Subgroups — C₅×#5595 #11755980

Order 960.

504 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#120	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#953	C₂	1	2	yes	maximal, complemented
3	C₅×#940	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#939	C₂	1	2	yes	maximal, complemented
5	C₅×#938	C₂	1	2	yes	maximal, complemented
6	C₅×#935	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#884	C₂	1	2	yes	maximal, complemented
8	C₅×#883	C₂	1	2	yes	maximal, complemented
9	C₅×#879	C₂	1	2	yes	maximal, complemented
10	C₅×#560	—	3	1	no	maximal, Hall
11	C₂⁴×C₃×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₅×D₄	C₂²	1	2	yes	—
14	C₂×C₅×#67	C₂²	1	2	yes	—
15	C₂×C₅×#67	C₂²	1	2	yes	—
16	C₂×C₅×#67	C₂²	1	2	yes	—
17	C₂×C₅×#67	C₂²	1	2	yes	complemented
18	C₂²×C₅×Dic₃	C₂²	1	1	yes	characteristic, complemented
19	C₂×C₅×D₁₂	C₂²	1	1	yes	characteristic, complemented
20	C₂×C₄×C₅×D₃	C₂²	1	2	yes	complemented
21	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₃×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
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, complemented
27	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	complemented
28	C₅×C₄⋊Dic₃	C₂²	1	2	yes	complemented
29	C₅×Dic₃⋊C₄	C₂²	1	2	yes	complemented
30	C₅×Dic₃⋊C₄	C₂²	1	2	yes	complemented
31	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic, complemented
32	#5595	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
33	C₅×#127	—	3	1	no	—
34	C₅×#126	—	3	2	no	—
35	C₅×#123	—	3	1	no	—
36	C₅×#123	—	3	2	no	—
37	C₅×#121	—	3	2	no	—
38	C₅×#121	—	3	2	no	—
39	C₅×#121	—	3	2	no	—
40	C₅×#120	—	3	2	no	—
41	C₅×#120	D₃	1	1	yes	characteristic, complemented
42	C₂³×C₃×C₅	—	4	2	no	—
43	C₂³×C₃×C₅	—	4	1	no	—
44	C₂³×C₃×C₅	C₂³	1	2	yes	—
45	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
46	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
47	C₂²×C₅×D₃	C₂³	1	2	yes	—
48	C₃×C₅×D₄	—	4	2	no	—
49	C₃×C₅×D₄	—	4	1	no	—
50	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
52	C₅×#67	—	4	2	no	—
53	C₅×#67	—	4	2	no	—
54	C₅×#67	—	4	2	no	—
55	C₅×#67	—	4	2	no	—
56	C₂×C₅×Dic₃	—	4	1	no	—
57	C₂×C₅×Dic₃	C₂³	1	2	yes	—
58	C₂×C₅×Dic₃	C₂³	1	2	yes	—
59	C₂×C₅×Dic₃	C₂³	1	2	yes	—
60	C₅×D₁₂	—	4	1	no	—
61	C₄×C₅×D₃	—	4	2	no	—
62	C₃×#120	C₂×C₅	1	1	yes	characteristic, complemented
63	#953	C₂×C₅	1	2	yes	complemented
64	#940	C₂×C₅	1	1	yes	characteristic, complemented
65	#939	C₂×C₅	1	2	yes	complemented
66	#938	C₂×C₅	1	2	yes	complemented
67	#935	C₂×C₅	1	1	yes	characteristic, complemented
68	#884	C₂×C₅	1	2	yes	complemented
69	#883	C₂×C₅	1	2	yes	complemented
70	#879	C₂×C₅	1	2	yes	complemented
71	C₂⁴×C₅	C₂×D₃	1	1	yes	characteristic
