Subgroups — #11755326

Order 960.

516 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1010	C₂	1	1	yes	maximal, characteristic, complemented
2	#92019	C₂	1	1	yes	maximal, characteristic, complemented
3	#91838	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4096	C₂	1	1	yes	maximal, characteristic, complemented
5	#91520	C₂	1	2	yes	maximal, complemented
6	D₃×#688	C₂	1	2	yes	maximal, complemented
7	#91495	C₂	1	2	yes	maximal, complemented
8	#91480	C₂	1	2	yes	maximal, complemented
9	C₂×#6660	C₂	1	1	yes	maximal, characteristic, complemented
10	#91307	C₂	1	1	yes	maximal, characteristic, complemented
11	#91286	C₂	1	1	yes	maximal, characteristic, complemented
12	#66171	—	3	1	no	maximal, Hall
13	C₃×C₅×#41	C₂²	1	1	yes	characteristic, complemented
14	C₂×D₆₀	C₂²	1	1	yes	characteristic
15	C₅×D₃×Q₈	C₂²	1	2	yes	complemented
16	C₅×#237	—	2	2	no	—
17	C₅×#237	C₂²	1	2	yes	complemented
18	C₅×#235	—	2	1	no	—
19	C₅×#235	C₂²	1	1	yes	characteristic
20	C₂×C₅×Dic₆	—	2	1	no	—
21	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic, complemented
22	C₃×#709	C₂²	1	1	yes	characteristic
23	#6790	—	2	2	no	—
24	D₃×D₂₀	C₂²	1	2	yes	—
25	D₅×D₁₂	—	2	1	no	—
26	#6773	—	2	1	no	—
27	#6769	C₂²	1	2	yes	—
28	#6717	C₂²	1	2	yes	complemented
29	#6715	C₂²	1	2	yes	complemented
30	#6710	C₂²	1	1	yes	characteristic, complemented
31	C₃×#689	C₂²	1	1	yes	characteristic, complemented
32	C₃×#688	C₂²	1	1	yes	characteristic, complemented
33	C₃×#687	C₂²	1	1	yes	characteristic, complemented
34	C₃×#686	C₂²	1	1	yes	characteristic, complemented
35	C₂×C₃×#168	C₂²	1	1	yes	characteristic, complemented
36	#6661	C₂²	1	2	yes	complemented
37	#6660	C₂²	1	2	yes	complemented
38	#6660	C₂²	1	2	yes	complemented
39	#6655	C₂²	1	2	yes	complemented
40	#6650	C₂²	1	2	yes	complemented
41	D₃×#168	C₂²	1	2	yes	complemented
42	#5770	—	5	1	no	maximal, Hall
43	C₅×2₋¹⁺⁴	—	3	1	no	—
44	#4149	—	3	1	no	—
45	#4096	—	3	2	no	—
46	#4096	D₃	1	1	yes	characteristic, complemented
47	#4095	—	3	1	no	—
48	#4095	—	3	2	no	—
49	#4094	—	3	1	no	—
50	#4088	—	3	1	no	—
51	#4088	—	3	2	no	—
52	C₂×#688	—	3	2	no	—
53	C₂×#688	—	3	1	no	—
54	C₂²×D₁₅	—	2	1	no	—
55	C₂×D₃×D₅	—	2	2	no	—
56	C₃×C₅×Q₈	C₂³	1	1	yes	characteristic
57	C₃×C₅×D₄	—	2	1	no	—
58	C₃×C₅×D₄	C₂³	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	—	2	1	no	—
60	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
61	D₆₀	—	2	1	no	—
62	D₆₀	C₂³	1	2	yes	—
