Subgroups — C₅×#71661 #14301573

Order 1920.

688 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#519	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#947	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#836	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5001	C₂	1	2	yes	maximal
5	C₅×#4532	C₂	1	2	yes	maximal, complemented
6	C₅×#1888	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
8	C₂²×C₅×D₁₂	C₂²	1	1	yes	characteristic
9	C₃×C₄×C₅×Q₈	—	2	4	no	—
10	C₃×C₅×#117	—	2	2	no	—
11	C₃×C₅×#117	C₂²	1	2	yes	—
12	C₅×#947	—	2	2	no	—
13	C₅×#947	C₄	1	4	yes	complemented
14	C₂×C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
15	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
16	C₂×C₅×#208	C₂²	1	2	yes	—
17	C₅×#836	—	2	2	no	—
18	C₅×#809	—	2	4	no	—
19	#71661	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₂×C₅×#124	—	3	1	no	—
21	C₅×#519	D₃	1	1	yes	characteristic, complemented
22	C₅×#418	—	3	2	no	—
23	C₂×C₅×#101	—	3	1	no	—
24	C₅×#339	—	3	2	no	—
25	C₂³×C₅×D₃	—	2	1	no	—
26	C₂×C₃×C₅×Q₈	—	2	2	no	—
27	C₂×C₃×C₅×Q₈	—	2	2	no	—
28	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₅×Q₈	D₄	1	2	yes	—
30	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
31	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂²×C₅×Dic₃	—	2	1	no	—
33	C₂×C₅×D₁₂	—	2	2	no	—
34	C₂×C₅×D₁₂	—	2	2	no	—
35	C₂×C₅×D₁₂	—	2	1	no	—
36	C₂×C₅×D₁₂	C₂×C₄	1	2	yes	—
37	C₃×C₅×C₄⋊C₄	—	2	4	no	—
38	C₃×C₅×C₄⋊C₄	D₄	1	4	yes	—
39	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
40	C₃×C₄²×C₅	—	2	4	no	—
41	C₃×C₄²×C₅	—	2	2	no	—
42	C₅×(C₂×D₃)⋊C₄	—	2	8	no	—
43	C₄×C₅×Dic₃	—	2	1	no	—
44	C₄×C₅×Dic₃	C₂×C₄	1	2	yes	—
45	C₅×#208	—	2	2	no	—
46	C₅×#208	—	2	2	no	—
47	C₂×C₅×C₃⋊C₈	—	2	2	no	—
48	C₃×#519	C₂×C₅	1	1	yes	characteristic, complemented
49	C₂×#947	C₂×C₅	1	1	yes	characteristic
50	C₂×#836	C₂×C₅	1	1	yes	characteristic, complemented
51	#5001	C₂×C₅	1	2	yes	—
52	#4532	C₂×C₅	1	2	yes	complemented
53	C₂²×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
54	C₂²×C₅×D₄	—	3	1	no	—
55	C₂×C₅×M₁₆	—	3	2	no	—
56	C₅×#124	—	3	4	no	—
57	C₅×#124	—	6	2	no	—
58	C₄×C₅×Q₈	—	2	4	no	—
59	C₅×#117	—	2	2	no	—
60	C₅×#117	C₂×D₃	1	2	yes	—
61	C₂×C₅×C₂²⋊C₄	—	6	2	no	—
62	C₂×C₄²×C₅	—	3	1	no	—
63	C₅×#102	—	6	4	no	—
64	C₅×#101	—	6	2	no	—
65	#1888	—	3	1	no	Sylow
66	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₅×D₃	—	2	2	no	—
68	C₂²×C₅×D₃	—	4	4	no	—
69	C₂²×C₅×D₃	—	4	1	no	—
70	C₂²×C₅×D₃	—	4	1	no	—
71	C₃×C₅×Q₈	—	2	4	no	—
72	C₃×C₅×Q₈	—	2	4	no	—
73	C₂×C₃×C₄×C₅	—	2	2	no	—
74	C₂×C₃×C₄×C₅	—	2	2	no	—
75	C₂×C₃×C₄×C₅	—	2	4	no	—
76	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
