Subgroups — C₂×C₅×#4062 #13106915

Order 1600.

600 conjugacy classes of subgroups.

The subgroup lattice has 303 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#713	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#711	C₂	1	1	yes	maximal, characteristic, complemented
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×#686	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4062	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₅×#677	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×C₈×D₅	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×#713	—	2	2	no	—
14	C₅×#713	C₂²	1	4	yes	complemented
15	C₅×#711	—	2	2	no	—
16	C₅×#711	C₂²	1	4	yes	complemented
17	C₂×C₅×D₂₀	—	2	1	no	—
18	C₂×C₅×D₂₀	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₅×D₅	—	2	1	no	—
20	C₂×C₄×C₅×D₅	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×#686	C₂²	1	4	yes	complemented
26	C₂×C₅×#168	C₂²	1	1	yes	characteristic
27	C₅×#677	C₂²	1	4	yes	complemented
28	C₅×C₈×D₅	C₂²	1	4	yes	complemented
29	C₂×C₅×#135	—	5	1	no	maximal
30	C₂×#4062	C₅	1	1	yes	maximal, characteristic, direct product factor
31	C₂³×C₅²	—	2	1	no	—
32	C₂²×C₅×D₅	—	2	1	no	—
33	C₂²×C₅×D₅	D₄	1	1	yes	characteristic
34	C₅²×Q₈	—	2	1	no	—
35	C₅²×Q₈	C₂³	1	2	yes	—
36	C₅²×D₄	—	2	1	no	—
37	C₅²×D₄	C₂³	1	2	yes	—
38	C₂×C₄×C₅²	—	2	1	no	—
39	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
40	C₅×C₂²⋊D₅	—	2	4	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	C₅×D₂₀	—	2	4	no	—
45	C₅×D₂₀	—	2	1	no	—
46	C₅×D₂₀	C₂³	1	2	yes	—
47	C₄×C₅×D₅	—	2	4	no	—
48	C₄×C₅×D₅	D₄	1	2	yes	—
49	C₄×C₅×D₅	C₂³	1	2	yes	—
50	C₅×Dic₁₀	—	2	1	no	—
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	C₂×C₅×#41	—	5	1	no	—
56	C₂×#713	C₂×C₅	1	1	yes	characteristic, complemented
57	C₂×#711	C₂×C₅	1	1	yes	characteristic, complemented
58	C₅×#135	—	5	8	no	—
59	C₂×C₅×Q₁₆	—	5	1	no	—
60	C₂×C₅×SD₁₆	—	2	2	no	—
61	C₂×C₅×SD₁₆	—	5	1	no	—
62	C₂×C₅×SD₁₆	D₅	1	1	yes	characteristic, complemented
63	C₂×C₅×SD₁₆	C₂×C₅	1	1	yes	characteristic, complemented
64	C₂×C₅×D₈	—	5	1	no	—
65	C₂²×C₅×C₈	—	5	1	no	—
66	C₂×#689	C₂×C₅	1	1	yes	characteristic, complemented
67	C₂×#686	C₂×C₅	1	1	yes	characteristic, complemented
68	#4062	C₂×C₅	1	8	yes	direct product factor
69	C₂×#677	C₂×C₅	1	1	yes	characteristic, complemented
70	C₂×C₈×D₅	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	—	2	2	no	—
76	C₂×C₅×D₅	—	4	2	no	—
77	C₂×C₅×D₅	C₂×D₄	1	2	yes	—
78	C₄×C₅²	—	2	2	no	—
79	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
80	C₄×C₅²	C₂⁴	1	1	yes	characteristic
81	C₅×Dic₅	—	2	2	no	—
