Subgroups — C₅×#5616 #11756001

Order 960.

472 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#121	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#939	C₂	1	2	yes	maximal, complemented
3	C₅×#938	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#936	C₂	1	2	yes	maximal, complemented
5	C₅×D₄×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×C₅×#67	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#926	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#892	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#888	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#885	C₂	1	2	yes	maximal, complemented
11	C₅×#877	C₂	1	2	yes	maximal, complemented
12	C₅×#555	—	3	1	no	maximal, Hall
13	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₅×D₄	C₂²	1	2	yes	—
15	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
16	C₂×C₅×#67	C₂²	1	1	yes	characteristic
17	C₂×C₅×#67	C₂²	1	2	yes	—
18	C₂²×C₅×Dic₃	C₂²	1	2	yes	—
19	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
20	C₂×C₅×Dic₆	C₂²	1	2	yes	—
21	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₅×#217	C₂²	1	2	yes	—
24	C₅×#217	C₂²	1	2	yes	—
25	C₅×#217	C₂²	1	2	yes	—
26	C₅×#217	C₂²	1	1	yes	characteristic
27	C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
28	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
29	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
30	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
31	C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
32	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
33	C₄×C₅×Dic₃	C₂²	1	2	yes	—
34	C₄×C₅×Dic₃	C₂²	1	2	yes	—
35	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
36	#5616	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
37	C₅×C₄⋊Q₈	—	3	1	no	—
38	C₅×#124	—	3	2	no	—
39	C₅×#124	—	3	2	no	—
40	C₅×#121	—	3	1	no	—
41	C₅×#121	—	3	2	no	—
42	C₅×#121	D₃	1	1	yes	characteristic, complemented
43	C₄×C₅×D₄	—	3	1	no	—
44	C₄×C₅×D₄	—	3	1	no	—
45	C₅×#117	—	3	1	no	—
46	C₅×#117	—	3	1	no	—
47	C₅×#117	—	3	2	no	—
48	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
49	C₂³×C₃×C₅	C₂³	1	2	yes	—
50	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
51	C₃×C₅×D₄	—	2	2	no	—
52	C₃×C₅×D₄	—	4	2	no	—
53	C₂×C₃×C₄×C₅	—	2	2	no	—
54	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
56	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
57	C₅×#67	—	2	2	no	—
58	C₅×#67	—	4	2	no	—
59	C₂×C₅×Dic₃	—	2	4	no	—
60	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
61	C₂×C₅×Dic₃	C₂³	1	2	yes	—
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	C₄×C₅×D₃	—	2	2	no	—
66	C₅×Dic₆	—	4	2	no	—
67	C₃×#121	C₂×C₅	1	1	yes	characteristic, complemented
68	#939	C₂×C₅	1	2	yes	complemented
69	#938	C₂×C₅	1	1	yes	characteristic, complemented
70	#936	C₂×C₅	1	2	yes	complemented
71	D₄×Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
