Subgroups — C₅×#8618 #11931338

Order 1280.

454 conjugacy classes of subgroups.

The subgroup lattice has 288 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×#1800	C₂	1	2	yes	maximal
3	C₅×#1567	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1566	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1292	C₂	1	2	yes	maximal, complemented
6	C₂²×C₅×M₁₆	C₄	1	2	yes	—
7	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
8	C₅×#436	—	2	1	no	—
9	C₅×#436	—	2	1	no	—
10	C₅×#436	—	2	1	no	—
11	C₅×#436	—	2	1	no	—
12	C₅×#423	C₂²	1	2	yes	—
13	C₅×#423	C₂²	1	2	yes	—
14	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₅×#100	—	2	2	no	—
16	C₂×C₅×C₂²⋊C₈	C₄	1	2	yes	—
17	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₅×#343	—	2	2	no	—
19	C₅×#335	—	2	1	no	—
20	C₅×#335	—	2	1	no	—
21	C₅×#334	—	2	1	no	—
22	C₅×#334	—	2	1	no	—
23	#8618	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂²×C₅×D₄	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
26	C₂×C₅×M₁₆	—	2	2	no	—
27	C₂×C₅×M₁₆	—	2	2	no	—
28	C₂×C₅×M₁₆	—	2	2	no	—
29	C₂×C₅×M₁₆	—	2	2	no	—
30	C₂×C₅×M₁₆	—	4	4	no	—
31	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
33	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
34	C₄×C₅×D₄	—	2	1	no	—
35	C₄×C₅×D₄	—	2	1	no	—
36	C₄×C₅×D₄	—	2	1	no	—
37	C₄×C₅×D₄	—	2	1	no	—
38	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
39	C₂×C₅×C₂²⋊C₄	Q₈	1	2	yes	—
40	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
41	C₅×C₄⋊C₈	—	2	1	no	—
42	C₅×C₄⋊C₈	—	2	1	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	—	2	2	no	—
47	C₅×#100	—	4	4	no	—
48	C₅×C₂²⋊C₈	—	2	2	no	—
49	C₅×C₂²⋊C₈	—	2	2	no	—
50	C₄×C₅×C₈	D₄	1	1	yes	characteristic
51	C₄×C₅×C₈	D₄	1	1	yes	characteristic
52	C₄×C₅×C₈	D₄	1	1	yes	characteristic
53	C₄×C₅×C₈	D₄	1	1	yes	characteristic
54	C₂×#436	C₂×C₅	1	1	yes	characteristic
55	#1800	C₂×C₅	1	2	yes	—
56	#1567	C₂×C₅	1	1	yes	characteristic, complemented
57	#1566	C₂×C₅	1	1	yes	characteristic, complemented
58	#1292	C₂×C₅	1	2	yes	complemented
59	C₂⁴×C₅	C₄⋊C₄	1	2	yes	—
60	C₂×C₅×D₄	—	2	1	no	—
61	C₂×C₅×D₄	—	2	1	no	—
62	C₂×C₅×D₄	—	2	2	no	—
63	C₂×C₅×D₄	—	2	2	no	—
64	C₂²×C₄×C₅	—	2	2	no	—
65	C₂²×C₄×C₅	—	2	2	no	—
66	C₂²×C₄×C₅	—	4	2	no	—
67	C₂²×C₄×C₅	—	4	2	no	—
68	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
72	C₅×M₁₆	—	4	4	no	—
73	C₅×M₁₆	—	4	4	no	—
74	C₅×M₁₆	—	4	2	no	—
75	C₅×M₁₆	—	4	2	no	—
76	C₅×M₁₆	—	4	2	no	—
77	C₅×M₁₆	—	4	2	no	—
78	C₂×C₅×C₈	—	2	2	no	—
79	C₂×C₅×C₈	—	2	2	no	—
80	C₂×C₅×C₈	—	4	1	no	—
81	C₂×C₅×C₈	—	4	2	no	—
82	C₂×C₅×C₈	—	4	2	no	—
83	C₂×C₅×C₈	—	4	2	no	—
84	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
88	C₅×C₄⋊C₄	—	2	1	no	—
89	C₅×C₄⋊C₄	—	2	1	no	—
90	C₅×C₂²⋊C₄	—	2	2	no	—
91	C₅×C₂²⋊C₄	—	2	2	no	—
92	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₅	#41	1	1	yes	characteristic
95	C₄²×C₅	#41	1	1	yes	characteristic
96	C₂²×M₁₆	C₄×C₅	1	2	yes	—
97	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
