Subgroups — C₂×D₅×#713 #13107829

Order 1600.

1944 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#87904	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#713	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×Q₈×D₅	C₂	1	1	yes	maximal, characteristic, complemented
4	D₅×#713	C₂	1	16	yes	maximal, direct product factor
5	C₂×D₅×D₂₀	C₂	1	3	yes	maximal, complemented
6	C₂×C₄×D₅²	C₂	1	3	yes	maximal, complemented
7	C₂×#87874	C₂	1	3	yes	maximal, complemented
8	C₂×#87871	C₂	1	3	yes	maximal, complemented
9	C₂²×D₅²	C₂²	1	3	yes	—
10	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic, complemented
11	#87904	C₂²	1	8	yes	complemented
12	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	3	yes	complemented
13	C₂×C₄×C₅⋊D₅	C₂²	1	3	yes	complemented
14	C₅×#713	C₂²	1	8	yes	complemented
15	C₅×Q₈×D₅	C₂²	1	8	yes	complemented
16	C₂×C₅×D₂₀	C₂²	1	3	yes	complemented
17	C₂×C₄×C₅×D₅	C₂²	1	3	yes	—
18	C₂×C₄×C₅×D₅	C₂²	1	3	yes	complemented
19	C₂×C₅×Dic₁₀	C₂²	1	3	yes	complemented
20	C₂×#6038	C₂²	1	6	yes	complemented
21	C₂×#6036	C₂²	1	3	yes	complemented
22	C₂×D₅×Dic₅	C₂²	1	1	yes	characteristic
23	C₂×D₅×Dic₅	C₂²	1	3	yes	complemented
24	D₅×D₂₀	C₂²	1	24	yes	complemented
25	C₄×D₅²	C₂²	1	24	yes	complemented
26	#87874	C₂²	1	24	yes	complemented
27	#87871	C₂²	1	24	yes	complemented
28	C₂×D₅×#41	—	5	1	no	maximal
29	C₂²×#713	—	5	1	no	maximal
30	C₂²×C₅⋊D₅	C₂³	1	3	yes	—
31	C₂²×C₅×D₅	C₂³	1	3	yes	—
32	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
33	C₂×D₅²	—	2	12	no	—
34	C₂×D₅²	C₂³	1	12	yes	—
35	C₅²×Q₈	C₂³	1	4	yes	complemented
36	C₂×C₄×C₅²	C₂³	1	3	yes	—
37	C₂×#1039	C₂³	1	1	yes	characteristic
38	C₄⋊(C₅⋊D₅)	C₂³	1	12	yes	complemented
39	C₄×C₅⋊D₅	C₂³	1	12	yes	complemented
40	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
41	C₂×C₅×Dic₅	C₂³	1	3	yes	—
42	C₅×D₂₀	C₂³	1	12	yes	complemented
43	C₄×C₅×D₅	C₂³	1	12	yes	—
44	C₄×C₅×D₅	C₂³	1	12	yes	complemented
45	C₅×Dic₁₀	C₂³	1	12	yes	complemented
46	#6038	C₂³	1	24	yes	complemented
47	#6036	C₂³	1	12	yes	complemented
48	D₅×Dic₅	C₂³	1	4	yes	—
49	D₅×Dic₅	C₂³	1	12	yes	complemented
50	C₂×C₅×#41	—	5	1	no	—
