Subgroups — C₅×#19281 #11942001

Order 1280.

658 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3170	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#3077	C₂	1	2	yes	maximal
3	C₂×C₅×#436	C₂	1	1	yes	maximal, characteristic
4	C₅×#1703	C₂	1	4	yes	maximal, complemented
5	C₅×#1695	C₂	1	2	yes	maximal, complemented
6	C₅×#1689	C₂	1	2	yes	maximal, complemented
7	C₅×#1566	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1550	C₂	1	2	yes	maximal, complemented
9	C₂²×C₅×M₁₆	—	2	1	no	—
10	C₅×#533	C₂²	1	2	yes	—
11	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
12	C₅×#500	C₂²	1	1	yes	characteristic, complemented
13	C₅×#494	C₂²	1	1	yes	characteristic, complemented
14	C₅×#487	C₂²	1	2	yes	complemented
15	C₅×#478	—	2	4	no	—
16	C₅×#476	C₂²	1	4	yes	—
17	C₅×#474	C₂²	1	4	yes	—
18	C₅×#467	—	2	1	no	—
19	C₅×#467	—	2	1	no	—
20	C₄×C₅×Q₁₆	—	2	1	no	—
21	C₄×C₅×Q₁₆	—	2	1	no	—
22	C₅×#436	—	2	1	no	—
23	C₅×#436	—	2	1	no	—
24	C₅×#436	C₂²	1	2	yes	—
25	C₅×#436	C₂²	1	2	yes	—
26	C₅×#428	—	2	2	no	—
27	C₅×#423	C₂²	1	2	yes	—
28	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₅×#418	—	2	2	no	—
30	C₂×C₅×C₂²⋊C₈	—	2	1	no	—
31	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
32	C₅×#334	C₂²	1	2	yes	complemented
33	C₅×#334	C₂²	1	2	yes	complemented
34	C₅×#333	C₂²	1	4	yes	complemented
35	C₅×#332	C₂²	1	4	yes	complemented
36	#19281	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂²×C₅×D₄	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₅	—	2	1	no	—
39	C₂×C₅×Q₁₆	—	2	1	no	—
40	C₂×C₅×Q₁₆	—	2	1	no	—
41	C₂×C₅×M₁₆	—	2	2	no	—
42	C₂×C₅×M₁₆	—	2	2	no	—
43	C₂×C₅×M₁₆	—	2	2	no	—
44	C₂×C₅×M₁₆	—	4	4	no	—
45	C₂²×C₅×C₈	—	2	1	no	—
46	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
47	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
48	C₅×C₄⋊Q₈	C₂³	1	2	yes	—
49	C₅×C₄⋊Q₈	C₂³	1	2	yes	—
50	C₅×#125	—	2	2	no	—
51	C₅×C₂²⋊Q₈	—	2	2	no	—
52	C₅×C₂²⋊Q₈	—	2	2	no	—
53	C₄×C₅×Q₈	—	2	2	no	—
54	C₄×C₅×Q₈	—	2	2	no	—
55	C₄×C₅×D₄	—	2	1	no	—
56	C₄×C₅×D₄	—	2	1	no	—
57	C₄×C₅×D₄	D₄	1	2	yes	—
58	C₄×C₅×D₄	C₂³	1	2	yes	—
59	C₅×#117	—	2	2	no	—
60	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
61	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
62	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
63	C₅×#107	—	2	1	no	—
64	C₅×#107	—	2	1	no	—
65	C₅×#107	—	2	4	no	—
66	C₅×#106	—	2	4	no	—
67	C₅×C₄⋊C₈	—	2	1	no	—
68	C₅×C₄⋊C₈	—	2	2	no	—
69	C₅×C₄⋊C₈	C₂³	1	2	yes	—
70	C₅×C₄⋊C₈	C₂³	1	4	yes	—
71	C₅×#103	—	2	2	no	—
72	C₅×#103	—	2	2	no	—
73	C₅×#102	—	2	4	no	—
74	C₅×#102	—	2	4	no	—
75	C₅×C₂²⋊C₈	—	2	2	no	—
76	C₅×C₂²⋊C₈	—	2	2	no	—
77	C₄×C₅×C₈	D₄	1	1	yes	characteristic
78	C₄×C₅×C₈	D₄	1	1	yes	characteristic
79	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
80	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
81	#3170	C₂×C₅	1	1	yes	characteristic, complemented
82	#3077	C₂×C₅	1	2	yes	—
83	C₂×#436	C₂×C₅	1	1	yes	characteristic
84	#1703	C₂×C₅	1	4	yes	complemented
85	#1695	C₂×C₅	1	2	yes	complemented
