Subgroups — #13105028

Order 1600.

480 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11717189	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#4018	C₂	1	1	yes	maximal, characteristic
3	C₅×#4016	C₂	1	1	yes	maximal, characteristic, complemented
4	#11717036	C₂	1	1	yes	maximal, characteristic, complemented
5	#11717015	C₂	1	2	yes	maximal
6	D₅×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic
7	#11717007	C₂	1	2	yes	maximal
8	#11716988	C₂	1	1	yes	maximal, characteristic, complemented
9	#11716963	C₂	1	2	yes	maximal, complemented
10	#11716959	C₂	1	1	yes	maximal, characteristic
11	#11716958	C₂	1	2	yes	maximal, complemented
12	C₂×#6045	C₂²	1	2	yes	—
13	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
14	C₂×#6039	C₂²	1	2	yes	—
15	C₂×D₅×Dic₅	C₂²	1	2	yes	—
16	C₄²×C₅²	C₂²	1	1	yes	characteristic
17	C₄⋊#1039	C₂²	1	1	yes	characteristic
18	C₄⋊#1039	C₂²	1	2	yes	—
19	C₄⋊#1039	C₂²	1	1	yes	characteristic
20	C₅×#685	C₂²	1	1	yes	characteristic
21	C₅×#685	C₂²	1	1	yes	characteristic
22	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
23	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
24	C₅×#683	C₂²	1	1	yes	characteristic
25	C₅×#683	C₂²	1	2	yes	—
26	C₅×#683	C₂²	1	1	yes	characteristic
27	C₅×#683	C₂²	1	2	yes	—
28	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
29	#87781	C₂²	1	2	yes	—
30	#87780	C₂²	1	2	yes	—
31	#87780	C₂²	1	2	yes	—
32	#87780	C₂²	1	2	yes	—
33	#87778	C₂²	1	2	yes	—
34	Dic₅²	C₂²	1	2	yes	—
35	#66023	—	5	1	no	maximal
36	#65853	—	5	1	no	maximal
37	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
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	2	yes	—
42	C₂×#1039	C₂³	1	2	yes	—
43	#6045	—	4	2	no	—
44	C₂×C₅×Dic₅	C₂³	1	2	yes	—
45	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
46	C₂×C₅×Dic₅	C₂³	1	2	yes	—
47	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
48	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
49	C₄×C₅×D₅	Q₈	1	4	yes	complemented
50	#6039	—	4	2	no	—
51	D₅×Dic₅	—	4	2	no	—
52	C₅×#125	—	5	1	no	—
53	C₅×#117	—	5	1	no	—
54	#4072	—	5	1	no	—
55	C₂×C₄⋊Dic₅	—	5	1	no	—
56	#4043	—	5	2	no	—
57	#4042	—	5	2	no	—
58	#4038	—	5	1	no	—
59	D₅×C₄⋊C₄	—	5	1	no	—
60	#4036	—	5	2	no	—
61	#4036	—	5	1	no	—
62	#4035	—	5	2	no	—
63	#4034	—	5	2	no	—
64	#4034	—	5	2	no	—
65	#4034	—	5	1	no	—
66	#4024	—	5	2	no	—
67	#4024	—	5	2	no	—
