Subgroups — C₅×#12647 #11935367

Order 1280.

486 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#442	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#361	C₂	1	1	yes	maximal, characteristic
3	C₅×#2140	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#347	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1773	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1386	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1383	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₅×D₄	—	2	1	no	—
10	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
12	C₅×#442	—	2	2	no	—
13	C₅×#442	C₄	1	4	yes	complemented
14	C₂×C₅×#108	C₂²	1	1	yes	characteristic
15	C₅×#427	C₂²	1	1	yes	characteristic
16	C₂×C₅×#106	C₂²	1	1	yes	characteristic
17	C₂×C₅×#102	—	2	1	no	—
18	C₅×#405	—	2	1	no	—
19	C₂×C₅×#97	C₂²	1	1	yes	characteristic
20	C₅×#361	C₄	1	4	yes	complemented
21	C₅×#357	C₄	1	4	yes	complemented
22	C₅×#347	—	2	2	no	—
23	C₅×#339	—	2	1	no	—
24	#12647	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂²×C₅×D₄	—	2	1	no	—
26	C₂³×C₄×C₅	—	4	1	no	—
27	C₂×C₅×D₈	—	2	1	no	—
28	C₂×C₅×D₈	—	2	1	no	—
29	C₂×C₅×D₈	—	2	1	no	—
30	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
31	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
32	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
33	C₂×C₅×M₁₆	—	2	1	no	—
34	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
35	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₄×C₅×D₄	—	2	4	no	—
37	C₄×C₅×D₄	—	4	2	no	—
38	C₅×#117	—	2	1	no	—
39	C₂×C₅×C₄⋊C₄	—	2	1	no	—
40	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
41	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₅×M₃₂	—	2	1	no	—
43	C₅×M₃₂	—	2	1	no	—
44	C₅×M₃₂	—	2	1	no	—
45	C₅×M₃₂	C₂×C₄	1	2	yes	complemented
46	C₂×C₅×C₁₆	—	2	1	no	—
47	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	complemented
48	C₅×#108	—	2	1	no	—
49	C₅×#108	C₂×C₄	1	2	yes	complemented
50	C₅×#107	C₂×C₄	1	2	yes	complemented
51	C₅×#106	—	2	1	no	—
52	C₅×#106	—	2	1	no	—
53	C₅×#106	C₂×C₄	1	2	yes	—
54	C₅×#102	—	2	4	no	—
55	C₅×#97	—	4	1	no	—
56	C₅×#97	C₂×C₄	1	2	yes	—
57	C₅×#97	D₄	1	2	yes	—
58	C₄×C₅×C₈	—	4	1	no	—
59	C₂×#442	C₂×C₅	1	1	yes	characteristic
60	C₂×#361	C₂×C₅	1	1	yes	characteristic
61	#2140	C₂×C₅	1	1	yes	characteristic
62	C₂×#347	C₂×C₅	1	1	yes	characteristic, complemented
63	#1773	C₂×C₅	1	1	yes	characteristic, complemented
64	#1386	C₂×C₅	1	1	yes	characteristic, complemented
65	#1383	C₂×C₅	1	1	yes	characteristic, complemented
66	C₂⁴×C₅	—	4	1	no	—
67	C₂×C₅×D₄	—	2	2	no	—
68	C₂×C₅×D₄	—	2	2	no	—
69	C₂×C₅×D₄	—	2	2	no	—
70	C₂×C₅×D₄	—	4	1	no	—
71	C₂×C₅×D₄	—	4	1	no	—
72	C₂×C₅×D₄	—	4	1	no	—
73	C₂²×C₄×C₅	—	2	1	no	—
74	C₂²×C₄×C₅	—	4	4	no	—
75	C₂²×C₄×C₅	—	4	1	no	—
76	C₂²×C₄×C₅	—	8	4	no	—
77	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₅×D₈	—	2	2	no	—
79	C₅×D₈	—	2	2	no	—
80	C₅×D₈	—	2	2	no	—
81	C₅×D₈	C₄²	1	4	yes	—
82	C₅×M₁₆	—	2	2	no	—
83	C₅×M₁₆	—	4	1	no	—
84	C₂×C₅×C₈	—	2	1	no	—
85	C₂×C₅×C₈	—	2	1	no	—
86	C₂×C₅×C₈	—	4	1	no	—
87	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
88	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
92	C₂×C₅×C₈	#41	1	1	yes	characteristic
93	C₂×C₅×C₈	#41	1	1	yes	characteristic
94	C₅×C₄⋊C₄	—	2	2	no	—
95	C₅×C₄⋊C₄	—	2	2	no	—
96	C₅×C₄⋊C₄	—	4	1	no	—
97	C₅×C₂²⋊C₄	—	4	4	no	—
98	C₅×C₂²⋊C₄	—	4	1	no	—
99	C₄²×C₅	—	2	1	no	—
100	C₄²×C₅	—	4	1	no	—
