Subgroups — C₅×#8645 #11931365

Order 1280.

454 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₅×#1845	C₂	1	1	yes	maximal, characteristic
3	C₅×#1803	C₂	1	2	yes	maximal
4	C₂×C₅×#335	C₂	1	1	yes	maximal, characteristic
5	C₅×#1296	C₂	1	2	yes	maximal
6	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
7	C₅×C₈×Q₈	—	2	2	no	—
8	C₅×C₈×Q₈	C₄	1	4	yes	complemented
9	C₂×C₅×#107	—	2	1	no	—
10	C₂×C₅×#107	—	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	1	yes	characteristic
14	C₂×C₅×#103	—	2	2	no	—
15	C₂×C₄×C₅×C₈	—	2	1	no	—
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₅×#384	C₂²	1	2	yes	—
18	C₅×#340	—	2	2	no	—
19	C₅×#335	—	2	2	no	—
20	#8645	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₅×Q₈	D₄	1	1	yes	characteristic
22	C₂²×C₅×C₈	—	2	1	no	—
23	C₂²×C₅×C₈	—	2	2	no	—
24	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
26	C₄×C₅×Q₈	—	2	1	no	—
27	C₄×C₅×Q₈	—	2	1	no	—
28	C₄×C₅×Q₈	C₂×C₄	1	2	yes	—
29	C₄×C₅×Q₈	Q₈	1	2	yes	—
30	C₂×C₅×C₄⋊C₄	—	2	1	no	—
31	C₂×C₅×C₄⋊C₄	—	2	2	no	—
32	C₂×C₅×C₄⋊C₄	—	2	2	no	—
33	C₂×C₅×C₄⋊C₄	—	2	2	no	—
34	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₅	—	2	1	no	—
36	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
37	C₅×#107	—	4	2	no	—
38	C₅×#107	—	4	2	no	—
39	C₅×C₄⋊C₈	—	2	4	no	—
40	C₅×C₄⋊C₈	—	2	1	no	—
41	C₅×C₄⋊C₈	—	2	2	no	—
42	C₅×C₄⋊C₈	—	2	2	no	—
43	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
44	C₅×#103	—	4	4	no	—
45	C₄×C₅×C₈	—	2	4	no	—
46	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
47	C₄×C₅×C₈	D₄	1	2	yes	—
48	C₂×C₈×Q₈	C₂×C₅	1	1	yes	characteristic
49	#1845	C₂×C₅	1	1	yes	characteristic
50	#1803	C₂×C₅	1	2	yes	—
51	C₂×#335	C₂×C₅	1	1	yes	characteristic
52	#1296	C₂×C₅	1	2	yes	—
53	C₂×C₅×Q₈	—	2	1	no	—
54	C₂×C₅×Q₈	—	2	1	no	—
55	C₂×C₅×Q₈	—	2	1	no	—
56	C₂×C₅×Q₈	C₄⋊C₄	1	2	yes	—
57	C₂×C₅×Q₈	D₈	1	2	yes	—
58	C₂×C₅×Q₈	Q₁₆	1	2	yes	—
59	C₂²×C₄×C₅	—	2	1	no	—
60	C₂²×C₄×C₅	—	2	1	no	—
61	C₂²×C₄×C₅	—	4	2	no	—
62	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅	#41	1	1	yes	characteristic
65	C₂×C₅×C₈	—	2	1	no	—
66	C₂×C₅×C₈	—	2	1	no	—
67	C₂×C₅×C₈	—	2	2	no	—
68	C₂×C₅×C₈	—	4	1	no	—
69	C₂×C₅×C₈	—	4	1	no	—
70	C₂×C₅×C₈	—	4	2	no	—
71	C₂×C₅×C₈	—	4	2	no	—
72	C₂×C₅×C₈	—	4	2	no	—
73	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
74	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
75	C₂×C₅×C₈	SD₁₆	1	2	yes	—
76	C₂×C₅×C₈	Q₁₆	1	2	yes	—
77	C₅×C₄⋊C₄	—	2	2	no	—
78	C₅×C₄⋊C₄	—	2	1	no	—
79	C₅×C₄⋊C₄	—	2	2	no	—
80	C₅×C₄⋊C₄	—	4	2	no	—
81	C₅×C₄⋊C₄	—	4	2	no	—
82	C₅×C₄⋊C₄	—	4	2	no	—
83	C₅×C₄⋊C₄	—	4	2	no	—
84	C₅×C₄⋊C₄	—	8	2	no	—
85	C₅×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
86	C₄²×C₅	—	2	2	no	—
87	C₄²×C₅	—	2	2	no	—
88	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
91	C₄²×C₅	#41	1	1	yes	characteristic
92	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
93	C₈×Q₈	—	2	2	no	—
94	C₈×Q₈	C₄×C₅	1	4	yes	complemented
95	C₂×#107	—	2	1	no	—
96	C₂×#107	—	2	1	no	—
97	C₂×C₄⋊C₈	—	2	1	no	—
98	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
99	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
100	C₂×#103	—	2	2	no	—
101	C₂×C₄×C₈	—	2	1	no	—
