—	2	4	no	—
44	#6766	—	4	2	no	—
45	#6765	—	2	4	no	—
46	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
47	Dic₆₀	C₂³	1	2	yes	—
48	D₁₂₀	C₂³	1	2	yes	—
49	#6706	—	2	2	no	—
50	C₅×#208	D₄	1	1	yes	characteristic
51	C₅×ΓL(1,𝔽₂₅)	D₄	1	2	yes	—
52	C₅×D₃×C₈	C₂³	1	2	yes	—
53	C₃×D₄₀	—	2	2	no	—
54	C₃×D₄₀	C₂³	1	4	yes	—
55	#6657	—	2	4	no	—
56	#6653	—	2	4	no	—
57	C₃×C₅×C₁₆	C₂³	1	1	yes	characteristic
58	C₃×C₅×C₁₆	C₂³	1	1	yes	characteristic
59	C₅×C₃⋊C₁₆	C₂³	1	2	yes	—
60	#5757	—	5	2	no	—
61	C₂×C₃×D₁₆	—	5	1	no	—
62	#5147	—	5	2	no	—
63	#4912	—	5	2	no	—
64	#4911	—	5	4	no	—
65	D₃×D₁₆	—	5	2	no	—
66	#4904	—	5	1	no	—
67	#4901	D₅	1	1	yes	characteristic, complemented
68	#4149	—	6	2	no	—
69	C₅×#131	—	3	1	no	—
70	#4054	—	6	4	no	—
71	#4054	—	6	2	no	—
72	#4050	—	3	2	no	—
73	#4050	—	3	4	no	—
74	C₂×D₄₀	—	3	4	no	—
75	C₂×D₄₀	C₂×D₃	1	2	yes	complemented
76	C₅×M₃₂	—	3	1	no	—
77	C₅×M₃₂	—	3	2	no	—
78	C₂×C₅×C₁₆	—	3	2	no	—
79	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic, complemented
80	Dic₄₀	—	3	1	no	—
81	#3932	—	3	4	no	—
82	#3932	—	3	2	no	—
83	D₈₀	—	3	4	no	—
84	D₈₀	—	3	1	no	—
85	D₈₀	C₂×D₃	1	2	yes	complemented
86	D₈₀	C₂×D₃	1	2	yes	complemented
87	#3415	—	15	1	no	Sylow
88	C₂²×C₃×D₅	—	4	2	no	—
89	C₂×D₃×D₅	—	4	4	no	—
90	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
91	#1220	—	4	2	no	—
92	D₆₀	—	2	2	no	—
93	C₄×D₁₅	—	4	2	no	—
94	Dic₃₀	—	2	2	no	—
95	C₅×#67	D₈	1	2	yes	—
96	C₅×D₁₂	D₈	1	1	yes	characteristic
97	C₄×C₅×D₃	C₂×D₄	1	2	yes	—
98	C₅×Dic₆	D₈	1	1	yes	characteristic
99	C₃×D₂₀	—	2	4	no	—
100	C₃×D₂₀	—	4	2	no	—
101	#1202	—	4	4	no	—
102	#1201	—	4	4	no	—
103	D₅×Dic₃	—	4	4	no	—
104	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
105	C₃×C₅×C₈	C₂⁴	1	1	yes	characteristic
106	C₅×C₃⋊C₈	C₂×D₄	1	2	yes	—
107	#1004	—	10	2	no	—
108	C₂×C₃×D₈	—	5	2	no	—
109	#932	—	10	2	no	—
110	#912	—	5	4	no	—
111	#911	—	10	4	no	—
112	D₃×D₈	—	5	4	no	—
113	#904	—	5	2	no	—
114	#901	C₂×D₅	1	1	yes	characteristic
115	C₃×D₁₆	—	5	2	no	—
116	C₃×D₁₆	—	5	2	no	—
117	C₂×C₃×C₁₆	C₂×D₅	1	1	yes	characteristic, complemented
118	#827	—	5	4	no	—
119	#826	—	5	4	no	—
120	#812	C₂×D₅	1	1	yes	characteristic, complemented
121	Dic₂₄	—	5	1	no	—
122	#800	—	5	2	no	—
123	D₄₈	—	5	1	no	—
124	#798	C₂×D₅	1	2	yes	complemented
125	D₃×C₁₆	C₂×D₅	1	2	yes	complemented
126	C₅×#41	—	3	1	no	—
127	#713	—	12	2	no	—
128	D₄×D₅	—	12	2	no	—
129	D₄×D₅	—	12	4	no	—