68	#4018	D₅	1	1	yes	characteristic
69	#4016	—	5	1	no	—
70	#4016	D₅	1	1	yes	characteristic, complemented
71	#4015	—	5	1	no	—
72	#4015	—	5	1	no	—
73	#4015	—	5	1	no	—
74	#4015	—	10	2	no	—
75	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
76	C₂×C₅×D₅	—	2	1	no	—
77	C₂×C₅×D₅	—	2	1	no	—
78	C₂×C₅×D₅	C₂×Q₈	1	2	yes	—
79	C₄×C₅²	—	2	1	no	—
80	C₄×C₅²	—	2	1	no	—
81	C₄×C₅²	C₂×Q₈	1	2	yes	—
82	#1039	—	2	2	no	—
83	#1039	—	2	2	no	—
84	C₅×Dic₅	—	2	2	no	—
85	C₅×Dic₅	—	2	1	no	—
86	C₅×Dic₅	—	2	2	no	—
87	C₅×Dic₅	—	2	1	no	—
88	C₅×Dic₅	C₂×Q₈	1	2	yes	—
89	C₂²×C₄×C₅	—	5	1	no	—
90	C₂²×Dic₅	—	5	2	no	—
91	C₂×C₄×D₅	—	5	2	no	—
92	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
93	C₂×Dic₁₀	—	5	2	no	—
94	C₂×Dic₁₀	—	5	2	no	—
95	C₂×Dic₁₀	—	5	2	no	—
96	C₂×Dic₁₀	—	5	2	no	—
97	C₂×Dic₁₀	—	10	4	no	—
98	C₅×C₄⋊C₄	—	5	1	no	—
99	C₅×C₄⋊C₄	—	5	1	no	—
100	C₅×C₄⋊C₄	—	5	1	no	—
101	C₅×C₄⋊C₄	—	5	2	no	—
102	C₅×C₄⋊C₄	—	5	2	no	—
103	C₅×C₄⋊C₄	—	5	1	no	—
104	C₅×C₂²⋊C₄	—	5	1	no	—
105	C₅×C₂²⋊C₄	—	5	1	no	—
106	C₄²×C₅	—	2	2	no	—
107	C₄²×C₅	—	5	1	no	—
108	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
109	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
110	C₂²⋊Dic₅	—	5	2	no	—
111	#685	—	5	2	no	—
112	#685	C₂×D₅	1	1	yes	characteristic
113	#685	C₂×D₅	1	1	yes	characteristic
114	C₄⋊Dic₅	—	5	1	no	—
115	C₄⋊Dic₅	—	5	1	no	—
116	C₄⋊Dic₅	—	5	2	no	—
117	C₄⋊Dic₅	—	5	1	no	—
118	C₄⋊Dic₅	—	5	2	no	—
119	C₄⋊Dic₅	—	5	1	no	—
120	C₄⋊Dic₅	—	5	2	no	—
121	C₄⋊Dic₅	—	5	2	no	—
122	C₄⋊Dic₅	—	5	2	no	—
123	C₄⋊Dic₅	—	10	2	no	—
124	C₄⋊Dic₅	—	10	2	no	—
125	C₄⋊Dic₅	—	10	4	no	—
126	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
127	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
128	#683	—	5	2	no	—
129	#683	—	5	2	no	—
130	#683	—	5	2	no	—
131	#683	—	5	2	no	—
132	#683	—	5	2	no	—
133	#683	C₂×D₅	1	2	yes	—
134	#683	C₂×D₅	1	2	yes	—
135	#683	C₂×D₅	1	1	yes	characteristic
136	#683	C₂×D₅	1	1	yes	characteristic
137	C₄×Dic₅	—	5	2	no	—
138	C₄×Dic₅	—	5	2	no	—
139	C₄×Dic₅	C₂×D₅	1	1	yes	characteristic
140	#544	—	25	1	no	Sylow
141	C₂×C₅²	C₂²×Q₈	1	1	yes	characteristic
142	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
143	C₂×C₅²	2₋¹⁺⁴	1	1	yes	characteristic
144	C₅×D₅	—	2	2	no	—
145	C₂³×C₅	—	5	1	no	—
146	C₂²×D₅	C₂²×D₅	1	1	yes	characteristic
147	C₂×C₄×C₅	—	2	2	no	—
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	2	2	no	—
