Subgroups — C₂×C₅×#4029 #13106643

Order 1600.

816 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₅×D₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅²×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×C₂²⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×#685	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×#683	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#4029	C₂	1	8	yes	maximal, direct product factor
9	C₂⁴×C₅²	C₂²	1	1	yes	characteristic
10	C₂³×C₅×D₅	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
13	C₂×C₅×C₂²⋊D₅	—	2	4	no	—
14	C₂×C₅×C₂²⋊D₅	C₂²	1	4	yes	—
15	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
16	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
17	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₅×D₅	—	2	4	no	—
19	C₂×C₄×C₅×D₅	C₂²	1	4	yes	complemented
20	C₅²×C₂²⋊C₄	C₂²	1	4	yes	complemented
21	C₅×C₂²⋊Dic₅	C₂²	1	4	yes	complemented
22	C₅×#685	C₂²	1	4	yes	complemented
23	C₅×C₄⋊Dic₅	C₂²	1	4	yes	complemented
24	C₅×#683	C₂²	1	4	yes	complemented
25	C₂×C₅×#123	—	5	1	no	maximal
26	C₂×#4029	C₅	1	1	yes	maximal, characteristic, direct product factor
27	C₂³×C₅²	—	2	2	no	—
28	C₂³×C₅²	—	2	2	no	—
29	C₂³×C₅²	—	2	2	no	—
30	C₂³×C₅²	C₂³	1	2	yes	—
31	C₂³×C₅²	C₂³	1	1	yes	characteristic
32	C₂²×C₅×D₅	—	2	2	no	—
33	C₂²×C₅×D₅	—	2	2	no	—
34	C₂²×C₅×D₅	D₄	1	4	yes	—
35	C₂²×C₅×D₅	C₂³	1	2	yes	—
36	C₂×C₄×C₅²	—	2	2	no	—
37	C₂×C₄×C₅²	—	2	2	no	—
38	C₂×C₄×C₅²	C₂³	1	2	yes	—
39	C₂×C₄×C₅²	C₂³	1	2	yes	—
40	C₅×C₂²⋊D₅	—	2	8	no	—
41	C₂×C₅×Dic₅	—	2	2	no	—
42	C₂×C₅×Dic₅	—	2	2	no	—
43	C₂×C₅×Dic₅	—	2	2	no	—
44	C₂×C₅×Dic₅	C₂³	1	2	yes	—
45	C₂×C₅×Dic₅	C₂³	1	2	yes	—
46	C₂×C₅×Dic₅	C₂³	1	2	yes	—
47	C₄×C₅×D₅	—	2	8	no	—
48	C₂²×C₅×D₄	—	5	1	no	—
49	C₂³×C₄×C₅	—	5	1	no	—
50	C₂²×C₂²⋊D₅	C₂×C₅	1	1	yes	characteristic
51	C₂²×C₄×D₅	C₂×C₅	1	1	yes	characteristic, complemented
52	C₅×#123	—	5	8	no	—
53	C₂×C₅×C₄⋊C₄	—	5	1	no	—
54	C₂×C₅×C₄⋊C₄	—	5	1	no	—
55	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
56	C₂×C₅×C₂²⋊C₄	—	5	1	no	—
57	C₂×C₅×C₂²⋊C₄	—	5	1	no	—
58	C₂×C₅×C₂²⋊C₄	D₅	1	1	yes	characteristic, complemented
59	C₂×C₅×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
60	C₂×C₂²⋊Dic₅	C₂×C₅	1	1	yes	characteristic, complemented
