Subgroups — C₅×#11886 #11934606

Order 1280.

470 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#466	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#1808	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1347	C₂	1	4	yes	maximal, complemented
5	C₂²×C₅×D₈	C₄	1	1	yes	characteristic, complemented
6	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₅×#121	—	2	1	no	—
8	C₂²×C₅×C₁₆	—	2	1	no	—
9	C₅×#466	—	2	1	no	—
10	C₅×#466	—	2	1	no	—
11	C₅×#466	C₄	1	2	yes	complemented
12	C₅×#466	C₂²	1	2	yes	—
13	C₂×C₅×#107	—	2	1	no	—
14	C₂×C₅×#107	C₄	1	1	yes	characteristic, complemented
15	C₂×C₅×#102	—	2	1	no	—
16	C₅×#380	—	2	1	no	—
17	C₅×C₂²⋊C₁₆	—	2	1	no	—
18	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	complemented
19	C₅×#340	C₂²	1	2	yes	complemented
20	#11886	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₅×D₄	—	2	1	no	—
22	C₂²×C₅×D₄	—	4	1	no	—
23	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
24	C₂×C₅×D₈	—	2	1	no	—
25	C₂×C₅×D₈	—	4	2	no	—
26	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
27	C₂²×C₅×C₈	—	4	2	no	—
28	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
31	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
33	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
34	C₅×#121	—	2	2	no	—
35	C₅×#121	—	2	2	no	—
36	C₅×#121	—	4	1	no	—
37	C₅×#121	—	4	1	no	—
38	C₂×C₅×C₄⋊C₄	—	2	1	no	—
39	C₂×C₅×C₄⋊C₄	—	2	2	no	—
40	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
41	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
42	C₂×C₅×C₁₆	—	2	2	no	—
43	C₂×C₅×C₁₆	—	4	2	no	—
44	C₅×#107	—	2	1	no	—
45	C₅×#107	—	4	2	no	—
46	C₅×#107	C₂×C₄	1	2	yes	—
47	C₅×#102	—	2	2	no	—
48	C₅×#102	—	2	2	no	—
49	C₅×#95	—	4	1	no	—
50	C₂×#466	C₂×C₅	1	1	yes	characteristic
51	C₂×C₂²⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
52	#1808	C₂×C₅	1	1	yes	characteristic, complemented
53	#1347	C₂×C₅	1	4	yes	complemented
54	C₂⁴×C₅	—	4	1	no	—
55	C₂⁴×C₅	D₈	1	1	yes	characteristic
56	C₂×C₅×D₄	—	2	2	no	—
57	C₂×C₅×D₄	—	4	2	no	—
58	C₂×C₅×D₄	—	4	2	no	—
59	C₂×C₅×D₄	—	4	2	no	—
60	C₂×C₅×D₄	—	4	2	no	—
61	C₂×C₅×D₄	—	4	1	no	—
62	C₂×C₅×D₄	—	8	2	no	—
63	C₂×C₅×D₄	—	8	2	no	—
64	C₂²×C₄×C₅	—	4	1	no	—
65	C₂²×C₄×C₅	—	4	1	no	—
66	C₂²×C₄×C₅	—	4	1	no	—
67	C₂²×C₄×C₅	—	4	2	no	—
68	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
71	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
72	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
74	C₅×D₈	—	4	2	no	—
75	C₅×D₈	—	4	2	no	—
76	C₂×C₅×C₈	—	2	2	no	—
77	C₂×C₅×C₈	—	2	2	no	—
78	C₂×C₅×C₈	—	2	2	no	—
79	C₂×C₅×C₈	—	4	1	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	1	yes	characteristic
84	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
86	C₅×C₄⋊C₄	—	2	2	no	—
87	C₅×C₄⋊C₄	—	4	2	no	—
88	C₅×C₄⋊C₄	—	4	2	no	—
89	C₅×C₄⋊C₄	—	4	1	no	—
90	C₅×C₂²⋊C₄	—	4	2	no	—
91	C₅×C₂²⋊C₄	—	4	2	no	—
92	C₅×C₂²⋊C₄	—	8	2	no	—
93	C₅×C₁₆	—	4	2	no	—
94	C₂²×D₈	C₄×C₅	1	1	yes	characteristic, complemented
95	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
96	C₂×#121	—	2	1	no	—
97	C₂²×C₁₆	—	2	1	no	—
98	#466	—	2	1	no	—
99	#466	—	2	1	no	—
100	#466	C₄×C₅	1	2	yes	complemented
101	#466	C₂²×C₅	1	2	yes	—
102	C₂×#107	—	2	1	no	—
