Subgroups — C₅×#10866 #11933586

Order 1280.

746 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	C₂×C₅×#427	C₂	1	2	yes	maximal
2	C₂×C₅×#405	C₂	1	1	yes	maximal, characteristic
3	C₅×#1740	C₂	1	2	yes	maximal
4	C₂×C₅×H(3,ℤ/4)	C₂	1	2	yes	maximal
5	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
6	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₅×#117	C₂²	1	4	yes	—
8	C₂×C₅×#117	C₂²	1	1	yes	characteristic
9	C₅×#427	—	2	4	no	—
10	C₅×#427	C₄	1	8	yes	complemented
11	C₂×C₅×#107	—	2	2	no	—
12	C₂×C₅×#106	—	2	2	no	—
13	C₅×#405	—	2	2	no	—
14	C₅×#405	—	2	2	no	—
15	C₂×C₅×#97	C₄	1	2	yes	complemented
16	C₂×C₄×C₅×C₈	C₄	1	2	yes	complemented
17	C₅×#342	—	2	4	no	—
18	C₅×H(3,ℤ/4)	—	2	4	no	—
19	#10866	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
21	C₂×C₅×M₁₆	—	2	2	no	—
22	C₂×C₅×M₁₆	—	2	2	no	—
23	C₂×C₅×M₁₆	D₄	1	2	yes	—
24	C₂×C₅×M₁₆	Q₈	1	2	yes	—
25	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
29	C₂²×C₅×C₈	D₄	1	2	yes	—
30	C₂²×C₅×C₈	D₄	1	2	yes	—
31	C₂²×C₅×C₈	D₄	1	2	yes	—
32	C₂²×C₅×C₈	Q₈	1	2	yes	—
33	C₅×#117	—	2	2	no	—
34	C₅×#117	—	2	2	no	—
35	C₅×#117	—	2	4	no	—
36	C₅×#117	—	2	4	no	—
37	C₅×#117	—	2	4	no	—
38	C₅×#117	C₂×C₄	1	8	yes	—
39	C₂×C₅×C₄⋊C₄	—	2	4	no	—
40	C₂×C₅×C₄⋊C₄	Q₈	1	2	yes	—
41	C₂×C₅×C₂²⋊C₄	—	2	4	no	—
42	C₂×C₅×C₂²⋊C₄	D₄	1	2	yes	—
43	C₂×C₄²×C₅	—	2	4	no	—
44	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
45	C₅×#107	—	2	8	no	—
46	C₅×#106	—	2	8	no	—
47	C₅×#97	—	2	4	no	—
48	C₄×C₅×C₈	—	2	2	no	—
49	C₄×C₅×C₈	—	2	2	no	—
50	C₂×#427	C₂×C₅	1	2	yes	—
51	C₂×#405	C₂×C₅	1	1	yes	characteristic
52	#1740	C₂×C₅	1	2	yes	—
53	C₂×H(3,ℤ/4)	C₂×C₅	1	2	yes	—
54	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅	—	4	4	no	—
56	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
57	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₅	#41	1	2	yes	—
66	C₂²×C₄×C₅	#41	1	2	yes	—
67	C₅×M₁₆	—	2	4	no	—
68	C₅×M₁₆	—	2	4	no	—
69	C₂×C₅×C₈	—	2	2	no	—
70	C₂×C₅×C₈	—	2	2	no	—
71	C₂×C₅×C₈	C₄²	1	2	yes	—
72	C₂×C₅×C₈	C₄²	1	2	yes	—
73	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
76	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
77	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
78	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
79	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
80	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
81	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
82	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
83	C₅×C₄⋊C₄	—	2	4	no	—
84	C₅×C₄⋊C₄	—	2	8	no	—
85	C₅×C₄⋊C₄	—	4	4	no	—
86	C₅×C₂²⋊C₄	—	2	4	no	—
87	C₅×C₂²⋊C₄	—	2	8	no	—
88	C₅×C₂²⋊C₄	—	4	4	no	—
89	C₄²×C₅	—	2	2	no	—
90	C₄²×C₅	—	2	2	no	—
91	C₄²×C₅	—	2	8	no	—
92	C₄²×C₅	—	4	4	no	—
93	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
94	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
95	C₂×#117	C₂²×C₅	1	4	yes	—
96	C₂×#117	C₂²×C₅	1	1	yes	characteristic
97	#427	—	2	4	no	—
98	#427	C₄×C₅	1	8	yes	complemented
99	C₂×#107	—	2	2	no	—
100	C₂×#106	—	2	2	no	—
101	#405	—	2	2	no	—
102	#405	—	2	2	no	—
103	C₂×#97	C₄×C₅	1	2	yes	complemented
104	C₂×C₄×C₈	C₄×C₅	1	2	yes	complemented
105	#342	—	2	4	no	—
106	H(3,ℤ/4)	—	2	4	no	—
107	C₂³×C₅	—	4	2	no	—
108	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂³×C₅	C₄×D₄	1	2	yes	—
110	C₂³×C₅	#121	1	2	yes	—
111	C₂³×C₅	#127	1	1	yes	characteristic
