Subgroups — C₅×#8608 #11931328

Order 1280.

550 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₈×D₄	C₂	1	1	yes	maximal, characteristic
2	C₅×#1807	C₂	1	2	yes	maximal, complemented
3	C₅×#1799	C₂	1	2	yes	maximal
4	C₅×#1289	C₂	1	2	yes	maximal, complemented
5	C₂³×C₅×C₈	C₄	1	1	yes	characteristic, complemented
6	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
8	C₅×C₈×D₄	—	2	2	no	—
9	C₅×C₈×D₄	—	2	2	no	—
10	C₂×C₅×#106	—	2	2	no	—
11	C₂×C₅×C₄⋊C₈	C₄	1	1	yes	characteristic, complemented
12	C₂×C₅×#99	—	2	2	no	—
13	C₂×C₅×C₂²⋊C₈	C₄	1	1	yes	characteristic, complemented
14	C₂×C₅×C₂²⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic, complemented
16	C₅×#380	C₂²	1	4	yes	—
17	C₅×#342	—	2	2	no	—
18	C₅×#340	—	2	2	no	—
19	#8608	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂²×C₅×D₄	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₂²×C₅×C₈	—	2	1	no	—
24	C₂²×C₅×C₈	—	2	1	no	—
25	C₂²×C₅×C₈	—	4	2	no	—
26	C₂²×C₅×C₈	—	4	2	no	—
27	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₅×C₈	D₄	1	2	yes	—
32	C₂²×C₅×C₈	D₄	1	2	yes	—
33	C₄×C₅×D₄	—	2	2	no	—
34	C₄×C₅×D₄	—	2	2	no	—
35	C₂×C₅×C₄⋊C₄	—	2	4	no	—
36	C₂×C₅×C₄⋊C₄	Q₈	1	1	yes	characteristic
37	C₂×C₅×C₂²⋊C₄	—	2	4	no	—
38	C₂×C₅×C₂²⋊C₄	D₄	1	1	yes	characteristic
39	C₂×C₅×C₂²⋊C₄	Q₈	1	1	yes	characteristic
40	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
41	C₅×#106	—	4	4	no	—
42	C₅×C₄⋊C₈	—	2	2	no	—
43	C₅×#99	—	4	4	no	—
44	C₅×C₂²⋊C₈	—	2	2	no	—
45	C₅×C₂²⋊C₈	—	2	2	no	—
46	C₄×C₅×C₈	—	2	1	no	—
47	C₄×C₅×C₈	—	2	1	no	—
48	C₅×#95	—	2	4	no	—
49	C₂×C₈×D₄	C₂×C₅	1	1	yes	characteristic
50	#1807	C₂×C₅	1	2	yes	complemented
51	#1799	C₂×C₅	1	2	yes	—
52	#1289	C₂×C₅	1	2	yes	complemented
53	C₂⁴×C₅	C₄⋊C₄	1	1	yes	characteristic
54	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂×C₅×D₄	—	2	2	no	—
56	C₂×C₅×D₄	—	2	2	no	—
57	C₂×C₅×D₄	SD₁₆	1	4	yes	—
58	C₂²×C₄×C₅	—	2	1	no	—
59	C₂²×C₄×C₅	—	2	1	no	—
60	C₂²×C₄×C₅	—	4	1	no	—
61	C₂²×C₄×C₅	—	4	1	no	—
62	C₂²×C₄×C₅	—	4	1	no	—
63	C₂²×C₄×C₅	—	4	1	no	—
64	C₂²×C₄×C₅	—	4	4	no	—
65	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
71	C₂²×C₄×C₅	#41	1	2	yes	—
72	C₂×C₅×C₈	—	2	2	no	—
73	C₂×C₅×C₈	—	2	4	no	—
74	C₂×C₅×C₈	—	2	1	no	—
75	C₂×C₅×C₈	—	2	1	no	—
76	C₂×C₅×C₈	—	4	1	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	1	no	—
81	C₂×C₅×C₈	—	4	1	no	—
82	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
84	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
85	C₅×C₄⋊C₄	—	2	2	no	—
86	C₅×C₄⋊C₄	—	4	4	no	—
87	C₅×C₄⋊C₄	—	4	4	no	—
88	C₅×C₂²⋊C₄	—	2	2	no	—
89	C₅×C₂²⋊C₄	—	2	2	no	—
90	C₅×C₂²⋊C₄	—	4	4	no	—
91	C₅×C₂²⋊C₄	—	4	4	no	—
92	C₄²×C₅	—	2	1	no	—
93	C₄²×C₅	—	2	1	no	—
94	C₂³×C₈	C₄×C₅	1	1	yes	characteristic, complemented
95	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
96	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
97	C₈×D₄	—	2	2	no	—
98	C₈×D₄	—	2	2	no	—
99	C₂×#106	—	2	2	no	—
100	C₂×C₄⋊C₈	C₄×C₅	1	1	yes	characteristic, complemented
101	C₂×#99	—	2	2	no	—
102	C₂×C₂²⋊C₈	C₄×C₅	1	1	yes	characteristic, complemented
103	C₂×C₂²⋊C₈	C₂²×C₅	1	1	yes	characteristic
104	C₂×C₄×C₈	C₄×C₅	1	1	yes	characteristic, complemented
105	#380	C₂²×C₅	1	4	yes	—
106	#342	—	2	2	no	—