63	C₅×#67	—	2	2	no	—
64	C₅×#67	D₄	1	2	yes	—
65	C₂×C₅×Dic₃	—	2	1	no	—
66	C₂×C₅×Dic₃	—	2	2	no	—
67	C₅×D₁₂	D₄	1	1	yes	characteristic
68	C₄×C₅×D₃	—	2	2	no	—
69	C₄×C₅×D₃	C₂³	1	2	yes	—
70	C₅×Dic₆	—	2	1	no	—
71	C₅×Dic₆	—	2	2	no	—
72	C₅×Dic₆	D₄	1	1	yes	characteristic
73	C₅×Dic₆	C₂³	1	2	yes	—
74	C₃×C₂²⋊D₅	—	2	1	no	—
75	C₃×D₂₀	C₂³	1	1	yes	characteristic
76	C₃×C₄×D₅	—	2	1	no	—
77	C₃×Dic₁₀	C₂³	1	1	yes	characteristic
78	#1203	—	2	2	no	—
79	#1202	—	2	2	no	—
80	#1200	—	2	2	no	—
81	C₃⋊#168	C₂³	1	2	yes	—
82	C₃×#168	C₂³	1	1	yes	characteristic
83	C₃×#168	C₂³	1	1	yes	characteristic
84	#1010	D₅	1	1	yes	characteristic, complemented
85	#1009	—	5	1	no	—
86	C₃×#135	—	5	1	no	—
87	#951	—	5	1	no	—
88	#949	—	5	1	no	—
89	#918	—	5	2	no	—
90	#916	—	5	2	no	—
91	D₃×SD₁₆	—	5	2	no	—
92	#911	—	5	2	no	—
93	C₂×#204	—	5	1	no	—
94	#901	—	5	1	no	—
95	C₅×#41	—	3	1	no	—
96	C₅×#41	—	3	2	no	—
97	C₅×#41	—	6	1	no	—
98	C₅×#41	—	6	2	no	—
99	C₅×#41	C₂×D₃	1	1	yes	characteristic, complemented
100	C₂×C₅×Q₈	—	3	2	no	—
101	C₂×C₅×Q₈	—	3	1	no	—
102	C₂×C₅×Q₈	—	6	1	no	—
103	#713	—	6	1	no	—
104	D₄×D₅	—	6	1	no	—
105	D₄×D₅	—	6	2	no	—
106	#709	—	3	2	no	—
107	#709	C₂×D₃	1	1	yes	characteristic
108	C₂×D₂₀	—	3	1	no	—
109	C₂×D₂₀	—	3	2	no	—
110	#689	—	3	2	no	—
111	#689	C₂×D₃	1	1	yes	characteristic, complemented
112	#688	—	3	2	no	—
113	#688	—	3	2	no	—
114	#688	—	3	2	no	—
115	#688	—	3	2	no	—
116	#688	C₂×D₃	1	1	yes	characteristic, complemented
117	#687	C₂×D₃	1	1	yes	characteristic, complemented
118	#686	—	3	2	no	—
119	#686	C₂×D₃	1	1	yes	characteristic, complemented
120	#681	—	3	1	no	—
121	#681	—	3	2	no	—
122	C₂×#168	—	3	2	no	—
123	C₂×#168	C₂×D₃	1	1	yes	characteristic, complemented
124	#577	—	15	1	no	Sylow
125	C₂²×C₃×C₅	—	2	1	no	—
126	C₂²×C₃×C₅	C₂×D₄	1	1	yes	characteristic
127	C₂×D₁₅	—	2	2	no	—
128	C₂×D₁₅	—	8	1	no	—
129	C₂×C₅×D₃	C₂×D₄	1	2	yes	—
130	C₂×C₃×D₅	—	2	1	no	—
131	D₃×D₅	—	8	2	no	—
132	C₃×C₄×C₅	—	2	1	no	—
133	C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
134	C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
135	C₃×Dic₅	—	2	1	no	—
136	C₅×Dic₃	—	2	2	no	—
137	C₅×Dic₃	C₂×D₄	1	2	yes	—
138	C₃×#41	—	5	1	no	—
139	C₃×#41	C₂×D₅	1	1	yes	characteristic, complemented
140	#239	—	5	2	no	—
141	D₃×Q₈	C₂×D₅	1	2	yes	complemented
142	#237	—	2	2	no	—
143	#237	C₂×D₅	1	2	yes	complemented
144	D₃×D₄	—	5	2	no	—
145	D₃×D₄	—	10	2	no	—
146	#235	—	2	1	no	—
147	#235	—	10	1	no	—
