Subgroups — C₅×#65016 #13101989

Order 1600.

366 conjugacy classes of subgroups.

The subgroup lattice has 287 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#709	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#676	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×C₈×D₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅²×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×C₂²⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#3952	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#3952	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
9	C₂×C₅×C₂²⋊D₅	C₄	1	1	yes	characteristic
10	C₂×C₅×C₂²⋊D₅	C₄	1	1	yes	characteristic
11	C₅×#709	—	2	2	no	—
12	C₅×#709	—	2	2	no	—
13	C₂×C₅×D₂₀	C₄	1	1	yes	characteristic
14	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
16	C₂×C₅×Dic₁₀	C₄	1	1	yes	characteristic
17	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
18	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₅×#168	C₂²	1	1	yes	characteristic
20	C₂×C₅×#168	C₂²	1	1	yes	characteristic
21	C₅×#676	—	2	2	no	—
22	C₅×C₈×D₅	—	2	2	no	—
23	C₅×#408	—	5	1	no	maximal
24	#65016	C₅	1	1	yes	maximal, characteristic, direct product factor
25	C₂³×C₅²	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₅×D₅	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₅×D₅	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×C₅²	—	2	1	no	—
29	C₂×C₄×C₅²	—	2	1	no	—
30	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
32	C₅×C₂²⋊D₅	—	2	2	no	—
33	C₅×C₂²⋊D₅	—	2	2	no	—
34	C₂×C₅×Dic₅	C₂×C₄	1	1	yes	characteristic
35	C₂×C₅×Dic₅	C₂×C₄	1	1	yes	characteristic
36	C₅×D₂₀	—	2	2	no	—
37	C₄×C₅×D₅	—	2	1	no	—
38	C₄×C₅×D₅	—	2	1	no	—
39	C₄×C₅×D₅	D₄	1	2	yes	—
40	C₄×C₅×D₅	D₄	1	2	yes	—
41	C₅×Dic₁₀	—	2	2	no	—
42	C₅²×C₈	—	2	1	no	—
43	C₅²×C₈	—	2	1	no	—
44	C₅×#168	—	2	1	no	—
45	C₅×#168	—	2	1	no	—
46	C₂×C₅×#41	—	5	1	no	—
47	C₂×#709	C₂×C₅	1	1	yes	characteristic
48	C₂×C₅×M₁₆	—	5	1	no	—
49	C₂²×C₅×C₈	—	5	1	no	—
50	C₂×#676	C₂×C₅	1	1	yes	characteristic, complemented
51	C₂×C₈×D₅	C₂×C₅	1	1	yes	characteristic, complemented
52	C₅×C₂²⋊C₈	—	2	2	no	—
53	C₅×C₂²⋊C₈	—	5	1	no	—
54	C₅×C₂²⋊C₈	—	5	1	no	—
55	C₅×C₂²⋊C₈	—	5	1	no	—
56	C₅×C₂²⋊C₈	D₅	1	1	yes	characteristic, complemented
57	C₅×C₂²⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
58	C₂²⋊#168	C₂×C₅	1	1	yes	characteristic, complemented
59	#3952	C₂×C₅	1	1	yes	characteristic, complemented
60	#3952	C₂×C₅	1	1	yes	characteristic, complemented
