Subgroups — #13105008

Order 1600.

526 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#6045	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₅×D₂₀	C₂	1	1	yes	maximal, characteristic
3	C₅×#4016	C₂	1	1	yes	maximal, characteristic, complemented
4	#11717037	C₂	1	1	yes	maximal, characteristic, complemented
5	#11717017	C₂	1	1	yes	maximal, characteristic
6	#11717017	C₂	1	1	yes	maximal, characteristic
7	#11717015	C₂	1	2	yes	maximal, complemented
8	#11717007	C₂	1	2	yes	maximal, complemented
9	D₅×#683	C₂	1	2	yes	maximal, complemented
10	#11716991	C₂	1	1	yes	maximal, characteristic, complemented
11	#11716969	C₂	1	2	yes	maximal, complemented
12	C₂×#6045	C₂²	1	1	yes	characteristic
13	C₂×C₅×D₂₀	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅×D₅	C₂²	1	2	yes	—
15	C₂×#6039	C₂²	1	2	yes	—
16	C₂×D₅×Dic₅	C₂²	1	2	yes	—
17	C₂×D₅×Dic₅	C₂²	1	2	yes	—
18	C₄²×C₅²	C₂²	1	1	yes	characteristic
19	C₄⋊#1039	C₂²	1	1	yes	characteristic
20	#87806	C₂²	1	1	yes	characteristic
21	#87806	C₂²	1	1	yes	characteristic
22	C₄×#1039	C₂²	1	1	yes	characteristic
23	C₄×#1039	C₂²	1	1	yes	characteristic
24	C₅×#685	C₂²	1	2	yes	—
25	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
26	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
27	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
28	C₅×#683	C₂²	1	2	yes	—
29	C₅×#683	C₂²	1	2	yes	—
30	#87781	C₂²	1	2	yes	—
31	#87780	C₂²	1	2	yes	—
32	#87780	C₂²	1	2	yes	—
33	#87778	C₂²	1	2	yes	—
34	#87778	C₂²	1	2	yes	—
35	#66024	—	5	1	no	maximal
36	#65873	—	5	1	no	maximal
37	C₂²×C₅×D₅	C₂³	1	2	yes	—
38	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
41	C₂×#1039	C₂³	1	1	yes	characteristic
42	C₂×#1039	C₂³	1	1	yes	characteristic
43	C₂×#1039	C₂³	1	1	yes	characteristic
44	C₂×#1039	C₂³	1	1	yes	characteristic
45	#6045	—	2	2	no	—
46	C₂×C₅×Dic₅	C₂³	1	2	yes	—
47	C₂×C₅×Dic₅	C₂³	1	2	yes	—
48	C₂×C₅×Dic₅	C₂³	1	2	yes	—
49	C₅×D₂₀	Q₈	1	4	yes	complemented
50	C₄×C₅×D₅	—	4	2	no	—
51	#6039	—	4	2	no	—
52	D₅×Dic₅	—	2	4	no	—
53	D₅×Dic₅	—	4	2	no	—
54	C₅×#125	—	5	1	no	—
55	C₄×C₅×D₄	—	5	1	no	—
56	D₄×Dic₅	—	5	1	no	—
57	D₄×Dic₅	—	5	1	no	—
58	#4070	—	5	2	no	—
59	C₂×#683	—	5	2	no	—
60	#4043	—	5	2	no	—
61	#4042	—	5	2	no	—
62	#4042	—	5	2	no	—
63	#4039	—	5	1	no	—
64	#4039	—	5	1	no	—
65	D₅×C₄⋊C₄	—	5	2	no	—
66	#4036	—	5	1	no	—
67	#4036	—	5	1	no	—
68	#4034	—	5	1	no	—
69	#4034	—	5	1	no	—
70	#4024	—	5	2	no	—
