Subgroups — C₅×#5603 #11755988

Order 960.

432 conjugacy classes of subgroups.

The subgroup lattice has 296 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
2	C₅×#936	C₂	1	2	yes	maximal, complemented
3	C₅×#936	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#935	C₂	1	2	yes	maximal, complemented
5	C₅×D₄×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#926	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#924	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#888	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#887	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#879	C₂	1	2	yes	maximal, complemented
11	C₅×#878	C₂	1	2	yes	maximal, complemented
12	C₅×#542	—	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	C₂²×C₅×Dic₃	C₂²	1	2	yes	—
17	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
18	C₂×C₅×Dic₆	C₂²	1	2	yes	—
19	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₅×#217	C₂²	1	2	yes	—
22	C₅×#217	C₂²	1	2	yes	—
23	C₅×#217	C₂²	1	2	yes	—
24	C₅×#217	C₂²	1	2	yes	—
25	C₅×#217	C₂²	1	1	yes	characteristic
26	C₅×#217	C₂²	1	1	yes	characteristic
27	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
28	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
29	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
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	C₄×C₅×Dic₃	C₂²	1	2	yes	—
34	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
35	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
36	#5603	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
37	C₅×C₄⋊Q₈	—	3	1	no	—
38	C₅×#126	—	3	2	no	—
39	C₅×#124	—	3	1	no	—
40	C₅×#124	—	3	2	no	—
41	C₅×#123	—	3	2	no	—
42	C₅×C₂²⋊Q₈	—	3	1	no	—
43	C₅×C₂²⋊Q₈	—	3	2	no	—
44	C₅×#121	D₃	1	1	yes	characteristic
45	C₄×C₅×Q₈	—	3	1	no	—
46	C₄×C₅×D₄	—	3	1	no	—
47	C₅×#117	—	3	1	no	—
48	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
49	C₂³×C₃×C₅	C₂³	1	2	yes	—
50	C₃×C₅×D₄	—	2	2	no	—
51	C₃×C₅×D₄	—	4	2	no	—
52	C₂×C₃×C₄×C₅	—	2	2	no	—
53	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
54	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
56	C₂×C₅×Dic₃	—	4	2	no	—
57	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
58	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
59	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
60	C₂×C₅×Dic₃	C₂³	1	2	yes	—
61	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
62	C₂×C₅×Dic₃	C₂³	1	2	yes	—
63	C₅×Dic₆	—	2	2	no	—
64	C₅×Dic₆	—	4	2	no	—
65	C₃×#121	C₂×C₅	1	1	yes	characteristic
66	#936	C₂×C₅	1	2	yes	complemented
67	#936	C₂×C₅	1	1	yes	characteristic, complemented
68	#935	C₂×C₅	1	2	yes	complemented
69	D₄×Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
70	#926	C₂×C₅	1	1	yes	characteristic, complemented
71	#924	C₂×C₅	1	1	yes	characteristic, complemented
72	#888	C₂×C₅	1	1	yes	characteristic, complemented
73	#887	C₂×C₅	1	1	yes	characteristic, complemented
74	#879	C₂×C₅	1	2	yes	complemented
75	#878	C₂×C₅	1	2	yes	complemented
76	C₂×C₅×Q₈	—	3	1	no	—
77	C₂×C₅×Q₈	—	3	2	no	—
78	C₂×C₅×D₄	C₂×D₃	1	1	yes	characteristic
79	C₂×C₅×D₄	C₂×D₃	1	2	yes	—
80	C₂²×C₄×C₅	—	3	2	no	—
81	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
82	C₅×C₄⋊C₄	—	3	1	no	—
83	C₅×C₄⋊C₄	—	3	1	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	C₂×D₃	1	1	yes	characteristic
89	C₅×C₂²⋊C₄	—	3	1	no	—
90	C₅×C₂²⋊C₄	—	3	1	no	—
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	C₂×D₃	1	2	yes	—
96	C₄²×C₅	—	3	1	no	—
97	C₄²×C₅	—	3	1	no	—
