Subgroups — C₂×C₅×#4024 #13106635

Order 1600.

776 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×Dic₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅²×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×C₂²⋊Dic₅	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×#683	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×#683	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#4024	C₂	1	8	yes	maximal, direct product factor
9	C₂⁴×C₅²	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
12	C₂²×C₅×Dic₅	—	2	4	no	—
13	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
14	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
15	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
16	C₂²×C₅×Dic₅	C₂²	1	4	yes	—
17	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
18	C₂×C₅×Dic₁₀	—	2	4	no	—
19	C₂×C₅×Dic₁₀	C₂²	1	4	yes	complemented
20	C₅²×C₂²⋊C₄	C₂²	1	4	yes	—
21	C₅×C₂²⋊Dic₅	C₂²	1	4	yes	—
22	C₅×C₄⋊Dic₅	C₂²	1	4	yes	complemented
23	C₅×#683	C₂²	1	4	yes	complemented
24	C₅×#683	C₂²	1	4	yes	complemented
25	C₂×C₅×C₂²⋊Q₈	—	5	1	no	maximal
26	C₂×#4024	C₅	1	1	yes	maximal, characteristic, direct product factor
27	C₂³×C₅²	—	2	2	no	—
28	C₂³×C₅²	—	2	2	no	—
29	C₂³×C₅²	Q₈	1	4	yes	complemented
30	C₂³×C₅²	C₂³	1	2	yes	—
31	C₂³×C₅²	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₅²	—	2	2	no	—
33	C₂×C₄×C₅²	—	2	2	no	—
34	C₂×C₄×C₅²	C₂³	1	2	yes	—
35	C₂×C₄×C₅²	C₂³	1	2	yes	—
36	C₂×C₅×Dic₅	—	2	2	no	—
37	C₂×C₅×Dic₅	—	2	2	no	—
38	C₂×C₅×Dic₅	—	2	2	no	—
39	C₂×C₅×Dic₅	—	2	8	no	—
40	C₂×C₅×Dic₅	D₄	1	4	yes	—
41	C₂×C₅×Dic₅	C₂³	1	2	yes	—
42	C₂×C₅×Dic₅	C₂³	1	2	yes	—
43	C₂×C₅×Dic₅	C₂³	1	2	yes	—
44	C₂×C₅×Dic₅	C₂³	1	2	yes	—
45	C₅×Dic₁₀	—	2	8	no	—
46	C₂²×C₅×Q₈	—	5	1	no	—
47	C₂³×C₄×C₅	—	5	1	no	—
48	C₂³×Dic₅	C₂×C₅	1	1	yes	characteristic
49	C₂²×Dic₁₀	C₂×C₅	1	1	yes	characteristic, complemented
50	C₅×C₂²⋊Q₈	—	5	8	no	—
51	C₂×C₅×C₄⋊C₄	—	5	1	no	—
52	C₂×C₅×C₄⋊C₄	—	5	1	no	—
53	C₂×C₅×C₄⋊C₄	—	5	1	no	—
54	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
55	C₂×C₅×C₂²⋊C₄	—	5	1	no	—
56	C₂×C₅×C₂²⋊C₄	D₅	1	1	yes	characteristic
57	C₂×C₅×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic
58	C₂×C₂²⋊Dic₅	C₂×C₅	1	1	yes	characteristic
59	C₂×C₄⋊Dic₅	C₂×C₅	1	1	yes	characteristic, complemented
60	C₂×#683	C₂×C₅	1	1	yes	characteristic, complemented
61	C₂×#683	C₂×C₅	1	1	yes	characteristic, complemented
62	#4024	C₂×C₅	1	8	yes	direct product factor
63	C₂²×C₅²	—	2	8	no	—
64	C₂²×C₅²	—	2	2	no	—
65	C₂²×C₅²	—	2	2	no	—
66	C₂²×C₅²	C₂×D₄	1	2	yes	—
67	C₂²×C₅²	C₂×Q₈	1	2	yes	—
68	C₂²×C₅²	C₂×Q₈	1	4	yes	complemented
69	C₂²×C₅²	#41	1	2	yes	—
70	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