71	#4024	—	5	2	no	—
72	C₄×D₂₀	D₅	1	1	yes	characteristic
73	#4016	D₅	1	1	yes	characteristic, complemented
74	C₄×Dic₁₀	—	5	1	no	—
75	C₄×Dic₁₀	—	5	1	no	—
76	C₄×Dic₁₀	—	10	2	no	—
77	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
78	C₂×C₅×D₅	—	2	2	no	—
79	C₂×C₅×D₅	—	2	2	no	—
80	C₂×C₅×D₅	C₂×Q₈	1	4	yes	—
81	C₄×C₅²	—	2	1	no	—
82	C₄×C₅²	—	2	1	no	—
83	C₄×C₅²	C₂×Q₈	1	2	yes	—
84	#1039	—	2	1	no	—
85	#1039	—	2	1	no	—
86	#1039	—	2	1	no	—
87	#1039	#41	1	2	yes	—
88	C₅×Dic₅	—	2	2	no	—
89	C₅×Dic₅	—	2	2	no	—
90	C₅×Dic₅	—	2	2	no	—
91	C₂×C₅×D₄	—	5	1	no	—
92	C₂²×C₄×C₅	—	5	2	no	—
93	C₂²×Dic₅	—	5	2	no	—
94	C₂²×Dic₅	—	5	2	no	—
95	C₂×D₂₀	C₂×D₅	1	1	yes	characteristic
96	C₂×C₄×D₅	—	5	2	no	—
97	C₂×C₄×D₅	—	5	2	no	—
98	C₂×C₄×D₅	C₂×D₅	1	2	yes	—
99	C₂×Dic₁₀	—	5	1	no	—
100	C₂×Dic₁₀	—	5	1	no	—
101	C₂×Dic₁₀	—	5	2	no	—
102	C₂×Dic₁₀	—	5	2	no	—
103	C₂×Dic₁₀	—	10	2	no	—
104	C₅×C₄⋊C₄	—	5	1	no	—
105	C₅×C₄⋊C₄	—	5	1	no	—
106	C₅×C₄⋊C₄	—	5	1	no	—
107	C₅×C₄⋊C₄	—	5	2	no	—
108	C₅×C₄⋊C₄	—	5	2	no	—
109	C₅×C₂²⋊C₄	—	5	2	no	—
110	C₄²×C₅	—	2	2	no	—
111	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
112	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
113	C₂²⋊Dic₅	—	5	2	no	—
114	C₂²⋊Dic₅	—	5	2	no	—
115	#685	—	5	2	no	—
116	#685	—	5	2	no	—
117	#685	C₂×D₅	1	2	yes	—
118	C₄⋊Dic₅	—	5	1	no	—
119	C₄⋊Dic₅	—	5	1	no	—
120	C₄⋊Dic₅	—	5	2	no	—
121	C₄⋊Dic₅	—	5	2	no	—
122	C₄⋊Dic₅	—	10	2	no	—
123	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
124	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
125	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
126	#683	—	5	1	no	—
127	#683	—	5	1	no	—
128	#683	—	5	1	no	—
129	#683	—	5	1	no	—
130	#683	—	5	2	no	—
131	#683	—	5	2	no	—
132	#683	—	5	2	no	—
133	#683	—	5	2	no	—
134	#683	—	10	2	no	—
135	#683	—	10	2	no	—
136	#683	C₂×D₅	1	2	yes	—
137	#683	C₂×D₅	1	2	yes	—
138	C₄×Dic₅	—	5	1	no	—
139	C₄×Dic₅	—	5	1	no	—
140	C₄×Dic₅	—	5	1	no	—
141	C₄×Dic₅	—	5	1	no	—
142	C₄×Dic₅	—	10	2	no	—
143	C₄×Dic₅	—	10	2	no	—
144	#554	—	25	1	no	Sylow
145	C₂×C₅²	C₂²×Q₈	1	1	yes	characteristic
146	C₂×C₅²	C₂×#41	1	1	yes	characteristic
147	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
148	C₅×D₅	—	2	4	no	—
149	C₂³×C₅	—	5	2	no	—
150	C₂²×D₅	C₂²×D₅	1	2	yes	—
151	C₅×D₄	—	5	4	no	—
152	C₂×C₄×C₅	—	2	2	no	—
153	C₂×C₄×C₅	—	2	2	no	—
154	C₂×C₄×C₅	—	2	2	no	—
155	C₂×C₄×C₅	—	5	2	no	—
156	C₂×C₄×C₅	—	5	2	no	—
157	C₂×C₄×C₅	—	5	2	no	—