77	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
79	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	#41	1	2	yes	—
81	C₂×C₅×Dic₃	—	2	2	no	—
82	C₂×C₅×Dic₃	—	4	2	no	—
83	C₅×D₁₂	—	2	4	no	—
84	C₅×D₁₂	—	4	2	no	—
85	C₅×C₃⋊C₈	—	4	2	no	—
86	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
87	C₂²×D₁₂	C₂²×C₅	1	1	yes	characteristic
88	C₃×C₄×Q₈	—	2	4	no	—
89	C₃×#117	—	2	2	no	—
90	C₃×#117	C₂²×C₅	1	2	yes	—
91	#947	—	2	2	no	—
92	#947	C₄×C₅	1	4	yes	complemented
93	C₂×(C₂×D₃)⋊C₄	—	2	2	no	—
94	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
95	C₂×#208	C₂²×C₅	1	2	yes	—
96	#836	—	2	2	no	—
97	#809	—	2	4	no	—
98	C₂⁴×C₅	—	6	1	no	—
99	C₂×C₅×Q₈	—	2	2	no	—
100	C₂×C₅×Q₈	—	2	2	no	—
101	C₂×C₅×Q₈	C₄×D₃	1	2	yes	—
102	C₂×C₅×Q₈	D₁₂	1	2	yes	—
103	C₂×C₅×D₄	—	3	2	no	—
104	C₂×C₅×D₄	—	6	1	no	—
105	C₂×C₅×D₄	—	6	2	no	—
106	C₂×C₅×D₄	—	6	2	no	—
107	C₂²×C₄×C₅	—	6	1	no	—
108	C₂²×C₄×C₅	#67	1	2	yes	—
109	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
110	C₅×M₁₆	—	6	2	no	—
111	C₅×M₁₆	—	6	2	no	—
112	C₂×C₅×C₈	—	6	2	no	—
113	C₅×C₄⋊C₄	—	2	4	no	—
114	C₅×C₄⋊C₄	#67	1	4	yes	—
115	C₅×C₂²⋊C₄	—	2	2	no	—
116	C₅×C₂²⋊C₄	—	6	8	no	—
117	C₄²×C₅	—	2	4	no	—
118	C₄²×C₅	—	2	2	no	—
119	C₄²×C₅	—	3	2	no	—
120	C₄²×C₅	—	6	1	no	—
121	C₂×#124	—	3	1	no	—
122	#519	C₅×D₃	1	1	yes	characteristic, complemented
123	#418	—	3	2	no	—
124	C₂×#101	—	3	1	no	—
125	#339	—	3	2	no	—
126	C₂²×C₃×C₅	—	2	2	no	—
127	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂²×C₃×C₅	#120	1	1	yes	characteristic
129	C₂²×C₃×C₅	#123	1	1	yes	characteristic
130	C₂×C₅×D₃	—	4	4	no	—
131	C₂×C₅×D₃	—	4	2	no	—
132	C₂×C₅×D₃	—	8	2	no	—
133	C₂×C₅×D₃	—	8	2	no	—
134	C₃×C₄×C₅	—	2	4	no	—
135	C₃×C₄×C₅	—	4	4	no	—
136	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
137	C₃×C₄×C₅	#121	1	2	yes	—
138	C₅×Dic₃	—	4	2	no	—
139	C₂³×D₃	—	2	1	no	—
140	C₂×C₃×Q₈	—	2	2	no	—
141	C₂×C₃×Q₈	—	2	2	no	—
142	C₂×C₃×Q₈	C₂×C₄×C₅	1	2	yes	—
143	C₂×C₃×Q₈	C₅×D₄	1	2	yes	—
144	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
145	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
146	C₂²×Dic₃	—	2	1	no	—
147	C₂×D₁₂	—	2	2	no	—
148	C₂×D₁₂	—	2	2	no	—
149	C₂×D₁₂	—	2	1	no	—
150	C₂×D₁₂	C₂×C₄×C₅	1	2	yes	—
151	C₃×C₄⋊C₄	—	2	4	no	—
152	C₃×C₄⋊C₄	C₅×D₄	1	4	yes	—
153	C₃×C₂²⋊C₄	—	2	2	no	—
154	C₃×C₄²	—	2	4	no	—
155	C₃×C₄²	—	2	2	no	—
156	(C₂×D₃)⋊C₄	—	2	8	no	—
157	C₄×Dic₃	—	2	1	no	—
158	C₄×Dic₃	C₂×C₄×C₅	1	2	yes	—
159	#208	—	2	2	no	—
160	#208	—	2	2	no	—
161	C₂×C₃⋊C₈	—	2	2	no	—
162	C₂³×C₅	—	6	2	no	—
163	C₂³×C₅	—	12	4	no	—
164	C₂³×C₅	—	12	1	no	—
165	C₂³×C₅	—	12	1	no	—
166	C₂³×C₅	C₂×#67	1	1	yes	characteristic
167	C₅×Q₈	—	2	4	no	—
168	C₅×Q₈	—	2	4	no	—
169	C₅×D₄	—	6	4	no	—
