Subgroups — #13106049

Order 1600.

752 conjugacy classes of subgroups.

The subgroup lattice has 273 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#87900	C₂	1	1	yes	maximal, characteristic
2	C₅×#4072	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	#11717070	C₂	1	4	yes	maximal, complemented
5	#11717018	C₂	1	2	yes	maximal, complemented
6	#11717013	C₂	1	2	yes	maximal, complemented
7	#11716967	C₂	1	2	yes	maximal, complemented
8	#11716962	C₂	1	2	yes	maximal, complemented
9	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊(C₅⋊D₅)	C₂²	1	2	yes	—
11	#87900	C₄	1	8	yes	complemented
12	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅⋊D₅	C₂²	1	2	yes	—
14	C₂×#6045	C₂²	1	1	yes	characteristic
15	C₂²×C₅×Dic₅	C₂²	1	2	yes	—
16	C₂×#6036	C₂²	1	4	yes	—
17	C₅×C₂²⋊Dic₅	C₂²	1	2	yes	—
18	C₅×C₄⋊Dic₅	C₂²	1	2	yes	—
19	C₅×C₄⋊Dic₅	C₂²	1	2	yes	—
20	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
21	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
22	C₄×C₅×Dic₅	C₂²	1	2	yes	—
23	#87781	C₂²	1	4	yes	—
24	#87779	C₂²	1	4	yes	—
25	Dic₅²	C₂²	1	4	yes	—
26	#65857	—	5	1	no	maximal
27	#65838	—	5	1	no	maximal
28	C₂³×C₅²	C₂³	1	1	yes	characteristic
29	C₂²×C₅⋊D₅	C₂³	1	2	yes	—
30	C₂×C₄×C₅²	C₂×C₄	1	4	yes	complemented
31	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
33	C₂²⋊(C₅⋊D₅)	C₂×C₄	1	8	yes	complemented
34	C₂×#1039	C₂³	1	2	yes	—
35	C₄⋊(C₅⋊D₅)	C₂×C₄	1	4	yes	complemented
36	C₄×C₅⋊D₅	C₂×C₄	1	8	yes	complemented
37	#6045	C₂×C₄	1	4	yes	complemented
38	C₂×C₅×Dic₅	—	4	2	no	—
39	C₂×C₅×Dic₅	C₂³	1	2	yes	—
40	C₂×C₅×Dic₅	C₂³	1	4	yes	—
41	C₂×C₅×Dic₅	C₂³	1	2	yes	—
42	#6036	—	4	4	no	—
43	C₂×#709	—	5	1	no	—
44	C₂×#709	—	5	1	no	—
45	C₂×#709	—	10	2	no	—
46	C₅×#117	—	5	1	no	—
47	C₂×C₅×C₄⋊C₄	—	5	1	no	—
48	C₄×C₂²⋊D₅	—	5	4	no	—
49	#4072	D₅	1	1	yes	characteristic, complemented
50	C₂×C₄⋊Dic₅	D₅	1	1	yes	characteristic, complemented
51	#4039	—	5	2	no	—
52	#4038	—	5	2	no	—
53	#4038	—	5	2	no	—
54	D₅×C₄⋊C₄	—	5	2	no	—
55	D₅×C₄⋊C₄	—	5	2	no	—
56	#4033	—	5	2	no	—
57	#4027	—	5	4	no	—
58	C₄×D₂₀	—	5	2	no	—
59	C₄×Dic₁₀	—	5	2	no	—
60	C₂²×C₅²	—	2	1	no	—
61	C₂²×C₅²	—	2	1	no	—
62	C₂²×C₅²	C₂²×C₄	1	2	yes	—
63	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
