Subgroups — D₅×#4072 #13105992

Order 1600.

736 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	C₂²×C₄×C₅×D₅	C₂	1	1	yes	maximal, characteristic
2	#11717246	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4072	C₂	1	1	yes	maximal, characteristic, complemented
4	D₅×C₂²⋊Dic₅	C₂	1	2	yes	maximal
5	#11717073	C₂	1	2	yes	maximal, complemented
6	D₅×C₄⋊Dic₅	C₂	1	2	yes	maximal, complemented
7	C₄×D₅×Dic₅	C₂	1	2	yes	maximal, complemented
8	#11716973	C₂	1	2	yes	maximal, complemented
9	#11716959	C₂	1	2	yes	maximal, complemented
10	C₂³×C₅×D₅	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
12	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×D₅	C₄	1	8	yes	complemented
14	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
18	C₂×D₅×Dic₅	C₂²	1	4	yes	—
19	C₂²⋊#1039	C₂²	1	2	yes	—
20	C₄⋊#1039	C₂²	1	2	yes	complemented
21	C₄×#1039	C₂²	1	2	yes	complemented
22	C₅×C₂²⋊Dic₅	C₂²	1	2	yes	—
23	C₅×C₄⋊Dic₅	C₂²	1	2	yes	complemented
24	C₄×C₅×Dic₅	C₂²	1	2	yes	complemented
25	#87780	C₂²	1	4	yes	complemented
26	#87778	C₂²	1	4	yes	—
27	Dic₅²	C₂²	1	4	yes	complemented
28	C₂×#4072	—	5	1	no	maximal
29	D₅×#117	—	5	1	no	maximal
30	C₂³×C₅²	C₂³	1	1	yes	characteristic
31	C₂²×C₅×D₅	—	2	4	no	—
32	C₂²×C₅×D₅	C₂×C₄	1	4	yes	—
33	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
34	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₅²	C₂×C₄	1	4	yes	complemented
36	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
38	C₂×#1039	C₂³	1	4	yes	—
39	C₂×C₅×Dic₅	C₂×C₄	1	4	yes	complemented
40	C₂×C₅×Dic₅	C₂³	1	4	yes	—
41	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
42	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
43	C₄×C₅×D₅	C₂×C₄	1	4	yes	—
44	C₄×C₅×D₅	C₂×C₄	1	4	yes	—
45	C₄×C₅×D₅	C₂×C₄	1	4	yes	complemented
46	C₄×C₅×D₅	C₂×C₄	1	4	yes	complemented
47	D₅×Dic₅	—	2	8	no	—
48	C₂³×C₄×C₅	—	5	1	no	—
49	C₂²×C₄×D₅	D₅	1	1	yes	characteristic
50	C₅×#117	—	5	1	no	—
51	C₂×C₂²⋊Dic₅	—	5	2	no	—
52	#4072	—	5	1	no	—
53	#4072	—	5	1	no	—
54	#4072	—	5	2	no	—
55	#4072	—	5	2	no	—
56	#4072	—	5	2	no	—
57	#4072	—	10	2	no	—
58	#4072	D₅	1	1	yes	characteristic, direct product factor
59	C₂×C₄⋊Dic₅	—	5	2	no	—
60	C₂×C₄×Dic₅	—	5	2	no	—
61	#4038	—	5	2	no	—
62	D₅×C₄⋊C₄	—	5	2	no	—
63	D₅×C₂²⋊C₄	—	5	2	no	—
64	#4023	—	5	2	no	—
65	#4018	—	5	2	no	—
66	C₄²×D₅	—	5	2	no	—
67	C₂²×C₅²	—	2	2	no	—
68	C₂²×C₅²	C₂²×C₄	1	2	yes	—
69	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
70	C₂×C₅×D₅	—	2	8	no	—
71	C₂×C₅×D₅	—	2	2	no	—