130	#709	—	6	4	no	—
131	#709	—	6	2	no	—
132	C₂×D₂₀	—	2	2	no	—
133	C₂×D₂₀	—	6	4	no	—
134	C₅×M₁₆	—	3	1	no	—
135	C₅×M₁₆	—	3	2	no	—
136	C₂×C₅×C₈	—	3	2	no	—
137	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
138	Dic₂₀	—	3	2	no	—
139	D₄₀	—	2	2	no	—
140	D₄₀	—	3	2	no	—
141	D₄₀	—	6	4	no	—
142	D₄₀	C₂²×D₃	1	4	yes	—
143	#677	—	6	4	no	—
144	#677	—	6	2	no	—
145	C₅×C₁₆	—	3	2	no	—
146	C₅×C₁₆	C₂²×D₃	1	1	yes	characteristic
147	C₅×C₁₆	C₂²×D₃	1	1	yes	characteristic
148	#576	—	15	2	no	—
149	#509	—	15	4	no	—
150	#509	—	15	2	no	—
151	#508	—	15	2	no	—
152	#508	—	15	1	no	—
153	C₂×D₁₆	—	5	1	no	—
154	C₂×D₁₆	—	15	2	no	—
155	#504	—	3	1	no	—
156	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
157	C₂×D₁₅	—	4	2	no	—
158	C₂×C₅×D₃	C₂×D₈	1	2	yes	—
159	C₂×C₃×D₅	—	4	4	no	—
160	C₂×C₃×D₅	—	16	2	no	—
161	D₃×D₅	—	16	4	no	—
162	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
163	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
164	Dic₁₅	—	4	2	no	—
165	C₅×Dic₃	C₂×D₈	1	2	yes	—
166	C₂×C₃×D₄	—	10	2	no	—
167	C₂×#67	—	20	4	no	—
168	#237	—	10	4	no	—
169	#237	—	20	2	no	—
170	D₃×D₄	—	10	4	no	—
171	D₃×D₄	—	20	2	no	—
172	#235	—	10	2	no	—
173	#235	D₂₀	1	1	yes	characteristic
174	C₃×D₈	—	5	4	no	—
175	C₃×D₈	—	10	2	no	—
176	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
177	#214	—	10	4	no	—
178	#213	—	10	4	no	—
179	#208	D₂₀	1	1	yes	characteristic
180	Dic₁₂	—	5	2	no	—
181	D₂₄	—	5	2	no	—
182	#204	—	10	2	no	—
183	ΓL(1,𝔽₂₅)	D₂₀	1	2	yes	—
184	D₃×C₈	C₂²×D₅	1	2	yes	—
185	C₃×C₁₆	C₂²×D₅	1	1	yes	characteristic
186	C₃×C₁₆	C₂²×D₅	1	1	yes	characteristic
187	C₃⋊C₁₆	C₂²×D₅	1	2	yes	—
188	C₂²×D₅	—	4	2	no	—
189	C₂²×D₅	—	12	4	no	—
190	C₅×Q₈	—	3	1	no	—
191	C₅×D₄	—	3	2	no	—
192	C₅×D₄	—	3	1	no	—
193	C₂×C₄×C₅	—	3	2	no	—
194	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
195	C₂²⋊D₅	—	12	4	no	—
196	C₂²⋊D₅	—	12	2	no	—
197	D₂₀	—	2	4	no	—
198	D₂₀	—	4	2	no	—
199	D₂₀	—	6	2	no	—
200	D₂₀	—	12	4	no	—
201	C₄×D₅	—	12	4	no	—
202	C₄×D₅	—	12	2	no	—
203	Dic₁₀	—	6	2	no	—
204	C₅×C₈	—	3	2	no	—
205	C₅×C₈	D₃×D₄	1	1	yes	characteristic
206	C₅×C₈	C₂³×D₃	1	1	yes	characteristic
207	H(5,𝔽₂)	—	30	2	no	—
208	AGL(1,ℤ/8)	—	30	4	no	—
209	AGL(1,ℤ/8)	—	30	2	no	—
210	#135	—	15	4	no	—
211	#135	—	15	2	no	—
212	C₂×D₈	—	5	2	no	—
213	C₂×D₈	—	15	4	no	—
214	#131	—	3	1	no	—
215	Q₃₂	—	15	1	no	—
216	SD₃₂	—	15	4	no	—
217	SD₃₂	—	15	2	no	—