61	C₂²×C₅²	—	2	1	no	—
62	C₂²×C₅²	—	2	1	no	—
63	C₂²×C₅²	—	2	1	no	—
64	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₅×D₅	—	2	1	no	—
66	C₂×C₅×D₅	—	2	1	no	—
67	C₂×C₅×D₅	—	2	1	no	—
68	C₂×C₅×D₅	—	2	1	no	—
69	C₂×C₅×D₅	—	2	1	no	—
70	C₂×C₅×D₅	C₂²⋊C₄	1	2	yes	—
71	C₄×C₅²	—	2	1	no	—
72	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
73	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
74	C₅×Dic₅	—	2	1	no	—
75	C₅×Dic₅	C₂²⋊C₄	1	2	yes	—
76	C₅×#41	—	10	2	no	—
77	C₅×#41	—	10	2	no	—
78	C₂×C₅×Q₈	—	5	1	no	—
79	C₂×C₅×D₄	—	5	1	no	—
80	C₂×C₅×D₄	—	5	1	no	—
81	C₂×C₅×D₄	—	5	1	no	—
82	C₂²×C₄×C₅	—	2	2	no	—
83	C₂²×C₄×C₅	—	5	1	no	—
84	C₂²×C₄×C₅	—	5	1	no	—
85	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
87	C₂×C₂²⋊D₅	C₄×C₅	1	1	yes	characteristic
88	C₂×C₂²⋊D₅	C₄×C₅	1	1	yes	characteristic
89	#709	—	2	2	no	—
90	#709	—	2	2	no	—
91	C₂×D₂₀	C₄×C₅	1	1	yes	characteristic
92	C₂×C₄×D₅	C₂²×C₅	1	1	yes	characteristic
93	C₂×C₄×D₅	C₂²×C₅	1	1	yes	characteristic
94	C₂×Dic₁₀	C₄×C₅	1	1	yes	characteristic
95	C₅×M₁₆	—	10	2	no	—
96	C₂×C₅×C₈	—	2	2	no	—
97	C₂×C₅×C₈	—	2	2	no	—
98	C₂×C₅×C₈	—	5	1	no	—
99	C₂×C₅×C₈	—	5	1	no	—
100	C₂×C₅×C₈	—	10	2	no	—
101	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic
102	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic
103	C₂×C₅×C₈	C₂²×C₅	1	1	yes	characteristic
104	C₂×C₅×C₈	C₂²×C₅	1	1	yes	characteristic
105	C₂×#168	C₂²×C₅	1	1	yes	characteristic
106	C₂×#168	C₂²×C₅	1	1	yes	characteristic
107	#676	—	2	2	no	—
108	C₈×D₅	—	2	2	no	—
109	#408	—	5	1	no	Sylow
110	C₂×C₅²	—	4	1	no	—
111	C₂×C₅²	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₅²	#131	1	1	yes	characteristic
113	C₂×C₅²	#131	1	1	yes	characteristic
114	C₅×D₅	—	2	2	no	—
115	C₅×D₅	—	4	1	no	—
116	C₂³×C₅	—	2	2	no	—
117	C₂³×C₅	—	5	1	no	—
118	C₂³×C₅	—	5	1	no	—
119	C₂³×C₅	C₄×D₅	1	1	yes	characteristic
120	C₂³×C₅	C₂×C₄×C₅	1	1	yes	characteristic
121	C₂²×D₅	C₂×C₄×C₅	1	1	yes	characteristic
122	C₂²×D₅	C₂×C₄×C₅	1	1	yes	characteristic
123	C₅×Q₈	—	10	2	no	—
124	C₅×D₄	—	10	2	no	—
125	C₅×D₄	—	10	2	no	—
126	C₅×D₄	—	10	2	no	—
127	C₂×C₄×C₅	—	2	1	no	—
128	C₂×C₄×C₅	—	2	1	no	—
129	C₂×C₄×C₅	—	2	1	no	—
130	C₂×C₄×C₅	—	2	1	no	—
131	C₂×C₄×C₅	—	2	2	no	—
132	C₂×C₄×C₅	—	2	2	no	—
133	C₂×C₄×C₅	—	4	2	no	—
134	C₂×C₄×C₅	—	4	2	no	—
135	C₂×C₄×C₅	—	5	1	no	—
136	C₂×C₄×C₅	—	5	2	no	—
137	C₂×C₄×C₅	—	5	2	no	—
138	C₂×C₄×C₅	—	5	1	no	—
139	C₂×C₄×C₅	—	10	1	no	—
140	C₂×C₄×C₅	—	10	1	no	—