71	C₄×C₅²	—	2	2	no	—
72	C₄×C₅²	—	2	2	no	—
73	C₅×Dic₅	—	2	2	no	—
74	C₅×Dic₅	—	2	2	no	—
75	C₅×Dic₅	—	2	2	no	—
76	C₅×Dic₅	C₂×D₄	1	4	yes	—
77	C₂⁴×C₅	—	2	2	no	—
78	C₂⁴×C₅	C₂×D₅	1	1	yes	characteristic
79	C₂⁴×C₅	C₂²×C₅	1	1	yes	characteristic
80	C₂×C₅×Q₈	—	5	4	no	—
81	C₂×C₅×Q₈	—	10	4	no	—
82	C₂²×C₄×C₅	—	2	2	no	—
83	C₂²×C₄×C₅	—	2	2	no	—
84	C₂²×C₄×C₅	—	5	1	no	—
85	C₂²×C₄×C₅	—	5	4	no	—
86	C₂²×C₄×C₅	—	5	1	no	—
87	C₂²×C₄×C₅	—	5	1	no	—
88	C₂²×C₄×C₅	—	5	1	no	—
89	C₂²×C₄×C₅	—	10	4	no	—
90	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
91	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
92	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
93	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
94	C₂²×Dic₅	—	2	4	no	—
95	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
96	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
97	C₂²×Dic₅	C₂²×C₅	1	4	yes	—
98	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
99	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
100	C₂×Dic₁₀	—	2	4	no	—
101	C₂×Dic₁₀	C₂²×C₅	1	4	yes	complemented
102	C₅×C₄⋊C₄	—	5	4	no	—
103	C₅×C₄⋊C₄	—	5	4	no	—
104	C₅×C₄⋊C₄	—	5	4	no	—
105	C₅×C₂²⋊C₄	—	2	8	no	—
106	C₅×C₂²⋊C₄	—	5	4	no	—
107	C₅×C₂²⋊C₄	C₂×D₅	1	4	yes	—
108	C₅×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
109	C₂²⋊Dic₅	C₂²×C₅	1	4	yes	—
110	C₄⋊Dic₅	C₂²×C₅	1	4	yes	complemented
111	#683	C₂²×C₅	1	4	yes	complemented
112	#683	C₂²×C₅	1	4	yes	complemented
113	C₂×C₂²⋊Q₈	—	5	1	no	Sylow
114	C₂×C₅²	—	2	4	no	—
115	C₂×C₅²	C₂²⋊Q₈	1	4	yes	complemented
116	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₅²	C₂²×Q₈	1	1	yes	characteristic
118	C₂×C₅²	C₂×#41	1	1	yes	characteristic
119	C₂³×C₅	—	2	2	no	—
120	C₂³×C₅	—	2	2	no	—
121	C₂³×C₅	—	2	2	no	—
122	C₂³×C₅	—	2	2	no	—
123	C₂³×C₅	—	2	8	no	—
124	C₂³×C₅	—	2	2	no	—
125	C₂³×C₅	—	2	4	no	—
126	C₂³×C₅	—	4	4	no	—
127	C₂³×C₅	—	4	4	no	—
128	C₂³×C₅	Dic₁₀	1	4	yes	complemented
129	C₂³×C₅	C₅×Q₈	1	4	yes	complemented
130	C₂³×C₅	C₂²×D₅	1	2	yes	—
131	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
132	C₂³×C₅	C₂³×C₅	1	2	yes	—
133	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
134	C₅×Q₈	—	10	8	no	—
135	C₂×C₄×C₅	—	2	2	no	—
136	C₂×C₄×C₅	—	2	2	no	—
137	C₂×C₄×C₅	—	2	2	no	—
138	C₂×C₄×C₅	—	2	2	no	—
139	C₂×C₄×C₅	—	2	4	no	—
140	C₂×C₄×C₅	—	2	4	no	—
141	C₂×C₄×C₅	—	4	4	no	—
142	C₂×C₄×C₅	—	4	4	no	—
143	C₂×C₄×C₅	—	5	4	no	—
144	C₂×C₄×C₅	—	5	2	no	—
145	C₂×C₄×C₅	—	5	2	no	—
146	C₂×C₄×C₅	—	5	2	no	—
147	C₂×C₄×C₅	—	5	2	no	—
148	C₂×C₄×C₅	—	10	8	no	—
149	C₂×C₄×C₅	—	10	2	no	—
150	C₂×C₄×C₅	—	10	2	no	—
151	C₂×C₄×C₅	—	10	2	no	—
152	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
