Subgroups — C₅×#19286 #11942006

Order 1280.

690 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×#3169	C₂	1	1	yes	maximal, characteristic
2	C₅×#3037	C₂	1	2	yes	maximal, complemented
3	C₂×C₅×#446	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1702	C₂	1	4	yes	maximal, complemented
5	C₅×#1694	C₂	1	2	yes	maximal, complemented
6	C₅×#1688	C₂	1	2	yes	maximal, complemented
7	C₅×#1566	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1547	C₂	1	2	yes	maximal, complemented
9	C₂×C₅×#137	—	2	2	no	—
10	C₅×#532	C₂²	1	2	yes	—
11	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
12	C₅×#500	C₂²	1	1	yes	characteristic
13	C₅×#493	C₂²	1	1	yes	characteristic
14	C₅×#486	C₂²	1	2	yes	—
15	C₅×#466	—	2	1	no	—
16	C₅×#466	—	2	1	no	—
17	C₅×#464	—	2	4	no	—
18	C₅×#462	C₂²	1	4	yes	—
19	C₅×#460	C₂²	1	4	yes	—
20	C₅×#446	—	2	1	no	—
21	C₅×#446	—	2	1	no	—
22	C₅×#446	C₂²	1	2	yes	—
23	C₅×#446	C₂²	1	2	yes	—
24	C₄×C₅×D₈	—	2	1	no	—
25	C₄×C₅×D₈	—	2	1	no	—
26	C₅×#423	C₂²	1	2	yes	—
27	C₂×C₅×C₄⋊C₈	—	2	1	no	—
28	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₅×#417	—	2	2	no	—
30	C₂×C₅×#97	—	2	1	no	—
31	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
32	C₅×#334	C₂²	1	2	yes	—
33	C₅×#334	C₂²	1	2	yes	—
34	C₅×#332	C₂²	1	4	yes	—
35	C₅×#331	C₂²	1	4	yes	—
36	#19286	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂×C₅×#41	—	2	2	no	—
38	C₂²×C₅×Q₈	D₄	1	1	yes	characteristic
39	C₅×#137	—	4	8	no	—
40	C₂×C₅×Q₁₆	—	2	4	no	—
41	C₂×C₅×SD₁₆	—	2	4	no	—
42	C₂×C₅×D₈	—	2	1	no	—
43	C₂×C₅×D₈	—	2	1	no	—
44	C₂×C₅×M₁₆	—	2	2	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×#127	C₂³	1	2	yes	—
50	C₅×#124	—	2	2	no	—
51	C₅×#121	—	2	2	no	—
52	C₅×#121	—	2	2	no	—
53	C₄×C₅×Q₈	—	2	1	no	—
54	C₄×C₅×Q₈	—	2	1	no	—
55	C₄×C₅×Q₈	D₄	1	2	yes	—
56	C₄×C₅×Q₈	C₂³	1	2	yes	—
57	C₄×C₅×D₄	—	2	2	no	—
58	C₄×C₅×D₄	—	2	2	no	—
59	C₂×C₅×C₄⋊C₄	—	2	1	no	—
60	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
61	C₂×C₄²×C₅	—	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×C₄⋊C₈	—	2	2	no	—
66	C₅×C₄⋊C₈	—	2	2	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	4	no	—
72	C₅×#102	—	2	2	no	—
73	C₅×#102	—	2	2	no	—
74	C₅×#102	—	2	4	no	—
75	C₅×#97	—	2	2	no	—
76	C₅×#97	—	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	#3169	C₂×C₅	1	1	yes	characteristic
82	#3037	C₂×C₅	1	2	yes	complemented
83	C₂×#446	C₂×C₅	1	1	yes	characteristic, complemented
84	#1702	C₂×C₅	1	4	yes	complemented
85	#1694	C₂×C₅	1	2	yes	complemented
86	#1688	C₂×C₅	1	2	yes	complemented
87	#1566	C₂×C₅	1	1	yes	characteristic, complemented
88	#1547	C₂×C₅	1	2	yes	complemented
