Subgroups — C₅×#65416 #13102388

Order 1600.

316 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₅×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×C₈×D₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×C₂²⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#3952	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#3947	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#3946	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×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
11	C₅²×M₁₆	—	2	2	no	—
12	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
14	C₅×C₂²⋊Dic₅	C₄	1	1	yes	characteristic
15	C₅×#685	C₄	1	1	yes	characteristic
16	C₅×#683	C₄	1	1	yes	characteristic
17	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
18	C₂×C₅×#168	C₂²	1	1	yes	characteristic
19	C₂×C₅×#168	C₂²	1	1	yes	characteristic
20	C₅×C₈×D₅	—	2	2	no	—
21	C₅×#435	—	5	1	no	maximal
22	#65416	C₅	1	1	yes	maximal, characteristic, direct product factor
23	C₂³×C₅²	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₅×D₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₅²	—	2	1	no	—
26	C₂×C₄×C₅²	—	2	1	no	—
27	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
29	C₅×C₂²⋊D₅	—	2	2	no	—
30	C₂×C₅×Dic₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₅×Dic₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₅×Dic₅	C₂×C₄	1	1	yes	characteristic
33	C₄×C₅×D₅	—	2	1	no	—
34	C₄×C₅×D₅	—	2	1	no	—
35	C₅²×C₈	—	2	1	no	—
36	C₅²×C₈	D₄	1	2	yes	complemented
37	C₅×#168	—	2	1	no	—
38	C₅×#168	—	2	1	no	—
39	C₂×C₅×M₁₆	—	2	2	no	—
40	C₂×C₅×M₁₆	D₅	1	1	yes	characteristic, complemented
41	C₂×C₅×M₁₆	C₂×C₅	1	1	yes	characteristic, complemented
42	C₂²×C₅×C₈	—	5	1	no	—
43	C₄×C₅×D₄	—	5	1	no	—
44	C₄×C₂²⋊D₅	C₂×C₅	1	1	yes	characteristic
45	C₂×C₈×D₅	C₂×C₅	1	1	yes	characteristic, complemented
46	C₅×C₄⋊C₈	—	5	1	no	—
47	C₅×C₂²⋊C₈	—	5	1	no	—
48	C₅×C₂²⋊C₈	—	5	1	no	—
49	C₅×#97	—	5	1	no	—
50	C₂²⋊#168	C₂×C₅	1	1	yes	characteristic, complemented
51	#3952	C₂×C₅	1	1	yes	characteristic, complemented
52	#3947	C₂×C₅	1	1	yes	characteristic, complemented
53	#3946	C₂×C₅	1	1	yes	characteristic, complemented
54	C₂²×C₅²	—	2	1	no	—
55	C₂²×C₅²	—	2	1	no	—
56	C₂²×C₅²	—	2	1	no	—
57	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₅×D₅	—	2	1	no	—
59	C₂×C₅×D₅	—	2	1	no	—
60	C₂×C₅×D₅	M₁₆	1	2	yes	complemented
61	C₄×C₅²	—	2	1	no	—
62	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
63	C₄×C₅²	#41	1	1	yes	characteristic
64	C₅×Dic₅	—	2	1	no	—
65	C₅×Dic₅	—	2	1	no	—
66	C₅×Dic₅	—	2	1	no	—
67	C₂×C₅×D₄	—	5	1	no	—
68	C₂²×C₄×C₅	—	2	2	no	—
69	C₂²×C₄×C₅	—	5	1	no	—
70	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
72	C₂×C₂²⋊D₅	C₄×C₅	1	1	yes	characteristic
73	C₂×C₄×D₅	C₂²×C₅	1	1	yes	characteristic
74	C₅×M₁₆	—	2	8	no	—
75	C₅×M₁₆	—	2	2	no	—
76	C₅×M₁₆	—	2	2	no	—
77	C₂×C₅×C₈	—	2	2	no	—
78	C₂×C₅×C₈	—	2	2	no	—
79	C₂×C₅×C₈	—	5	1	no	—
80	C₂×C₅×C₈	—	5	1	no	—
81	C₂×C₅×C₈	—	10	2	no	—
82	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic
