Subgroups — C₅×#8656 #11931376

Order 1280.

494 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#436	C₂	1	1	yes	maximal, characteristic
2	C₅×#1834	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#1802	C₂	1	2	yes	maximal
4	C₂×C₅×#334	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1296	C₂	1	2	yes	maximal, complemented
6	C₂²×C₅×M₁₆	—	2	1	no	—
7	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
8	C₅×#436	—	2	2	no	—
9	C₅×#436	C₄	1	4	yes	complemented
10	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₅×#102	—	2	2	no	—
13	C₂×C₅×C₂²⋊C₈	—	2	1	no	—
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₅×#384	C₂²	1	2	yes	—
16	C₅×#340	—	2	1	no	—
17	C₅×#340	—	2	1	no	—
18	C₅×#340	—	2	2	no	—
19	C₅×#334	—	2	2	no	—
20	#8656	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₅×D₄	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₅	—	2	1	no	—
23	C₂×C₅×M₁₆	—	2	4	no	—
24	C₂×C₅×M₁₆	—	4	4	no	—
25	C₂²×C₅×C₈	—	2	1	no	—
26	C₂²×C₅×C₈	—	2	2	no	—
27	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
28	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
29	C₄×C₅×D₄	—	2	1	no	—
30	C₄×C₅×D₄	—	2	1	no	—
31	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
32	C₄×C₅×D₄	Q₈	1	2	yes	—
33	C₂×C₅×C₄⋊C₄	—	2	2	no	—
34	C₂×C₅×C₄⋊C₄	—	2	2	no	—
35	C₂×C₅×C₄⋊C₄	—	2	2	no	—
36	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
37	C₂×C₅×C₂²⋊C₄	—	2	1	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	—	2	2	no	—
42	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
43	C₅×#102	—	4	4	no	—
44	C₅×C₂²⋊C₈	—	2	4	no	—
45	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
46	C₄×C₅×C₈	D₄	1	2	yes	—
47	C₂×#436	C₂×C₅	1	1	yes	characteristic
48	#1834	C₂×C₅	1	1	yes	characteristic, complemented
49	#1802	C₂×C₅	1	2	yes	—
50	C₂×#334	C₂×C₅	1	1	yes	characteristic, complemented
51	#1296	C₂×C₅	1	2	yes	complemented
52	C₂⁴×C₅	—	2	1	no	—
53	C₂×C₅×D₄	—	2	1	no	—
54	C₂×C₅×D₄	—	2	2	no	—
55	C₂×C₅×D₄	—	2	2	no	—
56	C₂×C₅×D₄	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₅	—	2	1	no	—
58	C₂²×C₄×C₅	—	2	2	no	—
59	C₂²×C₄×C₅	—	2	2	no	—
60	C₂²×C₄×C₅	—	4	2	no	—
61	C₂²×C₄×C₅	—	4	2	no	—
62	C₂²×C₄×C₅	—	4	2	no	—
63	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	#41	1	1	yes	characteristic
66	C₅×M₁₆	—	4	8	no	—
67	C₂×C₅×C₈	—	2	2	no	—
68	C₂×C₅×C₈	—	2	2	no	—
69	C₂×C₅×C₈	—	2	2	no	—
70	C₂×C₅×C₈	—	4	2	no	—
71	C₂×C₅×C₈	—	4	2	no	—
72	C₂×C₅×C₈	—	4	2	no	—
73	C₂×C₅×C₈	—	4	2	no	—
74	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
75	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
76	C₅×C₄⋊C₄	—	2	1	no	—
77	C₅×C₄⋊C₄	—	4	2	no	—
78	C₅×C₄⋊C₄	—	4	2	no	—
79	C₅×C₄⋊C₄	—	4	2	no	—
80	C₅×C₄⋊C₄	—	4	2	no	—
81	C₅×C₄⋊C₄	—	8	2	no	—
82	C₅×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
83	C₅×C₂²⋊C₄	—	2	2	no	—
84	C₅×C₂²⋊C₄	—	2	2	no	—
85	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