158	C₂×C₄×C₅	—	20	2	no	—
159	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
161	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
164	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
165	C₂×Dic₅	—	5	1	no	—
166	C₂×Dic₅	—	5	1	no	—
167	C₂×Dic₅	—	5	1	no	—
168	C₂×Dic₅	—	5	1	no	—
169	C₂×Dic₅	—	5	1	no	—
170	C₂×Dic₅	—	5	1	no	—
171	C₂×Dic₅	—	5	1	no	—
172	C₂×Dic₅	—	5	1	no	—
173	C₂×Dic₅	—	10	2	no	—
174	C₂×Dic₅	—	10	2	no	—
175	C₂×Dic₅	—	10	2	no	—
176	C₂×Dic₅	—	10	2	no	—
177	C₂×Dic₅	—	10	4	no	—
178	C₂×Dic₅	—	20	2	no	—
179	C₂×Dic₅	C₂²×D₅	1	2	yes	—
180	C₂×Dic₅	C₂²×D₅	1	2	yes	—
181	C₂×Dic₅	C₂²×D₅	1	2	yes	—
182	D₂₀	Dic₁₀	1	4	yes	complemented
183	C₄×D₅	—	4	2	no	—
184	C₄×D₅	—	10	4	no	—
185	C₄×D₅	—	20	2	no	—
186	Dic₁₀	—	10	2	no	—
187	Dic₁₀	—	10	2	no	—
188	Dic₁₀	—	10	8	no	—
189	Dic₁₀	—	20	2	no	—
190	Dic₁₀	—	20	2	no	—
191	C₄⋊Q₈	—	25	1	no	—
192	#125	—	5	1	no	—
193	#125	—	25	1	no	—
194	C₂²⋊Q₈	—	25	2	no	—
195	C₂²⋊Q₈	—	25	2	no	—
196	C₂²⋊Q₈	—	25	2	no	—
197	C₄×Q₈	—	25	1	no	—
198	C₄×D₄	—	5	1	no	—
199	C₄×D₄	—	25	1	no	—
200	C₄×D₄	—	25	1	no	—
201	C₂×C₄⋊C₄	—	25	2	no	—
202	C₅²	#554	1	1	yes	characteristic, complemented, Sylow
203	C₂²×C₅	—	2	2	no	—
204	C₂²×C₅	—	5	4	no	—
205	C₂²×C₅	—	10	2	no	—
206	C₂²×C₅	—	10	2	no	—
207	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
208	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
209	C₂×D₅	—	2	2	no	—
210	C₂×D₅	—	2	2	no	—
211	C₂×D₅	C₂×Dic₁₀	1	4	yes	—
212	C₄×C₅	—	2	4	no	—
213	C₄×C₅	—	2	1	no	—
214	C₄×C₅	—	2	1	no	—
215	C₄×C₅	—	2	1	no	—
216	C₄×C₅	—	2	1	no	—
217	C₄×C₅	—	2	4	no	—
218	C₄×C₅	—	4	2	no	—
219	C₄×C₅	—	10	2	no	—
220	C₄×C₅	—	10	2	no	—
221	C₄×C₅	—	10	2	no	—
222	C₄×C₅	C₂×Dic₁₀	1	2	yes	—
223	C₄×C₅	Q₈×D₅	1	2	yes	—
224	Dic₅	—	2	2	no	—
225	Dic₅	—	2	2	no	—
226	Dic₅	—	2	2	no	—
227	Dic₅	—	5	2	no	—
228	Dic₅	—	5	2	no	—
229	Dic₅	—	10	4	no	—
230	Dic₅	—	10	4	no	—
231	Dic₅	—	10	1	no	—
232	Dic₅	—	10	1	no	—
233	Dic₅	—	10	1	no	—
234	Dic₅	—	10	1	no	—
235	Dic₅	—	10	1	no	—
236	Dic₅	—	10	1	no	—
237	Dic₅	—	20	2	no	—
238	Dic₅	—	20	2	no	—
239	C₂×Q₈	—	25	2	no	—
240	C₂×Q₈	—	25	1	no	—
241	C₂×D₄	—	5	1	no	—
242	C₂²×C₄	—	5	2	no	—
243	C₂²×C₄	—	25	2	no	—
244	C₂²×C₄	—	25	2	no	—
245	C₄⋊C₄	—	5	2	no	—
246	C₄⋊C₄	—	5	1	no	—
247	C₄⋊C₄	—	5	1	no	—
248	C₄⋊C₄	—	5	1	no	—
249	C₄⋊C₄	—	5	2	no	—
250	C₄⋊C₄	—	25	2	no	—
251	C₄⋊C₄	—	25	2	no	—
252	C₄⋊C₄	—	25	2	no	—
253	C₄⋊C₄	—	25	1	no	—
254	C₄⋊C₄	—	25	1	no	—
255	C₄⋊C₄	—	25	1	no	—
