Subgroups — #14478487

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	#11756516	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₅×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#11751248	C₂	1	2	yes	maximal, complemented
4	#11751013	C₂	1	2	yes	maximal, complemented
5	#11751012	C₂	1	4	yes	maximal, complemented
6	D₁₅×D₁₆	C₂	1	2	yes	maximal, complemented
7	#11751005	C₂	1	1	yes	maximal, characteristic, complemented
8	#11751002	C₂	1	1	yes	maximal, characteristic, complemented
9	#10613750	—	3	1	no	maximal, Hall
10	#92095	—	2	2	no	—
11	C₂×C₃×C₅×D₈	C₂²	1	2	yes	complemented
12	#91821	—	2	2	no	—
13	#91801	C₂²	1	4	yes	complemented
14	#91800	—	2	4	no	—
15	D₈×D₁₅	C₂²	1	4	yes	complemented
16	#91793	C₂²	1	2	yes	complemented
17	#91790	C₂²	1	1	yes	characteristic
18	C₃×C₅×D₁₆	C₂²	1	2	yes	complemented
19	C₃×C₅×D₁₆	C₂²	1	2	yes	complemented
20	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic, complemented
21	#91111	C₂²	1	4	yes	complemented
22	#91110	C₂²	1	4	yes	complemented
23	#91096	C₂²	1	1	yes	characteristic, complemented
24	Dic₁₂₀	C₂²	1	1	yes	characteristic, complemented
25	#91084	C₂²	1	2	yes	complemented
26	D₂₄₀	C₂²	1	1	yes	characteristic, complemented
27	#91082	C₂²	1	2	yes	complemented
28	C₁₆×D₁₅	C₂²	1	2	yes	complemented
29	#82019	—	5	1	no	maximal, Hall
30	#66091	—	3	2	no	—
31	C₂×C₅×D₁₆	D₃	1	1	yes	characteristic, complemented
32	#65436	—	3	2	no	—
33	#65201	—	3	2	no	—
34	#65200	—	3	4	no	—
35	D₅×D₁₆	—	3	2	no	—
36	#65193	—	3	1	no	—
37	#65190	—	3	1	no	—
38	C₂×C₃×C₅×D₄	—	2	2	no	—
39	C₂×#1220	—	4	4	no	—
40	#6819	—	2	4	no	—
41	#6819	—	4	2	no	—
42	D₄×D₁₅	—	2	4	no	—
43	D₄×D₁₅	—	4	2	no	—
44	#6817	—	2	2	no	—
45	#6817	D₄	1	1	yes	characteristic
46	C₃×C₅×D₈	—	2	2	no	—
47	C₃×C₅×D₈	C₂³	1	4	yes	—
48	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
49	#6716	—	2	4	no	—
50	#6715	—	2	4	no	—
51	#6710	D₄	1	1	yes	characteristic
52	Dic₆₀	C₂³	1	2	yes	—
53	D₁₂₀	C₂³	1	2	yes	—
54	#6706	—	2	2	no	—
55	#6705	D₄	1	2	yes	—
56	C₈×D₁₅	C₂³	1	2	yes	—
57	C₃×C₅×C₁₆	C₂³	1	1	yes	characteristic
58	C₃×C₅×C₁₆	C₂³	1	1	yes	characteristic
59	C₃⋊#672	C₂³	1	2	yes	—
60	#5757	—	5	2	no	—
61	C₂×C₃×D₁₆	D₅	1	1	yes	characteristic, complemented
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	—	5	1	no	—
68	#4144	—	6	2	no	—
69	C₂×C₅×D₈	C₂×D₃	1	2	yes	complemented
70	#4078	—	6	2	no	—
71	#4058	—	3	4	no	—
72	#4057	—	6	4	no	—
73	D₅×D₈	—	3	4	no	—
74	#4050	—	3	2	no	—
75	#4047	—	3	1	no	—
76	C₅×D₁₆	C₂×D₃	1	2	yes	complemented
77	C₅×D₁₆	C₂×D₃	1	2	yes	complemented
78	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic, complemented
79	#3959	—	3	4	no	—
80	#3958	—	3	4	no	—
81	#3944	—	3	1	no	—
82	Dic₄₀	—	3	1	no	—
83	#3932	—	3	2	no	—
84	D₈₀	—	3	1	no	—
85	#3930	—	3	2	no	—
86	D₅×C₁₆	—	3	2	no	—
87	#3415	—	15	1	no	Sylow
88	C₂³×C₃×C₅	—	4	2	no	—
89	C₂²×D₁₅	—	4	4	no	—
90	C₃×C₅×D₄	—	2	4	no	—
91	C₃×C₅×D₄	—	4	2	no	—
92	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
93	#1220	—	4	2	no	—
94	#1220	—	4	4	no	—
95	#1220	—	4	4	no	—
96	#1220	D₈	1	2	yes	—
97	C₂×Dic₁₅	—	4	4	no	—
98	D₆₀	—	2	2	no	—
99	D₆₀	D₈	1	1	yes	characteristic