102	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
103	#384	C₂²×C₅	1	2	yes	—
104	#340	—	2	2	no	—
105	#335	—	2	2	no	—
106	C₂³×C₅	#121	1	1	yes	characteristic
107	C₅×Q₈	—	2	2	no	—
108	C₅×Q₈	—	2	2	no	—
109	C₅×Q₈	—	2	2	no	—
110	C₅×Q₈	#107	1	4	yes	complemented
111	C₂×C₄×C₅	—	2	2	no	—
112	C₂×C₄×C₅	—	2	2	no	—
113	C₂×C₄×C₅	—	2	2	no	—
114	C₂×C₄×C₅	—	2	1	no	—
115	C₂×C₄×C₅	—	2	1	no	—
116	C₂×C₄×C₅	—	2	2	no	—
117	C₂×C₄×C₅	—	4	1	no	—
118	C₂×C₄×C₅	—	4	1	no	—
119	C₂×C₄×C₅	—	8	2	no	—
120	C₂×C₄×C₅	—	8	2	no	—
121	C₂×C₄×C₅	—	8	2	no	—
122	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₅	#117	1	1	yes	characteristic
125	C₂×C₄×C₅	#121	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
132	C₂×C₄×C₅	#135	1	1	yes	characteristic
133	C₂×C₄×C₅	#135	1	1	yes	characteristic
134	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₂×C₄×C₅	#137	1	1	yes	characteristic
136	C₅×C₈	—	2	2	no	—
137	C₅×C₈	—	4	2	no	—
138	C₅×C₈	—	8	2	no	—
139	C₅×C₈	#103	1	4	yes	complemented
140	C₂²×Q₈	C₅×D₄	1	1	yes	characteristic
141	C₂²×C₈	—	2	1	no	—
142	C₂²×C₈	—	2	2	no	—
143	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
144	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
145	C₄×Q₈	—	2	1	no	—
146	C₄×Q₈	—	2	1	no	—
147	C₄×Q₈	C₂×C₄×C₅	1	2	yes	—
148	C₄×Q₈	C₅×Q₈	1	2	yes	—
149	C₂×C₄⋊C₄	—	2	1	no	—
150	C₂×C₄⋊C₄	—	2	2	no	—
151	C₂×C₄⋊C₄	—	2	2	no	—
152	C₂×C₄⋊C₄	—	2	2	no	—
153	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
154	C₂×C₄²	—	2	1	no	—
155	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
156	#107	—	4	2	no	—
157	#107	—	4	2	no	—
158	C₄⋊C₈	—	2	4	no	—
159	C₄⋊C₈	—	2	1	no	—
160	C₄⋊C₈	—	2	2	no	—
161	C₄⋊C₈	—	2	2	no	—
162	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
163	#103	—	4	4	no	—
164	C₄×C₈	—	2	4	no	—
165	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
166	C₄×C₈	C₅×D₄	1	2	yes	—
167	C₂²×C₅	#380	1	1	yes	characteristic
168	C₂²×C₅	#460	1	1	yes	characteristic
169	C₂²×C₅	#462	1	1	yes	characteristic
170	C₂²×C₅	#466	1	1	yes	characteristic
171	C₂²×C₅	#467	1	1	yes	characteristic
172	C₂²×C₅	#472	1	1	yes	characteristic
173	C₂²×C₅	#473	1	1	yes	characteristic
174	C₄×C₅	—	2	4	no	—
175	C₄×C₅	—	2	1	no	—
176	C₄×C₅	—	2	1	no	—
177	C₄×C₅	—	4	2	no	—
178	C₄×C₅	—	16	2	no	—
179	C₄×C₅	C₂×#103	1	1	yes	characteristic
180	C₄×C₅	#417	1	1	yes	characteristic
181	C₄×C₅	C₂×#107	1	1	yes	characteristic
182	C₄×C₅	#427	1	1	yes	characteristic
183	C₄×C₅	#466	1	1	yes	characteristic
184	C₄×C₅	#467	1	1	yes	characteristic
185	C₄×C₅	#475	1	1	yes	characteristic
186	C₄×C₅	#477	1	1	yes	characteristic
187	C₂×Q₈	—	2	1	no	—
188	C₂×Q₈	—	2	1	no	—
189	C₂×Q₈	—	2	1	no	—
190	C₂×Q₈	C₅×C₄⋊C₄	1	2	yes	—
191	C₂×Q₈	C₅×D₈	1	2	yes	—
192	C₂×Q₈	C₅×Q₁₆	1	2	yes	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
198	C₂²×C₄	C₅×#41	1	1	yes	characteristic
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	4	1	no	—
203	C₂×C₈	—	4	1	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
208	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
209	C₂×C₈	C₅×SD₁₆	1	2	yes	—
210	C₂×C₈	C₅×Q₁₆	1	2	yes	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	1	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄⋊C₄	—	8	2	no	—