82	C₅×Dic₅	C₂×D₄	1	2	yes	—
83	C₅×#41	—	5	4	no	—
84	C₅×#41	—	5	4	no	—
85	C₅×#41	—	10	2	no	—
86	C₅×#41	—	10	2	no	—
87	C₂×C₅×Q₈	—	2	2	no	—
88	C₂×C₅×Q₈	—	5	1	no	—
89	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic
90	C₂×C₅×Q₈	C₂²×C₅	1	1	yes	characteristic
91	C₂×C₅×D₄	—	2	2	no	—
92	C₂×C₅×D₄	—	5	1	no	—
93	C₂×C₅×D₄	—	10	1	no	—
94	C₂×C₅×D₄	—	10	1	no	—
95	C₂×C₅×D₄	C₂×D₅	1	1	yes	characteristic
96	C₂×C₅×D₄	C₂²×C₅	1	1	yes	characteristic
97	C₂²×C₄×C₅	—	5	1	no	—
98	C₂²×C₄×C₅	—	10	1	no	—
99	C₂²×C₄×C₅	—	10	1	no	—
100	C₂×C₂²⋊D₅	—	2	1	no	—
101	C₂²×Dic₅	—	2	1	no	—
102	#713	—	2	2	no	—
103	#713	C₂²×C₅	1	4	yes	complemented
104	#711	—	2	2	no	—
105	#711	C₂²×C₅	1	4	yes	complemented
106	C₂×D₂₀	—	2	1	no	—
107	C₂×D₂₀	C₂²×C₅	1	1	yes	characteristic
108	C₂×C₄×D₅	—	2	1	no	—
109	C₂×C₄×D₅	C₂²×C₅	1	1	yes	characteristic
110	C₂×Dic₁₀	C₂²×C₅	1	1	yes	characteristic
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×D₈	—	5	4	no	—
117	C₂×C₅×C₈	—	2	2	no	—
118	C₂×C₅×C₈	—	5	1	no	—
119	C₂×C₅×C₈	—	5	4	no	—
120	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic
121	C₂×C₅×C₈	C₂²×C₅	1	1	yes	characteristic
122	#689	C₂²×C₅	1	4	yes	complemented
123	#686	C₂²×C₅	1	4	yes	complemented
124	C₂×#168	C₂²×C₅	1	1	yes	characteristic
125	#677	C₂²×C₅	1	4	yes	complemented
126	C₈×D₅	C₂²×C₅	1	4	yes	complemented
127	C₂×#135	—	5	1	no	Sylow
128	C₂×C₅²	—	4	2	no	—
129	C₂×C₅²	#135	1	2	yes	complemented
130	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
131	C₅×D₅	—	2	2	no	—
132	C₅×D₅	—	4	2	no	—
133	C₂³×C₅	—	2	1	no	—
134	C₂³×C₅	—	2	1	no	—
135	C₂³×C₅	—	4	2	no	—
136	C₂³×C₅	—	5	1	no	—
137	C₂³×C₅	—	10	1	no	—
138	C₂²×D₅	—	2	1	no	—
139	C₂²×D₅	C₅×D₄	1	1	yes	characteristic
140	C₅×Q₈	—	2	1	no	—
141	C₅×Q₈	—	2	1	no	—
142	C₅×Q₈	—	2	4	no	—
143	C₅×Q₈	—	4	2	no	—
144	C₅×Q₈	—	5	2	no	—
145	C₅×Q₈	—	10	1	no	—
146	C₅×Q₈	C₂²×D₅	1	2	yes	—
147	C₅×Q₈	C₂³×C₅	1	2	yes	—
148	C₅×D₄	—	2	1	no	—
149	C₅×D₄	—	2	1	no	—
150	C₅×D₄	—	2	4	no	—
151	C₅×D₄	—	4	2	no	—
152	C₅×D₄	—	5	2	no	—
153	C₅×D₄	—	10	4	no	—
154	C₅×D₄	—	10	4	no	—
155	C₅×D₄	—	10	1	no	—
156	C₅×D₄	C₂²×D₅	1	2	yes	—
157	C₅×D₄	C₂³×C₅	1	2	yes	—
158	C₂×C₄×C₅	—	2	1	no	—
159	C₂×C₄×C₅	—	2	1	no	—
160	C₂×C₄×C₅	—	2	2	no	—
161	C₂×C₄×C₅	—	4	2	no	—
162	C₂×C₄×C₅	—	5	1	no	—
163	C₂×C₄×C₅	—	5	2	no	—
164	C₂×C₄×C₅	—	5	2	no	—
165	C₂×C₄×C₅	—	10	4	no	—
166	C₂×C₄×C₅	—	10	4	no	—
167	C₂×C₄×C₅	—	10	1	no	—
168	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
169	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
170	C₂²⋊D₅	—	2	4	no	—
171	C₂×Dic₅	—	2	1	no	—
172	C₂×Dic₅	—	2	4	no	—