86	#1689	C₂×C₅	1	2	yes	complemented
87	#1566	C₂×C₅	1	1	yes	characteristic, complemented
88	#1550	C₂×C₅	1	2	yes	complemented
89	C₂⁴×C₅	—	2	1	no	—
90	C₂×C₅×Q₈	—	2	2	no	—
91	C₂×C₅×Q₈	—	2	2	no	—
92	C₂×C₅×D₄	—	2	1	no	—
93	C₂×C₅×D₄	—	2	4	no	—
94	C₂×C₅×D₄	C₂×D₄	1	2	yes	—
95	C₂²×C₄×C₅	—	2	1	no	—
96	C₂²×C₄×C₅	—	2	2	no	—
97	C₂²×C₄×C₅	—	2	2	no	—
98	C₂²×C₄×C₅	—	4	2	no	—
99	C₂²×C₄×C₅	—	4	2	no	—
100	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₅	#41	1	1	yes	characteristic
103	C₅×Q₁₆	—	4	2	no	—
104	C₅×Q₁₆	—	4	2	no	—
105	C₅×M₁₆	—	4	4	no	—
106	C₅×M₁₆	—	4	4	no	—
107	C₅×M₁₆	—	4	4	no	—
108	C₂×C₅×C₈	—	2	2	no	—
109	C₂×C₅×C₈	—	2	2	no	—
110	C₂×C₅×C₈	—	2	4	no	—
111	C₂×C₅×C₈	—	4	1	no	—
112	C₂×C₅×C₈	—	4	2	no	—
113	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
114	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
115	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
116	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
117	C₅×C₄⋊C₄	—	2	1	no	—
118	C₅×C₄⋊C₄	—	2	2	no	—
119	C₅×C₄⋊C₄	—	2	2	no	—
120	C₅×C₄⋊C₄	—	2	2	no	—
121	C₅×C₄⋊C₄	—	2	2	no	—
122	C₅×C₄⋊C₄	—	4	2	no	—
123	C₅×C₄⋊C₄	—	4	2	no	—
124	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
125	C₅×C₂²⋊C₄	—	2	2	no	—
126	C₅×C₂²⋊C₄	—	2	2	no	—
127	C₅×C₂²⋊C₄	—	4	2	no	—
128	C₄²×C₅	—	4	2	no	—
129	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
131	C₄²×C₅	#41	1	1	yes	characteristic
132	C₄²×C₅	C₂⁴	1	1	yes	characteristic
133	C₂²×M₁₆	—	2	1	no	—
134	#533	C₂²×C₅	1	2	yes	—
135	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
136	#500	C₂²×C₅	1	1	yes	characteristic, complemented
137	#494	C₂²×C₅	1	1	yes	characteristic, complemented
138	#487	C₂²×C₅	1	2	yes	complemented
139	#478	—	2	4	no	—
140	#476	C₂²×C₅	1	4	yes	—
141	#474	C₂²×C₅	1	4	yes	—
142	#467	—	2	1	no	—
143	#467	—	2	1	no	—
144	C₄×Q₁₆	—	2	1	no	—
145	C₄×Q₁₆	—	2	1	no	—
146	#436	—	2	1	no	—
147	#436	—	2	1	no	—
148	#436	C₂²×C₅	1	2	yes	—
149	#436	C₂²×C₅	1	2	yes	—
150	#428	—	2	2	no	—
151	#423	C₂²×C₅	1	2	yes	—
152	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
153	#418	—	2	2	no	—
154	C₂×C₂²⋊C₈	—	2	1	no	—
155	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
156	#334	C₂²×C₅	1	2	yes	complemented
157	#334	C₂²×C₅	1	2	yes	complemented
158	#333	C₂²×C₅	1	4	yes	complemented
159	#332	C₂²×C₅	1	4	yes	complemented
160	C₂³×C₅	—	2	2	no	—
161	C₂³×C₅	—	4	2	no	—
162	C₂³×C₅	—	4	2	no	—
163	C₂³×C₅	—	4	2	no	—
164	C₂³×C₅	#121	1	1	yes	characteristic
165	C₅×Q₈	—	4	2	no	—
166	C₅×Q₈	—	4	2	no	—
167	C₅×Q₈	—	8	2	no	—
168	C₅×D₄	—	2	4	no	—
169	C₅×D₄	—	4	2	no	—
170	C₂×C₄×C₅	—	2	1	no	—
171	C₂×C₄×C₅	—	2	1	no	—
172	C₂×C₄×C₅	—	2	2	no	—
173	C₂×C₄×C₅	—	4	1	no	—
174	C₂×C₄×C₅	—	4	4	no	—
175	C₂×C₄×C₅	—	4	2	no	—
176	C₂×C₄×C₅	—	4	2	no	—
177	C₂×C₄×C₅	—	4	2	no	—
178	C₂×C₄×C₅	—	4	2	no	—
179	C₂×C₄×C₅	—	4	2	no	—
180	C₂×C₄×C₅	#121	1	1	yes	characteristic
181	C₂×C₄×C₅	#121	1	1	yes	characteristic
182	C₂×C₄×C₅	#121	1	1	yes	characteristic
183	C₂×C₄×C₅	#137	1	4	yes	—
184	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