141	C₂×C₄×C₅	C₄×D₅	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×C₄×C₅	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
145	C₂²⋊D₅	—	2	2	no	—
146	C₂²⋊D₅	—	2	2	no	—
147	C₂×Dic₅	C₂×C₄×C₅	1	1	yes	characteristic
148	C₂×Dic₅	C₂×C₄×C₅	1	1	yes	characteristic
149	D₂₀	—	2	2	no	—
150	C₄×D₅	—	2	1	no	—
151	C₄×D₅	—	2	1	no	—
152	C₄×D₅	C₅×D₄	1	2	yes	—
153	C₄×D₅	C₅×D₄	1	2	yes	—
154	Dic₁₀	—	2	2	no	—
155	C₅×C₈	—	2	1	no	—
156	C₅×C₈	—	2	1	no	—
157	C₅×C₈	—	2	1	no	—
158	C₅×C₈	—	2	1	no	—
159	C₅×C₈	—	4	2	no	—
160	C₅×C₈	—	4	2	no	—
161	C₅×C₈	—	10	1	no	—
162	C₅×C₈	—	10	1	no	—
163	#168	—	2	1	no	—
164	#168	—	2	1	no	—
165	C₂×#41	—	5	1	no	—
166	C₂×M₁₆	—	5	1	no	—
167	C₂²×C₈	—	5	1	no	—
168	C₂²⋊C₈	—	5	1	no	—
169	C₂²⋊C₈	—	5	1	no	—
170	C₂²⋊C₈	—	5	1	no	—
171	C₂²⋊C₈	C₅×D₅	1	1	yes	characteristic, complemented
172	C₅²	#408	1	1	yes	characteristic, complemented, Sylow
173	C₂²×C₅	—	2	4	no	—
174	C₂²×C₅	—	2	1	no	—
175	C₂²×C₅	—	2	1	no	—
176	C₂²×C₅	—	2	1	no	—
177	C₂²×C₅	—	2	1	no	—
178	C₂²×C₅	—	2	1	no	—
179	C₂²×C₅	—	2	1	no	—
180	C₂²×C₅	—	2	2	no	—
181	C₂²×C₅	—	4	2	no	—
182	C₂²×C₅	—	4	2	no	—
183	C₂²×C₅	—	5	2	no	—
184	C₂²×C₅	—	10	1	no	—
185	C₂²×C₅	—	10	1	no	—
186	C₂²×C₅	—	10	1	no	—
187	C₂²×C₅	—	10	1	no	—
188	C₂²×C₅	—	10	1	no	—
189	C₂²×C₅	C₂×C₄×D₅	1	1	yes	characteristic
190	C₂²×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
191	C₂×D₅	—	2	1	no	—
192	C₂×D₅	—	2	1	no	—
193	C₂×D₅	—	2	1	no	—
194	C₂×D₅	—	2	1	no	—
195	C₂×D₅	—	2	1	no	—
196	C₂×D₅	C₅×C₂²⋊C₄	1	2	yes	—
197	C₄×C₅	—	2	1	no	—
198	C₄×C₅	—	2	1	no	—
199	C₄×C₅	—	2	2	no	—
200	C₄×C₅	—	2	2	no	—
201	C₄×C₅	—	4	2	no	—
202	C₄×C₅	—	5	2	no	—
203	C₄×C₅	—	10	1	no	—
204	C₄×C₅	D₄×D₅	1	1	yes	characteristic
205	C₄×C₅	D₄×D₅	1	1	yes	characteristic
206	C₄×C₅	C₂×C₅×D₄	1	1	yes	characteristic
207	C₄×C₅	C₂×C₅×D₄	1	1	yes	characteristic
208	Dic₅	—	2	1	no	—
209	Dic₅	C₅×C₂²⋊C₄	1	2	yes	—
210	#41	—	10	2	no	—
211	#41	—	10	2	no	—
212	C₂×Q₈	—	5	1	no	—
213	C₂×D₄	—	5	1	no	—
214	C₂×D₄	—	5	1	no	—
215	C₂×D₄	—	5	1	no	—
216	C₂²×C₄	—	5	1	no	—
217	C₂²×C₄	—	5	1	no	—
218	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
219	M₁₆	—	10	2	no	—
220	C₂×C₈	—	5	1	no	—
221	C₂×C₈	—	5	1	no	—
222	C₂×C₈	—	10	2	no	—
223	C₂×C₈	C₂×C₅×D₅	1	1	yes	characteristic
224	C₂×C₈	C₂×C₅×D₅	1	1	yes	characteristic
225	C₂×C₅	—	2	2	no	—
226	C₂×C₅	—	2	2	no	—
227	C₂×C₅	—	2	2	no	—
228	C₂×C₅	—	4	4	no	—
229	C₂×C₅	—	4	1	no	—
230	C₂×C₅	—	4	1	no	—
231	C₂×C₅	—	10	2	no	—
232	C₂×C₅	—	20	1	no	—
233	C₂×C₅	D₅×C₂²⋊C₄	1	1	yes	characteristic
234	C₂×C₅	#4047	1	1	yes	characteristic
