Subgroups — C₅×#19440 #11942160

Order 1280.

656 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3226	C₂	1	1	yes	maximal, characteristic
2	C₅×#3037	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#462	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2983	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1702	C₂	1	4	yes	maximal, complemented
6	C₅×#1694	C₂	1	4	yes	maximal, complemented
7	C₅×#1570	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1546	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1540	C₂	1	1	yes	maximal, characteristic
10	C₂×C₅×#137	—	2	1	no	—
11	C₂²×C₅×Q₁₆	—	2	1	no	—
12	C₅×#532	C₂²	1	1	yes	characteristic
13	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
15	C₅×#500	C₂²	1	2	yes	—
16	C₅×#486	C₂²	1	2	yes	—
17	C₅×#480	—	2	2	no	—
18	C₅×#474	—	2	2	no	—
19	C₅×#464	—	2	2	no	—
20	C₅×#462	—	2	2	no	—
21	C₅×#462	C₂²	1	4	yes	—
22	C₅×#462	C₂²	1	2	yes	—
23	C₅×#460	C₂²	1	2	yes	—
24	C₅×#446	—	2	2	no	—
25	C₅×#446	C₂²	1	4	yes	—
26	C₅×#432	—	2	2	no	—
27	C₅×#431	C₂²	1	1	yes	characteristic
28	C₅×#431	C₂²	1	1	yes	characteristic
29	C₅×#423	C₂²	1	1	yes	characteristic
30	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₅×#417	—	2	1	no	—
32	C₂×C₅×#103	—	2	1	no	—
33	C₅×#334	C₂²	1	4	yes	—
34	C₅×#332	C₂²	1	2	yes	—
35	C₅×#332	C₂²	1	4	yes	—
36	C₅×#331	C₂²	1	2	yes	—
37	#19440	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₂×C₅×#41	—	2	1	no	—
39	C₂²×C₅×Q₈	—	2	1	no	—
40	C₂²×C₅×Q₈	D₄	1	1	yes	characteristic
41	C₅×#137	—	4	4	no	—
42	C₂×C₅×Q₁₆	—	2	4	no	—
43	C₂×C₅×Q₁₆	—	2	2	no	—
44	C₂×C₅×Q₁₆	—	4	4	no	—
45	C₂×C₅×SD₁₆	—	2	2	no	—
46	C₂×C₅×M₁₆	—	2	1	no	—
47	C₂²×C₅×C₈	—	2	1	no	—
48	C₅×C₄⋊Q₈	—	2	1	no	—
49	C₅×C₄⋊Q₈	—	2	1	no	—
50	C₅×C₄⋊Q₈	—	2	1	no	—
51	C₅×C₄⋊Q₈	C₂³	1	2	yes	—
52	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
53	C₅×#127	C₂³	1	1	yes	characteristic
54	C₅×#124	—	2	1	no	—
55	C₅×#121	—	2	1	no	—
56	C₅×#121	—	2	1	no	—
57	C₄×C₅×Q₈	—	2	1	no	—
58	C₄×C₅×Q₈	—	2	1	no	—
59	C₄×C₅×Q₈	D₄	1	2	yes	—
60	C₄×C₅×Q₈	C₂³	1	2	yes	—
61	C₄×C₅×D₄	—	2	1	no	—
62	C₄×C₅×D₄	—	2	1	no	—
63	C₂×C₅×C₄⋊C₄	—	2	1	no	—
64	C₂×C₅×C₄⋊C₄	—	2	1	no	—
65	C₂×C₅×C₄⋊C₄	—	2	1	no	—
66	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
67	C₂×C₄²×C₅	—	2	1	no	—
68	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
69	C₅×#107	—	2	4	no	—
70	C₅×C₄⋊C₈	—	2	4	no	—
71	C₅×C₄⋊C₈	—	2	1	no	—
