Subgroups — C₂×C₅×#4042 #13106651

Order 1600.

736 conjugacy classes of subgroups.

The subgroup lattice has 280 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	C₂²×C₅×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅²×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#685	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅×#685	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×#683	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×#683	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#4042	C₂	1	8	yes	maximal, direct product factor
9	C₂³×C₅×D₅	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×C₄×C₅²	C₂²	1	1	yes	characteristic
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	C₂×C₄×C₅×D₅	—	2	4	no	—
17	C₂×C₄×C₅×D₅	C₂²	1	4	yes	—
18	C₂×C₅×Dic₁₀	—	2	4	no	—
19	C₂×C₅×Dic₁₀	C₂²	1	4	yes	complemented
20	C₅²×C₄⋊C₄	C₂²	1	4	yes	complemented
21	C₅×#685	C₂²	1	4	yes	—
22	C₅×#685	C₂²	1	4	yes	—
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×#4042	C₅	1	1	yes	maximal, characteristic, direct product factor
27	C₂³×C₅²	C₂³	1	1	yes	characteristic
28	C₂²×C₅×D₅	—	2	2	no	—
29	C₂²×C₅×D₅	—	2	2	no	—
30	C₂²×C₅×D₅	Q₈	1	4	yes	complemented
31	C₂²×C₅×D₅	C₂³	1	2	yes	—
32	C₂×C₄×C₅²	—	2	2	no	—
33	C₂×C₄×C₅²	—	2	2	no	—
34	C₂×C₄×C₅²	—	2	2	no	—
35	C₂×C₄×C₅²	C₂³	1	2	yes	—
36	C₂×C₄×C₅²	C₂³	1	2	yes	—
37	C₂×C₄×C₅²	C₂³	1	2	yes	—
38	C₂×C₅×Dic₅	—	2	2	no	—
39	C₂×C₅×Dic₅	—	2	2	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	C₄×C₅×D₅	—	2	8	no	—
45	C₅×Dic₁₀	—	2	8	no	—
46	C₂²×C₅×Q₈	—	5	1	no	—
47	C₂³×C₄×C₅	—	5	1	no	—
48	C₂²×C₄×D₅	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	—	2	2	no	—
52	C₂×C₅×C₄⋊C₄	—	5	1	no	—
53	C₂×C₅×C₄⋊C₄	—	5	1	no	—
54	C₂×C₅×C₄⋊C₄	D₅	1	1	yes	characteristic, complemented
55	C₂×C₅×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
56	C₂×C₅×C₂²⋊C₄	—	5	1	no	—
57	C₂×C₅×C₂²⋊C₄	—	5	1	no	—
58	C₂×#685	C₂×C₅	1	1	yes	characteristic
59	C₂×#685	C₂×C₅	1	1	yes	characteristic
60	C₂×#683	C₂×C₅	1	1	yes	characteristic, complemented
61	C₂×#683	C₂×C₅	1	1	yes	characteristic, complemented
62	#4042	C₂×C₅	1	8	yes	direct product factor
63	C₂²×C₅²	C₂×D₄	1	2	yes	—
64	C₂²×C₅²	C₂×Q₈	1	2	yes	—
65	C₂²×C₅²	#41	1	2	yes	—
66	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
67	C₂×C₅×D₅	—	2	8	no	—
68	C₂×C₅×D₅	—	2	2	no	—
69	C₂×C₅×D₅	—	2	2	no	—
70	C₂×C₅×D₅	C₂×Q₈	1	4	yes	complemented
71	C₄×C₅²	—	2	2	no	—
72	C₄×C₅²	—	2	2	no	—
73	C₄×C₅²	—	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	—	5	1	no	—
78	C₂³×D₅	C₂²×C₅	1	1	yes	characteristic
79	C₂×C₅×Q₈	—	5	4	no	—
80	C₂×C₅×Q₈	—	10	4	no	—
81	C₂²×C₄×C₅	—	2	2	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	1	no	—
86	C₂²×C₄×C₅	—	5	1	no	—
87	C₂²×C₄×C₅	—	5	4	no	—
88	C₂²×C₄×C₅	—	10	4	no	—
89	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
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×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
95	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
96	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
97	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
98	C₂×C₄×D₅	—	2	4	no	—
99	C₂×C₄×D₅	C₂²×C₅	1	4	yes	—
100	C₂×Dic₁₀	—	2	4	no	—
