Subgroups — C₅×#5539 #11755924

Order 960.

456 conjugacy classes of subgroups.

The subgroup lattice has 284 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×D₄×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₅×D₄×Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₅×#924	C₂	1	2	yes	maximal, complemented
5	C₂×C₅×Dic₃⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₅×#890	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#888	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#878	C₂	1	2	yes	maximal, complemented
9	C₅×#878	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×#554	—	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	C₂²×C₅×Dic₃	C₂²	1	2	yes	—
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	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
22	C₅×#217	C₂²	1	2	yes	—
23	C₅×#217	C₂²	1	2	yes	—
24	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
25	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
26	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
27	C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
28	C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
29	C₅×Dic₃⋊C₄	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	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
34	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
35	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
36	#5539	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
37	C₅×C₄⋊Q₈	—	3	1	no	—
38	C₅×#125	—	3	1	no	—
39	C₅×#125	—	3	1	no	—
40	C₅×C₂²⋊Q₈	—	3	2	no	—
41	C₅×C₂²⋊Q₈	—	3	2	no	—
42	C₅×C₂²⋊Q₈	—	3	2	no	—
43	C₄×C₅×Q₈	—	3	1	no	—
44	C₄×C₅×D₄	—	3	1	no	—
45	C₄×C₅×D₄	—	3	1	no	—
46	C₄×C₅×D₄	D₃	1	1	yes	characteristic
47	C₂×C₅×C₄⋊C₄	—	3	2	no	—
48	C₂³×C₃×C₅	C₂³	1	2	yes	—
49	C₃×C₅×D₄	Q₈	1	4	yes	complemented
50	C₂×C₃×C₄×C₅	—	4	2	no	—
51	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
54	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
55	C₂×C₅×Dic₃	—	2	4	no	—
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	2	yes	—
62	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
63	C₅×Dic₆	—	2	2	no	—
64	C₅×Dic₆	—	4	2	no	—
65	C₃×C₄×D₄	C₂×C₅	1	1	yes	characteristic
66	D₄×Dic₃	C₂×C₅	1	1	yes	characteristic
67	D₄×Dic₃	C₂×C₅	1	1	yes	characteristic
68	#924	C₂×C₅	1	2	yes	complemented
69	C₂×Dic₃⋊C₄	C₂×C₅	1	2	yes	complemented
70	#890	C₂×C₅	1	1	yes	characteristic, complemented
71	#888	C₂×C₅	1	1	yes	characteristic, complemented
72	#878	C₂×C₅	1	2	yes	complemented
73	#878	C₂×C₅	1	2	yes	complemented
74	#870	C₂×C₅	1	1	yes	characteristic, complemented
75	C₄×Dic₆	C₂×C₅	1	1	yes	characteristic, 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	C₂²×C₄×C₅	—	3	2	no	—
80	C₂²×C₄×C₅	—	3	2	no	—
81	C₂²×C₄×C₅	C₂×D₃	1	2	yes	—
82	C₅×C₄⋊C₄	—	3	1	no	—
83	C₅×C₄⋊C₄	—	3	2	no	—
84	C₅×C₄⋊C₄	—	3	2	no	—
85	C₅×C₄⋊C₄	—	3	2	no	—
86	C₅×C₄⋊C₄	—	3	1	no	—
87	C₅×C₄⋊C₄	—	3	1	no	—
88	C₅×C₄⋊C₄	—	3	1	no	—
89	C₅×C₄⋊C₄	—	3	1	no	—
90	C₅×C₄⋊C₄	—	3	2	no	—
91	C₅×C₄⋊C₄	—	3	2	no	—
92	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
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	C₂×D₃	1	1	yes	characteristic
99	#554	—	3	1	no	Sylow
100	C₂²×C₃×C₅	—	2	2	no	—
101	C₂²×C₃×C₅	—	2	2	no	—
102	C₂²×C₃×C₅	C₂×Q₈	1	4	yes	—
103	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
