Subgroups — C₅×#5690 #11756075

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×C₅×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#897	C₂	1	2	yes	maximal, complemented
3	C₅×#896	C₂	1	2	yes	maximal, complemented
4	C₅×#896	C₂	1	2	yes	maximal, complemented
5	C₅×#893	C₂	1	1	yes	maximal, characteristic
6	C₅×#893	C₂	1	1	yes	maximal, characteristic
7	C₅×D₃×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
8	C₅×#890	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#888	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₅×D₁₂	C₂	1	1	yes	maximal, characteristic
11	C₄×C₅×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#554	—	3	1	no	maximal, Hall
13	C₂×C₅×D₁₂	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
15	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
16	C₂×C₄×C₅×D₃	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×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
22	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
23	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
24	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
25	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
26	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
27	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
28	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
29	C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
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	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	#5690	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	D₃	1	1	yes	characteristic, complemented
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	—	3	1	no	—
47	C₂×C₅×C₄⋊C₄	—	3	2	no	—
48	C₂²×C₅×D₃	C₂³	1	2	yes	—
49	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
51	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
52	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
54	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
55	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
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	C₅×D₁₂	Q₈	1	4	yes	complemented
60	C₄×C₅×D₃	—	2	4	no	—
61	C₄×C₅×D₃	—	4	2	no	—
62	C₄×C₅×D₃	—	4	2	no	—
63	C₅×Dic₆	—	2	2	no	—
64	C₅×Dic₆	—	4	2	no	—
65	C₃×#125	C₂×C₅	1	1	yes	characteristic, complemented
66	#897	C₂×C₅	1	2	yes	complemented
67	#896	C₂×C₅	1	2	yes	complemented
68	#896	C₂×C₅	1	2	yes	complemented
69	#893	C₂×C₅	1	1	yes	characteristic
70	#893	C₂×C₅	1	1	yes	characteristic
71	D₃×C₄⋊C₄	C₂×C₅	1	2	yes	complemented
72	#890	C₂×C₅	1	1	yes	characteristic, complemented
73	#888	C₂×C₅	1	1	yes	characteristic, complemented
74	C₄×D₁₂	C₂×C₅	1	1	yes	characteristic
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	—	3	1	no	—
79	C₂²×C₄×C₅	—	3	2	no	—
80	C₂²×C₄×C₅	—	3	2	no	—
81	C₂²×C₄×C₅	—	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	1	no	—
88	C₅×C₄⋊C₄	—	3	2	no	—
89	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
90	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
91	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
92	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
93	C₅×C₂²⋊C₄	—	3	2	no	—
94	C₅×C₂²⋊C₄	—	3	2	no	—
95	C₅×C₂²⋊C₄	—	3	2	no	—
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	C₂⁴	1	1	yes	characteristic
101	C₂×C₅×D₃	—	2	2	no	—
102	C₂×C₅×D₃	—	2	2	no	—
103	C₂×C₅×D₃	C₂×Q₈	1	4	yes	—
104	C₃×C₄×C₅	—	2	1	no	—
105	C₃×C₄×C₅	—	2	2	no	—
106	C₃×C₄×C₅	—	2	2	no	—
107	C₃×C₄×C₅	—	2	1	no	—
108	C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
109	C₅×Dic₃	—	2	1	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×D₁₂	C₂²×C₅	1	1	yes	characteristic
115	C₂×C₄×D₃	C₂²×C₅	1	2	yes	—
116	C₂×C₄×D₃	C₂²×C₅	1	2	yes	—
117	C₂×C₄×D₃	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×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
122	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
123	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
124	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
125	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
126	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
127	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
128	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
129	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
130	C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
131	Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
132	Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
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	—	3	2	no	—
138	C₅×Q₈	—	6	2	no	—
139	C₅×Q₈	—	12	2	no	—
140	C₅×D₄	—	3	4	no	—
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	—	6	4	no	—
147	C₂×C₄×C₅	—	12	2	no	—
148	C₂×C₄×C₅	—	12	2	no	—
149	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
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	C₅×D₃	1	1	yes	characteristic, complemented
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	—	3	1	no	—
164	C₂×C₄⋊C₄	—	3	2	no	—
165	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
166	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
167	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
168	C₅×D₃	—	2	4	no	—
169	C₂²×D₃	C₂³×C₅	1	2	yes	—
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	2	yes	—
173	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
175	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
176	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
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	D₁₂	C₅×Q₈	1	4	yes	complemented
181	C₄×D₃	—	2	4	no	—
182	C₄×D₃	—	4	2	no	—
183	C₄×D₃	—	4	2	no	—
184	Dic₆	—	2	2	no	—
185	Dic₆	—	4	2	no	—
186	C₂²×C₅	—	3	4	no	—
187	C₂²×C₅	—	6	2	no	—
188	C₂²×C₅	—	6	2	no	—
189	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
190	C₄×C₅	—	2	2	no	—
191	C₄×C₅	—	2	1	no	—
192	C₄×C₅	—	2	1	no	—
193	C₄×C₅	—	2	2	no	—
194	C₄×C₅	—	3	2	no	—
195	C₄×C₅	—	6	1	no	—
196	C₄×C₅	—	6	1	no	—
197	C₄×C₅	—	6	2	no	—
198	C₄×C₅	—	6	1	no	—
199	C₄×C₅	D₃×Q₈	1	2	yes	—
200	C₂×Q₈	—	3	1	no	—
201	C₂×Q₈	—	3	2	no	—
202	C₂×D₄	—	3	1	no	—
203	C₂²×C₄	—	3	2	no	—
204	C₂²×C₄	—	3	2	no	—
205	C₂²×C₄	—	3	2	no	—
206	C₄⋊C₄	—	3	1	no	—
207	C₄⋊C₄	—	3	2	no	—
208	C₄⋊C₄	—	3	2	no	—
209	C₄⋊C₄	—	3	1	no	—
210	C₄⋊C₄	—	3	1	no	—
211	C₄⋊C₄	—	3	1	no	—
212	C₄⋊C₄	—	3	2	no	—
213	C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
214	C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
215	C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
216	C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
217	C₂²⋊C₄	—	3	2	no	—
218	C₂²⋊C₄	—	3	2	no	—
219	C₂²⋊C₄	—	3	2	no	—
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	C₂⁴×C₅	1	1	yes	characteristic
225	C₂×D₃	—	2	2	no	—
226	C₂×D₃	—	2	2	no	—
227	C₂×D₃	C₂×C₅×Q₈	1	4	yes	—
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	2	2	no	—
230	C₃×C₄	—	2	2	no	—
231	C₃×C₄	—	2	1	no	—
232	C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
233	Dic₃	—	2	1	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	—	6	4	no	—
239	C₂×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
240	C₂×C₅	D₃×#41	1	1	yes	characteristic
241	C₂×C₅	#1009	1	1	yes	characteristic
242	C₂³	—	3	2	no	—
243	Q₈	—	6	2	no	—
244	Q₈	—	12	2	no	—
245	D₄	—	3	4	no	—
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	—	6	4	no	—
252	C₂×C₄	—	12	2	no	—
253	C₂×C₄	—	12	2	no	—
254	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
255	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
256	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
257	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
258	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
259	C₂×C₃	C₂²×C₅×Q₈	1	1	yes	characteristic
260	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
261	C₂×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
262	D₃	—	2	4	no	—
263	C₅	#5690	1	1	yes	characteristic, direct product factor, Sylow
264	C₂²	—	3	4	no	—
265	C₂²	—	6	2	no	—
266	C₂²	—	6	2	no	—
267	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	2	no	—
272	C₄	—	3	2	no	—
273	C₄	—	6	1	no	—
274	C₄	—	6	1	no	—
275	C₄	—	6	2	no	—
276	C₄	—	6	1	no	—
277	C₄	C₅×D₃×Q₈	1	2	yes	—
278	C₃	C₅×#554	1	1	yes	characteristic, complemented, Sylow
279	C₂	—	6	4	no	—
280	C₂	C₂×C₅×D₃×Q₈	1	1	yes	characteristic
281	C₂	C₅×D₃×#41	1	1	yes	characteristic
282	C₂	C₅×#1009	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×#125	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂×C₅×D₁₂	C₂²
hypercenter	C₅×#125	D₃
hypocenter	C₃	C₅×#554
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#125