100	C₄×D₁₅	—	4	2	no	—
101	C₄×D₁₅	C₂×D₄	1	2	yes	—
102	Dic₃₀	—	2	2	no	—
103	Dic₃₀	D₈	1	1	yes	characteristic
104	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
105	C₃×C₅×C₈	C₂⁴	1	1	yes	characteristic
106	C₃⋊#168	C₂×D₄	1	2	yes	—
107	#1004	—	10	2	no	—
108	C₂×C₃×D₈	C₂×D₅	1	2	yes	complemented
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	—	5	1	no	—
115	C₃×D₁₆	C₂×D₅	1	2	yes	complemented
116	C₃×D₁₆	C₂×D₅	1	2	yes	complemented
117	C₂×C₃×C₁₆	C₂×D₅	1	1	yes	characteristic, complemented
118	#827	—	5	4	no	—
119	#826	—	5	4	no	—
120	#812	—	5	1	no	—
121	Dic₂₄	—	5	1	no	—
122	#800	—	5	2	no	—
123	D₄₈	—	5	1	no	—
124	#798	—	5	2	no	—
125	D₃×C₁₆	—	5	2	no	—
126	C₂×C₅×D₄	—	2	2	no	—
127	C₂×C₂²⋊D₅	—	12	4	no	—
128	#711	—	6	4	no	—
129	#711	—	12	2	no	—
130	D₄×D₅	—	6	4	no	—
131	D₄×D₅	—	12	2	no	—
132	#709	—	3	1	no	—
133	#709	—	6	2	no	—
134	C₅×D₈	—	2	2	no	—
135	C₅×D₈	C₂²×D₃	1	4	yes	—
136	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
137	#687	—	6	4	no	—
138	#686	—	6	4	no	—
139	#681	—	3	1	no	—
140	Dic₂₀	—	3	2	no	—
141	D₄₀	—	3	2	no	—
142	#677	—	6	2	no	—
143	#676	—	3	2	no	—
144	C₈×D₅	—	3	2	no	—
145	C₅×C₁₆	C₂²×D₃	1	1	yes	characteristic
146	C₅×C₁₆	C₂²×D₃	1	1	yes	characteristic
147	#672	—	3	2	no	—
148	#576	—	15	2	no	—
149	#509	—	15	2	no	—
150	#509	—	15	4	no	—
151	#508	—	15	2	no	—
152	#508	—	15	1	no	—
153	C₂×D₁₆	—	15	2	no	—
154	C₂×D₁₆	D₁₅	1	1	yes	characteristic, complemented
155	#504	—	15	1	no	—
156	C₂²×C₃×C₅	—	4	4	no	—
157	C₂²×C₃×C₅	—	16	2	no	—
158	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
159	C₂×D₁₅	—	4	2	no	—
160	C₂×D₁₅	—	16	4	no	—
161	C₂×D₁₅	C₂×D₈	1	2	yes	—
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	Dic₁₅	C₂×D₈	1	2	yes	—
166	C₂×C₃×D₄	—	2	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	—	5	1	no	—
173	#235	—	10	2	no	—
174	C₃×D₈	—	2	2	no	—
175	C₃×D₈	C₂²×D₅	1	4	yes	—
176	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
177	#214	—	10	4	no	—
178	#213	—	10	4	no	—
179	#208	—	5	1	no	—
180	Dic₁₂	—	5	2	no	—
181	D₂₄	—	5	2	no	—
182	#204	—	10	2	no	—
183	ΓL(1,𝔽₂₅)	—	5	2	no	—
184	D₃×C₈	—	5	2	no	—
185	C₃×C₁₆	C₂²×D₅	1	1	yes	characteristic
186	C₃×C₁₆	C₂²×D₅	1	1	yes	characteristic
187	C₃⋊C₁₆	—	5	2	no	—
188	C₂³×C₅	—	4	2	no	—
189	C₂²×D₅	—	12	4	no	—
190	C₅×D₄	—	2	4	no	—
191	C₅×D₄	—	4	2	no	—
192	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
193	C₂²⋊D₅	—	3	2	no	—
194	C₂²⋊D₅	—	12	4	no	—
195	C₂²⋊D₅	—	12	4	no	—
196	C₂²⋊D₅	—	12	2	no	—
197	C₂×Dic₅	—	12	4	no	—
198	D₂₀	—	3	1	no	—
199	D₂₀	—	6	2	no	—
200	C₄×D₅	—	3	2	no	—
201	C₄×D₅	—	12	2	no	—
202	Dic₁₀	—	3	1	no	—
203	Dic₁₀	—	6	2	no	—
204	C₅×C₈	D₃×D₄	1	1	yes	characteristic
205	C₅×C₈	C₂³×D₃	1	1	yes	characteristic
206	#168	—	3	2	no	—
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	—	15	4	no	—
213	C₂×D₈	C₂×D₁₅	1	2	yes	complemented
214	#131	—	15	1	no	—
215	Q₃₂	—	15	1	no	—
216	SD₃₂	—	15	4	no	—
217	SD₃₂	—	15	2	no	—
218	D₁₆	—	15	4	no	—
219	D₁₆	—	15	1	no	—
220	D₁₆	C₂×D₁₅	1	2	yes	complemented
221	D₁₆	C₂×D₁₅	1	2	yes	complemented
222	M₃₂	—	15	1	no	—
223	M₃₂	—	15	2	no	—
224	C₂×C₁₆	—	15	2	no	—
225	C₂×C₁₆	C₂×D₁₅	1	1	yes	characteristic, complemented