98	C₄²×C₅	—	3	2	no	—
99	#542	—	3	1	no	Sylow
100	C₂²×C₃×C₅	—	2	1	no	—
101	C₂²×C₃×C₅	—	2	2	no	—
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	C₂⁴	1	1	yes	characteristic
107	C₃×C₄×C₅	—	2	1	no	—
108	C₃×C₄×C₅	—	2	2	no	—
109	C₃×C₄×C₅	#41	1	2	yes	—
110	C₅×Dic₃	—	2	2	no	—
111	C₅×Dic₃	—	2	1	no	—
112	C₅×Dic₃	—	2	2	no	—
113	C₅×Dic₃	—	2	1	no	—
114	C₅×Dic₃	—	2	1	no	—
115	C₅×Dic₃	—	2	1	no	—
116	C₂×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
117	C₂×C₃×D₄	C₂²×C₅	1	2	yes	—
118	C₂²×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
119	C₂²×Dic₃	C₂²×C₅	1	2	yes	—
120	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
121	C₂×Dic₆	C₂²×C₅	1	2	yes	—
122	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
123	C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
124	#217	C₂²×C₅	1	2	yes	—
125	#217	C₂²×C₅	1	2	yes	—
126	#217	C₂²×C₅	1	2	yes	—
127	#217	C₂²×C₅	1	2	yes	—
128	#217	C₂²×C₅	1	1	yes	characteristic
129	#217	C₂²×C₅	1	1	yes	characteristic
130	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
131	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
132	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
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	1	yes	characteristic
138	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
139	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
140	C₂³×C₅	C₂²×D₃	1	2	yes	—
141	C₅×Q₈	—	6	2	no	—
142	C₅×Q₈	—	12	2	no	—
143	C₅×D₄	—	2	2	no	—
144	C₅×D₄	—	4	2	no	—
145	C₂×C₄×C₅	—	2	2	no	—
146	C₂×C₄×C₅	—	3	1	no	—
147	C₂×C₄×C₅	—	3	1	no	—
148	C₂×C₄×C₅	—	3	2	no	—
149	C₂×C₄×C₅	—	3	1	no	—
150	C₂×C₄×C₅	—	3	1	no	—
151	C₂×C₄×C₅	—	3	2	no	—
152	C₂×C₄×C₅	—	12	2	no	—
153	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
154	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
155	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
156	C₄⋊Q₈	—	3	1	no	—
157	#126	—	3	2	no	—
158	#124	—	3	1	no	—
159	#124	—	3	2	no	—
160	#123	—	3	2	no	—
161	C₂²⋊Q₈	—	3	1	no	—
162	C₂²⋊Q₈	—	3	2	no	—
163	#121	C₅×D₃	1	1	yes	characteristic
164	C₄×Q₈	—	3	1	no	—
165	C₄×D₄	—	3	1	no	—
166	#117	—	3	1	no	—
167	C₂×C₃×C₅	—	4	1	no	—
168	C₂×C₃×C₅	—	4	2	no	—
169	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
170	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
171	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
172	C₂³×C₃	C₂³×C₅	1	1	yes	characteristic
173	C₂³×C₃	C₂³×C₅	1	2	yes	—
174	C₃×D₄	—	2	2	no	—
175	C₃×D₄	—	4	2	no	—
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
178	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
180	C₂×Dic₃	—	4	2	no	—
181	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
182	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
183	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
184	C₂×Dic₃	C₂³×C₅	1	2	yes	—
185	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
186	C₂×Dic₃	C₂³×C₅	1	2	yes	—
187	Dic₆	—	2	2	no	—
188	Dic₆	—	4	2	no	—
189	C₂²×C₅	—	2	1	no	—
190	C₂²×C₅	—	2	2	no	—
191	C₂²×C₅	—	2	2	no	—
192	C₂²×C₅	—	2	1	no	—
193	C₂²×C₅	—	2	1	no	—
194	C₂²×C₅	—	2	2	no	—
195	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
196	C₄×C₅	—	2	1	no	—
197	C₄×C₅	—	2	2	no	—
198	C₄×C₅	—	6	2	no	—
199	C₄×C₅	—	6	1	no	—
200	C₄×C₅	—	6	2	no	—
201	C₄×C₅	—	6	1	no	—
202	C₄×C₅	—	6	1	no	—
203	C₄×C₅	—	6	1	no	—
204	C₄×C₅	#237	1	2	yes	—
205	C₂×Q₈	—	3	1	no	—
206	C₂×Q₈	—	3	2	no	—
207	C₂×D₄	C₂×C₅×D₃	1	1	yes	characteristic
208	C₂×D₄	C₂×C₅×D₃	1	2	yes	—
209	C₂²×C₄	—	3	2	no	—
210	C₂²×C₄	C₂×C₅×D₃	1	1	yes	characteristic
