Subgroups — C₂×C₅×#4061 #13106914

Order 1600.

584 conjugacy classes of subgroups.

The subgroup lattice has 295 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×Q₈×D₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#711	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅²×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#689	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×#687	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4061	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₅×Dic₂₀	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#676	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic
11	C₂×C₅×C₂²⋊D₅	—	2	1	no	—
12	C₂²×C₅×Dic₅	—	2	1	no	—
13	C₅×Q₈×D₅	—	2	2	no	—
14	C₅×Q₈×D₅	C₂²	1	4	yes	complemented
15	C₅×#711	—	2	2	no	—
16	C₅×#711	C₂²	1	4	yes	—
17	C₂×C₄×C₅×D₅	—	2	1	no	—
18	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
19	C₂×C₅×Dic₁₀	—	2	1	no	—
20	C₂×C₅×Dic₁₀	C₂²	1	1	yes	characteristic
21	C₂×C₅×Dic₁₀	C₂²	1	1	yes	characteristic
22	C₅²×SD₁₆	C₂²	1	4	yes	complemented
23	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
24	C₅×#689	C₂²	1	4	yes	complemented
25	C₅×#687	C₂²	1	4	yes	complemented
26	C₂×C₅×#168	C₂²	1	1	yes	characteristic
27	C₅×Dic₂₀	C₂²	1	4	yes	complemented
28	C₅×#676	C₂²	1	4	yes	complemented
29	C₂×C₅×#137	—	5	1	no	maximal
30	C₂×#4061	C₅	1	1	yes	maximal, characteristic, direct product factor
31	C₂³×C₅²	—	2	1	no	—
32	C₂²×C₅×D₅	D₄	1	1	yes	characteristic
33	C₅²×Q₈	—	2	1	no	—
34	C₅²×Q₈	C₂³	1	2	yes	—
35	C₅²×D₄	—	2	1	no	—
36	C₅²×D₄	C₂³	1	2	yes	—
37	C₂×C₄×C₅²	—	2	1	no	—
38	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
39	C₅×C₂²⋊D₅	—	2	4	no	—
40	C₂×C₅×Dic₅	—	2	1	no	—
41	C₂×C₅×Dic₅	—	2	1	no	—
42	C₂×C₅×Dic₅	—	2	4	no	—
43	C₂×C₅×Dic₅	D₄	1	1	yes	characteristic
44	C₄×C₅×D₅	—	2	4	no	—
45	C₄×C₅×D₅	D₄	1	2	yes	—
46	C₄×C₅×D₅	C₂³	1	2	yes	—
47	C₅×Dic₁₀	—	2	1	no	—
48	C₅×Dic₁₀	—	2	1	no	—
49	C₅×Dic₁₀	—	2	4	no	—
50	C₅×Dic₁₀	C₂³	1	2	yes	—
51	C₅×Dic₁₀	C₂³	1	2	yes	—
52	C₅²×C₈	C₂³	1	2	yes	—
53	C₅×#168	C₂³	1	2	yes	—
54	C₂×C₅×#41	—	5	1	no	—
55	C₂²×C₅×Q₈	—	5	1	no	—
56	C₂×Q₈×D₅	C₂×C₅	1	1	yes	characteristic, complemented