150	C₂×C₄×C₅	—	5	1	no	—
151	C₂×C₄×C₅	—	5	4	no	—
152	C₂×C₄×C₅	—	5	2	no	—
153	C₂×C₄×C₅	—	5	1	no	—
154	C₂×C₄×C₅	—	5	1	no	—
155	C₂×C₄×C₅	—	5	2	no	—
156	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
158	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
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×Dic₅	—	5	2	no	—
163	C₂×Dic₅	—	5	2	no	—
164	C₂×Dic₅	—	5	2	no	—
165	C₂×Dic₅	—	5	2	no	—
166	C₂×Dic₅	—	10	4	no	—
167	C₂×Dic₅	—	10	4	no	—
168	C₂×Dic₅	—	20	2	no	—
169	C₂×Dic₅	C₂²×D₅	1	2	yes	—
170	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
171	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
172	C₂×Dic₅	C₂²×D₅	1	2	yes	—
173	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
174	C₄×D₅	—	20	2	no	—
175	C₄×D₅	Dic₁₀	1	4	yes	complemented
176	Dic₁₀	—	20	2	no	—
177	Dic₁₀	—	20	2	no	—
178	Dic₁₀	—	20	2	no	—
179	Dic₁₀	—	20	2	no	—
180	Dic₁₀	—	20	8	no	—
181	C₄⋊Q₈	—	25	1	no	—
182	C₄⋊Q₈	—	25	2	no	—
183	C₄⋊Q₈	—	25	1	no	—
184	#125	—	5	1	no	—
185	#125	—	25	1	no	—
186	#125	—	25	2	no	—
187	C₂²⋊Q₈	—	25	2	no	—
188	C₂²⋊Q₈	—	25	2	no	—
189	#117	—	5	1	no	—
190	#117	—	25	1	no	—
191	C₂×C₄⋊C₄	—	25	1	no	—
192	C₅²	#544	1	1	yes	characteristic, complemented, Sylow
193	C₂²×C₅	—	2	2	no	—
194	C₂²×C₅	—	5	2	no	—
195	C₂²×C₅	—	10	1	no	—
196	C₂²×C₅	—	10	1	no	—
197	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
198	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
199	C₂×D₅	—	2	1	no	—
200	C₂×D₅	—	2	1	no	—
201	C₂×D₅	C₂×Dic₁₀	1	2	yes	—
202	C₄×C₅	—	2	4	no	—
203	C₄×C₅	—	2	4	no	—
204	C₄×C₅	—	2	1	no	—
205	C₄×C₅	—	2	1	no	—
206	C₄×C₅	—	2	1	no	—
207	C₄×C₅	—	2	1	no	—
208	C₄×C₅	—	2	4	no	—
209	C₄×C₅	—	5	2	no	—
210	C₄×C₅	—	10	1	no	—
211	C₄×C₅	—	10	2	no	—
212	C₄×C₅	—	10	2	no	—
213	C₄×C₅	—	10	1	no	—
214	C₄×C₅	C₂×Dic₁₀	1	2	yes	—
215	C₄×C₅	Q₈×D₅	1	2	yes	—
216	Dic₅	—	2	1	no	—
217	Dic₅	—	2	2	no	—
218	Dic₅	—	2	1	no	—
219	Dic₅	—	2	2	no	—
220	Dic₅	—	10	2	no	—
221	Dic₅	—	10	2	no	—
222	Dic₅	—	10	2	no	—
223	Dic₅	—	10	2	no	—
224	Dic₅	—	20	4	no	—
225	Dic₅	—	20	4	no	—
226	Dic₅	C₂×Dic₁₀	1	2	yes	—
227	C₂×Q₈	—	25	2	no	—
228	C₂×Q₈	—	25	2	no	—
229	C₂²×C₄	—	5	1	no	—
230	C₂²×C₄	—	25	2	no	—
231	C₄⋊C₄	—	5	1	no	—
232	C₄⋊C₄	—	5	1	no	—
233	C₄⋊C₄	—	5	1	no	—
234	C₄⋊C₄	—	5	1	no	—
235	C₄⋊C₄	—	5	2	no	—
236	C₄⋊C₄	—	5	2	no	—
237	C₄⋊C₄	—	25	2	no	—
238	C₄⋊C₄	—	25	2	no	—
239	C₄⋊C₄	—	25	2	no	—
240	C₄⋊C₄	—	25	2	no	—