218	D₁₆	—	5	2	no	—
219	D₁₆	—	5	2	no	—
220	D₁₆	—	15	4	no	—
221	D₁₆	—	15	1	no	—
222	M₃₂	—	3	1	no	—
223	M₃₂	—	3	2	no	—
224	C₂×C₁₆	—	3	2	no	—
225	C₂×C₁₆	D₃×D₅	1	1	yes	characteristic, complemented
226	C₂×C₃×C₅	—	2	1	no	—
227	C₂×C₃×C₅	C₂²×D₈	1	1	yes	characteristic
228	D₁₅	—	8	2	no	—
229	C₃×D₅	—	8	4	no	—
230	C₅×D₃	—	2	2	no	—
231	C₂³×C₃	—	20	2	no	—
232	C₂²×D₃	—	20	4	no	—
233	C₃×D₄	—	10	4	no	—
234	C₃×D₄	—	20	2	no	—
235	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
236	#67	—	20	2	no	—
237	#67	—	20	4	no	—
238	#67	—	20	4	no	—
239	#67	D₄₀	1	2	yes	—
240	C₂×Dic₃	—	20	4	no	—
241	D₁₂	—	10	2	no	—
242	D₁₂	D₄₀	1	1	yes	characteristic
243	C₄×D₃	—	20	2	no	—
244	C₄×D₃	C₂×D₂₀	1	2	yes	—
245	Dic₆	—	10	2	no	—
246	Dic₆	D₄₀	1	1	yes	characteristic
247	C₃×C₈	C₂×D₂₀	1	1	yes	characteristic
248	C₃×C₈	C₂³×D₅	1	1	yes	characteristic
249	C₃⋊C₈	C₂×D₂₀	1	2	yes	—
250	C₂²×C₅	—	3	2	no	—
251	C₂²×C₅	D₃×D₈	1	1	yes	characteristic
252	C₂×D₅	—	4	4	no	—
253	C₂×D₅	—	12	2	no	—
254	C₂×D₅	—	16	2	no	—
255	C₂×D₅	—	48	4	no	—
256	C₄×C₅	—	3	2	no	—
257	C₄×C₅	D₃×D₈	1	1	yes	characteristic
258	C₄×C₅	C₂×D₃×D₄	1	1	yes	characteristic
259	Dic₅	—	12	2	no	—
260	#41	—	3	1	no	—
261	#41	—	30	4	no	—
262	#41	—	30	2	no	—
263	#41	—	60	2	no	—
264	C₂×D₄	—	10	2	no	—
265	C₂×D₄	—	30	4	no	—
266	C₂×D₄	—	60	2	no	—
267	C₂×D₄	—	60	4	no	—
268	Q₁₆	—	15	2	no	—
269	SD₁₆	—	30	4	no	—
270	SD₁₆	—	30	2	no	—
271	D₈	—	5	4	no	—
272	D₈	—	10	2	no	—
273	D₈	—	15	2	no	—
274	D₈	—	30	4	no	—
275	M₁₆	—	3	1	no	—
276	M₁₆	—	3	2	no	—
277	C₂×C₈	—	3	2	no	—
278	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
279	C₁₆	—	3	2	no	—
280	C₁₆	C₂×D₃×D₅	1	1	yes	characteristic
281	C₁₆	C₂×D₃×D₅	1	1	yes	characteristic
282	C₃×C₅	#3415	1	1	yes	characteristic, complemented, Hall
283	C₂²×C₃	—	20	4	no	—
284	C₂²×C₃	—	80	2	no	—
285	C₂²×C₃	C₂×D₄₀	1	1	yes	characteristic
286	C₂×D₃	—	20	2	no	—
287	C₂×D₃	—	80	4	no	—
288	C₂×D₃	C₂×D₄₀	1	2	yes	—
289	C₃×C₄	C₂×D₄₀	1	1	yes	characteristic
290	C₃×C₄	C₂²×D₂₀	1	1	yes	characteristic
291	Dic₃	—	20	2	no	—
292	Dic₃	C₂×D₄₀	1	2	yes	—
293	C₂×C₅	—	2	1	no	—
294	C₂×C₅	—	6	2	no	—
295	C₂×C₅	C₂×D₃×D₈	1	1	yes	characteristic
296	D₅	—	8	4	no	—
297	D₅	—	24	2	no	—
298	C₂³	—	20	2	no	—
299	C₂³	—	60	4	no	—
300	Q₈	—	3	1	no	—
301	Q₈	—	30	2	no	—
302	D₄	—	3	2	no	—
303	D₄	—	3	1	no	—
304	D₄	—	10	4	no	—
305	D₄	—	20	2	no	—
306	D₄	—	30	2	no	—
307	D₄	—	60	4	no	—