148	#235	C₂×D₅	1	1	yes	characteristic
149	C₂×D₁₂	—	5	1	no	—
150	C₂×D₁₂	—	10	1	no	—
151	C₂×Dic₆	—	2	1	no	—
152	C₂×Dic₆	C₂×D₅	1	1	yes	characteristic, complemented
153	C₃×Q₁₆	—	5	1	no	—
154	C₃×SD₁₆	—	5	1	no	—
155	C₃×SD₁₆	—	5	1	no	—
156	C₃×D₈	—	5	1	no	—
157	C₂×C₃×C₈	—	5	1	no	—
158	#216	—	5	2	no	—
159	#215	—	5	2	no	—
160	#214	—	5	2	no	—
161	#213	—	5	2	no	—
162	#208	—	5	1	no	—
163	#204	—	5	2	no	—
164	#204	—	5	2	no	—
165	ΓL(1,𝔽₂₅)	—	5	2	no	—
166	D₃×C₈	—	5	2	no	—
167	C₂²×D₅	—	6	2	no	—
168	C₂²×D₅	—	6	1	no	—
169	C₅×Q₈	—	3	1	no	—
170	C₅×Q₈	—	3	2	no	—
171	C₅×Q₈	—	6	2	no	—
172	C₅×Q₈	—	6	1	no	—
173	C₅×Q₈	C₂²×D₃	1	1	yes	characteristic
174	C₅×D₄	—	2	1	no	—
175	C₅×D₄	—	3	2	no	—
176	C₅×D₄	—	3	1	no	—
177	C₅×D₄	—	6	2	no	—
178	C₅×D₄	C₂²×D₃	1	1	yes	characteristic
179	C₂×C₄×C₅	—	2	1	no	—
180	C₂×C₄×C₅	—	3	2	no	—
181	C₂×C₄×C₅	—	6	2	no	—
182	C₂×C₄×C₅	—	6	2	no	—
183	C₂×C₄×C₅	—	6	1	no	—
184	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
185	C₂²⋊D₅	—	2	1	no	—
186	C₂²⋊D₅	—	6	2	no	—
187	D₂₀	—	3	2	no	—
188	D₂₀	—	6	2	no	—
189	D₂₀	—	6	1	no	—
190	D₂₀	C₂²×D₃	1	1	yes	characteristic
191	C₄×D₅	—	2	1	no	—
192	C₄×D₅	—	6	2	no	—
193	Dic₁₀	C₂²×D₃	1	1	yes	characteristic
194	#168	—	3	2	no	—
195	#168	C₂²×D₃	1	1	yes	characteristic
196	#168	C₂²×D₃	1	1	yes	characteristic
197	2₋¹⁺⁴	—	3	1	no	—
198	H(5,𝔽₂)	—	15	1	no	—
199	#137	—	15	2	no	—
200	#137	—	15	1	no	—
201	AGL(1,ℤ/8)	—	15	2	no	—
202	AGL(1,ℤ/8)	—	15	1	no	—
203	#135	—	5	1	no	—
204	#135	—	15	2	no	—
205	C₂×SD₁₆	—	15	1	no	—
206	C₂×SD₁₆	—	15	2	no	—
207	#131	—	15	1	no	—
208	C₂×C₃×C₅	—	2	1	no	—
209	C₂×C₃×C₅	—	4	1	no	—
210	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
211	D₁₅	—	4	2	no	—
212	C₃×D₅	—	4	1	no	—
213	C₅×D₃	—	2	2	no	—
214	C₂²×D₃	—	10	1	no	—
215	C₂²×D₃	—	10	2	no	—
216	C₃×Q₈	—	5	1	no	—
217	C₃×Q₈	C₂²×D₅	1	1	yes	characteristic
218	C₃×D₄	—	2	1	no	—
219	C₃×D₄	—	5	1	no	—
220	C₃×D₄	—	10	1	no	—
221	C₃×D₄	C₂²×D₅	1	1	yes	characteristic
222	C₂×C₃×C₄	—	2	1	no	—
223	C₂×C₃×C₄	—	10	1	no	—
224	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
225	#67	—	2	2	no	—
226	#67	—	10	2	no	—
227	#67	C₂²⋊D₅	1	2	yes	—
228	C₂×Dic₃	—	2	1	no	—
229	C₂×Dic₃	—	2	2	no	—
230	D₁₂	—	5	2	no	—
231	D₁₂	—	10	2	no	—
232	D₁₂	—	10	1	no	—
233	D₁₂	C₂²⋊D₅	1	1	yes	characteristic
234	C₄×D₃	—	2	2	no	—
235	C₄×D₃	—	10	2	no	—
236	C₄×D₃	C₂²×D₅	1	2	yes	—
237	Dic₆	—	2	1	no	—
238	Dic₆	—	2	2	no	—
239	Dic₆	C₂²⋊D₅	1	1	yes	characteristic