51	C₂²×C₅×Q₈	—	5	1	no	—
52	D₅×#41	—	5	16	no	—
53	C₂×#713	—	5	1	no	—
54	C₂×#713	—	5	1	no	—
55	C₂×#713	—	5	16	no	—
56	C₂×#713	—	5	3	no	—
57	C₂×#713	—	5	3	no	—
58	C₂×#713	—	10	2	no	—
59	C₂×#713	D₅	1	1	yes	characteristic, direct product factor
60	C₂×Q₈×D₅	D₅	1	1	yes	characteristic, complemented
61	C₂×#711	—	5	3	no	—
62	C₂×D₄×D₅	—	5	3	no	—
63	C₂×#709	—	5	3	no	—
64	C₂²×D₂₀	—	5	3	no	—
65	C₂²×C₄×D₅	—	5	3	no	—
66	C₂²×C₄×D₅	—	5	3	no	—
67	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
68	C₂×C₅⋊D₅	—	2	6	no	—
69	C₂×C₅⋊D₅	C₂⁴	1	6	yes	—
70	C₂×C₅×D₅	—	2	6	no	—
71	C₂×C₅×D₅	#41	1	4	yes	complemented
72	C₂×C₅×D₅	C₂⁴	1	6	yes	—
73	C₂×C₅×D₅	C₂⁴	1	2	yes	—
74	D₅²	—	2	24	no	—
75	C₄×C₅²	C₂⁴	1	6	yes	—
76	#1039	C₂⁴	1	2	yes	—
77	C₅×Dic₅	C₂⁴	1	2	yes	—
78	C₅×Dic₅	C₂⁴	1	6	yes	—
79	C₂³×D₅	—	5	3	no	—
80	C₂³×D₅	—	5	3	no	—
81	C₅×#41	—	5	8	no	—
82	C₂×C₅×Q₈	—	2	2	no	—
83	C₂×C₅×Q₈	—	5	8	no	—
84	C₂×C₅×Q₈	—	5	3	no	—
85	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic, complemented
86	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic, complemented
87	C₂×C₅×D₄	—	5	3	no	—
88	C₂²×C₄×C₅	—	5	3	no	—
89	C₂²×C₄×C₅	—	5	3	no	—
90	C₂×C₂²⋊D₅	—	5	6	no	—
91	C₂²×Dic₅	—	5	1	no	—
92	C₂²×Dic₅	—	5	3	no	—
93	#713	—	5	8	no	—
94	#713	—	5	8	no	—
95	#713	—	5	24	no	—
96	#713	—	5	24	no	—
97	#713	—	10	16	no	—
98	#713	C₂×D₅	1	8	yes	direct product factor
99	Q₈×D₅	C₂×D₅	1	8	yes	complemented
100	#711	—	5	24	no	—
101	D₄×D₅	—	5	24	no	—
102	#709	—	5	24	no	—
103	C₂×D₂₀	—	5	6	no	—
104	C₂×D₂₀	—	5	3	no	—
105	C₂×D₂₀	—	5	3	no	—
106	C₂×D₂₀	—	5	24	no	—
107	C₂×D₂₀	—	10	6	no	—
108	C₂×D₂₀	C₂×D₅	1	3	yes	complemented
109	C₂×C₄×D₅	—	5	1	no	—
110	C₂×C₄×D₅	—	5	3	no	—
111	C₂×C₄×D₅	—	5	3	no	—
112	C₂×C₄×D₅	—	5	3	no	—
113	C₂×C₄×D₅	—	5	3	no	—
114	C₂×C₄×D₅	—	5	3	no	—
115	C₂×C₄×D₅	—	5	24	no	—
116	C₂×C₄×D₅	—	5	24	no	—
117	C₂×C₄×D₅	—	10	6	no	—
118	C₂×C₄×D₅	C₂×D₅	1	3	yes	—
119	C₂×C₄×D₅	C₂×D₅	1	3	yes	complemented
120	C₂×Dic₁₀	C₂×D₅	1	3	yes	complemented
121	C₂²×#41	—	25	1	no	Sylow
122	C₂×C₅²	C₂×#41	1	2	yes	complemented
123	C₂×C₅²	C₂⁵	1	1	yes	characteristic
124	C₅⋊D₅	—	2	6	no	—
125	C₅×D₅	—	2	6	no	—