98	#436	—	2	1	no	—
99	#436	—	2	1	no	—
100	#436	—	2	1	no	—
101	#436	—	2	1	no	—
102	#423	C₂²×C₅	1	2	yes	—
103	#423	C₂²×C₅	1	2	yes	—
104	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
105	C₂×#100	—	2	2	no	—
106	C₂×C₂²⋊C₈	C₄×C₅	1	2	yes	—
107	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
108	#343	—	2	2	no	—
109	#335	—	2	1	no	—
110	#335	—	2	1	no	—
111	#334	—	2	1	no	—
112	#334	—	2	1	no	—
113	C₂³×C₅	—	2	2	no	—
114	C₂³×C₅	—	2	4	no	—
115	C₂³×C₅	—	4	2	no	—
116	C₂³×C₅	—	4	2	no	—
117	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₅×D₄	—	4	2	no	—
119	C₅×D₄	—	4	1	no	—
120	C₅×D₄	—	4	1	no	—
121	C₂×C₄×C₅	—	2	1	no	—
122	C₂×C₄×C₅	—	2	1	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	4	1	no	—
125	C₂×C₄×C₅	—	4	4	no	—
126	C₂×C₄×C₅	—	4	2	no	—
127	C₂×C₄×C₅	—	4	2	no	—
128	C₂×C₄×C₅	—	4	2	no	—
129	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₅	#117	1	1	yes	characteristic
131	C₂×C₄×C₅	#121	1	1	yes	characteristic
132	C₂×C₄×C₅	#121	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
135	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
136	C₂×C₄×C₅	#137	1	2	yes	—
137	C₅×C₈	—	2	1	no	—
138	C₅×C₈	—	2	1	no	—
139	C₅×C₈	—	2	1	no	—
140	C₅×C₈	—	4	2	no	—
141	C₅×C₈	—	8	2	no	—
142	C₅×C₈	—	8	2	no	—
143	C₅×C₈	#100	1	2	yes	complemented
144	C₂²×D₄	C₅×D₄	1	1	yes	characteristic
145	C₂³×C₄	C₂×C₄×C₅	1	2	yes	—
146	C₂×M₁₆	—	2	2	no	—
147	C₂×M₁₆	—	2	2	no	—
148	C₂×M₁₆	—	2	2	no	—
149	C₂×M₁₆	—	2	2	no	—
150	C₂×M₁₆	—	4	4	no	—
151	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
152	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
153	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
154	C₄×D₄	—	2	1	no	—
155	C₄×D₄	—	2	1	no	—
156	C₄×D₄	—	2	1	no	—
157	C₄×D₄	—	2	1	no	—
158	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
159	C₂×C₂²⋊C₄	C₅×Q₈	1	2	yes	—
160	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
161	C₄⋊C₈	—	2	1	no	—
162	C₄⋊C₈	—	2	1	no	—
163	C₄⋊C₈	—	2	2	no	—
164	C₄⋊C₈	—	2	2	no	—
165	C₄⋊C₈	—	2	2	no	—
166	C₄⋊C₈	—	2	2	no	—
167	#100	—	4	4	no	—
168	C₂²⋊C₈	—	2	2	no	—
169	C₂²⋊C₈	—	2	2	no	—
170	C₄×C₈	C₅×D₄	1	1	yes	characteristic
171	C₄×C₈	C₅×D₄	1	1	yes	characteristic
172	C₄×C₈	C₅×D₄	1	1	yes	characteristic
173	C₄×C₈	C₅×D₄	1	1	yes	characteristic
174	C₂²×C₅	—	2	1	no	—
175	C₂²×C₅	—	2	1	no	—
176	C₂²×C₅	—	4	2	no	—
177	C₂²×C₅	—	4	2	no	—
178	C₂²×C₅	—	4	2	no	—
179	C₂²×C₅	—	8	4	no	—
180	C₂²×C₅	#380	1	1	yes	characteristic
181	C₂²×C₅	#428	1	2	yes	—
182	C₄×C₅	—	2	1	no	—
183	C₄×C₅	—	2	1	no	—
184	C₄×C₅	—	2	1	no	—
185	C₄×C₅	—	4	2	no	—
186	C₄×C₅	C₂×#100	1	1	yes	characteristic
187	C₄×C₅	#413	1	1	yes	characteristic
188	C₄×C₅	#468	1	2	yes	—
189	C₄×C₅	#469	1	1	yes	characteristic
190	C₄×C₅	#470	1	1	yes	characteristic
191	C₂⁴	C₅×C₄⋊C₄	1	2	yes	—
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	—	2	1	no	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