57	C₂×#711	C₂×C₅	1	1	yes	characteristic
58	C₅×#137	—	5	8	no	—
59	C₂×C₅×Q₁₆	—	5	1	no	—
60	C₂×C₅×Q₁₆	—	5	1	no	—
61	C₂×C₅×SD₁₆	—	2	2	no	—
62	C₂×C₅×SD₁₆	—	5	1	no	—
63	C₂×C₅×SD₁₆	D₅	1	1	yes	characteristic, complemented
64	C₂×C₅×SD₁₆	C₂×C₅	1	1	yes	characteristic, complemented
65	C₂×C₅×M₁₆	—	5	1	no	—
66	C₂×#689	C₂×C₅	1	1	yes	characteristic, complemented
67	C₂×#687	C₂×C₅	1	1	yes	characteristic, complemented
68	#4061	C₂×C₅	1	8	yes	direct product factor
69	C₂×Dic₂₀	C₂×C₅	1	1	yes	characteristic, complemented
70	C₂×#676	C₂×C₅	1	1	yes	characteristic, complemented
71	C₂²×C₅²	—	2	2	no	—
72	C₂²×C₅²	—	4	2	no	—
73	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
74	C₂×C₅×D₅	—	2	2	no	—
75	C₂×C₅×D₅	C₂×D₄	1	2	yes	—
76	C₄×C₅²	—	2	2	no	—
77	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
78	C₄×C₅²	C₂⁴	1	1	yes	characteristic
79	C₅×Dic₅	—	2	2	no	—
80	C₅×Dic₅	—	2	2	no	—
81	C₅×Dic₅	C₂×D₄	1	2	yes	—
82	C₅×#41	—	5	4	no	—
83	C₅×#41	—	10	2	no	—
84	C₂×C₅×Q₈	—	2	2	no	—
85	C₂×C₅×Q₈	—	5	4	no	—
86	C₂×C₅×Q₈	—	5	1	no	—
87	C₂×C₅×Q₈	—	5	1	no	—
88	C₂×C₅×Q₈	—	10	1	no	—
89	C₂×C₅×Q₈	—	10	2	no	—
90	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic
91	C₂×C₅×Q₈	C₂²×C₅	1	1	yes	characteristic
92	C₂×C₅×D₄	—	2	2	no	—
93	C₂×C₅×D₄	—	10	1	no	—
94	C₂×C₅×D₄	C₂×D₅	1	1	yes	characteristic
95	C₂×C₅×D₄	C₂²×C₅	1	1	yes	characteristic
96	C₂²×C₄×C₅	—	5	1	no	—
97	C₂²×C₄×C₅	—	10	1	no	—
98	C₂²×C₄×C₅	—	10	1	no	—
99	C₂×C₂²⋊D₅	—	2	1	no	—
100	C₂²×Dic₅	—	2	1	no	—
101	Q₈×D₅	—	2	2	no	—
102	Q₈×D₅	C₂²×C₅	1	4	yes	complemented
103	#711	—	2	2	no	—
104	#711	C₂²×C₅	1	4	yes	—
105	C₂×C₄×D₅	—	2	1	no	—
106	C₂×C₄×D₅	C₂²×C₅	1	1	yes	characteristic
107	C₂×Dic₁₀	—	2	1	no	—
108	C₂×Dic₁₀	C₂²×C₅	1	1	yes	characteristic
109	C₂×Dic₁₀	C₂²×C₅	1	1	yes	characteristic
110	C₅×Q₁₆	—	5	4	no	—
111	C₅×Q₁₆	—	5	4	no	—
112	C₅×SD₁₆	—	2	8	no	—
113	C₅×SD₁₆	—	5	4	no	—
114	C₅×SD₁₆	C₂×D₅	1	4	yes	complemented
115	C₅×SD₁₆	C₂²×C₅	1	4	yes	complemented
116	C₅×M₁₆	—	5	4	no	—
117	C₂×C₅×C₈	—	2	2	no	—
118	C₂×C₅×C₈	—	5	1	no	—
119	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic
120	C₂×C₅×C₈	C₂²×C₅	1	1	yes	characteristic
121	#689	C₂²×C₅	1	4	yes	complemented