61	C₂×#685	C₂×C₅	1	1	yes	characteristic, complemented
62	C₂×C₄⋊Dic₅	C₂×C₅	1	1	yes	characteristic, complemented
63	C₂×#683	C₂×C₅	1	1	yes	characteristic, complemented
64	#4029	C₂×C₅	1	8	yes	direct product factor
65	C₂²×C₅²	—	2	2	no	—
66	C₂²×C₅²	—	2	2	no	—
67	C₂²×C₅²	—	2	2	no	—
68	C₂²×C₅²	—	4	4	no	—
69	C₂²×C₅²	C₂×D₄	1	2	yes	—
70	C₂²×C₅²	#41	1	2	yes	—
71	C₂²×C₅²	#41	1	2	yes	—
72	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
73	C₂×C₅×D₅	—	2	8	no	—
74	C₂×C₅×D₅	—	2	2	no	—
75	C₂×C₅×D₅	—	2	2	no	—
76	C₂×C₅×D₅	C₂×D₄	1	4	yes	—
77	C₄×C₅²	—	2	2	no	—
78	C₄×C₅²	—	2	2	no	—
79	C₅×Dic₅	—	2	2	no	—
80	C₅×Dic₅	—	2	2	no	—
81	C₅×Dic₅	—	2	2	no	—
82	C₂⁴×C₅	—	2	2	no	—
83	C₂⁴×C₅	—	5	1	no	—
84	C₂⁴×C₅	C₂×D₅	1	1	yes	characteristic
85	C₂⁴×C₅	C₂²×C₅	1	1	yes	characteristic
86	C₂³×D₅	C₂²×C₅	1	1	yes	characteristic
87	C₂×C₅×D₄	—	5	4	no	—
88	C₂×C₅×D₄	—	10	4	no	—
89	C₂²×C₄×C₅	—	2	2	no	—
90	C₂²×C₄×C₅	—	2	2	no	—
91	C₂²×C₄×C₅	—	5	1	no	—
92	C₂²×C₄×C₅	—	5	1	no	—
93	C₂²×C₄×C₅	—	5	1	no	—
94	C₂²×C₄×C₅	—	5	4	no	—
95	C₂²×C₄×C₅	—	10	4	no	—
96	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
97	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
98	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
99	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
100	C₂×C₂²⋊D₅	—	2	4	no	—
101	C₂×C₂²⋊D₅	C₂²×C₅	1	4	yes	—
102	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
103	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
104	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
105	C₂×C₄×D₅	—	2	4	no	—
106	C₂×C₄×D₅	C₂²×C₅	1	4	yes	complemented
107	C₅×C₄⋊C₄	—	5	4	no	—
108	C₅×C₄⋊C₄	—	5	4	no	—
109	C₅×C₂²⋊C₄	—	2	8	no	—
110	C₅×C₂²⋊C₄	—	5	4	no	—
111	C₅×C₂²⋊C₄	—	5	4	no	—
112	C₅×C₂²⋊C₄	C₂×D₅	1	4	yes	complemented
113	C₅×C₂²⋊C₄	C₂²×C₅	1	4	yes	complemented
114	C₂²⋊Dic₅	C₂²×C₅	1	4	yes	complemented
115	#685	C₂²×C₅	1	4	yes	complemented
116	C₄⋊Dic₅	C₂²×C₅	1	4	yes	complemented
117	#683	C₂²×C₅	1	4	yes	complemented
118	C₂×#123	—	5	1	no	Sylow
119	C₂×C₅²	—	4	2	no	—
120	C₂×C₅²	#123	1	4	yes	complemented
121	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₅²	C₂×#41	1	1	yes	characteristic
123	C₂×C₅²	C₂×#41	1	1	yes	characteristic
124	C₅×D₅	—	2	4	no	—
125	C₂³×C₅	—	2	8	no	—
126	C₂³×C₅	—	2	2	no	—
127	C₂³×C₅	—	2	2	no	—
128	C₂³×C₅	—	2	2	no	—