226	C₂×C₃×C₅	—	2	1	no	—
227	C₂×C₃×C₅	—	8	4	no	—
228	C₂×C₃×C₅	C₂²×D₈	1	1	yes	characteristic
229	D₁₅	—	2	2	no	—
230	D₁₅	—	8	2	no	—
231	C₂³×C₃	—	4	2	no	—
232	C₂²×D₃	—	20	4	no	—
233	C₃×D₄	—	2	4	no	—
234	C₃×D₄	—	4	2	no	—
235	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
236	#67	—	5	2	no	—
237	#67	—	20	4	no	—
238	#67	—	20	4	no	—
239	#67	—	20	2	no	—
240	C₂×Dic₃	—	20	4	no	—
241	D₁₂	—	5	1	no	—
242	D₁₂	—	10	2	no	—
243	C₄×D₃	—	5	2	no	—
244	C₄×D₃	—	20	2	no	—
245	Dic₆	—	5	1	no	—
246	Dic₆	—	10	2	no	—
247	C₃×C₈	D₄×D₅	1	1	yes	characteristic
248	C₃×C₈	C₂³×D₅	1	1	yes	characteristic
249	C₃⋊C₈	—	5	2	no	—
250	C₂²×C₅	—	4	4	no	—
251	C₂²×C₅	—	16	2	no	—
252	C₂²×C₅	D₃×D₈	1	1	yes	characteristic
253	C₂×D₅	—	3	2	no	—
254	C₂×D₅	—	12	2	no	—
255	C₂×D₅	—	48	4	no	—
256	C₄×C₅	D₃×D₈	1	1	yes	characteristic
257	C₄×C₅	C₂×D₃×D₄	1	1	yes	characteristic
258	Dic₅	—	3	2	no	—
259	Dic₅	—	12	2	no	—
260	#41	—	15	1	no	—
261	#41	—	30	4	no	—
262	#41	—	30	2	no	—
263	#41	—	60	2	no	—
264	C₂×D₄	—	2	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	—	2	2	no	—
272	D₈	—	15	2	no	—
273	D₈	—	30	4	no	—
274	D₈	C₂²×D₁₅	1	4	yes	—
275	M₁₆	—	15	1	no	—
276	M₁₆	—	15	2	no	—
277	C₂×C₈	—	15	2	no	—
278	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
279	C₁₆	—	15	2	no	—
280	C₁₆	C₂²×D₁₅	1	1	yes	characteristic
281	C₁₆	C₂²×D₁₅	1	1	yes	characteristic
282	C₃×C₅	#3415	1	1	yes	characteristic, complemented, Hall
283	C₂²×C₃	—	4	4	no	—
284	C₂²×C₃	—	16	2	no	—
285	C₂²×C₃	D₅×D₈	1	1	yes	characteristic
286	C₂×D₃	—	5	2	no	—
287	C₂×D₃	—	20	2	no	—
288	C₂×D₃	—	80	4	no	—
289	C₃×C₄	D₅×D₈	1	1	yes	characteristic
290	C₃×C₄	C₂×D₄×D₅	1	1	yes	characteristic
291	Dic₃	—	5	2	no	—
292	Dic₃	—	20	2	no	—
293	C₂×C₅	—	2	1	no	—
294	C₂×C₅	—	8	4	no	—
295	C₂×C₅	C₂×D₃×D₈	1	1	yes	characteristic
296	D₅	—	6	2	no	—
297	D₅	—	24	2	no	—
298	C₂³	—	4	2	no	—
299	C₂³	—	60	4	no	—
300	Q₈	—	15	1	no	—
301	Q₈	—	30	2	no	—
302	D₄	—	2	4	no	—
303	D₄	—	4	2	no	—
304	D₄	—	15	2	no	—
305	D₄	—	15	1	no	—
306	D₄	—	30	2	no	—
307	D₄	—	60	4	no	—
308	D₄	—	60	4	no	—
309	D₄	—	60	2	no	—
310	C₂×C₄	—	15	2	no	—
311	C₂×C₄	—	60	4	no	—
312	C₂×C₄	—	60	2	no	—
313	C₂×C₄	D₄×D₁₅	1	1	yes	characteristic
314	C₈	—	15	2	no	—
315	C₈	D₄×D₁₅	1	1	yes	characteristic
316	C₈	C₂³×D₁₅	1	1	yes	characteristic
317	C₂×C₃	—	2	1	no	—
318	C₂×C₃	—	8	4	no	—
319	C₂×C₃	C₂×D₅×D₈	1	1	yes	characteristic
320	D₃	—	10	2	no	—
321	D₃	—	40	2	no	—
322	C₅	#82019	1	1	yes	characteristic, complemented, Sylow
323	C₂²	—	4	4	no	—
324	C₂²	—	15	2	no	—
325	C₂²	—	16	2	no	—
326	C₂²	—	60	2	no	—
327	C₂²	—	240	4	no	—
328	C₂²	D₈×D₁₅	1	1	yes	characteristic
329	C₄	—	15	2	no	—
330	C₄	—	60	2	no	—
331	C₄	D₈×D₁₅	1	1	yes	characteristic
332	C₄	C₂×D₄×D₁₅	1	1	yes	characteristic
333	C₃	#10613750	1	1	yes	characteristic, complemented, Sylow
334	C₂	—	2	1	no	—
335	C₂	—	8	4	no	—
336	C₂	—	30	2	no	—
337	C₂	—	120	2	no	—
338	C₂	C₂×D₈×D₁₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₈×D₁₅
commutator, cosocle	C₃×C₅×C₈	C₂⁴
Frattini	C₈	C₂³×D₁₅
Fitting	C₂×C₃×C₅×D₁₆	C₂
socle	C₂×C₃×C₅	C₂²×D₈
hypercenter	C₂×D₁₆	D₁₅
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×D₁₆