89	C₅×#41	—	4	4	no	—
90	C₅×#41	—	4	4	no	—
91	C₂×C₅×Q₈	—	2	1	no	—
92	C₂×C₅×Q₈	—	2	4	no	—
93	C₂×C₅×Q₈	—	2	2	no	—
94	C₂×C₅×Q₈	C₂×D₄	1	2	yes	—
95	C₂×C₅×D₄	—	2	2	no	—
96	C₂×C₅×D₄	—	2	2	no	—
97	C₂×C₅×D₄	—	2	2	no	—
98	C₂×C₅×D₄	—	4	2	no	—
99	C₂²×C₄×C₅	—	2	1	no	—
100	C₂²×C₄×C₅	—	2	1	no	—
101	C₂²×C₄×C₅	—	4	2	no	—
102	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₅	#41	1	1	yes	characteristic
105	C₅×Q₁₆	—	4	8	no	—
106	C₅×SD₁₆	—	4	8	no	—
107	C₅×D₈	—	4	2	no	—
108	C₅×D₈	—	4	2	no	—
109	C₅×M₁₆	—	4	4	no	—
110	C₂×C₅×C₈	—	2	2	no	—
111	C₂×C₅×C₈	—	2	2	no	—
112	C₂×C₅×C₈	—	2	4	no	—
113	C₂×C₅×C₈	—	4	1	no	—
114	C₂×C₅×C₈	—	4	2	no	—
115	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
116	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
117	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
118	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
119	C₅×C₄⋊C₄	—	2	2	no	—
120	C₅×C₄⋊C₄	—	2	1	no	—
121	C₅×C₄⋊C₄	—	2	2	no	—
122	C₅×C₄⋊C₄	—	2	2	no	—
123	C₅×C₄⋊C₄	—	2	2	no	—
124	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
125	C₅×C₂²⋊C₄	—	4	2	no	—
126	C₅×C₂²⋊C₄	—	4	2	no	—
127	C₄²×C₅	—	2	2	no	—
128	C₄²×C₅	—	2	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	C₂×#137	—	2	2	no	—
134	#532	C₂²×C₅	1	2	yes	—
135	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
136	#500	C₂²×C₅	1	1	yes	characteristic
137	#493	C₂²×C₅	1	1	yes	characteristic
138	#486	C₂²×C₅	1	2	yes	—
139	#466	—	2	1	no	—
140	#466	—	2	1	no	—
141	#464	—	2	4	no	—
142	#462	C₂²×C₅	1	4	yes	—
143	#460	C₂²×C₅	1	4	yes	—
144	#446	—	2	1	no	—
145	#446	—	2	1	no	—
146	#446	C₂²×C₅	1	2	yes	—
147	#446	C₂²×C₅	1	2	yes	—
148	C₄×D₈	—	2	1	no	—
149	C₄×D₈	—	2	1	no	—
150	#423	C₂²×C₅	1	2	yes	—
151	C₂×C₄⋊C₈	—	2	1	no	—
152	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
153	#417	—	2	2	no	—
154	C₂×#97	—	2	1	no	—
155	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
156	#334	C₂²×C₅	1	2	yes	—
157	#334	C₂²×C₅	1	2	yes	—
158	#332	C₂²×C₅	1	4	yes	—
159	#331	C₂²×C₅	1	4	yes	—
160	C₂³×C₅	—	4	2	no	—
161	C₂³×C₅	#121	1	1	yes	characteristic
162	C₅×Q₈	—	2	4	no	—
163	C₅×Q₈	—	4	2	no	—
164	C₅×Q₈	—	4	2	no	—
165	C₅×Q₈	—	4	2	no	—
166	C₅×D₄	—	4	2	no	—
167	C₅×D₄	—	4	2	no	—
168	C₅×D₄	—	4	2	no	—
169	C₅×D₄	—	4	2	no	—
170	C₅×D₄	—	8	2	no	—
171	C₅×D₄	—	8	4	no	—
172	C₂×C₄×C₅	—	2	1	no	—
173	C₂×C₄×C₅	—	2	1	no	—
174	C₂×C₄×C₅	—	2	2	no	—
175	C₂×C₄×C₅	—	2	2	no	—
176	C₂×C₄×C₅	—	4	1	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	—	8	4	no	—
181	C₂×C₄×C₅	#121	1	1	yes	characteristic
182	C₂×C₄×C₅	#121	1	1	yes	characteristic
183	C₂×C₄×C₅	#121	1	1	yes	characteristic
184	C₂×C₄×C₅	AGL(1,ℤ/8)	1	4	yes	—
