Subgroups — C₅×#5677 #11756062

Order 960.

580 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×D₃×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#237	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#946	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#938	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#885	C₂	1	2	yes	maximal, complemented
7	C₅×#884	C₂	1	2	yes	maximal, complemented
8	C₅×#67⋊C₄	C₂	1	2	yes	maximal, complemented
9	C₅×#878	C₂	1	2	yes	maximal, complemented
10	C₄×C₅×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₅×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#548	—	3	1	no	maximal, Hall
13	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
15	C₂×C₅×#67	C₂²	1	2	yes	—
16	C₂×C₅×#67	C₂²	1	2	yes	—
17	C₂²×C₅×Dic₃	C₂²	1	2	yes	—
18	C₅×D₃×Q₈	—	2	4	no	—
19	C₅×#237	—	2	4	no	—
20	C₂×C₅×D₁₂	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
24	C₂×C₅×Dic₆	C₂²	1	2	yes	—
25	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
26	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
27	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
28	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
29	C₅×#217	C₂²	1	2	yes	—
30	C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
31	C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
32	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
33	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
34	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
35	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
36	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
37	C₄×C₅×Dic₃	C₂²	1	2	yes	—
38	#5677	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
39	C₂×C₅×#41	—	3	1	no	—
40	C₂²×C₅×Q₈	—	3	1	no	—
41	C₅×#124	—	3	2	no	—
42	C₅×#124	D₃	1	1	yes	characteristic, complemented
43	C₅×C₂²⋊Q₈	—	3	1	no	—
44	C₅×C₂²⋊Q₈	—	3	2	no	—
45	C₅×#121	—	3	1	no	—
46	C₅×#121	—	3	2	no	—
47	C₄×C₅×Q₈	—	3	1	no	—
48	C₄×C₅×D₄	—	3	2	no	—
49	C₄×C₅×D₄	—	3	1	no	—
50	C₂³×C₃×C₅	C₂³	1	2	yes	—
51	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
52	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
53	C₃×C₅×Q₈	—	2	2	no	—
54	C₃×C₅×D₄	—	2	2	no	—
55	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
58	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
59	C₅×#67	—	2	4	no	—
60	C₅×#67	—	2	4	no	—
61	C₂×C₅×Dic₃	—	2	4	no	—
62	C₂×C₅×Dic₃	C₂³	1	2	yes	—
63	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
64	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
65	C₂×C₅×Dic₃	C₂³	1	2	yes	—
66	C₅×D₁₂	—	2	2	no	—
67	C₄×C₅×D₃	—	2	2	no	—
68	C₄×C₅×D₃	—	2	2	no	—
69	C₄×C₅×D₃	—	2	4	no	—
70	C₅×Dic₆	—	2	4	no	—
71	C₅×Dic₆	D₄	1	4	yes	complemented
72	C₂×D₃×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
73	C₂×#237	C₂×C₅	1	1	yes	characteristic, complemented
74	C₃×#124	C₂×C₅	1	1	yes	characteristic, complemented
75	#946	C₂×C₅	1	1	yes	characteristic, complemented
76	#938	C₂×C₅	1	1	yes	characteristic, complemented
77	#885	C₂×C₅	1	2	yes	complemented
78	#884	C₂×C₅	1	2	yes	complemented
79	#67⋊C₄	C₂×C₅	1	2	yes	complemented
