Subgroups — C₅×#5533 #11755918

Order 960.

424 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#117	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₅×#67	C₂	1	1	yes	maximal, characteristic
3	C₅×#924	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#897	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#892	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#888	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#887	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#885	C₂	1	2	yes	maximal
9	C₅×#879	C₂	1	2	yes	maximal, complemented
10	C₅×#876	C₂	1	2	yes	maximal, complemented
11	C₄×C₅×Dic₆	C₂	1	2	yes	maximal, complemented
12	C₅×#556	—	3	1	no	maximal, Hall
13	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
14	C₂×C₅×#67	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
16	C₂×C₅×Dic₆	C₂²	1	2	yes	—
17	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
18	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
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	2	yes	—
22	C₅×#217	C₂²	1	1	yes	characteristic
23	C₅×#217	C₂²	1	2	yes	—
24	C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
25	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
26	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
27	C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
28	C₅×C₄⋊Dic₃	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	C₄×C₅×Dic₃	C₂²	1	2	yes	—
34	C₄×C₅×Dic₃	C₂²	1	2	yes	—
35	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
36	#5533	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	2	no	—
41	C₅×C₂²⋊Q₈	—	3	1	no	—
42	C₅×C₂²⋊Q₈	—	3	1	no	—
43	C₄×C₅×Q₈	—	3	1	no	—
44	C₄×C₅×Q₈	—	3	2	no	—
45	C₄×C₅×D₄	—	3	1	no	—
46	C₅×#117	—	3	1	no	—
47	C₅×#117	D₃	1	1	yes	characteristic, complemented
48	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
49	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	—	2	2	no	—
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	2	yes	—
54	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
55	C₅×#67	—	2	2	no	—
56	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
57	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
58	C₂×C₅×Dic₃	C₂³	1	2	yes	—
59	C₂×C₅×Dic₃	C₂³	1	2	yes	—
60	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
61	C₄×C₅×D₃	—	2	2	no	—
62	C₅×Dic₆	—	2	2	no	—
63	C₅×Dic₆	—	2	4	no	—
64	C₃×#117	C₂×C₅	1	1	yes	characteristic, complemented
65	C₄×#67	C₂×C₅	1	1	yes	characteristic
66	#924	C₂×C₅	1	1	yes	characteristic, complemented
67	#897	C₂×C₅	1	1	yes	characteristic, complemented
68	#892	C₂×C₅	1	1	yes	characteristic, complemented
69	#888	C₂×C₅	1	1	yes	characteristic, complemented
70	#887	C₂×C₅	1	1	yes	characteristic, complemented
71	#885	C₂×C₅	1	2	yes	—
72	#879	C₂×C₅	1	2	yes	complemented
73	#876	C₂×C₅	1	2	yes	complemented
74	C₄×Dic₆	C₂×C₅	1	2	yes	complemented
75	C₂×C₅×Q₈	—	3	1	no	—
76	C₂×C₅×Q₈	—	3	2	no	—
77	C₂×C₅×D₄	—	3	1	no	—
78	C₂²×C₄×C₅	—	3	1	no	—
79	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
80	C₅×C₄⋊C₄	—	3	1	no	—
81	C₅×C₄⋊C₄	—	3	1	no	—
82	C₅×C₄⋊C₄	—	3	2	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	C₂×D₃	1	1	yes	characteristic
87	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
88	C₅×C₂²⋊C₄	—	3	1	no	—
89	C₅×C₂²⋊C₄	—	3	2	no	—
90	C₅×C₂²⋊C₄	—	3	2	no	—
91	C₅×C₂²⋊C₄	—	3	1	no	—
92	C₅×C₂²⋊C₄	—	3	2	no	—
93	C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
94	C₄²×C₅	—	3	2	no	—
95	C₄²×C₅	—	3	1	no	—
96	C₄²×C₅	—	3	2	no	—
97	C₄²×C₅	C₂×D₃	1	2	yes	—
98	#556	—	3	1	no	Sylow
99	C₂²×C₃×C₅	—	2	1	no	—
100	C₂²×C₃×C₅	—	2	1	no	—
101	C₂²×C₃×C₅	—	2	1	no	—
102	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
103	C₂×C₅×D₃	—	2	1	no	—
104	C₂×C₅×D₃	—	2	1	no	—
105	C₂×C₅×D₃	—	2	1	no	—
106	C₃×C₄×C₅	—	2	2	no	—
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	2	no	—
112	C₅×Dic₃	—	2	1	no	—
113	C₅×Dic₃	—	2	1	no	—
114	C₅×Dic₃	#41	1	2	yes	—
115	C₂²×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
116	C₂×#67	C₂²×C₅	1	1	yes	characteristic
117	C₂×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
118	C₂×Dic₆	C₂²×C₅	1	2	yes	—
119	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
120	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
121	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
122	C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
123	C₃×C₄²	C₂²×C₅	1	2	yes	—
124	#217	C₂²×C₅	1	1	yes	characteristic
125	#217	C₂²×C₅	1	2	yes	—
126	(C₂×D₃)⋊C₄	C₂²×C₅	1	1	yes	characteristic
127	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
128	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
129	C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
130	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