122	#687	C₂²×C₅	1	4	yes	complemented
123	C₂×#168	C₂²×C₅	1	1	yes	characteristic
124	Dic₂₀	C₂²×C₅	1	4	yes	complemented
125	#676	C₂²×C₅	1	4	yes	complemented
126	C₂×#137	—	5	1	no	Sylow
127	C₂×C₅²	—	4	2	no	—
128	C₂×C₅²	#137	1	2	yes	complemented
129	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
130	C₅×D₅	—	2	2	no	—
131	C₂³×C₅	—	2	1	no	—
132	C₂³×C₅	—	2	1	no	—
133	C₂³×C₅	—	4	2	no	—
134	C₂³×C₅	—	5	1	no	—
135	C₂²×D₅	C₅×D₄	1	1	yes	characteristic
136	C₅×Q₈	—	2	1	no	—
137	C₅×Q₈	—	2	1	no	—
138	C₅×Q₈	—	2	4	no	—
139	C₅×Q₈	—	4	2	no	—
140	C₅×Q₈	—	5	2	no	—
141	C₅×Q₈	—	5	2	no	—
142	C₅×Q₈	—	10	4	no	—
143	C₅×Q₈	—	10	1	no	—
144	C₅×Q₈	—	10	1	no	—
145	C₅×Q₈	C₂²×D₅	1	2	yes	—
146	C₅×Q₈	C₂³×C₅	1	2	yes	—
147	C₅×D₄	—	2	1	no	—
148	C₅×D₄	—	2	1	no	—
149	C₅×D₄	—	2	4	no	—
150	C₅×D₄	—	4	2	no	—
151	C₅×D₄	—	10	4	no	—
152	C₅×D₄	C₂²×D₅	1	2	yes	—
153	C₅×D₄	C₂³×C₅	1	2	yes	—
154	C₂×C₄×C₅	—	2	1	no	—
155	C₂×C₄×C₅	—	2	1	no	—
156	C₂×C₄×C₅	—	2	2	no	—
157	C₂×C₄×C₅	—	4	2	no	—
158	C₂×C₄×C₅	—	5	1	no	—
159	C₂×C₄×C₅	—	5	2	no	—
160	C₂×C₄×C₅	—	5	2	no	—
161	C₂×C₄×C₅	—	10	4	no	—
162	C₂×C₄×C₅	—	10	4	no	—
163	C₂×C₄×C₅	—	10	1	no	—
164	C₂×C₄×C₅	—	10	1	no	—
165	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
166	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
167	C₂²⋊D₅	—	2	4	no	—
168	C₂×Dic₅	—	2	1	no	—
169	C₂×Dic₅	—	2	1	no	—
170	C₂×Dic₅	—	2	4	no	—
171	C₂×Dic₅	C₅×D₄	1	1	yes	characteristic
172	C₄×D₅	—	2	4	no	—
173	C₄×D₅	C₅×D₄	1	2	yes	—
174	C₄×D₅	C₂³×C₅	1	2	yes	—
175	Dic₁₀	—	2	1	no	—
176	Dic₁₀	—	2	1	no	—
177	Dic₁₀	—	2	4	no	—
178	Dic₁₀	C₂³×C₅	1	2	yes	—
179	Dic₁₀	C₂³×C₅	1	2	yes	—
180	C₅×C₈	—	2	4	no	—
181	C₅×C₈	—	5	2	no	—
182	C₅×C₈	C₂²×D₅	1	2	yes	—
183	C₅×C₈	C₂³×C₅	1	2	yes	—
184	#168	C₂³×C₅	1	2	yes	—
185	C₂×#41	—	5	1	no	—
186	C₂²×Q₈	—	5	1	no	—
187	#137	—	5	8	no	—
188	C₂×Q₁₆	—	5	1	no	—
189	C₂×Q₁₆	—	5	1	no	—
190	C₂×SD₁₆	—	5	1	no	—
191	C₂×SD₁₆	C₅×D₅	1	1	yes	characteristic, complemented
192	C₂×M₁₆	—	5	1	no	—
193	C₅²	C₂×#137	1	1	yes	characteristic, complemented, Sylow
194	C₂²×C₅	—	2	2	no	—
195	C₂²×C₅	—	2	2	no	—
196	C₂²×C₅	—	2	2	no	—
197	C₂²×C₅	—	4	4	no	—
198	C₂²×C₅	—	4	2	no	—
199	C₂²×C₅	—	4	2	no	—
200	C₂²×C₅	—	5	2	no	—