211	C₄⋊C₄	—	3	1	no	—
212	C₄⋊C₄	—	3	1	no	—
213	C₄⋊C₄	—	3	1	no	—
214	C₄⋊C₄	—	3	2	no	—
215	C₄⋊C₄	—	3	2	no	—
216	C₄⋊C₄	—	3	2	no	—
217	C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
218	C₂²⋊C₄	—	3	1	no	—
219	C₂²⋊C₄	—	3	1	no	—
220	C₂²⋊C₄	—	3	2	no	—
221	C₂²⋊C₄	—	3	2	no	—
222	C₂²⋊C₄	—	3	2	no	—
223	C₂²⋊C₄	—	3	2	no	—
224	C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
225	C₄²	—	3	1	no	—
226	C₄²	—	3	1	no	—
227	C₄²	—	3	2	no	—
228	C₃×C₅	#542	1	1	yes	characteristic, complemented, Hall
229	C₂²×C₃	—	2	1	no	—
230	C₂²×C₃	—	2	2	no	—
231	C₂²×C₃	—	2	2	no	—
232	C₂²×C₃	—	2	1	no	—
233	C₂²×C₃	—	2	1	no	—
234	C₂²×C₃	—	2	2	no	—
235	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
236	C₃×C₄	—	2	1	no	—
237	C₃×C₄	—	2	2	no	—
238	C₃×C₄	C₅×#41	1	2	yes	—
239	Dic₃	—	2	2	no	—
240	Dic₃	—	2	1	no	—
241	Dic₃	—	2	2	no	—
242	Dic₃	—	2	1	no	—
243	Dic₃	—	2	1	no	—
244	Dic₃	—	2	1	no	—
245	C₂×C₅	—	4	1	no	—
246	C₂×C₅	—	4	2	no	—
247	C₂×C₅	C₂×#237	1	1	yes	characteristic
248	C₂×C₅	#1004	1	1	yes	characteristic
249	C₂×C₅	#1010	1	1	yes	characteristic
250	C₂³	C₂²×C₅×D₃	1	1	yes	characteristic
251	C₂³	C₂²×C₅×D₃	1	2	yes	—
252	Q₈	—	6	2	no	—
253	Q₈	—	12	2	no	—
254	D₄	—	2	2	no	—
255	D₄	—	4	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	3	1	no	—
258	C₂×C₄	—	3	1	no	—
259	C₂×C₄	—	3	2	no	—
260	C₂×C₄	—	3	1	no	—
261	C₂×C₄	—	3	1	no	—
262	C₂×C₄	—	3	2	no	—
263	C₂×C₄	—	12	2	no	—
264	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
265	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
266	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
267	C₂×C₃	—	4	1	no	—
268	C₂×C₃	—	4	2	no	—
269	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
270	C₂×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
271	C₂×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
272	C₅	#5603	1	1	yes	characteristic, direct product factor, Sylow
273	C₂²	—	2	1	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	2	no	—
279	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
280	C₄	—	2	1	no	—
281	C₄	—	2	2	no	—
282	C₄	—	6	2	no	—
283	C₄	—	6	1	no	—
284	C₄	—	6	2	no	—
285	C₄	—	6	1	no	—
286	C₄	—	6	1	no	—
287	C₄	—	6	1	no	—
288	C₄	C₅×#237	1	2	yes	—
289	C₃	C₅×#542	1	1	yes	characteristic, complemented, Sylow
290	C₂	—	4	1	no	—
291	C₂	—	4	2	no	—
292	C₂	C₂×C₅×#237	1	1	yes	characteristic
293	C₂	C₅×#1004	1	1	yes	characteristic
294	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, prime radical	C₃×C₅×#121	C₂
socle	C₂²×C₃×C₅	C₂⁴
hypercenter	C₅×#121	D₃
hypocenter	C₃	C₅×#542
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
2. C₅×#936	C₂	semidirect
3. C₅×#936	C₂	semidirect
4. C₅×#935	C₂	semidirect
5. C₅×D₄×Dic₃	C₂	semidirect
6. C₅×#926	C₂	semidirect
7. C₅×#924	C₂	semidirect
8. C₅×#888	C₂	semidirect
9. C₅×#887	C₂	semidirect
10. C₅×#879	C₂	semidirect
11. C₅×#878	C₂	semidirect
36. #5603	C₅	direct (N × H)
66. #936	C₂×C₅	semidirect
67. #936	C₂×C₅	semidirect
68. #935	C₂×C₅	semidirect
69. D₄×Dic₃	C₂×C₅	semidirect
70. #926	C₂×C₅	semidirect
71. #924	C₂×C₅	semidirect
72. #888	C₂×C₅	semidirect
73. #887	C₂×C₅	semidirect
74. #879	C₂×C₅	semidirect
75. #878	C₂×C₅	semidirect
228. C₃×C₅	#542	semidirect
272. C₅	#5603	direct (N × H)
289. C₃	C₅×#542	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×#936	2
C₅×#935	2
C₅×D₄×Dic₃	2
C₅×#926	2
C₅×#924	2
C₅×#888	2
C₅×#887	2
C₅×#879	2
C₅×#878	2
C₅×#542	3
#5603	5

Subgroups — C5×#5603 #11755988

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5603 #11755988