Subgroups — C₅×#71587 #14294779

Order 1920.

776 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#117	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×#208	C₂	1	2	yes	maximal
3	C₅×#4530	C₂	1	4	yes	maximal, complemented
4	C₅×#1742	—	3	1	no	maximal, Hall
5	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
6	C₃×C₅×#117	—	2	2	no	—
7	C₃×C₅×#117	—	2	2	no	—
8	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₃×C₅×C₂²⋊C₄	C₄	1	2	yes	—
10	C₂×C₃×C₄²×C₅	C₄	1	2	yes	—
11	C₂×C₅×#208	—	2	4	no	—
12	C₂×C₅×#208	—	2	2	no	—
13	C₂×C₅×#208	C₄	1	8	yes	complemented
14	C₂×C₅×#208	C₂²	1	4	yes	—
15	C₂²×C₅×C₃⋊C₈	—	2	2	no	—
16	#71587	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂²×C₅×M₁₆	—	3	2	no	—
18	C₂×C₅×#117	D₃	1	1	yes	characteristic
19	C₅×#343	—	3	4	no	—
20	C₂⁴×C₃×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₃×C₄×C₅	C₂×C₄	1	4	yes	—
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₃×C₄×C₅	C₂×C₄	1	4	yes	—
24	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
25	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
26	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
27	C₂²×C₃×C₄×C₅	Q₈	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₃×C₅×C₄⋊C₄	—	2	4	no	—
30	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
31	C₃×C₄²×C₅	—	2	2	no	—
32	C₃×C₄²×C₅	—	2	2	no	—
33	C₅×#208	—	2	8	no	—
34	C₅×#208	—	2	4	no	—
35	C₅×#208	C₂×C₄	1	8	yes	—
36	C₂×C₅×C₃⋊C₈	—	2	8	no	—
37	C₂×C₅×C₃⋊C₈	—	2	4	no	—
38	C₂×C₃×#117	C₂×C₅	1	1	yes	characteristic
39	C₂²×#208	C₂×C₅	1	2	yes	—
40	#4530	C₂×C₅	1	4	yes	complemented
41	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
42	C₂×C₅×M₁₆	—	3	8	no	—
43	C₂×C₅×M₁₆	—	3	4	no	—
44	C₂×C₅×M₁₆	—	6	4	no	—
45	C₂×C₅×M₁₆	—	6	2	no	—
46	C₂²×C₅×C₈	—	6	2	no	—
47	C₅×#117	—	2	2	no	—
48	C₅×#117	—	2	2	no	—
49	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
50	C₂×C₅×C₂²⋊C₄	Dic₃	1	2	yes	—
51	C₂×C₄²×C₅	Dic₃	1	2	yes	—
52	#1742	—	3	1	no	Sylow
53	C₂³×C₃×C₅	—	4	2	no	—
54	C₂³×C₃×C₅	C₄²	1	2	yes	—
55	C₂³×C₃×C₅	C₂²⋊C₄	1	2	yes	—
56	C₂³×C₃×C₅	C₄⋊C₄	1	2	yes	—
57	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	—	2	4	no	—
59	C₂×C₃×C₄×C₅	—	4	4	no	—
60	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
61	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
64	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
65	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
66	C₂×C₃×C₄×C₅	C₄⋊C₄	1	4	yes	—
67	C₂×C₃×C₄×C₅	C₄⋊C₄	1	4	yes	—
68	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
69	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
71	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
72	C₅×C₃⋊C₈	—	2	8	no	—
73	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
74	C₃×#117	—	2	2	no	—
