Subgroups — C₅×#8203 #11930923

Order 1280.

482 conjugacy classes of subgroups.

The subgroup lattice has 302 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×#1770	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×#103	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅×#322	C₂	1	1	yes	maximal, characteristic
6	C₅×#1279	C₂	1	1	yes	maximal, characteristic
7	C₅×#1277	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
9	C₅×C₈×Q₈	—	2	2	no	—
10	C₅×C₈×Q₈	C₄	1	4	yes	complemented
11	C₂×C₅×C₄⋊C₈	—	2	1	no	—
12	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₅×#103	C₄	1	2	yes	—
15	C₂×C₅×#97	—	2	1	no	—
16	C₂×C₄×C₅×C₈	—	2	1	no	—
17	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₅×#340	C₄	1	2	yes	—
22	C₅×#336	—	2	1	no	—
23	C₅×Q₈⋊C₈	C₄	1	4	yes	complemented
24	C₅×#322	—	2	2	no	—
25	#8203	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂²×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	—	2	1	no	—
28	C₂²×C₅×C₈	—	2	1	no	—
29	C₂²×C₅×C₈	—	2	1	no	—
30	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
31	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
33	C₄×C₅×Q₈	—	2	1	no	—
34	C₄×C₅×Q₈	—	2	1	no	—
35	C₄×C₅×Q₈	C₂×C₄	1	2	yes	—
36	C₄×C₅×Q₈	C₂×C₄	1	2	yes	—
37	C₂×C₅×C₄⋊C₄	—	2	1	no	—
38	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
39	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅	—	2	1	no	—
41	C₂×C₄²×C₅	—	2	1	no	—
42	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₅×C₄⋊C₈	—	2	4	no	—
44	C₅×C₄⋊C₈	—	2	1	no	—
45	C₅×C₄⋊C₈	—	2	1	no	—
46	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
47	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
48	C₅×#103	—	2	4	no	—
49	C₅×#97	—	4	2	no	—
50	C₄×C₅×C₈	—	2	4	no	—
51	C₄×C₅×C₈	—	2	1	no	—
52	C₄×C₅×C₈	—	2	1	no	—
53	C₄×C₅×C₈	—	2	1	no	—
54	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
55	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
56	C₄×C₅×C₈	D₄	1	2	yes	—
57	C₅×#95	—	2	1	no	—
58	C₂×C₈×Q₈	C₂×C₅	1	1	yes	characteristic
59	#1770	C₂×C₅	1	1	yes	characteristic
60	C₄×#103	C₂×C₅	1	1	yes	characteristic
61	C₂×Q₈⋊C₈	C₂×C₅	1	1	yes	characteristic
62	C₂×#322	C₂×C₅	1	1	yes	characteristic
63	#1279	C₂×C₅	1	1	yes	characteristic
64	#1277	C₂×C₅	1	1	yes	characteristic
65	C₂×C₅×Q₈	—	2	1	no	—
66	C₂×C₅×Q₈	—	2	2	no	—
67	C₂×C₅×Q₈	C₄²	1	2	yes	—
68	C₂×C₅×Q₈	M₁₆	1	4	yes	—
69	C₂²×C₄×C₅	—	2	1	no	—
70	C₂²×C₄×C₅	—	2	1	no	—
71	C₂²×C₄×C₅	—	2	2	no	—
72	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₅	#41	1	1	yes	characteristic
75	C₂×C₅×C₈	—	2	1	no	—
76	C₂×C₅×C₈	—	2	1	no	—
77	C₂×C₅×C₈	—	2	2	no	—
78	C₂×C₅×C₈	—	2	2	no	—
79	C₂×C₅×C₈	—	2	2	no	—
80	C₂×C₅×C₈	—	2	2	no	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₂×C₅×C₈	—	2	2	no	—
83	C₂×C₅×C₈	—	4	2	no	—
84	C₂×C₅×C₈	—	4	1	no	—
85	C₂×C₅×C₈	—	4	1	no	—
86	C₂×C₅×C₈	—	4	1	no	—
87	C₂×C₅×C₈	—	4	1	no	—
88	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
89	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
90	C₂×C₅×C₈	SD₁₆	1	2	yes	—
91	C₂×C₅×C₈	Q₁₆	1	2	yes	—
92	C₅×C₄⋊C₄	—	2	2	no	—
93	C₅×C₄⋊C₄	—	2	2	no	—
94	C₅×C₄⋊C₄	—	2	1	no	—
95	C₅×C₄⋊C₄	—	2	2	no	—
96	C₅×C₄⋊C₄	—	2	2	no	—
97	C₅×C₄⋊C₄	C₄²	1	2	yes	—
98	C₄²×C₅	—	2	2	no	—
99	C₄²×C₅	—	2	2	no	—
100	C₄²×C₅	—	4	1	no	—
101	C₄²×C₅	—	4	1	no	—