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#125	C₂	semidirect
2. C₅×#897	C₂	semidirect
3. C₅×#896	C₂	semidirect
4. C₅×#896	C₂	semidirect
7. C₅×D₃×C₄⋊C₄	C₂	semidirect
8. C₅×#890	C₂	semidirect
9. C₅×#888	C₂	semidirect
11. C₄×C₅×Dic₆	C₂	semidirect
36. #5690	C₅	direct (N × H)
39. C₅×#125	D₃	semidirect
59. C₅×D₁₂	Q₈	semidirect
65. C₃×#125	C₂×C₅	semidirect
66. #897	C₂×C₅	semidirect
67. #896	C₂×C₅	semidirect
68. #896	C₂×C₅	semidirect
71. D₃×C₄⋊C₄	C₂×C₅	semidirect
72. #890	C₂×C₅	semidirect
73. #888	C₂×C₅	semidirect
75. C₄×Dic₆	C₂×C₅	semidirect
156. #125	C₅×D₃	semidirect
180. D₁₂	C₅×Q₈	semidirect
223. C₃×C₅	#554	semidirect
263. C₅	#5690	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×C₅×#125	2
C₅×#897	2
C₅×#896	2
C₅×#893	2
C₅×D₃×C₄⋊C₄	2
C₅×#890	2
C₅×#888	2
C₄×C₅×D₁₂	2
C₄×C₅×Dic₆	2
C₅×#554	3
#5690	5

Subgroups — C5×#5690 #11756075

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5690 #11756075