219	C₄⋊C₄	C₅×C₄⋊C₄	1	2	yes	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
223	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
224	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
225	C₄²	C₅×#41	1	1	yes	characteristic
226	C₂×C₅	#1668	1	1	yes	characteristic
227	C₂×C₅	#1670	1	1	yes	characteristic
228	C₂×C₅	#1674	1	1	yes	characteristic
229	C₂×C₅	#1676	1	1	yes	characteristic
230	C₂×C₅	#1803	1	1	yes	characteristic
231	C₂×C₅	#1808	1	1	yes	characteristic
232	C₂×C₅	#1816	1	1	yes	characteristic
233	C₂³	C₅×#121	1	1	yes	characteristic
234	Q₈	—	2	2	no	—
235	Q₈	—	2	2	no	—
236	Q₈	—	2	2	no	—
237	Q₈	C₅×#107	1	4	yes	complemented
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	8	2	no	—
247	C₂×C₄	—	8	2	no	—
248	C₂×C₄	—	8	2	no	—
249	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
250	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
251	C₂×C₄	C₅×#117	1	1	yes	characteristic
252	C₂×C₄	C₅×#121	1	1	yes	characteristic
253	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
254	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
255	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
256	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
257	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
258	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
259	C₂×C₄	C₅×#135	1	1	yes	characteristic
260	C₂×C₄	C₅×#135	1	1	yes	characteristic
261	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
262	C₂×C₄	C₅×#137	1	1	yes	characteristic
263	C₈	—	2	2	no	—
264	C₈	—	4	2	no	—
265	C₈	—	8	2	no	—
266	C₈	C₅×#103	1	4	yes	complemented
267	C₅	#8645	1	1	yes	characteristic, direct product factor, Sylow
268	C₂²	C₅×#380	1	1	yes	characteristic
269	C₂²	C₅×#460	1	1	yes	characteristic
270	C₂²	C₅×#462	1	1	yes	characteristic
271	C₂²	C₅×#466	1	1	yes	characteristic
272	C₂²	C₅×#467	1	1	yes	characteristic
273	C₂²	C₅×#472	1	1	yes	characteristic
274	C₂²	C₅×#473	1	1	yes	characteristic
275	C₄	—	2	4	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	4	2	no	—
279	C₄	—	16	2	no	—
280	C₄	C₂×C₅×#103	1	1	yes	characteristic
281	C₄	C₅×#417	1	1	yes	characteristic
282	C₄	C₂×C₅×#107	1	1	yes	characteristic
283	C₄	C₅×#427	1	1	yes	characteristic
284	C₄	C₅×#466	1	1	yes	characteristic
285	C₄	C₅×#467	1	1	yes	characteristic
286	C₄	C₅×#475	1	1	yes	characteristic
287	C₄	C₅×#477	1	1	yes	characteristic
288	C₂	C₅×#1668	1	1	yes	characteristic
289	C₂	C₅×#1670	1	1	yes	characteristic
290	C₂	C₅×#1674	1	1	yes	characteristic
291	C₂	C₅×#1676	1	1	yes	characteristic
292	C₂	C₅×#1803	1	1	yes	characteristic
293	C₂	C₅×#1808	1	1	yes	characteristic
294	C₂	C₅×#1816	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	#121
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₅×C₈×Q₈	C₄	semidirect
20. #8645	C₅	direct (N × H)
94. C₈×Q₈	C₄×C₅	semidirect
110. C₅×Q₈	#107	semidirect
139. C₅×C₈	#103	semidirect
237. Q₈	C₅×#107	semidirect
266. C₈	C₅×#103	semidirect
267. C₅	#8645	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×Q₈	2
C₅×#1845	2
C₅×#1803	2
C₂×C₅×#335	2
C₅×#1296	2
#8645	5

Subgroups — C5×#8645 #11931365

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8645 #11931365