Subgroups — C₅×#5541 #11755926

Order 960.

428 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic
2	C₅×#935	C₂	1	2	yes	maximal
3	C₅×#926	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#924	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#924	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#889	C₂	1	2	yes	maximal, complemented
8	C₅×#879	C₂	1	2	yes	maximal, complemented
9	C₅×#877	C₂	1	2	yes	maximal, complemented
10	C₅×#870	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₅×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#552	—	3	1	no	maximal, Hall
13	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
16	C₂²×C₅×Dic₃	C₂²	1	2	yes	—
17	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
18	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₃×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
20	C₃×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
21	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
22	C₅×#217	C₂²	1	2	yes	—
23	C₅×#217	C₂²	1	2	yes	—
24	C₅×#217	C₂²	1	2	yes	—
25	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
26	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
27	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
28	C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
29	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
30	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
31	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
32	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
33	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
34	C₄×C₅×Dic₃	C₂²	1	2	yes	—
35	C₄×C₅×Dic₃	C₂²	1	2	yes	—
36	#5541	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
37	C₅×#126	—	3	2	no	—
38	C₅×#125	—	3	2	no	—
39	C₅×#125	—	3	1	no	—
40	C₅×#123	—	3	2	no	—
41	C₅×C₂²⋊Q₈	—	3	1	no	—
42	C₅×C₂²⋊Q₈	—	3	1	no	—
43	C₄×C₅×Q₈	—	3	1	no	—
44	C₄×C₅×D₄	D₃	1	1	yes	characteristic
45	C₅×#117	—	3	1	no	—
46	C₅×#117	—	3	2	no	—
47	C₂×C₅×C₄⋊C₄	—	3	1	no	—
48	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
49	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
50	C₃×C₅×D₄	—	2	2	no	—
51	C₂×C₃×C₄×C₅	—	2	2	no	—
52	C₂×C₃×C₄×C₅	—	2	2	no	—
53	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
58	C₂×C₅×Dic₃	—	2	4	no	—
59	C₂×C₅×Dic₃	C₂³	1	2	yes	—
60	C₂×C₅×Dic₃	C₂³	1	2	yes	—
61	C₂×C₅×Dic₃	C₂³	1	2	yes	—
62	C₂×C₅×Dic₃	C₂³	1	2	yes	—
63	C₅×Dic₆	—	2	2	no	—
64	C₃×C₄×D₄	C₂×C₅	1	1	yes	characteristic
65	#935	C₂×C₅	1	2	yes	—
66	#926	C₂×C₅	1	1	yes	characteristic, complemented
67	C₂×C₄⋊Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
68	#924	C₂×C₅	1	1	yes	characteristic, complemented
69	#924	C₂×C₅	1	1	yes	characteristic, complemented
70	#889	C₂×C₅	1	2	yes	complemented
71	#879	C₂×C₅	1	2	yes	complemented
72	#877	C₂×C₅	1	2	yes	complemented
73	#870	C₂×C₅	1	1	yes	characteristic, complemented
74	C₄×Dic₆	C₂×C₅	1	1	yes	characteristic, complemented
75	C₂×C₅×Q₈	—	3	1	no	—
76	C₂×C₅×D₄	C₂×D₃	1	1	yes	characteristic
77	C₂²×C₄×C₅	—	3	2	no	—
78	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
80	C₅×C₄⋊C₄	—	3	1	no	—
81	C₅×C₄⋊C₄	—	3	1	no	—
82	C₅×C₄⋊C₄	—	3	2	no	—
83	C₅×C₄⋊C₄	—	3	2	no	—
84	C₅×C₄⋊C₄	—	3	1	no	—
85	C₅×C₄⋊C₄	—	3	2	no	—
86	C₅×C₄⋊C₄	—	3	2	no	—
87	C₅×C₄⋊C₄	—	3	2	no	—
88	C₅×C₄⋊C₄	—	3	2	no	—
89	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
90	C₅×C₂²⋊C₄	—	3	2	no	—
91	C₅×C₂²⋊C₄	—	3	2	no	—
92	C₅×C₂²⋊C₄	—	3	2	no	—
93	C₅×C₂²⋊C₄	C₂×D₃	1	1	yes	characteristic
94	C₅×C₂²⋊C₄	C₂×D₃	1	1	yes	characteristic
95	C₄²×C₅	—	3	2	no	—
96	C₄²×C₅	—	3	2	no	—
97	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
98	#552	—	3	1	no	Sylow
99	C₂²×C₃×C₅	—	2	1	no	—
100	C₂²×C₃×C₅	—	2	1	no	—
101	C₂²×C₃×C₅	—	2	1	no	—
102	C₂²×C₃×C₅	—	2	1	no	—
103	C₂²×C₃×C₅	—	2	1	no	—
104	C₂²×C₃×C₅	#41	1	2	yes	—
105	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
106	C₃×C₄×C₅	—	2	1	no	—
107	C₃×C₄×C₅	—	2	1	no	—
108	C₃×C₄×C₅	—	2	1	no	—
109	C₃×C₄×C₅	—	2	1	no	—
110	C₃×C₄×C₅	#41	1	2	yes	—
111	C₅×Dic₃	—	2	2	no	—
112	C₅×Dic₃	—	2	2	no	—
113	C₅×Dic₃	—	2	2	no	—
114	C₅×Dic₃	—	2	2	no	—
115	C₂×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
116	C₂²×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
117	C₂²×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
118	C₂²×Dic₃	C₂²×C₅	1	2	yes	—
119	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
120	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
121	C₃×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
122	C₃×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
123	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
124	#217	C₂²×C₅	1	2	yes	—
125	#217	C₂²×C₅	1	2	yes	—
126	#217	C₂²×C₅	1	2	yes	—
127	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
128	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
129	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