64	C₂×C₅⋊D₅	—	2	2	no	—
65	C₂×C₅⋊D₅	—	2	2	no	—
66	C₂×C₅⋊D₅	C₂²×C₄	1	4	yes	—
67	C₄×C₅²	C₂²×C₄	1	2	yes	—
68	C₄×C₅²	C₂²×C₄	1	2	yes	—
69	#1039	C₂²×C₄	1	4	yes	—
70	C₅×Dic₅	—	2	2	no	—
71	C₅×Dic₅	—	2	4	no	—
72	C₅×Dic₅	—	2	2	no	—
73	C₂²×C₄×C₅	—	2	2	no	—
74	C₂²×C₄×C₅	—	5	2	no	—
75	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
77	C₂×C₂²⋊D₅	—	5	2	no	—
78	C₂×C₂²⋊D₅	—	5	2	no	—
79	C₂×C₂²⋊D₅	—	10	4	no	—
80	C₂²×Dic₅	C₂×D₅	1	2	yes	—
81	#709	—	5	8	no	—
82	#709	—	5	8	no	—
83	#709	—	10	16	no	—
84	C₂×D₂₀	—	5	1	no	—
85	C₂×D₂₀	—	5	1	no	—
86	C₂×D₂₀	—	10	2	no	—
87	C₂×C₄×D₅	—	5	2	no	—
88	C₂×C₄×D₅	—	5	2	no	—
89	C₂×C₄×D₅	—	5	4	no	—
90	C₂×C₄×D₅	—	5	4	no	—
91	C₂×C₄×D₅	—	10	4	no	—
92	C₂×Dic₁₀	—	5	1	no	—
93	C₂×Dic₁₀	—	5	1	no	—
94	C₂×Dic₁₀	—	10	2	no	—
95	C₅×C₄⋊C₄	—	5	2	no	—
96	C₅×C₄⋊C₄	—	5	2	no	—
97	C₅×C₄⋊C₄	—	5	1	no	—
98	C₅×C₄⋊C₄	—	5	1	no	—
99	C₅×C₂²⋊C₄	—	5	2	no	—
100	C₄²×C₅	—	5	2	no	—
101	C₂²⋊Dic₅	C₂×D₅	1	2	yes	—
102	#685	—	5	4	no	—
103	#685	—	5	4	no	—
104	C₄⋊Dic₅	C₂×D₅	1	2	yes	—
105	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
106	C₄⋊Dic₅	C₂×D₅	1	2	yes	—
107	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
108	#683	—	5	4	no	—
109	#683	—	5	4	no	—
110	C₄×Dic₅	—	5	4	no	—
111	C₄×Dic₅	—	5	4	no	—
112	C₄×Dic₅	C₂×D₅	1	2	yes	—
113	#520	—	25	1	no	Sylow
114	C₂×C₅²	—	2	2	no	—
115	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
117	C₂×C₅²	2₋¹⁺⁴	1	1	yes	characteristic
118	C₅⋊D₅	—	2	4	no	—
119	C₂³×C₅	—	2	2	no	—
120	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
121	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
122	C₂²×D₅	—	5	2	no	—
123	C₂²×D₅	—	5	2	no	—
124	C₂²×D₅	—	10	4	no	—
125	C₂×C₄×C₅	—	2	8	no	—
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	—	2	2	no	—
128	C₂×C₄×C₅	—	5	2	no	—
129	C₂×C₄×C₅	—	5	4	no	—
130	C₂×C₄×C₅	—	5	2	no	—
131	C₂×C₄×C₅	—	20	2	no	—
132	C₂×C₄×C₅	C₄×D₅	1	4	yes	complemented
133	C₂×C₄×C₅	C₄×D₅	1	4	yes	complemented
134	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
138	C₂²⋊D₅	—	5	8	no	—
139	C₂²⋊D₅	—	5	8	no	—
140	C₂²⋊D₅	—	10	16	no	—
141	C₂×Dic₅	—	4	2	no	—
142	C₂×Dic₅	—	5	2	no	—
143	C₂×Dic₅	—	5	2	no	—
144	C₂×Dic₅	—	10	4	no	—
145	C₂×Dic₅	C₂²×D₅	1	2	yes	—
