Subgroups — #14478520

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	#11753963	C₂	1	2	yes	maximal, complemented
2	C₂×D₂₄₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4904	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65190	C₂	1	1	yes	maximal, characteristic, complemented
5	#11747548	C₂	1	2	yes	maximal, complemented
6	#11746834	C₂	1	2	yes	maximal, complemented
7	#11746817	C₂	1	4	yes	maximal, complemented
8	D₅×D₄₈	C₂	1	2	yes	maximal, complemented
9	#10613762	—	3	1	no	maximal, Hall
10	#92019	—	2	2	no	—
11	C₂×D₁₂₀	C₂²	1	2	yes	complemented
12	C₅×#904	C₂²	1	2	yes	complemented
13	C₃×#4047	C₂²	1	1	yes	characteristic
14	#91305	—	2	2	no	—
15	#91271	C₂²	1	4	yes	complemented
16	#91249	—	2	4	no	—
17	D₅×D₂₄	C₂²	1	4	yes	complemented
18	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic, complemented
19	D₂₄₀	C₂²	1	2	yes	complemented
20	D₂₄₀	C₂²	1	2	yes	complemented
21	C₅×Dic₂₄	C₂²	1	1	yes	characteristic, complemented
22	C₅×#800	C₂²	1	2	yes	complemented
23	C₅×D₄₈	C₂²	1	1	yes	characteristic, complemented
24	C₃×#3944	C₂²	1	1	yes	characteristic, complemented
25	C₃×#3930	C₂²	1	2	yes	complemented
26	C₃×D₅×C₁₆	C₂²	1	2	yes	complemented
27	#90945	C₂²	1	4	yes	complemented
28	#90939	C₂²	1	4	yes	complemented
29	#82013	—	5	1	no	maximal, Hall
30	#66103	—	3	2	no	—
31	C₅×#508	—	3	1	no	—
32	#65482	—	3	2	no	—
33	#65208	—	3	2	no	—
34	#65203	—	3	4	no	—
35	D₅×D₁₆	—	3	2	no	—
36	C₂×D₈₀	—	3	1	no	—
37	#65190	D₃	1	1	yes	characteristic, complemented
38	C₂×D₆₀	—	2	2	no	—
39	C₅×#235	—	2	2	no	—
40	C₃×#709	D₄	1	1	yes	characteristic
41	#6790	—	4	4	no	—
42	D₃×D₂₀	—	4	2	no	—
43	D₅×D₁₂	—	2	4	no	—
44	#6773	—	2	4	no	—
45	#6769	—	4	2	no	—
46	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
47	D₁₂₀	—	2	2	no	—
48	D₁₂₀	C₂³	1	4	yes	—
49	C₅×Dic₁₂	C₂³	1	2	yes	—
50	C₅×D₂₄	C₂³	1	2	yes	—
51	C₅×#204	—	2	2	no	—
52	C₃×#681	D₄	1	1	yes	characteristic
53	C₃×#676	D₄	1	2	yes	—
54	C₃×C₈×D₅	C₂³	1	2	yes	—
55	#6660	—	2	4	no	—
56	#6654	—	2	4	no	—
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	#5752	—	5	2	no	—
61	C₃×#504	—	5	1	no	—
62	#4908	—	5	2	no	—
63	#4908	—	5	4	no	—
64	#4904	—	5	2	no	—
65	#4904	D₅	1	1	yes	characteristic, complemented
66	C₂×D₄₈	—	5	1	no	—
67	C₂×D₄₈	—	5	2	no	—
68	#4149	—	6	2	no	—
69	C₅×#135	—	3	2	no	—
70	#4095	—	6	2	no	—
71	#4065	—	3	4	no	—
72	#4060	—	6	4	no	—
73	D₅×D₈	—	3	4	no	—
74	C₂×D₄₀	—	3	2	no	—
75	#4047	C₂×D₃	1	1	yes	characteristic
76	C₅×Q₃₂	—	3	1	no	—
77	C₅×SD₃₂	—	3	2	no	—
78	C₅×D₁₆	—	3	1	no	—
79	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic, complemented
80	#3960	—	3	4	no	—
81	#3958	—	3	4	no	—