72	C₅×C₄⋊C₈	—	2	1	no	—
73	C₅×C₄⋊C₈	D₄	1	2	yes	—
74	C₅×C₄⋊C₈	C₂³	1	2	yes	—
75	C₅×C₄⋊C₈	C₂³	1	2	yes	—
76	C₅×C₄⋊C₈	C₂³	1	2	yes	—
77	C₅×#103	—	2	4	no	—
78	C₅×#103	—	2	2	no	—
79	C₅×#102	—	2	4	no	—
80	C₅×#102	—	2	2	no	—
81	C₅×C₂²⋊C₈	—	2	1	no	—
82	C₅×C₂²⋊C₈	D₄	1	2	yes	—
83	C₅×#97	—	2	4	no	—
84	C₄×C₅×C₈	—	2	1	no	—
85	C₄×C₅×C₈	C₂³	1	2	yes	—
86	#3226	C₂×C₅	1	1	yes	characteristic
87	#3037	C₂×C₅	1	1	yes	characteristic
88	C₂×#462	C₂×C₅	1	1	yes	characteristic, complemented
89	#2983	C₂×C₅	1	1	yes	characteristic, complemented
90	#1702	C₂×C₅	1	4	yes	complemented
91	#1694	C₂×C₅	1	4	yes	complemented
92	#1570	C₂×C₅	1	1	yes	characteristic, complemented
93	#1546	C₂×C₅	1	1	yes	characteristic, complemented
94	#1540	C₂×C₅	1	1	yes	characteristic
95	C₅×#41	—	4	2	no	—
96	C₅×#41	—	4	2	no	—
97	C₂×C₅×Q₈	—	2	1	no	—
98	C₂×C₅×Q₈	—	2	2	no	—
99	C₂×C₅×Q₈	—	2	2	no	—
100	C₂×C₅×Q₈	—	2	1	no	—
101	C₂×C₅×Q₈	—	2	2	no	—
102	C₂×C₅×Q₈	—	4	1	no	—
103	C₂×C₅×Q₈	—	4	2	no	—
104	C₂×C₅×Q₈	—	4	2	no	—
105	C₂×C₅×Q₈	C₂×D₄	1	2	yes	—
106	C₂×C₅×D₄	—	2	1	no	—
107	C₂×C₅×D₄	—	2	1	no	—
108	C₂×C₅×D₄	—	2	1	no	—
109	C₂×C₅×D₄	—	4	1	no	—
110	C₂²×C₄×C₅	—	2	1	no	—
111	C₂²×C₄×C₅	—	2	1	no	—
112	C₂²×C₄×C₅	—	4	1	no	—
113	C₂²×C₄×C₅	—	4	1	no	—
114	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₅	#41	1	1	yes	characteristic
117	C₅×Q₁₆	—	4	8	no	—
118	C₅×Q₁₆	—	4	4	no	—
119	C₅×SD₁₆	—	4	4	no	—
120	C₅×M₁₆	—	4	2	no	—
121	C₂×C₅×C₈	—	2	2	no	—
122	C₂×C₅×C₈	—	2	2	no	—
123	C₂×C₅×C₈	—	2	2	no	—
124	C₂×C₅×C₈	—	4	2	no	—
125	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
126	C₅×C₄⋊C₄	—	2	2	no	—
127	C₅×C₄⋊C₄	—	2	1	no	—
128	C₅×C₄⋊C₄	—	2	2	no	—
129	C₅×C₄⋊C₄	—	2	2	no	—
130	C₅×C₄⋊C₄	—	2	1	no	—
131	C₅×C₄⋊C₄	—	2	2	no	—
132	C₅×C₄⋊C₄	—	2	1	no	—
133	C₅×C₄⋊C₄	—	4	1	no	—
134	C₅×C₄⋊C₄	—	4	1	no	—
135	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
136	C₅×C₂²⋊C₄	—	4	1	no	—
137	C₅×C₂²⋊C₄	—	4	1	no	—
138	C₄²×C₅	—	2	2	no	—
139	C₄²×C₅	—	2	2	no	—
140	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
141	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
142	C₄²×C₅	#41	1	1	yes	characteristic
143	C₄²×C₅	C₂⁴	1	1	yes	characteristic
144	C₂×#137	—	2	1	no	—
145	C₂²×Q₁₆	—	2	1	no	—
146	#532	C₂²×C₅	1	1	yes	characteristic
147	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
148	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
149	#500	C₂²×C₅	1	2	yes	—
150	#486	C₂²×C₅	1	2	yes	—