241	C₄⋊C₄	—	25	1	no	—
242	C₄⋊C₄	—	25	1	no	—
243	C₄⋊C₄	—	25	2	no	—
244	C₂²⋊C₄	—	5	1	no	—
245	C₂²⋊C₄	—	5	1	no	—
246	C₂²⋊C₄	—	25	2	no	—
247	C₄²	—	5	1	no	—
248	C₄²	—	25	2	no	—
249	C₄²	D₅²	1	1	yes	characteristic
250	C₂×C₅	—	2	2	no	—
251	C₂×C₅	—	2	2	no	—
252	C₂×C₅	—	2	2	no	—
253	C₂×C₅	—	10	2	no	—
254	C₂×C₅	C₂²×Dic₁₀	1	1	yes	characteristic
255	C₂×C₅	C₂×Q₈×D₅	1	1	yes	characteristic
256	C₂×C₅	#4149	1	1	yes	characteristic
257	C₂×C₅	#4149	1	1	yes	characteristic
258	C₂×C₅	#4150	1	1	yes	characteristic
259	C₂×C₅	#4150	1	1	yes	characteristic
260	D₅	—	2	2	no	—
261	C₂³	—	5	1	no	—
262	Q₈	—	100	2	no	—
263	Q₈	—	100	2	no	—
264	C₂×C₄	—	5	1	no	—
265	C₂×C₄	—	5	4	no	—
266	C₂×C₄	—	5	1	no	—
267	C₂×C₄	—	5	1	no	—
268	C₂×C₄	—	5	2	no	—
269	C₂×C₄	—	5	2	no	—
270	C₂×C₄	—	25	2	no	—
271	C₂×C₄	—	25	2	no	—
272	C₂×C₄	—	100	2	no	—
273	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
274	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
275	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
276	C₅	—	2	2	no	—
277	C₅	#65853	1	1	yes	characteristic, complemented
278	C₅	#66023	1	1	yes	characteristic, complemented
279	C₂²	—	5	2	no	—
280	C₂²	—	10	1	no	—
281	C₂²	—	10	1	no	—
282	C₂²	C₂²×D₅²	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	5	2	no	—
286	C₄	—	10	1	no	—
287	C₄	—	10	2	no	—
288	C₄	—	10	2	no	—
289	C₄	—	10	1	no	—
290	C₄	—	50	2	no	—
291	C₄	—	50	2	no	—
292	C₄	D₅×Dic₁₀	1	2	yes	—
293	C₂	—	10	2	no	—
294	C₂	C₂×D₅×Dic₁₀	1	1	yes	characteristic
295	C₂	#11717463	1	1	yes	characteristic
296	C₂	#11717473	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅²
commutator, socle, cosocle	C₂²×C₅²	C₂⁴
Fitting	C₄²×C₅²	C₂²
prime radical	C₂²×C₅×D₅	C₂³
hypercenter	C₄²	D₅²
hypocenter	C₅²	#544
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. #11717189	C₂	semidirect
3. C₅×#4016	C₂	semidirect
4. #11717036	C₂	semidirect
8. #11716988	C₂	semidirect
9. #11716963	C₂	semidirect
11. #11716958	C₂	semidirect
49. C₄×C₅×D₅	Q₈	semidirect
70. #4016	D₅	semidirect
175. C₄×D₅	Dic₁₀	semidirect
192. C₅²	#544	semidirect
277. C₅	#65853	semidirect
278. C₅	#66023	semidirect

Maximal subgroups

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

subgroup	index
#11717189	2
C₅×#4018	2
C₅×#4016	2
#11717036	2
#11717015	2
D₅×C₄⋊Dic₅	2
#11717007	2
#11716988	2
#11716963	2
#11716959	2
#11716958	2
#66023	5
#65853	5