170	C₅×D₄	—	12	2	no	—
171	C₂×C₄×C₅	—	2	2	no	—
172	C₂×C₄×C₅	—	2	2	no	—
173	C₂×C₄×C₅	—	2	4	no	—
174	C₂×C₄×C₅	—	6	2	no	—
175	C₂×C₄×C₅	—	12	2	no	—
176	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	4	yes	—
177	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
178	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
179	C₂×C₄×C₅	#235	1	2	yes	—
180	C₂×C₄×C₅	C₂×#67	1	2	yes	—
181	C₅×C₈	—	12	2	no	—
182	C₂²×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
183	C₂²×D₄	—	3	1	no	—
184	C₂×M₁₆	—	3	2	no	—
185	#124	—	3	4	no	—
186	#124	—	6	2	no	—
187	C₄×Q₈	—	2	4	no	—
188	#117	—	2	2	no	—
189	#117	C₂×C₅×D₃	1	2	yes	—
190	C₂×C₂²⋊C₄	—	6	2	no	—
191	C₂×C₄²	—	3	1	no	—
192	#102	—	6	4	no	—
193	#101	—	6	2	no	—
194	C₂×C₃×C₅	—	2	1	no	—
195	C₂×C₃×C₅	—	2	1	no	—
196	C₂×C₃×C₅	#386	1	1	yes	characteristic
197	C₂×C₃×C₅	#454	1	2	yes	—
198	C₅×D₃	—	8	2	no	—
199	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
200	C₂²×D₃	—	2	2	no	—
201	C₂²×D₃	—	4	4	no	—
202	C₂²×D₃	—	4	1	no	—
203	C₂²×D₃	—	4	1	no	—
204	C₃×Q₈	—	2	4	no	—
205	C₃×Q₈	—	2	4	no	—
206	C₂×C₃×C₄	—	2	2	no	—
207	C₂×C₃×C₄	—	2	2	no	—
208	C₂×C₃×C₄	—	2	4	no	—
209	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	4	yes	—
210	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
211	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
213	C₂×C₃×C₄	C₅×#41	1	2	yes	—
214	C₂×Dic₃	—	2	2	no	—
215	C₂×Dic₃	—	4	2	no	—
216	D₁₂	—	2	4	no	—
217	D₁₂	—	4	2	no	—
218	C₃⋊C₈	—	4	2	no	—
219	C₂²×C₅	—	2	2	no	—
220	C₂²×C₅	—	12	4	no	—
221	C₂²×C₅	—	12	2	no	—
222	C₂²×C₅	—	24	2	no	—
223	C₂²×C₅	—	24	2	no	—
224	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
225	C₂²×C₅	#929	1	1	yes	characteristic
226	C₂²×C₅	#953	1	1	yes	characteristic
227	C₄×C₅	—	2	4	no	—
228	C₄×C₅	—	4	4	no	—
229	C₄×C₅	—	12	2	no	—
230	C₄×C₅	C₄×#67	1	2	yes	—
231	C₄×C₅	#930	1	2	yes	—
232	C₂⁴	—	6	1	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	C₄×C₅×D₃	1	2	yes	—
236	C₂×Q₈	C₅×D₁₂	1	2	yes	—
237	C₂×D₄	—	3	2	no	—
238	C₂×D₄	—	6	1	no	—
239	C₂×D₄	—	6	2	no	—
240	C₂×D₄	—	6	2	no	—
241	C₂²×C₄	—	6	1	no	—
242	C₂²×C₄	C₅×#67	1	2	yes	—
243	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
244	M₁₆	—	6	2	no	—
245	M₁₆	—	6	2	no	—
246	C₂×C₈	—	6	2	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	C₅×#67	1	4	yes	—
249	C₂²⋊C₄	—	2	2	no	—
250	C₂²⋊C₄	—	6	8	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	3	2	no	—
254	C₄²	—	6	1	no	—
255	C₃×C₅	#1888	1	1	yes	characteristic, complemented, Hall
256	C₂²×C₃	—	2	2	no	—
257	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
258	C₂²×C₃	C₅×#120	1	1	yes	characteristic
259	C₂²×C₃	C₅×#123	1	1	yes	characteristic