75	C₃×#117	—	2	2	no	—
76	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
77	C₂×C₃×C₂²⋊C₄	C₄×C₅	1	2	yes	—
78	C₂×C₃×C₄²	C₄×C₅	1	2	yes	—
79	C₂×#208	—	2	4	no	—
80	C₂×#208	—	2	2	no	—
81	C₂×#208	C₄×C₅	1	8	yes	complemented
82	C₂×#208	C₂²×C₅	1	4	yes	—
83	C₂²×C₃⋊C₈	—	2	2	no	—
84	C₂⁴×C₅	C₂×Dic₃	1	1	yes	characteristic
85	C₂²×C₄×C₅	Dic₆	1	2	yes	—
86	C₂²×C₄×C₅	C₄×D₃	1	4	yes	—
87	C₂²×C₄×C₅	D₁₂	1	2	yes	—
88	C₂²×C₄×C₅	C₂×Dic₃	1	4	yes	—
89	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
90	C₂²×C₄×C₅	#67	1	2	yes	—
91	C₂²×C₄×C₅	#67	1	2	yes	—
92	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
93	C₅×M₁₆	—	3	8	no	—
94	C₅×M₁₆	—	6	8	no	—
95	C₅×M₁₆	—	6	4	no	—
96	C₂×C₅×C₈	—	6	8	no	—
97	C₂×C₅×C₈	—	6	4	no	—
98	C₅×C₄⋊C₄	—	2	4	no	—
99	C₅×C₂²⋊C₄	—	2	4	no	—
100	C₄²×C₅	—	2	2	no	—
101	C₄²×C₅	—	2	2	no	—
102	C₂²×M₁₆	—	3	2	no	—
103	C₂×#117	C₅×D₃	1	1	yes	characteristic
104	#343	—	3	4	no	—
105	C₂²×C₃×C₅	—	2	2	no	—
106	C₂²×C₃×C₅	—	2	4	no	—
107	C₂²×C₃×C₅	—	4	2	no	—
108	C₂²×C₃×C₅	—	4	2	no	—
109	C₂²×C₃×C₅	#95	1	4	yes	—
110	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
111	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₃×C₄×C₅	—	4	4	no	—
114	C₃×C₄×C₅	#95	1	2	yes	—
115	C₃×C₄×C₅	#95	1	2	yes	—
116	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
117	C₃×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
118	C₂⁴×C₃	C₂×C₄×C₅	1	1	yes	characteristic
119	C₂²×C₃×C₄	C₂×C₄×C₅	1	4	yes	—
120	C₂²×C₃×C₄	C₂×C₄×C₅	1	4	yes	—
121	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
122	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
123	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
124	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
125	C₂²×C₃×C₄	C₅×Q₈	1	2	yes	—
126	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
127	C₃×C₄⋊C₄	—	2	4	no	—
128	C₃×C₂²⋊C₄	—	2	4	no	—
129	C₃×C₄²	—	2	2	no	—
130	C₃×C₄²	—	2	2	no	—
131	#208	—	2	8	no	—
132	#208	—	2	4	no	—
133	#208	C₂×C₄×C₅	1	8	yes	—
134	C₂×C₃⋊C₈	—	2	8	no	—
135	C₂×C₃⋊C₈	—	2	4	no	—
136	C₂³×C₅	—	4	2	no	—
137	C₂³×C₅	C₄×Dic₃	1	2	yes	—
138	C₂³×C₅	C₄⋊Dic₃	1	2	yes	—
139	C₂³×C₅	#217	1	2	yes	—
140	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
141	C₂×C₄×C₅	—	2	4	no	—
142	C₂×C₄×C₅	—	4	4	no	—
143	C₂×C₄×C₅	C₄×Dic₃	1	2	yes	—
144	C₂×C₄×C₅	Dic₃⋊C₄	1	4	yes	—
145	C₂×C₄×C₅	Dic₃⋊C₄	1	4	yes	—
146	C₂×C₄×C₅	C₄⋊Dic₃	1	2	yes	—
147	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	4	yes	—
148	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	4	yes	—
149	C₂×C₄×C₅	#217	1	1	yes	characteristic
150	C₂×C₄×C₅	#217	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
153	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂×#67	1	2	yes	—
155	C₅×C₈	—	6	8	no	—