103	C₂×#107	C₄×C₅	1	1	yes	characteristic, complemented
104	C₂×#102	—	2	1	no	—
105	#380	—	2	1	no	—
106	C₂²⋊C₁₆	—	2	1	no	—
107	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	complemented
108	#340	C₂²×C₅	1	2	yes	complemented
109	C₂³×C₅	—	4	1	no	—
110	C₂³×C₅	—	4	1	no	—
111	C₂³×C₅	—	4	2	no	—
112	C₂³×C₅	—	8	2	no	—
113	C₂³×C₅	—	8	2	no	—
114	C₂³×C₅	—	8	2	no	—
115	C₂³×C₅	#102	1	1	yes	characteristic
116	C₂³×C₅	#102	1	1	yes	characteristic
117	C₂³×C₅	D₁₆	1	2	yes	—
118	C₂³×C₅	SD₃₂	1	2	yes	—
119	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
120	C₅×D₄	—	4	2	no	—
121	C₅×D₄	—	4	2	no	—
122	C₅×D₄	—	8	2	no	—
123	C₅×D₄	—	8	4	no	—
124	C₅×D₄	—	8	2	no	—
125	C₅×D₄	—	8	2	no	—
126	C₂×C₄×C₅	—	2	1	no	—
127	C₂×C₄×C₅	—	2	1	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	—	4	1	no	—
133	C₂×C₄×C₅	—	4	1	no	—
134	C₂×C₄×C₅	—	4	1	no	—
135	C₂×C₄×C₅	—	4	2	no	—
136	C₂×C₄×C₅	—	8	2	no	—
137	C₂×C₄×C₅	—	8	2	no	—
138	C₂×C₄×C₅	—	8	2	no	—
139	C₂×C₄×C₅	—	8	2	no	—
140	C₂×C₄×C₅	#102	1	1	yes	characteristic
141	C₂×C₄×C₅	#102	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
144	C₅×C₈	—	2	1	no	—
145	C₅×C₈	—	2	1	no	—
146	C₅×C₈	—	2	1	no	—
147	C₅×C₈	#99	1	2	yes	—
148	C₂²×D₄	—	2	1	no	—
149	C₂²×D₄	—	4	1	no	—
150	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
151	C₂×D₈	—	2	1	no	—
152	C₂×D₈	—	4	2	no	—
153	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
154	C₂²×C₈	—	4	2	no	—
155	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
156	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
157	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
158	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
159	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
160	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
161	#121	—	2	2	no	—
162	#121	—	2	2	no	—
163	#121	—	4	1	no	—
164	#121	—	4	1	no	—
165	C₂×C₄⋊C₄	—	2	1	no	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₂²⋊C₄	—	4	1	no	—
168	C₂×C₂²⋊C₄	—	4	1	no	—
169	C₂×C₁₆	—	2	2	no	—
170	C₂×C₁₆	—	4	2	no	—
171	#107	—	2	1	no	—
172	#107	—	4	2	no	—
173	#107	C₂×C₄×C₅	1	2	yes	—
174	#102	—	2	2	no	—
175	#102	—	2	2	no	—
176	#95	—	4	1	no	—
177	C₂²×C₅	—	2	1	no	—
178	C₂²×C₅	—	2	1	no	—
179	C₂²×C₅	—	2	1	no	—
180	C₂²×C₅	—	2	1	no	—
181	C₂²×C₅	—	2	1	no	—
182	C₂²×C₅	—	2	1	no	—
183	C₂²×C₅	—	4	2	no	—
184	C₂²×C₅	—	4	2	no	—
185	C₂²×C₅	—	8	2	no	—
186	C₂²×C₅	—	8	2	no	—
187	C₂²×C₅	—	8	2	no	—
188	C₂²×C₅	—	16	4	no	—
189	C₂²×C₅	#357	1	2	yes	—
190	C₂²×C₅	#357	1	2	yes	—
191	C₂²×C₅	C₂×#102	1	1	yes	characteristic
192	C₂²×C₅	C₂×D₁₆	1	1	yes	characteristic
193	C₂²×C₅	C₂×SD₃₂	1	1	yes	characteristic
194	C₄×C₅	—	2	1	no	—
195	C₄×C₅	—	2	1	no	—
196	C₄×C₅	—	2	1	no	—
197	C₄×C₅	—	8	2	no	—
198	C₄×C₅	—	16	2	no	—
199	C₄×C₅	C₂×#99	1	1	yes	characteristic
200	C₄×C₅	#421	1	1	yes	characteristic
201	C₂⁴	—	4	1	no	—
202	C₂⁴	C₅×D₈	1	1	yes	characteristic
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	4	2	no	—
205	C₂×D₄	—	4	2	no	—
206	C₂×D₄	—	4	2	no	—
207	C₂×D₄	—	4	2	no	—
208	C₂×D₄	—	4	1	no	—
209	C₂×D₄	—	8	2	no	—
210	C₂×D₄	—	8	2	no	—
211	C₂²×C₄	—	4	1	no	—
212	C₂²×C₄	—	4	1	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