146	C₂×Dic₅	C₂²×D₅	1	4	yes	—
147	C₂×Dic₅	C₂²×D₅	1	2	yes	—
148	D₂₀	—	5	4	no	—
149	D₂₀	—	5	4	no	—
150	D₂₀	—	10	8	no	—
151	C₄×D₅	—	5	8	no	—
152	C₄×D₅	—	5	8	no	—
153	C₄×D₅	—	10	16	no	—
154	C₄×D₅	—	20	4	no	—
155	C₄×D₅	—	20	4	no	—
156	Dic₁₀	—	5	4	no	—
157	Dic₁₀	—	5	4	no	—
158	Dic₁₀	—	10	8	no	—
159	C₂×#41	—	25	1	no	—
160	C₄×Q₈	—	25	2	no	—
161	C₄×D₄	—	25	2	no	—
162	C₄×D₄	—	25	4	no	—
163	#117	—	5	1	no	—
164	#117	—	25	2	no	—
165	C₂×C₄⋊C₄	—	5	1	no	—
166	C₂×C₄⋊C₄	—	25	2	no	—
167	C₅²	#520	1	1	yes	characteristic, complemented, Sylow
168	C₂²×C₅	—	2	1	no	—
169	C₂²×C₅	—	2	1	no	—
170	C₂²×C₅	—	2	1	no	—
171	C₂²×C₅	—	2	1	no	—
172	C₂²×C₅	—	2	4	no	—
173	C₂²×C₅	—	2	2	no	—
174	C₂²×C₅	—	4	2	no	—
175	C₂²×C₅	—	4	2	no	—
176	C₂²×C₅	C₂×C₄×D₅	1	2	yes	—
177	C₂²×C₅	C₂×C₄×D₅	1	2	yes	—
178	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
179	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
180	C₂×D₅	—	5	4	no	—
181	C₂×D₅	—	5	4	no	—
182	C₂×D₅	—	10	8	no	—
183	C₂×D₅	—	10	2	no	—
184	C₂×D₅	—	10	2	no	—
185	C₂×D₅	—	10	2	no	—
186	C₂×D₅	—	10	2	no	—
187	C₂×D₅	—	20	4	no	—
188	C₂×D₅	—	20	4	no	—
189	C₄×C₅	—	2	4	no	—
190	C₄×C₅	—	2	4	no	—
191	C₄×C₅	—	10	2	no	—
192	C₄×C₅	—	10	2	no	—
193	C₄×C₅	—	10	4	no	—
194	C₄×C₅	C₂×C₄×D₅	1	2	yes	—
195	C₄×C₅	C₂×C₄×D₅	1	2	yes	—
196	C₄×C₅	C₂×C₄×D₅	1	2	yes	—
197	C₄×C₅	C₂×C₄×D₅	1	2	yes	—
198	Dic₅	—	2	4	no	—
199	Dic₅	—	2	2	no	—
200	Dic₅	—	2	2	no	—
201	Dic₅	—	5	4	no	—
202	Dic₅	—	5	4	no	—
203	Dic₅	—	10	8	no	—
204	#41	—	25	8	no	—
205	C₂×Q₈	—	25	1	no	—
206	C₂×D₄	—	25	2	no	—
207	C₂×D₄	—	25	1	no	—
208	C₂²×C₄	—	5	2	no	—
209	C₂²×C₄	—	25	4	no	—
210	C₂²×C₄	—	25	2	no	—
211	C₂²×C₄	D₅²	1	1	yes	characteristic
212	C₄⋊C₄	—	5	2	no	—
213	C₄⋊C₄	—	5	2	no	—
214	C₄⋊C₄	—	5	1	no	—
215	C₄⋊C₄	—	5	1	no	—
216	C₄⋊C₄	—	25	4	no	—
217	C₂²⋊C₄	—	5	2	no	—
218	C₂²⋊C₄	—	25	4	no	—
219	C₄²	—	5	2	no	—
220	C₄²	—	25	4	no	—
221	C₂×C₅	—	2	2	no	—
222	C₂×C₅	—	2	2	no	—
223	C₂×C₅	—	2	2	no	—
224	C₂×C₅	—	2	2	no	—
225	C₂×C₅	—	2	2	no	—
226	C₂×C₅	—	4	4	no	—
227	C₂×C₅	C₂²×C₄×D₅	1	1	yes	characteristic
228	C₂×C₅	C₂²×C₄×D₅	1	1	yes	characteristic
229	C₂×C₅	#4144	1	1	yes	characteristic
230	C₂×C₅	#4147	1	1	yes	characteristic
231	C₂×C₅	#4149	1	1	yes	characteristic
