Subgroups — C₅×#68488 #14285359

Order 1920.

404 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5728	C₂	1	1	yes	maximal, characteristic
2	C₅×#4837	C₂	1	1	yes	maximal, characteristic
3	C₅×#4835	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#333	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4546	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4486	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4482	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1559	—	3	1	no	maximal, Hall
9	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₅×#946	C₄	1	2	yes	—
11	C₅×Q₈×Dic₃	C₄	1	2	yes	—
12	C₂×C₅×ΓL(1,𝔽₂₅)	—	2	1	no	—
13	C₅×#897	—	2	1	no	—
14	C₅×#892	—	2	1	no	—
15	C₅×#890	—	2	1	no	—
16	C₅×#887	—	2	1	no	—
17	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
18	C₅×#869	C₂²	1	1	yes	characteristic
19	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₅×(C₂×D₃)⋊C₈	—	2	1	no	—
22	C₅×#815	—	2	1	no	—
23	C₅×Dic₃⋊C₈	—	2	1	no	—
24	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
25	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
26	#68488	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
27	C₅×#533	—	3	1	no	—
28	C₅×#432	—	3	1	no	—
29	C₅×#423	—	3	1	no	—
30	C₅×#333	—	3	1	no	—
31	C₅×#333	D₃	1	1	yes	characteristic, complemented
32	C₅×#332	—	3	1	no	—
33	C₅×#332	—	3	1	no	—
34	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
35	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
37	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
38	C₂×C₅×Dic₆	—	2	1	no	—
39	C₂×C₃×C₅×C₈	—	2	1	no	—
40	C₂×C₃×C₅×C₈	—	2	1	no	—
41	C₃×C₅×C₄⋊C₄	—	2	1	no	—
42	C₃×C₅×C₄⋊C₄	—	2	1	no	—
43	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
45	C₅×C₄⋊Dic₃	—	2	1	no	—
46	C₅×C₄⋊Dic₃	—	2	2	no	—
47	C₅×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
48	C₅×Dic₃⋊C₄	—	2	2	no	—
49	C₄×C₅×Dic₃	—	2	2	no	—
50	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
52	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
53	C₂×C₅×C₃⋊C₈	—	2	1	no	—
54	C₂×C₅×C₃⋊C₈	—	2	1	no	—
55	C₅×ΓL(1,𝔽₂₅)	—	4	2	no	—
56	#5728	C₂×C₅	1	1	yes	characteristic
57	#4837	C₂×C₅	1	1	yes	characteristic
58	#4835	C₂×C₅	1	1	yes	characteristic
59	C₃×#333	C₂×C₅	1	1	yes	characteristic, complemented
60	#4546	C₂×C₅	1	1	yes	characteristic, complemented
61	#4486	C₂×C₅	1	1	yes	characteristic, complemented
62	#4482	C₂×C₅	1	1	yes	characteristic, complemented
63	C₂×C₅×M₁₆	—	6	1	no	—
64	C₅×C₄⋊Q₈	—	3	1	no	—
65	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
66	C₅×#125	—	6	1	no	—
67	C₅×C₂²⋊Q₈	—	3	2	no	—
68	C₅×C₂²⋊Q₈	—	6	1	no	—
69	C₄×C₅×Q₈	—	3	2	no	—
70	C₄×C₅×Q₈	—	6	1	no	—
71	C₅×#117	—	6	1	no	—
72	C₂×C₄²×C₅	—	3	1	no	—
73	C₅×C₄⋊C₈	—	3	1	no	—
74	C₅×C₄⋊C₈	—	3	1	no	—
75	C₅×C₄⋊C₈	—	6	1	no	—
76	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
77	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
78	C₅×C₂²⋊C₈	—	6	1	no	—
79	C₅×#97	—	6	1	no	—
80	#1559	—	3	1	no	Sylow