107	#340	—	2	2	no	—
108	C₂³×C₅	—	2	1	no	—
109	C₂³×C₅	—	2	1	no	—
110	C₂³×C₅	—	4	1	no	—
111	C₂³×C₅	—	4	1	no	—
112	C₂³×C₅	—	4	1	no	—
113	C₂³×C₅	—	4	1	no	—
114	C₂³×C₅	#106	1	2	yes	complemented
115	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂³×C₅	#121	1	2	yes	—
117	C₂³×C₅	C₂×SD₁₆	1	2	yes	—
118	C₂³×C₅	#135	1	2	yes	—
119	C₅×D₄	—	2	8	no	—
120	C₂×C₄×C₅	—	2	1	no	—
121	C₂×C₄×C₅	—	2	1	no	—
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	—	2	1	no	—
124	C₂×C₄×C₅	—	2	2	no	—
125	C₂×C₄×C₅	—	2	1	no	—
126	C₂×C₄×C₅	—	2	1	no	—
127	C₂×C₄×C₅	—	2	1	no	—
128	C₂×C₄×C₅	—	4	1	no	—
129	C₂×C₄×C₅	—	4	2	no	—
130	C₂×C₄×C₅	—	4	2	no	—
131	C₂×C₄×C₅	—	4	1	no	—
132	C₂×C₄×C₅	—	4	1	no	—
133	C₂×C₄×C₅	—	4	1	no	—
134	C₂×C₄×C₅	—	4	1	no	—
135	C₂×C₄×C₅	—	8	4	no	—
136	C₂×C₄×C₅	—	8	4	no	—
137	C₂×C₄×C₅	—	8	4	no	—
138	C₂×C₄×C₅	#106	1	2	yes	complemented
139	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₅	#117	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
142	C₅×C₈	—	2	1	no	—
143	C₅×C₈	—	2	2	no	—
144	C₅×C₈	—	4	1	no	—
145	C₅×C₈	—	4	1	no	—
146	C₅×C₈	#99	1	2	yes	complemented
147	C₂²×D₄	C₅×D₄	1	1	yes	characteristic
148	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
149	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
150	C₂²×C₈	—	2	1	no	—
151	C₂²×C₈	—	2	1	no	—
152	C₂²×C₈	—	4	2	no	—
153	C₂²×C₈	—	4	2	no	—
154	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
155	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
156	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
157	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
158	C₂²×C₈	C₅×D₄	1	2	yes	—
159	C₂²×C₈	C₅×D₄	1	2	yes	—
160	C₄×D₄	—	2	2	no	—
161	C₄×D₄	—	2	2	no	—
162	C₂×C₄⋊C₄	—	2	4	no	—
163	C₂×C₄⋊C₄	C₅×Q₈	1	1	yes	characteristic
164	C₂×C₂²⋊C₄	—	2	4	no	—
165	C₂×C₂²⋊C₄	C₅×D₄	1	1	yes	characteristic
166	C₂×C₂²⋊C₄	C₅×Q₈	1	1	yes	characteristic
167	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
168	#106	—	4	4	no	—
169	C₄⋊C₈	—	2	2	no	—
170	#99	—	4	4	no	—
171	C₂²⋊C₈	—	2	2	no	—
172	C₂²⋊C₈	—	2	2	no	—
173	C₄×C₈	—	2	1	no	—
174	C₄×C₈	—	2	1	no	—
175	#95	—	2	4	no	—
176	C₂²×C₅	—	2	1	no	—
177	C₂²×C₅	—	2	2	no	—
178	C₂²×C₅	—	2	1	no	—
179	C₂²×C₅	—	2	2	no	—
180	C₂²×C₅	—	2	2	no	—
181	C₂²×C₅	—	4	2	no	—
182	C₂²×C₅	—	4	2	no	—
183	C₂²×C₅	—	4	2	no	—
184	C₂²×C₅	—	4	2	no	—
185	C₂²×C₅	—	4	1	no	—
186	C₂²×C₅	—	4	1	no	—
187	C₂²×C₅	#380	1	1	yes	characteristic
188	C₂²×C₅	C₂×#106	1	1	yes	characteristic
189	C₂²×C₅	#427	1	1	yes	characteristic
190	C₂²×C₅	#465	1	4	yes	—
191	C₄×C₅	—	2	1	no	—
192	C₄×C₅	—	2	2	no	—
193	C₄×C₅	—	2	2	no	—
194	C₄×C₅	—	4	1	no	—
195	C₄×C₅	—	4	1	no	—
196	C₄×C₅	—	16	4	no	—
197	C₄×C₅	C₂×#99	1	1	yes	characteristic
198	C₄×C₅	#410	1	1	yes	characteristic
199	C₂⁴	C₅×C₄⋊C₄	1	1	yes	characteristic
200	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	C₅×SD₁₆	1	4	yes	—
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	4	1	no	—
207	C₂²×C₄	—	4	1	no	—
208	C₂²×C₄	—	4	1	no	—
209	C₂²×C₄	—	4	1	no	—
210	C₂²×C₄	—	4	4	no	—
211	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
212	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
213	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
