Subgroups — C₅×#5559 #11755944

Order 960.

472 conjugacy classes of subgroups.

The subgroup lattice has 304 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, complemented
2	C₅×#940	C₂	1	1	yes	maximal, characteristic
3	C₅×#936	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#929	C₂	1	2	yes	maximal, complemented
5	C₄×C₅×#67	C₂	1	2	yes	maximal, complemented
6	C₅×#893	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#887	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#885	C₂	1	2	yes	maximal, complemented
9	C₅×#883	C₂	1	2	yes	maximal, complemented
10	C₅×#875	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₅×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#559	—	3	1	no	maximal, Hall
13	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₄×C₅	C₂²	1	2	yes	—
15	C₂×C₅×#67	C₂²	1	2	yes	—
16	C₂×C₅×#67	C₂²	1	2	yes	—
17	C₂×C₅×D₁₂	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
19	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
20	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
21	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
24	C₅×#217	C₂²	1	2	yes	—
25	C₅×#217	C₂²	1	2	yes	—
26	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
27	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
28	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
29	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
30	C₅×Dic₃⋊C₄	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	1	yes	characteristic
34	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
35	C₄×C₅×Dic₃	C₂²	1	2	yes	—
36	#5559	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
37	C₅×#127	—	3	1	no	—
38	C₅×#124	—	3	1	no	—
39	C₅×#124	—	3	2	no	—
40	C₅×#124	—	3	1	no	—
41	C₅×#123	—	3	2	no	—
42	C₅×#123	—	3	2	no	—
43	C₄×C₅×Q₈	—	3	1	no	—
44	C₄×C₅×Q₈	—	3	1	no	—
45	C₄×C₅×D₄	—	3	2	no	—
46	C₄×C₅×D₄	—	3	1	no	—
47	C₄×C₅×D₄	D₃	1	1	yes	characteristic, complemented
48	C₂³×C₃×C₅	C₂³	1	2	yes	—
49	C₂²×C₅×D₃	C₂³	1	2	yes	—
50	C₃×C₅×D₄	—	2	2	no	—
51	C₂×C₃×C₄×C₅	—	4	2	no	—
52	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
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	C₅×#67	—	2	4	no	—
57	C₅×#67	—	4	2	no	—
58	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
59	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
60	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
61	C₂×C₅×Dic₃	C₂³	1	2	yes	—
62	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
63	C₅×D₁₂	—	2	2	no	—
64	C₄×C₅×D₃	—	4	2	no	—
65	C₅×Dic₆	—	2	2	no	—
66	C₅×Dic₆	—	2	2	no	—
67	C₃×C₄×D₄	C₂×C₅	1	1	yes	characteristic, complemented
68	#940	C₂×C₅	1	1	yes	characteristic
69	#936	C₂×C₅	1	1	yes	characteristic, complemented
70	#929	C₂×C₅	1	2	yes	complemented
71	C₄×#67	C₂×C₅	1	2	yes	complemented
72	#893	C₂×C₅	1	1	yes	characteristic, complemented
73	#887	C₂×C₅	1	1	yes	characteristic, complemented
74	#885	C₂×C₅	1	2	yes	complemented
75	#883	C₂×C₅	1	2	yes	complemented
76	#875	C₂×C₅	1	1	yes	characteristic, complemented
77	C₄×Dic₆	C₂×C₅	1	1	yes	characteristic, complemented
78	C₂×C₅×Q₈	—	3	1	no	—
79	C₂×C₅×Q₈	—	3	1	no	—
80	C₂×C₅×D₄	—	3	2	no	—
81	C₂×C₅×D₄	—	3	2	no	—
82	C₂×C₅×D₄	—	3	1	no	—
83	C₂×C₅×D₄	C₂×D₃	1	1	yes	characteristic
84	C₂²×C₄×C₅	—	3	2	no	—
85	C₂²×C₄×C₅	C₂×D₃	1	2	yes	—
86	C₅×C₄⋊C₄	—	3	1	no	—
87	C₅×C₄⋊C₄	—	3	2	no	—
88	C₅×C₄⋊C₄	—	3	1	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	2	no	—
94	C₅×C₂²⋊C₄	—	3	2	no	—
95	C₅×C₂²⋊C₄	—	3	2	no	—
96	C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
97	C₄²×C₅	—	3	2	no	—
98	C₄²×C₅	—	3	1	no	—
99	C₄²×C₅	—	3	1	no	—
100	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
101	#559	—	3	1	no	Sylow
102	C₂²×C₃×C₅	—	2	2	no	—
