Subgroups — C₅×#79041 #14415229

Order 1920.

1000 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#907	C₂	1	2	yes	maximal, complemented
2	C₅×#5477	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#423	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5129	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4838	C₂	1	4	yes	maximal, complemented
7	C₅×#4834	C₂	1	4	yes	maximal, complemented
8	C₅×#2931	—	3	1	no	maximal, Hall
9	C₂×C₅×#235	C₂²	1	2	yes	—
10	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₅×#930	C₄	1	4	yes	—
13	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₅×#907	—	2	8	no	—
16	C₂×C₅×ΓL(1,𝔽₂₅)	C₂²	1	4	yes	—
17	C₂×C₅×D₃×C₈	C₂²	1	4	yes	—
18	C₅×#875	C₄	1	2	yes	—
19	C₅×#874	C₄	1	1	yes	characteristic
20	C₄×C₅×D₁₂	C₂²	1	4	yes	—
21	C₅×#869	C₄	1	1	yes	characteristic
22	C₃×C₅×C₄⋊C₈	C₂²	1	4	yes	—
23	C₅×(C₂×D₃)⋊C₈	C₂²	1	8	yes	—
24	C₄×C₅×C₃⋊C₈	C₂²	1	2	yes	—
25	C₄×C₅×C₃⋊C₈	C₂²	1	2	yes	—
26	#79041	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
27	C₂×C₅×#131	—	3	2	no	—
28	C₅×#532	—	3	1	no	—
29	C₅×#436	—	3	4	no	—
30	C₅×C₈×D₄	—	3	4	no	—
31	C₅×#423	D₃	1	1	yes	characteristic, complemented
32	C₅×#408	—	3	2	no	—
33	C₂×C₄×C₅×C₈	—	3	1	no	—
34	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
35	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₂×C₅×#67	C₂×C₄	1	4	yes	—
37	C₅×#235	—	2	4	no	—
38	C₅×#235	—	2	4	no	—
39	C₂×C₅×D₁₂	C₂×C₄	1	4	yes	—
40	C₂×C₅×D₁₂	C₂×C₄	1	2	yes	—
41	C₂×C₄×C₅×D₃	C₂³	1	4	yes	—
42	C₂×C₅×Dic₆	C₂×C₄	1	2	yes	—
43	C₃×C₅×M₁₆	—	2	4	no	—
44	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
45	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
46	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
47	C₃×C₄²×C₅	C₂³	1	2	yes	—
48	C₅×(C₂×D₃)⋊C₄	C₂×C₄	1	8	yes	—
49	C₅×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
50	C₂×C₅×C₃⋊C₈	—	2	2	no	—
51	C₂×C₅×C₃⋊C₈	D₄	1	4	yes	complemented
52	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
53	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
54	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
55	C₅×ΓL(1,𝔽₂₅)	—	2	8	no	—
56	C₅×D₃×C₈	—	2	8	no	—
57	C₂×#907	C₂×C₅	1	2	yes	complemented
58	#5477	C₂×C₅	1	1	yes	characteristic
59	C₃×#423	C₂×C₅	1	1	yes	characteristic, complemented
60	#5129	C₂×C₅	1	2	yes	complemented
61	C₂×C₄×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
62	#4838	C₂×C₅	1	4	yes	complemented
63	#4834	C₂×C₅	1	4	yes	complemented
64	C₂×C₅×#41	—	3	2	no	—
65	C₅×#131	—	6	8	no	—
66	C₂×C₅×M₁₆	—	3	4	no	—
67	C₂×C₅×M₁₆	C₂×D₃	1	2	yes	—
68	C₂²×C₅×C₈	—	3	1	no	—
69	C₂²×C₅×C₈	—	3	1	no	—
70	C₂²×C₅×C₈	—	3	4	no	—
71	C₅×C₄⋊Q₈	—	3	1	no	—
72	C₅×#127	—	3	1	no	—
73	C₅×#124	—	3	2	no	—
74	C₅×#121	—	3	4	no	—
75	C₄×C₅×D₄	—	3	4	no	—
76	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
77	C₅×C₄⋊C₈	C₂×D₃	1	4	yes	—
78	C₅×C₂²⋊C₈	—	3	8	no	—
79	C₄×C₅×C₈	—	3	2	no	—
80	C₄×C₅×C₈	—	3	2	no	—
81	#2931	—	3	1	no	Sylow
82	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₅×D₃	C₂²×C₄	1	4	yes	—
84	C₂×C₃×C₄×C₅	—	2	4	no	—
85	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
86	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
