Subgroups — C₂×D₅×#681 #13105902

Order 1600.

664 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×#87804	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#681	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#87768	C₂	1	2	yes	maximal, complemented
5	D₅×#681	C₂	1	8	yes	maximal, direct product factor
6	C₂×D₅×#168	C₂	1	2	yes	maximal, complemented
7	C₂³×C₅×D₅	C₄	1	1	yes	characteristic
8	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
9	C₂²×C₅×Dic₅	C₄	1	1	yes	characteristic
10	C₂×C₄×C₅×D₅	C₄	1	1	yes	characteristic
11	C₂×C₄×C₅×D₅	C₄	1	4	yes	—
12	C₂×C₄×C₅×D₅	C₄	1	1	yes	characteristic
13	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅×D₅	C₂²	1	4	yes	—
15	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
16	#87804	C₂²	1	4	yes	complemented
17	C₂×#6029	C₂²	1	2	yes	complemented
18	C₅×#681	C₂²	1	4	yes	complemented
19	C₂×C₅×#168	C₂²	1	2	yes	complemented
20	#87768	C₂²	1	8	yes	complemented
21	D₅×#168	C₂²	1	8	yes	complemented
22	C₂²×#681	—	5	1	no	maximal
23	C₂×D₅×M₁₆	—	5	1	no	maximal
24	C₂³×C₅²	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₅×D₅	—	2	4	no	—
26	C₂²×C₅×D₅	C₂×C₄	1	4	yes	—
27	C₂²×C₅×D₅	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₅×D₅	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
30	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₅²	C₂³	1	2	yes	—
33	C₂×C₅×Dic₅	C₂×C₄	1	4	yes	—
34	C₂×C₅×Dic₅	C₂×C₄	1	1	yes	characteristic
35	C₂×C₅×Dic₅	C₂×C₄	1	1	yes	characteristic
36	C₄×C₅×D₅	C₂×C₄	1	4	yes	—
37	C₄×C₅×D₅	C₂×C₄	1	2	yes	—
38	C₄×C₅×D₅	C₂×C₄	1	2	yes	—
39	C₄×C₅×D₅	C₂×C₄	1	4	yes	—
40	C₄×C₅×D₅	C₂³	1	2	yes	—
41	C₄×C₅×D₅	C₂³	1	2	yes	—
42	#6029	C₂³	1	4	yes	complemented
43	C₅×#168	C₂³	1	4	yes	complemented
44	C₂³×C₄×C₅	—	5	1	no	—
45	C₂²×C₄×D₅	D₅	1	1	yes	characteristic
46	C₂×C₅×M₁₆	—	5	1	no	—
47	C₂×#681	—	5	1	no	—
48	C₂×#681	—	5	1	no	—
49	C₂×#681	—	5	8	no	—
50	C₂×#681	—	5	2	no	—
51	C₂×#681	—	10	2	no	—
52	C₂×#681	D₅	1	1	yes	characteristic, direct product factor
53	C₂²×#168	—	5	2	no	—
54	D₅×M₁₆	—	5	8	no	—
55	C₂×#676	—	5	2	no	—
56	C₂×C₈×D₅	—	5	2	no	—
57	C₂²×C₅²	—	2	2	no	—
58	C₂²×C₅²	C₂²×C₄	1	2	yes	—
59	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₅×D₅	—	2	8	no	—
61	C₂×C₅×D₅	—	2	2	no	—
62	C₂×C₅×D₅	M₁₆	1	4	yes	complemented
63	C₂×C₅×D₅	C₂²×C₄	1	2	yes	—
64	C₂×C₅×D₅	C₂²×C₄	1	2	yes	—