201	C₂²×C₅	—	8	4	no	—
202	C₂²×C₅	—	10	2	no	—
203	C₂²×C₅	D₄×D₅	1	1	yes	characteristic
204	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
205	C₂×D₅	—	2	2	no	—
206	C₂×D₅	C₂×C₅×D₄	1	2	yes	—
207	C₄×C₅	—	2	2	no	—
208	C₄×C₅	—	2	2	no	—
209	C₄×C₅	—	2	2	no	—
210	C₄×C₅	—	2	2	no	—
211	C₄×C₅	—	4	4	no	—
212	C₄×C₅	—	5	2	no	—
213	C₄×C₅	—	10	2	no	—
214	C₄×C₅	—	10	2	no	—
215	C₄×C₅	D₄×D₅	1	1	yes	characteristic
216	C₄×C₅	C₂×C₅×D₄	1	1	yes	characteristic
217	C₄×C₅	C₂³×D₅	1	1	yes	characteristic
218	C₄×C₅	C₂⁴×C₅	1	1	yes	characteristic
219	Dic₅	—	2	2	no	—
220	Dic₅	—	2	2	no	—
221	Dic₅	C₂×C₅×D₄	1	2	yes	—
222	#41	—	5	4	no	—
223	#41	—	10	2	no	—
224	C₂×Q₈	—	5	4	no	—
225	C₂×Q₈	—	5	1	no	—
226	C₂×Q₈	—	5	1	no	—
227	C₂×Q₈	—	10	1	no	—
228	C₂×Q₈	—	10	2	no	—
229	C₂×Q₈	C₂×C₅×D₅	1	1	yes	characteristic
230	C₂×D₄	—	10	1	no	—
231	C₂×D₄	C₂×C₅×D₅	1	1	yes	characteristic
232	C₂²×C₄	—	5	1	no	—
233	C₂²×C₄	—	10	1	no	—
234	C₂²×C₄	—	10	1	no	—
235	Q₁₆	—	5	4	no	—
236	Q₁₆	—	5	4	no	—
237	SD₁₆	—	5	4	no	—
238	SD₁₆	C₂×C₅×D₅	1	4	yes	complemented
239	M₁₆	—	5	4	no	—
240	C₂×C₈	—	5	1	no	—
241	C₂×C₈	C₂×C₅×D₅	1	1	yes	characteristic
242	C₂×C₅	—	2	4	no	—
243	C₂×C₅	—	2	2	no	—
244	C₂×C₅	—	4	2	no	—
245	C₂×C₅	—	4	2	no	—
246	C₂×C₅	—	8	4	no	—
247	C₂×C₅	—	10	2	no	—
248	C₂×C₅	#4061	1	2	yes	direct product factor
249	C₂×C₅	C₅×#137	1	2	yes	complemented
250	C₂×C₅	C₂×D₄×D₅	1	1	yes	characteristic
251	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
252	D₅	—	2	2	no	—
253	C₂³	—	2	1	no	—
254	C₂³	—	5	1	no	—
255	Q₈	—	2	1	no	—
256	Q₈	—	5	2	no	—
257	Q₈	—	5	2	no	—
258	Q₈	—	10	4	no	—
259	Q₈	—	10	1	no	—
260	Q₈	—	10	1	no	—
261	Q₈	C₂²×C₅×D₅	1	2	yes	—
262	D₄	—	2	1	no	—
263	D₄	—	10	4	no	—
264	D₄	C₂²×C₅×D₅	1	2	yes	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	5	1	no	—
267	C₂×C₄	—	5	2	no	—
268	C₂×C₄	—	5	2	no	—
269	C₂×C₄	—	10	4	no	—
270	C₂×C₄	—	10	4	no	—
271	C₂×C₄	—	10	1	no	—
272	C₂×C₄	—	10	1	no	—
273	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
274	C₈	—	5	2	no	—
275	C₈	C₂²×C₅×D₅	1	2	yes	—
276	C₅	—	2	2	no	—
277	C₅	C₂×#4061	1	1	yes	characteristic, direct product factor
278	C₅	C₂×C₅×#137	1	1	yes	characteristic, complemented