187	C₅×C₈	—	2	1	no	—
188	C₅×C₈	—	2	1	no	—
189	C₅×C₈	—	2	1	no	—
190	C₅×C₈	—	4	2	no	—
191	C₅×C₈	—	4	4	no	—
192	C₅×C₈	AGL(1,ℤ/8)	1	2	yes	—
193	C₂²×D₄	C₅×D₄	1	1	yes	characteristic
194	C₂³×C₄	—	2	1	no	—
195	C₂×Q₁₆	—	2	1	no	—
196	C₂×Q₁₆	—	2	1	no	—
197	C₂×M₁₆	—	2	2	no	—
198	C₂×M₁₆	—	2	2	no	—
199	C₂×M₁₆	—	2	2	no	—
200	C₂×M₁₆	—	4	4	no	—
201	C₂²×C₈	—	2	1	no	—
202	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
203	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
204	C₄⋊Q₈	C₂³×C₅	1	2	yes	—
205	C₄⋊Q₈	C₂³×C₅	1	2	yes	—
206	#125	—	2	2	no	—
207	C₂²⋊Q₈	—	2	2	no	—
208	C₂²⋊Q₈	—	2	2	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₄×Q₈	—	2	2	no	—
211	C₄×D₄	—	2	1	no	—
212	C₄×D₄	—	2	1	no	—
213	C₄×D₄	C₅×D₄	1	2	yes	—
214	C₄×D₄	C₂³×C₅	1	2	yes	—
215	#117	—	2	2	no	—
216	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
217	C₂×C₂²⋊C₄	—	2	1	no	—
218	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
219	#107	—	2	1	no	—
220	#107	—	2	1	no	—
221	#107	—	2	4	no	—
222	#106	—	2	4	no	—
223	C₄⋊C₈	—	2	1	no	—
224	C₄⋊C₈	—	2	2	no	—
225	C₄⋊C₈	C₂³×C₅	1	2	yes	—
226	C₄⋊C₈	C₂³×C₅	1	4	yes	—
227	#103	—	2	2	no	—
228	#103	—	2	2	no	—
229	#102	—	2	4	no	—
230	#102	—	2	4	no	—
231	C₂²⋊C₈	—	2	2	no	—
232	C₂²⋊C₈	—	2	2	no	—
233	C₄×C₈	C₅×D₄	1	1	yes	characteristic
234	C₄×C₈	C₅×D₄	1	1	yes	characteristic
235	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
236	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
237	C₂²×C₅	—	2	1	no	—
238	C₂²×C₅	—	2	1	no	—
239	C₂²×C₅	—	4	2	no	—
240	C₂²×C₅	—	4	2	no	—
241	C₂²×C₅	—	4	2	no	—
242	C₂²×C₅	—	8	4	no	—
243	C₂²×C₅	#470	1	2	yes	—
244	C₂²×C₅	C₂×#121	1	1	yes	characteristic
245	C₄×C₅	—	2	1	no	—
246	C₄×C₅	—	2	1	no	—
247	C₄×C₅	—	2	1	no	—
248	C₄×C₅	—	4	2	no	—
249	C₄×C₅	—	8	2	no	—
250	C₄×C₅	—	8	2	no	—
251	C₄×C₅	#470	1	2	yes	—
252	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
253	C₄×C₅	C₂×#137	1	2	yes	—
254	C₄×C₅	#575	1	1	yes	characteristic
255	C₂⁴	—	2	1	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×D₄	—	2	1	no	—
259	C₂×D₄	—	2	4	no	—
260	C₂×D₄	C₂×C₅×D₄	1	2	yes	—
261	C₂²×C₄	—	2	1	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	4	2	no	—
265	C₂²×C₄	—	4	2	no	—
266	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
267	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
268	C₂²×C₄	C₅×#41	1	1	yes	characteristic
269	Q₁₆	—	4	2	no	—
270	Q₁₆	—	4	2	no	—
271	M₁₆	—	4	4	no	—
272	M₁₆	—	4	4	no	—
273	M₁₆	—	4	4	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	4	no	—
277	C₂×C₈	—	4	1	no	—
278	C₂×C₈	—	4	2	no	—
279	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
280	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
281	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
282	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
283	C₄⋊C₄	—	2	1	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	4	2	no	—
289	C₄⋊C₄	—	4	2	no	—
290	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
291	C₂²⋊C₄	—	2	2	no	—