101	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
102	C₅×C₁₆	—	2	2	no	—
103	C₅×C₁₆	—	2	2	no	—
104	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
105	C₂×C₄×D₄	—	2	1	no	—
106	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
107	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
108	#442	—	2	2	no	—
109	#442	C₄×C₅	1	4	yes	complemented
110	C₂×#108	C₂²×C₅	1	1	yes	characteristic
111	#427	C₂²×C₅	1	1	yes	characteristic
112	C₂×#106	C₂²×C₅	1	1	yes	characteristic
113	C₂×#102	—	2	1	no	—
114	#405	—	2	1	no	—
115	C₂×#97	C₂²×C₅	1	1	yes	characteristic
116	#361	C₄×C₅	1	4	yes	complemented
117	#357	C₄×C₅	1	4	yes	complemented
118	#347	—	2	2	no	—
119	#339	—	2	1	no	—
120	C₂³×C₅	—	4	2	no	—
121	C₂³×C₅	—	4	2	no	—
122	C₂³×C₅	—	4	2	no	—
123	C₂³×C₅	—	8	2	no	—
124	C₂³×C₅	—	8	2	no	—
125	C₂³×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₅×D₄	—	2	4	no	—
127	C₅×D₄	—	4	2	no	—
128	C₅×D₄	—	4	2	no	—
129	C₅×D₄	—	4	2	no	—
130	C₂×C₄×C₅	—	2	2	no	—
131	C₂×C₄×C₅	—	4	2	no	—
132	C₂×C₄×C₅	—	4	2	no	—
133	C₂×C₄×C₅	—	4	2	no	—
134	C₂×C₄×C₅	—	8	8	no	—
135	C₂×C₄×C₅	—	8	2	no	—
136	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₂×C₄×C₅	#137	1	1	yes	characteristic
142	C₅×C₈	—	2	2	no	—
143	C₅×C₈	—	4	2	no	—
144	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
145	C₅×C₈	#117	1	1	yes	characteristic
146	C₅×C₈	C₄×D₄	1	1	yes	characteristic
147	C₅×C₈	C₄×D₄	1	1	yes	characteristic
148	C₂²×D₄	—	2	1	no	—
149	C₂³×C₄	—	4	1	no	—
150	C₂×D₈	—	2	1	no	—
151	C₂×D₈	—	2	1	no	—
152	C₂×D₈	—	2	1	no	—
153	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
154	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
155	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
156	C₂×M₁₆	—	2	1	no	—
157	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
158	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
159	C₄×D₄	—	2	4	no	—
160	C₄×D₄	—	4	2	no	—
161	#117	—	2	1	no	—
162	C₂×C₄⋊C₄	—	2	1	no	—
163	C₂×C₂²⋊C₄	—	4	1	no	—
164	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
165	M₃₂	—	2	1	no	—
166	M₃₂	—	2	1	no	—
167	M₃₂	—	2	1	no	—
168	M₃₂	C₂×C₄×C₅	1	2	yes	complemented
169	C₂×C₁₆	—	2	1	no	—
170	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	complemented
171	#108	—	2	1	no	—
172	#108	C₂×C₄×C₅	1	2	yes	complemented
173	#107	C₂×C₄×C₅	1	2	yes	complemented
174	#106	—	2	1	no	—
175	#106	—	2	1	no	—
176	#106	C₂×C₄×C₅	1	2	yes	—
177	#102	—	2	4	no	—
178	#97	—	4	1	no	—
179	#97	C₂×C₄×C₅	1	2	yes	—
180	#97	C₅×D₄	1	2	yes	—
181	C₄×C₈	—	4	1	no	—
182	C₂²×C₅	—	2	2	no	—
183	C₂²×C₅	—	4	4	no	—
184	C₂²×C₅	—	8	2	no	—
185	C₂²×C₅	—	8	2	no	—
186	C₂²×C₅	—	8	4	no	—
187	C₂²×C₅	—	8	4	no	—
188	C₂²×C₅	#418	1	1	yes	characteristic
189	C₂²×C₅	#440	1	1	yes	characteristic
190	C₂²×C₅	#442	1	1	yes	characteristic
191	C₄×C₅	—	4	2	no	—
192	C₄×C₅	—	8	2	no	—
193	C₄×C₅	—	8	2	no	—
194	C₄×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
195	C₄×C₅	#417	1	1	yes	characteristic
196	C₄×C₅	#440	1	1	yes	characteristic
197	C₄×C₅	#442	1	1	yes	characteristic
198	C₂⁴	—	4	1	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	—	4	1	no	—
203	C₂×D₄	—	4	1	no	—
204	C₂×D₄	—	4	1	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	4	4	no	—
207	C₂²×C₄	—	4	1	no	—
208	C₂²×C₄	—	8	4	no	—
209	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
210	D₈	—	2	2	no	—
211	D₈	—	2	2	no	—
212	D₈	—	2	2	no	—
213	D₈	C₄²×C₅	1	4	yes	—
214	M₁₆	—	2	2	no	—
215	M₁₆	—	4	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