80	#878	C₂×C₅	1	2	yes	complemented
81	C₄×D₁₂	C₂×C₅	1	1	yes	characteristic, complemented
82	C₄×Dic₆	C₂×C₅	1	1	yes	characteristic, complemented
83	C₅×#41	—	6	4	no	—
84	C₂×C₅×Q₈	—	3	1	no	—
85	C₂×C₅×Q₈	—	3	2	no	—
86	C₂×C₅×Q₈	—	6	4	no	—
87	C₂×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
88	C₂×C₅×D₄	—	3	2	no	—
89	C₂×C₅×D₄	—	3	2	no	—
90	C₂×C₅×D₄	—	3	1	no	—
91	C₂×C₅×D₄	C₂×D₃	1	1	yes	characteristic
92	C₂²×C₄×C₅	—	3	2	no	—
93	C₂²×C₄×C₅	—	3	2	no	—
94	C₂²×C₄×C₅	—	3	1	no	—
95	C₂²×C₄×C₅	—	3	1	no	—
96	C₅×C₄⋊C₄	—	3	2	no	—
97	C₅×C₄⋊C₄	—	3	1	no	—
98	C₅×C₄⋊C₄	—	3	2	no	—
99	C₅×C₄⋊C₄	—	3	1	no	—
100	C₅×C₂²⋊C₄	—	3	2	no	—
101	C₅×C₂²⋊C₄	—	3	1	no	—
102	C₅×C₂²⋊C₄	—	3	1	no	—
103	C₅×C₂²⋊C₄	—	3	2	no	—
104	C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
105	C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
106	C₄²×C₅	—	3	2	no	—
107	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
108	#548	—	3	1	no	Sylow
109	C₂²×C₃×C₅	—	2	2	no	—
110	C₂²×C₃×C₅	—	2	2	no	—
111	C₂²×C₃×C₅	—	2	2	no	—
112	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
113	C₂×C₅×D₃	—	2	1	no	—
114	C₂×C₅×D₃	—	2	1	no	—
115	C₂×C₅×D₃	—	2	1	no	—
116	C₂×C₅×D₃	—	2	1	no	—
117	C₂×C₅×D₃	—	2	1	no	—
118	C₂×C₅×D₃	#41	1	2	yes	complemented
119	C₃×C₄×C₅	—	2	2	no	—
120	C₃×C₄×C₅	—	2	1	no	—
121	C₃×C₄×C₅	—	2	1	no	—
122	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
123	C₅×Dic₃	—	2	1	no	—
124	C₅×Dic₃	—	2	1	no	—
125	C₅×Dic₃	—	2	2	no	—
126	C₅×Dic₃	C₂×D₄	1	4	yes	—
127	C₂×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
128	C₂×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
129	C₂×#67	C₂²×C₅	1	2	yes	—
130	C₂×#67	C₂²×C₅	1	2	yes	—
131	C₂²×Dic₃	C₂²×C₅	1	2	yes	—
132	D₃×Q₈	—	2	4	no	—
133	#237	—	2	4	no	—
134	C₂×D₁₂	C₂²×C₅	1	1	yes	characteristic
135	C₂×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
136	C₂×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
137	C₂×C₄×D₃	C₂²×C₅	1	2	yes	—
138	C₂×Dic₆	C₂²×C₅	1	2	yes	—
139	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
140	C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
141	C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
142	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
143	#217	C₂²×C₅	1	2	yes	—
144	(C₂×D₃)⋊C₄	C₂²×C₅	1	1	yes	characteristic
145	(C₂×D₃)⋊C₄	C₂²×C₅	1	1	yes	characteristic
146	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
147	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
148	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
149	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
150	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
151	C₄×Dic₃	C₂²×C₅	1	2	yes	—
152	C₂³×C₅	—	3	1	no	—
153	C₂³×C₅	—	3	1	no	—
154	C₂³×C₅	C₂²×D₃	1	2	yes	—
155	C₅×Q₈	—	2	2	no	—
156	C₅×Q₈	—	3	4	no	—
157	C₅×Q₈	—	6	4	no	—
158	C₅×D₄	—	2	2	no	—
159	C₅×D₄	—	6	2	no	—
160	C₅×D₄	—	6	4	no	—
161	C₅×D₄	—	6	4	no	—
162	C₂×C₄×C₅	—	3	2	no	—
163	C₂×C₄×C₅	—	3	2	no	—
164	C₂×C₄×C₅	—	3	1	no	—
165	C₂×C₄×C₅	—	3	1	no	—
166	C₂×C₄×C₅	—	6	4	no	—
167	C₂×C₄×C₅	—	6	4	no	—
168	C₂×C₄×C₅	—	6	2	no	—
169	C₂×C₄×C₅	—	6	2	no	—