88	C₄²×C₅	#41	1	1	yes	characteristic
89	C₂²×M₁₆	—	2	1	no	—
90	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
91	#436	—	2	2	no	—
92	#436	C₄×C₅	1	4	yes	complemented
93	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
94	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
95	C₂×#102	—	2	2	no	—
96	C₂×C₂²⋊C₈	—	2	1	no	—
97	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
98	#384	C₂²×C₅	1	2	yes	—
99	#340	—	2	1	no	—
100	#340	—	2	1	no	—
101	#340	—	2	2	no	—
102	#334	—	2	2	no	—
103	C₂³×C₅	—	2	2	no	—
104	C₂³×C₅	—	4	2	no	—
105	C₂³×C₅	—	4	1	no	—
106	C₂³×C₅	—	4	1	no	—
107	C₂³×C₅	—	4	2	no	—
108	C₂³×C₅	#121	1	1	yes	characteristic
109	C₅×D₄	—	2	4	no	—
110	C₅×D₄	—	4	2	no	—
111	C₂×C₄×C₅	—	2	2	no	—
112	C₂×C₄×C₅	—	2	2	no	—
113	C₂×C₄×C₅	—	4	2	no	—
114	C₂×C₄×C₅	—	4	2	no	—
115	C₂×C₄×C₅	—	4	2	no	—
116	C₂×C₄×C₅	—	4	4	no	—
117	C₂×C₄×C₅	—	8	2	no	—
118	C₂×C₄×C₅	—	8	2	no	—
119	C₂×C₄×C₅	—	8	2	no	—
120	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₅	#117	1	1	yes	characteristic
123	C₂×C₄×C₅	#121	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
126	C₂×C₄×C₅	#135	1	2	yes	—
127	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
128	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
129	C₂×C₄×C₅	#137	1	2	yes	—
130	C₅×C₈	—	2	2	no	—
131	C₅×C₈	—	2	2	no	—
132	C₅×C₈	—	4	2	no	—
133	C₅×C₈	—	8	2	no	—
134	C₂²×D₄	C₅×D₄	1	1	yes	characteristic
135	C₂³×C₄	—	2	1	no	—
136	C₂×M₁₆	—	2	4	no	—
137	C₂×M₁₆	—	4	4	no	—
138	C₂²×C₈	—	2	1	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×D₄	—	2	1	no	—
143	C₄×D₄	—	2	1	no	—
144	C₄×D₄	C₂×C₄×C₅	1	2	yes	—
145	C₄×D₄	C₅×Q₈	1	2	yes	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₄⋊C₄	—	2	2	no	—
149	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
150	C₂×C₂²⋊C₄	—	2	1	no	—
151	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
152	C₄⋊C₈	—	2	1	no	—
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	—	2	2	no	—
155	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
156	#102	—	4	4	no	—
157	C₂²⋊C₈	—	2	4	no	—
158	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
159	C₄×C₈	C₅×D₄	1	2	yes	—
160	C₂²×C₅	—	4	2	no	—
161	C₂²×C₅	—	4	2	no	—
162	C₂²×C₅	—	4	2	no	—
163	C₂²×C₅	—	8	2	no	—
164	C₂²×C₅	—	8	2	no	—
165	C₂²×C₅	#380	1	1	yes	characteristic
166	C₂²×C₅	#463	1	2	yes	—
167	C₂²×C₅	#469	1	1	yes	characteristic
168	C₂²×C₅	#470	1	1	yes	characteristic
169	C₂²×C₅	#471	1	2	yes	—
170	C₄×C₅	—	2	1	no	—
171	C₄×C₅	—	2	1	no	—
172	C₄×C₅	—	4	2	no	—
173	C₄×C₅	—	16	2	no	—
174	C₄×C₅	#416	1	1	yes	characteristic
175	C₄×C₅	#418	1	1	yes	characteristic
176	C₄×C₅	#428	1	2	yes	—
177	C₄×C₅	#468	1	2	yes	—
178	C₄×C₅	#478	1	2	yes	—
179	C₂⁴	—	2	1	no	—
180	C₂×D₄	—	2	1	no	—
181	C₂×D₄	—	2	2	no	—
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	C₅×C₄⋊C₄	1	2	yes	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
192	C₂²×C₄	C₅×#41	1	1	yes	characteristic