279	C₂²	—	2	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	5	2	no	—
282	C₂²	—	10	2	no	—
283	C₂²	C₅×D₄×D₅	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	5	2	no	—
286	C₄	—	10	2	no	—
287	C₄	—	10	2	no	—
288	C₄	C₅×D₄×D₅	1	1	yes	characteristic
289	C₄	C₂³×C₅×D₅	1	1	yes	characteristic
290	C₂	—	4	2	no	—
291	C₂	—	10	2	no	—
292	C₂	C₅×#4061	1	2	yes	direct product factor
293	C₂	C₂×C₅×D₄×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	D₄×D₅
commutator, cosocle	C₄×C₅	C₂⁴×C₅
Frattini	C₄	C₂³×C₅×D₅
Fitting	C₂×C₅²×SD₁₆	C₂
socle	C₂²×C₅²	C₂×D₄
prime radical	C₂×C₅×#711	C₂
hypercenter	C₂×C₅×SD₁₆	D₅
hypocenter	C₅	C₂×C₅×#137
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×SD₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×Q₈×D₅	C₂	semidirect
3. C₂×C₅²×SD₁₆	C₂	semidirect
4. C₂×C₅×#689	C₂	semidirect
5. C₂×C₅×#687	C₂	semidirect
6. C₅×#4061	C₂	direct (N × H)
7. C₂×C₅×Dic₂₀	C₂	semidirect
8. C₂×C₅×#676	C₂	semidirect
14. C₅×Q₈×D₅	C₂²	semidirect
22. C₅²×SD₁₆	C₂²	semidirect
24. C₅×#689	C₂²	semidirect
25. C₅×#687	C₂²	semidirect
27. C₅×Dic₂₀	C₂²	semidirect
28. C₅×#676	C₂²	semidirect
30. C₂×#4061	C₅	direct (N × H)
56. C₂×Q₈×D₅	C₂×C₅	semidirect
63. C₂×C₅×SD₁₆	D₅	semidirect
64. C₂×C₅×SD₁₆	C₂×C₅	semidirect
66. C₂×#689	C₂×C₅	semidirect
67. C₂×#687	C₂×C₅	semidirect
68. #4061	C₂×C₅	direct (N × H)
69. C₂×Dic₂₀	C₂×C₅	semidirect
70. C₂×#676	C₂×C₅	semidirect
102. Q₈×D₅	C₂²×C₅	semidirect
114. C₅×SD₁₆	C₂×D₅	semidirect
115. C₅×SD₁₆	C₂²×C₅	semidirect
121. #689	C₂²×C₅	semidirect
122. #687	C₂²×C₅	semidirect
124. Dic₂₀	C₂²×C₅	semidirect
125. #676	C₂²×C₅	semidirect
128. C₂×C₅²	#137	semidirect
191. C₂×SD₁₆	C₅×D₅	semidirect
193. C₅²	C₂×#137	semidirect
238. SD₁₆	C₂×C₅×D₅	semidirect
248. C₂×C₅	#4061	direct (N × H)
249. C₂×C₅	C₅×#137	semidirect
277. C₅	C₂×#4061	direct (N × H)
278. C₅	C₂×C₅×#137	semidirect
292. C₂	C₅×#4061	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×Q₈×D₅	2
C₂×C₅×#711	2
C₂×C₅²×SD₁₆	2
C₂×C₅×#689	2
C₂×C₅×#687	2
C₅×#4061	2
C₂×C₅×Dic₂₀	2
C₂×C₅×#676	2
C₂×C₅×#137	5
C₂×#4061	5

Subgroups — C2×C5×#4061 #13106914

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4061 #13106914