129	C₂³×C₅	—	2	2	no	—
130	C₂³×C₅	—	2	2	no	—
131	C₂³×C₅	—	2	2	no	—
132	C₂³×C₅	—	2	2	no	—
133	C₂³×C₅	—	2	4	no	—
134	C₂³×C₅	—	4	4	no	—
135	C₂³×C₅	—	4	4	no	—
136	C₂³×C₅	—	5	4	no	—
137	C₂³×C₅	—	5	2	no	—
138	C₂³×C₅	—	10	2	no	—
139	C₂³×C₅	—	10	2	no	—
140	C₂³×C₅	C₂²×D₅	1	2	yes	—
141	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
142	C₂³×C₅	C₂³×C₅	1	2	yes	—
143	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
144	C₂²×D₅	—	2	2	no	—
145	C₂²×D₅	—	2	2	no	—
146	C₂²×D₅	C₅×D₄	1	4	yes	—
147	C₂²×D₅	C₂³×C₅	1	2	yes	—
148	C₅×D₄	—	10	8	no	—
149	C₂×C₄×C₅	—	2	2	no	—
150	C₂×C₄×C₅	—	2	2	no	—
151	C₂×C₄×C₅	—	2	2	no	—
152	C₂×C₄×C₅	—	2	2	no	—
153	C₂×C₄×C₅	—	2	4	no	—
154	C₂×C₄×C₅	—	2	4	no	—
155	C₂×C₄×C₅	—	4	4	no	—
156	C₂×C₄×C₅	—	4	4	no	—
157	C₂×C₄×C₅	—	5	2	no	—
158	C₂×C₄×C₅	—	5	2	no	—
159	C₂×C₄×C₅	—	5	2	no	—
160	C₂×C₄×C₅	—	10	8	no	—
161	C₂×C₄×C₅	—	10	2	no	—
162	C₂×C₄×C₅	—	10	2	no	—
163	C₂×C₄×C₅	—	10	2	no	—
164	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
165	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
166	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
167	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
168	C₂²⋊D₅	—	2	8	no	—
169	C₂×Dic₅	—	2	2	no	—
170	C₂×Dic₅	—	2	2	no	—
171	C₂×Dic₅	—	2	2	no	—
172	C₂×Dic₅	C₂³×C₅	1	2	yes	—
173	C₂×Dic₅	C₂³×C₅	1	2	yes	—
174	C₂×Dic₅	C₂³×C₅	1	2	yes	—
175	C₄×D₅	—	2	8	no	—
176	C₂²×D₄	—	5	1	no	—
177	C₂³×C₄	—	5	1	no	—
178	#123	—	5	8	no	—
179	C₂×C₄⋊C₄	—	5	1	no	—
180	C₂×C₄⋊C₄	—	5	1	no	—
181	C₂×C₂²⋊C₄	—	5	1	no	—
182	C₂×C₂²⋊C₄	—	5	1	no	—
183	C₂×C₂²⋊C₄	C₅×D₅	1	1	yes	characteristic, complemented
184	C₅²	C₂×#123	1	1	yes	characteristic, complemented, Sylow
185	C₂²×C₅	—	2	8	no	—
186	C₂²×C₅	—	2	2	no	—
187	C₂²×C₅	—	2	2	no	—
188	C₂²×C₅	—	2	2	no	—
189	C₂²×C₅	—	2	2	no	—
190	C₂²×C₅	—	2	2	no	—
191	C₂²×C₅	—	2	2	no	—
192	C₂²×C₅	—	2	4	no	—
193	C₂²×C₅	—	2	4	no	—
194	C₂²×C₅	—	2	4	no	—
195	C₂²×C₅	—	2	2	no	—
196	C₂²×C₅	—	4	4	no	—
197	C₂²×C₅	—	4	4	no	—
198	C₂²×C₅	—	4	16	no	—
199	C₂²×C₅	—	4	4	no	—
200	C₂²×C₅	—	4	4	no	—
201	C₂²×C₅	—	5	4	no	—
202	C₂²×C₅	—	10	2	no	—
203	C₂²×C₅	—	10	2	no	—
204	C₂²×C₅	—	10	8	no	—
205	C₂²×C₅	#709	1	2	yes	—
206	C₂²×C₅	D₄×D₅	1	2	yes	—
207	C₂²×C₅	#711	1	2	yes	—
208	C₂²×C₅	C₂×C₅×D₄	1	2	yes	—
209	C₂²×C₅	C₅×#41	1	2	yes	—