104	C₃×C₄×C₅	—	2	1	no	—
105	C₃×C₄×C₅	—	2	2	no	—
106	C₃×C₄×C₅	—	2	1	no	—
107	C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
108	C₅×Dic₃	—	2	1	no	—
109	C₅×Dic₃	—	2	2	no	—
110	C₅×Dic₃	—	2	1	no	—
111	C₅×Dic₃	—	2	2	no	—
112	C₅×Dic₃	—	2	1	no	—
113	C₅×Dic₃	#41	1	2	yes	—
114	C₂×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
115	C₂²×C₃×C₄	C₂²×C₅	1	2	yes	—
116	C₂²×Dic₃	C₂²×C₅	1	2	yes	—
117	C₂²×Dic₃	C₂²×C₅	1	2	yes	—
118	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
119	C₂×Dic₆	C₂²×C₅	1	2	yes	—
120	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
121	C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
122	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
123	#217	C₂²×C₅	1	2	yes	—
124	#217	C₂²×C₅	1	2	yes	—
125	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
126	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
127	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
128	C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
129	Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
130	Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
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	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
136	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
137	C₂³×C₅	C₂²×D₃	1	2	yes	—
138	C₅×Q₈	—	6	2	no	—
139	C₅×Q₈	—	12	2	no	—
140	C₅×D₄	Dic₆	1	4	yes	complemented
141	C₂×C₄×C₅	—	3	1	no	—
142	C₂×C₄×C₅	—	3	1	no	—
143	C₂×C₄×C₅	—	3	1	no	—
144	C₂×C₄×C₅	—	3	1	no	—
145	C₂×C₄×C₅	—	3	2	no	—
146	C₂×C₄×C₅	—	3	2	no	—
147	C₂×C₄×C₅	—	4	2	no	—
148	C₂×C₄×C₅	—	6	4	no	—
149	C₂×C₄×C₅	—	12	2	no	—
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	2	yes	—
153	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
154	C₄⋊Q₈	—	3	1	no	—
155	#125	—	3	1	no	—
156	#125	—	3	1	no	—
157	C₂²⋊Q₈	—	3	2	no	—
158	C₂²⋊Q₈	—	3	2	no	—
159	C₂²⋊Q₈	—	3	2	no	—
160	C₄×Q₈	—	3	1	no	—
161	C₄×D₄	—	3	1	no	—
162	C₄×D₄	—	3	1	no	—
163	C₄×D₄	C₅×D₃	1	1	yes	characteristic
164	C₂×C₄⋊C₄	—	3	2	no	—
165	C₂×C₃×C₅	—	2	4	no	—
166	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
167	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
168	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
169	C₂³×C₃	C₂³×C₅	1	2	yes	—
170	C₃×D₄	C₅×Q₈	1	4	yes	complemented
171	C₂×C₃×C₄	—	4	2	no	—
172	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
175	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
176	C₂×Dic₃	—	2	4	no	—
177	C₂×Dic₃	—	4	2	no	—
178	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
179	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
180	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
181	C₂×Dic₃	C₂³×C₅	1	2	yes	—
182	C₂×Dic₃	C₂³×C₅	1	2	yes	—
183	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
184	Dic₆	—	2	2	no	—
185	Dic₆	—	4	2	no	—
186	C₂²×C₅	—	2	2	no	—
187	C₂²×C₅	—	2	2	no	—
188	C₂²×C₅	C₂×Dic₆	1	4	yes	—
189	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
190	C₄×C₅	—	2	1	no	—
191	C₄×C₅	—	2	2	no	—
192	C₄×C₅	—	2	1	no	—
193	C₄×C₅	—	3	2	no	—
194	C₄×C₅	—	6	1	no	—
195	C₄×C₅	—	6	2	no	—
196	C₄×C₅	—	6	1	no	—
197	C₄×C₅	—	6	2	no	—
198	C₄×C₅	—	6	1	no	—
199	C₄×C₅	C₂×Dic₆	1	2	yes	—
200	C₂×Q₈	—	3	1	no	—
201	C₂×Q₈	—	3	2	no	—
202	C₂×D₄	C₂×C₅×D₃	1	1	yes	characteristic
203	C₂²×C₄	—	3	2	no	—
204	C₂²×C₄	—	3	2	no	—
205	C₂²×C₄	C₂×C₅×D₃	1	2	yes	—