193	M₁₆	—	4	8	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
202	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
203	C₄⋊C₄	—	2	1	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	—	8	2	no	—
209	C₄⋊C₄	C₅×C₄⋊C₄	1	2	yes	—
210	C₂²⋊C₄	—	2	2	no	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
213	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
214	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
215	C₄²	C₅×#41	1	1	yes	characteristic
216	C₂×C₅	—	8	2	no	—
217	C₂×C₅	#1701	1	2	yes	—
218	C₂×C₅	#1703	1	2	yes	—
219	C₂×C₅	#1804	1	1	yes	characteristic
220	C₂×C₅	#1809	1	1	yes	characteristic
221	C₂×C₅	#1815	1	1	yes	characteristic
222	C₂³	—	2	2	no	—
223	C₂³	—	4	2	no	—
224	C₂³	—	4	1	no	—
225	C₂³	—	4	1	no	—
226	C₂³	—	4	2	no	—
227	C₂³	C₅×#121	1	1	yes	characteristic
228	D₄	—	2	4	no	—
229	D₄	—	4	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	8	2	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
240	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
241	C₂×C₄	C₅×#117	1	1	yes	characteristic
242	C₂×C₄	C₅×#121	1	1	yes	characteristic
243	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
244	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
245	C₂×C₄	C₅×#135	1	2	yes	—
246	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
247	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
248	C₂×C₄	C₅×#137	1	2	yes	—
249	C₈	—	2	2	no	—
250	C₈	—	2	2	no	—
251	C₈	—	4	2	no	—
252	C₈	—	8	2	no	—
253	C₅	#8656	1	1	yes	characteristic, direct product factor, Sylow
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	8	2	no	—
258	C₂²	—	8	2	no	—
259	C₂²	C₅×#380	1	1	yes	characteristic
260	C₂²	C₅×#463	1	2	yes	—
261	C₂²	C₅×#469	1	1	yes	characteristic
262	C₂²	C₅×#470	1	1	yes	characteristic
263	C₂²	C₅×#471	1	2	yes	—
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	4	2	no	—
267	C₄	—	16	2	no	—
268	C₄	C₅×#416	1	1	yes	characteristic
269	C₄	C₅×#418	1	1	yes	characteristic
270	C₄	C₅×#428	1	2	yes	—
271	C₄	C₅×#468	1	2	yes	—
272	C₄	C₅×#478	1	2	yes	—
273	C₂	—	8	2	no	—
274	C₂	C₅×#1701	1	2	yes	—
275	C₂	C₅×#1703	1	2	yes	—
276	C₂	C₅×#1804	1	1	yes	characteristic
277	C₂	C₅×#1809	1	1	yes	characteristic
278	C₂	C₅×#1815	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	#121
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂²×C₅×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#1834	C₂	semidirect
4. C₂×C₅×#334	C₂	semidirect
5. C₅×#1296	C₂	semidirect
9. C₅×#436	C₄	semidirect
20. #8656	C₅	direct (N × H)
48. #1834	C₂×C₅	semidirect
50. C₂×#334	C₂×C₅	semidirect
51. #1296	C₂×C₅	semidirect
92. #436	C₄×C₅	semidirect
253. C₅	#8656	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×#436	2
C₅×#1834	2
C₅×#1802	2
C₂×C₅×#334	2
C₅×#1296	2
#8656	5

Subgroups — C5×#8656 #11931376

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8656 #11931376