72	C₄×#67	C₂×C₅	1	1	yes	characteristic, complemented
73	#926	C₂×C₅	1	1	yes	characteristic, complemented
74	#892	C₂×C₅	1	1	yes	characteristic, complemented
75	#888	C₂×C₅	1	1	yes	characteristic, complemented
76	#885	C₂×C₅	1	2	yes	complemented
77	#877	C₂×C₅	1	2	yes	complemented
78	C₂×C₅×Q₈	—	3	2	no	—
79	C₂×C₅×D₄	—	3	1	no	—
80	C₂×C₅×D₄	—	3	2	no	—
81	C₂×C₅×D₄	C₂×D₃	1	1	yes	characteristic
82	C₂×C₅×D₄	C₂×D₃	1	2	yes	—
83	C₂²×C₄×C₅	—	3	2	no	—
84	C₂²×C₄×C₅	—	3	1	no	—
85	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
86	C₅×C₄⋊C₄	—	3	1	no	—
87	C₅×C₄⋊C₄	—	3	1	no	—
88	C₅×C₄⋊C₄	—	3	2	no	—
89	C₅×C₄⋊C₄	—	3	1	no	—
90	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
91	C₅×C₂²⋊C₄	—	3	2	no	—
92	C₅×C₂²⋊C₄	—	3	2	no	—
93	C₅×C₂²⋊C₄	—	3	1	no	—
94	C₅×C₂²⋊C₄	—	3	2	no	—
95	C₅×C₂²⋊C₄	—	3	2	no	—
96	C₅×C₂²⋊C₄	—	3	1	no	—
97	C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
98	C₄²×C₅	—	3	2	no	—
99	C₄²×C₅	—	3	2	no	—
100	C₄²×C₅	—	3	1	no	—
101	#555	—	3	1	no	Sylow
102	C₂²×C₃×C₅	—	2	2	no	—
103	C₂²×C₃×C₅	—	2	1	no	—
104	C₂²×C₃×C₅	—	2	1	no	—
105	C₂²×C₃×C₅	—	2	2	no	—
106	C₂²×C₃×C₅	—	2	2	no	—
107	C₂²×C₃×C₅	—	2	1	no	—
108	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
109	C₂×C₅×D₃	—	2	1	no	—
110	C₂×C₅×D₃	—	2	1	no	—
111	C₂×C₅×D₃	—	2	1	no	—
112	C₃×C₄×C₅	—	2	1	no	—
113	C₃×C₄×C₅	—	2	2	no	—
114	C₃×C₄×C₅	#41	1	2	yes	—
115	C₅×Dic₃	—	2	1	no	—
116	C₅×Dic₃	—	2	2	no	—
117	C₅×Dic₃	—	2	1	no	—
118	C₅×Dic₃	—	2	2	no	—
119	C₅×Dic₃	#41	1	2	yes	—
120	C₂×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
121	C₂×C₃×D₄	C₂²×C₅	1	2	yes	—
122	C₂²×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
123	C₂×#67	C₂²×C₅	1	1	yes	characteristic
124	C₂×#67	C₂²×C₅	1	2	yes	—
125	C₂²×Dic₃	C₂²×C₅	1	2	yes	—
126	C₂×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
127	C₂×Dic₆	C₂²×C₅	1	2	yes	—
128	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
129	C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
130	#217	C₂²×C₅	1	2	yes	—
131	#217	C₂²×C₅	1	2	yes	—
132	#217	C₂²×C₅	1	2	yes	—
133	#217	C₂²×C₅	1	1	yes	characteristic
134	(C₂×D₃)⋊C₄	C₂²×C₅	1	1	yes	characteristic
135	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
136	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
137	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
138	Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
139	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
140	C₄×Dic₃	C₂²×C₅	1	2	yes	—
141	C₄×Dic₃	C₂²×C₅	1	2	yes	—
142	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
143	C₂³×C₅	—	3	1	no	—
144	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
145	C₂³×C₅	C₂²×D₃	1	2	yes	—
146	C₅×Q₈	—	12	2	no	—
147	C₅×D₄	—	2	2	no	—
148	C₅×D₄	—	4	2	no	—
149	C₅×D₄	—	6	2	no	—
150	C₅×D₄	—	12	2	no	—
151	C₂×C₄×C₅	—	2	2	no	—
152	C₂×C₄×C₅	—	3	1	no	—
153	C₂×C₄×C₅	—	3	1	no	—
154	C₂×C₄×C₅	—	3	2	no	—
155	C₂×C₄×C₅	—	3	1	no	—
156	C₂×C₄×C₅	—	3	2	no	—