102	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₅	#41	1	1	yes	characteristic
106	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
107	C₈×Q₈	—	2	2	no	—
108	C₈×Q₈	C₄×C₅	1	4	yes	complemented
109	C₂×C₄⋊C₈	—	2	1	no	—
110	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
111	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
112	C₂×#103	C₄×C₅	1	2	yes	—
113	C₂×#97	—	2	1	no	—
114	C₂×C₄×C₈	—	2	1	no	—
115	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
116	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
117	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
118	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
119	#340	C₄×C₅	1	2	yes	—
120	#336	—	2	1	no	—
121	Q₈⋊C₈	C₄×C₅	1	4	yes	complemented
122	#322	—	2	2	no	—
123	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
124	C₅×Q₈	—	2	2	no	—
125	C₅×Q₈	—	2	4	no	—
126	C₅×Q₈	#97	1	4	yes	complemented
127	C₂×C₄×C₅	—	2	2	no	—
128	C₂×C₄×C₅	—	2	4	no	—
129	C₂×C₄×C₅	—	2	2	no	—
130	C₂×C₄×C₅	—	2	2	no	—
131	C₂×C₄×C₅	—	4	2	no	—
132	C₂×C₄×C₅	—	4	2	no	—
133	C₂×C₄×C₅	—	4	2	no	—
134	C₂×C₄×C₅	—	4	2	no	—
135	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₅	#117	1	1	yes	characteristic
138	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
142	C₂×C₄×C₅	#131	1	2	yes	—
143	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
145	C₂×C₄×C₅	#135	1	1	yes	characteristic
146	C₂×C₄×C₅	#135	1	1	yes	characteristic
147	C₅×C₈	—	2	2	no	—
148	C₅×C₈	—	4	2	no	—
149	C₅×C₈	—	4	2	no	—
150	C₅×C₈	—	4	2	no	—
151	C₅×C₈	—	4	2	no	—
152	C₅×C₈	#103	1	4	yes	complemented
153	C₂²×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
154	C₂²×C₈	—	2	1	no	—
155	C₂²×C₈	—	2	1	no	—
156	C₂²×C₈	—	2	1	no	—
157	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
158	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
159	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
160	C₄×Q₈	—	2	1	no	—
161	C₄×Q₈	—	2	1	no	—
162	C₄×Q₈	C₂×C₄×C₅	1	2	yes	—
163	C₄×Q₈	C₂×C₄×C₅	1	2	yes	—
164	C₂×C₄⋊C₄	—	2	1	no	—
165	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
166	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
167	C₂×C₄²	—	2	1	no	—
168	C₂×C₄²	—	2	1	no	—
169	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
170	C₄⋊C₈	—	2	4	no	—
171	C₄⋊C₈	—	2	1	no	—
172	C₄⋊C₈	—	2	1	no	—
173	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
174	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
175	#103	—	2	4	no	—
176	#97	—	4	2	no	—
177	C₄×C₈	—	2	4	no	—
178	C₄×C₈	—	2	1	no	—
179	C₄×C₈	—	2	1	no	—
180	C₄×C₈	—	2	1	no	—
181	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
182	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
183	C₄×C₈	C₅×D₄	1	2	yes	—
184	#95	—	2	1	no	—
185	C₂²×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
186	C₂²×C₅	#435	1	2	yes	—
187	C₂²×C₅	C₄×SD₁₆	1	1	yes	characteristic
188	C₂²×C₅	C₄×Q₁₆	1	1	yes	characteristic
189	C₂²×C₅	#444	1	2	yes	—
190	C₄×C₅	—	2	4	no	—
191	C₄×C₅	—	2	1	no	—
192	C₄×C₅	—	2	1	no	—
193	C₄×C₅	—	4	2	no	—
194	C₄×C₅	—	8	2	no	—
195	C₄×C₅	C₂×#97	1	1	yes	characteristic
196	C₄×C₅	#405	1	1	yes	characteristic
197	C₄×C₅	C₂×#103	1	1	yes	characteristic
198	C₄×C₅	#416	1	1	yes	characteristic
199	C₄×C₅	#435	1	2	yes	—
200	C₄×C₅	C₄×SD₁₆	1	1	yes	characteristic