240	Dic₆	C₂²×D₅	1	2	yes	—
241	C₃×C₈	—	5	1	no	—
242	C₃×C₈	—	5	1	no	—
243	C₃⋊C₈	—	5	2	no	—
244	C₂²×C₅	—	2	1	no	—
245	C₂²×C₅	—	3	2	no	—
246	C₂²×C₅	D₃×D₄	1	1	yes	characteristic
247	C₂×D₅	—	2	1	no	—
248	C₂×D₅	—	6	2	no	—
249	C₂×D₅	—	24	2	no	—
250	C₂×D₅	—	24	1	no	—
251	C₄×C₅	—	2	1	no	—
252	C₄×C₅	—	3	2	no	—
253	C₄×C₅	—	6	2	no	—
254	C₄×C₅	D₃×D₄	1	1	yes	characteristic
255	C₄×C₅	C₂³×D₃	1	1	yes	characteristic
256	Dic₅	—	2	1	no	—
257	#41	—	3	1	no	—
258	#41	—	3	2	no	—
259	#41	—	5	1	no	—
260	#41	—	6	1	no	—
261	#41	—	6	2	no	—
262	#41	—	15	2	no	—
263	#41	—	30	1	no	—
264	#41	D₃×D₅	1	1	yes	characteristic, complemented
265	C₂×Q₈	—	3	2	no	—
266	C₂×Q₈	—	3	1	no	—
267	C₂×Q₈	—	6	1	no	—
268	C₂×D₄	—	15	2	no	—
269	C₂×D₄	—	15	1	no	—
270	C₂×D₄	—	30	1	no	—
271	C₂×D₄	—	30	2	no	—
272	Q₁₆	—	5	1	no	—
273	Q₁₆	—	15	2	no	—
274	SD₁₆	—	5	1	no	—
275	SD₁₆	—	5	1	no	—
276	SD₁₆	—	15	2	no	—
277	SD₁₆	—	15	2	no	—
278	SD₁₆	—	15	2	no	—
279	SD₁₆	—	15	2	no	—
280	D₈	—	5	1	no	—
281	D₈	—	15	2	no	—
282	M₁₆	—	15	2	no	—
283	M₁₆	—	15	1	no	—
284	C₂×C₈	—	5	1	no	—
285	C₂×C₈	—	15	2	no	—
286	C₃×C₅	#577	1	1	yes	characteristic, complemented, Hall
287	C₂²×C₃	—	2	1	no	—
288	C₂²×C₃	—	10	1	no	—
289	C₂²×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
290	C₂×D₃	—	10	2	no	—
291	C₂×D₃	—	40	2	no	—
292	C₂×D₃	—	40	1	no	—
293	C₂×D₃	C₂×C₂²⋊D₅	1	2	yes	—
294	C₃×C₄	—	2	1	no	—
295	C₃×C₄	—	10	1	no	—
296	C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
297	C₃×C₄	C₂³×D₅	1	1	yes	characteristic
298	Dic₃	—	2	2	no	—
299	Dic₃	C₂×C₂²⋊D₅	1	2	yes	—
300	C₂×C₅	—	2	1	no	—
301	C₂×C₅	—	4	1	no	—
302	C₂×C₅	—	6	2	no	—
303	C₂×C₅	C₂×D₃×D₄	1	1	yes	characteristic
304	D₅	—	4	1	no	—
305	D₅	—	12	2	no	—
306	C₂³	—	30	2	no	—
307	C₂³	—	30	1	no	—
308	Q₈	—	3	1	no	—
309	Q₈	—	3	2	no	—
310	Q₈	—	5	1	no	—
311	Q₈	—	6	2	no	—
312	Q₈	—	6	1	no	—
313	Q₈	C₂×D₃×D₅	1	1	yes	characteristic
314	D₄	—	2	1	no	—
315	D₄	—	3	2	no	—
316	D₄	—	3	1	no	—
317	D₄	—	5	1	no	—
318	D₄	—	6	2	no	—
319	D₄	—	10	1	no	—
320	D₄	—	15	2	no	—
321	D₄	—	30	2	no	—
322	D₄	—	30	2	no	—
323	D₄	—	30	1	no	—
324	D₄	C₂×D₃×D₅	1	1	yes	characteristic
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	3	2	no	—
327	C₂×C₄	—	6	2	no	—
328	C₂×C₄	—	6	2	no	—
329	C₂×C₄	—	6	1	no	—
330	C₂×C₄	—	10	1	no	—
331	C₂×C₄	—	30	2	no	—
332	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
333	C₈	—	5	1	no	—
334	C₈	—	5	1	no	—
335	C₈	—	15	2	no	—
336	C₂×C₃	—	2	1	no	—