256	C₂²⋊C₄	—	5	2	no	—
257	C₂²⋊C₄	—	25	2	no	—
258	C₂²⋊C₄	—	25	2	no	—
259	C₄²	—	25	1	no	—
260	C₄²	—	25	1	no	—
261	C₄²	D₅²	1	1	yes	characteristic
262	C₂×C₅	—	2	2	no	—
263	C₂×C₅	—	2	2	no	—
264	C₂×C₅	—	2	2	no	—
265	C₂×C₅	—	10	4	no	—
266	C₂×C₅	C₂²×Dic₁₀	1	1	yes	characteristic
267	C₂×C₅	#4144	1	1	yes	characteristic
268	C₂×C₅	C₂×Q₈×D₅	1	1	yes	characteristic
269	C₂×C₅	D₅×#41	1	1	yes	characteristic
270	C₂×C₅	D₅×#41	1	1	yes	characteristic
271	C₂×C₅	#4149	1	1	yes	characteristic
272	D₅	—	2	4	no	—
273	C₂³	—	5	2	no	—
274	Q₈	—	50	2	no	—
275	Q₈	—	100	2	no	—
276	D₄	—	5	4	no	—
277	C₂×C₄	—	5	2	no	—
278	C₂×C₄	—	5	2	no	—
279	C₂×C₄	—	5	2	no	—
280	C₂×C₄	—	20	2	no	—
281	C₂×C₄	—	25	1	no	—
282	C₂×C₄	—	25	1	no	—
283	C₂×C₄	—	25	1	no	—
284	C₂×C₄	—	25	1	no	—
285	C₂×C₄	—	50	4	no	—
286	C₂×C₄	—	100	2	no	—
287	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
288	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
289	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
290	C₅	—	2	2	no	—
291	C₅	#65873	1	1	yes	characteristic, complemented
292	C₅	#66024	1	1	yes	characteristic, complemented
293	C₂²	—	5	4	no	—
294	C₂²	—	10	2	no	—
295	C₂²	—	10	2	no	—
296	C₂²	C₂²×D₅²	1	1	yes	characteristic
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	10	2	no	—
300	C₄	—	10	2	no	—
301	C₄	—	10	2	no	—
302	C₄	—	25	2	no	—
303	C₄	—	50	1	no	—
304	C₄	—	50	1	no	—
305	C₄	—	50	1	no	—
306	C₄	D₅×Dic₁₀	1	2	yes	—
307	C₂	—	10	4	no	—
308	C₂	C₂×D₅×Dic₁₀	1	1	yes	characteristic
309	C₂	#11717471	1	1	yes	characteristic
310	C₂	#11717472	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅²
commutator, socle, cosocle	C₂²×C₅²	C₂⁴
Fitting	C₄²×C₅²	C₂²
prime radical	C₂×C₅×D₂₀	C₂²
hypercenter	C₄²	D₅²
hypocenter	C₅²	#554
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
1. C₄×#6045	C₂	semidirect
3. C₅×#4016	C₂	semidirect
4. #11717037	C₂	semidirect
7. #11717015	C₂	semidirect
8. #11717007	C₂	semidirect
9. D₅×#683	C₂	semidirect
10. #11716991	C₂	semidirect
11. #11716969	C₂	semidirect
49. C₅×D₂₀	Q₈	semidirect
73. #4016	D₅	semidirect
182. D₂₀	Dic₁₀	semidirect
202. C₅²	#554	semidirect
291. C₅	#65873	semidirect
292. C₅	#66024	semidirect

Maximal subgroups

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

subgroup	index
C₄×#6045	2
C₄×C₅×D₂₀	2
C₅×#4016	2
#11717037	2
#11717017	2
#11717015	2
#11717007	2
D₅×#683	2
#11716991	2
#11716969	2
#66024	5
#65873	5