81	C₂²×C₅×D₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₃×C₅×Q₈	—	4	2	no	—
83	C₂×C₃×C₄×C₅	—	2	2	no	—
84	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂×C₅×Dic₃	—	2	2	no	—
88	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
91	C₄×C₅×D₃	—	2	1	no	—
92	C₄×C₅×D₃	—	2	1	no	—
93	C₄×C₅×D₃	—	2	1	no	—
94	C₄×C₅×D₃	—	2	1	no	—
95	C₄×C₅×D₃	—	4	1	no	—
96	C₅×Dic₆	—	4	1	no	—
97	C₅×Dic₆	—	4	1	no	—
98	C₃×C₅×C₈	—	4	1	no	—
99	C₃×C₅×C₈	—	4	1	no	—
100	C₅×C₃⋊C₈	—	4	1	no	—
101	C₅×C₃⋊C₈	—	4	1	no	—
102	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
103	#946	C₄×C₅	1	2	yes	—
104	Q₈×Dic₃	C₄×C₅	1	2	yes	—
105	C₂×ΓL(1,𝔽₂₅)	—	2	1	no	—
106	#897	—	2	1	no	—
107	#892	—	2	1	no	—
108	#890	—	2	1	no	—
109	#887	—	2	1	no	—
110	C₄²×D₃	C₂²×C₅	1	1	yes	characteristic
111	#869	C₂²×C₅	1	1	yes	characteristic
112	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
113	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
114	(C₂×D₃)⋊C₈	—	2	1	no	—
115	#815	—	2	1	no	—
116	Dic₃⋊C₈	—	2	1	no	—
117	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
118	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
119	C₂×C₅×Q₈	—	6	1	no	—
120	C₂×C₅×Q₈	C₄×D₃	1	2	yes	—
121	C₂²×C₄×C₅	—	3	1	no	—
122	C₂²×C₄×C₅	—	3	1	no	—
123	C₂²×C₄×C₅	—	3	1	no	—
124	C₅×M₁₆	—	12	2	no	—
125	C₂×C₅×C₈	—	2	1	no	—
126	C₂×C₅×C₈	—	2	1	no	—
127	C₂×C₅×C₈	—	6	1	no	—
128	C₂×C₅×C₈	—	6	1	no	—
129	C₅×C₄⋊C₄	—	2	1	no	—
130	C₅×C₄⋊C₄	—	2	1	no	—
131	C₅×C₄⋊C₄	—	3	2	no	—
132	C₅×C₄⋊C₄	—	6	2	no	—
133	C₅×C₄⋊C₄	—	6	2	no	—
134	C₅×C₄⋊C₄	—	6	1	no	—
135	C₅×C₂²⋊C₄	—	6	2	no	—
136	C₄²×C₅	—	3	1	no	—
137	C₄²×C₅	—	3	1	no	—
138	C₄²×C₅	—	3	1	no	—
139	C₄²×C₅	—	6	2	no	—
140	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
141	#533	—	3	1	no	—
142	#432	—	3	1	no	—
143	#423	—	3	1	no	—
144	#333	—	3	1	no	—
145	#333	C₅×D₃	1	1	yes	characteristic, complemented
146	#332	—	3	1	no	—
147	#332	—	3	1	no	—
148	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₅×D₃	—	2	1	no	—
150	C₂×C₅×D₃	—	2	1	no	—
151	C₂×C₅×D₃	—	2	1	no	—
152	C₃×C₄×C₅	—	2	1	no	—
153	C₃×C₄×C₅	—	4	2	no	—
154	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
155	C₃×C₄×C₅	#137	1	1	yes	characteristic
156	C₃×C₄×C₅	#137	1	1	yes	characteristic
157	C₃×C₄×C₅	#137	1	1	yes	characteristic
158	C₅×Dic₃	—	2	1	no	—
159	C₅×Dic₃	—	2	1	no	—
160	C₅×Dic₃	—	2	1	no	—
161	C₅×Dic₃	—	4	2	no	—
162	C₂×C₃×Q₈	C₂×C₄×C₅	1	2	yes	—
163	C₂×C₄×D₃	C₂×C₄×C₅	1	1	yes	characteristic
164	C₂×C₄×D₃	C₅×D₄	1	1	yes	characteristic
165	C₂×C₄×D₃	C₅×D₄	1	1	yes	characteristic
166	C₂×Dic₆	—	2	1	no	—
167	C₂×C₃×C₈	—	2	1	no	—
168	C₂×C₃×C₈	—	2	1	no	—
169	C₃×C₄⋊C₄	—	2	1	no	—
170	C₃×C₄⋊C₄	—	2	1	no	—
171	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
172	(C₂×D₃)⋊C₄	—	2	2	no	—
173	C₄⋊Dic₃	—	2	1	no	—
174	C₄⋊Dic₃	—	2	2	no	—
175	C₄⋊Dic₃	C₂×C₄×C₅	1	2	yes	—
176	Dic₃⋊C₄	—	2	2	no	—
177	C₄×Dic₃	—	2	2	no	—
178	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
179	C₄×Dic₃	C₅×D₄	1	1	yes	characteristic
180	C₄×Dic₃	C₅×D₄	1	1	yes	characteristic
181	C₂×C₃⋊C₈	—	2	1	no	—