185	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
187	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
188	C₅×C₈	—	2	1	no	—
189	C₅×C₈	—	2	1	no	—
190	C₅×C₈	—	2	1	no	—
191	C₅×C₈	—	4	4	no	—
192	C₅×C₈	—	4	2	no	—
193	C₅×C₈	#137	1	2	yes	complemented
194	C₂×#41	—	2	2	no	—
195	C₂²×Q₈	C₅×D₄	1	1	yes	characteristic
196	#137	—	4	8	no	—
197	C₂×Q₁₆	—	2	4	no	—
198	C₂×SD₁₆	—	2	4	no	—
199	C₂×D₈	—	2	1	no	—
200	C₂×D₈	—	2	1	no	—
201	C₂×M₁₆	—	2	2	no	—
202	C₂²×C₈	—	2	1	no	—
203	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
204	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
205	C₄⋊Q₈	C₂³×C₅	1	2	yes	—
206	#127	C₂³×C₅	1	2	yes	—
207	#124	—	2	2	no	—
208	#121	—	2	2	no	—
209	#121	—	2	2	no	—
210	C₄×Q₈	—	2	1	no	—
211	C₄×Q₈	—	2	1	no	—
212	C₄×Q₈	C₅×D₄	1	2	yes	—
213	C₄×Q₈	C₂³×C₅	1	2	yes	—
214	C₄×D₄	—	2	2	no	—
215	C₄×D₄	—	2	2	no	—
216	C₂×C₄⋊C₄	—	2	1	no	—
217	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
218	C₂×C₄²	—	2	1	no	—
219	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
220	#107	—	2	1	no	—
221	#107	—	2	1	no	—
222	C₄⋊C₈	—	2	2	no	—
223	C₄⋊C₈	—	2	2	no	—
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	2	2	no	—
226	C₄⋊C₈	C₂³×C₅	1	2	yes	—
227	C₄⋊C₈	C₂³×C₅	1	4	yes	—
228	#103	—	2	4	no	—
229	#102	—	2	2	no	—
230	#102	—	2	2	no	—
231	#102	—	2	4	no	—
232	#97	—	2	2	no	—
233	#97	—	2	2	no	—
234	C₄×C₈	C₅×D₄	1	1	yes	characteristic
235	C₄×C₈	C₅×D₄	1	1	yes	characteristic
236	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
237	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
238	C₂²×C₅	—	2	1	no	—
239	C₂²×C₅	—	2	1	no	—
240	C₂²×C₅	—	8	2	no	—
241	C₂²×C₅	—	8	2	no	—
242	C₂²×C₅	—	8	2	no	—
243	C₂²×C₅	#469	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	—	4	2	no	—
250	C₄×C₅	—	8	2	no	—
251	C₄×C₅	#469	1	2	yes	—
252	C₄×C₅	C₂×AGL(1,ℤ/8)	1	2	yes	—
253	C₄×C₅	C₂×#137	1	1	yes	characteristic
254	C₄×C₅	#575	1	1	yes	characteristic
255	#41	—	4	4	no	—
256	#41	—	4	4	no	—
257	C₂×Q₈	—	2	1	no	—
258	C₂×Q₈	—	2	4	no	—
259	C₂×Q₈	—	2	2	no	—
260	C₂×Q₈	C₂×C₅×D₄	1	2	yes	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	—	4	2	no	—
265	C₂²×C₄	—	2	1	no	—
266	C₂²×C₄	—	2	1	no	—
267	C₂²×C₄	—	4	2	no	—
268	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
269	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
270	C₂²×C₄	C₅×#41	1	1	yes	characteristic
271	Q₁₆	—	4	8	no	—
272	SD₁₆	—	4	8	no	—
273	D₈	—	4	2	no	—
274	D₈	—	4	2	no	—
275	M₁₆	—	4	4	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	4	no	—
279	C₂×C₈	—	4	1	no	—
280	C₂×C₈	—	4	2	no	—
281	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