210	C₂²×C₅	C₅×#41	1	2	yes	—
211	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
212	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
213	C₂×D₅	—	2	8	no	—
214	C₂×D₅	—	2	2	no	—
215	C₂×D₅	—	2	2	no	—
216	C₂×D₅	C₂×C₅×D₄	1	4	yes	—
217	C₄×C₅	—	2	2	no	—
218	C₄×C₅	—	2	2	no	—
219	C₄×C₅	—	2	2	no	—
220	C₄×C₅	—	2	2	no	—
221	C₄×C₅	—	4	4	no	—
222	C₄×C₅	—	4	4	no	—
223	C₄×C₅	—	10	2	no	—
224	C₄×C₅	—	10	2	no	—
225	C₄×C₅	—	10	2	no	—
226	Dic₅	—	2	2	no	—
227	Dic₅	—	2	2	no	—
228	Dic₅	—	2	2	no	—
229	C₂⁴	—	5	1	no	—
230	C₂⁴	C₂×C₅×D₅	1	1	yes	characteristic
231	C₂×D₄	—	5	4	no	—
232	C₂×D₄	—	10	4	no	—
233	C₂²×C₄	—	5	1	no	—
234	C₂²×C₄	—	5	1	no	—
235	C₂²×C₄	—	5	1	no	—
236	C₂²×C₄	—	5	4	no	—
237	C₂²×C₄	—	10	4	no	—
238	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
240	C₄⋊C₄	—	5	4	no	—
241	C₄⋊C₄	—	5	4	no	—
242	C₂²⋊C₄	—	5	4	no	—
243	C₂²⋊C₄	—	5	4	no	—
244	C₂²⋊C₄	C₂×C₅×D₅	1	4	yes	complemented
245	C₂×C₅	—	2	2	no	—
246	C₂×C₅	—	2	2	no	—
247	C₂×C₅	—	2	2	no	—
248	C₂×C₅	—	2	8	no	—
249	C₂×C₅	—	4	8	no	—
250	C₂×C₅	—	4	2	no	—
251	C₂×C₅	—	4	2	no	—
252	C₂×C₅	—	10	4	no	—
253	C₂×C₅	#4029	1	4	yes	direct product factor
254	C₂×C₅	C₅×#123	1	4	yes	complemented
255	C₂×C₅	C₂×#709	1	1	yes	characteristic
256	C₂×C₅	C₂×D₄×D₅	1	1	yes	characteristic
257	C₂×C₅	C₂×#711	1	1	yes	characteristic
258	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
259	C₂×C₅	C₂×C₅×#41	1	1	yes	characteristic
260	C₂×C₅	C₂×C₅×#41	1	1	yes	characteristic
261	D₅	—	2	4	no	—
262	C₂³	—	2	2	no	—
263	C₂³	—	2	2	no	—
264	C₂³	—	2	2	no	—
265	C₂³	—	5	4	no	—
266	C₂³	—	5	2	no	—
267	C₂³	—	10	2	no	—
268	C₂³	—	10	2	no	—
269	C₂³	C₂²×C₅×D₅	1	2	yes	—
270	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
271	D₄	—	10	8	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	5	2	no	—
275	C₂×C₄	—	5	2	no	—
276	C₂×C₄	—	5	2	no	—
277	C₂×C₄	—	10	8	no	—
278	C₂×C₄	—	10	2	no	—
279	C₂×C₄	—	10	2	no	—
280	C₂×C₄	—	10	2	no	—
281	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
282	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
283	C₅	—	2	2	no	—
284	C₅	C₂×#4029	1	1	yes	characteristic, direct product factor
285	C₅	C₂×C₅×#123	1	1	yes	characteristic, complemented
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	4	no	—
290	C₂²	—	5	4	no	—