72	C₂×C₅×D₅	C₂²×C₄	1	2	yes	—
73	C₂×C₅×D₅	C₂²×C₄	1	2	yes	—
74	C₂×C₅×D₅	#41	1	4	yes	—
75	C₄×C₅²	C₂²×C₄	1	2	yes	—
76	C₄×C₅²	C₂²×C₄	1	2	yes	—
77	#1039	—	2	4	no	—
78	C₅×Dic₅	—	2	4	no	—
79	C₅×Dic₅	C₂²×C₄	1	2	yes	—
80	C₅×Dic₅	C₂²×C₄	1	2	yes	—
81	C₂⁴×C₅	—	5	1	no	—
82	C₂³×D₅	C₂×D₅	1	1	yes	characteristic
83	C₂²×C₄×C₅	—	2	2	no	—
84	C₂²×C₄×C₅	—	5	1	no	—
85	C₂²×C₄×C₅	—	5	8	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	—	5	1	no	—
90	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
91	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
92	C₂²×Dic₅	—	5	4	no	—
93	C₂²×Dic₅	C₂×D₅	1	1	yes	characteristic
94	C₂×C₄×D₅	—	5	4	no	—
95	C₂×C₄×D₅	Dic₅	1	8	yes	complemented
96	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
97	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
98	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
99	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
100	C₅×C₄⋊C₄	—	5	2	no	—
101	C₅×C₂²⋊C₄	—	5	2	no	—
102	C₄²×C₅	—	5	2	no	—
103	C₂²⋊Dic₅	—	5	2	no	—
104	C₂²⋊Dic₅	—	5	2	no	—
105	C₂²⋊Dic₅	—	5	4	no	—
106	C₂²⋊Dic₅	—	10	4	no	—
107	C₂²⋊Dic₅	C₂×D₅	1	2	yes	—
108	#685	—	5	4	no	—
109	C₄⋊Dic₅	—	5	2	no	—
110	C₄⋊Dic₅	—	5	2	no	—
111	C₄⋊Dic₅	—	5	4	no	—
112	C₄⋊Dic₅	—	10	4	no	—
113	C₄⋊Dic₅	C₂×D₅	1	2	yes	complemented
114	#683	—	5	4	no	—
115	C₄×Dic₅	—	5	2	no	—
116	C₄×Dic₅	—	5	2	no	—
117	C₄×Dic₅	—	5	4	no	—
118	C₄×Dic₅	—	5	4	no	—
119	C₄×Dic₅	—	10	4	no	—
120	C₄×Dic₅	C₂×D₅	1	2	yes	complemented
121	C₂×#117	—	25	1	no	Sylow
122	C₂×C₅²	—	2	2	no	—
123	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₅²	C₂×#41	1	2	yes	—
125	C₅×D₅	—	2	2	no	—
126	C₅×D₅	#117	1	4	yes	complemented
127	C₂³×C₅	—	2	2	no	—
128	C₂³×C₅	—	5	1	no	—
129	C₂³×C₅	—	5	4	no	—
130	C₂³×C₅	—	5	1	no	—
131	C₂³×C₅	—	10	4	no	—
132	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
133	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
134	C₂²×D₅	—	2	4	no	—
135	C₂²×D₅	C₂×Dic₅	1	4	yes	—
136	C₂²×D₅	C₂²×D₅	1	1	yes	characteristic
137	C₂²×D₅	C₂²×D₅	1	1	yes	characteristic
138	C₂×C₄×C₅	—	2	8	no	—
139	C₂×C₄×C₅	—	2	2	no	—
140	C₂×C₄×C₅	—	2	2	no	—
141	C₂×C₄×C₅	—	5	4	no	—
142	C₂×C₄×C₅	—	5	1	no	—
143	C₂×C₄×C₅	—	5	1	no	—
144	C₂×C₄×C₅	—	5	4	no	—
145	C₂×C₄×C₅	—	5	4	no	—
146	C₂×C₄×C₅	—	5	4	no	—
147	C₂×C₄×C₅	—	5	4	no	—
148	C₂×C₄×C₅	—	5	4	no	—
149	C₂×C₄×C₅	C₄×D₅	1	4	yes	complemented
150	C₂×C₄×C₅	C₂×Dic₅	1	4	yes	complemented
151	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
155	C₂×Dic₅	—	5	4	no	—
156	C₂×Dic₅	—	5	4	no	—