103	C₂²×C₃×C₅	—	2	2	no	—
104	C₂²×C₃×C₅	—	2	2	no	—
105	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
106	C₂×C₅×D₃	—	2	2	no	—
107	C₂×C₅×D₃	—	2	2	no	—
108	C₂×C₅×D₃	—	2	2	no	—
109	C₃×C₄×C₅	—	2	1	no	—
110	C₃×C₄×C₅	—	2	2	no	—
111	C₃×C₄×C₅	—	2	1	no	—
112	C₃×C₄×C₅	#41	1	2	yes	—
113	C₅×Dic₃	—	2	2	no	—
114	C₅×Dic₃	—	2	1	no	—
115	C₅×Dic₃	—	2	1	no	—
116	C₅×Dic₃	—	2	1	no	—
117	C₅×Dic₃	#41	1	2	yes	—
118	C₂×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
119	C₂²×C₃×C₄	C₂²×C₅	1	2	yes	—
120	C₂×#67	C₂²×C₅	1	2	yes	—
121	C₂×#67	C₂²×C₅	1	2	yes	—
122	C₂×D₁₂	C₂²×C₅	1	1	yes	characteristic
123	C₂×C₄×D₃	C₂²×C₅	1	2	yes	—
124	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
125	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
126	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
127	C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
128	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
129	#217	C₂²×C₅	1	2	yes	—
130	#217	C₂²×C₅	1	2	yes	—
131	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
132	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
133	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
134	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
135	Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
136	Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
137	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
138	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
139	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
140	C₄×Dic₃	C₂²×C₅	1	2	yes	—
141	C₂³×C₅	—	3	2	no	—
142	C₂³×C₅	C₂²×D₃	1	2	yes	—
143	C₅×Q₈	—	6	2	no	—
144	C₅×Q₈	—	6	2	no	—
145	C₅×D₄	—	2	2	no	—
146	C₅×D₄	—	6	4	no	—
147	C₅×D₄	—	6	2	no	—
148	C₅×D₄	—	12	2	no	—
149	C₂×C₄×C₅	—	3	1	no	—
150	C₂×C₄×C₅	—	3	2	no	—
151	C₂×C₄×C₅	—	3	1	no	—
152	C₂×C₄×C₅	—	3	1	no	—
153	C₂×C₄×C₅	—	3	1	no	—
154	C₂×C₄×C₅	—	4	2	no	—
155	C₂×C₄×C₅	—	12	2	no	—
156	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
157	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
158	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
160	#127	—	3	1	no	—
161	#124	—	3	1	no	—
162	#124	—	3	2	no	—
163	#124	—	3	1	no	—
164	#123	—	3	2	no	—
165	#123	—	3	2	no	—
166	C₄×Q₈	—	3	1	no	—
167	C₄×Q₈	—	3	1	no	—
168	C₄×D₄	—	3	2	no	—
169	C₄×D₄	—	3	1	no	—
170	C₄×D₄	C₅×D₃	1	1	yes	characteristic, complemented
171	C₂×C₃×C₅	—	4	2	no	—
172	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
173	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
174	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
175	C₅×D₃	—	4	2	no	—
176	C₂³×C₃	C₂³×C₅	1	2	yes	—
177	C₂²×D₃	C₂³×C₅	1	2	yes	—
178	C₃×D₄	—	2	2	no	—
179	C₂×C₃×C₄	—	4	2	no	—
180	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
181	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
184	#67	—	2	4	no	—
185	#67	—	4	2	no	—
186	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
187	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
188	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
189	C₂×Dic₃	C₂³×C₅	1	2	yes	—
190	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
191	D₁₂	—	2	2	no	—
192	C₄×D₃	—	4	2	no	—
193	Dic₆	—	2	2	no	—
194	Dic₆	—	2	2	no	—
195	C₂²×C₅	—	2	2	no	—
196	C₂²×C₅	—	2	2	no	—
197	C₂²×C₅	—	2	2	no	—
198	C₂²×C₅	—	6	2	no	—
199	C₂²×C₅	—	6	2	no	—
200	C₂²×C₅	—	6	2	no	—
201	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
202	C₄×C₅	—	2	1	no	—
203	C₄×C₅	—	2	2	no	—
204	C₄×C₅	—	2	1	no	—
205	C₄×C₅	—	3	2	no	—
206	C₄×C₅	—	6	2	no	—
207	C₄×C₅	—	6	1	no	—
208	C₄×C₅	—	6	1	no	—
209	C₄×C₅	—	6	1	no	—
210	C₄×C₅	#235	1	2	yes	—
211	C₂×Q₈	—	3	1	no	—
212	C₂×Q₈	—	3	1	no	—
213	C₂×D₄	—	3	2	no	—
214	C₂×D₄	—	3	2	no	—
215	C₂×D₄	—	3	1	no	—
