Subgroups — C₅×#22617 #11945337

Order 1280.

866 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#107	C₂	1	1	yes	maximal, characteristic
2	C₅×#2344	C₂	1	2	yes	maximal
3	C₅×#1943	C₂	1	2	yes	maximal, complemented
4	C₅×#1921	C₂	1	4	yes	maximal, complemented
5	C₅×#1823	C₂	1	2	yes	maximal
6	C₅×#1807	C₂	1	1	yes	maximal, characteristic
7	C₅×#1776	C₂	1	2	yes	maximal, complemented
8	C₅×#1752	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₅×#117	—	2	2	no	—
11	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₅×#107	C₄	1	8	yes	complemented
13	C₂×C₅×#107	C₂²	1	1	yes	characteristic
14	C₂×C₅×#107	C₂²	1	2	yes	—
15	C₂×C₅×#107	C₂²	1	1	yes	characteristic
16	C₂×C₅×#106	C₂²	1	2	yes	—
17	C₂×C₅×C₂²⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₅×#97	C₂²	1	2	yes	—
19	C₅×#389	C₂²	1	4	yes	—
20	C₅×#387	—	2	4	no	—
21	C₅×#381	—	2	2	no	—
22	C₅×#380	C₂²	1	2	yes	—
23	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
24	C₄×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
25	C₅×#340	C₂²	1	4	yes	—
26	C₅×#340	C₂²	1	2	yes	—
27	C₅×#340	C₂²	1	4	yes	—
28	C₅×#336	C₂²	1	2	yes	—
29	#22617	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂³×C₄×C₅	—	2	2	no	—
31	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂²×C₅×C₈	—	2	4	no	—
33	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
34	C₂²×C₅×C₈	C₂³	1	2	yes	—
35	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
37	C₅×#117	—	4	8	no	—
38	C₂×C₅×C₄⋊C₄	—	2	2	no	—
39	C₂×C₅×C₄⋊C₄	—	2	4	no	—
40	C₂×C₅×C₄⋊C₄	—	2	4	no	—
41	C₂×C₅×C₄⋊C₄	C₂×C₄	1	8	yes	—
42	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
43	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
44	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
45	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
46	C₂×C₅×C₂²⋊C₄	D₄	1	2	yes	—
47	C₂×C₄²×C₅	—	2	4	no	—
48	C₂×C₄²×C₅	C₂³	1	2	yes	—
49	C₅×#107	C₂×C₄	1	8	yes	—
50	C₅×#107	C₂×C₄	1	4	yes	—
51	C₅×#107	C₂×C₄	1	4	yes	—
52	C₅×#106	—	2	4	no	—
53	C₅×C₂²⋊C₈	—	2	2	no	—
54	C₅×#97	—	2	4	no	—
55	C₅×#95	—	2	4	no	—
56	C₅×#95	—	2	2	no	—
57	C₅×#95	—	2	4	no	—
58	C₅×#95	D₄	1	2	yes	—
59	C₂²×#107	C₂×C₅	1	1	yes	characteristic
60	#2344	C₂×C₅	1	2	yes	—
61	#1943	C₂×C₅	1	2	yes	complemented
62	#1921	C₂×C₅	1	4	yes	complemented
63	#1823	C₂×C₅	1	2	yes	—
64	#1807	C₂×C₅	1	1	yes	characteristic
65	#1776	C₂×C₅	1	2	yes	complemented
66	#1752	C₂×C₅	1	1	yes	characteristic
67	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	—	2	2	no	—
69	C₂²×C₄×C₅	—	2	2	no	—
70	C₂²×C₄×C₅	—	2	8	no	—
71	C₂²×C₄×C₅	—	2	2	no	—
72	C₂²×C₄×C₅	—	2	2	no	—
73	C₂²×C₄×C₅	—	2	4	no	—
74	C₂²×C₄×C₅	—	4	4	no	—
75	C₂²×C₄×C₅	—	4	4	no	—
76	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
78	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	#41	1	2	yes	—
80	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
81	C₂×C₅×C₈	—	2	2	no	—
82	C₂×C₅×C₈	—	2	8	no	—
83	C₂×C₅×C₈	—	2	2	no	—
84	C₂×C₅×C₈	—	2	2	no	—
85	C₂×C₅×C₈	—	2	2	no	—
86	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
87	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
88	C₂×C₅×C₈	#41	1	4	yes	—
89	C₅×C₄⋊C₄	—	2	8	no	—
90	C₅×C₄⋊C₄	—	2	2	no	—