157	C₂×Dic₅	—	10	8	no	—
158	C₂×Dic₅	—	10	8	no	—
159	C₂×Dic₅	C₂×Dic₅	1	4	yes	complemented
160	C₂×Dic₅	C₂²×D₅	1	4	yes	—
161	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
162	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
163	C₄×D₅	—	10	8	no	—
164	C₄×D₅	C₂×Dic₅	1	4	yes	—
165	C₄×D₅	C₂×Dic₅	1	4	yes	—
166	C₄×D₅	C₂×Dic₅	1	4	yes	complemented
167	C₄×D₅	C₂×Dic₅	1	4	yes	complemented
168	C₂³×C₄	—	5	1	no	—
169	#117	—	5	1	no	—
170	#117	—	25	2	no	—
171	#117	—	25	2	no	—
172	#117	—	25	2	no	—
173	#117	—	25	1	no	—
174	C₂×C₄⋊C₄	—	25	2	no	—
175	C₂×C₂²⋊C₄	—	25	2	no	—
176	C₂×C₄²	—	25	2	no	—
177	C₅²	C₂×#117	1	1	yes	characteristic, complemented, Sylow
178	C₂²×C₅	—	2	2	no	—
179	C₂²×C₅	—	2	2	no	—
180	C₂²×C₅	—	2	4	no	—
181	C₂²×C₅	—	2	2	no	—
182	C₂²×C₅	—	4	4	no	—
183	C₂²×C₅	—	5	2	no	—
184	C₂²×C₅	—	5	4	no	—
185	C₂²×C₅	—	5	2	no	—
186	C₂²×C₅	—	10	8	no	—
187	C₂²×C₅	—	10	2	no	—
188	C₂²×C₅	C₂×C₄×D₅	1	2	yes	—
189	C₂²×C₅	C₂²×Dic₅	1	2	yes	—
190	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
191	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
192	C₂×D₅	—	2	2	no	—
193	C₂×D₅	—	2	8	no	—
194	C₂×D₅	#709	1	4	yes	—
195	C₂×D₅	C₂²×Dic₅	1	2	yes	—
196	C₂×D₅	C₂²×Dic₅	1	2	yes	—
197	C₄×C₅	—	2	4	no	—
198	C₄×C₅	—	2	4	no	—
199	C₄×C₅	—	5	2	no	—
200	C₄×C₅	—	5	2	no	—
201	C₄×C₅	—	10	4	no	—
202	C₄×C₅	C₂×C₄×D₅	1	2	yes	—
203	C₄×C₅	C₂×C₄×D₅	1	2	yes	—
204	C₄×C₅	C₂²×Dic₅	1	2	yes	—
205	C₄×C₅	C₂²×Dic₅	1	2	yes	—
206	Dic₅	—	2	4	no	—
207	Dic₅	—	10	4	no	—
208	Dic₅	—	10	4	no	—
209	Dic₅	—	20	8	no	—
210	Dic₅	C₂²×Dic₅	1	2	yes	—
211	Dic₅	C₂²×Dic₅	1	2	yes	—
212	C₂⁴	—	5	1	no	—
213	C₂²×C₄	—	5	1	no	—
214	C₂²×C₄	—	5	8	no	—
215	C₂²×C₄	—	5	1	no	—
216	C₂²×C₄	—	5	1	no	—
217	C₂²×C₄	—	5	1	no	—
218	C₂²×C₄	—	5	1	no	—
219	C₂²×C₄	—	25	4	no	—
220	C₂²×C₄	D₅²	1	1	yes	characteristic
221	C₄⋊C₄	—	5	2	no	—
222	C₄⋊C₄	—	25	4	no	—
223	C₄⋊C₄	—	25	2	no	—
224	C₂²⋊C₄	—	5	2	no	—
225	C₂²⋊C₄	—	25	4	no	—
226	C₂²⋊C₄	—	25	2	no	—
227	C₄²	—	5	2	no	—
228	C₄²	—	25	4	no	—
229	C₄²	—	25	2	no	—
230	C₂×C₅	—	2	2	no	—
231	C₂×C₅	—	2	2	no	—
232	C₂×C₅	—	2	4	no	—
233	C₂×C₅	—	2	2	no	—
234	C₂×C₅	—	4	4	no	—
235	C₂×C₅	—	5	4	no	—
236	C₂×C₅	—	10	2	no	—
237	C₂×C₅	C₂²×C₄×D₅	1	1	yes	characteristic
238	C₂×C₅	C₂×#709	1	2	yes	—
239	C₂×C₅	D₅×#41	1	2	yes	—
240	C₂×C₅	C₂³×Dic₅	1	1	yes	characteristic
241	D₅	—	2	2	no	—
242	D₅	#4072	1	4	yes	direct product factor
243	C₂³	—	5	1	no	—
244	C₂³	—	5	4	no	—
245	C₂³	—	5	1	no	—
246	C₂³	—	10	4	no	—
247	C₂³	C₂×D₅²	1	1	yes	characteristic