83	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic
84	C₂×C₅×C₈	C₂²×C₅	1	1	yes	characteristic
85	C₂×C₅×C₈	C₂²×C₅	1	1	yes	characteristic
86	C₅×C₄⋊C₄	—	5	1	no	—
87	C₅×C₂²⋊C₄	—	5	1	no	—
88	C₅×C₂²⋊C₄	—	5	1	no	—
89	C₄²×C₅	—	5	1	no	—
90	C₂²⋊Dic₅	C₄×C₅	1	1	yes	characteristic
91	#685	C₄×C₅	1	1	yes	characteristic
92	#683	C₄×C₅	1	1	yes	characteristic
93	C₄×Dic₅	C₂²×C₅	1	1	yes	characteristic
94	C₂×#168	C₂²×C₅	1	1	yes	characteristic
95	C₂×#168	C₂²×C₅	1	1	yes	characteristic
96	C₈×D₅	—	2	2	no	—
97	#435	—	5	1	no	Sylow
98	C₂×C₅²	—	4	1	no	—
99	C₂×C₅²	C₄×D₄	1	1	yes	characteristic
100	C₂×C₅²	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₅²	#131	1	1	yes	characteristic
102	C₅×D₅	—	2	2	no	—
103	C₂³×C₅	—	2	2	no	—
104	C₂³×C₅	—	5	1	no	—
105	C₂³×C₅	C₄×D₅	1	1	yes	characteristic
106	C₂³×C₅	C₂×C₄×C₅	1	1	yes	characteristic
107	C₂²×D₅	C₂×C₄×C₅	1	1	yes	characteristic
108	C₅×D₄	—	10	2	no	—
109	C₂×C₄×C₅	—	2	4	no	—
110	C₂×C₄×C₅	—	2	4	no	—
111	C₂×C₄×C₅	—	2	1	no	—
112	C₂×C₄×C₅	—	2	1	no	—
113	C₂×C₄×C₅	—	2	1	no	—
114	C₂×C₄×C₅	—	2	1	no	—
115	C₂×C₄×C₅	—	2	2	no	—
116	C₂×C₄×C₅	—	2	2	no	—
117	C₂×C₄×C₅	—	5	1	no	—
118	C₂×C₄×C₅	—	5	1	no	—
119	C₂×C₄×C₅	—	5	1	no	—
120	C₂×C₄×C₅	—	10	1	no	—
121	C₂×C₄×C₅	—	10	1	no	—
122	C₂×C₄×C₅	C₄×D₅	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂×C₄×C₅	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
126	C₂²⋊D₅	—	2	2	no	—
127	C₂×Dic₅	C₂×C₄×C₅	1	1	yes	characteristic
128	C₂×Dic₅	C₂×C₄×C₅	1	1	yes	characteristic
129	C₂×Dic₅	C₂×C₄×C₅	1	1	yes	characteristic
130	C₄×D₅	—	2	1	no	—
131	C₄×D₅	—	2	1	no	—
132	C₅×C₈	—	2	4	no	—
133	C₅×C₈	—	2	1	no	—
134	C₅×C₈	—	2	1	no	—
135	C₅×C₈	—	2	4	no	—
136	C₅×C₈	—	10	1	no	—
137	C₅×C₈	—	10	1	no	—
138	C₅×C₈	C₂²⋊D₅	1	2	yes	complemented
139	C₅×C₈	C₅×D₄	1	2	yes	complemented
140	#168	—	2	1	no	—
141	#168	—	2	1	no	—
142	C₂×M₁₆	C₅×D₅	1	1	yes	characteristic, complemented
143	C₂²×C₈	—	5	1	no	—
144	C₄×D₄	—	5	1	no	—
145	C₄⋊C₈	—	5	1	no	—
146	C₂²⋊C₈	—	5	1	no	—
147	C₂²⋊C₈	—	5	1	no	—
148	#97	—	5	1	no	—
149	C₅²	#435	1	1	yes	characteristic, complemented, Sylow
150	C₂²×C₅	—	2	4	no	—
151	C₂²×C₅	—	2	1	no	—
152	C₂²×C₅	—	2	1	no	—
153	C₂²×C₅	—	2	1	no	—
154	C₂²×C₅	—	2	1	no	—
155	C₂²×C₅	—	2	1	no	—
156	C₂²×C₅	—	2	1	no	—
157	C₂²×C₅	—	2	2	no	—
158	C₂²×C₅	—	4	2	no	—
159	C₂²×C₅	—	4	2	no	—
160	C₂²×C₅	—	5	2	no	—
161	C₂²×C₅	—	10	1	no	—
162	C₂²×C₅	—	10	1	no	—
163	C₂²×C₅	C₂×C₄×D₅	1	1	yes	characteristic
164	C₂²×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
165	C₂×D₅	—	2	1	no	—
166	C₂×D₅	—	2	1	no	—
167	C₂×D₅	C₅×M₁₆	1	2	yes	complemented
168	C₄×C₅	—	2	4	no	—
169	C₄×C₅	—	2	1	no	—
170	C₄×C₅	—	2	1	no	—
171	C₄×C₅	—	2	2	no	—
172	C₄×C₅	—	2	2	no	—
173	C₄×C₅	—	10	1	no	—
174	C₄×C₅	—	10	1	no	—
175	C₄×C₅	—	10	1	no	—
176	C₄×C₅	#709	1	1	yes	characteristic
177	C₄×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
178	C₄×C₅	C₂×C₅×D₄	1	1	yes	characteristic
179	C₄×C₅	C₅×#41	1	1	yes	characteristic
180	Dic₅	—	2	1	no	—
181	Dic₅	—	2	1	no	—
182	Dic₅	—	2	1	no	—
183	C₂×D₄	—	5	1	no	—
184	C₂²×C₄	—	5	1	no	—