170	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
172	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
173	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
174	C₂×#41	—	3	1	no	—
175	C₂²×Q₈	—	3	1	no	—
176	#124	—	3	2	no	—
177	#124	C₅×D₃	1	1	yes	characteristic, complemented
178	C₂²⋊Q₈	—	3	1	no	—
179	C₂²⋊Q₈	—	3	2	no	—
180	#121	—	3	1	no	—
181	#121	—	3	2	no	—
182	C₄×Q₈	—	3	1	no	—
183	C₄×D₄	—	3	2	no	—
184	C₄×D₄	—	3	1	no	—
185	C₂×C₃×C₅	—	4	2	no	—
186	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
187	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
188	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
189	C₅×D₃	—	2	2	no	—
190	C₅×D₃	—	4	1	no	—
191	C₂³×C₃	C₂³×C₅	1	2	yes	—
192	C₂²×D₃	C₂³×C₅	1	1	yes	characteristic
193	C₂²×D₃	C₂³×C₅	1	1	yes	characteristic
194	C₃×Q₈	—	2	2	no	—
195	C₃×D₄	—	2	2	no	—
196	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
199	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
200	#67	—	2	4	no	—
201	#67	—	2	4	no	—
202	C₂×Dic₃	—	2	4	no	—
203	C₂×Dic₃	C₂³×C₅	1	2	yes	—
204	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
205	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
206	C₂×Dic₃	C₂³×C₅	1	2	yes	—
207	D₁₂	—	2	2	no	—
208	C₄×D₃	—	2	2	no	—
209	C₄×D₃	—	2	2	no	—
210	C₄×D₃	—	2	4	no	—
211	Dic₆	—	2	4	no	—
212	Dic₆	C₅×D₄	1	4	yes	complemented
213	C₂²×C₅	—	2	2	no	—
214	C₂²×C₅	—	2	2	no	—
215	C₂²×C₅	—	2	2	no	—
216	C₂²×C₅	—	3	2	no	—
217	C₂²×C₅	—	6	1	no	—
218	C₂²×C₅	—	6	1	no	—
219	C₂²×C₅	—	6	1	no	—
220	C₂²×C₅	—	6	1	no	—
221	C₂²×C₅	—	6	1	no	—
222	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
223	C₄×C₅	—	2	2	no	—
224	C₄×C₅	—	2	1	no	—
225	C₄×C₅	—	2	1	no	—
226	C₄×C₅	—	3	4	no	—
227	C₄×C₅	—	6	1	no	—
228	C₄×C₅	—	6	1	no	—
229	C₄×C₅	—	6	2	no	—
230	C₄×C₅	D₃×D₄	1	2	yes	—
231	#41	—	6	4	no	—
232	C₂×Q₈	—	3	1	no	—
233	C₂×Q₈	—	3	2	no	—
234	C₂×Q₈	—	6	4	no	—
235	C₂×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
236	C₂×D₄	—	3	2	no	—
237	C₂×D₄	—	3	2	no	—
238	C₂×D₄	—	3	1	no	—
239	C₂×D₄	C₂×C₅×D₃	1	1	yes	characteristic
240	C₂²×C₄	—	3	2	no	—
241	C₂²×C₄	—	3	2	no	—
242	C₂²×C₄	—	3	1	no	—
243	C₂²×C₄	—	3	1	no	—
244	C₄⋊C₄	—	3	2	no	—
245	C₄⋊C₄	—	3	1	no	—
246	C₄⋊C₄	—	3	2	no	—
247	C₄⋊C₄	—	3	1	no	—
248	C₂²⋊C₄	—	3	2	no	—
249	C₂²⋊C₄	—	3	1	no	—
250	C₂²⋊C₄	—	3	1	no	—
251	C₂²⋊C₄	—	3	2	no	—
252	C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
253	C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
254	C₄²	—	3	2	no	—
255	C₄²	C₂×C₅×D₃	1	1	yes	characteristic
256	C₃×C₅	#548	1	1	yes	characteristic, complemented, Hall
257	C₂²×C₃	—	2	2	no	—
258	C₂²×C₃	—	2	2	no	—
259	C₂²×C₃	—	2	2	no	—
260	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
261	C₂×D₃	—	2	1	no	—
262	C₂×D₃	—	2	1	no	—
263	C₂×D₃	—	2	1	no	—
264	C₂×D₃	—	2	1	no	—
265	C₂×D₃	—	2	1	no	—
266	C₂×D₃	C₅×#41	1	2	yes	complemented
267	C₃×C₄	—	2	2	no	—
268	C₃×C₄	—	2	1	no	—
269	C₃×C₄	—	2	1	no	—
270	C₃×C₄	C₂×C₅×D₄	1	2	yes	—
271	Dic₃	—	2	1	no	—
272	Dic₃	—	2	1	no	—
273	Dic₃	—	2	2	no	—
274	Dic₃	C₂×C₅×D₄	1	4	yes	—
275	C₂×C₅	—	4	2	no	—
276	C₂×C₅	—	6	2	no	—
277	C₂×C₅	—	12	1	no	—