91	C₅×C₄⋊C₄	—	2	2	no	—
92	C₅×C₄⋊C₄	—	2	4	no	—
93	C₅×C₄⋊C₄	—	2	4	no	—
94	C₅×C₄⋊C₄	—	4	8	no	—
95	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
96	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
97	C₅×C₂²⋊C₄	—	2	4	no	—
98	C₅×C₂²⋊C₄	—	4	4	no	—
99	C₄²×C₅	—	2	2	no	—
100	C₄²×C₅	—	2	2	no	—
101	C₄²×C₅	—	4	8	no	—
102	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
103	C₂×#117	—	2	2	no	—
104	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	—
105	C₂×#107	C₄×C₅	1	8	yes	complemented
106	C₂×#107	C₂²×C₅	1	2	yes	—
107	C₂×#107	C₂²×C₅	1	1	yes	characteristic
108	C₂×#107	C₂²×C₅	1	1	yes	characteristic
109	C₂×#106	C₂²×C₅	1	2	yes	—
110	C₂×C₂²⋊C₈	C₂²×C₅	1	1	yes	characteristic
111	C₂×#97	C₂²×C₅	1	2	yes	—
112	#389	C₂²×C₅	1	4	yes	—
113	#387	—	2	4	no	—
114	#381	—	2	2	no	—
115	#380	C₂²×C₅	1	2	yes	—
116	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
117	C₄×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
118	#340	C₂²×C₅	1	2	yes	—
119	#340	C₂²×C₅	1	4	yes	—
120	#340	C₂²×C₅	1	4	yes	—
121	#336	C₂²×C₅	1	2	yes	—
122	C₂³×C₅	—	2	2	no	—
123	C₂³×C₅	—	2	2	no	—
124	C₂³×C₅	C₄×D₄	1	2	yes	—
125	C₂³×C₅	AGL(1,ℤ/8)	1	4	yes	—
126	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₅	—	2	2	no	—
128	C₂×C₄×C₅	—	2	2	no	—
129	C₂×C₄×C₅	—	2	2	no	—
130	C₂×C₄×C₅	—	2	4	no	—
131	C₂×C₄×C₅	—	2	4	no	—
132	C₂×C₄×C₅	—	2	4	no	—
133	C₂×C₄×C₅	—	2	4	no	—
134	C₂×C₄×C₅	—	2	4	no	—
135	C₂×C₄×C₅	—	2	4	no	—
136	C₂×C₄×C₅	—	4	2	no	—
137	C₂×C₄×C₅	—	4	2	no	—
138	C₂×C₄×C₅	—	4	2	no	—
139	C₂×C₄×C₅	—	4	2	no	—
140	C₂×C₄×C₅	—	4	16	no	—
141	C₂×C₄×C₅	—	4	4	no	—
142	C₂×C₄×C₅	—	4	4	no	—
143	C₂×C₄×C₅	—	4	4	no	—
144	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
145	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂×#41	1	2	yes	—
147	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
148	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
149	C₅×C₈	—	2	2	no	—
150	C₅×C₈	—	4	2	no	—
151	C₅×C₈	#117	1	4	yes	complemented
152	C₂³×C₄	—	2	2	no	—
153	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
154	C₂²×C₈	—	2	4	no	—
155	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
156	C₂²×C₈	C₂³×C₅	1	2	yes	—
157	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
158	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
159	#117	—	4	8	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	2	4	no	—
162	C₂×C₄⋊C₄	—	2	4	no	—
163	C₂×C₄⋊C₄	C₂×C₄×C₅	1	8	yes	—
164	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
165	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
166	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
167	C₂×C₂²⋊C₄	—	2	2	no	—
168	C₂×C₂²⋊C₄	C₅×D₄	1	2	yes	—
169	C₂×C₄²	—	2	4	no	—
170	C₂×C₄²	C₂³×C₅	1	2	yes	—
171	#107	C₂×C₄×C₅	1	8	yes	—
172	#107	C₂×C₄×C₅	1	4	yes	—
173	#107	C₂×C₄×C₅	1	4	yes	—
174	#106	—	2	4	no	—
175	C₂²⋊C₈	—	2	2	no	—
176	#97	—	2	4	no	—
177	#95	—	2	4	no	—
178	#95	—	2	2	no	—
179	#95	—	2	4	no	—
180	#95	C₅×D₄	1	2	yes	—
181	C₂²×C₅	—	2	8	no	—
182	C₂²×C₅	—	2	2	no	—
183	C₂²×C₅	—	2	2	no	—
184	C₂²×C₅	#442	1	4	yes	—
185	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
186	C₂²×C₅	#547	1	2	yes	—
187	C₂²×C₅	C₂×AGL(1,ℤ/8)	1	2	yes	—
