Subgroups — C₅×#8686 #11931406

Order 1280.

414 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2931	C₂	1	1	yes	maximal, characteristic
2	C₅×#1837	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#334	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1547	C₂	1	2	yes	maximal
5	C₅×#1292	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#131	—	2	1	no	—
7	C₅×#532	C₂²	1	1	yes	characteristic
8	C₅×#436	—	2	1	no	—
9	C₅×#436	C₄	1	2	yes	—
10	C₅×C₈×D₄	—	2	1	no	—
11	C₅×C₈×D₄	C₄	1	2	yes	—
12	C₅×#423	C₂²	1	2	yes	—
13	C₅×#423	C₂²	1	1	yes	characteristic
14	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
15	C₅×#408	—	2	1	no	—
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₅×#341	—	2	1	no	—
18	C₅×#341	—	2	1	no	—
19	C₅×#334	—	2	1	no	—
20	C₅×#334	—	2	1	no	—
21	C₅×#332	—	2	2	no	—
22	C₅×#331	—	2	2	no	—
23	#8686	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂×C₅×#41	—	2	1	no	—
25	C₅×#131	—	4	4	no	—
26	C₂×C₅×M₁₆	—	2	1	no	—
27	C₂×C₅×M₁₆	—	2	2	no	—
28	C₂×C₅×M₁₆	—	2	2	no	—
29	C₂²×C₅×C₈	—	2	2	no	—
30	C₂²×C₅×C₈	—	2	2	no	—
31	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
33	C₅×C₄⋊Q₈	D₄	1	1	yes	characteristic
34	C₅×#127	D₄	1	1	yes	characteristic
35	C₅×#124	—	2	1	no	—
36	C₅×#121	—	2	1	no	—
37	C₅×#121	Q₈	1	2	yes	—
38	C₄×C₅×D₄	—	2	1	no	—
39	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
41	C₅×C₄⋊C₈	—	2	1	no	—
42	C₅×C₄⋊C₈	—	2	2	no	—
43	C₅×C₄⋊C₈	—	2	2	no	—
44	C₅×C₄⋊C₈	—	2	2	no	—
45	C₅×C₄⋊C₈	—	2	2	no	—
46	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
47	C₅×C₂²⋊C₈	—	2	2	no	—
48	C₅×C₂²⋊C₈	—	2	2	no	—
49	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
50	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
51	C₄×C₅×C₈	D₄	1	1	yes	characteristic
52	C₄×C₅×C₈	D₄	1	1	yes	characteristic
53	#2931	C₂×C₅	1	1	yes	characteristic
54	#1837	C₂×C₅	1	1	yes	characteristic, complemented
55	C₂×#334	C₂×C₅	1	1	yes	characteristic, complemented
56	#1547	C₂×C₅	1	2	yes	—
57	#1292	C₂×C₅	1	2	yes	complemented
58	C₅×#41	—	4	2	no	—
59	C₅×#41	—	4	2	no	—
60	C₂×C₅×Q₈	—	2	1	no	—
61	C₂×C₅×D₄	—	2	2	no	—
62	C₂×C₅×D₄	—	2	1	no	—
63	C₂×C₅×D₄	—	2	1	no	—
64	C₂×C₅×D₄	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₅	—	2	2	no	—
66	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₅	#41	1	1	yes	characteristic
69	C₅×M₁₆	—	4	4	no	—
70	C₅×M₁₆	—	4	2	no	—
71	C₅×M₁₆	—	4	2	no	—
72	C₅×M₁₆	—	4	2	no	—
73	C₂×C₅×C₈	—	2	2	no	—
74	C₂×C₅×C₈	—	2	2	no	—
75	C₂×C₅×C₈	—	2	2	no	—
76	C₂×C₅×C₈	—	4	4	no	—
77	C₂×C₅×C₈	—	4	2	no	—
78	C₂×C₅×C₈	—	4	2	no	—
79	C₂×C₅×C₈	—	8	2	no	—
80	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
84	C₅×C₄⋊C₄	—	2	1	no	—
85	C₅×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
86	C₅×C₂²⋊C₄	—	2	2	no	—
87	C₅×C₂²⋊C₄	—	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×D₄	1	1	yes	characteristic
91	C₄²×C₅	#41	1	1	yes	characteristic
92	C₂×#131	—	2	1	no	—
93	#532	C₂²×C₅	1	1	yes	characteristic
94	#436	—	2	1	no	—
95	#436	C₄×C₅	1	2	yes	—
96	C₈×D₄	—	2	1	no	—
97	C₈×D₄	C₄×C₅	1	2	yes	—
98	#423	C₂²×C₅	1	2	yes	—
99	#423	C₂²×C₅	1	1	yes	characteristic
100	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
101	#408	—	2	1	no	—
102	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
103	#341	—	2	1	no	—
104	#341	—	2	1	no	—
105	#334	—	2	1	no	—
106	#334	—	2	1	no	—
107	#332	—	2	2	no	—
108	#331	—	2	2	no	—
109	C₂³×C₅	—	2	2	no	—
110	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
111	C₅×Q₈	—	4	2	no	—
112	C₅×D₄	—	4	2	no	—
113	C₅×D₄	—	4	4	no	—
114	C₅×D₄	—	4	2	no	—
115	C₅×D₄	—	4	2	no	—
116	C₂×C₄×C₅	—	2	2	no	—
117	C₂×C₄×C₅	—	2	2	no	—
118	C₂×C₄×C₅	—	4	1	no	—
119	C₂×C₄×C₅	—	4	2	no	—
120	C₂×C₄×C₅	—	4	2	no	—
121	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₅	#117	1	1	yes	characteristic
124	C₂×C₄×C₅	#121	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	#135	1	2	yes	—
128	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
129	C₅×C₈	—	2	1	no	—
130	C₅×C₈	—	2	1	no	—
131	C₅×C₈	—	2	1	no	—
132	C₅×C₈	—	2	1	no	—
133	C₅×C₈	—	4	2	no	—
134	C₅×C₈	—	8	2	no	—
135	C₅×C₈	—	8	2	no	—
136	C₂×#41	—	2	1	no	—