130	C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
131	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
132	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
133	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
134	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
135	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
136	C₄×Dic₃	C₂²×C₅	1	2	yes	—
137	C₄×Dic₃	C₂²×C₅	1	2	yes	—
138	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
139	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
140	C₅×Q₈	—	6	2	no	—
141	C₅×D₄	—	2	2	no	—
142	C₂×C₄×C₅	—	2	2	no	—
143	C₂×C₄×C₅	—	2	2	no	—
144	C₂×C₄×C₅	—	3	2	no	—
145	C₂×C₄×C₅	—	3	2	no	—
146	C₂×C₄×C₅	—	3	2	no	—
147	C₂×C₄×C₅	—	3	2	no	—
148	C₂×C₄×C₅	—	6	4	no	—
149	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
154	#126	—	3	2	no	—
155	#125	—	3	2	no	—
156	#125	—	3	1	no	—
157	#123	—	3	2	no	—
158	C₂²⋊Q₈	—	3	1	no	—
159	C₂²⋊Q₈	—	3	1	no	—
160	C₄×Q₈	—	3	1	no	—
161	C₄×D₄	C₅×D₃	1	1	yes	characteristic
162	#117	—	3	1	no	—
163	#117	—	3	2	no	—
164	C₂×C₄⋊C₄	—	3	1	no	—
165	C₂×C₃×C₅	—	2	2	no	—
166	C₂×C₃×C₅	—	4	1	no	—
167	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
168	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
169	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
170	C₂³×C₃	C₂³×C₅	1	1	yes	characteristic
171	C₂³×C₃	C₂³×C₅	1	1	yes	characteristic
172	C₃×D₄	—	2	2	no	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	2	2	no	—
175	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
180	C₂×Dic₃	—	2	4	no	—
181	C₂×Dic₃	C₂³×C₅	1	2	yes	—
182	C₂×Dic₃	C₂³×C₅	1	2	yes	—
183	C₂×Dic₃	C₂³×C₅	1	2	yes	—
184	C₂×Dic₃	C₂³×C₅	1	2	yes	—
185	Dic₆	—	2	2	no	—
186	C₂²×C₅	—	2	1	no	—
187	C₂²×C₅	—	2	1	no	—
188	C₂²×C₅	—	2	1	no	—
189	C₂²×C₅	—	2	1	no	—
190	C₂²×C₅	—	2	1	no	—
191	C₂²×C₅	#235	1	2	yes	—
192	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
193	C₄×C₅	—	2	1	no	—
194	C₄×C₅	—	2	1	no	—
195	C₄×C₅	—	2	1	no	—
196	C₄×C₅	—	2	1	no	—
197	C₄×C₅	—	6	2	no	—
198	C₄×C₅	—	6	2	no	—
199	C₄×C₅	—	6	2	no	—
200	C₄×C₅	—	6	2	no	—
201	C₄×C₅	#237	1	2	yes	—
202	C₂×Q₈	—	3	1	no	—
203	C₂×D₄	C₂×C₅×D₃	1	1	yes	characteristic
204	C₂²×C₄	—	3	2	no	—
205	C₂²×C₄	C₂×C₅×D₃	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₅×D₃	1	1	yes	characteristic
207	C₄⋊C₄	—	3	1	no	—
208	C₄⋊C₄	—	3	1	no	—
209	C₄⋊C₄	—	3	2	no	—
210	C₄⋊C₄	—	3	2	no	—
211	C₄⋊C₄	—	3	1	no	—
212	C₄⋊C₄	—	3	2	no	—
213	C₄⋊C₄	—	3	2	no	—
214	C₄⋊C₄	—	3	2	no	—
215	C₄⋊C₄	—	3	2	no	—
216	C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
217	C₂²⋊C₄	—	3	2	no	—
218	C₂²⋊C₄	—	3	2	no	—
219	C₂²⋊C₄	—	3	2	no	—
220	C₂²⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
221	C₂²⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
222	C₄²	—	3	2	no	—
223	C₄²	—	3	2	no	—
224	C₄²	C₂×C₅×D₃	1	1	yes	characteristic
225	C₃×C₅	#552	1	1	yes	characteristic, complemented, Hall
226	C₂²×C₃	—	2	1	no	—
227	C₂²×C₃	—	2	1	no	—
228	C₂²×C₃	—	2	1	no	—
229	C₂²×C₃	—	2	1	no	—
230	C₂²×C₃	—	2	1	no	—
231	C₂²×C₃	C₅×#41	1	2	yes	—
232	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
233	C₃×C₄	—	2	1	no	—
234	C₃×C₄	—	2	1	no	—
235	C₃×C₄	—	2	1	no	—
236	C₃×C₄	—	2	1	no	—
237	C₃×C₄	C₅×#41	1	2	yes	—
238	Dic₃	—	2	2	no	—
239	Dic₃	—	2	2	no	—
240	Dic₃	—	2	2	no	—
241	Dic₃	—	2	2	no	—
242	C₂×C₅	—	2	2	no	—
243	C₂×C₅	—	4	1	no	—
244	C₂×C₅	C₂×#235	1	1	yes	characteristic
245	C₂×C₅	C₂×#237	1	1	yes	characteristic
246	C₂×C₅	#1010	1	1	yes	characteristic
247	C₂³	C₂²×C₅×D₃	1	1	yes	characteristic
248	C₂³	C₂²×C₅×D₃	1	1	yes	characteristic
249	Q₈	—	6	2	no	—
250	D₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	3	2	no	—
254	C₂×C₄	—	3	2	no	—
255	C₂×C₄	—	3	2	no	—
256	C₂×C₄	—	3	2	no	—
257	C₂×C₄	—	6	4	no	—
258	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
260	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
261	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
262	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
263	C₂×C₃	—	2	2	no	—
264	C₂×C₃	—	4	1	no	—
265	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
266	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
267	C₂×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
268	C₅	#5541	1	1	yes	characteristic, direct product factor, Sylow
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	C₅×#235	1	2	yes	—
275	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	6	2	no	—
281	C₄	—	6	2	no	—
282	C₄	—	6	2	no	—
283	C₄	—	6	2	no	—
284	C₄	C₅×#237	1	2	yes	—
285	C₃	C₅×#552	1	1	yes	characteristic, complemented, Sylow
286	C₂	—	2	2	no	—
287	C₂	—	4	1	no	—
288	C₂	C₂×C₅×#235	1	1	yes	characteristic
289	C₂	C₂×C₅×#237	1	1	yes	characteristic
290	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×C₄×C₅×D₄	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂×C₃×C₅×D₄	C₂²
hypercenter	C₄×C₅×D₄	D₃
hypocenter	C₃	C₅×#552
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄×C₅×D₄