216	C₂×D₄	C₂×C₅×D₃	1	1	yes	characteristic
217	C₂²×C₄	—	3	2	no	—
218	C₂²×C₄	C₂×C₅×D₃	1	2	yes	—
219	C₄⋊C₄	—	3	1	no	—
220	C₄⋊C₄	—	3	2	no	—
221	C₄⋊C₄	—	3	1	no	—
222	C₄⋊C₄	—	3	1	no	—
223	C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
224	C₂²⋊C₄	—	3	2	no	—
225	C₂²⋊C₄	—	3	2	no	—
226	C₂²⋊C₄	—	3	2	no	—
227	C₂²⋊C₄	—	3	2	no	—
228	C₂²⋊C₄	—	3	2	no	—
229	C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
230	C₄²	—	3	2	no	—
231	C₄²	—	3	1	no	—
232	C₄²	—	3	1	no	—
233	C₄²	C₂×C₅×D₃	1	1	yes	characteristic
234	C₃×C₅	#559	1	1	yes	characteristic, complemented, Hall
235	C₂²×C₃	—	2	2	no	—
236	C₂²×C₃	—	2	2	no	—
237	C₂²×C₃	—	2	2	no	—
238	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
239	C₂×D₃	—	2	2	no	—
240	C₂×D₃	—	2	2	no	—
241	C₂×D₃	—	2	2	no	—
242	C₃×C₄	—	2	1	no	—
243	C₃×C₄	—	2	2	no	—
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	C₅×#41	1	2	yes	—
246	Dic₃	—	2	2	no	—
247	Dic₃	—	2	1	no	—
248	Dic₃	—	2	1	no	—
249	Dic₃	—	2	1	no	—
250	Dic₃	C₅×#41	1	2	yes	—
251	C₂×C₅	—	4	2	no	—
252	C₂×C₅	—	12	2	no	—
253	C₂×C₅	C₂×#235	1	1	yes	characteristic
254	C₂×C₅	#1004	1	1	yes	characteristic
255	C₂×C₅	D₃×#41	1	1	yes	characteristic
256	C₂³	—	3	2	no	—
257	C₂³	C₂²×C₅×D₃	1	2	yes	—
258	Q₈	—	6	2	no	—
259	Q₈	—	6	2	no	—
260	D₄	—	2	2	no	—
261	D₄	—	6	4	no	—
262	D₄	—	6	2	no	—
263	D₄	—	12	2	no	—
264	C₂×C₄	—	3	1	no	—
265	C₂×C₄	—	3	2	no	—
266	C₂×C₄	—	3	1	no	—
267	C₂×C₄	—	3	1	no	—
268	C₂×C₄	—	3	1	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	12	2	no	—
271	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
272	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
273	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
275	C₂×C₃	—	4	2	no	—
276	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
277	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
278	C₂×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
279	D₃	—	4	2	no	—
280	C₅	#5559	1	1	yes	characteristic, direct product factor, Sylow
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	6	2	no	—
285	C₂²	—	6	2	no	—
286	C₂²	—	6	2	no	—
287	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	2	no	—
290	C₄	—	2	1	no	—
291	C₄	—	3	2	no	—
292	C₄	—	6	2	no	—
293	C₄	—	6	1	no	—
294	C₄	—	6	1	no	—
295	C₄	—	6	1	no	—
296	C₄	C₅×#235	1	2	yes	—
297	C₃	C₅×#559	1	1	yes	characteristic, complemented, Sylow
298	C₂	—	4	2	no	—
299	C₂	—	12	2	no	—
300	C₂	C₂×C₅×#235	1	1	yes	characteristic
301	C₂	C₅×#1004	1	1	yes	characteristic
302	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×C₄×C₅×D₄	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₅×#940	C₂
hypercenter	C₄×C₅×D₄	D₃
hypocenter	C₃	C₅×#559
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
1. C₃×C₄×C₅×D₄	C₂	semidirect
3. C₅×#936	C₂	semidirect
4. C₅×#929	C₂	semidirect
5. C₄×C₅×#67	C₂	semidirect
6. C₅×#893	C₂	semidirect
7. C₅×#887	C₂	semidirect
8. C₅×#885	C₂	semidirect
9. C₅×#883	C₂	semidirect
10. C₅×#875	C₂	semidirect
11. C₄×C₅×Dic₆	C₂	semidirect
36. #5559	C₅	direct (N × H)
47. C₄×C₅×D₄	D₃	semidirect
67. C₃×C₄×D₄	C₂×C₅	semidirect
69. #936	C₂×C₅	semidirect
70. #929	C₂×C₅	semidirect
71. C₄×#67	C₂×C₅	semidirect
72. #893	C₂×C₅	semidirect
73. #887	C₂×C₅	semidirect
74. #885	C₂×C₅	semidirect
75. #883	C₂×C₅	semidirect
76. #875	C₂×C₅	semidirect
77. C₄×Dic₆	C₂×C₅	semidirect
170. C₄×D₄	C₅×D₃	semidirect
234. C₃×C₅	#559	semidirect
280. C₅	#5559	direct (N × H)
297. C₃	C₅×#559	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×#940	2
C₅×#936	2
C₅×#929	2
C₄×C₅×#67	2
C₅×#893	2
C₅×#887	2
C₅×#885	2
C₅×#883	2
C₅×#875	2
C₄×C₅×Dic₆	2
C₅×#559	3
#5559	5

Subgroups — C5×#5559 #11755944

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5559 #11755944