201	C₄×C₅	C₄×Q₁₆	1	1	yes	characteristic
202	C₂×Q₈	—	2	1	no	—
203	C₂×Q₈	—	2	2	no	—
204	C₂×Q₈	C₄²×C₅	1	2	yes	—
205	C₂×Q₈	C₅×M₁₆	1	4	yes	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
211	C₂²×C₄	C₅×#41	1	1	yes	characteristic
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
226	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
227	C₂×C₈	C₅×SD₁₆	1	2	yes	—
228	C₂×C₈	C₅×Q₁₆	1	2	yes	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	C₄²×C₅	1	2	yes	—
235	C₄²	—	2	2	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	4	1	no	—
238	C₄²	—	4	1	no	—
239	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
240	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
241	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
242	C₄²	C₅×#41	1	1	yes	characteristic
243	C₂×C₅	#1583	1	2	yes	—
244	C₂×C₅	#1584	1	2	yes	—
245	C₂×C₅	#1752	1	1	yes	characteristic
246	C₂×C₅	C₄×#103	1	1	yes	characteristic
247	C₂×C₅	#1765	1	1	yes	characteristic
248	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
249	Q₈	—	2	2	no	—
250	Q₈	—	2	4	no	—
251	Q₈	C₅×#97	1	4	yes	complemented
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
261	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
262	C₂×C₄	C₅×#117	1	1	yes	characteristic
263	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
264	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
265	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
266	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
267	C₂×C₄	C₅×#131	1	2	yes	—
268	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
269	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
270	C₂×C₄	C₅×#135	1	1	yes	characteristic
271	C₂×C₄	C₅×#135	1	1	yes	characteristic
272	C₈	—	2	2	no	—
273	C₈	—	4	2	no	—
274	C₈	—	4	2	no	—
275	C₈	—	4	2	no	—
276	C₈	—	4	2	no	—
277	C₈	C₅×#103	1	4	yes	complemented
278	C₅	#8203	1	1	yes	characteristic, direct product factor, Sylow
279	C₂²	C₄×C₅×C₂²⋊C₄	1	1	yes	characteristic
280	C₂²	C₅×#435	1	2	yes	—
281	C₂²	C₄×C₅×SD₁₆	1	1	yes	characteristic
282	C₂²	C₄×C₅×Q₁₆	1	1	yes	characteristic
283	C₂²	C₅×#444	1	2	yes	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	4	no	—
287	C₄	—	4	2	no	—
288	C₄	—	8	2	no	—
289	C₄	C₂×C₅×#97	1	1	yes	characteristic
290	C₄	C₅×#405	1	1	yes	characteristic
291	C₄	C₂×C₅×#103	1	1	yes	characteristic
292	C₄	C₅×#416	1	1	yes	characteristic
293	C₄	C₅×#435	1	2	yes	—
294	C₄	C₄×C₅×SD₁₆	1	1	yes	characteristic
295	C₄	C₄×C₅×Q₁₆	1	1	yes	characteristic
296	C₂	C₅×#1583	1	2	yes	—
297	C₂	C₅×#1584	1	2	yes	—
298	C₂	C₅×#1752	1	1	yes	characteristic
299	C₂	C₄×C₅×#103	1	1	yes	characteristic
300	C₂	C₅×#1765	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle, prime radical	C₂³×C₅	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₅×C₈×Q₈	C₄	semidirect
23. C₅×Q₈⋊C₈	C₄	semidirect
25. #8203	C₅	direct (N × H)
108. C₈×Q₈	C₄×C₅	semidirect
121. Q₈⋊C₈	C₄×C₅	semidirect
126. C₅×Q₈	#97	semidirect
152. C₅×C₈	#103	semidirect
251. Q₈	C₅×#97	semidirect
277. C₈	C₅×#103	semidirect
278. C₅	#8203	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×#1770	2
C₄×C₅×#103	2
C₂×C₅×Q₈⋊C₈	2
C₂×C₅×#322	2
C₅×#1279	2
C₅×#1277	2
#8203	5

Subgroups — C5×#8203 #11930923

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8203 #11930923