82	#3944	C₂×D₃	1	1	yes	characteristic, complemented
83	D₈₀	—	3	2	no	—
84	D₈₀	—	3	2	no	—
85	#3930	C₂×D₃	1	2	yes	complemented
86	D₅×C₁₆	C₂×D₃	1	2	yes	complemented
87	#3415	—	15	1	no	Sylow
88	C₂²×D₁₅	—	4	2	no	—
89	C₂×D₃×D₅	—	4	4	no	—
90	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
91	D₆₀	—	2	4	no	—
92	D₆₀	—	4	2	no	—
93	C₅×#67	—	4	2	no	—
94	C₅×D₁₂	—	2	2	no	—
95	C₄×C₅×D₃	—	4	2	no	—
96	C₅×Dic₆	—	2	2	no	—
97	C₃×C₂²⋊D₅	D₈	1	2	yes	—
98	C₃×D₂₀	D₈	1	1	yes	characteristic
99	C₃×C₄×D₅	C₂×D₄	1	2	yes	—
100	C₃×Dic₁₀	D₈	1	1	yes	characteristic
101	#1203	—	4	4	no	—
102	#1202	—	4	4	no	—
103	#1200	—	4	4	no	—
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	#1009	—	10	2	no	—
108	C₃×#131	—	5	1	no	—
109	#908	—	10	4	no	—
110	#908	—	10	2	no	—
111	#904	—	5	4	no	—
112	#904	C₂×D₅	1	2	yes	complemented
113	C₂×D₂₄	—	5	2	no	—
114	C₂×D₂₄	—	5	4	no	—
115	C₃×M₃₂	—	5	1	no	—
116	C₃×M₃₂	—	5	2	no	—
117	C₂×C₃×C₁₆	—	5	2	no	—
118	C₂×C₃×C₁₆	C₂×D₅	1	1	yes	characteristic, complemented
119	Dic₂₄	C₂×D₅	1	1	yes	characteristic, complemented
120	#800	—	5	4	no	—
121	#800	C₂×D₅	1	2	yes	complemented
122	D₄₈	—	5	4	no	—
123	D₄₈	—	5	2	no	—
124	D₄₈	—	5	2	no	—
125	D₄₈	C₂×D₅	1	1	yes	characteristic, complemented
126	C₅×#41	—	6	2	no	—
127	#713	—	6	4	no	—
128	D₄×D₅	—	6	4	no	—
129	D₄×D₅	—	12	4	no	—
130	#709	—	12	2	no	—
131	#709	D₁₂	1	1	yes	characteristic
132	C₂×D₂₀	—	6	2	no	—
133	C₂×D₂₀	—	12	2	no	—
134	C₅×Q₁₆	—	3	2	no	—
135	C₅×SD₁₆	—	6	2	no	—
136	C₅×D₈	—	3	2	no	—
137	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
138	#688	—	6	4	no	—
139	#686	—	6	4	no	—
140	#681	D₁₂	1	1	yes	characteristic
141	D₄₀	—	3	4	no	—
142	D₄₀	—	6	2	no	—
143	#676	D₁₂	1	2	yes	—
144	C₈×D₅	C₂²×D₃	1	2	yes	—
145	C₅×C₁₆	C₂²×D₃	1	1	yes	characteristic
146	C₅×C₁₆	C₂²×D₃	1	1	yes	characteristic
147	#672	C₂²×D₃	1	2	yes	—
148	#576	—	15	2	no	—
149	#509	—	15	4	no	—
150	#509	—	15	2	no	—
151	#508	—	3	1	no	—
152	#508	—	15	2	no	—
153	C₂×D₁₆	—	15	2	no	—
154	C₂×D₁₆	—	15	1	no	—
155	#504	—	5	1	no	—
156	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
157	C₂×D₁₅	—	4	4	no	—
158	C₂×D₁₅	—	16	2	no	—
159	C₂×C₅×D₃	—	4	2	no	—
160	C₂×C₃×D₅	C₂×D₈	1	2	yes	—
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	C₃×Dic₅	C₂×D₈	1	2	yes	—
165	C₅×Dic₃	—	4	2	no	—
166	C₃×#41	—	5	1	no	—
167	#239	—	20	2	no	—
168	D₃×D₄	—	20	2	no	—
169	D₃×D₄	—	20	4	no	—
170	#235	—	2	2	no	—
171	#235	—	10	4	no	—
172	C₂×D₁₂	—	10	4	no	—
173	C₂×D₁₂	—	10	2	no	—
174	C₃×M₁₆	—	5	1	no	—
175	C₃×M₁₆	—	5	2	no	—
176	C₂×C₃×C₈	—	5	2	no	—
177	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
178	Dic₁₂	C₂²×D₅	1	2	yes	—
179	D₂₄	—	5	4	no	—
180	D₂₄	—	10	4	no	—
181	D₂₄	—	10	2	no	—