292	C₂²⋊C₄	—	2	2	no	—
293	C₂²⋊C₄	—	4	2	no	—
294	C₄²	—	4	2	no	—
295	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
296	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
297	C₄²	C₅×#41	1	1	yes	characteristic
298	C₄²	C₂⁴×C₅	1	1	yes	characteristic
299	C₂×C₅	—	2	2	no	—
300	C₂×C₅	—	8	2	no	—
301	C₂×C₅	#3036	1	1	yes	characteristic
302	C₂×C₅	C₂×#470	1	1	yes	characteristic
303	C₂×C₅	#3067	1	1	yes	characteristic
304	C₂³	—	2	2	no	—
305	C₂³	—	4	2	no	—
306	C₂³	—	4	2	no	—
307	C₂³	—	4	2	no	—
308	C₂³	C₅×#121	1	1	yes	characteristic
309	Q₈	—	4	2	no	—
310	Q₈	—	4	2	no	—
311	Q₈	—	8	2	no	—
312	D₄	—	2	4	no	—
313	D₄	—	4	2	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	4	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	C₅×#121	1	1	yes	characteristic
325	C₂×C₄	C₅×#121	1	1	yes	characteristic
326	C₂×C₄	C₅×#121	1	1	yes	characteristic
327	C₂×C₄	C₅×#137	1	4	yes	—
328	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
329	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
330	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
331	C₈	—	2	1	no	—
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	4	2	no	—
335	C₈	—	4	4	no	—
336	C₈	C₅×AGL(1,ℤ/8)	1	2	yes	—
337	C₅	#19281	1	1	yes	characteristic, direct product factor, Sylow
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	4	2	no	—
341	C₂²	—	4	2	no	—
342	C₂²	—	4	2	no	—
343	C₂²	—	8	4	no	—
344	C₂²	C₅×#470	1	2	yes	—
345	C₂²	C₂×C₅×#121	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	4	2	no	—
350	C₄	—	8	2	no	—
351	C₄	—	8	2	no	—
352	C₄	C₅×#470	1	2	yes	—
353	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
354	C₄	C₂×C₅×#137	1	2	yes	—
355	C₄	C₅×#575	1	1	yes	characteristic
356	C₂	—	2	2	no	—
357	C₂	—	8	2	no	—
358	C₂	C₅×#3036	1	1	yes	characteristic
359	C₂	C₂×C₅×#470	1	1	yes	characteristic
360	C₂	C₅×#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
prime radical	C₂²×C₅×D₄	D₄
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
1. C₅×#3170	C₂	semidirect
4. C₅×#1703	C₂	semidirect
5. C₅×#1695	C₂	semidirect
6. C₅×#1689	C₂	semidirect
7. C₅×#1566	C₂	semidirect
8. C₅×#1550	C₂	semidirect
12. C₅×#500	C₂²	semidirect
13. C₅×#494	C₂²	semidirect
14. C₅×#487	C₂²	semidirect
32. C₅×#334	C₂²	semidirect
33. C₅×#334	C₂²	semidirect
34. C₅×#333	C₂²	semidirect
35. C₅×#332	C₂²	semidirect
36. #19281	C₅	direct (N × H)
81. #3170	C₂×C₅	semidirect
84. #1703	C₂×C₅	semidirect
85. #1695	C₂×C₅	semidirect
86. #1689	C₂×C₅	semidirect
87. #1566	C₂×C₅	semidirect
88. #1550	C₂×C₅	semidirect
136. #500	C₂²×C₅	semidirect
137. #494	C₂²×C₅	semidirect
138. #487	C₂²×C₅	semidirect
156. #334	C₂²×C₅	semidirect
157. #334	C₂²×C₅	semidirect
158. #333	C₂²×C₅	semidirect
159. #332	C₂²×C₅	semidirect
337. C₅	#19281	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×#3170	2
C₅×#3077	2
C₂×C₅×#436	2
C₅×#1703	2
C₅×#1695	2
C₅×#1689	2
C₅×#1566	2
C₅×#1550	2
#19281	5

Subgroups — C5×#19281 #11942001

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#19281 #11942001