Subgroups — C₅×#5691 #11756076

Order 960.

468 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×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#898	C₂	1	2	yes	maximal, complemented
3	C₅×#897	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#895	C₂	1	1	yes	maximal, characteristic
5	C₅×#894	C₂	1	2	yes	maximal, complemented
6	C₅×#893	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#893	C₂	1	2	yes	maximal, complemented
8	C₅×#892	C₂	1	2	yes	maximal, complemented
9	C₅×#887	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#875	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄²×C₅×D₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#529	—	3	1	no	maximal, Hall
13	C₂×C₅×D₁₂	C₂²	1	1	yes	characteristic
14	C₂×C₅×D₁₂	C₂²	1	2	yes	—
15	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
19	C₂×C₅×Dic₆	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	1	yes	characteristic
23	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
24	C₃×C₄²×C₅	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×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
28	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
29	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
30	C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
31	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
32	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
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	#5691	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
37	C₅×#126	—	3	2	no	—
38	C₅×#125	D₃	1	1	yes	characteristic, complemented
39	C₅×#124	—	3	1	no	—
40	C₅×#123	—	3	2	no	—
41	C₅×C₂²⋊Q₈	—	3	1	no	—
42	C₅×#121	—	3	1	no	—
43	C₄×C₅×Q₈	—	3	1	no	—
44	C₄×C₅×D₄	—	3	1	no	—
45	C₄×C₅×D₄	—	3	2	no	—
46	C₅×#117	—	3	2	no	—
47	C₂×C₄²×C₅	—	3	1	no	—
48	C₂²×C₅×D₃	C₂³	1	2	yes	—
49	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
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	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	—	2	4	no	—
60	C₅×D₁₂	—	2	2	no	—
61	C₄×C₅×D₃	—	2	2	no	—
62	C₄×C₅×D₃	—	2	2	no	—
63	C₄×C₅×D₃	—	2	2	no	—
64	C₄×C₅×D₃	—	2	4	no	—
65	C₅×Dic₆	—	2	2	no	—
66	C₃×#125	C₂×C₅	1	1	yes	characteristic, complemented
67	#898	C₂×C₅	1	2	yes	complemented
68	#897	C₂×C₅	1	1	yes	characteristic, complemented
69	#895	C₂×C₅	1	1	yes	characteristic
70	#894	C₂×C₅	1	2	yes	complemented
71	#893	C₂×C₅	1	1	yes	characteristic, complemented
72	#893	C₂×C₅	1	2	yes	complemented
73	#892	C₂×C₅	1	2	yes	complemented
74	#887	C₂×C₅	1	1	yes	characteristic, complemented
75	#875	C₂×C₅	1	1	yes	characteristic, complemented
76	C₄²×D₃	C₂×C₅	1	1	yes	characteristic, complemented
77	C₂×C₅×Q₈	—	3	1	no	—
78	C₂×C₅×D₄	—	3	2	no	—
79	C₂×C₅×D₄	—	3	1	no	—
80	C₂²×C₄×C₅	—	3	1	no	—
81	C₂²×C₄×C₅	—	3	1	no	—
82	C₂²×C₄×C₅	—	3	1	no	—
83	C₂²×C₄×C₅	—	3	2	no	—
84	C₅×C₄⋊C₄	—	3	2	no	—
85	C₅×C₄⋊C₄	—	3	2	no	—
86	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
87	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
88	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
89	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
90	C₅×C₂²⋊C₄	—	3	2	no	—
91	C₅×C₂²⋊C₄	—	3	2	no	—
92	C₅×C₂²⋊C₄	—	3	2	no	—
93	C₅×C₂²⋊C₄	—	3	2	no	—
94	C₅×C₂²⋊C₄	—	3	2	no	—
95	C₄²×C₅	—	3	1	no	—
96	C₄²×C₅	—	3	1	no	—
97	C₄²×C₅	—	3	1	no	—
98	C₄²×C₅	—	3	2	no	—
99	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
100	#529	—	3	1	no	Sylow
101	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
102	C₂×C₅×D₃	—	2	1	no	—
103	C₂×C₅×D₃	—	2	1	no	—
104	C₂×C₅×D₃	—	2	2	no	—
105	C₂×C₅×D₃	—	2	2	no	—
106	C₂×C₅×D₃	—	2	2	no	—
107	C₂×C₅×D₃	#41	1	2	yes	—
108	C₃×C₄×C₅	—	2	1	no	—