131	Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
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	2	yes	—
136	C₄×Dic₃	C₂²×C₅	1	2	yes	—
137	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
138	C₂³×C₅	—	3	1	no	—
139	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
140	C₅×Q₈	—	6	4	no	—
141	C₅×Q₈	—	6	2	no	—
142	C₅×D₄	—	6	2	no	—
143	C₂×C₄×C₅	—	2	2	no	—
144	C₂×C₄×C₅	—	3	1	no	—
145	C₂×C₄×C₅	—	3	1	no	—
146	C₂×C₄×C₅	—	3	1	no	—
147	C₂×C₄×C₅	—	3	2	no	—
148	C₂×C₄×C₅	—	3	2	no	—
149	C₂×C₄×C₅	—	6	2	no	—
150	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
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	#126	—	3	2	no	—
156	#126	—	3	2	no	—
157	#124	—	3	2	no	—
158	C₂²⋊Q₈	—	3	1	no	—
159	C₂²⋊Q₈	—	3	1	no	—
160	C₄×Q₈	—	3	1	no	—
161	C₄×Q₈	—	3	2	no	—
162	C₄×D₄	—	3	1	no	—
163	#117	—	3	1	no	—
164	#117	C₅×D₃	1	1	yes	characteristic, complemented
165	C₂×C₃×C₅	—	4	1	no	—
166	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
167	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
168	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
169	C₅×D₃	—	4	1	no	—
170	C₂³×C₃	C₂³×C₅	1	1	yes	characteristic
171	C₂²×D₃	C₂³×C₅	1	1	yes	characteristic
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
174	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
176	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
177	#67	—	2	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	2	yes	—
181	C₂×Dic₃	C₂³×C₅	1	2	yes	—
182	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
183	C₄×D₃	—	2	2	no	—
184	Dic₆	—	2	2	no	—
185	Dic₆	—	2	4	no	—
186	C₂²×C₅	—	2	1	no	—
187	C₂²×C₅	—	2	1	no	—
188	C₂²×C₅	—	2	1	no	—
189	C₂²×C₅	—	6	1	no	—
190	C₂²×C₅	—	6	1	no	—
191	C₂²×C₅	—	6	1	no	—
192	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
193	C₄×C₅	—	2	1	no	—
194	C₄×C₅	—	2	2	no	—
195	C₄×C₅	—	2	2	no	—
196	C₄×C₅	—	3	2	no	—
197	C₄×C₅	—	6	1	no	—
198	C₄×C₅	—	6	2	no	—
199	C₄×C₅	—	6	2	no	—
200	C₄×C₅	—	6	1	no	—
201	C₄×C₅	#235	1	2	yes	—
202	C₂×Q₈	—	3	1	no	—
203	C₂×Q₈	—	3	2	no	—
204	C₂×D₄	—	3	1	no	—
205	C₂²×C₄	—	3	1	no	—
206	C₂²×C₄	C₂×C₅×D₃	1	1	yes	characteristic
207	C₄⋊C₄	—	3	1	no	—
208	C₄⋊C₄	—	3	1	no	—
209	C₄⋊C₄	—	3	2	no	—
210	C₄⋊C₄	—	3	2	no	—
211	C₄⋊C₄	—	3	2	no	—
212	C₄⋊C₄	—	3	2	no	—
213	C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
214	C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
215	C₂²⋊C₄	—	3	1	no	—
216	C₂²⋊C₄	—	3	2	no	—
217	C₂²⋊C₄	—	3	2	no	—
218	C₂²⋊C₄	—	3	1	no	—
219	C₂²⋊C₄	—	3	2	no	—
220	C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
221	C₄²	—	3	2	no	—
222	C₄²	—	3	1	no	—
223	C₄²	—	3	2	no	—
224	C₄²	C₂×C₅×D₃	1	2	yes	—
225	C₃×C₅	#556	1	1	yes	characteristic, complemented, Hall
226	C₂²×C₃	—	2	1	no	—
227	C₂²×C₃	—	2	1	no	—
228	C₂²×C₃	—	2	1	no	—
229	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
230	C₂×D₃	—	2	1	no	—
231	C₂×D₃	—	2	1	no	—
232	C₂×D₃	—	2	1	no	—
233	C₃×C₄	—	2	2	no	—
234	C₃×C₄	—	2	1	no	—
235	C₃×C₄	—	2	2	no	—
236	C₃×C₄	C₅×#41	1	2	yes	—
237	Dic₃	—	2	2	no	—
238	Dic₃	—	2	2	no	—
239	Dic₃	—	2	1	no	—
240	Dic₃	—	2	1	no	—
241	Dic₃	C₅×#41	1	2	yes	—
242	C₂×C₅	—	4	1	no	—
243	C₂×C₅	—	12	1	no	—
244	C₂×C₅	C₂×#235	1	1	yes	characteristic
245	C₂×C₅	D₃×#41	1	1	yes	characteristic
246	C₂×C₅	#1010	1	1	yes	characteristic
247	C₂³	—	3	1	no	—
248	C₂³	C₂²×C₅×D₃	1	1	yes	characteristic
249	Q₈	—	6	4	no	—
250	Q₈	—	6	2	no	—
251	D₄	—	6	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	3	1	no	—
254	C₂×C₄	—	3	1	no	—
255	C₂×C₄	—	3	1	no	—
256	C₂×C₄	—	3	2	no	—
257	C₂×C₄	—	3	2	no	—
258	C₂×C₄	—	6	2	no	—
259	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
260	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
261	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
262	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
263	C₂×C₃	—	4	1	no	—
264	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
265	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
266	C₂×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
267	D₃	—	4	1	no	—
268	C₅	#5533	1	1	yes	characteristic, direct product factor, Sylow
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	6	1	no	—
273	C₂²	—	6	1	no	—
274	C₂²	—	6	1	no	—
275	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	3	2	no	—
280	C₄	—	6	1	no	—
281	C₄	—	6	2	no	—
282	C₄	—	6	2	no	—
283	C₄	—	6	1	no	—
284	C₄	C₅×#235	1	2	yes	—
285	C₃	C₅×#556	1	1	yes	characteristic, complemented, Sylow
286	C₂	—	4	1	no	—
287	C₂	—	12	1	no	—
288	C₂	C₂×C₅×#235	1	1	yes	characteristic
289	C₂	C₅×D₃×#41	1	1	yes	characteristic
290	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×#117	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂×C₅×#67	C₂²
hypercenter	C₅×#117	D₃
hypocenter	C₃	C₅×#556
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#117