188	C₂²×C₅	#578	1	2	yes	—
189	C₄×C₅	—	2	2	no	—
190	C₄×C₅	—	4	4	no	—
191	C₄×C₅	—	4	4	no	—
192	C₄×C₅	—	4	4	no	—
193	C₄×C₅	—	8	4	no	—
194	C₄×C₅	C₂×#117	1	1	yes	characteristic
195	C₄×C₅	#519	1	1	yes	characteristic
196	C₄×C₅	#552	1	2	yes	—
197	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	8	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	4	4	no	—
205	C₂²×C₄	—	4	4	no	—
206	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
207	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
208	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
209	C₂²×C₄	C₅×#41	1	2	yes	—
210	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	8	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
217	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
218	C₂×C₈	C₅×#41	1	4	yes	—
219	C₄⋊C₄	—	2	8	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	4	8	no	—
225	C₄⋊C₄	C₂²×C₄×C₅	1	4	yes	—
226	C₄⋊C₄	C₂²×C₄×C₅	1	4	yes	—
227	C₂²⋊C₄	—	2	4	no	—
228	C₂²⋊C₄	—	4	4	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	4	8	no	—
232	C₂×C₅	—	2	4	no	—
233	C₂×C₅	#2344	1	1	yes	characteristic
234	C₂×C₅	C₂×#442	1	1	yes	characteristic
235	C₂×C₅	#2950	1	1	yes	characteristic
236	C₂×C₅	#3343	1	4	yes	—
237	C₂³	—	2	2	no	—
238	C₂³	—	2	2	no	—
239	C₂³	C₄×C₅×D₄	1	2	yes	—
240	C₂³	C₅×AGL(1,ℤ/8)	1	4	yes	—
241	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	4	16	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
260	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
261	C₂×C₄	C₂×C₅×#41	1	2	yes	—
262	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
263	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
264	C₈	—	2	2	no	—
265	C₈	—	4	2	no	—
266	C₈	C₅×#117	1	4	yes	complemented
267	C₅	#22617	1	1	yes	characteristic, direct product factor, Sylow
268	C₂²	—	2	8	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	C₅×#442	1	4	yes	—
272	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
273	C₂²	C₅×#547	1	2	yes	—
274	C₂²	C₂×C₅×AGL(1,ℤ/8)	1	2	yes	—
275	C₂²	C₅×#578	1	2	yes	—
276	C₄	—	2	2	no	—
277	C₄	—	4	4	no	—
278	C₄	—	4	4	no	—
279	C₄	—	4	4	no	—
280	C₄	—	8	4	no	—
281	C₄	C₂×C₅×#117	1	1	yes	characteristic
282	C₄	C₅×#519	1	1	yes	characteristic
283	C₄	C₅×#552	1	2	yes	—
284	C₂	—	2	4	no	—
285	C₂	C₅×#2344	1	1	yes	characteristic
286	C₂	C₂×C₅×#442	1	1	yes	characteristic
287	C₂	C₅×#2950	1	1	yes	characteristic
288	C₂	C₅×#3343	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂⁴×C₅	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄×C₅×C₂²⋊C₄ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1943	C₂	semidirect
4. C₅×#1921	C₂	semidirect
7. C₅×#1776	C₂	semidirect
12. C₂×C₅×#107	C₄	semidirect
29. #22617	C₅	direct (N × H)
61. #1943	C₂×C₅	semidirect
62. #1921	C₂×C₅	semidirect
65. #1776	C₂×C₅	semidirect
105. C₂×#107	C₄×C₅	semidirect
151. C₅×C₈	#117	semidirect
266. C₈	C₅×#117	semidirect
267. C₅	#22617	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×#107	2
C₅×#2344	2
C₅×#1943	2
C₅×#1921	2
C₅×#1823	2
C₅×#1807	2
C₅×#1776	2
C₅×#1752	2
#22617	5

Subgroups — C5×#22617 #11945337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22617 #11945337