126	C₅×D₅	C₂×#41	1	4	yes	complemented
127	C₂³×C₅	—	5	1	no	—
128	C₂³×C₅	—	5	3	no	—
129	C₂²×D₅	—	5	3	no	—
130	C₂²×D₅	—	5	3	no	—
131	C₂²×D₅	—	5	12	no	—
132	C₂²×D₅	—	5	12	no	—
133	C₂²×D₅	—	10	6	no	—
134	C₂²×D₅	—	10	12	no	—
135	C₂²×D₅	—	10	12	no	—
136	C₂²×D₅	C₂²×D₅	1	3	yes	—
137	C₂²×D₅	C₂²×D₅	1	1	yes	characteristic
138	C₅×Q₈	—	2	8	no	—
139	C₅×Q₈	—	5	12	no	—
140	C₅×Q₈	C₂²×D₅	1	4	yes	complemented
141	C₅×Q₈	C₂²×D₅	1	4	yes	complemented
142	C₅×D₄	—	5	12	no	—
143	C₂×C₄×C₅	—	2	6	no	—
144	C₂×C₄×C₅	—	5	1	no	—
145	C₂×C₄×C₅	—	5	3	no	—
146	C₂×C₄×C₅	—	5	12	no	—
147	C₂×C₄×C₅	—	5	12	no	—
148	C₂×C₄×C₅	C₂²×D₅	1	3	yes	—
149	C₂×C₄×C₅	C₂²×D₅	1	3	yes	—
150	C₂²⋊D₅	—	5	24	no	—
151	C₂×Dic₅	—	5	1	no	—
152	C₂×Dic₅	—	5	1	no	—
153	C₂×Dic₅	—	5	4	no	—
154	C₂×Dic₅	—	5	12	no	—
155	C₂×Dic₅	—	10	2	no	—
156	C₂×Dic₅	C₂²×D₅	1	3	yes	—
157	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
158	D₂₀	—	5	24	no	—
159	D₂₀	—	5	12	no	—
160	D₂₀	—	5	12	no	—
161	D₂₀	—	10	24	no	—
162	D₂₀	C₂²×D₅	1	12	yes	complemented
163	C₄×D₅	—	5	4	no	—
164	C₄×D₅	—	5	12	no	—
165	C₄×D₅	—	5	12	no	—
166	C₄×D₅	—	5	12	no	—
167	C₄×D₅	—	5	12	no	—
168	C₄×D₅	—	5	12	no	—
169	C₄×D₅	—	10	24	no	—
170	C₄×D₅	C₂²×D₅	1	12	yes	—
171	C₄×D₅	C₂²×D₅	1	12	yes	complemented
172	Dic₁₀	C₂²×D₅	1	12	yes	complemented
173	C₂×#41	—	5	1	no	—
174	C₂×#41	—	25	3	no	—
175	C₂×#41	—	25	3	no	—
176	C₂×#41	—	25	16	no	—
177	C₂×#41	—	25	1	no	—
178	C₂²×Q₈	—	5	1	no	—
179	C₂²×D₄	—	25	3	no	—
180	C₂³×C₄	—	25	3	no	—
181	C₅²	C₂²×#41	1	1	yes	characteristic, complemented, Sylow
182	C₂²×C₅	—	2	2	no	—
183	C₂²×C₅	—	5	4	no	—
184	C₂²×C₅	—	5	2	no	—
185	C₂²×C₅	—	5	6	no	—
186	C₂²×C₅	—	10	6	no	—
187	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
188	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
189	C₂×D₅	—	2	6	no	—
190	C₂×D₅	—	5	6	no	—
191	C₂×D₅	—	5	6	no	—
192	C₂×D₅	—	10	12	no	—
193	C₂×D₅	—	10	24	no	—
194	C₂×D₅	—	10	6	no	—
195	C₂×D₅	—	10	6	no	—
196	C₂×D₅	—	10	24	no	—
197	C₂×D₅	—	20	12	no	—
198	C₂×D₅	#713	1	4	yes	direct product factor
199	C₂×D₅	C₂³×D₅	1	6	yes	—
200	C₂×D₅	C₂³×D₅	1	2	yes	—
201	C₄×C₅	—	2	12	no	—
202	C₄×C₅	—	5	2	no	—
203	C₄×C₅	—	5	6	no	—