151	#480	—	2	2	no	—
152	#474	—	2	2	no	—
153	#464	—	2	2	no	—
154	#462	—	2	2	no	—
155	#462	C₂²×C₅	1	2	yes	—
156	#462	C₂²×C₅	1	4	yes	—
157	#460	C₂²×C₅	1	2	yes	—
158	#446	—	2	2	no	—
159	#446	C₂²×C₅	1	4	yes	—
160	#432	—	2	2	no	—
161	#431	C₂²×C₅	1	1	yes	characteristic
162	#431	C₂²×C₅	1	1	yes	characteristic
163	#423	C₂²×C₅	1	1	yes	characteristic
164	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
165	#417	—	2	1	no	—
166	C₂×#103	—	2	1	no	—
167	#334	C₂²×C₅	1	4	yes	—
168	#332	C₂²×C₅	1	2	yes	—
169	#332	C₂²×C₅	1	4	yes	—
170	#331	C₂²×C₅	1	2	yes	—
171	C₂³×C₅	—	4	1	no	—
172	C₂³×C₅	#121	1	1	yes	characteristic
173	C₅×Q₈	—	2	2	no	—
174	C₅×Q₈	—	2	2	no	—
175	C₅×Q₈	—	4	2	no	—
176	C₅×Q₈	—	4	2	no	—
177	C₅×Q₈	—	4	1	no	—
178	C₅×Q₈	—	4	2	no	—
179	C₅×Q₈	—	4	1	no	—
180	C₅×Q₈	—	8	2	no	—
181	C₅×D₄	—	4	1	no	—
182	C₅×D₄	—	4	1	no	—
183	C₅×D₄	—	4	2	no	—
184	C₅×D₄	—	8	1	no	—
185	C₅×D₄	—	8	2	no	—
186	C₂×C₄×C₅	—	2	2	no	—
187	C₂×C₄×C₅	—	2	2	no	—
188	C₂×C₄×C₅	—	2	2	no	—
189	C₂×C₄×C₅	—	4	1	no	—
190	C₂×C₄×C₅	—	4	2	no	—
191	C₂×C₄×C₅	—	4	1	no	—
192	C₂×C₄×C₅	—	4	2	no	—
193	C₂×C₄×C₅	—	8	1	no	—
194	C₂×C₄×C₅	—	8	2	no	—
195	C₂×C₄×C₅	—	8	2	no	—
196	C₂×C₄×C₅	#121	1	1	yes	characteristic
197	C₂×C₄×C₅	#121	1	1	yes	characteristic
198	C₂×C₄×C₅	#121	1	1	yes	characteristic
199	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
200	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
201	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
202	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
203	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
204	C₅×C₈	—	2	4	no	—
205	C₅×C₈	—	4	2	no	—
206	C₅×C₈	—	4	2	no	—
207	C₅×C₈	—	4	2	no	—
208	C₂×#41	—	2	1	no	—
209	C₂²×Q₈	—	2	1	no	—
210	C₂²×Q₈	C₅×D₄	1	1	yes	characteristic
211	#137	—	4	4	no	—
212	C₂×Q₁₆	—	2	4	no	—
213	C₂×Q₁₆	—	2	2	no	—
214	C₂×Q₁₆	—	4	4	no	—
215	C₂×SD₁₆	—	2	2	no	—
216	C₂×M₁₆	—	2	1	no	—
217	C₂²×C₈	—	2	1	no	—
218	C₄⋊Q₈	—	2	1	no	—
219	C₄⋊Q₈	—	2	1	no	—
220	C₄⋊Q₈	—	2	1	no	—
221	C₄⋊Q₈	C₂³×C₅	1	2	yes	—
222	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
223	#127	C₂³×C₅	1	1	yes	characteristic
224	#124	—	2	1	no	—
225	#121	—	2	1	no	—
226	#121	—	2	1	no	—
227	C₄×Q₈	—	2	1	no	—
228	C₄×Q₈	—	2	1	no	—
229	C₄×Q₈	C₅×D₄	1	2	yes	—
230	C₄×Q₈	C₂³×C₅	1	2	yes	—
231	C₄×D₄	—	2	1	no	—
232	C₄×D₄	—	2	1	no	—
233	C₂×C₄⋊C₄	—	2	1	no	—
234	C₂×C₄⋊C₄	—	2	1	no	—
235	C₂×C₄⋊C₄	—	2	1	no	—