72	C₂×C₅×D₄	—	3	2	no	—
73	C₂×C₅×D₄	—	3	2	no	—
74	C₂×C₅×D₄	—	3	2	no	—
75	C₂×C₅×D₄	—	3	2	no	—
76	C₂×C₅×D₄	—	3	1	no	—
77	C₂×C₅×D₄	C₂×D₃	1	2	yes	—
78	C₂×C₅×D₄	C₂×D₃	1	1	yes	characteristic
79	C₂²×C₄×C₅	—	3	1	no	—
80	C₂²×C₄×C₅	—	3	2	no	—
81	C₅×C₄⋊C₄	—	3	2	no	—
82	C₅×C₄⋊C₄	—	3	2	no	—
83	C₅×C₄⋊C₄	—	3	2	no	—
84	C₅×C₂²⋊C₄	—	3	2	no	—
85	C₅×C₂²⋊C₄	—	3	2	no	—
86	C₅×C₂²⋊C₄	—	3	2	no	—
87	C₅×C₂²⋊C₄	—	3	1	no	—
88	C₅×C₂²⋊C₄	—	3	2	no	—
89	C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
90	C₅×C₂²⋊C₄	C₂×D₃	1	1	yes	characteristic
91	C₄²×C₅	—	3	1	no	—
92	#560	—	3	1	no	Sylow
93	C₂²×C₃×C₅	—	2	1	no	—
94	C₂²×C₃×C₅	—	2	2	no	—
95	C₂²×C₃×C₅	—	2	2	no	—
96	C₂²×C₃×C₅	—	2	2	no	—
97	C₂²×C₃×C₅	—	2	2	no	—
98	C₂²×C₃×C₅	—	2	2	no	—
99	C₂²×C₃×C₅	—	2	1	no	—
100	C₂²×C₃×C₅	—	4	4	no	—
101	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
102	C₂×C₅×D₃	—	2	2	no	—
103	C₂×C₅×D₃	—	2	2	no	—
104	C₂×C₅×D₃	—	2	2	no	—
105	C₃×C₄×C₅	—	2	2	no	—
106	C₃×C₄×C₅	—	2	1	no	—
107	C₅×Dic₃	—	2	2	no	—
108	C₅×Dic₃	—	2	2	no	—
109	C₅×Dic₃	—	2	2	no	—
110	C₂⁴×C₃	C₂²×C₅	1	1	yes	characteristic
111	C₂×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
112	C₂×C₃×D₄	C₂²×C₅	1	2	yes	—
113	C₂×#67	C₂²×C₅	1	2	yes	—
114	C₂×#67	C₂²×C₅	1	2	yes	—
115	C₂×#67	C₂²×C₅	1	2	yes	—
116	C₂×#67	C₂²×C₅	1	2	yes	complemented
117	C₂²×Dic₃	C₂²×C₅	1	1	yes	characteristic, complemented
118	C₂×D₁₂	C₂²×C₅	1	1	yes	characteristic, complemented
119	C₂×C₄×D₃	C₂²×C₅	1	2	yes	complemented
120	C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
121	C₃×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
122	#217	C₂²×C₅	1	2	yes	—
123	#217	C₂²×C₅	1	2	yes	—
124	#217	C₂²×C₅	1	2	yes	—
125	#217	C₂²×C₅	1	1	yes	characteristic, complemented
126	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	complemented
127	C₄⋊Dic₃	C₂²×C₅	1	2	yes	complemented
128	Dic₃⋊C₄	C₂²×C₅	1	2	yes	complemented
129	Dic₃⋊C₄	C₂²×C₅	1	2	yes	complemented
130	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic, complemented
131	C₂³×C₅	—	3	2	no	—
132	C₂³×C₅	—	4	2	no	—
133	C₂³×C₅	—	4	1	no	—
134	C₂³×C₅	C₂²×D₃	1	2	yes	—
135	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
136	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
137	C₅×D₄	—	4	2	no	—
138	C₅×D₄	—	4	1	no	—
139	C₅×D₄	—	12	1	no	—
140	C₅×D₄	—	12	2	no	—
141	C₅×D₄	—	12	2	no	—
142	C₅×D₄	—	12	2	no	—
143	C₅×D₄	—	12	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	—	12	2	no	—
148	C₂×C₄×C₅	—	12	1	no	—
149	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
151	#127	—	3	1	no	—
152	#126	—	3	2	no	—
153	#123	—	3	1	no	—
154	#123	—	3	2	no	—
155	#121	—	3	2	no	—
156	#121	—	3	2	no	—
157	#121	—	3	2	no	—
158	#120	—	3	2	no	—