153	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
154	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
155	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
156	C₂×Dic₅	—	2	2	no	—
157	C₂×Dic₅	—	2	2	no	—
158	C₂×Dic₅	—	2	2	no	—
159	C₂×Dic₅	—	2	8	no	—
160	C₂×Dic₅	C₅×D₄	1	4	yes	—
161	C₂×Dic₅	C₂³×C₅	1	2	yes	—
162	C₂×Dic₅	C₂³×C₅	1	2	yes	—
163	C₂×Dic₅	C₂³×C₅	1	2	yes	—
164	C₂×Dic₅	C₂³×C₅	1	2	yes	—
165	Dic₁₀	—	2	8	no	—
166	C₂²×Q₈	—	5	1	no	—
167	C₂³×C₄	—	5	1	no	—
168	C₂²⋊Q₈	—	5	8	no	—
169	C₂×C₄⋊C₄	—	5	1	no	—
170	C₂×C₄⋊C₄	—	5	1	no	—
171	C₂×C₄⋊C₄	—	5	1	no	—
172	C₂×C₂²⋊C₄	—	5	1	no	—
173	C₂×C₂²⋊C₄	C₅×D₅	1	1	yes	characteristic
174	C₅²	C₂×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
175	C₂²×C₅	—	2	8	no	—
176	C₂²×C₅	—	2	2	no	—
177	C₂²×C₅	—	2	2	no	—
178	C₂²×C₅	—	2	2	no	—
179	C₂²×C₅	—	2	2	no	—
180	C₂²×C₅	—	2	8	no	—
181	C₂²×C₅	—	2	4	no	—
182	C₂²×C₅	—	2	8	no	—
183	C₂²×C₅	—	2	4	no	—
184	C₂²×C₅	—	2	4	no	—
185	C₂²×C₅	—	2	2	no	—
186	C₂²×C₅	—	4	4	no	—
187	C₂²×C₅	—	4	4	no	—
188	C₂²×C₅	—	4	16	no	—
189	C₂²×C₅	C₂×Dic₁₀	1	2	yes	—
190	C₂²×C₅	C₂×Dic₁₀	1	4	yes	complemented
191	C₂²×C₅	D₄×D₅	1	2	yes	—
192	C₂²×C₅	#711	1	2	yes	—
193	C₂²×C₅	C₂×C₅×D₄	1	2	yes	—
194	C₂²×C₅	C₂×C₅×Q₈	1	2	yes	—
195	C₂²×C₅	C₂×C₅×Q₈	1	4	yes	complemented
196	C₂²×C₅	C₅×#41	1	2	yes	—
197	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
198	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
199	C₄×C₅	—	2	2	no	—
200	C₄×C₅	—	2	2	no	—
201	C₄×C₅	—	2	2	no	—
202	C₄×C₅	—	2	2	no	—
203	C₄×C₅	—	4	4	no	—
204	C₄×C₅	—	4	4	no	—
205	C₄×C₅	—	5	4	no	—
206	C₄×C₅	—	10	2	no	—
207	C₄×C₅	—	10	2	no	—
208	C₄×C₅	—	10	2	no	—
209	Dic₅	—	2	2	no	—
210	Dic₅	—	2	2	no	—
211	Dic₅	—	2	2	no	—
212	Dic₅	C₂×C₅×D₄	1	4	yes	—
213	C₂⁴	C₂×C₅×D₅	1	1	yes	characteristic
214	C₂×Q₈	—	5	4	no	—
215	C₂×Q₈	—	10	4	no	—
216	C₂²×C₄	—	5	1	no	—
217	C₂²×C₄	—	5	1	no	—
218	C₂²×C₄	—	5	1	no	—
219	C₂²×C₄	—	5	1	no	—
220	C₂²×C₄	—	5	4	no	—
221	C₂²×C₄	—	10	4	no	—
222	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
224	C₄⋊C₄	—	5	4	no	—
225	C₄⋊C₄	—	5	4	no	—
226	C₄⋊C₄	—	5	4	no	—
227	C₂²⋊C₄	—	5	4	no	—
228	C₂²⋊C₄	C₂×C₅×D₅	1	4	yes	—
229	C₂×C₅	—	2	4	no	—
230	C₂×C₅	—	2	4	no	—
231	C₂×C₅	—	2	2	no	—
232	C₂×C₅	—	2	2	no	—
233	C₂×C₅	—	2	2	no	—
234	C₂×C₅	—	2	8	no	—
235	C₂×C₅	—	4	8	no	—
236	C₂×C₅	#4024	1	4	yes	direct product factor
237	C₂×C₅	C₅×C₂²⋊Q₈	1	4	yes	complemented
238	C₂×C₅	C₂²×Dic₁₀	1	1	yes	characteristic
239	C₂×C₅	C₂×D₄×D₅	1	1	yes	characteristic
240	C₂×C₅	C₂×#711	1	1	yes	characteristic
241	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
242	C₂×C₅	C₂²×C₅×Q₈	1	1	yes	characteristic
243	C₂×C₅	C₂×C₅×#41	1	1	yes	characteristic
244	C₂³	—	2	2	no	—