156	C₂³×C₄	C₂×C₅×D₃	1	1	yes	characteristic
157	C₂×M₁₆	—	3	8	no	—
158	C₂×M₁₆	—	3	4	no	—
159	C₂×M₁₆	—	6	4	no	—
160	C₂×M₁₆	—	6	2	no	—
161	C₂²×C₈	—	6	2	no	—
162	#117	—	2	2	no	—
163	#117	—	2	2	no	—
164	C₂×C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
165	C₂×C₂²⋊C₄	C₅×Dic₃	1	2	yes	—
166	C₂×C₄²	C₅×Dic₃	1	2	yes	—
167	C₂×C₃×C₅	—	2	4	no	—
168	C₂×C₃×C₅	—	2	1	no	—
169	C₂×C₃×C₅	—	2	1	no	—
170	C₂×C₃×C₅	C₂×#95	1	1	yes	characteristic
171	C₂×C₃×C₅	#413	1	2	yes	—
172	C₂³×C₃	—	4	2	no	—
173	C₂³×C₃	C₄²×C₅	1	2	yes	—
174	C₂³×C₃	C₅×C₂²⋊C₄	1	2	yes	—
175	C₂³×C₃	C₅×C₄⋊C₄	1	2	yes	—
176	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
177	C₂×C₃×C₄	—	2	4	no	—
178	C₂×C₃×C₄	—	4	4	no	—
179	C₂×C₃×C₄	C₄²×C₅	1	2	yes	—
180	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	4	yes	—
181	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	4	yes	—
182	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
183	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
184	C₂×C₃×C₄	C₅×C₄⋊C₄	1	4	yes	—
185	C₂×C₃×C₄	C₅×C₄⋊C₄	1	4	yes	—
186	C₂×C₃×C₄	C₅×C₄⋊C₄	1	2	yes	—
187	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
188	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
189	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
190	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
191	C₃⋊C₈	—	2	8	no	—
192	C₂²×C₅	—	2	2	no	—
193	C₂²×C₅	—	2	4	no	—
194	C₂²×C₅	—	4	2	no	—
195	C₂²×C₅	—	4	2	no	—
196	C₂²×C₅	C₃⋊#95	1	4	yes	—
197	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
198	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
199	C₂²×C₅	C₂×#217	1	1	yes	characteristic
200	C₄×C₅	—	4	4	no	—
201	C₄×C₅	C₃⋊#95	1	2	yes	—
202	C₄×C₅	C₃⋊#95	1	2	yes	—
203	C₄×C₅	C₂×Dic₃⋊C₄	1	2	yes	—
204	C₄×C₅	C₂×(C₂×D₃)⋊C₄	1	2	yes	—
205	C₂⁴	C₂×C₅×Dic₃	1	1	yes	characteristic
206	C₂²×C₄	C₅×Dic₆	1	2	yes	—
207	C₂²×C₄	C₄×C₅×D₃	1	4	yes	—
208	C₂²×C₄	C₅×D₁₂	1	2	yes	—
209	C₂²×C₄	C₂×C₅×Dic₃	1	4	yes	—
210	C₂²×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
211	C₂²×C₄	C₅×#67	1	2	yes	—
212	C₂²×C₄	C₅×#67	1	2	yes	—
213	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
214	M₁₆	—	3	8	no	—
215	M₁₆	—	6	8	no	—
216	M₁₆	—	6	4	no	—
217	C₂×C₈	—	6	8	no	—
218	C₂×C₈	—	6	4	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₂²⋊C₄	—	2	4	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	2	2	no	—
223	C₃×C₅	#1742	1	1	yes	characteristic, complemented, Hall
224	C₂²×C₃	—	2	2	no	—
225	C₂²×C₃	—	2	4	no	—
226	C₂²×C₃	—	4	2	no	—
227	C₂²×C₃	—	4	2	no	—
228	C₂²×C₃	C₅×#95	1	4	yes	—
229	C₂²×C₃	C₂×C₄²×C₅	1	1	yes	characteristic
230	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
231	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
232	C₃×C₄	—	4	4	no	—
233	C₃×C₄	C₅×#95	1	2	yes	—
