Subgroups — C₅×#8678 #11931398

Order 1280.

418 conjugacy classes of subgroups.

The subgroup lattice has 270 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×#1835	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#1564	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1517	C₂	1	2	yes	maximal
5	C₅×#1286	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	complemented
10	C₅×C₈×D₄	—	2	1	no	—
11	C₅×C₈×D₄	C₄	1	2	yes	complemented
12	C₅×#431	C₂²	1	2	yes	—
13	C₅×#423	C₂²	1	1	yes	characteristic
14	C₅×#408	—	2	1	no	—
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₅×#382	C₂²	1	2	yes	—
17	C₅×#340	—	2	1	no	—
18	C₅×#340	—	2	1	no	—
19	C₅×#335	—	2	1	no	—
20	C₅×#334	—	2	1	no	—
21	C₅×#327	—	2	4	no	—
22	#8678	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂×C₅×#41	—	2	1	no	—
24	C₅×#131	—	4	4	no	—
25	C₂×C₅×M₁₆	—	2	1	no	—
26	C₂×C₅×M₁₆	—	2	2	no	—
27	C₂²×C₅×C₈	—	2	2	no	—
28	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
30	C₅×C₄⋊Q₈	Q₈	1	1	yes	characteristic
31	C₅×#127	Q₈	1	1	yes	characteristic
32	C₅×#124	—	2	1	no	—
33	C₅×#121	—	2	1	no	—
34	C₅×#121	D₄	1	2	yes	—
35	C₄×C₅×D₄	—	2	1	no	—
36	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
37	C₂×C₅×C₄⋊C₄	—	2	2	no	—
38	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
39	C₅×C₄⋊C₈	—	2	1	no	—
40	C₅×C₄⋊C₈	—	2	2	no	—
41	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
42	C₅×C₂²⋊C₈	—	2	2	no	—
43	C₅×C₂²⋊C₈	—	2	2	no	—
44	C₅×C₂²⋊C₈	—	2	2	no	—
45	C₄×C₅×C₈	—	2	2	no	—
46	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
47	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
48	C₄×C₅×C₈	D₄	1	1	yes	characteristic
49	C₄×C₅×C₈	D₄	1	1	yes	characteristic
50	C₅×#95	—	2	2	no	—
51	C₅×#95	—	2	2	no	—
52	#2931	C₂×C₅	1	1	yes	characteristic
53	#1835	C₂×C₅	1	1	yes	characteristic, complemented
54	#1564	C₂×C₅	1	1	yes	characteristic, complemented
55	#1517	C₂×C₅	1	2	yes	—
56	#1286	C₂×C₅	1	2	yes	complemented
57	C₅×#41	—	4	2	no	—
58	C₅×#41	—	4	2	no	—
59	C₂×C₅×Q₈	—	2	1	no	—
60	C₂×C₅×D₄	—	2	2	no	—
61	C₂×C₅×D₄	—	2	1	no	—
62	C₂×C₅×D₄	—	2	1	no	—
63	C₂×C₅×D₄	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₅	—	2	2	no	—
65	C₂²×C₄×C₅	—	4	2	no	—
66	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂×D₄	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	C₂×C₅×C₈	—	2	2	no	—
72	C₂×C₅×C₈	—	2	2	no	—
73	C₂×C₅×C₈	—	2	2	no	—
74	C₂×C₅×C₈	—	4	4	no	—
75	C₂×C₅×C₈	—	4	2	no	—
76	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
80	C₅×C₄⋊C₄	—	2	1	no	—
81	C₅×C₄⋊C₄	—	4	2	no	—
82	C₅×C₄⋊C₄	—	4	2	no	—
83	C₅×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
84	C₅×C₂²⋊C₄	—	2	2	no	—
85	C₅×C₂²⋊C₄	—	2	2	no	—
86	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
89	C₄²×C₅	#41	1	1	yes	characteristic
90	C₂×#131	—	2	1	no	—
91	#532	C₂²×C₅	1	1	yes	characteristic
92	#436	—	2	1	no	—
93	#436	C₄×C₅	1	2	yes	complemented
94	C₈×D₄	—	2	1	no	—
95	C₈×D₄	C₄×C₅	1	2	yes	complemented
96	#431	C₂²×C₅	1	2	yes	—
97	#423	C₂²×C₅	1	1	yes	characteristic
98	#408	—	2	1	no	—
99	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
100	#382	C₂²×C₅	1	2	yes	—
101	#340	—	2	1	no	—
102	#340	—	2	1	no	—
103	#335	—	2	1	no	—
104	#334	—	2	1	no	—
105	#327	—	2	4	no	—
106	C₂³×C₅	—	2	2	no	—
107	C₂³×C₅	#121	1	1	yes	characteristic
108	C₅×Q₈	—	4	2	no	—
109	C₅×D₄	—	4	2	no	—
110	C₅×D₄	—	4	2	no	—
111	C₅×D₄	—	4	4	no	—
112	C₅×D₄	—	4	2	no	—
113	C₂×C₄×C₅	—	2	2	no	—
114	C₂×C₄×C₅	—	4	1	no	—
115	C₂×C₄×C₅	—	4	1	no	—
116	C₂×C₄×C₅	—	4	2	no	—
117	C₂×C₄×C₅	—	4	2	no	—
118	C₂×C₄×C₅	—	8	2	no	—
119	C₂×C₄×C₅	—	8	2	no	—
120	C₂×C₄×C₅	—	8	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	C₂²⋊Q₈	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	C₂×#41	—	2	1	no	—
136	#131	—	4	4	no	—
137	C₂×M₁₆	—	2	1	no	—
138	C₂×M₁₆	—	2	2	no	—
139	C₂²×C₈	—	2	2	no	—
140	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