65	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
66	C₄×C₅²	C₂²×C₄	1	2	yes	—
67	C₄×C₅²	C₂⁴	1	1	yes	characteristic
68	C₅×Dic₅	C₂²×C₄	1	2	yes	—
69	C₅×Dic₅	C₂²×C₄	1	2	yes	—
70	C₂⁴×C₅	—	5	1	no	—
71	C₂³×D₅	Dic₅	1	1	yes	characteristic
72	C₂²×C₄×C₅	—	2	2	no	—
73	C₂²×C₄×C₅	—	5	1	no	—
74	C₂²×C₄×C₅	—	5	1	no	—
75	C₂²×C₄×C₅	—	5	4	no	—
76	C₂²×C₄×C₅	—	5	1	no	—
77	C₂²×C₄×C₅	—	5	1	no	—
78	C₂²×C₄×C₅	—	5	4	no	—
79	C₂²×C₄×C₅	—	5	1	no	—
80	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
82	C₂²×Dic₅	Dic₅	1	1	yes	characteristic
83	C₂×C₄×D₅	Dic₅	1	4	yes	—
84	C₂×C₄×D₅	Dic₅	1	1	yes	characteristic
85	C₂×C₄×D₅	Dic₅	1	1	yes	characteristic
86	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
87	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
88	C₂×C₄×D₅	C₂×D₅	1	4	yes	—
89	C₅×M₁₆	—	5	4	no	—
90	C₂×C₅×C₈	—	5	2	no	—
91	#681	—	5	4	no	—
92	#681	—	5	4	no	—
93	#681	—	5	8	no	—
94	#681	—	10	8	no	—
95	#681	C₂×D₅	1	4	yes	direct product factor
96	C₂×#168	—	5	2	no	—
97	C₂×#168	—	5	2	no	—
98	C₂×#168	—	5	8	no	—
99	C₂×#168	—	10	4	no	—
100	C₂×#168	C₂×D₅	1	2	yes	complemented
101	#676	—	5	8	no	—
102	C₈×D₅	—	5	8	no	—
103	C₂²×M₁₆	—	25	1	no	Sylow
104	C₂×C₅²	—	2	2	no	—
105	C₂×C₅²	C₂×M₁₆	1	2	yes	complemented
106	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
107	C₅×D₅	—	2	2	no	—
108	C₅×D₅	C₂×M₁₆	1	4	yes	complemented
109	C₂³×C₅	—	2	2	no	—
110	C₂³×C₅	—	5	1	no	—
111	C₂³×C₅	—	5	4	no	—
112	C₂³×C₅	—	5	1	no	—
113	C₂³×C₅	—	10	4	no	—
114	C₂³×C₅	C₄×D₅	1	1	yes	characteristic
115	C₂³×C₅	C₂×Dic₅	1	1	yes	characteristic
116	C₂²×D₅	—	2	4	no	—
117	C₂²×D₅	C₂×Dic₅	1	1	yes	characteristic
118	C₂²×D₅	C₂×Dic₅	1	4	yes	—
119	C₂²×D₅	C₂×Dic₅	1	1	yes	characteristic
120	C₂×C₄×C₅	—	2	2	no	—
121	C₂×C₄×C₅	—	2	4	no	—
122	C₂×C₄×C₅	—	2	4	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	5	4	no	—
125	C₂×C₄×C₅	—	5	1	no	—
126	C₂×C₄×C₅	—	5	1	no	—
127	C₂×C₄×C₅	—	5	2	no	—
128	C₂×C₄×C₅	—	5	4	no	—
129	C₂×C₄×C₅	—	5	4	no	—
130	C₂×C₄×C₅	—	5	2	no	—
131	C₂×C₄×C₅	—	5	2	no	—
132	C₂×C₄×C₅	—	5	2	no	—
133	C₂×C₄×C₅	C₄×D₅	1	2	yes	—
134	C₂×C₄×C₅	C₄×D₅	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂×Dic₅	1	2	yes	—
136	C₂×C₄×C₅	C₂×Dic₅	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
138	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
139	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
141	C₂×Dic₅	C₂×Dic₅	1	4	yes	—