Complements

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

N	H ≅ G/N	type
3. C₅×#926	C₂	semidirect
4. C₂×C₅×C₄⋊Dic₃	C₂	semidirect
5. C₅×#924	C₂	semidirect
6. C₅×#924	C₂	semidirect
7. C₅×#889	C₂	semidirect
8. C₅×#879	C₂	semidirect
9. C₅×#877	C₂	semidirect
10. C₅×#870	C₂	semidirect
11. C₄×C₅×Dic₆	C₂	semidirect
36. #5541	C₅	direct (N × H)
66. #926	C₂×C₅	semidirect
67. C₂×C₄⋊Dic₃	C₂×C₅	semidirect
68. #924	C₂×C₅	semidirect
69. #924	C₂×C₅	semidirect
70. #889	C₂×C₅	semidirect
71. #879	C₂×C₅	semidirect
72. #877	C₂×C₅	semidirect
73. #870	C₂×C₅	semidirect
74. C₄×Dic₆	C₂×C₅	semidirect
225. C₃×C₅	#552	semidirect
268. C₅	#5541	direct (N × H)
285. C₃	C₅×#552	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₅×D₄	2
C₅×#935	2
C₅×#926	2
C₂×C₅×C₄⋊Dic₃	2
C₅×#924	2
C₅×#889	2
C₅×#879	2
C₅×#877	2
C₅×#870	2
C₄×C₅×Dic₆	2
C₅×#552	3
#5541	5

Subgroups — C5×#5541 #11755926

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5541 #11755926