88	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
89	C₂×C₃×C₄×C₅	#41	1	2	yes	—
90	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
91	C₅×#67	—	2	8	no	—
92	C₂×C₅×Dic₃	C₂²×C₄	1	4	yes	—
93	C₅×D₁₂	—	2	4	no	—
94	C₅×D₁₂	—	2	8	no	—
95	C₄×C₅×D₃	—	2	4	no	—
96	C₄×C₅×D₃	—	2	4	no	—
97	C₅×Dic₆	—	2	4	no	—
98	C₃×C₅×C₈	—	2	4	no	—
99	C₅×C₃⋊C₈	—	2	2	no	—
100	C₅×C₃⋊C₈	C₂×D₄	1	2	yes	—
101	C₅×C₃⋊C₈	C₂×D₄	1	2	yes	—
102	C₂×#235	C₂²×C₅	1	2	yes	—
103	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
104	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
105	#930	C₄×C₅	1	4	yes	—
106	C₂²×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
107	C₂²×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
108	#907	—	2	8	no	—
109	C₂×ΓL(1,𝔽₂₅)	C₂²×C₅	1	4	yes	—
110	C₂×D₃×C₈	C₂²×C₅	1	4	yes	—
111	#875	C₄×C₅	1	2	yes	—
112	#874	C₄×C₅	1	1	yes	characteristic
113	C₄×D₁₂	C₂²×C₅	1	4	yes	—
114	#869	C₄×C₅	1	1	yes	characteristic
115	C₃×C₄⋊C₈	C₂²×C₅	1	4	yes	—
116	(C₂×D₃)⋊C₈	C₂²×C₅	1	8	yes	—
117	C₄×C₃⋊C₈	C₂²×C₅	1	2	yes	—
118	C₄×C₃⋊C₈	C₂²×C₅	1	2	yes	—
119	C₅×#41	—	6	4	no	—
120	C₅×#41	—	6	4	no	—
121	C₂×C₅×Q₈	—	3	2	no	—
122	C₂×C₅×D₄	—	3	4	no	—
123	C₂×C₅×D₄	—	3	4	no	—
124	C₂×C₅×D₄	—	3	2	no	—
125	C₂²×C₄×C₅	—	3	4	no	—
126	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
127	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
128	C₅×M₁₆	—	2	4	no	—
129	C₅×M₁₆	—	6	8	no	—
130	C₂×C₅×C₈	—	3	4	no	—
131	C₂×C₅×C₈	—	3	1	no	—
132	C₂×C₅×C₈	—	3	1	no	—
133	C₂×C₅×C₈	—	3	2	no	—
134	C₂×C₅×C₈	—	6	8	no	—
135	C₂×C₅×C₈	—	6	2	no	—
136	C₂×C₅×C₈	C₂²×D₃	1	4	yes	—
137	C₅×C₄⋊C₄	—	3	4	no	—
138	C₅×C₂²⋊C₄	—	3	8	no	—
139	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
140	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
141	C₄²×C₅	C₂²×D₃	1	2	yes	—
142	C₂×#131	—	3	2	no	—
143	#532	—	3	1	no	—
144	#436	—	3	4	no	—
145	C₈×D₄	—	3	4	no	—
146	#423	C₅×D₃	1	1	yes	characteristic, complemented
147	#408	—	3	2	no	—
148	C₂×C₄×C₈	—	3	1	no	—
149	C₂²×C₃×C₅	—	2	2	no	—
150	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
151	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
152	C₂×C₅×D₃	—	2	4	no	—
153	C₂×C₅×D₃	—	2	8	no	—
154	C₃×C₄×C₅	—	2	2	no	—
155	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
156	C₃×C₄×C₅	#131	1	4	yes	complemented
157	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
158	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
159	C₅×Dic₃	—	2	4	no	—
160	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
161	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
162	C₂×#67	C₂×C₄×C₅	1	4	yes	—
163	#235	—	2	4	no	—
164	#235	—	2	4	no	—
165	C₂×D₁₂	C₂×C₄×C₅	1	4	yes	—
166	C₂×D₁₂	C₂×C₄×C₅	1	2	yes	—
167	C₂×C₄×D₃	C₂³×C₅	1	4	yes	—
168	C₂×Dic₆	C₂×C₄×C₅	1	2	yes	—
169	C₃×M₁₆	—	2	4	no	—
170	C₂×C₃×C₈	C₂³×C₅	1	4	yes	—
171	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
172	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
173	C₃×C₄²	C₂³×C₅	1	2	yes	—
174	(C₂×D₃)⋊C₄	C₂×C₄×C₅	1	8	yes	—
175	C₄⋊Dic₃	C₂×C₄×C₅	1	4	yes	—
176	C₂×C₃⋊C₈	—	2	2	no	—
177	C₂×C₃⋊C₈	C₅×D₄	1	4	yes	complemented
178	C₂×C₃⋊C₈	C₂³×C₅	1	2	yes	—
