Subgroups — C₅×#71865 #14302785

Order 1920.

624 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#536	C₂	1	1	yes	maximal, characteristic
2	C₅×#5819	C₂	1	1	yes	maximal, characteristic
3	C₅×#5240	C₂	1	1	yes	maximal, characteristic
4	C₅×#5225	C₂	1	2	yes	maximal, complemented
5	C₅×#4531	C₂	1	2	yes	maximal, complemented
6	C₅×#1909	—	3	1	no	maximal, Hall
7	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
9	C₃×C₅×C₄⋊Q₈	C₄	1	2	yes	complemented
10	C₃×C₅×#124	C₄	1	2	yes	complemented
11	C₃×C₅×#123	—	2	2	no	—
12	C₃×C₅×C₂²⋊Q₈	—	2	2	no	—
13	C₃×C₅×#117	C₂²	1	1	yes	characteristic
14	C₅×#948	—	2	4	no	—
15	C₅×#926	C₂²	1	2	yes	—
16	C₂×C₅×#208	—	2	2	no	—
17	C₂×C₅×#208	C₂²	1	2	yes	—
18	C₅×#835	—	2	4	no	—
19	C₅×#834	—	2	2	no	—
20	#71865	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₅×#568	—	3	1	no	—
22	C₅×#536	D₃	1	1	yes	characteristic
23	C₅×#419	—	3	2	no	—
24	C₅×#410	—	3	1	no	—
25	C₅×H(3,ℤ/4)	—	3	2	no	—
26	C₃×C₅×#41	—	2	2	no	—
27	C₃×C₅×#41	—	2	2	no	—
28	C₂×C₃×C₅×Q₈	—	2	1	no	—
29	C₂×C₃×C₅×Q₈	—	2	2	no	—
30	C₂×C₃×C₅×Q₈	—	2	2	no	—
31	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
32	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃×C₅×D₄	D₄	1	2	yes	—
35	C₂²×C₃×C₄×C₅	—	2	1	no	—
36	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
37	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
38	C₃×C₅×C₄⋊C₄	—	2	4	no	—
39	C₃×C₅×C₄⋊C₄	—	2	1	no	—
40	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
41	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
42	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
43	C₅×#217	—	2	2	no	—
44	C₅×C₄⋊Dic₃	—	2	2	no	—
45	C₄×C₅×Dic₃	—	2	2	no	—
46	C₅×#208	—	2	4	no	—
47	C₅×#208	—	2	2	no	—
48	C₅×#208	—	2	2	no	—
49	C₂×C₅×C₃⋊C₈	—	2	4	no	—
50	C₂×C₅×C₃⋊C₈	D₄	1	4	yes	—
51	C₃×#536	C₂×C₅	1	1	yes	characteristic
52	#5819	C₂×C₅	1	1	yes	characteristic
53	#5240	C₂×C₅	1	1	yes	characteristic
54	#5225	C₂×C₅	1	2	yes	complemented
55	#4531	C₂×C₅	1	2	yes	complemented
56	C₂×C₅×#41	C₂×D₃	1	1	yes	characteristic
57	C₂²×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
58	C₅×#131	—	6	4	no	—
59	C₂×C₅×M₁₆	—	3	2	no	—
60	C₂×C₅×M₁₆	—	6	2	no	—
61	C₅×C₄⋊Q₈	Dic₃	1	2	yes	complemented
62	C₅×#124	Dic₃	1	2	yes	complemented
63	C₅×#123	—	2	2	no	—
64	C₅×C₂²⋊Q₈	—	2	2	no	—
65	C₅×#117	—	3	2	no	—
66	C₅×#117	C₂×D₃	1	1	yes	characteristic
67	C₅×#103	—	6	4	no	—
68	C₅×#99	—	6	2	no	—
69	#1909	—	3	1	no	Sylow
70	C₂³×C₃×C₅	C₂²⋊C₄	1	2	yes	—
71	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
72	C₃×C₅×Q₈	—	2	2	no	—
73	C₃×C₅×Q₈	—	2	4	no	—
74	C₃×C₅×Q₈	—	4	1	no	—
75	C₃×C₅×Q₈	—	4	1	no	—
76	C₃×C₅×D₄	—	2	4	no	—
77	C₃×C₅×D₄	—	2	2	no	—
78	C₂×C₃×C₄×C₅	—	2	2	no	—
79	C₂×C₃×C₄×C₅	—	2	2	no	—
80	C₂×C₃×C₄×C₅	—	2	2	no	—
81	C₂×C₃×C₄×C₅	—	2	2	no	—