137	#131	—	4	4	no	—
138	C₂×M₁₆	—	2	1	no	—
139	C₂×M₁₆	—	2	2	no	—
140	C₂×M₁₆	—	2	2	no	—
141	C₂²×C₈	—	2	2	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	C₅×D₄	1	1	yes	characteristic
146	#127	C₅×D₄	1	1	yes	characteristic
147	#124	—	2	1	no	—
148	#121	—	2	1	no	—
149	#121	C₅×Q₈	1	2	yes	—
150	C₄×D₄	—	2	1	no	—
151	C₄×D₄	C₂×C₄×C₅	1	2	yes	—
152	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
153	C₄⋊C₈	—	2	1	no	—
154	C₄⋊C₈	—	2	2	no	—
155	C₄⋊C₈	—	2	2	no	—
156	C₄⋊C₈	—	2	2	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
159	C₂²⋊C₈	—	2	2	no	—
160	C₂²⋊C₈	—	2	2	no	—
161	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
162	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
163	C₄×C₈	C₅×D₄	1	1	yes	characteristic
164	C₄×C₈	C₅×D₄	1	1	yes	characteristic
165	C₂²×C₅	—	2	1	no	—
166	C₂²×C₅	—	2	1	no	—
167	C₂²×C₅	—	4	2	no	—
168	C₂²×C₅	—	4	2	no	—
169	C₂²×C₅	—	4	2	no	—
170	C₂²×C₅	#380	1	1	yes	characteristic
171	C₂²×C₅	#478	1	2	yes	—
172	C₄×C₅	—	2	1	no	—
173	C₄×C₅	—	2	1	no	—
174	C₄×C₅	—	2	1	no	—
175	C₄×C₅	—	4	2	no	—
176	C₄×C₅	#416	1	1	yes	characteristic
177	C₄×C₅	#418	1	1	yes	characteristic
178	C₄×C₅	#463	1	2	yes	—
179	C₄×C₅	#471	1	2	yes	—
180	#41	—	4	2	no	—
181	#41	—	4	2	no	—
182	C₂×Q₈	—	2	1	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	—	2	1	no	—
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	C₅×C₄⋊C₄	1	2	yes	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
190	C₂²×C₄	C₅×#41	1	1	yes	characteristic
191	M₁₆	—	4	4	no	—
192	M₁₆	—	4	2	no	—
193	M₁₆	—	4	2	no	—
194	M₁₆	—	4	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	4	4	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	8	2	no	—
202	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
203	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
204	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
205	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
206	C₄⋊C₄	—	2	1	no	—
207	C₄⋊C₄	C₅×C₄⋊C₄	1	2	yes	—
208	C₂²⋊C₄	—	2	2	no	—
209	C₂²⋊C₄	—	2	2	no	—
210	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
211	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
212	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
213	C₄²	C₅×#41	1	1	yes	characteristic
214	C₂×C₅	—	2	2	no	—
215	C₂×C₅	—	8	2	no	—
216	C₂×C₅	#1804	1	1	yes	characteristic
217	C₂×C₅	#1813	1	1	yes	characteristic
218	C₂×C₅	#1815	1	1	yes	characteristic
219	C₂³	—	2	2	no	—
220	C₂³	C₅×C₂²⋊Q₈	1	1	yes	characteristic
221	Q₈	—	4	2	no	—
222	D₄	—	4	2	no	—
223	D₄	—	4	4	no	—
224	D₄	—	4	2	no	—
225	D₄	—	4	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
232	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
233	C₂×C₄	C₅×#117	1	1	yes	characteristic
234	C₂×C₄	C₅×#121	1	1	yes	characteristic
235	C₂×C₄	C₅×#121	1	1	yes	characteristic
236	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
237	C₂×C₄	C₅×#135	1	2	yes	—
238	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	4	2	no	—
244	C₈	—	8	2	no	—
245	C₈	—	8	2	no	—
246	C₅	#8686	1	1	yes	characteristic, direct product factor, Sylow
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	C₅×#380	1	1	yes	characteristic
253	C₂²	C₅×#478	1	2	yes	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	4	2	no	—
258	C₄	C₅×#416	1	1	yes	characteristic
259	C₄	C₅×#418	1	1	yes	characteristic
260	C₄	C₅×#463	1	2	yes	—
261	C₄	C₅×#471	1	2	yes	—
262	C₂	—	2	2	no	—
263	C₂	—	8	2	no	—
264	C₂	C₅×#1804	1	1	yes	characteristic
265	C₂	C₅×#1813	1	1	yes	characteristic
266	C₂	C₅×#1815	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	#380
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₅×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#1837	C₂	semidirect
3. C₂×C₅×#334	C₂	semidirect
5. C₅×#1292	C₂	semidirect
23. #8686	C₅	direct (N × H)
54. #1837	C₂×C₅	semidirect
55. C₂×#334	C₂×C₅	semidirect
57. #1292	C₂×C₅	semidirect
246. C₅	#8686	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#2931	2
C₅×#1837	2
C₂×C₅×#334	2
C₅×#1547	2
C₅×#1292	2
#8686	5

Subgroups — C5×#8686 #11931406

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8686 #11931406