Subgroups — #11755319

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	C₃×#4150	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	C₅×#950	C₂	1	1	yes	maximal, characteristic, complemented
5	#91515	C₂	1	2	yes	maximal, complemented
6	D₅×#215	C₂	1	2	yes	maximal, complemented
7	#91492	C₂	1	2	yes	maximal, complemented
8	#91478	C₂	1	2	yes	maximal, complemented
9	C₂×#6658	C₂	1	1	yes	maximal, characteristic, complemented
10	#91309	C₂	1	1	yes	maximal, characteristic, complemented
11	#91283	C₂	1	1	yes	maximal, characteristic, complemented
12	#66104	—	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×#235	C₂²	1	1	yes	characteristic
16	C₃×Q₈×D₅	C₂²	1	2	yes	complemented
17	C₃×#711	—	2	2	no	—
18	C₃×#711	C₂²	1	2	yes	complemented
19	C₃×#709	—	2	1	no	—
20	C₃×#709	C₂²	1	1	yes	characteristic
21	C₂×C₃×Dic₁₀	—	2	1	no	—
22	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic, complemented
23	#6790	—	2	2	no	—
24	D₃×D₂₀	—	2	1	no	—
25	D₅×D₁₂	C₂²	1	2	yes	—
26	#6773	C₂²	1	2	yes	—
27	#6769	—	2	1	no	—
28	#6717	C₂²	1	2	yes	complemented
29	#6715	C₂²	1	2	yes	complemented
30	#6710	C₂²	1	1	yes	characteristic, complemented
31	C₅×#216	C₂²	1	1	yes	characteristic, complemented
32	C₅×#215	C₂²	1	1	yes	characteristic, complemented
33	C₅×#214	C₂²	1	1	yes	characteristic, complemented
34	C₅×#213	C₂²	1	1	yes	characteristic, complemented
35	C₂×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
36	#6661	C₂²	1	2	yes	complemented
37	#6658	C₂²	1	2	yes	complemented
38	#6658	C₂²	1	2	yes	complemented
39	#6656	C₂²	1	2	yes	complemented
40	#6649	C₂²	1	2	yes	complemented
41	D₅×C₃⋊C₈	C₂²	1	2	yes	complemented
42	#5837	—	5	1	no	maximal, Hall
43	#4150	D₃	1	1	yes	characteristic, complemented
44	#4149	—	3	1	no	—
45	C₅×#135	—	3	1	no	—
46	#4097	—	3	1	no	—
47	#4095	—	3	1	no	—
48	#4064	—	3	2	no	—
49	#4062	—	3	2	no	—
50	D₅×SD₁₆	—	3	2	no	—
51	#4057	—	3	2	no	—
52	C₂×#677	—	3	1	no	—
53	#4047	—	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	1	no	—
64	C₅×D₁₂	C₂³	1	1	yes	characteristic
65	C₄×C₅×D₃	—	2	1	no	—
66	C₅×Dic₆	C₂³	1	1	yes	characteristic
67	C₃×C₂²⋊D₅	—	2	2	no	—
68	C₃×C₂²⋊D₅	D₄	1	2	yes	—
69	C₂×C₃×Dic₅	—	2	1	no	—
70	C₂×C₃×Dic₅	—	2	2	no	—
71	C₃×D₂₀	D₄	1	1	yes	characteristic
72	C₃×C₄×D₅	—	2	2	no	—
73	C₃×C₄×D₅	C₂³	1	2	yes	—
74	C₃×Dic₁₀	—	2	1	no	—
75	C₃×Dic₁₀	—	2	2	no	—
76	C₃×Dic₁₀	D₄	1	1	yes	characteristic
77	C₃×Dic₁₀	C₂³	1	2	yes	—
78	#1203	—	2	2	no	—
79	#1202	—	2	2	no	—
80	#1200	—	2	2	no	—
81	C₃⋊#168	C₂³	1	2	yes	—
82	C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
83	C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
84	C₃×2₋¹⁺⁴	—	5	1	no	—
85	#1009	—	5	1	no	—
86	#950	—	5	2	no	—