109	C₃×C₄×C₅	—	2	1	no	—
110	C₃×C₄×C₅	—	2	2	no	—
111	C₃×C₄×C₅	—	2	2	no	—
112	C₃×C₄×C₅	#41	1	2	yes	—
113	C₅×Dic₃	—	2	2	no	—
114	C₅×Dic₃	—	2	1	no	—
115	C₅×Dic₃	—	2	1	no	—
116	C₅×Dic₃	#41	1	2	yes	—
117	C₂×D₁₂	C₂²×C₅	1	1	yes	characteristic
118	C₂×D₁₂	C₂²×C₅	1	2	yes	—
119	C₂×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
120	C₂×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
121	C₂×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
122	C₂×C₄×D₃	C₂²×C₅	1	2	yes	—
123	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
124	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
125	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
126	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
127	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
128	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
129	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
130	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
131	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
132	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
133	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
134	C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
135	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
136	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
137	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
138	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
139	C₄×Dic₃	C₂²×C₅	1	2	yes	—
140	C₂³×C₅	—	3	1	no	—
141	C₂³×C₅	—	3	2	no	—
142	C₅×Q₈	—	6	2	no	—
143	C₅×D₄	—	6	4	no	—
144	C₅×D₄	—	6	2	no	—
145	C₂×C₄×C₅	—	3	1	no	—
146	C₂×C₄×C₅	—	3	1	no	—
147	C₂×C₄×C₅	—	3	1	no	—
148	C₂×C₄×C₅	—	3	2	no	—
149	C₂×C₄×C₅	—	6	2	no	—
150	C₂×C₄×C₅	—	6	4	no	—
151	C₂×C₄×C₅	—	6	2	no	—
152	C₂×C₄×C₅	—	6	2	no	—
153	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
155	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
156	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
157	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
158	#126	—	3	2	no	—
159	#125	C₅×D₃	1	1	yes	characteristic, complemented
160	#124	—	3	1	no	—
161	#123	—	3	2	no	—
162	C₂²⋊Q₈	—	3	1	no	—
163	#121	—	3	1	no	—
164	C₄×Q₈	—	3	1	no	—
165	C₄×D₄	—	3	1	no	—
166	C₄×D₄	—	3	2	no	—
167	#117	—	3	2	no	—
168	C₂×C₄²	—	3	1	no	—
169	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
170	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
171	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
172	C₅×D₃	—	2	2	no	—
173	C₅×D₃	—	4	2	no	—
174	C₂²×D₃	C₂³×C₅	1	2	yes	—
175	C₂²×D₃	C₂³×C₅	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
177	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
180	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
181	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
182	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
183	C₂×Dic₃	C₂³×C₅	1	2	yes	—
184	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
185	D₁₂	—	2	4	no	—
186	D₁₂	—	2	2	no	—
187	C₄×D₃	—	2	2	no	—
188	C₄×D₃	—	2	2	no	—
189	C₄×D₃	—	2	2	no	—
190	C₄×D₃	—	2	4	no	—
191	Dic₆	—	2	2	no	—
192	C₂²×C₅	—	3	2	no	—
193	C₂²×C₅	—	6	1	no	—
194	C₂²×C₅	—	6	1	no	—
195	C₂²×C₅	—	6	2	no	—
196	C₂²×C₅	—	6	2	no	—
197	C₂²×C₅	—	6	2	no	—
198	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
199	C₄×C₅	—	2	1	no	—
200	C₄×C₅	—	2	2	no	—
201	C₄×C₅	—	2	1	no	—
202	C₄×C₅	—	2	2	no	—
203	C₄×C₅	—	3	2	no	—
204	C₄×C₅	—	6	1	no	—
205	C₄×C₅	—	6	2	no	—
206	C₄×C₅	—	6	1	no	—
207	C₄×C₅	#239	1	2	yes	—
208	C₂×Q₈	—	3	1	no	—
209	C₂×D₄	—	3	2	no	—
210	C₂×D₄	—	3	1	no	—
211	C₂²×C₄	—	3	1	no	—
212	C₂²×C₄	—	3	1	no	—
213	C₂²×C₄	—	3	1	no	—
214	C₂²×C₄	—	3	2	no	—