248	C₂×C₄	—	5	4	no	—
249	C₂×C₄	—	5	1	no	—
250	C₂×C₄	—	5	1	no	—
251	C₂×C₄	—	5	4	no	—
252	C₂×C₄	—	5	4	no	—
253	C₂×C₄	—	5	4	no	—
254	C₂×C₄	—	5	4	no	—
255	C₂×C₄	—	5	4	no	—
256	C₂×C₄	—	25	4	no	—
257	C₂×C₄	—	50	8	no	—
258	C₂×C₄	D₅×Dic₅	1	4	yes	complemented
259	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
260	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
261	C₅	—	2	2	no	—
262	C₅	D₅×#117	1	1	yes	characteristic, complemented
263	C₅	C₂×#4072	1	1	yes	characteristic, complemented
264	C₂²	—	2	2	no	—
265	C₂²	—	5	4	no	—
266	C₂²	—	5	2	no	—
267	C₂²	—	5	2	no	—
268	C₂²	—	10	8	no	—
269	C₂²	—	10	2	no	—
270	C₂²	C₂×D₅×Dic₅	1	2	yes	—
271	C₂²	C₂²×D₅²	1	1	yes	characteristic
272	C₄	—	5	2	no	—
273	C₄	—	5	2	no	—
274	C₄	—	10	4	no	—
275	C₄	—	50	4	no	—
276	C₄	C₂×D₅×Dic₅	1	2	yes	—
277	C₄	C₂×D₅×Dic₅	1	2	yes	—
278	C₂	—	2	2	no	—
279	C₂	—	5	4	no	—
280	C₂	—	10	2	no	—
281	C₂	D₅×#709	1	2	yes	—
282	C₂	C₂²×D₅×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₅²
commutator	C₂×C₅²	C₂³×C₄
Frattini	C₂²	C₂²×D₅²
Fitting	C₂²×C₄×C₅²	C₂²
socle, cosocle	C₂²×C₅²	C₂⁴
prime radical	C₂³×C₅×D₅	C₂²
hypercenter	C₂²×C₄	D₅²
hypocenter	C₅²	C₂×#117
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. #11717246	C₂	semidirect
3. C₅×#4072	C₂	semidirect
5. #11717073	C₂	semidirect
6. D₅×C₄⋊Dic₅	C₂	semidirect
7. C₄×D₅×Dic₅	C₂	semidirect
8. #11716973	C₂	semidirect
9. #11716959	C₂	semidirect
13. C₂×C₄×C₅×D₅	C₄	semidirect
20. C₄⋊#1039	C₂²	semidirect
21. C₄×#1039	C₂²	semidirect
23. C₅×C₄⋊Dic₅	C₂²	semidirect
24. C₄×C₅×Dic₅	C₂²	semidirect
25. #87780	C₂²	semidirect
27. Dic₅²	C₂²	semidirect
35. C₂×C₄×C₅²	C₂×C₄	semidirect
39. C₂×C₅×Dic₅	C₂×C₄	semidirect
45. C₄×C₅×D₅	C₂×C₄	semidirect
46. C₄×C₅×D₅	C₂×C₄	semidirect
58. #4072	D₅	direct (N × H)
95. C₂×C₄×D₅	Dic₅	semidirect
113. C₄⋊Dic₅	C₂×D₅	semidirect
120. C₄×Dic₅	C₂×D₅	semidirect
126. C₅×D₅	#117	semidirect
149. C₂×C₄×C₅	C₄×D₅	semidirect
150. C₂×C₄×C₅	C₂×Dic₅	semidirect
159. C₂×Dic₅	C₂×Dic₅	semidirect
166. C₄×D₅	C₂×Dic₅	semidirect
167. C₄×D₅	C₂×Dic₅	semidirect
177. C₅²	C₂×#117	semidirect
242. D₅	#4072	direct (N × H)
258. C₂×C₄	D₅×Dic₅	semidirect
262. C₅	D₅×#117	semidirect
263. C₅	C₂×#4072	semidirect

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
#11717246	2
C₅×#4072	2
D₅×C₂²⋊Dic₅	2
#11717073	2
D₅×C₄⋊Dic₅	2
C₄×D₅×Dic₅	2
#11716973	2
#11716959	2
C₂×#4072	5
D₅×#117	5

Subgroups — D5×#4072 #13105992

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×#4072 #13105992