182	D₂₄	C₂²×D₅	1	2	yes	—
183	#204	—	2	2	no	—
184	#204	—	10	4	no	—
185	C₃×C₁₆	—	5	2	no	—
186	C₃×C₁₆	C₂²×D₅	1	1	yes	characteristic
187	C₃×C₁₆	C₂²×D₅	1	1	yes	characteristic
188	C₂²×D₅	—	12	2	no	—
189	C₂²×D₅	—	12	4	no	—
190	C₅×Q₈	—	6	2	no	—
191	C₅×D₄	—	6	2	no	—
192	C₅×D₄	—	12	2	no	—
193	C₂×C₄×C₅	—	12	2	no	—
194	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
195	C₂²⋊D₅	—	12	4	no	—
196	C₂²⋊D₅	D₂₄	1	2	yes	—
197	D₂₀	—	6	4	no	—
198	D₂₀	—	12	4	no	—
199	D₂₀	—	12	2	no	—
200	D₂₀	D₂₄	1	1	yes	characteristic
201	C₄×D₅	—	12	4	no	—
202	C₄×D₅	C₂×D₁₂	1	2	yes	—
203	Dic₁₀	D₂₄	1	1	yes	characteristic
204	C₅×C₈	C₂×D₁₂	1	1	yes	characteristic
205	C₅×C₈	C₂³×D₃	1	1	yes	characteristic
206	#168	C₂×D₁₂	1	2	yes	—
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	—	3	2	no	—
211	#135	—	15	4	no	—
212	C₂×D₈	—	15	4	no	—
213	C₂×D₈	—	15	2	no	—
214	#131	—	5	1	no	—
215	Q₃₂	—	3	1	no	—
216	SD₃₂	—	3	2	no	—
217	SD₃₂	—	15	4	no	—
218	D₁₆	—	3	1	no	—
219	D₁₆	—	15	4	no	—
220	D₁₆	—	15	2	no	—
221	D₁₆	—	15	2	no	—
222	M₃₂	—	5	1	no	—
223	M₃₂	—	5	2	no	—
224	C₂×C₁₆	—	5	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	4	no	—
229	C₃×D₅	—	2	2	no	—
230	C₅×D₃	—	8	2	no	—
231	C₂²×D₃	—	20	4	no	—
232	C₂²×D₃	—	20	2	no	—
233	C₃×Q₈	—	5	1	no	—
234	C₃×D₄	—	5	2	no	—
235	C₃×D₄	—	5	1	no	—
236	C₂×C₃×C₄	—	5	2	no	—
237	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
238	#67	—	4	2	no	—
239	#67	—	20	4	no	—
240	D₁₂	—	2	2	no	—
241	D₁₂	—	10	4	no	—
242	D₁₂	—	20	4	no	—
243	D₁₂	—	20	2	no	—
244	C₄×D₃	—	4	2	no	—
245	C₄×D₃	—	20	4	no	—
246	Dic₆	—	2	2	no	—
247	C₃×C₈	—	5	2	no	—
248	C₃×C₈	D₄×D₅	1	1	yes	characteristic
249	C₃×C₈	C₂³×D₅	1	1	yes	characteristic
250	C₂²×C₅	—	12	2	no	—
251	C₂²×C₅	C₂×D₂₄	1	1	yes	characteristic
252	C₂×D₅	—	12	4	no	—
253	C₂×D₅	—	48	4	no	—
254	C₂×D₅	—	48	2	no	—
255	C₂×D₅	C₂×D₂₄	1	2	yes	—
256	C₄×C₅	—	12	2	no	—
257	C₄×C₅	C₂×D₂₄	1	1	yes	characteristic
258	C₄×C₅	C₂²×D₁₂	1	1	yes	characteristic
259	Dic₅	C₂×D₂₄	1	2	yes	—
260	#41	—	5	1	no	—
261	#41	—	6	2	no	—
262	#41	—	30	4	no	—
263	#41	—	60	2	no	—
264	C₂×D₄	—	30	4	no	—
265	C₂×D₄	—	30	2	no	—
266	C₂×D₄	—	60	2	no	—
267	C₂×D₄	—	60	4	no	—
268	Q₁₆	—	3	2	no	—
269	SD₁₆	—	6	2	no	—
270	SD₁₆	—	30	4	no	—
271	D₈	—	3	2	no	—
272	D₈	—	15	4	no	—
273	D₈	—	30	4	no	—
274	D₈	—	30	2	no	—
275	M₁₆	—	5	1	no	—
276	M₁₆	—	5	2	no	—
277	C₂×C₈	—	5	2	no	—
278	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
279	C₁₆	—	5	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	—	5	2	no	—
284	C₂²×C₃	D₅×D₈	1	1	yes	characteristic
285	C₂×D₃	—	4	2	no	—
286	C₂×D₃	—	20	4	no	—
287	C₂×D₃	—	80	4	no	—
288	C₂×D₃	—	80	2	no	—
289	C₃×C₄	—	5	2	no	—