87	#950	D₅	1	1	yes	characteristic, complemented
88	#949	—	5	1	no	—
89	#949	—	5	2	no	—
90	#948	—	5	1	no	—
91	#942	—	5	1	no	—
92	#942	—	5	2	no	—
93	C₂×#215	—	5	2	no	—
94	C₂×#215	—	5	1	no	—
95	C₅×#41	—	3	1	no	—
96	C₅×#41	C₂×D₃	1	1	yes	characteristic, complemented
97	#713	—	3	2	no	—
98	Q₈×D₅	C₂×D₃	1	2	yes	complemented
99	#711	—	2	2	no	—
100	#711	C₂×D₃	1	2	yes	complemented
101	D₄×D₅	—	3	2	no	—
102	D₄×D₅	—	6	2	no	—
103	#709	—	2	1	no	—
104	#709	—	6	1	no	—
105	#709	C₂×D₃	1	1	yes	characteristic
106	C₂×D₂₀	—	3	1	no	—
107	C₂×D₂₀	—	6	1	no	—
108	C₂×Dic₁₀	—	2	1	no	—
109	C₂×Dic₁₀	C₂×D₃	1	1	yes	characteristic, complemented
110	C₅×Q₁₆	—	3	1	no	—
111	C₅×SD₁₆	—	3	1	no	—
112	C₅×SD₁₆	—	3	1	no	—
113	C₅×D₈	—	3	1	no	—
114	C₂×C₅×C₈	—	3	1	no	—
115	#689	—	3	2	no	—
116	#688	—	3	2	no	—
117	#687	—	3	2	no	—
118	#686	—	3	2	no	—
119	#681	—	3	1	no	—
120	#677	—	3	2	no	—
121	#677	—	3	2	no	—
122	#676	—	3	2	no	—
123	C₈×D₅	—	3	2	no	—
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	—	2	1	no	—
130	C₂×C₃×D₅	C₂×D₄	1	2	yes	—
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	2	no	—
136	C₃×Dic₅	C₂×D₄	1	2	yes	—
137	C₅×Dic₃	—	2	1	no	—
138	C₃×#41	—	5	1	no	—
139	C₃×#41	—	5	2	no	—
140	C₃×#41	—	10	1	no	—
141	C₃×#41	—	10	2	no	—
142	C₃×#41	C₂×D₅	1	1	yes	characteristic, complemented
143	C₂×C₃×Q₈	—	5	2	no	—
144	C₂×C₃×Q₈	—	5	1	no	—
145	C₂×C₃×Q₈	—	10	1	no	—
146	#239	—	10	1	no	—
147	D₃×D₄	—	10	1	no	—
148	D₃×D₄	—	10	2	no	—
149	#235	—	5	2	no	—
150	#235	C₂×D₅	1	1	yes	characteristic
151	C₂×D₁₂	—	5	1	no	—
152	C₂×D₁₂	—	5	2	no	—
153	#216	—	5	2	no	—
154	#216	C₂×D₅	1	1	yes	characteristic, complemented
155	#215	—	5	2	no	—
156	#215	—	5	2	no	—
157	#215	—	5	2	no	—
158	#215	—	5	2	no	—
159	#215	C₂×D₅	1	1	yes	characteristic, complemented
160	#214	C₂×D₅	1	1	yes	characteristic, complemented
161	#213	—	5	2	no	—
162	#213	C₂×D₅	1	1	yes	characteristic, complemented
163	#208	—	5	1	no	—
164	#208	—	5	2	no	—
165	C₂×C₃⋊C₈	—	5	2	no	—
166	C₂×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic, complemented
167	C₂²×D₅	—	6	1	no	—
168	C₂²×D₅	—	6	2	no	—
169	C₅×Q₈	—	3	1	no	—
170	C₅×Q₈	C₂²×D₃	1	1	yes	characteristic
171	C₅×D₄	—	2	1	no	—
172	C₅×D₄	—	3	1	no	—
173	C₅×D₄	—	6	1	no	—
174	C₅×D₄	C₂²×D₃	1	1	yes	characteristic
175	C₂×C₄×C₅	—	2	1	no	—
176	C₂×C₄×C₅	—	6	1	no	—
177	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
178	C₂²⋊D₅	—	2	2	no	—
179	C₂²⋊D₅	—	6	2	no	—
180	C₂²⋊D₅	#67	1	2	yes	—
181	C₂×Dic₅	—	2	1	no	—
182	C₂×Dic₅	—	2	2	no	—
183	D₂₀	—	3	2	no	—
184	D₂₀	—	6	2	no	—
185	D₂₀	—	6	1	no	—
186	D₂₀	#67	1	1	yes	characteristic
187	C₄×D₅	—	2	2	no	—
188	C₄×D₅	—	6	2	no	—
189	C₄×D₅	C₂²×D₃	1	2	yes	—
190	Dic₁₀	—	2	1	no	—