Complements

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

N	H ≅ G/N	type
1. #11756516	C₂	semidirect
2. C₂×C₃×C₅×D₁₆	C₂	semidirect
3. #11751248	C₂	semidirect
4. #11751013	C₂	semidirect
5. #11751012	C₂	semidirect
6. D₁₅×D₁₆	C₂	semidirect
7. #11751005	C₂	semidirect
8. #11751002	C₂	semidirect
11. C₂×C₃×C₅×D₈	C₂²	semidirect
13. #91801	C₂²	semidirect
15. D₈×D₁₅	C₂²	semidirect
16. #91793	C₂²	semidirect
18. C₃×C₅×D₁₆	C₂²	semidirect
19. C₃×C₅×D₁₆	C₂²	semidirect
20. C₂×C₃×C₅×C₁₆	C₂²	semidirect
21. #91111	C₂²	semidirect
22. #91110	C₂²	semidirect
23. #91096	C₂²	semidirect
24. Dic₁₂₀	C₂²	semidirect
25. #91084	C₂²	semidirect
26. D₂₄₀	C₂²	semidirect
27. #91082	C₂²	semidirect
28. C₁₆×D₁₅	C₂²	semidirect
31. C₂×C₅×D₁₆	D₃	semidirect
61. C₂×C₃×D₁₆	D₅	semidirect
69. C₂×C₅×D₈	C₂×D₃	semidirect
76. C₅×D₁₆	C₂×D₃	semidirect
77. C₅×D₁₆	C₂×D₃	semidirect
78. C₂×C₅×C₁₆	C₂×D₃	semidirect
108. C₂×C₃×D₈	C₂×D₅	semidirect
115. C₃×D₁₆	C₂×D₅	semidirect
116. C₃×D₁₆	C₂×D₅	semidirect
117. C₂×C₃×C₁₆	C₂×D₅	semidirect
154. C₂×D₁₆	D₁₅	semidirect
213. C₂×D₈	C₂×D₁₅	semidirect
220. D₁₆	C₂×D₁₅	semidirect
221. D₁₆	C₂×D₁₅	semidirect
225. C₂×C₁₆	C₂×D₁₅	semidirect
282. C₃×C₅	#3415	semidirect
322. C₅	#82019	semidirect
333. C₃	#10613750	semidirect

Maximal subgroups

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

subgroup	index
#11756516	2
C₂×C₃×C₅×D₁₆	2
#11751248	2
#11751013	2
#11751012	2
D₁₅×D₁₆	2
#11751005	2
#11751002	2
#10613750	3
#82019	5