220	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
221	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
222	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
223	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
224	C₂×C₈	C₅×#41	1	1	yes	characteristic
225	C₂×C₈	C₅×#41	1	1	yes	characteristic
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	4	1	no	—
229	C₂²⋊C₄	—	4	4	no	—
230	C₂²⋊C₄	—	4	1	no	—
231	C₄²	—	2	1	no	—
232	C₄²	—	4	1	no	—
233	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
234	C₁₆	—	2	2	no	—
235	C₁₆	—	2	2	no	—
236	C₂×C₅	—	2	1	no	—
237	C₂×C₅	—	2	1	no	—
238	C₂×C₅	—	8	4	no	—
239	C₂×C₅	#1762	1	1	yes	characteristic
240	C₂×C₅	AGL(1,ℤ/16)	1	2	yes	—
241	C₂³	—	4	2	no	—
242	C₂³	—	4	2	no	—
243	C₂³	—	4	2	no	—
244	C₂³	—	8	2	no	—
245	C₂³	—	8	2	no	—
246	C₂³	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
247	D₄	—	2	4	no	—
248	D₄	—	4	2	no	—
249	D₄	—	4	2	no	—
250	D₄	—	4	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	8	8	no	—
256	C₂×C₄	—	8	2	no	—
257	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
258	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
259	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
260	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
261	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
262	C₂×C₄	C₅×#137	1	1	yes	characteristic
263	C₈	—	2	2	no	—
264	C₈	—	4	2	no	—
265	C₈	C₂×C₄²×C₅	1	1	yes	characteristic
266	C₈	C₅×#117	1	1	yes	characteristic
267	C₈	C₄×C₅×D₄	1	1	yes	characteristic
268	C₈	C₄×C₅×D₄	1	1	yes	characteristic
269	C₅	#12647	1	1	yes	characteristic, direct product factor, Sylow
270	C₂²	—	2	2	no	—
271	C₂²	—	4	4	no	—
272	C₂²	—	8	2	no	—
273	C₂²	—	8	2	no	—
274	C₂²	—	8	4	no	—
275	C₂²	—	8	4	no	—
276	C₂²	C₅×#418	1	1	yes	characteristic
277	C₂²	C₅×#440	1	1	yes	characteristic
278	C₂²	C₅×#442	1	1	yes	characteristic
279	C₄	—	4	2	no	—
280	C₄	—	8	2	no	—
281	C₄	—	8	2	no	—
282	C₄	C₄×C₅×C₂²⋊C₄	1	1	yes	characteristic
283	C₄	C₅×#417	1	1	yes	characteristic
284	C₄	C₅×#440	1	1	yes	characteristic
285	C₄	C₅×#442	1	1	yes	characteristic
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	—	8	4	no	—
289	C₂	C₅×#1762	1	1	yes	characteristic
290	C₂	C₅×AGL(1,ℤ/16)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	#442
commutator	C₈	C₂×C₄²×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂²×C₅×D₈	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#97 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₅×#347	C₂	semidirect
5. C₅×#1773	C₂	semidirect
6. C₅×#1386	C₂	semidirect
7. C₅×#1383	C₂	semidirect
13. C₅×#442	C₄	semidirect
20. C₅×#361	C₄	semidirect
21. C₅×#357	C₄	semidirect
24. #12647	C₅	direct (N × H)
45. C₅×M₃₂	C₂×C₄	semidirect
47. C₂×C₅×C₁₆	C₂×C₄	semidirect
49. C₅×#108	C₂×C₄	semidirect
50. C₅×#107	C₂×C₄	semidirect
62. C₂×#347	C₂×C₅	semidirect
63. #1773	C₂×C₅	semidirect
64. #1386	C₂×C₅	semidirect
65. #1383	C₂×C₅	semidirect
109. #442	C₄×C₅	semidirect
116. #361	C₄×C₅	semidirect
117. #357	C₄×C₅	semidirect
168. M₃₂	C₂×C₄×C₅	semidirect
170. C₂×C₁₆	C₂×C₄×C₅	semidirect
172. #108	C₂×C₄×C₅	semidirect
173. #107	C₂×C₄×C₅	semidirect
269. C₅	#12647	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×#442	2
C₂×C₅×#361	2
C₅×#2140	2
C₂×C₅×#347	2
C₅×#1773	2
C₅×#1386	2
C₅×#1383	2
#12647	5

Subgroups — C5×#12647 #11935367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12647 #11935367