142	C₂×Dic₅	C₂×Dic₅	1	1	yes	characteristic
143	C₂×Dic₅	C₂×Dic₅	1	1	yes	characteristic
144	C₄×D₅	C₂×Dic₅	1	2	yes	—
145	C₄×D₅	C₂×Dic₅	1	4	yes	—
146	C₄×D₅	C₂×Dic₅	1	2	yes	—
147	C₄×D₅	C₂×Dic₅	1	4	yes	—
148	C₄×D₅	C₂²×D₅	1	2	yes	—
149	C₄×D₅	C₂²×D₅	1	2	yes	—
150	C₅×C₈	—	5	4	no	—
151	#168	—	5	4	no	—
152	#168	—	5	4	no	—
153	#168	—	10	8	no	—
154	#168	C₂²×D₅	1	4	yes	complemented
155	C₂³×C₄	—	5	1	no	—
156	C₂×M₁₆	—	5	1	no	—
157	C₂×M₁₆	—	25	2	no	—
158	C₂×M₁₆	—	25	8	no	—
159	C₂×M₁₆	—	25	1	no	—
160	C₂²×C₈	—	25	2	no	—
161	C₅²	C₂²×M₁₆	1	1	yes	characteristic, complemented, Sylow
162	C₂²×C₅	—	2	2	no	—
163	C₂²×C₅	—	2	2	no	—
164	C₂²×C₅	—	2	2	no	—
165	C₂²×C₅	—	2	4	no	—
166	C₂²×C₅	—	4	4	no	—
167	C₂²×C₅	—	5	2	no	—
168	C₂²×C₅	—	5	4	no	—
169	C₂²×C₅	—	5	2	no	—
170	C₂²×C₅	—	10	8	no	—
171	C₂²×C₅	—	10	2	no	—
172	C₂²×C₅	C₂×C₄×D₅	1	2	yes	—
173	C₂²×C₅	C₂×C₄×D₅	1	1	yes	characteristic
174	C₂²×C₅	C₂²×Dic₅	1	2	yes	—
175	C₂²×C₅	C₂²×Dic₅	1	1	yes	characteristic
176	C₂×D₅	—	2	2	no	—
177	C₂×D₅	—	2	8	no	—
178	C₂×D₅	#681	1	4	yes	direct product factor
179	C₂×D₅	C₂²×Dic₅	1	2	yes	—
180	C₂×D₅	C₂²×Dic₅	1	2	yes	—
181	C₄×C₅	—	2	4	no	—
182	C₄×C₅	—	2	2	no	—
183	C₄×C₅	—	2	2	no	—
184	C₄×C₅	—	5	2	no	—
185	C₄×C₅	—	5	2	no	—
186	C₄×C₅	C₂×C₄×D₅	1	2	yes	—
187	C₄×C₅	C₂×C₄×D₅	1	1	yes	characteristic
188	C₄×C₅	C₂²×Dic₅	1	2	yes	—
189	C₄×C₅	C₂²×Dic₅	1	1	yes	characteristic
190	C₄×C₅	C₂³×D₅	1	1	yes	characteristic
191	C₄×C₅	C₂³×D₅	1	1	yes	characteristic
192	Dic₅	C₂²×Dic₅	1	2	yes	—
193	Dic₅	C₂²×Dic₅	1	2	yes	—
194	C₂⁴	—	5	1	no	—
195	C₂²×C₄	—	5	1	no	—
196	C₂²×C₄	—	5	1	no	—
197	C₂²×C₄	—	5	4	no	—
198	C₂²×C₄	—	5	1	no	—
199	C₂²×C₄	—	5	4	no	—
200	C₂²×C₄	—	5	1	no	—
201	C₂²×C₄	—	5	1	no	—
202	C₂²×C₄	D₅²	1	1	yes	characteristic
203	M₁₆	—	5	4	no	—
204	M₁₆	—	25	8	no	—
205	M₁₆	—	25	4	no	—
206	C₂×C₈	—	5	2	no	—
207	C₂×C₈	—	25	8	no	—
208	C₂×C₈	—	25	2	no	—
209	C₂×C₅	—	2	2	no	—
210	C₂×C₅	—	2	2	no	—
211	C₂×C₅	—	2	4	no	—
212	C₂×C₅	—	2	2	no	—
213	C₂×C₅	—	4	4	no	—
214	C₂×C₅	—	5	4	no	—
215	C₂×C₅	—	10	2	no	—
216	C₂×C₅	D₅×M₁₆	1	2	yes	complemented
217	C₂×C₅	C₂×#681	1	2	yes	complemented
218	C₂×C₅	C₂²×C₄×D₅	1	1	yes	characteristic
219	C₂×C₅	C₂³×Dic₅	1	1	yes	characteristic
220	D₅	—	2	2	no	—
221	D₅	C₂×#681	1	4	yes	direct product factor
222	C₂³	—	5	1	no	—
223	C₂³	—	5	4	no	—
224	C₂³	—	5	1	no	—
225	C₂³	—	10	4	no	—
226	C₂³	D₅×Dic₅	1	1	yes	characteristic