291	C₂²	—	10	2	no	—
292	C₂²	—	10	2	no	—
293	C₂²	—	10	8	no	—
294	C₂²	C₅×#709	1	2	yes	—
295	C₂²	C₅×D₄×D₅	1	2	yes	—
296	C₂²	C₅×#711	1	2	yes	—
297	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	10	2	no	—
301	C₄	—	10	2	no	—
302	C₄	—	10	2	no	—
303	C₂	—	4	2	no	—
304	C₂	—	10	4	no	—
305	C₂	C₅×#4029	1	4	yes	direct product factor
306	C₂	C₂×C₅×#709	1	1	yes	characteristic
307	C₂	C₂×C₅×D₄×D₅	1	1	yes	characteristic
308	C₂	C₂×C₅×#711	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂²×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₅
Fitting	C₂×C₅²×C₂²⋊C₄	C₂
socle	C₂³×C₅²	C₂³
prime radical	C₂²×C₅×C₂²⋊D₅	C₂
hypercenter	C₂×C₅×C₂²⋊C₄	D₅
hypocenter	C₅	C₂×C₅×#123
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₅×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×C₅×D₅	C₂	semidirect
3. C₂×C₅²×C₂²⋊C₄	C₂	semidirect
4. C₂×C₅×C₂²⋊Dic₅	C₂	semidirect
5. C₂×C₅×#685	C₂	semidirect
6. C₂×C₅×C₄⋊Dic₅	C₂	semidirect
7. C₂×C₅×#683	C₂	semidirect
8. C₅×#4029	C₂	direct (N × H)
19. C₂×C₄×C₅×D₅	C₂²	semidirect
20. C₅²×C₂²⋊C₄	C₂²	semidirect
21. C₅×C₂²⋊Dic₅	C₂²	semidirect
22. C₅×#685	C₂²	semidirect
23. C₅×C₄⋊Dic₅	C₂²	semidirect
24. C₅×#683	C₂²	semidirect
26. C₂×#4029	C₅	direct (N × H)
51. C₂²×C₄×D₅	C₂×C₅	semidirect
58. C₂×C₅×C₂²⋊C₄	D₅	semidirect
59. C₂×C₅×C₂²⋊C₄	C₂×C₅	semidirect
60. C₂×C₂²⋊Dic₅	C₂×C₅	semidirect
61. C₂×#685	C₂×C₅	semidirect
62. C₂×C₄⋊Dic₅	C₂×C₅	semidirect
63. C₂×#683	C₂×C₅	semidirect
64. #4029	C₂×C₅	direct (N × H)
106. C₂×C₄×D₅	C₂²×C₅	semidirect
112. C₅×C₂²⋊C₄	C₂×D₅	semidirect
113. C₅×C₂²⋊C₄	C₂²×C₅	semidirect
114. C₂²⋊Dic₅	C₂²×C₅	semidirect
115. #685	C₂²×C₅	semidirect
116. C₄⋊Dic₅	C₂²×C₅	semidirect
117. #683	C₂²×C₅	semidirect
120. C₂×C₅²	#123	semidirect
183. C₂×C₂²⋊C₄	C₅×D₅	semidirect
184. C₅²	C₂×#123	semidirect
244. C₂²⋊C₄	C₂×C₅×D₅	semidirect
253. C₂×C₅	#4029	direct (N × H)
254. C₂×C₅	C₅×#123	semidirect
284. C₅	C₂×#4029	direct (N × H)
285. C₅	C₂×C₅×#123	semidirect
305. C₂	C₅×#4029	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×C₅×C₂²⋊D₅	2
C₂²×C₄×C₅×D₅	2
C₂×C₅²×C₂²⋊C₄	2
C₂×C₅×C₂²⋊Dic₅	2
C₂×C₅×#685	2
C₂×C₅×C₄⋊Dic₅	2
C₂×C₅×#683	2
C₅×#4029	2
C₂×C₅×#123	5
C₂×#4029	5

Subgroups — C2×C5×#4029 #13106643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4029 #13106643