337	C₂×C₃	—	4	1	no	—
338	C₂×C₃	—	20	1	no	—
339	C₂×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
340	D₃	—	2	2	no	—
341	D₃	—	20	2	no	—
342	C₅	#5770	1	1	yes	characteristic, complemented, Sylow
343	C₂²	—	2	1	no	—
344	C₂²	—	3	2	no	—
345	C₂²	—	10	1	no	—
346	C₂²	—	30	2	no	—
347	C₂²	—	120	1	no	—
348	C₂²	—	120	2	no	—
349	C₂²	D₃×C₂²⋊D₅	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	3	2	no	—
352	C₄	—	6	2	no	—
353	C₄	—	10	1	no	—
354	C₄	D₃×C₂²⋊D₅	1	1	yes	characteristic
355	C₄	C₂²×D₃×D₅	1	1	yes	characteristic
356	C₃	#66171	1	1	yes	characteristic, complemented, Sylow
357	C₂	—	2	1	no	—
358	C₂	—	4	1	no	—
359	C₂	—	6	2	no	—
360	C₂	—	20	1	no	—
361	C₂	—	60	2	no	—
362	C₂	C₂×D₃×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₃×C₂²⋊D₅
commutator, cosocle	C₃×C₄×C₅	C₂⁴
Frattini	C₄	C₂²×D₃×D₅
Fitting	C₃×C₅×#41	C₂²
socle	C₂×C₃×C₅	C₂²×D₄
hypercenter	#41	D₃×D₅
hypocenter	C₃×C₅	#577
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
1. C₅×#1010	C₂	semidirect
2. #92019	C₂	semidirect
3. #91838	C₂	semidirect
4. C₃×#4096	C₂	semidirect
5. #91520	C₂	semidirect
6. D₃×#688	C₂	semidirect
7. #91495	C₂	semidirect
8. #91480	C₂	semidirect
9. C₂×#6660	C₂	semidirect
10. #91307	C₂	semidirect
11. #91286	C₂	semidirect
13. C₃×C₅×#41	C₂²	semidirect
15. C₅×D₃×Q₈	C₂²	semidirect
17. C₅×#237	C₂²	semidirect
21. C₂×C₅×Dic₆	C₂²	semidirect
28. #6717	C₂²	semidirect
29. #6715	C₂²	semidirect
30. #6710	C₂²	semidirect
31. C₃×#689	C₂²	semidirect
32. C₃×#688	C₂²	semidirect
33. C₃×#687	C₂²	semidirect
34. C₃×#686	C₂²	semidirect
35. C₂×C₃×#168	C₂²	semidirect
36. #6661	C₂²	semidirect
37. #6660	C₂²	semidirect
38. #6660	C₂²	semidirect
39. #6655	C₂²	semidirect
40. #6650	C₂²	semidirect
41. D₃×#168	C₂²	semidirect
46. #4096	D₃	semidirect
84. #1010	D₅	semidirect
99. C₅×#41	C₂×D₃	semidirect
111. #689	C₂×D₃	semidirect
116. #688	C₂×D₃	semidirect
117. #687	C₂×D₃	semidirect
119. #686	C₂×D₃	semidirect
123. C₂×#168	C₂×D₃	semidirect
139. C₃×#41	C₂×D₅	semidirect
141. D₃×Q₈	C₂×D₅	semidirect
143. #237	C₂×D₅	semidirect
152. C₂×Dic₆	C₂×D₅	semidirect
264. #41	D₃×D₅	semidirect
286. C₃×C₅	#577	semidirect
342. C₅	#5770	semidirect
356. C₃	#66171	semidirect

Maximal subgroups

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

subgroup	index
C₅×#1010	2
#92019	2
#91838	2
C₃×#4096	2
#91520	2
D₃×#688	2
#91495	2
#91480	2
C₂×#6660	2
#91307	2
#91286	2
#66171	3
#5770	5