215	C₄⋊C₄	—	3	2	no	—
216	C₄⋊C₄	—	3	2	no	—
217	C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
218	C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
219	C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
220	C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
221	C₂²⋊C₄	—	3	2	no	—
222	C₂²⋊C₄	—	3	2	no	—
223	C₂²⋊C₄	—	3	2	no	—
224	C₂²⋊C₄	—	3	2	no	—
225	C₂²⋊C₄	—	3	2	no	—
226	C₄²	—	3	1	no	—
227	C₄²	—	3	1	no	—
228	C₄²	—	3	1	no	—
229	C₄²	—	3	2	no	—
230	C₄²	C₂×C₅×D₃	1	1	yes	characteristic
231	C₃×C₅	#529	1	1	yes	characteristic, complemented, Hall
232	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
233	C₂×D₃	—	2	1	no	—
234	C₂×D₃	—	2	1	no	—
235	C₂×D₃	—	2	2	no	—
236	C₂×D₃	—	2	2	no	—
237	C₂×D₃	—	2	2	no	—
238	C₂×D₃	C₅×#41	1	2	yes	—
239	C₃×C₄	—	2	1	no	—
240	C₃×C₄	—	2	1	no	—
241	C₃×C₄	—	2	2	no	—
242	C₃×C₄	—	2	2	no	—
243	C₃×C₄	C₅×#41	1	2	yes	—
244	Dic₃	—	2	2	no	—
245	Dic₃	—	2	1	no	—
246	Dic₃	—	2	1	no	—
247	Dic₃	C₅×#41	1	2	yes	—
248	C₂×C₅	—	6	2	no	—
249	C₂×C₅	—	12	2	no	—
250	C₂×C₅	C₂×#239	1	1	yes	characteristic
251	C₂×C₅	D₃×#41	1	1	yes	characteristic
252	C₂×C₅	D₃×#41	1	1	yes	characteristic
253	C₂³	—	3	1	no	—
254	C₂³	—	3	2	no	—
255	Q₈	—	6	2	no	—
256	D₄	—	6	4	no	—
257	D₄	—	6	2	no	—
258	C₂×C₄	—	3	1	no	—
259	C₂×C₄	—	3	1	no	—
260	C₂×C₄	—	3	1	no	—
261	C₂×C₄	—	3	2	no	—
262	C₂×C₄	—	6	2	no	—
263	C₂×C₄	—	6	4	no	—
264	C₂×C₄	—	6	2	no	—
265	C₂×C₄	—	6	2	no	—
266	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
267	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
268	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
269	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
270	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
271	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
272	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
273	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
274	D₃	—	2	2	no	—
275	D₃	—	4	2	no	—
276	C₅	#5691	1	1	yes	characteristic, direct product factor, Sylow
277	C₂²	—	3	2	no	—
278	C₂²	—	6	1	no	—
279	C₂²	—	6	1	no	—
280	C₂²	—	6	2	no	—
281	C₂²	—	6	2	no	—
282	C₂²	—	6	2	no	—
283	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	2	no	—
288	C₄	—	3	2	no	—
289	C₄	—	6	1	no	—
290	C₄	—	6	2	no	—
291	C₄	—	6	1	no	—
292	C₄	C₅×#239	1	2	yes	—
293	C₃	C₅×#529	1	1	yes	characteristic, complemented, Sylow
294	C₂	—	6	2	no	—
295	C₂	—	12	2	no	—
296	C₂	C₂×C₅×#239	1	1	yes	characteristic
297	C₂	C₅×D₃×#41	1	1	yes	characteristic
298	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×C₅×#125	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₅×#895	C₂
hypercenter	C₅×#125	D₃
hypocenter	C₃	C₅×#529
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×#898	C₂	semidirect
3. C₅×#897	C₂	semidirect
5. C₅×#894	C₂	semidirect
6. C₅×#893	C₂	semidirect
7. C₅×#893	C₂	semidirect
8. C₅×#892	C₂	semidirect
9. C₅×#887	C₂	semidirect
10. C₅×#875	C₂	semidirect
11. C₄²×C₅×D₃	C₂	semidirect
36. #5691	C₅	direct (N × H)
38. C₅×#125	D₃	semidirect
66. C₃×#125	C₂×C₅	semidirect
67. #898	C₂×C₅	semidirect
68. #897	C₂×C₅	semidirect
70. #894	C₂×C₅	semidirect
71. #893	C₂×C₅	semidirect
72. #893	C₂×C₅	semidirect
73. #892	C₂×C₅	semidirect
74. #887	C₂×C₅	semidirect
75. #875	C₂×C₅	semidirect
76. C₄²×D₃	C₂×C₅	semidirect
159. #125	C₅×D₃	semidirect
231. C₃×C₅	#529	semidirect
276. C₅	#5691	direct (N × H)
293. C₃	C₅×#529	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×#898	2
C₅×#897	2
C₅×#895	2
C₅×#894	2
C₅×#893	2
C₅×#892	2
C₅×#887	2
C₅×#875	2
C₄²×C₅×D₃	2
C₅×#529	3
#5691	5

Subgroups — C5×#5691 #11756076

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5691 #11756076