206	C₄⋊C₄	—	3	1	no	—
207	C₄⋊C₄	—	3	2	no	—
208	C₄⋊C₄	—	3	2	no	—
209	C₄⋊C₄	—	3	2	no	—
210	C₄⋊C₄	—	3	1	no	—
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	C₂×C₅×D₃	1	1	yes	characteristic
217	C₂²⋊C₄	—	3	2	no	—
218	C₂²⋊C₄	—	3	2	no	—
219	C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
220	C₄²	—	3	1	no	—
221	C₄²	—	3	1	no	—
222	C₄²	C₂×C₅×D₃	1	1	yes	characteristic
223	C₃×C₅	#554	1	1	yes	characteristic, complemented, Hall
224	C₂²×C₃	—	2	2	no	—
225	C₂²×C₃	—	2	2	no	—
226	C₂²×C₃	C₂×C₅×Q₈	1	4	yes	—
227	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	2	2	no	—
230	C₃×C₄	—	2	1	no	—
231	C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
232	Dic₃	—	2	1	no	—
233	Dic₃	—	2	2	no	—
234	Dic₃	—	2	1	no	—
235	Dic₃	—	2	2	no	—
236	Dic₃	—	2	1	no	—
237	Dic₃	C₅×#41	1	2	yes	—
238	C₂×C₅	—	2	4	no	—
239	C₂×C₅	C₂²×Dic₆	1	1	yes	characteristic
240	C₂×C₅	#1004	1	1	yes	characteristic
241	C₂×C₅	D₃×#41	1	1	yes	characteristic
242	C₂³	C₂²×C₅×D₃	1	2	yes	—
243	Q₈	—	6	2	no	—
244	Q₈	—	12	2	no	—
245	D₄	C₅×Dic₆	1	4	yes	complemented
246	C₂×C₄	—	3	1	no	—
247	C₂×C₄	—	3	1	no	—
248	C₂×C₄	—	3	1	no	—
249	C₂×C₄	—	3	1	no	—
250	C₂×C₄	—	3	2	no	—
251	C₂×C₄	—	3	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	6	4	no	—
254	C₂×C₄	—	12	2	no	—
255	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
256	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
257	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
258	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
259	C₂×C₃	—	2	4	no	—
260	C₂×C₃	C₂²×C₅×Q₈	1	1	yes	characteristic
261	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
262	C₂×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
263	C₅	#5539	1	1	yes	characteristic, direct product factor, Sylow
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	C₂×C₅×Dic₆	1	4	yes	—
267	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	2	no	—
270	C₄	—	2	1	no	—
271	C₄	—	3	2	no	—
272	C₄	—	6	1	no	—
273	C₄	—	6	2	no	—
274	C₄	—	6	1	no	—
275	C₄	—	6	2	no	—
276	C₄	—	6	1	no	—
277	C₄	C₂×C₅×Dic₆	1	2	yes	—
278	C₃	C₅×#554	1	1	yes	characteristic, complemented, Sylow
279	C₂	—	2	4	no	—
280	C₂	C₂²×C₅×Dic₆	1	1	yes	characteristic
281	C₂	C₅×#1004	1	1	yes	characteristic
282	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	C₂×C₃×C₅×D₄	C₂²
hypercenter	C₄×C₅×D₄	D₃
hypocenter	C₃	C₅×#554
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
4. C₅×#924	C₂	semidirect
5. C₂×C₅×Dic₃⋊C₄	C₂	semidirect
6. C₅×#890	C₂	semidirect
7. C₅×#888	C₂	semidirect
8. C₅×#878	C₂	semidirect
9. C₅×#878	C₂	semidirect
10. C₅×#870	C₂	semidirect
11. C₄×C₅×Dic₆	C₂	semidirect
36. #5539	C₅	direct (N × H)
49. C₃×C₅×D₄	Q₈	semidirect
68. #924	C₂×C₅	semidirect
69. C₂×Dic₃⋊C₄	C₂×C₅	semidirect
70. #890	C₂×C₅	semidirect
71. #888	C₂×C₅	semidirect
72. #878	C₂×C₅	semidirect
73. #878	C₂×C₅	semidirect
74. #870	C₂×C₅	semidirect
75. C₄×Dic₆	C₂×C₅	semidirect
140. C₅×D₄	Dic₆	semidirect
170. C₃×D₄	C₅×Q₈	semidirect
223. C₃×C₅	#554	semidirect
245. D₄	C₅×Dic₆	semidirect
263. C₅	#5539	direct (N × H)
278. C₃	C₅×#554	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×D₄×Dic₃	2
C₅×#924	2
C₂×C₅×Dic₃⋊C₄	2
C₅×#890	2
C₅×#888	2
C₅×#878	2
C₅×#870	2
C₄×C₅×Dic₆	2
C₅×#554	3
#5539	5

Subgroups — C5×#5539 #11755924

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5539 #11755924