179	C₂×C₃⋊C₈	C₂³×C₅	1	1	yes	characteristic
180	C₂×C₃⋊C₈	C₂³×C₅	1	1	yes	characteristic
181	ΓL(1,𝔽₂₅)	—	2	8	no	—
182	D₃×C₈	—	2	8	no	—
183	C₂³×C₅	—	3	4	no	—
184	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
185	C₅×Q₈	—	6	4	no	—
186	C₅×D₄	—	6	8	no	—
187	C₅×D₄	—	6	4	no	—
188	C₅×D₄	—	6	8	no	—
189	C₂×C₄×C₅	—	2	4	no	—
190	C₂×C₄×C₅	—	3	4	no	—
191	C₂×C₄×C₅	—	6	4	no	—
192	C₂×C₄×C₅	—	6	4	no	—
193	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
194	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
195	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
196	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
197	C₂×C₄×C₅	#239	1	2	yes	—
198	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
199	C₅×C₈	—	2	4	no	—
200	C₅×C₈	—	3	2	no	—
201	C₅×C₈	—	3	2	no	—
202	C₅×C₈	—	6	2	no	—
203	C₂×#41	—	3	2	no	—
204	#131	—	6	8	no	—
205	C₂×M₁₆	—	3	4	no	—
206	C₂×M₁₆	C₂×C₅×D₃	1	2	yes	—
207	C₂²×C₈	—	3	1	no	—
208	C₂²×C₈	—	3	1	no	—
209	C₂²×C₈	—	3	4	no	—
210	C₄⋊Q₈	—	3	1	no	—
211	#127	—	3	1	no	—
212	#124	—	3	2	no	—
213	#121	—	3	4	no	—
214	C₄×D₄	—	3	4	no	—
215	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
216	C₄⋊C₈	C₂×C₅×D₃	1	4	yes	—
217	C₂²⋊C₈	—	3	8	no	—
218	C₄×C₈	—	3	2	no	—
219	C₄×C₈	—	3	2	no	—
220	C₂×C₃×C₅	—	2	2	no	—
221	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
222	C₂×C₃×C₅	C₂×#131	1	2	yes	—
223	C₅×D₃	—	4	4	no	—
224	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
225	C₂²×D₃	C₂²×C₄×C₅	1	4	yes	—
226	C₂×C₃×C₄	—	2	4	no	—
227	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
228	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
230	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
231	C₂×C₃×C₄	C₅×#41	1	2	yes	—
232	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
233	#67	—	2	8	no	—
234	C₂×Dic₃	C₂²×C₄×C₅	1	4	yes	—
235	D₁₂	—	2	4	no	—
236	D₁₂	—	2	8	no	—
237	C₄×D₃	—	2	4	no	—
238	C₄×D₃	—	2	4	no	—
239	Dic₆	—	2	4	no	—
240	C₃×C₈	—	2	4	no	—
241	C₃⋊C₈	—	2	2	no	—
242	C₃⋊C₈	C₂×C₅×D₄	1	2	yes	—
243	C₃⋊C₈	C₂×C₅×D₄	1	2	yes	—
244	C₂²×C₅	—	2	2	no	—
245	C₂²×C₅	—	6	8	no	—
246	C₂²×C₅	—	6	4	no	—
247	C₂²×C₅	#893	1	2	yes	—
248	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
249	C₄×C₅	—	2	2	no	—
250	C₄×C₅	—	6	4	no	—
251	C₄×C₅	#893	1	2	yes	—
252	C₄×C₅	#907	1	4	yes	complemented
253	C₄×C₅	C₂×D₃×D₄	1	1	yes	characteristic
254	C₄×C₅	C₂×#239	1	1	yes	characteristic
255	#41	—	6	4	no	—
256	#41	—	6	4	no	—
257	C₂×Q₈	—	3	2	no	—
258	C₂×D₄	—	3	4	no	—
259	C₂×D₄	—	3	4	no	—
260	C₂×D₄	—	3	2	no	—
261	C₂²×C₄	—	3	4	no	—
262	C₂²×C₄	C₄×C₅×D₃	1	2	yes	—
263	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
264	M₁₆	—	2	4	no	—
265	M₁₆	—	6	8	no	—
266	C₂×C₈	—	3	4	no	—
267	C₂×C₈	—	3	1	no	—
268	C₂×C₈	—	3	1	no	—
269	C₂×C₈	—	3	2	no	—
270	C₂×C₈	—	6	8	no	—
271	C₂×C₈	—	6	2	no	—
272	C₂×C₈	C₂²×C₅×D₃	1	4	yes	—
273	C₄⋊C₄	—	3	4	no	—
274	C₂²⋊C₄	—	3	8	no	—
275	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
276	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
277	C₄²	C₂²×C₅×D₃	1	2	yes	—
278	C₃×C₅	#2931	1	1	yes	characteristic, complemented, Hall
279	C₂²×C₃	—	2	2	no	—
280	C₂²×C₃	C₄×C₅×D₄	1	2	yes	—