245	C₂³	—	2	2	no	—
246	C₂³	C₅×Dic₁₀	1	4	yes	complemented
247	C₂³	C₂²×C₅×D₅	1	2	yes	—
248	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
249	Q₈	—	10	8	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	5	4	no	—
253	C₂×C₄	—	5	2	no	—
254	C₂×C₄	—	5	2	no	—
255	C₂×C₄	—	5	2	no	—
256	C₂×C₄	—	5	2	no	—
257	C₂×C₄	—	10	8	no	—
258	C₂×C₄	—	10	2	no	—
259	C₂×C₄	—	10	2	no	—
260	C₂×C₄	—	10	2	no	—
261	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
262	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
263	C₅	—	2	2	no	—
264	C₅	C₂×#4024	1	1	yes	characteristic, direct product factor
265	C₅	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic, complemented
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	8	no	—
269	C₂²	C₂×C₅×Dic₁₀	1	2	yes	—
270	C₂²	C₂×C₅×Dic₁₀	1	4	yes	complemented
271	C₂²	C₅×D₄×D₅	1	2	yes	—
272	C₂²	C₅×#711	1	2	yes	—
273	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	5	4	no	—
277	C₄	—	10	2	no	—
278	C₄	—	10	2	no	—
279	C₄	—	10	2	no	—
280	C₂	—	2	4	no	—
281	C₂	C₅×#4024	1	4	yes	direct product factor
282	C₂	C₂²×C₅×Dic₁₀	1	1	yes	characteristic
283	C₂	C₂×C₅×D₄×D₅	1	1	yes	characteristic
284	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	C₂⁴×C₅²	C₂²
hypercenter	C₂×C₅×C₂²⋊C₄	D₅
hypocenter	C₅	C₂×C₅×C₂²⋊Q₈
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×C₅×Dic₁₀	C₂	semidirect
5. C₂×C₅×C₄⋊Dic₅	C₂	semidirect
6. C₂×C₅×#683	C₂	semidirect
7. C₂×C₅×#683	C₂	semidirect
8. C₅×#4024	C₂	direct (N × H)
19. C₂×C₅×Dic₁₀	C₂²	semidirect
22. C₅×C₄⋊Dic₅	C₂²	semidirect
23. C₅×#683	C₂²	semidirect
24. C₅×#683	C₂²	semidirect
26. C₂×#4024	C₅	direct (N × H)
29. C₂³×C₅²	Q₈	semidirect
49. C₂²×Dic₁₀	C₂×C₅	semidirect
59. C₂×C₄⋊Dic₅	C₂×C₅	semidirect
60. C₂×#683	C₂×C₅	semidirect
61. C₂×#683	C₂×C₅	semidirect
62. #4024	C₂×C₅	direct (N × H)
68. C₂²×C₅²	C₂×Q₈	semidirect
101. C₂×Dic₁₀	C₂²×C₅	semidirect
110. C₄⋊Dic₅	C₂²×C₅	semidirect
111. #683	C₂²×C₅	semidirect
112. #683	C₂²×C₅	semidirect
115. C₂×C₅²	C₂²⋊Q₈	semidirect
128. C₂³×C₅	Dic₁₀	semidirect
129. C₂³×C₅	C₅×Q₈	semidirect
174. C₅²	C₂×C₂²⋊Q₈	semidirect
190. C₂²×C₅	C₂×Dic₁₀	semidirect
195. C₂²×C₅	C₂×C₅×Q₈	semidirect
236. C₂×C₅	#4024	direct (N × H)
237. C₂×C₅	C₅×C₂²⋊Q₈	semidirect
246. C₂³	C₅×Dic₁₀	semidirect
264. C₅	C₂×#4024	direct (N × H)
265. C₅	C₂×C₅×C₂²⋊Q₈	semidirect
270. C₂²	C₂×C₅×Dic₁₀	semidirect
281. C₂	C₅×#4024	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₅×Dic₅	2
C₂²×C₅×Dic₁₀	2
C₂×C₅²×C₂²⋊C₄	2
C₂×C₅×C₂²⋊Dic₅	2
C₂×C₅×C₄⋊Dic₅	2
C₂×C₅×#683	2
C₅×#4024	2
C₂×C₅×C₂²⋊Q₈	5
C₂×#4024	5

Subgroups — C2×C5×#4024 #13106635

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4024 #13106635