236	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
237	C₂×C₄²	—	2	1	no	—
238	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
239	#107	—	2	4	no	—
240	C₄⋊C₈	—	2	4	no	—
241	C₄⋊C₈	—	2	1	no	—
242	C₄⋊C₈	—	2	1	no	—
243	C₄⋊C₈	C₅×D₄	1	2	yes	—
244	C₄⋊C₈	C₂³×C₅	1	2	yes	—
245	C₄⋊C₈	C₂³×C₅	1	2	yes	—
246	C₄⋊C₈	C₂³×C₅	1	2	yes	—
247	#103	—	2	4	no	—
248	#103	—	2	2	no	—
249	#102	—	2	4	no	—
250	#102	—	2	2	no	—
251	C₂²⋊C₈	—	2	1	no	—
252	C₂²⋊C₈	C₅×D₄	1	2	yes	—
253	#97	—	2	4	no	—
254	C₄×C₈	—	2	1	no	—
255	C₄×C₈	C₂³×C₅	1	2	yes	—
256	C₂²×C₅	—	2	1	no	—
257	C₂²×C₅	—	2	1	no	—
258	C₂²×C₅	—	8	1	no	—
259	C₂²×C₅	—	8	1	no	—
260	C₂²×C₅	—	8	1	no	—
261	C₂²×C₅	#469	1	2	yes	—
262	C₂²×C₅	C₂×#121	1	1	yes	characteristic
263	C₄×C₅	—	2	1	no	—
264	C₄×C₅	—	2	1	no	—
265	C₄×C₅	—	2	1	no	—
266	C₄×C₅	—	4	2	no	—
267	C₄×C₅	—	4	2	no	—
268	C₄×C₅	—	8	1	no	—
269	C₄×C₅	—	8	2	no	—
270	C₄×C₅	#469	1	2	yes	—
271	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
272	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
273	C₄×C₅	#578	1	2	yes	—
274	#41	—	4	2	no	—
275	#41	—	4	2	no	—
276	C₂×Q₈	—	2	1	no	—
277	C₂×Q₈	—	2	2	no	—
278	C₂×Q₈	—	2	2	no	—
279	C₂×Q₈	—	2	1	no	—
280	C₂×Q₈	—	2	2	no	—
281	C₂×Q₈	—	4	1	no	—
282	C₂×Q₈	—	4	2	no	—
283	C₂×Q₈	—	4	2	no	—
284	C₂×Q₈	C₂×C₅×D₄	1	2	yes	—
285	C₂×D₄	—	2	1	no	—
286	C₂×D₄	—	2	1	no	—
287	C₂×D₄	—	2	1	no	—
288	C₂×D₄	—	4	1	no	—
289	C₂²×C₄	—	2	1	no	—
290	C₂²×C₄	—	2	1	no	—
291	C₂²×C₄	—	4	1	no	—
292	C₂²×C₄	—	4	1	no	—
293	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
294	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
295	C₂²×C₄	C₅×#41	1	1	yes	characteristic
296	Q₁₆	—	4	8	no	—
297	Q₁₆	—	4	4	no	—
298	SD₁₆	—	4	4	no	—
299	M₁₆	—	4	2	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	4	2	no	—
304	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	2	1	no	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	2	1	no	—
312	C₄⋊C₄	—	4	1	no	—
313	C₄⋊C₄	—	4	1	no	—
314	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
315	C₂²⋊C₄	—	4	1	no	—
316	C₂²⋊C₄	—	4	1	no	—
317	C₄²	—	2	2	no	—
318	C₄²	—	2	2	no	—
319	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
320	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
321	C₄²	C₅×#41	1	1	yes	characteristic
322	C₄²	C₂⁴×C₅	1	1	yes	characteristic
323	C₂×C₅	—	2	2	no	—
324	C₂×C₅	—	16	1	no	—
325	C₂×C₅	#3044	1	1	yes	characteristic