185	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
186	M₁₆	—	2	2	no	—
187	C₂×C₈	—	5	1	no	—
188	C₂×C₈	—	5	1	no	—
189	C₂×C₈	—	10	2	no	—
190	C₂×C₈	C₂×C₅×D₅	1	1	yes	characteristic
191	C₂×C₈	C₂×C₅×D₅	1	1	yes	characteristic
192	C₄⋊C₄	—	5	1	no	—
193	C₂²⋊C₄	—	5	1	no	—
194	C₂²⋊C₄	—	5	1	no	—
195	C₄²	—	5	1	no	—
196	C₂×C₅	—	2	2	no	—
197	C₂×C₅	—	2	2	no	—
198	C₂×C₅	—	2	2	no	—
199	C₂×C₅	—	4	4	no	—
200	C₂×C₅	—	4	1	no	—
201	C₂×C₅	—	4	1	no	—
202	C₂×C₅	—	10	2	no	—
203	C₂×C₅	D₅×M₁₆	1	1	yes	characteristic
204	C₂×C₅	#4053	1	1	yes	characteristic
205	C₂×C₅	C₄×C₂²⋊D₅	1	1	yes	characteristic
206	C₂×C₅	C₄×C₅×D₄	1	1	yes	characteristic
207	C₂×C₅	C₂×C₅×M₁₆	1	1	yes	characteristic
208	C₂×C₅	C₅×#131	1	1	yes	characteristic
209	D₅	—	2	2	no	—
210	C₂³	—	5	1	no	—
211	C₂³	C₄×C₅×D₅	1	1	yes	characteristic
212	D₄	—	10	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	5	1	no	—
216	C₂×C₄	—	5	1	no	—
217	C₂×C₄	—	5	1	no	—
218	C₂×C₄	—	10	1	no	—
219	C₂×C₄	—	10	1	no	—
220	C₂×C₄	C₄×C₅×D₅	1	1	yes	characteristic
221	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
222	C₈	—	2	1	no	—
223	C₈	—	10	1	no	—
224	C₈	—	10	1	no	—
225	C₈	C₅×C₂²⋊D₅	1	2	yes	complemented
226	C₅	—	2	2	no	—
227	C₅	#65416	1	1	yes	characteristic, direct product factor
228	C₅	C₅×#435	1	1	yes	characteristic, complemented
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	5	2	no	—
233	C₂²	—	10	1	no	—
234	C₂²	—	10	1	no	—
235	C₂²	C₂×C₄×C₅×D₅	1	1	yes	characteristic
236	C₄	—	2	1	no	—
237	C₄	—	10	1	no	—
238	C₄	—	10	1	no	—
239	C₄	—	10	1	no	—
240	C₄	C₅×#709	1	1	yes	characteristic
241	C₄	C₂×C₅×C₂²⋊D₅	1	1	yes	characteristic
242	C₂	—	4	1	no	—
243	C₂	—	10	2	no	—
244	C₂	C₅×D₅×M₁₆	1	1	yes	characteristic
245	C₂	C₅×#4053	1	1	yes	characteristic
246	C₂	C₄×C₅×C₂²⋊D₅	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	C₂×C₅²×M₁₆	C₂
socle	C₂²×C₅²	C₂²×C₄
prime radical	C₂×C₅×C₂²⋊D₅	C₄
cosocle	C₂×C₄×C₅	C₂³×C₅
hypercenter	C₂×C₅×M₁₆	D₅
hypocenter	C₅	C₅×#435
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₅×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×C₅²×M₁₆	C₂	semidirect
3. C₂×C₅×C₈×D₅	C₂	semidirect
4. C₅×C₂²⋊#168	C₂	semidirect
5. C₅×#3952	C₂	semidirect
6. C₅×#3947	C₂	semidirect
7. C₅×#3946	C₂	semidirect
22. #65416	C₅	direct (N × H)
36. C₅²×C₈	D₄	semidirect
40. C₂×C₅×M₁₆	D₅	semidirect
41. C₂×C₅×M₁₆	C₂×C₅	semidirect
45. C₂×C₈×D₅	C₂×C₅	semidirect
50. C₂²⋊#168	C₂×C₅	semidirect
51. #3952	C₂×C₅	semidirect
52. #3947	C₂×C₅	semidirect
53. #3946	C₂×C₅	semidirect
60. C₂×C₅×D₅	M₁₆	semidirect
138. C₅×C₈	C₂²⋊D₅	semidirect
139. C₅×C₈	C₅×D₄	semidirect
142. C₂×M₁₆	C₅×D₅	semidirect
149. C₅²	#435	semidirect
167. C₂×D₅	C₅×M₁₆	semidirect
225. C₈	C₅×C₂²⋊D₅	semidirect
227. C₅	#65416	direct (N × H)
228. C₅	C₅×#435	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅²×M₁₆	2
C₄×C₅×C₂²⋊D₅	2
C₂×C₅×C₈×D₅	2
C₅×C₂²⋊#168	2
C₅×#3952	2
C₅×#3947	2
C₅×#3946	2
C₅×#435	5
#65416	5

Subgroups — C5×#65416 #13102388

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65416 #13102388