235	C₂×C₅	#4053	1	1	yes	characteristic
236	C₂×C₅	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
237	C₂×C₅	C₅×#131	1	1	yes	characteristic
238	C₂×C₅	C₅×#131	1	1	yes	characteristic
239	D₅	—	2	2	no	—
240	D₅	—	4	1	no	—
241	C₂³	—	5	1	no	—
242	C₂³	—	5	1	no	—
243	C₂³	C₄×C₅×D₅	1	1	yes	characteristic
244	Q₈	—	10	2	no	—
245	D₄	—	10	2	no	—
246	D₄	—	10	2	no	—
247	D₄	—	10	2	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	5	1	no	—
251	C₂×C₄	—	5	2	no	—
252	C₂×C₄	—	5	2	no	—
253	C₂×C₄	—	5	1	no	—
254	C₂×C₄	—	10	1	no	—
255	C₂×C₄	—	10	1	no	—
256	C₂×C₄	C₄×C₅×D₅	1	1	yes	characteristic
257	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	10	1	no	—
261	C₈	—	10	1	no	—
262	C₅	—	2	2	no	—
263	C₅	#65016	1	1	yes	characteristic, direct product factor
264	C₅	C₅×#408	1	1	yes	characteristic, complemented
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	5	2	no	—
269	C₂²	—	10	1	no	—
270	C₂²	—	10	1	no	—
271	C₂²	—	10	1	no	—
272	C₂²	—	10	1	no	—
273	C₂²	—	10	1	no	—
274	C₂²	C₂×C₄×C₅×D₅	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	5	2	no	—
277	C₄	—	10	1	no	—
278	C₄	C₅×D₄×D₅	1	1	yes	characteristic
279	C₄	C₅×D₄×D₅	1	1	yes	characteristic
280	C₂	—	4	1	no	—
281	C₂	—	10	2	no	—
282	C₂	—	20	1	no	—
283	C₂	C₅×D₅×C₂²⋊C₄	1	1	yes	characteristic
284	C₂	C₅×#4047	1	1	yes	characteristic
285	C₂	C₅×#4053	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₅
commutator	C₂²×C₅	C₂²×C₄×C₅
Frattini	C₂×C₄	C₂²×C₅×D₅
Fitting	C₅²×C₂²⋊C₈	C₂
socle	C₂²×C₅²	C₂²×C₄
prime radical	C₂×C₅×#709	C₂
cosocle	C₂×C₄×C₅	C₂³×C₅
hypercenter	C₅×C₂²⋊C₈	D₅
hypocenter	C₅	C₅×#408
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₅×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#676	C₂	semidirect
3. C₂×C₅×C₈×D₅	C₂	semidirect
4. C₅²×C₂²⋊C₈	C₂	semidirect
5. C₅×C₂²⋊#168	C₂	semidirect
6. C₅×#3952	C₂	semidirect
7. C₅×#3952	C₂	semidirect
24. #65016	C₅	direct (N × H)
50. C₂×#676	C₂×C₅	semidirect
51. C₂×C₈×D₅	C₂×C₅	semidirect
56. C₅×C₂²⋊C₈	D₅	semidirect
57. C₅×C₂²⋊C₈	C₂×C₅	semidirect
58. C₂²⋊#168	C₂×C₅	semidirect
59. #3952	C₂×C₅	semidirect
60. #3952	C₂×C₅	semidirect
171. C₂²⋊C₈	C₅×D₅	semidirect
172. C₅²	#408	semidirect
263. C₅	#65016	direct (N × H)
264. C₅	C₅×#408	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₅×#709	2
C₂×C₅×#676	2
C₂×C₅×C₈×D₅	2
C₅²×C₂²⋊C₈	2
C₅×C₂²⋊#168	2
C₅×#3952	2
C₅×#408	5
#65016	5

Subgroups — C5×#65016 #13101989

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65016 #13101989