101	C₂×Dic₁₀	C₂²×C₅	1	4	yes	complemented
102	C₅×C₄⋊C₄	—	2	8	no	—
103	C₅×C₄⋊C₄	—	5	4	no	—
104	C₅×C₄⋊C₄	—	5	4	no	—
105	C₅×C₄⋊C₄	C₂×D₅	1	4	yes	complemented
106	C₅×C₄⋊C₄	C₂²×C₅	1	4	yes	complemented
107	C₅×C₂²⋊C₄	—	5	4	no	—
108	C₅×C₂²⋊C₄	—	5	4	no	—
109	#685	C₂²×C₅	1	4	yes	—
110	#685	C₂²×C₅	1	4	yes	—
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	C₂²⋊Q₈	1	4	yes	complemented
115	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₅²	C₂²×Q₈	1	1	yes	characteristic
117	C₂×C₅²	C₂×#41	1	1	yes	characteristic
118	C₅×D₅	—	2	4	no	—
119	C₂³×C₅	—	2	2	no	—
120	C₂³×C₅	—	5	4	no	—
121	C₂³×C₅	—	5	2	no	—
122	C₂³×C₅	—	10	2	no	—
123	C₂³×C₅	—	10	2	no	—
124	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
125	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
126	C₂²×D₅	—	2	2	no	—
127	C₂²×D₅	—	2	2	no	—
128	C₂²×D₅	C₅×Q₈	1	4	yes	complemented
129	C₂²×D₅	C₂³×C₅	1	2	yes	—
130	C₅×Q₈	—	10	8	no	—
131	C₂×C₄×C₅	—	2	8	no	—
132	C₂×C₄×C₅	—	2	2	no	—
133	C₂×C₄×C₅	—	2	2	no	—
134	C₂×C₄×C₅	—	2	2	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	4	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	—	10	8	no	—
148	C₂×C₄×C₅	—	10	2	no	—
149	C₂×C₄×C₅	—	10	2	no	—
150	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
151	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
152	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
153	C₂×C₄×C₅	C₂³×C₅	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	C₅×D₄	1	4	yes	—
159	C₂×Dic₅	C₂³×C₅	1	2	yes	—
160	C₂×Dic₅	C₂³×C₅	1	2	yes	—
161	C₂×Dic₅	C₂³×C₅	1	2	yes	—
162	C₄×D₅	—	2	8	no	—
163	Dic₁₀	—	2	8	no	—
164	C₂²×Q₈	—	5	1	no	—
165	C₂³×C₄	—	5	1	no	—
166	C₂²⋊Q₈	—	5	8	no	—
167	C₂×C₄⋊C₄	—	5	1	no	—
168	C₂×C₄⋊C₄	—	5	1	no	—
169	C₂×C₄⋊C₄	C₅×D₅	1	1	yes	characteristic, complemented
170	C₂×C₂²⋊C₄	—	5	1	no	—
171	C₂×C₂²⋊C₄	—	5	1	no	—
172	C₅²	C₂×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
173	C₂²×C₅	—	2	4	no	—
174	C₂²×C₅	—	2	4	no	—
175	C₂²×C₅	—	2	4	no	—
176	C₂²×C₅	—	2	2	no	—
177	C₂²×C₅	—	5	4	no	—
178	C₂²×C₅	—	10	2	no	—
179	C₂²×C₅	—	10	2	no	—
180	C₂²×C₅	—	10	8	no	—
181	C₂²×C₅	#709	1	2	yes	—
182	C₂²×C₅	D₄×D₅	1	2	yes	—
183	C₂²×C₅	Q₈×D₅	1	2	yes	—
184	C₂²×C₅	C₂×C₅×D₄	1	2	yes	—
185	C₂²×C₅	C₂×C₅×Q₈	1	2	yes	—
186	C₂²×C₅	C₅×#41	1	2	yes	—
187	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
188	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
189	C₂×D₅	—	2	8	no	—
190	C₂×D₅	—	2	2	no	—
191	C₂×D₅	—	2	2	no	—
192	C₂×D₅	C₂×C₅×Q₈	1	4	yes	complemented
193	C₄×C₅	—	2	8	no	—
194	C₄×C₅	—	2	2	no	—
195	C₄×C₅	—	2	2	no	—
196	C₄×C₅	—	2	2	no	—
197	C₄×C₅	—	2	2	no	—
198	C₄×C₅	—	2	2	no	—
199	C₄×C₅	—	2	2	no	—
200	C₄×C₅	—	4	4	no	—
201	C₄×C₅	—	4	4	no	—
202	C₄×C₅	—	5	4	no	—
203	C₄×C₅	—	10	2	no	—
204	C₄×C₅	—	10	2	no	—
205	Dic₅	—	2	2	no	—
206	Dic₅	—	2	2	no	—
207	Dic₅	C₂×C₅×D₄	1	4	yes	—
208	C₂⁴	—	5	1	no	—
209	C₂×Q₈	—	5	4	no	—
210	C₂×Q₈	—	10	4	no	—
211	C₂²×C₄	—	5	1	no	—
212	C₂²×C₄	—	5	1	no	—
213	C₂²×C₄	—	5	1	no	—
214	C₂²×C₄	—	5	4	no	—
215	C₂²×C₄	—	10	4	no	—
216	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
219	C₄⋊C₄	—	5	4	no	—
220	C₄⋊C₄	—	5	4	no	—
221	C₄⋊C₄	C₂×C₅×D₅	1	4	yes	complemented
222	C₂²⋊C₄	—	5	4	no	—
223	C₂²⋊C₄	—	5	4	no	—
224	C₂×C₅	—	2	8	no	—