214	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
216	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
217	C₂²×C₄	C₅×#41	1	2	yes	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
229	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
230	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	4	4	no	—
233	C₄⋊C₄	—	4	4	no	—
234	C₂²⋊C₄	—	2	2	no	—
235	C₂²⋊C₄	—	2	2	no	—
236	C₂²⋊C₄	—	4	4	no	—
237	C₂²⋊C₄	—	4	4	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	2	1	no	—
240	C₂×C₅	—	2	4	no	—
241	C₂×C₅	—	2	1	no	—
242	C₂×C₅	—	2	1	no	—
243	C₂×C₅	—	4	2	no	—
244	C₂×C₅	#1799	1	1	yes	characteristic
245	C₂×C₅	#1807	1	1	yes	characteristic
246	C₂×C₅	#1812	1	1	yes	characteristic
247	C₂³	—	2	1	no	—
248	C₂³	—	2	1	no	—
249	C₂³	—	4	1	no	—
250	C₂³	—	4	1	no	—
251	C₂³	—	4	1	no	—
252	C₂³	—	4	1	no	—
253	C₂³	C₅×#106	1	2	yes	complemented
254	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
255	C₂³	C₅×#121	1	2	yes	—
256	C₂³	C₂×C₅×SD₁₆	1	2	yes	—
257	C₂³	C₅×#135	1	2	yes	—
258	D₄	—	2	8	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	8	4	no	—
275	C₂×C₄	—	8	4	no	—
276	C₂×C₄	—	8	4	no	—
277	C₂×C₄	C₅×#106	1	2	yes	complemented
278	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
279	C₂×C₄	C₅×#117	1	1	yes	characteristic
280	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
281	C₈	—	2	1	no	—
282	C₈	—	2	2	no	—
283	C₈	—	4	1	no	—
284	C₈	—	4	1	no	—
285	C₈	C₅×#99	1	2	yes	complemented
286	C₅	#8608	1	1	yes	characteristic, direct product factor, Sylow
287	C₂²	—	2	1	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	4	2	no	—
293	C₂²	—	4	2	no	—
294	C₂²	—	4	2	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	1	no	—
298	C₂²	C₅×#380	1	1	yes	characteristic
299	C₂²	C₂×C₅×#106	1	1	yes	characteristic
300	C₂²	C₅×#427	1	1	yes	characteristic
301	C₂²	C₅×#465	1	4	yes	—
302	C₄	—	2	1	no	—
303	C₄	—	2	2	no	—
304	C₄	—	2	2	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	16	4	no	—
308	C₄	C₂×C₅×#99	1	1	yes	characteristic
309	C₄	C₅×#410	1	1	yes	characteristic
310	C₂	—	2	4	no	—
311	C₂	—	2	1	no	—
312	C₂	—	2	1	no	—
313	C₂	—	4	2	no	—
314	C₂	C₅×#1799	1	1	yes	characteristic
315	C₂	C₅×#1807	1	1	yes	characteristic
316	C₂	C₅×#1812	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	#380
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₂²×C₅×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#1807	C₂	semidirect
4. C₅×#1289	C₂	semidirect
5. C₂³×C₅×C₈	C₄	semidirect
11. C₂×C₅×C₄⋊C₈	C₄	semidirect
13. C₂×C₅×C₂²⋊C₈	C₄	semidirect
15. C₂×C₄×C₅×C₈	C₄	semidirect
19. #8608	C₅	direct (N × H)
50. #1807	C₂×C₅	semidirect
52. #1289	C₂×C₅	semidirect
94. C₂³×C₈	C₄×C₅	semidirect
100. C₂×C₄⋊C₈	C₄×C₅	semidirect
102. C₂×C₂²⋊C₈	C₄×C₅	semidirect
104. C₂×C₄×C₈	C₄×C₅	semidirect
114. C₂³×C₅	#106	semidirect
138. C₂×C₄×C₅	#106	semidirect
146. C₅×C₈	#99	semidirect
253. C₂³	C₅×#106	semidirect
277. C₂×C₄	C₅×#106	semidirect
285. C₈	C₅×#99	semidirect
286. C₅	#8608	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×C₈×D₄	2
C₅×#1807	2
C₅×#1799	2
C₅×#1289	2
#8608	5

Subgroups — C5×#8608 #11931328

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8608 #11931328