182	C₂×C₃⋊C₈	—	2	1	no	—
183	ΓL(1,𝔽₂₅)	—	4	2	no	—
184	C₂³×C₅	—	3	1	no	—
185	C₅×Q₈	—	4	2	no	—
186	C₅×Q₈	—	12	1	no	—
187	C₅×Q₈	—	12	1	no	—
188	C₂×C₄×C₅	—	2	2	no	—
189	C₂×C₄×C₅	—	3	1	no	—
190	C₂×C₄×C₅	—	3	1	no	—
191	C₂×C₄×C₅	—	3	1	no	—
192	C₂×C₄×C₅	—	6	1	no	—
193	C₂×C₄×C₅	—	6	1	no	—
194	C₂×C₄×C₅	—	6	1	no	—
195	C₂×C₄×C₅	—	6	1	no	—
196	C₂×C₄×C₅	—	6	2	no	—
197	C₂×C₄×C₅	—	12	1	no	—
198	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
199	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
200	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
201	C₅×C₈	—	4	1	no	—
202	C₅×C₈	—	4	1	no	—
203	C₅×C₈	—	12	1	no	—
204	C₅×C₈	—	12	1	no	—
205	C₂×M₁₆	—	6	1	no	—
206	C₄⋊Q₈	—	3	1	no	—
207	C₄⋊Q₈	C₂×C₅×D₃	1	1	yes	characteristic
208	#125	—	6	1	no	—
209	C₂²⋊Q₈	—	3	2	no	—
210	C₂²⋊Q₈	—	6	1	no	—
211	C₄×Q₈	—	3	2	no	—
212	C₄×Q₈	—	6	1	no	—
213	#117	—	6	1	no	—
214	C₂×C₄²	—	3	1	no	—
215	C₄⋊C₈	—	3	1	no	—
216	C₄⋊C₈	—	3	1	no	—
217	C₄⋊C₈	—	6	1	no	—
218	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
219	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
220	C₂²⋊C₈	—	6	1	no	—
221	#97	—	6	1	no	—
222	C₂×C₃×C₅	#413	1	1	yes	characteristic
223	C₂×C₃×C₅	#417	1	1	yes	characteristic
224	C₂×C₃×C₅	#419	1	1	yes	characteristic
225	C₅×D₃	—	4	1	no	—
226	C₂²×D₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
227	C₃×Q₈	—	4	2	no	—
228	C₂×C₃×C₄	—	2	2	no	—
229	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
230	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
231	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
232	C₂×Dic₃	—	2	2	no	—
233	C₂×Dic₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
234	C₂×Dic₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
235	C₂×Dic₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
236	C₄×D₃	—	2	1	no	—
237	C₄×D₃	—	2	1	no	—
238	C₄×D₃	—	2	1	no	—
239	C₄×D₃	—	2	1	no	—
240	C₄×D₃	—	4	1	no	—
241	Dic₆	—	4	1	no	—
242	Dic₆	—	4	1	no	—
243	C₃×C₈	—	4	1	no	—
244	C₃×C₈	—	4	1	no	—
245	C₃⋊C₈	—	4	1	no	—
246	C₃⋊C₈	—	4	1	no	—
247	C₂²×C₅	—	6	1	no	—
248	C₂²×C₅	—	6	1	no	—
249	C₂²×C₅	—	6	1	no	—
250	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
251	C₄×C₅	—	2	1	no	—
252	C₄×C₅	—	4	2	no	—
253	C₄×C₅	—	6	1	no	—
254	C₄×C₅	—	6	1	no	—
255	C₄×C₅	—	6	1	no	—
256	C₄×C₅	—	12	2	no	—
257	C₄×C₅	#914	1	1	yes	characteristic
258	C₄×C₅	#915	1	1	yes	characteristic
259	C₄×C₅	#918	1	1	yes	characteristic
260	C₄×C₅	#918	1	1	yes	characteristic
261	C₂×Q₈	—	6	1	no	—
262	C₂×Q₈	C₄×C₅×D₃	1	2	yes	—
263	C₂²×C₄	—	3	1	no	—
264	C₂²×C₄	—	3	1	no	—
265	C₂²×C₄	—	3	1	no	—
266	M₁₆	—	12	2	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	—	2	1	no	—
269	C₂×C₈	—	6	1	no	—
270	C₂×C₈	—	6	1	no	—
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	3	2	no	—
274	C₄⋊C₄	—	6	2	no	—
275	C₄⋊C₄	—	6	2	no	—
276	C₄⋊C₄	—	6	1	no	—
277	C₂²⋊C₄	—	6	2	no	—
278	C₄²	—	3	1	no	—
279	C₄²	—	3	1	no	—
280	C₄²	—	3	1	no	—
281	C₄²	—	6	2	no	—
282	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