141	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
142	C₄⋊Q₈	C₅×Q₈	1	1	yes	characteristic
143	#127	C₅×Q₈	1	1	yes	characteristic
144	#124	—	2	1	no	—
145	#121	—	2	1	no	—
146	#121	C₅×D₄	1	2	yes	—
147	C₄×D₄	—	2	1	no	—
148	C₄×D₄	C₂×C₄×C₅	1	2	yes	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
151	C₄⋊C₈	—	2	1	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
154	C₂²⋊C₈	—	2	2	no	—
155	C₂²⋊C₈	—	2	2	no	—
156	C₂²⋊C₈	—	2	2	no	—
157	C₄×C₈	—	2	2	no	—
158	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
159	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
160	C₄×C₈	C₅×D₄	1	1	yes	characteristic
161	C₄×C₈	C₅×D₄	1	1	yes	characteristic
162	#95	—	2	2	no	—
163	#95	—	2	2	no	—
164	C₂²×C₅	—	2	1	no	—
165	C₂²×C₅	—	2	1	no	—
166	C₂²×C₅	—	4	2	no	—
167	C₂²×C₅	—	4	2	no	—
168	C₂²×C₅	—	4	2	no	—
169	C₂²×C₅	#380	1	1	yes	characteristic
170	C₂²×C₅	#463	1	2	yes	—
171	C₄×C₅	—	2	1	no	—
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	—	16	2	no	—
177	C₄×C₅	#416	1	1	yes	characteristic
178	C₄×C₅	#418	1	1	yes	characteristic
179	C₄×C₅	#478	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	—	4	2	no	—
189	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
191	C₂²×C₄	C₅×#41	1	1	yes	characteristic
192	M₁₆	—	4	4	no	—
193	M₁₆	—	4	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	4	4	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
200	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
201	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
202	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
203	C₄⋊C₄	—	2	1	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	C₅×C₄⋊C₄	1	2	yes	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	—	2	2	no	—
209	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
210	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
211	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
212	C₄²	C₅×#41	1	1	yes	characteristic
213	C₂×C₅	—	2	2	no	—
214	C₂×C₅	—	8	2	no	—
215	C₂×C₅	#1804	1	1	yes	characteristic
216	C₂×C₅	#1812	1	1	yes	characteristic
217	C₂×C₅	#1814	1	1	yes	characteristic
218	C₂³	—	2	2	no	—
219	C₂³	C₅×#121	1	1	yes	characteristic
220	Q₈	—	4	2	no	—
221	D₄	—	4	2	no	—
222	D₄	—	4	2	no	—
223	D₄	—	4	4	no	—
224	D₄	—	4	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	—	8	2	no	—
233	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
234	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
235	C₂×C₄	C₅×#117	1	1	yes	characteristic
236	C₂×C₄	C₅×#121	1	1	yes	characteristic
237	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
238	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
239	C₂×C₄	C₅×#135	1	2	yes	—
240	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	4	2	no	—
246	C₈	—	8	2	no	—
247	C₅	#8678	1	1	yes	characteristic, direct product factor, Sylow
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	2	no	—
253	C₂²	C₅×#380	1	1	yes	characteristic
254	C₂²	C₅×#463	1	2	yes	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	4	2	no	—
260	C₄	—	16	2	no	—
261	C₄	C₅×#416	1	1	yes	characteristic
262	C₄	C₅×#418	1	1	yes	characteristic
263	C₄	C₅×#478	1	2	yes	—
264	C₂	—	2	2	no	—
265	C₂	—	8	2	no	—
266	C₂	C₅×#1804	1	1	yes	characteristic
267	C₂	C₅×#1812	1	1	yes	characteristic
268	C₂	C₅×#1814	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×#1835	C₂	semidirect
3. C₅×#1564	C₂	semidirect
5. C₅×#1286	C₂	semidirect
9. C₅×#436	C₄	semidirect
11. C₅×C₈×D₄	C₄	semidirect
22. #8678	C₅	direct (N × H)
53. #1835	C₂×C₅	semidirect
54. #1564	C₂×C₅	semidirect
56. #1286	C₂×C₅	semidirect
93. #436	C₄×C₅	semidirect
95. C₈×D₄	C₄×C₅	semidirect
247. C₅	#8678	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×#1835	2
C₅×#1564	2
C₅×#1517	2
C₅×#1286	2
#8678	5

Subgroups — C5×#8678 #11931398

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8678 #11931398