Subgroups — C₅×#8586 #11931306

Order 1280.

516 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₅×#1842	C₂	1	1	yes	maximal, characteristic
3	C₅×#1842	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#332	C₂	1	2	yes	maximal
5	C₅×#1296	C₂	1	2	yes	maximal
6	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₅×Q₈	—	2	1	no	—
8	C₂×C₅×C₄⋊C₈	—	2	1	no	—
9	C₂×C₅×C₄⋊C₈	—	2	1	no	—
10	C₂×C₅×C₄⋊C₈	—	2	1	no	—
11	C₂×C₅×C₄⋊C₈	—	2	1	no	—
12	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
14	C₅×#391	—	2	1	no	—
15	C₅×#389	—	2	1	no	—
16	C₅×#389	C₄	1	2	yes	—
17	C₅×#384	C₂²	1	1	yes	characteristic
18	C₄×C₅×C₄⋊C₄	—	2	1	no	—
19	C₄×C₅×C₄⋊C₄	C₄	1	2	yes	—
20	C₄³×C₅	C₂²	1	1	yes	characteristic
21	C₅×#332	—	2	4	no	—
22	#8586	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂²×C₅×Q₈	—	2	1	no	—
24	C₂²×C₅×C₈	—	2	2	no	—
25	C₂²×C₅×C₈	—	2	2	no	—
26	C₅×C₄⋊Q₈	—	2	1	no	—
27	C₅×C₄⋊Q₈	—	2	1	no	—
28	C₅×C₄⋊Q₈	D₄	1	2	yes	—
29	C₅×C₄⋊Q₈	Q₈	1	2	yes	—
30	C₄×C₅×Q₈	—	4	2	no	—
31	C₄×C₅×Q₈	—	4	2	no	—
32	C₂×C₅×C₄⋊C₄	—	2	1	no	—
33	C₂×C₅×C₄⋊C₄	—	2	2	no	—
34	C₂×C₅×C₄⋊C₄	—	2	1	no	—
35	C₂×C₅×C₄⋊C₄	—	2	1	no	—
36	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
37	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₅	—	2	2	no	—
39	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₅×C₄⋊C₈	—	2	2	no	—
47	C₅×C₄⋊C₈	—	2	2	no	—
48	C₅×C₄⋊C₈	—	2	2	no	—
49	C₅×C₄⋊C₈	—	2	2	no	—
50	C₅×C₄⋊C₈	—	4	2	no	—
51	C₅×C₄⋊C₈	—	4	2	no	—
52	C₅×C₄⋊C₈	—	4	2	no	—
53	C₅×C₄⋊C₈	—	4	2	no	—
54	C₅×#95	—	2	2	no	—
55	C₅×#95	—	2	2	no	—
56	C₄×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
57	#1842	C₂×C₅	1	1	yes	characteristic
58	#1842	C₂×C₅	1	1	yes	characteristic
59	C₂×#332	C₂×C₅	1	2	yes	—
60	#1296	C₂×C₅	1	2	yes	—
61	C₂×C₅×Q₈	—	2	2	no	—
62	C₂×C₅×Q₈	—	2	2	no	—
63	C₂×C₅×Q₈	—	4	1	no	—
64	C₂×C₅×Q₈	—	4	1	no	—
65	C₂×C₅×Q₈	—	4	1	no	—
66	C₂×C₅×Q₈	—	4	1	no	—
67	C₂²×C₄×C₅	—	2	2	no	—
68	C₂²×C₄×C₅	—	2	2	no	—
69	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₅×C₈	—	4	2	no	—
77	C₂×C₅×C₈	—	4	2	no	—
78	C₂×C₅×C₈	—	4	2	no	—
79	C₂×C₅×C₈	—	4	2	no	—
80	C₂×C₅×C₈	—	4	2	no	—
81	C₂×C₅×C₈	—	4	2	no	—
82	C₅×C₄⋊C₄	—	2	2	no	—
83	C₅×C₄⋊C₄	—	2	2	no	—
84	C₅×C₄⋊C₄	—	2	2	no	—
85	C₅×C₄⋊C₄	—	4	2	no	—
86	C₅×C₄⋊C₄	—	4	4	no	—
87	C₅×C₄⋊C₄	—	4	1	no	—
88	C₅×C₄⋊C₄	—	4	1	no	—
89	C₅×C₄⋊C₄	—	4	1	no	—
90	C₅×C₄⋊C₄	—	4	1	no	—
91	C₅×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
92	C₄²×C₅	—	2	2	no	—
93	C₄²×C₅	—	2	1	no	—
94	C₄²×C₅	—	2	1	no	—
95	C₄²×C₅	—	2	1	no	—
96	C₄²×C₅	—	2	1	no	—
97	C₄²×C₅	—	4	1	no	—
98	C₄²×C₅	—	4	4	no	—
99	C₄²×C₅	D₈	1	2	yes	—
100	C₄²×C₅	SD₁₆	1	2	yes	—
101	C₄²×C₅	SD₁₆	1	2	yes	—
102	C₄²×C₅	Q₁₆	1	2	yes	—
103	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
105	C₄²×C₅	#41	1	1	yes	characteristic
106	C₄²×C₅	#41	1	1	yes	characteristic
107	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
108	C₂×C₄×Q₈	—	2	1	no	—
109	C₂×C₄⋊C₈	—	2	1	no	—
110	C₂×C₄⋊C₈	—	2	1	no	—
111	C₂×C₄⋊C₈	—	2	1	no	—
112	C₂×C₄⋊C₈	—	2	1	no	—
113	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
114	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
115	#391	—	2	1	no	—
116	#389	—	2	1	no	—
117	#389	C₄×C₅	1	2	yes	—
118	#384	C₂²×C₅	1	1	yes	characteristic
119	C₄×C₄⋊C₄	—	2	1	no	—
120	C₄×C₄⋊C₄	C₄×C₅	1	2	yes	—
121	C₄³	C₂²×C₅	1	1	yes	characteristic
122	#332	—	2	4	no	—
123	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₅×Q₈	—	4	2	no	—