157	C₂×C₄×C₅	—	6	4	no	—
158	C₂×C₄×C₅	—	6	2	no	—
159	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
161	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
162	C₄⋊Q₈	—	3	1	no	—
163	#124	—	3	2	no	—
164	#124	—	3	2	no	—
165	#121	—	3	1	no	—
166	#121	—	3	2	no	—
167	#121	C₅×D₃	1	1	yes	characteristic, complemented
168	C₄×D₄	—	3	1	no	—
169	C₄×D₄	—	3	1	no	—
170	#117	—	3	1	no	—
171	#117	—	3	1	no	—
172	#117	—	3	2	no	—
173	C₂×C₃×C₅	—	4	2	no	—
174	C₂×C₃×C₅	—	4	1	no	—
175	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
176	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
177	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
178	C₅×D₃	—	4	1	no	—
179	C₂³×C₃	C₂³×C₅	1	1	yes	characteristic
180	C₂³×C₃	C₂³×C₅	1	2	yes	—
181	C₂²×D₃	C₂³×C₅	1	1	yes	characteristic
182	C₃×D₄	—	2	2	no	—
183	C₃×D₄	—	4	2	no	—
184	C₂×C₃×C₄	—	2	2	no	—
185	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
187	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
188	#67	—	2	2	no	—
189	#67	—	4	2	no	—
190	C₂×Dic₃	—	2	4	no	—
191	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
192	C₂×Dic₃	C₂³×C₅	1	2	yes	—
193	C₂×Dic₃	C₂³×C₅	1	2	yes	—
194	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
195	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
196	C₄×D₃	—	2	2	no	—
197	Dic₆	—	4	2	no	—
198	C₂²×C₅	—	2	1	no	—
199	C₂²×C₅	—	2	2	no	—
200	C₂²×C₅	—	2	1	no	—
201	C₂²×C₅	—	2	2	no	—
202	C₂²×C₅	—	2	2	no	—
203	C₂²×C₅	—	2	1	no	—
204	C₂²×C₅	—	6	1	no	—
205	C₂²×C₅	—	6	1	no	—
206	C₂²×C₅	—	6	1	no	—
207	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
208	C₄×C₅	—	2	1	no	—
209	C₄×C₅	—	2	2	no	—
210	C₄×C₅	—	3	2	no	—
211	C₄×C₅	—	6	1	no	—
212	C₄×C₅	—	6	2	no	—
213	C₄×C₅	—	6	1	no	—
214	C₄×C₅	—	6	2	no	—
215	C₄×C₅	#237	1	2	yes	—
216	C₂×Q₈	—	3	2	no	—
217	C₂×D₄	—	3	1	no	—
218	C₂×D₄	—	3	2	no	—
219	C₂×D₄	C₂×C₅×D₃	1	1	yes	characteristic
220	C₂×D₄	C₂×C₅×D₃	1	2	yes	—
221	C₂²×C₄	—	3	2	no	—
222	C₂²×C₄	—	3	1	no	—
223	C₂²×C₄	C₂×C₅×D₃	1	1	yes	characteristic
224	C₄⋊C₄	—	3	1	no	—
225	C₄⋊C₄	—	3	1	no	—
226	C₄⋊C₄	—	3	2	no	—
227	C₄⋊C₄	—	3	1	no	—
228	C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
229	C₂²⋊C₄	—	3	2	no	—
230	C₂²⋊C₄	—	3	2	no	—
231	C₂²⋊C₄	—	3	1	no	—
232	C₂²⋊C₄	—	3	2	no	—
233	C₂²⋊C₄	—	3	2	no	—
234	C₂²⋊C₄	—	3	1	no	—
235	C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
236	C₄²	—	3	2	no	—
237	C₄²	—	3	2	no	—
238	C₄²	—	3	1	no	—
239	C₃×C₅	#555	1	1	yes	characteristic, complemented, Hall
240	C₂²×C₃	—	2	2	no	—
241	C₂²×C₃	—	2	1	no	—
242	C₂²×C₃	—	2	1	no	—
243	C₂²×C₃	—	2	2	no	—
244	C₂²×C₃	—	2	2	no	—
245	C₂²×C₃	—	2	1	no	—
246	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
247	C₂×D₃	—	2	1	no	—
248	C₂×D₃	—	2	1	no	—
249	C₂×D₃	—	2	1	no	—
250	C₃×C₄	—	2	1	no	—
251	C₃×C₄	—	2	2	no	—
252	C₃×C₄	C₅×#41	1	2	yes	—
253	Dic₃	—	2	1	no	—
254	Dic₃	—	2	2	no	—
255	Dic₃	—	2	1	no	—
256	Dic₃	—	2	2	no	—