282	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
283	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
284	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	2	1	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
291	C₂²⋊C₄	—	4	2	no	—
292	C₂²⋊C₄	—	4	2	no	—
293	C₄²	—	2	2	no	—
294	C₄²	—	2	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	—	16	2	no	—
301	C₂×C₅	#3037	1	1	yes	characteristic
302	C₂×C₅	C₂×#469	1	1	yes	characteristic
303	C₂×C₅	#3067	1	1	yes	characteristic
304	C₂³	—	4	2	no	—
305	C₂³	C₅×#121	1	1	yes	characteristic
306	Q₈	—	2	4	no	—
307	Q₈	—	4	2	no	—
308	Q₈	—	4	2	no	—
309	Q₈	—	4	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	2	no	—
312	D₄	—	4	2	no	—
313	D₄	—	4	2	no	—
314	D₄	—	8	2	no	—
315	D₄	—	8	4	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	8	4	no	—
325	C₂×C₄	C₅×#121	1	1	yes	characteristic
326	C₂×C₄	C₅×#121	1	1	yes	characteristic
327	C₂×C₄	C₅×#121	1	1	yes	characteristic
328	C₂×C₄	C₅×AGL(1,ℤ/8)	1	4	yes	—
329	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
330	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
331	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	4	4	no	—
336	C₈	—	4	2	no	—
337	C₈	C₅×#137	1	2	yes	complemented
338	C₅	#19286	1	1	yes	characteristic, direct product factor, Sylow
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	8	2	no	—
344	C₂²	C₅×#469	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	—	4	2	no	—
351	C₄	—	8	2	no	—
352	C₄	C₅×#469	1	2	yes	—
353	C₄	C₂×C₅×AGL(1,ℤ/8)	1	2	yes	—
354	C₄	C₂×C₅×#137	1	1	yes	characteristic
355	C₄	C₅×#575	1	1	yes	characteristic
356	C₂	—	2	2	no	—
357	C₂	—	16	2	no	—
358	C₂	C₅×#3037	1	1	yes	characteristic
359	C₂	C₂×C₅×#469	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	C₅×#3169	C₂
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
2. C₅×#3037	C₂	semidirect
3. C₂×C₅×#446	C₂	semidirect
4. C₅×#1702	C₂	semidirect
5. C₅×#1694	C₂	semidirect
6. C₅×#1688	C₂	semidirect
7. C₅×#1566	C₂	semidirect
8. C₅×#1547	C₂	semidirect
36. #19286	C₅	direct (N × H)
82. #3037	C₂×C₅	semidirect
83. C₂×#446	C₂×C₅	semidirect
84. #1702	C₂×C₅	semidirect
85. #1694	C₂×C₅	semidirect
86. #1688	C₂×C₅	semidirect
87. #1566	C₂×C₅	semidirect
88. #1547	C₂×C₅	semidirect
193. C₅×C₈	#137	semidirect
337. C₈	C₅×#137	semidirect
338. C₅	#19286	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×#3169	2
C₅×#3037	2
C₂×C₅×#446	2
C₅×#1702	2
C₅×#1694	2
C₅×#1688	2
C₅×#1566	2
C₅×#1547	2
#19286	5

Subgroups — C5×#19286 #11942006

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#19286 #11942006