281	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
282	C₂×D₃	—	2	4	no	—
283	C₂×D₃	—	2	8	no	—
284	C₃×C₄	—	2	2	no	—
285	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
286	C₃×C₄	C₅×#131	1	4	yes	complemented
287	C₃×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
288	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
289	Dic₃	—	2	4	no	—
290	C₂×C₅	—	2	2	no	—
291	C₂×C₅	—	12	4	no	—
292	C₂×C₅	C₂×#893	1	1	yes	characteristic
293	C₂×C₅	C₂×#907	1	2	yes	—
294	C₂³	—	3	4	no	—
295	C₂³	C₂×C₄×C₅×D₃	1	1	yes	characteristic
296	Q₈	—	6	4	no	—
297	D₄	—	6	8	no	—
298	D₄	—	6	4	no	—
299	D₄	—	6	8	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	3	4	no	—
302	C₂×C₄	—	6	4	no	—
303	C₂×C₄	—	6	4	no	—
304	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
305	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
306	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
307	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
308	C₂×C₄	C₅×#239	1	2	yes	—
309	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
310	C₈	—	2	4	no	—
311	C₈	—	3	2	no	—
312	C₈	—	3	2	no	—
313	C₈	—	6	2	no	—
314	C₂×C₃	—	2	2	no	—
315	C₂×C₃	C₂×C₄×C₅×D₄	1	1	yes	characteristic
316	C₂×C₃	C₂×C₅×#131	1	2	yes	—
317	D₃	—	4	4	no	—
318	C₅	#79041	1	1	yes	characteristic, direct product factor, Sylow
319	C₂²	—	2	2	no	—
320	C₂²	—	6	8	no	—
321	C₂²	—	6	4	no	—
322	C₂²	C₅×#893	1	2	yes	—
323	C₂²	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	6	4	no	—
326	C₄	C₅×#893	1	2	yes	—
327	C₄	C₅×#907	1	4	yes	complemented
328	C₄	C₂×C₅×D₃×D₄	1	1	yes	characteristic
329	C₄	C₂×C₅×#239	1	1	yes	characteristic
330	C₃	C₅×#2931	1	1	yes	characteristic, complemented, Sylow
331	C₂	—	2	2	no	—
332	C₂	—	12	4	no	—
333	C₂	C₂×C₅×#893	1	1	yes	characteristic
334	C₂	C₂×C₅×#907	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	C₂³×D₃
commutator	C₂²×C₃	C₂³×C₄×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₃×C₅×#423	C₂
socle	C₂²×C₃×C₅	C₂³×C₄
prime radical	C₅×#5477	C₂
cosocle	C₂×C₃×C₄	C₂⁴×C₅
hypercenter	C₅×#423	D₃
hypocenter	C₃	C₅×#2931
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₅×#423

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#907	C₂	semidirect
3. C₃×C₅×#423	C₂	semidirect
4. C₅×#5129	C₂	semidirect
5. C₂×C₄×C₅×C₃⋊C₈	C₂	semidirect
6. C₅×#4838	C₂	semidirect
7. C₅×#4834	C₂	semidirect
26. #79041	C₅	direct (N × H)
31. C₅×#423	D₃	semidirect
51. C₂×C₅×C₃⋊C₈	D₄	semidirect
57. C₂×#907	C₂×C₅	semidirect
59. C₃×#423	C₂×C₅	semidirect
60. #5129	C₂×C₅	semidirect
61. C₂×C₄×C₃⋊C₈	C₂×C₅	semidirect
62. #4838	C₂×C₅	semidirect
63. #4834	C₂×C₅	semidirect
146. #423	C₅×D₃	semidirect
156. C₃×C₄×C₅	#131	semidirect
177. C₂×C₃⋊C₈	C₅×D₄	semidirect
252. C₄×C₅	#907	semidirect
278. C₃×C₅	#2931	semidirect
286. C₃×C₄	C₅×#131	semidirect
318. C₅	#79041	direct (N × H)
327. C₄	C₅×#907	semidirect
330. C₃	C₅×#2931	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#907	2
C₅×#5477	2
C₃×C₅×#423	2
C₅×#5129	2
C₂×C₄×C₅×C₃⋊C₈	2
C₅×#4838	2
C₅×#4834	2
C₅×#2931	3
#79041	5

Subgroups — C5×#79041 #14415229

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#79041 #14415229