260	C₂×D₃	—	4	4	no	—
261	C₂×D₃	—	4	2	no	—
262	C₂×D₃	—	8	2	no	—
263	C₂×D₃	—	8	2	no	—
264	C₃×C₄	—	2	4	no	—
265	C₃×C₄	—	4	4	no	—
266	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
267	C₃×C₄	C₅×#121	1	2	yes	—
268	Dic₃	—	4	2	no	—
269	C₂×C₅	—	2	1	no	—
270	C₂×C₅	—	2	1	no	—
271	C₂×C₅	—	24	2	no	—
272	C₂×C₅	#5213	1	1	yes	characteristic
273	C₂×C₅	#5245	1	2	yes	—
274	C₂³	—	6	2	no	—
275	C₂³	—	12	4	no	—
276	C₂³	—	12	1	no	—
277	C₂³	—	12	1	no	—
278	C₂³	C₂×C₅×#67	1	1	yes	characteristic
279	Q₈	—	2	4	no	—
280	Q₈	—	2	4	no	—
281	D₄	—	6	4	no	—
282	D₄	—	12	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	6	2	no	—
287	C₂×C₄	—	12	2	no	—
288	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	4	yes	—
289	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
290	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
291	C₂×C₄	C₅×#235	1	2	yes	—
292	C₂×C₄	C₂×C₅×#67	1	2	yes	—
293	C₈	—	12	2	no	—
294	C₂×C₃	—	2	1	no	—
295	C₂×C₃	—	2	1	no	—
296	C₂×C₃	C₅×#386	1	1	yes	characteristic
297	C₂×C₃	C₅×#454	1	2	yes	—
298	D₃	—	8	2	no	—
299	C₅	#71661	1	1	yes	characteristic, direct product factor, Sylow
300	C₂²	—	2	2	no	—
301	C₂²	—	12	4	no	—
302	C₂²	—	12	2	no	—
303	C₂²	—	24	2	no	—
304	C₂²	—	24	2	no	—
305	C₂²	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
306	C₂²	C₅×#929	1	1	yes	characteristic
307	C₂²	C₅×#953	1	1	yes	characteristic
308	C₄	—	2	4	no	—
309	C₄	—	4	4	no	—
310	C₄	—	12	2	no	—
311	C₄	C₄×C₅×#67	1	2	yes	—
312	C₄	C₅×#930	1	2	yes	—
313	C₃	C₅×#1888	1	1	yes	characteristic, complemented, Sylow
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	—	24	2	no	—
317	C₂	C₅×#5213	1	1	yes	characteristic
318	C₂	C₅×#5245	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅	#953
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#519	C₂
socle	C₂²×C₃×C₅	#120
prime radical	C₂²×C₅×D₁₂	C₂²
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#519	D₃
hypocenter	C₃	C₅×#1888
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#519

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#519	C₂	semidirect
3. C₂×C₅×#836	C₂	semidirect
5. C₅×#4532	C₂	semidirect
13. C₅×#947	C₄	semidirect
19. #71661	C₅	direct (N × H)
21. C₅×#519	D₃	semidirect
48. C₃×#519	C₂×C₅	semidirect
50. C₂×#836	C₂×C₅	semidirect
52. #4532	C₂×C₅	semidirect
92. #947	C₄×C₅	semidirect
122. #519	C₅×D₃	semidirect
255. C₃×C₅	#1888	semidirect
299. C₅	#71661	direct (N × H)
313. C₃	C₅×#1888	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#519	2
C₂×C₅×#947	2
C₂×C₅×#836	2
C₅×#5001	2
C₅×#4532	2
C₅×#1888	3
#71661	5

Subgroups — C5×#71661 #14301573

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71661 #14301573