326	C₂×C₅	C₂×#469	1	1	yes	characteristic
327	C₂×C₅	#3069	1	1	yes	characteristic
328	C₂³	—	4	1	no	—
329	C₂³	C₅×#121	1	1	yes	characteristic
330	Q₈	—	2	2	no	—
331	Q₈	—	2	2	no	—
332	Q₈	—	4	2	no	—
333	Q₈	—	4	2	no	—
334	Q₈	—	4	1	no	—
335	Q₈	—	4	2	no	—
336	Q₈	—	4	1	no	—
337	Q₈	—	8	2	no	—
338	D₄	—	4	1	no	—
339	D₄	—	4	1	no	—
340	D₄	—	4	2	no	—
341	D₄	—	8	1	no	—
342	D₄	—	8	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	8	1	no	—
351	C₂×C₄	—	8	2	no	—
352	C₂×C₄	—	8	2	no	—
353	C₂×C₄	C₅×#121	1	1	yes	characteristic
354	C₂×C₄	C₅×#121	1	1	yes	characteristic
355	C₂×C₄	C₅×#121	1	1	yes	characteristic
356	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
357	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
358	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
359	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
360	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
361	C₈	—	2	4	no	—
362	C₈	—	4	2	no	—
363	C₈	—	4	2	no	—
364	C₈	—	4	2	no	—
365	C₅	#19440	1	1	yes	characteristic, direct product factor, Sylow
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	8	1	no	—
369	C₂²	—	8	1	no	—
370	C₂²	—	8	1	no	—
371	C₂²	C₅×#469	1	2	yes	—
372	C₂²	C₂×C₅×#121	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	4	2	no	—
377	C₄	—	4	2	no	—
378	C₄	—	8	1	no	—
379	C₄	—	8	2	no	—
380	C₄	C₅×#469	1	2	yes	—
381	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
382	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
383	C₄	C₅×#578	1	2	yes	—
384	C₂	—	2	2	no	—
385	C₂	—	16	1	no	—
386	C₂	C₅×#3044	1	1	yes	characteristic
387	C₂	C₂×C₅×#469	1	1	yes	characteristic
388	C₂	C₅×#3069	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
prime radical	C₅×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₅×#462	C₂	semidirect
4. C₅×#2983	C₂	semidirect
5. C₅×#1702	C₂	semidirect
6. C₅×#1694	C₂	semidirect
7. C₅×#1570	C₂	semidirect
8. C₅×#1546	C₂	semidirect
37. #19440	C₅	direct (N × H)
88. C₂×#462	C₂×C₅	semidirect
89. #2983	C₂×C₅	semidirect
90. #1702	C₂×C₅	semidirect
91. #1694	C₂×C₅	semidirect
92. #1570	C₂×C₅	semidirect
93. #1546	C₂×C₅	semidirect
365. C₅	#19440	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#3226	2
C₅×#3037	2
C₂×C₅×#462	2
C₅×#2983	2
C₅×#1702	2
C₅×#1694	2
C₅×#1570	2
C₅×#1546	2
C₅×#1540	2
#19440	5

Subgroups — C5×#19440 #11942160

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#19440 #11942160