225	C₂×C₅	—	2	2	no	—
226	C₂×C₅	—	2	2	no	—
227	C₂×C₅	—	2	2	no	—
228	C₂×C₅	—	10	4	no	—
229	C₂×C₅	#4042	1	4	yes	direct product factor
230	C₂×C₅	C₅×C₂²⋊Q₈	1	4	yes	complemented
231	C₂×C₅	C₂×#709	1	1	yes	characteristic
232	C₂×C₅	C₂×D₄×D₅	1	1	yes	characteristic
233	C₂×C₅	C₂×Q₈×D₅	1	1	yes	characteristic
234	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
235	C₂×C₅	C₂²×C₅×Q₈	1	1	yes	characteristic
236	C₂×C₅	C₂×C₅×#41	1	1	yes	characteristic
237	D₅	—	2	4	no	—
238	C₂³	—	5	4	no	—
239	C₂³	—	5	2	no	—
240	C₂³	—	10	2	no	—
241	C₂³	—	10	2	no	—
242	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
243	Q₈	—	10	8	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	5	4	no	—
248	C₂×C₄	—	5	2	no	—
249	C₂×C₄	—	5	2	no	—
250	C₂×C₄	—	5	2	no	—
251	C₂×C₄	—	10	8	no	—
252	C₂×C₄	—	10	2	no	—
253	C₂×C₄	—	10	2	no	—
254	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
255	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
256	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
257	C₅	—	2	2	no	—
258	C₅	C₂×#4042	1	1	yes	characteristic, direct product factor
259	C₅	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic, complemented
260	C₂²	—	5	4	no	—
261	C₂²	—	10	8	no	—
262	C₂²	—	10	2	no	—
263	C₂²	—	10	2	no	—
264	C₂²	C₅×#709	1	2	yes	—
265	C₂²	C₅×D₄×D₅	1	2	yes	—
266	C₂²	C₅×Q₈×D₅	1	2	yes	—
267	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	5	4	no	—
272	C₄	—	10	2	no	—
273	C₄	—	10	2	no	—
274	C₂	—	10	4	no	—
275	C₂	C₅×#4042	1	4	yes	direct product factor
276	C₂	C₂×C₅×#709	1	1	yes	characteristic
277	C₂	C₂×C₅×D₄×D₅	1	1	yes	characteristic
278	C₂	C₂×C₅×Q₈×D₅	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×C₄⋊C₄	C₂
socle	C₂³×C₅²	C₂³
prime radical	C₂³×C₅×D₅	C₂²
hypercenter	C₂×C₅×C₄⋊C₄	D₅
hypocenter	C₅	C₂×C₅×C₂²⋊Q₈
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₅×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₅×Dic₁₀	C₂	semidirect
3. C₂×C₅²×C₄⋊C₄	C₂	semidirect
6. C₂×C₅×#683	C₂	semidirect
7. C₂×C₅×#683	C₂	semidirect
8. C₅×#4042	C₂	direct (N × H)
19. C₂×C₅×Dic₁₀	C₂²	semidirect
20. C₅²×C₄⋊C₄	C₂²	semidirect
23. C₅×#683	C₂²	semidirect
24. C₅×#683	C₂²	semidirect
26. C₂×#4042	C₅	direct (N × H)
30. C₂²×C₅×D₅	Q₈	semidirect
49. C₂²×Dic₁₀	C₂×C₅	semidirect
54. C₂×C₅×C₄⋊C₄	D₅	semidirect
55. C₂×C₅×C₄⋊C₄	C₂×C₅	semidirect
60. C₂×#683	C₂×C₅	semidirect
61. C₂×#683	C₂×C₅	semidirect
62. #4042	C₂×C₅	direct (N × H)
70. C₂×C₅×D₅	C₂×Q₈	semidirect
101. C₂×Dic₁₀	C₂²×C₅	semidirect
105. C₅×C₄⋊C₄	C₂×D₅	semidirect
106. C₅×C₄⋊C₄	C₂²×C₅	semidirect
111. #683	C₂²×C₅	semidirect
112. #683	C₂²×C₅	semidirect
114. C₂×C₅²	C₂²⋊Q₈	semidirect
128. C₂²×D₅	C₅×Q₈	semidirect
169. C₂×C₄⋊C₄	C₅×D₅	semidirect
172. C₅²	C₂×C₂²⋊Q₈	semidirect
192. C₂×D₅	C₂×C₅×Q₈	semidirect
221. C₄⋊C₄	C₂×C₅×D₅	semidirect
229. C₂×C₅	#4042	direct (N × H)
230. C₂×C₅	C₅×C₂²⋊Q₈	semidirect
258. C₅	C₂×#4042	direct (N × H)
259. C₅	C₂×C₅×C₂²⋊Q₈	semidirect
275. C₂	C₅×#4042	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
C₂²×C₅×Dic₁₀	2
C₂×C₅²×C₄⋊C₄	2
C₂×C₅×#685	2
C₂×C₅×#683	2
C₅×#4042	2
C₂×C₅×C₂²⋊Q₈	5
C₂×#4042	5

Subgroups — C2×C5×#4042 #13106651

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4042 #13106651