204	C₄×C₅	C₂³×D₅	1	6	yes	—
205	C₄×C₅	C₂³×D₅	1	6	yes	—
206	Dic₅	—	5	2	no	—
207	Dic₅	—	5	2	no	—
208	Dic₅	—	10	4	no	—
209	Dic₅	C₂³×D₅	1	6	yes	—
210	Dic₅	C₂³×D₅	1	2	yes	—
211	C₂⁴	—	25	3	no	—
212	#41	—	5	8	no	—
213	#41	—	25	24	no	—
214	#41	—	25	24	no	—
215	#41	—	25	8	no	—
216	C₂×Q₈	—	5	8	no	—
217	C₂×Q₈	—	5	3	no	—
218	C₂×Q₈	D₅²	1	1	yes	characteristic, complemented
219	C₂×D₄	—	5	3	no	—
220	C₂×D₄	—	25	24	no	—
221	C₂×D₄	—	25	6	no	—
222	C₂×D₄	—	25	3	no	—
223	C₂²×C₄	—	5	3	no	—
224	C₂²×C₄	—	5	3	no	—
225	C₂²×C₄	—	25	3	no	—
226	C₂²×C₄	—	25	24	no	—
227	C₂²×C₄	—	25	1	no	—
228	C₂²×C₄	—	25	3	no	—
229	C₂²×C₄	—	25	3	no	—
230	C₂×C₅	—	2	4	no	—
231	C₂×C₅	—	2	2	no	—
232	C₂×C₅	—	5	4	no	—
233	C₂×C₅	—	10	6	no	—
234	C₂×C₅	C₂×#713	1	2	yes	complemented
235	C₂×C₅	D₅×#41	1	2	yes	complemented
236	C₂×C₅	C₂⁴×D₅	1	1	yes	characteristic
237	C₂×C₅	C₂⁴×D₅	1	1	yes	characteristic
238	D₅	—	2	6	no	—
239	D₅	—	10	6	no	—
240	D₅	—	10	6	no	—
241	D₅	—	20	12	no	—
242	D₅	C₂×#713	1	4	yes	direct product factor
243	C₂³	—	5	1	no	—
244	C₂³	—	5	3	no	—
245	C₂³	—	25	12	no	—
246	C₂³	—	25	3	no	—
247	C₂³	—	50	12	no	—
248	Q₈	—	5	12	no	—
249	Q₈	C₂×D₅²	1	4	yes	complemented
250	D₄	—	5	12	no	—
251	D₄	—	25	24	no	—
252	D₄	—	25	12	no	—
253	C₂×C₄	—	5	1	no	—
254	C₂×C₄	—	5	3	no	—
255	C₂×C₄	—	5	12	no	—
256	C₂×C₄	—	5	12	no	—
257	C₂×C₄	—	25	12	no	—
258	C₂×C₄	—	25	4	no	—
259	C₂×C₄	—	25	12	no	—
260	C₂×C₄	—	25	1	no	—
261	C₂×C₄	—	25	12	no	—
262	C₂×C₄	C₂×D₅²	1	3	yes	—
263	C₅	—	2	2	no	—
264	C₅	C₂²×#713	1	1	yes	characteristic, complemented
265	C₅	C₂×D₅×#41	1	1	yes	characteristic, complemented
266	C₂²	—	5	4	no	—
267	C₂²	—	5	2	no	—
268	C₂²	—	5	6	no	—
269	C₂²	—	10	6	no	—
270	C₂²	—	25	6	no	—
271	C₂²	—	50	24	no	—
272	C₂²	—	50	6	no	—
273	C₂²	C₂²×D₅²	1	1	yes	characteristic
274	C₄	—	5	2	no	—
275	C₄	—	5	6	no	—
276	C₄	—	25	2	no	—
277	C₄	C₂²×D₅²	1	6	yes	—
278	C₂	—	5	4	no	—
279	C₂	—	10	6	no	—
280	C₂	—	50	6	no	—
281	C₂	D₅×#713	1	2	yes	direct product factor
282	C₂	C₂³×D₅²	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₅²
commutator, cosocle	C₂×C₅²	C₂⁵
Frattini	C₂	C₂³×D₅²
Fitting	C₂×C₅²×Q₈	C₂²
socle	C₂²×C₅²	C₂⁴