173	C₂×Dic₅	C₅×D₄	1	1	yes	characteristic
174	D₂₀	—	2	4	no	—
175	D₂₀	—	2	1	no	—
176	D₂₀	C₂³×C₅	1	2	yes	—
177	C₄×D₅	—	2	4	no	—
178	C₄×D₅	C₅×D₄	1	2	yes	—
179	C₄×D₅	C₂³×C₅	1	2	yes	—
180	Dic₁₀	—	2	1	no	—
181	Dic₁₀	C₂³×C₅	1	2	yes	—
182	C₅×C₈	—	2	4	no	—
183	C₅×C₈	—	5	2	no	—
184	C₅×C₈	C₂²×D₅	1	2	yes	—
185	C₅×C₈	C₂³×C₅	1	2	yes	—
186	#168	C₂³×C₅	1	2	yes	—
187	C₂×#41	—	5	1	no	—
188	C₂×#41	—	5	1	no	—
189	#135	—	5	8	no	—
190	C₂×Q₁₆	—	5	1	no	—
191	C₂×SD₁₆	—	5	1	no	—
192	C₂×SD₁₆	C₅×D₅	1	1	yes	characteristic, complemented
193	C₂×D₈	—	5	1	no	—
194	C₂²×C₈	—	5	1	no	—
195	C₅²	C₂×#135	1	1	yes	characteristic, complemented, Sylow
196	C₂²×C₅	—	2	2	no	—
197	C₂²×C₅	—	2	2	no	—
198	C₂²×C₅	—	2	2	no	—
199	C₂²×C₅	—	4	4	no	—
200	C₂²×C₅	—	4	2	no	—
201	C₂²×C₅	—	4	2	no	—
202	C₂²×C₅	—	5	2	no	—
203	C₂²×C₅	—	8	4	no	—
204	C₂²×C₅	—	10	2	no	—
205	C₂²×C₅	—	10	2	no	—
206	C₂²×C₅	—	20	2	no	—
207	C₂²×C₅	D₄×D₅	1	1	yes	characteristic
208	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
209	C₂×D₅	—	2	2	no	—
210	C₂×D₅	—	2	2	no	—
211	C₂×D₅	—	4	2	no	—
212	C₂×D₅	C₂×C₅×D₄	1	2	yes	—
213	C₄×C₅	—	2	2	no	—
214	C₄×C₅	—	2	2	no	—
215	C₄×C₅	—	2	2	no	—
216	C₄×C₅	—	2	2	no	—
217	C₄×C₅	—	4	4	no	—
218	C₄×C₅	—	5	2	no	—
219	C₄×C₅	—	10	2	no	—
220	C₄×C₅	D₄×D₅	1	1	yes	characteristic
221	C₄×C₅	C₂×C₅×D₄	1	1	yes	characteristic
222	C₄×C₅	C₂³×D₅	1	1	yes	characteristic
223	C₄×C₅	C₂⁴×C₅	1	1	yes	characteristic
224	Dic₅	—	2	2	no	—
225	Dic₅	C₂×C₅×D₄	1	2	yes	—
226	#41	—	5	4	no	—
227	#41	—	5	4	no	—
228	#41	—	10	2	no	—
229	#41	—	10	2	no	—
230	C₂×Q₈	—	5	1	no	—
231	C₂×Q₈	C₂×C₅×D₅	1	1	yes	characteristic
232	C₂×D₄	—	5	1	no	—
233	C₂×D₄	—	10	1	no	—
234	C₂×D₄	—	10	1	no	—
235	C₂×D₄	C₂×C₅×D₅	1	1	yes	characteristic
236	C₂²×C₄	—	5	1	no	—
237	C₂²×C₄	—	10	1	no	—
238	C₂²×C₄	—	10	1	no	—
239	Q₁₆	—	5	4	no	—
240	SD₁₆	—	5	4	no	—
241	SD₁₆	C₂×C₅×D₅	1	4	yes	complemented
242	D₈	—	5	4	no	—
243	C₂×C₈	—	5	1	no	—
244	C₂×C₈	—	5	4	no	—
245	C₂×C₈	C₂×C₅×D₅	1	1	yes	characteristic
246	C₂×C₅	—	2	4	no	—
247	C₂×C₅	—	2	2	no	—
248	C₂×C₅	—	4	2	no	—
249	C₂×C₅	—	4	2	no	—
250	C₂×C₅	—	8	4	no	—
251	C₂×C₅	—	10	2	no	—
252	C₂×C₅	—	20	2	no	—
253	C₂×C₅	#4062	1	2	yes	direct product factor
254	C₂×C₅	C₅×#135	1	2	yes	complemented
255	C₂×C₅	C₂×D₄×D₅	1	1	yes	characteristic
256	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
257	D₅	—	2	2	no	—
258	D₅	—	4	2	no	—
259	C₂³	—	2	1	no	—
260	C₂³	—	5	1	no	—
261	C₂³	—	10	1	no	—
262	Q₈	—	2	1	no	—
263	Q₈	—	5	2	no	—
264	Q₈	—	10	1	no	—
265	Q₈	C₂²×C₅×D₅	1	2	yes	—
266	D₄	—	2	1	no	—
267	D₄	—	5	2	no	—
268	D₄	—	10	4	no	—
269	D₄	—	10	4	no	—
270	D₄	—	10	1	no	—