125	C₅×Q₈	—	4	2	no	—
126	C₅×Q₈	—	4	2	no	—
127	C₅×Q₈	—	4	2	no	—
128	C₂×C₄×C₅	—	2	1	no	—
129	C₂×C₄×C₅	—	2	1	no	—
130	C₂×C₄×C₅	—	2	1	no	—
131	C₂×C₄×C₅	—	2	1	no	—
132	C₂×C₄×C₅	—	2	1	no	—
133	C₂×C₄×C₅	—	2	1	no	—
134	C₂×C₄×C₅	—	2	4	no	—
135	C₂×C₄×C₅	—	2	1	no	—
136	C₂×C₄×C₅	—	2	1	no	—
137	C₂×C₄×C₅	—	2	1	no	—
138	C₂×C₄×C₅	—	4	2	no	—
139	C₂×C₄×C₅	—	4	2	no	—
140	C₂×C₄×C₅	—	4	2	no	—
141	C₂×C₄×C₅	—	4	2	no	—
142	C₂×C₄×C₅	—	4	2	no	—
143	C₂×C₄×C₅	#101	1	2	yes	—
144	C₂×C₄×C₅	#102	1	2	yes	—
145	C₂×C₄×C₅	#102	1	2	yes	—
146	C₂×C₄×C₅	#103	1	2	yes	—
147	C₂×C₄×C₅	#103	1	2	yes	—
148	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₅	#117	1	1	yes	characteristic
150	C₂×C₄×C₅	#121	1	1	yes	characteristic
151	C₂×C₄×C₅	#121	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
156	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
158	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
159	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
160	C₂×C₄×C₅	#137	1	1	yes	characteristic
161	C₂×C₄×C₅	#137	1	1	yes	characteristic
162	C₅×C₈	—	8	2	no	—
163	C₅×C₈	—	8	2	no	—
164	C₂²×Q₈	—	2	1	no	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	—	2	2	no	—
167	C₄⋊Q₈	—	2	1	no	—
168	C₄⋊Q₈	—	2	1	no	—
169	C₄⋊Q₈	C₅×D₄	1	2	yes	—
170	C₄⋊Q₈	C₅×Q₈	1	2	yes	—
171	C₄×Q₈	—	4	2	no	—
172	C₄×Q₈	—	4	2	no	—
173	C₂×C₄⋊C₄	—	2	1	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	—	2	1	no	—
176	C₂×C₄⋊C₄	—	2	1	no	—
177	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
178	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
179	C₂×C₄²	—	2	2	no	—
180	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
181	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
182	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
183	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
184	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
185	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
186	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
187	C₄⋊C₈	—	2	2	no	—
188	C₄⋊C₈	—	2	2	no	—
189	C₄⋊C₈	—	2	2	no	—
190	C₄⋊C₈	—	2	2	no	—
191	C₄⋊C₈	—	4	2	no	—
192	C₄⋊C₈	—	4	2	no	—
193	C₄⋊C₈	—	4	2	no	—
194	C₄⋊C₈	—	4	2	no	—
195	#95	—	2	2	no	—
196	#95	—	2	2	no	—
197	C₂²×C₅	#380	1	1	yes	characteristic
198	C₂²×C₅	C₂×#101	1	1	yes	characteristic
199	C₂²×C₅	#413	1	1	yes	characteristic
200	C₂²×C₅	C₂×#102	1	1	yes	characteristic
201	C₂²×C₅	C₂×#103	1	1	yes	characteristic
202	C₂²×C₅	#417	1	1	yes	characteristic
203	C₂²×C₅	#417	1	1	yes	characteristic
204	C₄×C₅	—	2	1	no	—
205	C₄×C₅	—	2	1	no	—
206	C₄×C₅	—	2	2	no	—
207	C₄×C₅	—	2	2	no	—
208	C₄×C₅	—	4	4	no	—
209	C₄×C₅	—	4	1	no	—
210	C₄×C₅	—	8	2	no	—
211	C₄×C₅	#332	1	4	yes	complemented
212	C₄×C₅	#459	1	1	yes	characteristic
213	C₄×C₅	#460	1	1	yes	characteristic
214	C₄×C₅	#461	1	1	yes	characteristic
215	C₄×C₅	#462	1	1	yes	characteristic
216	C₄×C₅	#474	1	1	yes	characteristic
217	C₄×C₅	#475	1	1	yes	characteristic
218	C₄×C₅	#476	1	1	yes	characteristic
219	C₄×C₅	#477	1	1	yes	characteristic
220	C₂×Q₈	—	2	2	no	—
221	C₂×Q₈	—	2	2	no	—
222	C₂×Q₈	—	4	1	no	—
223	C₂×Q₈	—	4	1	no	—
224	C₂×Q₈	—	4	1	no	—
225	C₂×Q₈	—	4	1	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
229	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
230	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
231	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
232	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	4	1	no	—
