Subgroups — C₅×#5671 #11756056

Order 960.

424 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#124	C₂	1	1	yes	maximal, characteristic
2	C₅×Q₈×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#944	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#936	C₂	1	1	yes	maximal, characteristic
5	C₅×D₄×Dic₃	C₂	1	1	yes	maximal, characteristic
6	C₅×#879	C₂	1	2	yes	maximal, complemented
7	C₅×#879	C₂	1	2	yes	maximal, complemented
8	C₅×#878	C₂	1	2	yes	maximal, complemented
9	C₅×#877	C₂	1	2	yes	maximal, complemented
10	C₄×C₅×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₅×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#556	—	3	1	no	maximal, Hall
13	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
15	C₂²×C₅×Dic₃	C₂²	1	2	yes	—
16	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
17	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
18	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
19	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
21	C₅×#217	C₂²	1	2	yes	—
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	2	yes	—
27	C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
28	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
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	C₄×C₅×Dic₃	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	#5671	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×#126	—	3	2	no	—
40	C₅×#124	—	3	1	no	—
41	C₅×#124	D₃	1	1	yes	characteristic
42	C₅×C₂²⋊Q₈	—	3	2	no	—
43	C₄×C₅×Q₈	—	3	1	no	—
44	C₄×C₅×Q₈	—	3	1	no	—
45	C₄×C₅×Q₈	—	3	1	no	—
46	C₄×C₅×D₄	—	3	1	no	—
47	C₅×#117	—	3	2	no	—
48	C₂³×C₃×C₅	C₂³	1	2	yes	—
49	C₃×C₅×Q₈	—	2	2	no	—
50	C₃×C₅×D₄	—	2	2	no	—
51	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
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×C₅×Dic₃	—	2	4	no	—
56	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
57	C₂×C₅×Dic₃	C₂³	1	2	yes	—
58	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
59	C₂×C₅×Dic₃	C₂³	1	2	yes	—
60	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
61	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
62	C₅×Dic₆	—	2	2	no	—
63	C₅×Dic₆	—	2	2	no	—
64	C₃×#124	C₂×C₅	1	1	yes	characteristic
65	Q₈×Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
66	#944	C₂×C₅	1	1	yes	characteristic, complemented
67	#936	C₂×C₅	1	1	yes	characteristic
68	D₄×Dic₃	C₂×C₅	1	1	yes	characteristic
69	#879	C₂×C₅	1	2	yes	complemented
70	#879	C₂×C₅	1	2	yes	complemented
71	#878	C₂×C₅	1	2	yes	complemented
72	#877	C₂×C₅	1	2	yes	complemented
73	C₄×Dic₆	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×Q₈	—	3	1	no	—
77	C₂×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
78	C₂×C₅×D₄	C₂×D₃	1	1	yes	characteristic
79	C₂²×C₄×C₅	—	3	2	no	—
80	C₅×C₄⋊C₄	—	3	1	no	—
81	C₅×C₄⋊C₄	—	3	2	no	—
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	1	no	—
86	C₅×C₄⋊C₄	—	3	1	no	—
87	C₅×C₄⋊C₄	—	3	2	no	—
88	C₅×C₂²⋊C₄	—	3	2	no	—
89	C₅×C₂²⋊C₄	—	3	2	no	—
90	C₅×C₂²⋊C₄	—	3	2	no	—
91	C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
92	C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
93	C₄²×C₅	—	3	2	no	—
94	C₄²×C₅	—	3	2	no	—
95	C₄²×C₅	—	3	1	no	—
96	C₄²×C₅	—	3	1	no	—
97	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
98	#556	—	3	1	no	Sylow
99	C₂²×C₃×C₅	—	2	2	no	—
100	C₂²×C₃×C₅	—	2	2	no	—
101	C₂²×C₃×C₅	—	2	2	no	—
102	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
103	C₃×C₄×C₅	—	2	1	no	—
104	C₃×C₄×C₅	—	2	2	no	—
105	C₃×C₄×C₅	—	2	1	no	—
106	C₃×C₄×C₅	#41	1	2	yes	—
107	C₅×Dic₃	—	2	2	no	—
108	C₅×Dic₃	—	2	1	no	—
109	C₅×Dic₃	—	2	2	no	—
110	C₅×Dic₃	—	2	1	no	—
111	C₅×Dic₃	—	2	1	no	—
112	C₅×Dic₃	#41	1	2	yes	—
113	C₂×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
114	C₂×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
115	C₂²×Dic₃	C₂²×C₅	1	2	yes	—
116	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
117	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
118	C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
119	C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
120	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
121	#217	C₂²×C₅	1	2	yes	—
122	#217	C₂²×C₅	1	2	yes	—
123	#217	C₂²×C₅	1	2	yes	—
124	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
125	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
126	C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
127	Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
128	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
129	Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
130	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