278	C₂×C₅	C₂×D₃×D₄	1	1	yes	characteristic
279	C₂×C₅	D₃×#41	1	1	yes	characteristic
280	C₂×C₅	#1010	1	1	yes	characteristic
281	C₂³	—	3	1	no	—
282	C₂³	—	3	1	no	—
283	C₂³	C₂²×C₅×D₃	1	2	yes	—
284	Q₈	—	2	2	no	—
285	Q₈	—	3	4	no	—
286	Q₈	—	6	4	no	—
287	D₄	—	2	2	no	—
288	D₄	—	6	2	no	—
289	D₄	—	6	4	no	—
290	D₄	—	6	4	no	—
291	C₂×C₄	—	3	2	no	—
292	C₂×C₄	—	3	2	no	—
293	C₂×C₄	—	3	1	no	—
294	C₂×C₄	—	3	1	no	—
295	C₂×C₄	—	6	4	no	—
296	C₂×C₄	—	6	4	no	—
297	C₂×C₄	—	6	2	no	—
298	C₂×C₄	—	6	2	no	—
299	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
300	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
301	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
302	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
303	C₂×C₃	—	4	2	no	—
304	C₂×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
305	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
306	C₂×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
307	D₃	—	2	2	no	—
308	D₃	—	4	1	no	—
309	C₅	#5677	1	1	yes	characteristic, direct product factor, Sylow
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	3	2	no	—
314	C₂²	—	6	1	no	—
315	C₂²	—	6	1	no	—
316	C₂²	—	6	1	no	—
317	C₂²	—	6	1	no	—
318	C₂²	—	6	1	no	—
319	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
320	C₄	—	2	2	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	3	4	no	—
324	C₄	—	6	1	no	—
325	C₄	—	6	1	no	—
326	C₄	—	6	2	no	—
327	C₄	C₅×D₃×D₄	1	2	yes	—
328	C₃	C₅×#548	1	1	yes	characteristic, complemented, Sylow
329	C₂	—	4	2	no	—
330	C₂	—	6	2	no	—
331	C₂	—	12	1	no	—
332	C₂	C₂×C₅×D₃×D₄	1	1	yes	characteristic
333	C₂	C₅×D₃×#41	1	1	yes	characteristic
334	C₂	C₅×#1010	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₃
Fitting	C₃×C₅×#124	C₂
socle	C₂²×C₃×C₅	C₂⁴
hypercenter	C₅×#124	D₃
hypocenter	C₃	C₅×#548
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#124

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×D₃×Q₈	C₂	semidirect
2. C₂×C₅×#237	C₂	semidirect
3. C₃×C₅×#124	C₂	semidirect
4. C₅×#946	C₂	semidirect
5. C₅×#938	C₂	semidirect
6. C₅×#885	C₂	semidirect
7. C₅×#884	C₂	semidirect
8. C₅×#67⋊C₄	C₂	semidirect
9. C₅×#878	C₂	semidirect
10. C₄×C₅×D₁₂	C₂	semidirect
11. C₄×C₅×Dic₆	C₂	semidirect
38. #5677	C₅	direct (N × H)
42. C₅×#124	D₃	semidirect
71. C₅×Dic₆	D₄	semidirect
72. C₂×D₃×Q₈	C₂×C₅	semidirect
73. C₂×#237	C₂×C₅	semidirect
74. C₃×#124	C₂×C₅	semidirect
75. #946	C₂×C₅	semidirect
76. #938	C₂×C₅	semidirect
77. #885	C₂×C₅	semidirect
78. #884	C₂×C₅	semidirect
79. #67⋊C₄	C₂×C₅	semidirect
80. #878	C₂×C₅	semidirect
81. C₄×D₁₂	C₂×C₅	semidirect
82. C₄×Dic₆	C₂×C₅	semidirect
118. C₂×C₅×D₃	#41	semidirect
177. #124	C₅×D₃	semidirect
212. Dic₆	C₅×D₄	semidirect
256. C₃×C₅	#548	semidirect
266. C₂×D₃	C₅×#41	semidirect
309. C₅	#5677	direct (N × H)
328. C₃	C₅×#548	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×D₃×Q₈	2
C₂×C₅×#237	2
C₃×C₅×#124	2
C₅×#946	2
C₅×#938	2
C₅×#885	2
C₅×#884	2
C₅×#67⋊C₄	2
C₅×#878	2
C₄×C₅×D₁₂	2
C₄×C₅×Dic₆	2
C₅×#548	3
#5677	5

Subgroups — C5×#5677 #11756062

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5677 #11756062