191	Dic₁₀	—	2	2	no	—
192	Dic₁₀	#67	1	1	yes	characteristic
193	Dic₁₀	C₂²×D₃	1	2	yes	—
194	C₅×C₈	—	3	1	no	—
195	C₅×C₈	—	3	1	no	—
196	#168	—	3	2	no	—
197	2₋¹⁺⁴	—	5	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	—	3	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	—	2	2	no	—
213	C₅×D₃	—	4	1	no	—
214	C₂²×D₃	—	10	2	no	—
215	C₂²×D₃	—	10	1	no	—
216	C₃×Q₈	—	5	1	no	—
217	C₃×Q₈	—	5	2	no	—
218	C₃×Q₈	—	10	2	no	—
219	C₃×Q₈	—	10	1	no	—
220	C₃×Q₈	C₂²×D₅	1	1	yes	characteristic
221	C₃×D₄	—	2	1	no	—
222	C₃×D₄	—	5	2	no	—
223	C₃×D₄	—	5	1	no	—
224	C₃×D₄	—	10	2	no	—
225	C₃×D₄	C₂²×D₅	1	1	yes	characteristic
226	C₂×C₃×C₄	—	2	1	no	—
227	C₂×C₃×C₄	—	5	2	no	—
228	C₂×C₃×C₄	—	10	2	no	—
229	C₂×C₃×C₄	—	10	2	no	—
230	C₂×C₃×C₄	—	10	1	no	—
231	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
232	#67	—	2	1	no	—
233	#67	—	10	2	no	—
234	D₁₂	—	5	2	no	—
235	D₁₂	—	10	2	no	—
236	D₁₂	—	10	1	no	—
237	D₁₂	C₂²×D₅	1	1	yes	characteristic
238	C₄×D₃	—	2	1	no	—
239	C₄×D₃	—	10	2	no	—
240	Dic₆	C₂²×D₅	1	1	yes	characteristic
241	C₃⋊C₈	—	5	2	no	—
242	C₃⋊C₈	C₂²×D₅	1	1	yes	characteristic
243	C₃⋊C₈	C₂²×D₅	1	1	yes	characteristic
244	C₂²×C₅	—	2	1	no	—
245	C₂²×C₅	—	6	1	no	—
246	C₂²×C₅	C₂×#67	1	1	yes	characteristic
247	C₂×D₅	—	6	2	no	—
248	C₂×D₅	—	24	2	no	—
249	C₂×D₅	—	24	1	no	—
250	C₂×D₅	C₂×#67	1	2	yes	—
251	C₄×C₅	—	2	1	no	—
252	C₄×C₅	—	6	1	no	—
253	C₄×C₅	C₂×#67	1	1	yes	characteristic
254	C₄×C₅	C₂³×D₃	1	1	yes	characteristic
255	Dic₅	—	2	2	no	—
256	Dic₅	C₂×#67	1	2	yes	—
257	#41	—	3	1	no	—
258	#41	—	5	1	no	—
259	#41	—	5	2	no	—
260	#41	—	10	1	no	—
261	#41	—	10	2	no	—
262	#41	—	15	2	no	—
263	#41	—	30	1	no	—
264	#41	D₃×D₅	1	1	yes	characteristic, complemented
265	C₂×Q₈	—	5	2	no	—
266	C₂×Q₈	—	5	1	no	—
267	C₂×Q₈	—	10	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	—	3	1	no	—
273	Q₁₆	—	15	2	no	—
274	SD₁₆	—	3	1	no	—
275	SD₁₆	—	3	1	no	—
276	SD₁₆	—	15	2	no	—
277	SD₁₆	—	15	2	no	—
278	SD₁₆	—	15	2	no	—
279	SD₁₆	—	15	2	no	—
280	D₈	—	3	1	no	—
281	D₈	—	15	2	no	—
282	M₁₆	—	15	2	no	—
283	M₁₆	—	15	1	no	—
284	C₂×C₈	—	3	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	—	5	2	no	—
289	C₂²×C₃	D₄×D₅	1	1	yes	characteristic
290	C₂×D₃	—	2	1	no	—
291	C₂×D₃	—	10	2	no	—
292	C₂×D₃	—	40	2	no	—
293	C₂×D₃	—	40	1	no	—
294	C₃×C₄	—	2	1	no	—
295	C₃×C₄	—	5	2	no	—
296	C₃×C₄	—	10	2	no	—
297	C₃×C₄	D₄×D₅	1	1	yes	characteristic
298	C₃×C₄	C₂³×D₅	1	1	yes	characteristic
299	Dic₃	—	2	1	no	—
300	C₂×C₅	—	2	1	no	—
301	C₂×C₅	—	4	1	no	—
302	C₂×C₅	—	12	1	no	—
303	C₂×C₅	C₂²×#67	1	1	yes	characteristic
304	D₅	—	2	2	no	—
305	D₅	—	12	2	no	—
306	C₂³	—	30	2	no	—
307	C₂³	—	30	1	no	—
308	Q₈	—	3	1	no	—
309	Q₈	—	5	1	no	—
310	Q₈	—	5	2	no	—