131	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
132	C₄×Dic₃	C₂²×C₅	1	2	yes	—
133	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
134	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
135	C₄×Dic₃	C₂²×C₅	1	2	yes	—
136	C₂³×C₅	C₂²×D₃	1	2	yes	—
137	C₅×Q₈	—	2	2	no	—
138	C₅×Q₈	—	6	2	no	—
139	C₅×Q₈	—	6	2	no	—
140	C₅×D₄	—	2	2	no	—
141	C₂×C₄×C₅	—	3	1	no	—
142	C₂×C₄×C₅	—	3	1	no	—
143	C₂×C₄×C₅	—	3	2	no	—
144	C₂×C₄×C₅	—	3	1	no	—
145	C₂×C₄×C₅	—	3	2	no	—
146	C₂×C₄×C₅	—	3	1	no	—
147	C₂×C₄×C₅	—	6	4	no	—
148	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
150	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
152	C₄⋊Q₈	—	3	1	no	—
153	#126	—	3	2	no	—
154	#126	—	3	2	no	—
155	#124	—	3	1	no	—
156	#124	C₅×D₃	1	1	yes	characteristic
157	C₂²⋊Q₈	—	3	2	no	—
158	C₄×Q₈	—	3	1	no	—
159	C₄×Q₈	—	3	1	no	—
160	C₄×Q₈	—	3	1	no	—
161	C₄×D₄	—	3	1	no	—
162	#117	—	3	2	no	—
163	C₂×C₃×C₅	—	4	2	no	—
164	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
165	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
166	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
167	C₂³×C₃	C₂³×C₅	1	2	yes	—
168	C₃×Q₈	—	2	2	no	—
169	C₃×D₄	—	2	2	no	—
170	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
172	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
174	C₂×Dic₃	—	2	4	no	—
175	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
176	C₂×Dic₃	C₂³×C₅	1	2	yes	—
177	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
178	C₂×Dic₃	C₂³×C₅	1	2	yes	—
179	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
180	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
181	Dic₆	—	2	2	no	—
182	Dic₆	—	2	2	no	—
183	C₂²×C₅	—	2	2	no	—
184	C₂²×C₅	—	2	2	no	—
185	C₂²×C₅	—	2	2	no	—
186	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
187	C₄×C₅	—	2	2	no	—
188	C₄×C₅	—	2	1	no	—
189	C₄×C₅	—	2	1	no	—
190	C₄×C₅	—	3	2	no	—
191	C₄×C₅	—	6	2	no	—
192	C₄×C₅	—	6	2	no	—
193	C₄×C₅	—	6	1	no	—
194	C₄×C₅	—	6	1	no	—
195	C₄×C₅	—	6	1	no	—
196	C₄×C₅	#237	1	2	yes	—
197	C₂×Q₈	—	3	1	no	—
198	C₂×Q₈	—	3	1	no	—
199	C₂×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
200	C₂×D₄	C₂×C₅×D₃	1	1	yes	characteristic
201	C₂²×C₄	—	3	2	no	—
202	C₄⋊C₄	—	3	1	no	—
203	C₄⋊C₄	—	3	2	no	—
204	C₄⋊C₄	—	3	1	no	—
205	C₄⋊C₄	—	3	2	no	—
206	C₄⋊C₄	—	3	2	no	—
207	C₄⋊C₄	—	3	1	no	—
208	C₄⋊C₄	—	3	1	no	—
209	C₄⋊C₄	—	3	2	no	—
210	C₂²⋊C₄	—	3	2	no	—
211	C₂²⋊C₄	—	3	2	no	—
212	C₂²⋊C₄	—	3	2	no	—
213	C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
214	C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
215	C₄²	—	3	2	no	—
216	C₄²	—	3	2	no	—
217	C₄²	—	3	1	no	—
218	C₄²	—	3	1	no	—
219	C₄²	C₂×C₅×D₃	1	1	yes	characteristic
220	C₃×C₅	#556	1	1	yes	characteristic, complemented, Hall
221	C₂²×C₃	—	2	2	no	—
222	C₂²×C₃	—	2	2	no	—
223	C₂²×C₃	—	2	2	no	—
224	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
225	C₃×C₄	—	2	1	no	—
226	C₃×C₄	—	2	2	no	—
227	C₃×C₄	—	2	1	no	—
228	C₃×C₄	C₅×#41	1	2	yes	—
229	Dic₃	—	2	2	no	—
230	Dic₃	—	2	1	no	—
231	Dic₃	—	2	2	no	—
232	Dic₃	—	2	1	no	—
233	Dic₃	—	2	1	no	—
234	Dic₃	C₅×#41	1	2	yes	—
235	C₂×C₅	—	4	2	no	—
236	C₂×C₅	C₂×#237	1	1	yes	characteristic
237	C₂×C₅	D₃×#41	1	1	yes	characteristic
238	C₂×C₅	#1010	1	1	yes	characteristic
239	C₂³	C₂²×C₅×D₃	1	2	yes	—
240	Q₈	—	2	2	no	—
241	Q₈	—	6	2	no	—
242	Q₈	—	6	2	no	—
243	D₄	—	2	2	no	—
244	C₂×C₄	—	3	1	no	—
245	C₂×C₄	—	3	1	no	—
246	C₂×C₄	—	3	2	no	—
247	C₂×C₄	—	3	1	no	—
248	C₂×C₄	—	3	2	no	—
249	C₂×C₄	—	3	1	no	—
250	C₂×C₄	—	6	4	no	—
251	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
252	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
253	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
254	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
255	C₂×C₃	—	4	2	no	—
256	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
257	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
258	C₂×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
259	C₅	#5671	1	1	yes	characteristic, direct product factor, Sylow
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	3	2	no	—
268	C₄	—	6	2	no	—
269	C₄	—	6	2	no	—
270	C₄	—	6	1	no	—
271	C₄	—	6	1	no	—
272	C₄	—	6	1	no	—
273	C₄	C₅×#237	1	2	yes	—
274	C₃	C₅×#556	1	1	yes	characteristic, complemented, Sylow
275	C₂	—	4	2	no	—
276	C₂	C₂×C₅×#237	1	1	yes	characteristic
277	C₂	C₅×D₃×#41	1	1	yes	characteristic
278	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×C₅×#124	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂×C₃×C₅×D₄	C₂²
hypercenter	C₅×#124	D₃
hypocenter	C₃	C₅×#556
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#124