308	D₄	—	60	4	no	—
309	D₄	—	60	2	no	—
310	C₂×C₄	—	3	2	no	—
311	C₂×C₄	—	60	4	no	—
312	C₂×C₄	—	60	2	no	—
313	C₂×C₄	D₃×D₂₀	1	1	yes	characteristic
314	C₈	—	3	2	no	—
315	C₈	D₃×D₂₀	1	1	yes	characteristic
316	C₈	C₂²×D₃×D₅	1	1	yes	characteristic
317	C₂×C₃	—	2	1	no	—
318	C₂×C₃	—	40	4	no	—
319	C₂×C₃	C₂²×D₄₀	1	1	yes	characteristic
320	D₃	—	2	2	no	—
321	D₃	—	40	2	no	—
322	C₅	#82019	1	1	yes	characteristic, complemented, Sylow
323	C₂²	—	3	2	no	—
324	C₂²	—	20	4	no	—
325	C₂²	—	60	2	no	—
326	C₂²	—	80	2	no	—
327	C₂²	—	240	4	no	—
328	C₂²	D₃×D₄₀	1	1	yes	characteristic
329	C₄	—	3	2	no	—
330	C₄	—	60	2	no	—
331	C₄	D₃×D₄₀	1	1	yes	characteristic
332	C₄	C₂×D₃×D₂₀	1	1	yes	characteristic
333	C₃	#10613744	1	1	yes	characteristic, complemented, Sylow
334	C₂	—	2	1	no	—
335	C₂	—	6	2	no	—
336	C₂	—	40	4	no	—
337	C₂	—	120	2	no	—
338	C₂	C₂×D₃×D₄₀	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₃×D₄₀
commutator, cosocle	C₃×C₅×C₈	C₂⁴
Frattini	C₈	C₂²×D₃×D₅
Fitting	C₂×C₃×C₅×C₁₆	C₂²
socle	C₂×C₃×C₅	C₂²×D₈
hypercenter	C₂×C₁₆	D₃×D₅
hypocenter	C₃×C₅	#3415
LCS	G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
1. #11753971	C₂	semidirect
2. #11751005	C₂	semidirect
3. C₅×#4901	C₂	semidirect
4. C₂×C₃×D₈₀	C₂	semidirect
5. #11747543	C₂	semidirect
6. #11746835	C₂	semidirect
7. #11746820	C₂	semidirect
8. D₃×D₈₀	C₂	semidirect
11. #91793	C₂²	semidirect
13. C₂×C₃×D₄₀	C₂²	semidirect
15. #91273	C₂²	semidirect
17. D₃×D₄₀	C₂²	semidirect
18. C₂×C₃×C₅×C₁₆	C₂²	semidirect
19. Dic₁₂₀	C₂²	semidirect
20. #91084	C₂²	semidirect
21. D₂₄₀	C₂²	semidirect
22. C₅×#812	C₂²	semidirect
23. C₅×#798	C₂²	semidirect
24. C₅×D₃×C₁₆	C₂²	semidirect
25. C₃×D₈₀	C₂²	semidirect
26. C₃×D₈₀	C₂²	semidirect
27. #90942	C₂²	semidirect
28. #90938	C₂²	semidirect
37. C₂×D₈₀	D₃	semidirect
67. #4901	D₅	semidirect
75. C₂×D₄₀	C₂×D₃	semidirect
79. C₂×C₅×C₁₆	C₂×D₃	semidirect
85. D₈₀	C₂×D₃	semidirect
86. D₈₀	C₂×D₃	semidirect
117. C₂×C₃×C₁₆	C₂×D₅	semidirect
120. #812	C₂×D₅	semidirect
124. #798	C₂×D₅	semidirect
125. D₃×C₁₆	C₂×D₅	semidirect
225. C₂×C₁₆	D₃×D₅	semidirect
282. C₃×C₅	#3415	semidirect
322. C₅	#82019	semidirect
333. C₃	#10613744	semidirect

Maximal subgroups

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

subgroup	index
#11753971	2
#11751005	2
C₅×#4901	2
C₂×C₃×D₈₀	2
#11747543	2
#11746835	2
#11746820	2
D₃×D₈₀	2
#10613744	3
#82019	5