271	D₄	C₂²×C₅×D₅	1	2	yes	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	5	1	no	—
274	C₂×C₄	—	5	2	no	—
275	C₂×C₄	—	5	2	no	—
276	C₂×C₄	—	10	4	no	—
277	C₂×C₄	—	10	4	no	—
278	C₂×C₄	—	10	1	no	—
279	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
280	C₈	—	5	2	no	—
281	C₈	C₂²×C₅×D₅	1	2	yes	—
282	C₅	—	2	2	no	—
283	C₅	C₂×#4062	1	1	yes	characteristic, direct product factor
284	C₅	C₂×C₅×#135	1	1	yes	characteristic, complemented
285	C₂²	—	2	2	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	5	2	no	—
288	C₂²	—	10	2	no	—
289	C₂²	—	10	2	no	—
290	C₂²	—	20	2	no	—
291	C₂²	C₅×D₄×D₅	1	1	yes	characteristic
292	C₄	—	2	2	no	—
293	C₄	—	5	2	no	—
294	C₄	—	10	2	no	—
295	C₄	C₅×D₄×D₅	1	1	yes	characteristic
296	C₄	C₂³×C₅×D₅	1	1	yes	characteristic
297	C₂	—	4	2	no	—
298	C₂	—	10	2	no	—
299	C₂	—	20	2	no	—
300	C₂	C₅×#4062	1	2	yes	direct product factor
301	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
hypercenter	C₂×C₅×SD₁₆	D₅
hypocenter	C₅	C₂×C₅×#135
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×#713	C₂	semidirect
2. C₂×C₅×#711	C₂	semidirect
3. C₂×C₅²×SD₁₆	C₂	semidirect
4. C₂×C₅×#689	C₂	semidirect
5. C₂×C₅×#686	C₂	semidirect
6. C₅×#4062	C₂	direct (N × H)
7. C₂×C₅×#677	C₂	semidirect
8. C₂×C₅×C₈×D₅	C₂	semidirect
14. C₅×#713	C₂²	semidirect
16. C₅×#711	C₂²	semidirect
22. C₅²×SD₁₆	C₂²	semidirect
24. C₅×#689	C₂²	semidirect
25. C₅×#686	C₂²	semidirect
27. C₅×#677	C₂²	semidirect
28. C₅×C₈×D₅	C₂²	semidirect
30. C₂×#4062	C₅	direct (N × H)
56. C₂×#713	C₂×C₅	semidirect
57. C₂×#711	C₂×C₅	semidirect
62. C₂×C₅×SD₁₆	D₅	semidirect
63. C₂×C₅×SD₁₆	C₂×C₅	semidirect
66. C₂×#689	C₂×C₅	semidirect
67. C₂×#686	C₂×C₅	semidirect
68. #4062	C₂×C₅	direct (N × H)
69. C₂×#677	C₂×C₅	semidirect
70. C₂×C₈×D₅	C₂×C₅	semidirect
103. #713	C₂²×C₅	semidirect
105. #711	C₂²×C₅	semidirect
114. C₅×SD₁₆	C₂×D₅	semidirect
115. C₅×SD₁₆	C₂²×C₅	semidirect
122. #689	C₂²×C₅	semidirect
123. #686	C₂²×C₅	semidirect
125. #677	C₂²×C₅	semidirect
126. C₈×D₅	C₂²×C₅	semidirect
129. C₂×C₅²	#135	semidirect
192. C₂×SD₁₆	C₅×D₅	semidirect
195. C₅²	C₂×#135	semidirect
241. SD₁₆	C₂×C₅×D₅	semidirect
253. C₂×C₅	#4062	direct (N × H)
254. C₂×C₅	C₅×#135	semidirect
283. C₅	C₂×#4062	direct (N × H)
284. C₅	C₂×C₅×#135	semidirect
300. C₂	C₅×#4062	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×#713	2
C₂×C₅×#711	2
C₂×C₅²×SD₁₆	2
C₂×C₅×#689	2
C₂×C₅×#686	2
C₅×#4062	2
C₂×C₅×#677	2
C₂×C₅×C₈×D₅	2
C₂×C₅×#135	5
C₂×#4062	5

Subgroups — C2×C5×#4062 #13106915

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4062 #13106915