216	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
217	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
218	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
219	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
221	D₈	—	4	2	no	—
222	D₈	—	4	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
230	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
231	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
232	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	4	1	no	—
237	C₂²⋊C₄	—	4	2	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	8	2	no	—
240	C₁₆	—	4	2	no	—
241	C₂×C₅	—	2	1	no	—
242	C₂×C₅	—	2	2	no	—
243	C₂×C₅	—	2	2	no	—
244	C₂×C₅	—	2	1	no	—
245	C₂×C₅	—	16	2	no	—
246	C₂×C₅	#1519	1	1	yes	characteristic
247	C₂×C₅	C₂×#357	1	1	yes	characteristic
248	C₂×C₅	#2145	1	1	yes	characteristic
249	C₂³	—	4	1	no	—
250	C₂³	—	4	1	no	—
251	C₂³	—	4	2	no	—
252	C₂³	—	8	2	no	—
253	C₂³	—	8	2	no	—
254	C₂³	—	8	2	no	—
255	C₂³	C₅×#102	1	1	yes	characteristic
256	C₂³	C₅×#102	1	1	yes	characteristic
257	C₂³	C₅×D₁₆	1	2	yes	—
258	C₂³	C₅×SD₃₂	1	2	yes	—
259	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
260	D₄	—	4	2	no	—
261	D₄	—	4	2	no	—
262	D₄	—	8	2	no	—
263	D₄	—	8	4	no	—
264	D₄	—	8	2	no	—
265	D₄	—	8	2	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	8	2	no	—
277	C₂×C₄	—	8	2	no	—
278	C₂×C₄	—	8	2	no	—
279	C₂×C₄	—	8	2	no	—
280	C₂×C₄	C₅×#102	1	1	yes	characteristic
281	C₂×C₄	C₅×#102	1	1	yes	characteristic
282	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
283	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	C₅×#99	1	2	yes	—
288	C₅	#11886	1	1	yes	characteristic, direct product factor, Sylow
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	4	2	no	—
297	C₂²	—	8	2	no	—
298	C₂²	—	8	2	no	—
299	C₂²	—	8	2	no	—
300	C₂²	—	16	4	no	—
301	C₂²	C₅×#357	1	2	yes	—
302	C₂²	C₅×#357	1	2	yes	—
303	C₂²	C₂×C₅×#102	1	1	yes	characteristic
304	C₂²	C₂×C₅×D₁₆	1	1	yes	characteristic
305	C₂²	C₂×C₅×SD₃₂	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	8	2	no	—
310	C₄	—	16	2	no	—
311	C₄	C₂×C₅×#99	1	1	yes	characteristic
312	C₄	C₅×#421	1	1	yes	characteristic
313	C₂	—	2	1	no	—
314	C₂	—	2	2	no	—
315	C₂	—	2	2	no	—
316	C₂	—	2	1	no	—
317	C₂	—	16	2	no	—
318	C₂	C₅×#1519	1	1	yes	characteristic
319	C₂	C₂×C₅×#357	1	1	yes	characteristic
320	C₂	C₅×#2145	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂×C₅×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×C₂²⋊C₁₆	C₂	semidirect
3. C₅×#1808	C₂	semidirect
4. C₅×#1347	C₂	semidirect
5. C₂²×C₅×D₈	C₄	semidirect
11. C₅×#466	C₄	semidirect
14. C₂×C₅×#107	C₄	semidirect
18. C₅×C₂²⋊C₁₆	C₂²	semidirect
19. C₅×#340	C₂²	semidirect
20. #11886	C₅	direct (N × H)
51. C₂×C₂²⋊C₁₆	C₂×C₅	semidirect
52. #1808	C₂×C₅	semidirect
53. #1347	C₂×C₅	semidirect
94. C₂²×D₈	C₄×C₅	semidirect
100. #466	C₄×C₅	semidirect
103. C₂×#107	C₄×C₅	semidirect
107. C₂²⋊C₁₆	C₂²×C₅	semidirect
108. #340	C₂²×C₅	semidirect
288. C₅	#11886	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×#466	2
C₂×C₅×C₂²⋊C₁₆	2
C₅×#1808	2
C₅×#1347	2
#11886	5

Subgroups — C5×#11886 #11934606

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#11886 #11934606