232	C₂×C₅	#4150	1	1	yes	characteristic
233	D₅	—	10	4	no	—
234	D₅	—	10	4	no	—
235	D₅	—	20	8	no	—
236	C₂³	—	25	2	no	—
237	C₂³	C₂×D₅²	1	1	yes	characteristic
238	Q₈	—	25	4	no	—
239	D₄	—	25	8	no	—
240	D₄	—	25	4	no	—
241	C₂×C₄	—	5	2	no	—
242	C₂×C₄	—	5	2	no	—
243	C₂×C₄	—	5	4	no	—
244	C₂×C₄	—	20	2	no	—
245	C₂×C₄	—	25	2	no	—
246	C₂×C₄	—	25	8	no	—
247	C₂×C₄	—	100	4	no	—
248	C₂×C₄	#6036	1	4	yes	complemented
249	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
250	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
251	C₅	—	2	2	no	—
252	C₅	#65838	1	1	yes	characteristic, complemented
253	C₅	#65857	1	1	yes	characteristic, complemented
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	25	4	no	—
257	C₂²	—	50	2	no	—
258	C₂²	—	50	2	no	—
259	C₂²	C₂×#6036	1	2	yes	—
260	C₂²	C₂²×D₅²	1	1	yes	characteristic
261	C₄	—	10	2	no	—
262	C₄	—	10	4	no	—
263	C₄	—	10	2	no	—
264	C₄	—	25	4	no	—
265	C₄	C₂×#6036	1	2	yes	—
266	C₄	C₂×#6036	1	2	yes	—
267	C₂	—	2	2	no	—
268	C₂	—	50	4	no	—
269	C₂	#11717462	1	1	yes	characteristic
270	C₂	#11717472	1	1	yes	characteristic
271	C₂	C₂²×#6036	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅²
commutator	C₂×C₅²	C₂³×C₄
Fitting	C₂²×C₄×C₅²	C₂²
socle, cosocle	C₂²×C₅²	C₂⁴
prime radical	C₂×#87900	C₂
hypercenter	C₂²×C₄	D₅²
hypocenter	C₅²	#520
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₅×#4072	C₂	semidirect
3. C₂×C₅×C₄⋊Dic₅	C₂	semidirect
4. #11717070	C₂	semidirect
5. #11717018	C₂	semidirect
6. #11717013	C₂	semidirect
7. #11716967	C₂	semidirect
8. #11716962	C₂	semidirect
11. #87900	C₄	semidirect
30. C₂×C₄×C₅²	C₂×C₄	semidirect
33. C₂²⋊(C₅⋊D₅)	C₂×C₄	semidirect
35. C₄⋊(C₅⋊D₅)	C₂×C₄	semidirect
36. C₄×C₅⋊D₅	C₂×C₄	semidirect
37. #6045	C₂×C₄	semidirect
49. #4072	D₅	semidirect
50. C₂×C₄⋊Dic₅	D₅	semidirect
132. C₂×C₄×C₅	C₄×D₅	semidirect
133. C₂×C₄×C₅	C₄×D₅	semidirect
167. C₅²	#520	semidirect
248. C₂×C₄	#6036	semidirect
252. C₅	#65838	semidirect
253. C₅	#65857	semidirect

Maximal subgroups

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

subgroup	index
C₂×#87900	2
C₅×#4072	2
C₂×C₅×C₄⋊Dic₅	2
#11717070	2
#11717018	2
#11717013	2
#11716967	2
#11716962	2
#65857	5
#65838	5