82	C₂×C₃×C₄×C₅	—	4	2	no	—
83	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
84	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
86	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	#41	1	2	yes	—
88	C₂×C₅×Dic₃	—	4	2	no	—
89	C₅×C₃⋊C₈	—	2	4	no	—
90	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
91	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
92	C₃×C₄⋊Q₈	C₄×C₅	1	2	yes	complemented
93	C₃×#124	C₄×C₅	1	2	yes	complemented
94	C₃×#123	—	2	2	no	—
95	C₃×C₂²⋊Q₈	—	2	2	no	—
96	C₃×#117	C₂²×C₅	1	1	yes	characteristic
97	#948	—	2	4	no	—
98	#926	C₂²×C₅	1	2	yes	—
99	C₂×#208	—	2	2	no	—
100	C₂×#208	C₂²×C₅	1	2	yes	—
101	#835	—	2	4	no	—
102	#834	—	2	2	no	—
103	C₅×#41	—	2	2	no	—
104	C₅×#41	—	2	2	no	—
105	C₂×C₅×Q₈	—	2	1	no	—
106	C₂×C₅×Q₈	—	2	2	no	—
107	C₂×C₅×Q₈	—	2	2	no	—
108	C₂×C₅×Q₈	C₂×Dic₃	1	2	yes	—
109	C₂×C₅×Q₈	C₂×Dic₃	1	1	yes	characteristic
110	C₂×C₅×D₄	C₂×Dic₃	1	1	yes	characteristic
111	C₂×C₅×D₄	#67	1	2	yes	—
112	C₂²×C₄×C₅	—	2	1	no	—
113	C₂²×C₄×C₅	#67	1	2	yes	—
114	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
115	C₅×M₁₆	—	6	4	no	—
116	C₅×M₁₆	—	6	2	no	—
117	C₅×M₁₆	—	6	2	no	—
118	C₂×C₅×C₈	—	3	4	no	—
119	C₂×C₅×C₈	—	6	4	no	—
120	C₅×C₄⋊C₄	—	2	4	no	—
121	C₅×C₄⋊C₄	—	2	1	no	—
122	C₅×C₄⋊C₄	—	6	2	no	—
123	C₅×C₂²⋊C₄	—	2	4	no	—
124	C₅×C₂²⋊C₄	—	2	1	no	—
125	C₅×C₂²⋊C₄	—	6	2	no	—
126	C₄²×C₅	—	6	2	no	—
127	C₄²×C₅	C₂×Dic₃	1	2	yes	—
128	#568	—	3	1	no	—
129	#536	C₅×D₃	1	1	yes	characteristic
130	#419	—	3	2	no	—
131	#410	—	3	1	no	—
132	H(3,ℤ/4)	—	3	2	no	—
133	C₂²×C₃×C₅	—	2	2	no	—
134	C₂²×C₃×C₅	—	4	1	no	—
135	C₂²×C₃×C₅	—	4	2	no	—
136	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂²×C₃×C₅	#121	1	2	yes	—
138	C₃×C₄×C₅	—	2	2	no	—
139	C₃×C₄×C₅	—	4	2	no	—
140	C₃×C₄×C₅	—	4	2	no	—
141	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
142	C₃×C₄×C₅	#120	1	1	yes	characteristic
143	C₃×C₄×C₅	#123	1	1	yes	characteristic
144	C₅×Dic₃	—	8	2	no	—
145	C₃×#41	—	2	2	no	—
146	C₃×#41	—	2	2	no	—
147	C₂×C₃×Q₈	—	2	1	no	—
148	C₂×C₃×Q₈	—	2	2	no	—
149	C₂×C₃×Q₈	—	2	2	no	—
150	C₂×C₃×Q₈	C₂×C₄×C₅	1	2	yes	—
151	C₂×C₃×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
152	C₂×C₃×D₄	C₂×C₄×C₅	1	1	yes	characteristic
153	C₂×C₃×D₄	C₅×D₄	1	2	yes	—
154	C₂²×C₃×C₄	—	2	1	no	—
155	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
156	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
157	C₃×C₄⋊C₄	—	2	4	no	—
158	C₃×C₄⋊C₄	—	2	1	no	—
159	C₃×C₂²⋊C₄	—	2	4	no	—
160	C₃×C₂²⋊C₄	—	2	1	no	—
161	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
162	#217	—	2	2	no	—
163	C₄⋊Dic₃	—	2	2	no	—
164	C₄×Dic₃	—	2	2	no	—
165	#208	—	2	4	no	—
166	#208	—	2	2	no	—
167	#208	—	2	2	no	—
168	C₂×C₃⋊C₈	—	2	4	no	—
169	C₂×C₃⋊C₈	C₅×D₄	1	4	yes	—
170	C₂³×C₅	#217	1	2	yes	—
171	C₂³×C₅	C₂×#67	1	1	yes	characteristic