290	C₃×C₄	D₅×D₈	1	1	yes	characteristic
291	C₃×C₄	C₂×D₄×D₅	1	1	yes	characteristic
292	Dic₃	—	4	2	no	—
293	C₂×C₅	—	2	1	no	—
294	C₂×C₅	—	24	2	no	—
295	C₂×C₅	C₂²×D₂₄	1	1	yes	characteristic
296	D₅	—	2	2	no	—
297	D₅	—	24	4	no	—
298	C₂³	—	60	4	no	—
299	C₂³	—	60	2	no	—
300	Q₈	—	5	1	no	—
301	Q₈	—	6	2	no	—
302	D₄	—	5	2	no	—
303	D₄	—	5	1	no	—
304	D₄	—	6	2	no	—
305	D₄	—	12	2	no	—
306	D₄	—	30	4	no	—
307	D₄	—	60	4	no	—
308	D₄	—	60	4	no	—
309	D₄	—	60	2	no	—
310	C₂×C₄	—	5	2	no	—
311	C₂×C₄	—	12	2	no	—
312	C₂×C₄	—	60	4	no	—
313	C₂×C₄	D₅×D₁₂	1	1	yes	characteristic
314	C₈	—	5	2	no	—
315	C₈	D₅×D₁₂	1	1	yes	characteristic
316	C₈	C₂²×D₃×D₅	1	1	yes	characteristic
317	C₂×C₃	—	2	1	no	—
318	C₂×C₃	—	10	2	no	—
319	C₂×C₃	C₂×D₅×D₈	1	1	yes	characteristic
320	D₃	—	8	2	no	—
321	D₃	—	40	4	no	—
322	C₅	#82013	1	1	yes	characteristic, complemented, Sylow
323	C₂²	—	5	2	no	—
324	C₂²	—	12	2	no	—
325	C₂²	—	60	4	no	—
326	C₂²	—	240	2	no	—
327	C₂²	—	240	4	no	—
328	C₂²	D₅×D₂₄	1	1	yes	characteristic
329	C₄	—	5	2	no	—
330	C₄	—	12	2	no	—
331	C₄	D₅×D₂₄	1	1	yes	characteristic
332	C₄	C₂×D₅×D₁₂	1	1	yes	characteristic
333	C₃	#10613762	1	1	yes	characteristic, complemented, Sylow
334	C₂	—	2	1	no	—
335	C₂	—	10	2	no	—
336	C₂	—	24	2	no	—
337	C₂	—	120	4	no	—
338	C₂	C₂×D₅×D₂₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₅×D₂₄
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. #11753963	C₂	semidirect
2. C₂×D₂₄₀	C₂	semidirect
3. C₅×#4904	C₂	semidirect
4. C₃×#65190	C₂	semidirect
5. #11747548	C₂	semidirect
6. #11746834	C₂	semidirect
7. #11746817	C₂	semidirect
8. D₅×D₄₈	C₂	semidirect
11. C₂×D₁₂₀	C₂²	semidirect
12. C₅×#904	C₂²	semidirect
15. #91271	C₂²	semidirect
17. D₅×D₂₄	C₂²	semidirect
18. C₂×C₃×C₅×C₁₆	C₂²	semidirect
19. D₂₄₀	C₂²	semidirect
20. D₂₄₀	C₂²	semidirect
21. C₅×Dic₂₄	C₂²	semidirect
22. C₅×#800	C₂²	semidirect
23. C₅×D₄₈	C₂²	semidirect
24. C₃×#3944	C₂²	semidirect
25. C₃×#3930	C₂²	semidirect
26. C₃×D₅×C₁₆	C₂²	semidirect
27. #90945	C₂²	semidirect
28. #90939	C₂²	semidirect
37. #65190	D₃	semidirect
65. #4904	D₅	semidirect
79. C₂×C₅×C₁₆	C₂×D₃	semidirect
82. #3944	C₂×D₃	semidirect
85. #3930	C₂×D₃	semidirect
86. D₅×C₁₆	C₂×D₃	semidirect
112. #904	C₂×D₅	semidirect
118. C₂×C₃×C₁₆	C₂×D₅	semidirect
119. Dic₂₄	C₂×D₅	semidirect
121. #800	C₂×D₅	semidirect
125. D₄₈	C₂×D₅	semidirect
225. C₂×C₁₆	D₃×D₅	semidirect
282. C₃×C₅	#3415	semidirect
322. C₅	#82013	semidirect
333. C₃	#10613762	semidirect

Maximal subgroups

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

subgroup	index
#11753963	2
C₂×D₂₄₀	2
C₅×#4904	2
C₃×#65190	2
#11747548	2
#11746834	2
#11746817	2
D₅×D₄₈	2
#10613762	3
#82013	5