hypercenter	C₂×Q₈	D₅²
hypocenter	C₅²	C₂²×#41
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#87904	C₂	semidirect
2. C₂×C₅×#713	C₂	semidirect
3. C₂×C₅×Q₈×D₅	C₂	semidirect
4. D₅×#713	C₂	direct (N × H)
5. C₂×D₅×D₂₀	C₂	semidirect
6. C₂×C₄×D₅²	C₂	semidirect
7. C₂×#87874	C₂	semidirect
8. C₂×#87871	C₂	semidirect
10. C₂×C₅²×Q₈	C₂²	semidirect
11. #87904	C₂²	semidirect
12. C₂×C₄⋊(C₅⋊D₅)	C₂²	semidirect
13. C₂×C₄×C₅⋊D₅	C₂²	semidirect
14. C₅×#713	C₂²	semidirect
15. C₅×Q₈×D₅	C₂²	semidirect
16. C₂×C₅×D₂₀	C₂²	semidirect
18. C₂×C₄×C₅×D₅	C₂²	semidirect
19. C₂×C₅×Dic₁₀	C₂²	semidirect
20. C₂×#6038	C₂²	semidirect
21. C₂×#6036	C₂²	semidirect
23. C₂×D₅×Dic₅	C₂²	semidirect
24. D₅×D₂₀	C₂²	semidirect
25. C₄×D₅²	C₂²	semidirect
26. #87874	C₂²	semidirect
27. #87871	C₂²	semidirect
35. C₅²×Q₈	C₂³	semidirect
38. C₄⋊(C₅⋊D₅)	C₂³	semidirect
39. C₄×C₅⋊D₅	C₂³	semidirect
42. C₅×D₂₀	C₂³	semidirect
44. C₄×C₅×D₅	C₂³	semidirect
45. C₅×Dic₁₀	C₂³	semidirect
46. #6038	C₂³	semidirect
47. #6036	C₂³	semidirect
49. D₅×Dic₅	C₂³	semidirect
59. C₂×#713	D₅	direct (N × H)
60. C₂×Q₈×D₅	D₅	semidirect
71. C₂×C₅×D₅	#41	semidirect
85. C₂×C₅×Q₈	C₂×D₅	semidirect
86. C₂×C₅×Q₈	C₂×D₅	semidirect
98. #713	C₂×D₅	direct (N × H)
99. Q₈×D₅	C₂×D₅	semidirect
108. C₂×D₂₀	C₂×D₅	semidirect
119. C₂×C₄×D₅	C₂×D₅	semidirect
120. C₂×Dic₁₀	C₂×D₅	semidirect
122. C₂×C₅²	C₂×#41	semidirect
126. C₅×D₅	C₂×#41	semidirect
140. C₅×Q₈	C₂²×D₅	semidirect
141. C₅×Q₈	C₂²×D₅	semidirect
162. D₂₀	C₂²×D₅	semidirect
171. C₄×D₅	C₂²×D₅	semidirect
172. Dic₁₀	C₂²×D₅	semidirect
181. C₅²	C₂²×#41	semidirect
198. C₂×D₅	#713	direct (N × H)
218. C₂×Q₈	D₅²	semidirect
234. C₂×C₅	C₂×#713	semidirect
235. C₂×C₅	D₅×#41	semidirect
242. D₅	C₂×#713	direct (N × H)
249. Q₈	C₂×D₅²	semidirect
264. C₅	C₂²×#713	semidirect
265. C₅	C₂×D₅×#41	semidirect
281. C₂	D₅×#713	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×#87904	2
C₂×C₅×#713	2
C₂×C₅×Q₈×D₅	2
D₅×#713	2
C₂×D₅×D₂₀	2
C₂×C₄×D₅²	2
C₂×#87874	2
C₂×#87871	2
C₂×D₅×#41	5
C₂²×#713	5

Subgroups — C2×D5×#713 #13107829

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₅×#713 #13107829