234	C₃×C₄	C₅×#95	1	2	yes	—
235	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
236	C₃×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
237	C₂×C₅	—	2	4	no	—
238	C₂×C₅	—	2	1	no	—
239	C₂×C₅	—	2	1	no	—
240	C₂×C₅	C₂×C₃⋊#95	1	1	yes	characteristic
241	C₂×C₅	#5237	1	2	yes	—
242	C₂³	—	4	2	no	—
243	C₂³	C₄×C₅×Dic₃	1	2	yes	—
244	C₂³	C₅×C₄⋊Dic₃	1	2	yes	—
245	C₂³	C₅×#217	1	2	yes	—
246	C₂³	C₂²×C₅×Dic₃	1	1	yes	characteristic
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	C₄×C₅×Dic₃	1	2	yes	—
250	C₂×C₄	C₅×Dic₃⋊C₄	1	4	yes	—
251	C₂×C₄	C₅×Dic₃⋊C₄	1	4	yes	—
252	C₂×C₄	C₅×C₄⋊Dic₃	1	2	yes	—
253	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	4	yes	—
254	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	4	yes	—
255	C₂×C₄	C₅×#217	1	1	yes	characteristic
256	C₂×C₄	C₅×#217	1	1	yes	characteristic
257	C₂×C₄	C₂×C₅×Dic₆	1	1	yes	characteristic
258	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
259	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
260	C₂×C₄	C₂×C₅×#67	1	2	yes	—
261	C₈	—	6	8	no	—
262	C₂×C₃	—	2	1	no	—
263	C₂×C₃	—	2	1	no	—
264	C₂×C₃	—	2	4	no	—
265	C₂×C₃	C₂×C₅×#95	1	1	yes	characteristic
266	C₂×C₃	C₅×#413	1	2	yes	—
267	C₅	#71587	1	1	yes	characteristic, direct product factor, Sylow
268	C₂²	—	2	2	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	4	2	no	—
271	C₂²	—	4	2	no	—
272	C₂²	C₅×C₃⋊#95	1	4	yes	—
273	C₂²	C₂×C₄×C₅×Dic₃	1	1	yes	characteristic
274	C₂²	C₂×C₅×C₄⋊Dic₃	1	1	yes	characteristic
275	C₂²	C₂×C₅×#217	1	1	yes	characteristic
276	C₄	—	4	4	no	—
277	C₄	C₅×C₃⋊#95	1	2	yes	—
278	C₄	C₅×C₃⋊#95	1	2	yes	—
279	C₄	C₂×C₅×Dic₃⋊C₄	1	2	yes	—
280	C₄	C₂×C₅×(C₂×D₃)⋊C₄	1	2	yes	—
281	C₃	C₅×#1742	1	1	yes	characteristic, complemented, Sylow
282	C₂	—	2	1	no	—
283	C₂	—	2	1	no	—
284	C₂	—	2	4	no	—
285	C₂	C₂×C₅×C₃⋊#95	1	1	yes	characteristic
286	C₂	C₅×#5237	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	#217
commutator	C₂²×C₃	C₂×C₄²×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×#117	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₃×C₅	C₂×C₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₂×C₅×#117	D₃
hypocenter	C₃	C₅×#1742
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂³×C₄×C₅ ◁ C₂×C₅×#117

Complements

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

N	H ≅ G/N	type
3. C₅×#4530	C₂	semidirect
13. C₂×C₅×#208	C₄	semidirect
16. #71587	C₅	direct (N × H)
40. #4530	C₂×C₅	semidirect
81. C₂×#208	C₄×C₅	semidirect
223. C₃×C₅	#1742	semidirect
267. C₅	#71587	direct (N × H)
281. C₃	C₅×#1742	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×#117	2
C₂²×C₅×#208	2
C₅×#4530	2
C₅×#1742	3
#71587	5

Subgroups — C5×#71587 #14294779

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71587 #14294779