283	C₃×C₅	#1559	1	1	yes	characteristic, complemented, Hall
284	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
285	C₂×D₃	—	2	1	no	—
286	C₂×D₃	—	2	1	no	—
287	C₂×D₃	—	2	1	no	—
288	C₃×C₄	—	2	1	no	—
289	C₃×C₄	—	4	2	no	—
290	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
291	C₃×C₄	C₅×#137	1	1	yes	characteristic
292	C₃×C₄	C₅×#137	1	1	yes	characteristic
293	C₃×C₄	C₅×#137	1	1	yes	characteristic
294	Dic₃	—	2	1	no	—
295	Dic₃	—	2	1	no	—
296	Dic₃	—	2	1	no	—
297	Dic₃	—	4	2	no	—
298	C₂×C₅	—	12	1	no	—
299	C₂×C₅	#4745	1	1	yes	characteristic
300	C₂×C₅	#4802	1	1	yes	characteristic
301	C₂×C₅	#4803	1	1	yes	characteristic
302	C₂³	—	3	1	no	—
303	Q₈	—	4	2	no	—
304	Q₈	—	12	1	no	—
305	Q₈	—	12	1	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	3	1	no	—
308	C₂×C₄	—	3	1	no	—
309	C₂×C₄	—	3	1	no	—
310	C₂×C₄	—	6	1	no	—
311	C₂×C₄	—	6	1	no	—
312	C₂×C₄	—	6	1	no	—
313	C₂×C₄	—	6	1	no	—
314	C₂×C₄	—	6	2	no	—
315	C₂×C₄	—	12	1	no	—
316	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
317	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
318	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
319	C₈	—	4	1	no	—
320	C₈	—	4	1	no	—
321	C₈	—	12	1	no	—
322	C₈	—	12	1	no	—
323	C₂×C₃	C₅×#413	1	1	yes	characteristic
324	C₂×C₃	C₅×#417	1	1	yes	characteristic
325	C₂×C₃	C₅×#419	1	1	yes	characteristic
326	D₃	—	4	1	no	—
327	C₅	#68488	1	1	yes	characteristic, direct product factor, Sylow
328	C₂²	—	6	1	no	—
329	C₂²	—	6	1	no	—
330	C₂²	—	6	1	no	—
331	C₂²	C₅×D₃×C₂²⋊C₄	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	4	2	no	—
334	C₄	—	6	1	no	—
335	C₄	—	6	1	no	—
336	C₄	—	6	1	no	—
337	C₄	—	12	2	no	—
338	C₄	C₅×#914	1	1	yes	characteristic
339	C₄	C₅×#915	1	1	yes	characteristic
340	C₄	C₅×#918	1	1	yes	characteristic
341	C₄	C₅×#918	1	1	yes	characteristic
342	C₃	C₅×#1559	1	1	yes	characteristic, complemented, Sylow
343	C₂	—	12	1	no	—
344	C₂	C₅×#4745	1	1	yes	characteristic
345	C₂	C₅×#4802	1	1	yes	characteristic
346	C₂	C₅×#4803	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	D₃×C₂²⋊C₄
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#333	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×C₅×D₃	C₂²⋊C₄
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×#333	D₃
hypocenter	C₃	C₅×#1559
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#333

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×#333	C₂	semidirect
5. C₅×#4546	C₂	semidirect
6. C₅×#4486	C₂	semidirect
7. C₅×#4482	C₂	semidirect
26. #68488	C₅	direct (N × H)
31. C₅×#333	D₃	semidirect
59. C₃×#333	C₂×C₅	semidirect
60. #4546	C₂×C₅	semidirect
61. #4486	C₂×C₅	semidirect
62. #4482	C₂×C₅	semidirect
145. #333	C₅×D₃	semidirect
283. C₃×C₅	#1559	semidirect
327. C₅	#68488	direct (N × H)
342. C₃	C₅×#1559	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5728	2
C₅×#4837	2
C₅×#4835	2
C₃×C₅×#333	2
C₅×#4546	2
C₅×#4486	2
C₅×#4482	2
C₅×#1559	3
#68488	5

Subgroups — C5×#68488 #14285359

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68488 #14285359