257	Dic₃	C₅×#41	1	2	yes	—
258	C₂×C₅	—	4	1	no	—
259	C₂×C₅	—	4	2	no	—
260	C₂×C₅	—	12	1	no	—
261	C₂×C₅	C₂×#237	1	1	yes	characteristic
262	C₂×C₅	#1004	1	1	yes	characteristic
263	C₂×C₅	D₃×#41	1	1	yes	characteristic
264	C₂³	—	3	1	no	—
265	C₂³	C₂²×C₅×D₃	1	1	yes	characteristic
266	C₂³	C₂²×C₅×D₃	1	2	yes	—
267	Q₈	—	12	2	no	—
268	D₄	—	2	2	no	—
269	D₄	—	4	2	no	—
270	D₄	—	6	2	no	—
271	D₄	—	12	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	3	1	no	—
274	C₂×C₄	—	3	1	no	—
275	C₂×C₄	—	3	2	no	—
276	C₂×C₄	—	3	1	no	—
277	C₂×C₄	—	3	2	no	—
278	C₂×C₄	—	6	4	no	—
279	C₂×C₄	—	6	2	no	—
280	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
281	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
282	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
283	C₂×C₃	—	4	2	no	—
284	C₂×C₃	—	4	1	no	—
285	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
286	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
287	C₂×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
288	D₃	—	4	1	no	—
289	C₅	#5616	1	1	yes	characteristic, direct product factor, Sylow
290	C₂²	—	2	1	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	6	1	no	—
297	C₂²	—	6	1	no	—
298	C₂²	—	6	1	no	—
299	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	2	2	no	—
302	C₄	—	3	2	no	—
303	C₄	—	6	1	no	—
304	C₄	—	6	2	no	—
305	C₄	—	6	1	no	—
306	C₄	—	6	2	no	—
307	C₄	C₅×#237	1	2	yes	—
308	C₃	C₅×#555	1	1	yes	characteristic, complemented, Sylow
309	C₂	—	4	1	no	—
310	C₂	—	4	2	no	—
311	C₂	—	12	1	no	—
312	C₂	C₂×C₅×#237	1	1	yes	characteristic
313	C₂	C₅×#1004	1	1	yes	characteristic
314	C₂	C₅×D₃×#41	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×#121	C₂
socle	C₂²×C₃×C₅	C₂⁴
hypercenter	C₅×#121	D₃
hypocenter	C₃	C₅×#555
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#121

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#121	C₂	semidirect
2. C₅×#939	C₂	semidirect
3. C₅×#938	C₂	semidirect
4. C₅×#936	C₂	semidirect
5. C₅×D₄×Dic₃	C₂	semidirect
6. C₄×C₅×#67	C₂	semidirect
7. C₅×#926	C₂	semidirect
8. C₅×#892	C₂	semidirect
9. C₅×#888	C₂	semidirect
10. C₅×#885	C₂	semidirect
11. C₅×#877	C₂	semidirect
36. #5616	C₅	direct (N × H)
42. C₅×#121	D₃	semidirect
67. C₃×#121	C₂×C₅	semidirect
68. #939	C₂×C₅	semidirect
69. #938	C₂×C₅	semidirect
70. #936	C₂×C₅	semidirect
71. D₄×Dic₃	C₂×C₅	semidirect
72. C₄×#67	C₂×C₅	semidirect
73. #926	C₂×C₅	semidirect
74. #892	C₂×C₅	semidirect
75. #888	C₂×C₅	semidirect
76. #885	C₂×C₅	semidirect
77. #877	C₂×C₅	semidirect
167. #121	C₅×D₃	semidirect
239. C₃×C₅	#555	semidirect
289. C₅	#5616	direct (N × H)
308. C₃	C₅×#555	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#121	2
C₅×#939	2
C₅×#938	2
C₅×#936	2
C₅×D₄×Dic₃	2
C₄×C₅×#67	2
C₅×#926	2
C₅×#892	2
C₅×#888	2
C₅×#885	2
C₅×#877	2
C₅×#555	3
#5616	5

Subgroups — C5×#5616 #11756001

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5616 #11756001