159	#120	C₅×D₃	1	1	yes	characteristic, complemented
160	C₂×C₃×C₅	—	4	1	no	—
161	C₂×C₃×C₅	—	4	2	no	—
162	C₂×C₃×C₅	—	4	1	no	—
163	C₂×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
164	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
165	C₅×D₃	—	4	2	no	—
166	C₂³×C₃	—	4	2	no	—
167	C₂³×C₃	—	4	1	no	—
168	C₂³×C₃	C₂³×C₅	1	2	yes	—
169	C₂³×C₃	C₂³×C₅	1	1	yes	characteristic
170	C₂³×C₃	C₂³×C₅	1	1	yes	characteristic
171	C₂²×D₃	C₂³×C₅	1	2	yes	—
172	C₃×D₄	—	4	2	no	—
173	C₃×D₄	—	4	1	no	—
174	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
176	#67	—	4	2	no	—
177	#67	—	4	2	no	—
178	#67	—	4	2	no	—
179	#67	—	4	2	no	—
180	C₂×Dic₃	—	4	1	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	D₁₂	—	4	1	no	—
185	C₄×D₃	—	4	2	no	—
186	C₂²×C₅	—	2	1	no	—
187	C₂²×C₅	—	2	2	no	—
188	C₂²×C₅	—	2	2	no	—
189	C₂²×C₅	—	2	1	no	—
190	C₂²×C₅	—	2	2	no	—
191	C₂²×C₅	—	2	2	no	—
192	C₂²×C₅	—	2	2	no	—
193	C₂²×C₅	—	4	4	no	—
194	C₂²×C₅	—	6	2	no	—
195	C₂²×C₅	—	6	2	no	—
196	C₂²×C₅	—	6	2	no	—
197	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
198	C₄×C₅	—	2	2	no	—
199	C₄×C₅	—	2	1	no	—
200	C₄×C₅	—	6	2	no	—
201	C₄×C₅	—	6	2	no	—
202	C₄×C₅	—	6	2	no	—
203	C₂⁴	C₂×C₅×D₃	1	1	yes	characteristic
204	C₂×D₄	—	3	2	no	—
205	C₂×D₄	—	3	2	no	—
206	C₂×D₄	—	3	2	no	—
207	C₂×D₄	—	3	2	no	—
208	C₂×D₄	—	3	1	no	—
209	C₂×D₄	C₂×C₅×D₃	1	2	yes	—
210	C₂×D₄	C₂×C₅×D₃	1	1	yes	characteristic
211	C₂²×C₄	—	3	1	no	—
212	C₂²×C₄	—	3	2	no	—
213	C₄⋊C₄	—	3	2	no	—
214	C₄⋊C₄	—	3	2	no	—
215	C₄⋊C₄	—	3	2	no	—
216	C₂²⋊C₄	—	3	2	no	—
217	C₂²⋊C₄	—	3	2	no	—
218	C₂²⋊C₄	—	3	2	no	—
219	C₂²⋊C₄	—	3	1	no	—
220	C₂²⋊C₄	—	3	2	no	—
221	C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
222	C₂²⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
223	C₄²	—	3	1	no	—
224	C₃×C₅	#560	1	1	yes	characteristic, complemented, Hall
225	C₂²×C₃	—	2	1	no	—
226	C₂²×C₃	—	2	2	no	—
227	C₂²×C₃	—	2	2	no	—
228	C₂²×C₃	—	2	2	no	—
229	C₂²×C₃	—	2	2	no	—
230	C₂²×C₃	—	2	2	no	—
231	C₂²×C₃	—	2	1	no	—
232	C₂²×C₃	—	4	4	no	—
233	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
234	C₂×D₃	—	2	2	no	—
235	C₂×D₃	—	2	2	no	—
236	C₂×D₃	—	2	2	no	—
237	C₃×C₄	—	2	2	no	—
238	C₃×C₄	—	2	1	no	—
239	Dic₃	—	2	2	no	—
240	Dic₃	—	2	2	no	—
241	Dic₃	—	2	2	no	—
242	C₂×C₅	—	4	2	no	—
243	C₂×C₅	—	4	1	no	—
244	C₂×C₅	—	4	1	no	—
245	C₂×C₅	—	12	2	no	—
246	C₂×C₅	#1004	1	2	yes	—
247	C₂×C₅	#1004	1	1	yes	characteristic
248	C₂³	—	3	2	no	—
249	C₂³	—	4	2	no	—
250	C₂³	—	4	1	no	—
251	C₂³	C₂²×C₅×D₃	1	2	yes	—
252	C₂³	C₂²×C₅×D₃	1	1	yes	characteristic
253	C₂³	C₂²×C₅×D₃	1	1	yes	characteristic
254	D₄	—	4	2	no	—
255	D₄	—	4	1	no	—
256	D₄	—	12	1	no	—
257	D₄	—	12	2	no	—
258	D₄	—	12	2	no	—
259	D₄	—	12	2	no	—
260	D₄	—	12	2	no	—