172	C₅×Q₈	—	2	2	no	—
173	C₅×Q₈	—	2	4	no	—
174	C₅×Q₈	—	4	1	no	—
175	C₅×Q₈	—	4	1	no	—
176	C₅×D₄	—	2	4	no	—
177	C₅×D₄	—	2	2	no	—
178	C₂×C₄×C₅	—	2	2	no	—
179	C₂×C₄×C₅	—	2	2	no	—
180	C₂×C₄×C₅	—	2	2	no	—
181	C₂×C₄×C₅	—	2	2	no	—
182	C₂×C₄×C₅	—	4	2	no	—
183	C₂×C₄×C₅	—	12	2	no	—
184	C₂×C₄×C₅	#217	1	2	yes	—
185	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
186	C₂×C₄×C₅	#237	1	2	yes	—
187	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
188	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
189	C₅×C₈	—	6	4	no	—
190	C₂×#41	C₂×C₅×D₃	1	1	yes	characteristic
191	C₂²×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
192	#131	—	6	4	no	—
193	C₂×M₁₆	—	3	2	no	—
194	C₂×M₁₆	—	6	2	no	—
195	C₄⋊Q₈	C₅×Dic₃	1	2	yes	complemented
196	#124	C₅×Dic₃	1	2	yes	complemented
197	#123	—	2	2	no	—
198	C₂²⋊Q₈	—	2	2	no	—
199	#117	—	3	2	no	—
200	#117	C₂×C₅×D₃	1	1	yes	characteristic
201	#103	—	6	4	no	—
202	#99	—	6	2	no	—
203	C₂×C₃×C₅	—	2	2	no	—
204	C₂×C₃×C₅	—	2	1	no	—
205	C₂×C₃×C₅	—	4	2	no	—
206	C₂×C₃×C₅	#386	1	1	yes	characteristic
207	C₂³×C₃	C₅×C₂²⋊C₄	1	2	yes	—
208	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
209	C₃×Q₈	—	2	2	no	—
210	C₃×Q₈	—	2	4	no	—
211	C₃×Q₈	—	4	1	no	—
212	C₃×Q₈	—	4	1	no	—
213	C₃×D₄	—	2	4	no	—
214	C₃×D₄	—	2	2	no	—
215	C₂×C₃×C₄	—	2	2	no	—
216	C₂×C₃×C₄	—	2	2	no	—
217	C₂×C₃×C₄	—	2	2	no	—
218	C₂×C₃×C₄	—	2	2	no	—
219	C₂×C₃×C₄	—	4	2	no	—
220	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
221	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
222	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
223	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
224	C₂×C₃×C₄	C₅×#41	1	2	yes	—
225	C₂×Dic₃	—	4	2	no	—
226	C₃⋊C₈	—	2	4	no	—
227	C₂²×C₅	—	2	2	no	—
228	C₂²×C₅	—	4	1	no	—
229	C₂²×C₅	—	4	2	no	—
230	C₂²×C₅	#939	1	2	yes	—
231	C₂²×C₅	C₂×#217	1	1	yes	characteristic
232	C₄×C₅	—	2	2	no	—
233	C₄×C₅	—	4	2	no	—
234	C₄×C₅	—	4	2	no	—
235	C₄×C₅	—	24	2	no	—
236	C₄×C₅	D₄×Dic₃	1	2	yes	—
237	C₄×C₅	#935	1	1	yes	characteristic
238	C₄×C₅	#937	1	1	yes	characteristic
239	#41	—	2	2	no	—
240	#41	—	2	2	no	—
241	C₂×Q₈	—	2	1	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	C₂×C₅×Dic₃	1	2	yes	—
245	C₂×Q₈	C₂×C₅×Dic₃	1	1	yes	characteristic
246	C₂×D₄	C₂×C₅×Dic₃	1	1	yes	characteristic
247	C₂×D₄	C₅×#67	1	2	yes	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	C₅×#67	1	2	yes	—
250	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
251	M₁₆	—	6	4	no	—
252	M₁₆	—	6	2	no	—
253	M₁₆	—	6	2	no	—
254	C₂×C₈	—	3	4	no	—
255	C₂×C₈	—	6	4	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	6	2	no	—
259	C₂²⋊C₄	—	2	4	no	—
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	6	2	no	—
262	C₄²	—	6	2	no	—
263	C₄²	C₂×C₅×Dic₃	1	2	yes	—