311	Q₈	—	10	2	no	—
312	Q₈	—	10	1	no	—
313	Q₈	C₂×D₃×D₅	1	1	yes	characteristic
314	D₄	—	2	1	no	—
315	D₄	—	3	1	no	—
316	D₄	—	5	2	no	—
317	D₄	—	5	1	no	—
318	D₄	—	6	1	no	—
319	D₄	—	10	2	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	—	5	2	no	—
327	C₂×C₄	—	6	1	no	—
328	C₂×C₄	—	10	2	no	—
329	C₂×C₄	—	10	2	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	—	3	1	no	—
334	C₈	—	3	1	no	—
335	C₈	—	15	2	no	—
336	C₂×C₃	—	2	1	no	—
337	C₂×C₃	—	4	1	no	—
338	C₂×C₃	—	10	2	no	—
339	C₂×C₃	C₂×D₄×D₅	1	1	yes	characteristic
340	D₃	—	4	1	no	—
341	D₃	—	20	2	no	—
342	C₅	#5837	1	1	yes	characteristic, complemented, Sylow
343	C₂²	—	2	1	no	—
344	C₂²	—	5	2	no	—
345	C₂²	—	6	1	no	—
346	C₂²	—	30	2	no	—
347	C₂²	—	120	1	no	—
348	C₂²	—	120	2	no	—
349	C₂²	D₅×#67	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	5	2	no	—
352	C₄	—	6	1	no	—
353	C₄	—	10	2	no	—
354	C₄	D₅×#67	1	1	yes	characteristic
355	C₄	C₂²×D₃×D₅	1	1	yes	characteristic
356	C₃	#66104	1	1	yes	characteristic, complemented, Sylow
357	C₂	—	2	1	no	—
358	C₂	—	4	1	no	—
359	C₂	—	10	2	no	—
360	C₂	—	12	1	no	—
361	C₂	—	60	2	no	—
362	C₂	C₂×D₅×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₅×#67
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. C₃×#4150	C₂	semidirect
2. #92019	C₂	semidirect
3. #91838	C₂	semidirect
4. C₅×#950	C₂	semidirect
5. #91515	C₂	semidirect
6. D₅×#215	C₂	semidirect
7. #91492	C₂	semidirect
8. #91478	C₂	semidirect
9. C₂×#6658	C₂	semidirect
10. #91309	C₂	semidirect
11. #91283	C₂	semidirect
13. C₃×C₅×#41	C₂²	semidirect
16. C₃×Q₈×D₅	C₂²	semidirect
18. C₃×#711	C₂²	semidirect
22. C₂×C₃×Dic₁₀	C₂²	semidirect
28. #6717	C₂²	semidirect
29. #6715	C₂²	semidirect
30. #6710	C₂²	semidirect
31. C₅×#216	C₂²	semidirect
32. C₅×#215	C₂²	semidirect
33. C₅×#214	C₂²	semidirect
34. C₅×#213	C₂²	semidirect
35. C₂×C₅×C₃⋊C₈	C₂²	semidirect
36. #6661	C₂²	semidirect
37. #6658	C₂²	semidirect
38. #6658	C₂²	semidirect
39. #6656	C₂²	semidirect
40. #6649	C₂²	semidirect
41. D₅×C₃⋊C₈	C₂²	semidirect
43. #4150	D₃	semidirect
87. #950	D₅	semidirect
96. C₅×#41	C₂×D₃	semidirect
98. Q₈×D₅	C₂×D₃	semidirect
100. #711	C₂×D₃	semidirect
109. C₂×Dic₁₀	C₂×D₃	semidirect
142. C₃×#41	C₂×D₅	semidirect
154. #216	C₂×D₅	semidirect
159. #215	C₂×D₅	semidirect
160. #214	C₂×D₅	semidirect
162. #213	C₂×D₅	semidirect
166. C₂×C₃⋊C₈	C₂×D₅	semidirect
264. #41	D₃×D₅	semidirect
286. C₃×C₅	#577	semidirect
342. C₅	#5837	semidirect
356. C₃	#66104	semidirect

Maximal subgroups

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

subgroup	index
C₃×#4150	2
#92019	2
#91838	2
C₅×#950	2
#91515	2
D₅×#215	2
#91492	2
#91478	2
C₂×#6658	2
#91309	2
#91283	2
#66104	3
#5837	5