227	C₂×C₄	—	5	4	no	—
228	C₂×C₄	—	5	1	no	—
229	C₂×C₄	—	5	1	no	—
230	C₂×C₄	—	5	4	no	—
231	C₂×C₄	—	5	4	no	—
232	C₂×C₄	—	5	2	no	—
233	C₂×C₄	—	5	2	no	—
234	C₂×C₄	—	5	2	no	—
235	C₂×C₄	—	5	2	no	—
236	C₂×C₄	D₅×Dic₅	1	1	yes	characteristic
237	C₂×C₄	D₅×Dic₅	1	2	yes	—
238	C₂×C₄	C₂×D₅²	1	2	yes	—
239	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
240	C₈	—	5	4	no	—
241	C₈	—	25	4	no	—
242	C₅	—	2	2	no	—
243	C₅	C₂×D₅×M₁₆	1	1	yes	characteristic, complemented
244	C₅	C₂²×#681	1	1	yes	characteristic, complemented
245	C₂²	—	2	2	no	—
246	C₂²	—	5	4	no	—
247	C₂²	—	5	2	no	—
248	C₂²	—	5	2	no	—
249	C₂²	—	10	8	no	—
250	C₂²	—	10	2	no	—
251	C₂²	C₂×D₅×Dic₅	1	2	yes	—
252	C₂²	C₂×D₅×Dic₅	1	1	yes	characteristic
253	C₄	—	5	2	no	—
254	C₄	—	5	2	no	—
255	C₄	C₂×D₅×Dic₅	1	2	yes	—
256	C₄	C₂×D₅×Dic₅	1	1	yes	characteristic
257	C₄	C₂²×D₅²	1	1	yes	characteristic
258	C₂	—	2	2	no	—
259	C₂	—	5	4	no	—
260	C₂	—	10	2	no	—
261	C₂	D₅×#681	1	2	yes	direct product factor
262	C₂	C₂²×D₅×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₅²
commutator	C₂×C₅²	C₂³×C₄
Frattini	C₄	C₂²×D₅²
Fitting	C₂²×C₄×C₅²	C₂²
socle	C₂²×C₅²	C₂²×C₄
prime radical	C₂³×C₅×D₅	C₄
cosocle	C₄×C₅²	C₂⁴
hypercenter	C₂²×C₄	D₅²
hypocenter	C₅²	C₂²×M₁₆
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#87804	C₂	semidirect
3. C₂×C₅×#681	C₂	semidirect
4. C₂×#87768	C₂	semidirect
5. D₅×#681	C₂	direct (N × H)
6. C₂×D₅×#168	C₂	semidirect
16. #87804	C₂²	semidirect
17. C₂×#6029	C₂²	semidirect
18. C₅×#681	C₂²	semidirect
19. C₂×C₅×#168	C₂²	semidirect
20. #87768	C₂²	semidirect
21. D₅×#168	C₂²	semidirect
42. #6029	C₂³	semidirect
43. C₅×#168	C₂³	semidirect
52. C₂×#681	D₅	direct (N × H)
62. C₂×C₅×D₅	M₁₆	semidirect
95. #681	C₂×D₅	direct (N × H)
100. C₂×#168	C₂×D₅	semidirect
105. C₂×C₅²	C₂×M₁₆	semidirect
108. C₅×D₅	C₂×M₁₆	semidirect
154. #168	C₂²×D₅	semidirect
161. C₅²	C₂²×M₁₆	semidirect
178. C₂×D₅	#681	direct (N × H)
216. C₂×C₅	D₅×M₁₆	semidirect
217. C₂×C₅	C₂×#681	semidirect
221. D₅	C₂×#681	direct (N × H)
243. C₅	C₂×D₅×M₁₆	semidirect
244. C₅	C₂²×#681	semidirect
261. C₂	D₅×#681	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₅×D₅	2
C₂×#87804	2
C₂×C₅×#681	2
C₂×#87768	2
D₅×#681	2
C₂×D₅×#168	2
C₂²×#681	5
C₂×D₅×M₁₆	5

Subgroups — C2×D5×#681 #13105902

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₅×#681 #13105902