264	C₃×C₅	#1909	1	1	yes	characteristic, complemented, Hall
265	C₂²×C₃	—	2	2	no	—
266	C₂²×C₃	—	4	1	no	—
267	C₂²×C₃	—	4	2	no	—
268	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
269	C₂²×C₃	C₅×#121	1	2	yes	—
270	C₃×C₄	—	2	2	no	—
271	C₃×C₄	—	4	2	no	—
272	C₃×C₄	—	4	2	no	—
273	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
274	C₃×C₄	C₅×#120	1	1	yes	characteristic
275	C₃×C₄	C₅×#123	1	1	yes	characteristic
276	Dic₃	—	8	2	no	—
277	C₂×C₅	—	2	1	no	—
278	C₂×C₅	—	2	2	no	—
279	C₂×C₅	—	4	2	no	—
280	C₂×C₅	#5220	1	1	yes	characteristic
281	C₂³	C₅×#217	1	2	yes	—
282	C₂³	C₂×C₅×#67	1	1	yes	characteristic
283	Q₈	—	2	2	no	—
284	Q₈	—	2	4	no	—
285	Q₈	—	4	1	no	—
286	Q₈	—	4	1	no	—
287	D₄	—	2	4	no	—
288	D₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	12	2	no	—
295	C₂×C₄	C₅×#217	1	2	yes	—
296	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
297	C₂×C₄	C₅×#237	1	2	yes	—
298	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
299	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
300	C₈	—	6	4	no	—
301	C₂×C₃	—	2	2	no	—
302	C₂×C₃	—	2	1	no	—
303	C₂×C₃	—	4	2	no	—
304	C₂×C₃	C₅×#386	1	1	yes	characteristic
305	C₅	#71865	1	1	yes	characteristic, direct product factor, Sylow
306	C₂²	—	2	2	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	2	no	—
309	C₂²	C₅×#939	1	2	yes	—
310	C₂²	C₂×C₅×#217	1	1	yes	characteristic
311	C₄	—	2	2	no	—
312	C₄	—	4	2	no	—
313	C₄	—	4	2	no	—
314	C₄	—	24	2	no	—
315	C₄	C₅×D₄×Dic₃	1	2	yes	—
316	C₄	C₅×#935	1	1	yes	characteristic
317	C₄	C₅×#937	1	1	yes	characteristic
318	C₃	C₅×#1909	1	1	yes	characteristic, complemented, Sylow
319	C₂	—	2	1	no	—
320	C₂	—	2	2	no	—
321	C₂	—	4	2	no	—
322	C₂	C₅×#5220	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅	#5220
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#536	C₂
socle	C₂×C₃×C₅	#386
prime radical	C₂×C₃×C₅×#41	C₂²
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#536	D₃
hypocenter	C₃	C₅×#1909
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#536

Complements

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

N	H ≅ G/N	type
4. C₅×#5225	C₂	semidirect
5. C₅×#4531	C₂	semidirect
9. C₃×C₅×C₄⋊Q₈	C₄	semidirect
10. C₃×C₅×#124	C₄	semidirect
20. #71865	C₅	direct (N × H)
54. #5225	C₂×C₅	semidirect
55. #4531	C₂×C₅	semidirect
61. C₅×C₄⋊Q₈	Dic₃	semidirect
62. C₅×#124	Dic₃	semidirect
92. C₃×C₄⋊Q₈	C₄×C₅	semidirect
93. C₃×#124	C₄×C₅	semidirect
195. C₄⋊Q₈	C₅×Dic₃	semidirect
196. #124	C₅×Dic₃	semidirect
264. C₃×C₅	#1909	semidirect
305. C₅	#71865	direct (N × H)
318. C₃	C₅×#1909	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#536	2
C₅×#5819	2
C₅×#5240	2
C₅×#5225	2
C₅×#4531	2
C₅×#1909	3
#71865	5

Subgroups — C5×#71865 #14302785

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71865 #14302785