112	C₂³×C₅	C₄⋊Q₈	1	1	yes	characteristic
113	C₂×C₄×C₅	—	2	4	no	—
114	C₂×C₄×C₅	—	4	4	no	—
115	C₂×C₄×C₅	—	4	8	no	—
116	C₂×C₄×C₅	—	8	8	no	—
117	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
120	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₅	#117	1	1	yes	characteristic
124	C₂×C₄×C₅	#117	1	1	yes	characteristic
125	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
126	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
127	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
128	C₂×C₄×C₅	#121	1	2	yes	—
129	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
130	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
131	C₂×C₄×C₅	#124	1	2	yes	—
132	C₂×C₄×C₅	#125	1	2	yes	—
133	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
134	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
135	C₅×C₈	—	2	4	no	—
136	C₅×C₈	#95	1	2	yes	—
137	C₅×C₈	#95	1	2	yes	—
138	C₅×C₈	#95	1	4	yes	—
139	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
140	C₂×M₁₆	—	2	2	no	—
141	C₂×M₁₆	—	2	2	no	—
142	C₂×M₁₆	C₅×D₄	1	2	yes	—
143	C₂×M₁₆	C₅×Q₈	1	2	yes	—
144	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
145	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
146	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
147	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
148	C₂²×C₈	C₅×D₄	1	2	yes	—
149	C₂²×C₈	C₅×D₄	1	2	yes	—
150	C₂²×C₈	C₅×D₄	1	2	yes	—
151	C₂²×C₈	C₅×Q₈	1	2	yes	—
152	#117	—	2	2	no	—
153	#117	—	2	2	no	—
154	#117	—	2	4	no	—
155	#117	—	2	4	no	—
156	#117	—	2	4	no	—
157	#117	C₂×C₄×C₅	1	8	yes	—
158	C₂×C₄⋊C₄	—	2	4	no	—
159	C₂×C₄⋊C₄	C₅×Q₈	1	2	yes	—
160	C₂×C₂²⋊C₄	—	2	4	no	—
161	C₂×C₂²⋊C₄	C₅×D₄	1	2	yes	—
162	C₂×C₄²	—	2	4	no	—
163	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
164	#107	—	2	8	no	—
165	#106	—	2	8	no	—
166	#97	—	2	4	no	—
167	C₄×C₈	—	2	2	no	—
168	C₄×C₈	—	2	2	no	—
169	C₂²×C₅	—	2	2	no	—
170	C₂²×C₅	—	2	4	no	—
171	C₂²×C₅	—	4	2	no	—
172	C₂²×C₅	—	4	1	no	—
173	C₂²×C₅	—	4	1	no	—
174	C₂²×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
175	C₂²×C₅	#380	1	1	yes	characteristic
176	C₂²×C₅	#384	1	1	yes	characteristic
177	C₂²×C₅	#389	1	2	yes	—
178	C₂²×C₅	#390	1	2	yes	—
179	C₄×C₅	—	4	4	no	—
180	C₄×C₅	—	8	8	no	—
181	C₄×C₅	C₂×#95	1	1	yes	characteristic
182	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
183	C₄×C₅	#380	1	1	yes	characteristic
184	C₄×C₅	#382	1	1	yes	characteristic
185	C₄×C₅	#389	1	2	yes	—
186	C₄×C₅	#391	1	2	yes	—
187	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
188	C₂²×C₄	—	4	4	no	—
189	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
190	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
191	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
192	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
193	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
198	C₂²×C₄	C₅×#41	1	2	yes	—
199	C₂²×C₄	C₅×#41	1	2	yes	—
200	M₁₆	—	2	4	no	—
201	M₁₆	—	2	4	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	C₄²×C₅	1	2	yes	—
205	C₂×C₈	C₄²×C₅	1	2	yes	—
206	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
207	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
208	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
209	C₂×C₈	C₅×C₂²⋊C₄	1	4	yes	—
210	C₂×C₈	C₅×C₂²⋊C₄	1	4	yes	—
211	C₂×C₈	C₅×C₄⋊C₄	1	4	yes	—
212	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