246	C₄⋊C₄	—	4	1	no	—
247	C₄⋊C₄	—	4	1	no	—
248	C₄⋊C₄	—	4	1	no	—
249	C₄⋊C₄	—	4	4	no	—
250	C₄⋊C₄	C₅×C₄⋊C₄	1	4	yes	—
251	C₄²	—	2	1	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	4	1	no	—
257	C₄²	—	4	4	no	—
258	C₄²	C₅×D₈	1	2	yes	—
259	C₄²	C₅×SD₁₆	1	2	yes	—
260	C₄²	C₅×SD₁₆	1	2	yes	—
261	C₄²	C₅×Q₁₆	1	2	yes	—
262	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
263	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
264	C₄²	C₅×#41	1	1	yes	characteristic
265	C₄²	C₅×#41	1	1	yes	characteristic
266	C₂×C₅	C₂×#332	1	1	yes	characteristic
267	C₂×C₅	#1547	1	1	yes	characteristic
268	C₂×C₅	#1550	1	1	yes	characteristic
269	C₂×C₅	#1561	1	1	yes	characteristic
270	C₂×C₅	#1801	1	1	yes	characteristic
271	C₂×C₅	#1802	1	1	yes	characteristic
272	C₂×C₅	#1803	1	1	yes	characteristic
273	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
274	Q₈	—	4	2	no	—
275	Q₈	—	4	2	no	—
276	Q₈	—	4	2	no	—
277	Q₈	—	4	2	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	C₅×#101	1	2	yes	—
294	C₂×C₄	C₅×#102	1	2	yes	—
295	C₂×C₄	C₅×#102	1	2	yes	—
296	C₂×C₄	C₅×#103	1	2	yes	—
297	C₂×C₄	C₅×#103	1	2	yes	—
298	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
299	C₂×C₄	C₅×#117	1	1	yes	characteristic
300	C₂×C₄	C₅×#121	1	1	yes	characteristic
301	C₂×C₄	C₅×#121	1	1	yes	characteristic
302	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
303	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
304	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
305	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
306	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
307	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
308	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
309	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
310	C₂×C₄	C₅×#137	1	1	yes	characteristic
311	C₂×C₄	C₅×#137	1	1	yes	characteristic
312	C₈	—	8	2	no	—
313	C₈	—	8	2	no	—
314	C₅	#8586	1	1	yes	characteristic, direct product factor, Sylow
315	C₂²	C₅×#380	1	1	yes	characteristic
316	C₂²	C₂×C₅×#101	1	1	yes	characteristic
317	C₂²	C₅×#413	1	1	yes	characteristic
318	C₂²	C₂×C₅×#102	1	1	yes	characteristic
319	C₂²	C₂×C₅×#103	1	1	yes	characteristic
320	C₂²	C₅×#417	1	1	yes	characteristic
321	C₂²	C₅×#417	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	2	no	—
325	C₄	—	2	2	no	—
326	C₄	—	4	4	no	—
327	C₄	—	4	1	no	—
328	C₄	—	8	2	no	—
329	C₄	C₅×#332	1	4	yes	complemented
330	C₄	C₅×#459	1	1	yes	characteristic
331	C₄	C₅×#460	1	1	yes	characteristic
332	C₄	C₅×#461	1	1	yes	characteristic
333	C₄	C₅×#462	1	1	yes	characteristic
334	C₄	C₅×#474	1	1	yes	characteristic
335	C₄	C₅×#475	1	1	yes	characteristic
336	C₄	C₅×#476	1	1	yes	characteristic
337	C₄	C₅×#477	1	1	yes	characteristic
338	C₂	C₂×C₅×#332	1	1	yes	characteristic
339	C₂	C₅×#1547	1	1	yes	characteristic
340	C₂	C₅×#1550	1	1	yes	characteristic
341	C₂	C₅×#1561	1	1	yes	characteristic
342	C₂	C₅×#1801	1	1	yes	characteristic
343	C₂	C₅×#1802	1	1	yes	characteristic
344	C₂	C₅×#1803	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
22. #8586	C₅	direct (N × H)
211. C₄×C₅	#332	semidirect
314. C₅	#8586	direct (N × H)
329. C₄	C₅×#332	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×C₄⋊Q₈	2
C₅×#1842	2
C₂×C₅×#332	2
C₅×#1296	2
#8586	5

Subgroups — C5×#8586 #11931306

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8586 #11931306