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#117	C₂	semidirect
3. C₅×#924	C₂	semidirect
4. C₅×#897	C₂	semidirect
5. C₅×#892	C₂	semidirect
6. C₅×#888	C₂	semidirect
7. C₅×#887	C₂	semidirect
9. C₅×#879	C₂	semidirect
10. C₅×#876	C₂	semidirect
11. C₄×C₅×Dic₆	C₂	semidirect
36. #5533	C₅	direct (N × H)
47. C₅×#117	D₃	semidirect
64. C₃×#117	C₂×C₅	semidirect
66. #924	C₂×C₅	semidirect
67. #897	C₂×C₅	semidirect
68. #892	C₂×C₅	semidirect
69. #888	C₂×C₅	semidirect
70. #887	C₂×C₅	semidirect
72. #879	C₂×C₅	semidirect
73. #876	C₂×C₅	semidirect
74. C₄×Dic₆	C₂×C₅	semidirect
164. #117	C₅×D₃	semidirect
225. C₃×C₅	#556	semidirect
268. C₅	#5533	direct (N × H)
285. 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×#117	2
C₄×C₅×#67	2
C₅×#924	2
C₅×#897	2
C₅×#892	2
C₅×#888	2
C₅×#887	2
C₅×#885	2
C₅×#879	2
C₅×#876	2
C₄×C₅×Dic₆	2
C₅×#556	3
#5533	5

Subgroups — C5×#5533 #11755918

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5533 #11755918