213	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
214	C₂×C₈	C₅×C₄⋊C₄	1	4	yes	—
215	C₂×C₈	C₅×C₄⋊C₄	1	4	yes	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	2	8	no	—
218	C₄⋊C₄	—	4	4	no	—
219	C₂²⋊C₄	—	2	4	no	—
220	C₂²⋊C₄	—	2	8	no	—
221	C₂²⋊C₄	—	4	4	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	2	8	no	—
225	C₄²	—	4	4	no	—
226	C₂×C₅	—	2	2	no	—
227	C₂×C₅	—	2	1	no	—
228	C₂×C₅	—	2	1	no	—
229	C₂×C₅	—	2	2	no	—
230	C₂×C₅	C₄⋊#95	1	1	yes	characteristic
231	C₂×C₅	#1812	1	2	yes	—
232	C₂³	—	4	2	no	—
233	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
234	C₂³	C₄×C₅×D₄	1	2	yes	—
235	C₂³	C₅×#121	1	2	yes	—
236	C₂³	C₅×#127	1	1	yes	characteristic
237	C₂³	C₅×C₄⋊Q₈	1	1	yes	characteristic
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	8	no	—
241	C₂×C₄	—	8	8	no	—
242	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
243	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
244	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
245	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
246	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
247	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
248	C₂×C₄	C₅×#117	1	1	yes	characteristic
249	C₂×C₄	C₅×#117	1	1	yes	characteristic
250	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
251	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
252	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
253	C₂×C₄	C₅×#121	1	2	yes	—
254	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
255	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
256	C₂×C₄	C₅×#124	1	2	yes	—
257	C₂×C₄	C₅×#125	1	2	yes	—
258	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
259	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
260	C₈	—	2	4	no	—
261	C₈	C₅×#95	1	2	yes	—
262	C₈	C₅×#95	1	2	yes	—
263	C₈	C₅×#95	1	4	yes	—
264	C₅	#10866	1	1	yes	characteristic, direct product factor, Sylow
265	C₂²	—	2	2	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	1	no	—
269	C₂²	—	4	1	no	—
270	C₂²	C₄×C₅×C₄⋊C₄	1	1	yes	characteristic
271	C₂²	C₅×#380	1	1	yes	characteristic
272	C₂²	C₅×#384	1	1	yes	characteristic
273	C₂²	C₅×#389	1	2	yes	—
274	C₂²	C₅×#390	1	2	yes	—
275	C₄	—	4	4	no	—
276	C₄	—	8	8	no	—
277	C₄	C₂×C₅×#95	1	1	yes	characteristic
278	C₄	C₄×C₅×C₄⋊C₄	1	1	yes	characteristic
279	C₄	C₅×#380	1	1	yes	characteristic
280	C₄	C₅×#382	1	1	yes	characteristic
281	C₄	C₅×#389	1	2	yes	—
282	C₄	C₅×#391	1	2	yes	—
283	C₂	—	2	2	no	—
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	—	2	2	no	—
287	C₂	C₅×C₄⋊#95	1	1	yes	characteristic
288	C₂	C₅×#1812	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	#117
commutator	C₂×C₄	C₂×C₄²×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
socle	C₂²×C₅	#380
prime radical	C₂⁴×C₅	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₅×M₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₅×#427	C₄	semidirect
15. C₂×C₅×#97	C₄	semidirect
16. C₂×C₄×C₅×C₈	C₄	semidirect
19. #10866	C₅	direct (N × H)
98. #427	C₄×C₅	semidirect
103. C₂×#97	C₄×C₅	semidirect
104. C₂×C₄×C₈	C₄×C₅	semidirect
264. C₅	#10866	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×#427	2
C₂×C₅×#405	2
C₅×#1740	2
C₂×C₅×H(3,ℤ/4)	2
#10866	5

Subgroups — C5×#10866 #11933586

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10866 #11933586