201	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
204	M₁₆	—	4	4	no	—
205	M₁₆	—	4	4	no	—
206	M₁₆	—	4	2	no	—
207	M₁₆	—	4	2	no	—
208	M₁₆	—	4	2	no	—
209	M₁₆	—	4	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	4	1	no	—
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
217	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
218	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
219	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
220	C₄⋊C₄	—	2	1	no	—
221	C₄⋊C₄	—	2	1	no	—
222	C₂²⋊C₄	—	2	2	no	—
223	C₂²⋊C₄	—	2	2	no	—
224	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
225	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
226	C₄²	C₅×#41	1	1	yes	characteristic
227	C₄²	C₅×#41	1	1	yes	characteristic
228	C₂×C₅	—	2	2	no	—
229	C₂×C₅	—	8	2	no	—
230	C₂×C₅	#1800	1	1	yes	characteristic
231	C₂×C₅	#1809	1	1	yes	characteristic
232	C₂×C₅	#1813	1	1	yes	characteristic
233	C₂³	—	2	2	no	—
234	C₂³	—	2	4	no	—
235	C₂³	—	4	2	no	—
236	C₂³	—	4	2	no	—
237	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
238	D₄	—	4	2	no	—
239	D₄	—	4	1	no	—
240	D₄	—	4	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
250	C₂×C₄	C₅×#117	1	1	yes	characteristic
251	C₂×C₄	C₅×#121	1	1	yes	characteristic
252	C₂×C₄	C₅×#121	1	1	yes	characteristic
253	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
254	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
255	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
256	C₂×C₄	C₅×#137	1	2	yes	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	4	2	no	—
261	C₈	—	8	2	no	—
262	C₈	—	8	2	no	—
263	C₈	C₅×#100	1	2	yes	complemented
264	C₅	#8618	1	1	yes	characteristic, direct product factor, Sylow
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	8	4	no	—
271	C₂²	C₅×#380	1	1	yes	characteristic
272	C₂²	C₅×#428	1	2	yes	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	2	no	—
277	C₄	C₂×C₅×#100	1	1	yes	characteristic
278	C₄	C₅×#413	1	1	yes	characteristic
279	C₄	C₅×#468	1	2	yes	—
280	C₄	C₅×#469	1	1	yes	characteristic
281	C₄	C₅×#470	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	8	2	no	—
284	C₂	C₅×#1800	1	1	yes	characteristic
285	C₂	C₅×#1809	1	1	yes	characteristic
286	C₂	C₅×#1813	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	#380
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂²×C₅×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ 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
3. C₅×#1567	C₂	semidirect
4. C₅×#1566	C₂	semidirect
5. C₅×#1292	C₂	semidirect
23. #8618	C₅	direct (N × H)
56. #1567	C₂×C₅	semidirect
57. #1566	C₂×C₅	semidirect
58. #1292	C₂×C₅	semidirect
143. C₅×C₈	#100	semidirect
263. C₈	C₅×#100	semidirect
264. C₅	#8618	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×#1800	2
C₅×#1567	2
C₅×#1566	2
C₅×#1292	2
#8618	5

Subgroups — C5×#8618 #11931338

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8618 #11931338