Complements

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

N	H ≅ G/N	type
2. C₅×Q₈×Dic₃	C₂	semidirect
3. C₅×#944	C₂	semidirect
6. C₅×#879	C₂	semidirect
7. C₅×#879	C₂	semidirect
8. C₅×#878	C₂	semidirect
9. C₅×#877	C₂	semidirect
10. C₄×C₅×Dic₆	C₂	semidirect
11. C₄×C₅×Dic₆	C₂	semidirect
36. #5671	C₅	direct (N × H)
65. Q₈×Dic₃	C₂×C₅	semidirect
66. #944	C₂×C₅	semidirect
69. #879	C₂×C₅	semidirect
70. #879	C₂×C₅	semidirect
71. #878	C₂×C₅	semidirect
72. #877	C₂×C₅	semidirect
73. C₄×Dic₆	C₂×C₅	semidirect
74. C₄×Dic₆	C₂×C₅	semidirect
220. C₃×C₅	#556	semidirect
259. C₅	#5671	direct (N × H)
274. C₃	C₅×#556	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#124	2
C₅×Q₈×Dic₃	2
C₅×#944	2
C₅×#936	2
C₅×D₄×Dic₃	2
C₅×#879	2
C₅×#878	2
C₅×#877	2
C₄×C₅×Dic₆	2
C₅×#556	3
#5671	5

Subgroups — C5×#5671 #11756056

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5671 #11756056