261	C₂×C₄	—	3	2	no	—
262	C₂×C₄	—	3	2	no	—
263	C₂×C₄	—	3	2	no	—
264	C₂×C₄	—	12	2	no	—
265	C₂×C₄	—	12	1	no	—
266	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
267	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
268	C₂×C₃	—	4	2	no	—
269	C₂×C₃	—	4	1	no	—
270	C₂×C₃	—	4	1	no	—
271	C₂×C₃	C₅×H(5,𝔽₂)	1	2	yes	—
272	C₂×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
273	D₃	—	4	2	no	—
274	C₅	#5595	1	1	yes	characteristic, direct product factor, Sylow
275	C₂²	—	2	1	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	4	4	no	—
283	C₂²	—	6	2	no	—
284	C₂²	—	6	2	no	—
285	C₂²	—	6	2	no	—
286	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	2	1	no	—
289	C₄	—	6	2	no	—
290	C₄	—	6	2	no	—
291	C₄	—	6	2	no	—
292	C₃	C₅×#560	1	1	yes	characteristic, complemented, Sylow
293	C₂	—	4	2	no	—
294	C₂	—	4	1	no	—
295	C₂	—	4	1	no	—
296	C₂	—	12	2	no	—
297	C₂	C₅×#1004	1	2	yes	—
298	C₂	C₅×#1004	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×#120	C₂
socle	C₂²×C₃×C₅	C₂⁴
hypercenter	C₅×#120	D₃
hypocenter	C₃	C₅×#560
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#120

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#120	C₂	semidirect
2. C₅×#953	C₂	semidirect
3. C₅×#940	C₂	semidirect
4. C₅×#939	C₂	semidirect
5. C₅×#938	C₂	semidirect
6. C₅×#935	C₂	semidirect
7. C₅×#884	C₂	semidirect
8. C₅×#883	C₂	semidirect
9. C₅×#879	C₂	semidirect
17. C₂×C₅×#67	C₂²	semidirect
18. C₂²×C₅×Dic₃	C₂²	semidirect
19. C₂×C₅×D₁₂	C₂²	semidirect
20. C₂×C₄×C₅×D₃	C₂²	semidirect
26. C₅×#217	C₂²	semidirect
27. C₅×(C₂×D₃)⋊C₄	C₂²	semidirect
28. C₅×C₄⋊Dic₃	C₂²	semidirect
29. C₅×Dic₃⋊C₄	C₂²	semidirect
30. C₅×Dic₃⋊C₄	C₂²	semidirect
31. C₄×C₅×Dic₃	C₂²	semidirect
32. #5595	C₅	direct (N × H)
41. C₅×#120	D₃	semidirect
62. C₃×#120	C₂×C₅	semidirect
63. #953	C₂×C₅	semidirect
64. #940	C₂×C₅	semidirect
65. #939	C₂×C₅	semidirect
66. #938	C₂×C₅	semidirect
67. #935	C₂×C₅	semidirect
68. #884	C₂×C₅	semidirect
69. #883	C₂×C₅	semidirect
70. #879	C₂×C₅	semidirect
116. C₂×#67	C₂²×C₅	semidirect
117. C₂²×Dic₃	C₂²×C₅	semidirect
118. C₂×D₁₂	C₂²×C₅	semidirect
119. C₂×C₄×D₃	C₂²×C₅	semidirect
125. #217	C₂²×C₅	semidirect
126. (C₂×D₃)⋊C₄	C₂²×C₅	semidirect
127. C₄⋊Dic₃	C₂²×C₅	semidirect
128. Dic₃⋊C₄	C₂²×C₅	semidirect
129. Dic₃⋊C₄	C₂²×C₅	semidirect
130. C₄×Dic₃	C₂²×C₅	semidirect
159. #120	C₅×D₃	semidirect
224. C₃×C₅	#560	semidirect
274. C₅	#5595	direct (N × H)
292. C₃	C₅×#560	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#120	2
C₅×#953	2
C₅×#940	2
C₅×#939	2
C₅×#938	2
C₅×#935	2
C₅×#884	2
C₅×#883	2
C₅×#879	2
C₅×#560	3
#5595	5

Subgroups — C5×#5595 #11755980

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5595 #11755980