Subgroups — #13105491

Order 1600.

520 conjugacy classes of subgroups.

The subgroup lattice has 305 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11717212	C₂	1	1	yes	maximal, characteristic
2	C₅×#4036	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4034	C₂	1	1	yes	maximal, characteristic, complemented
4	#11717036	C₂	1	1	yes	maximal, characteristic, complemented
5	#11717009	C₂	1	2	yes	maximal
6	#11717008	C₂	1	2	yes	maximal
7	C₄×#6036	C₂	1	1	yes	maximal, characteristic
8	#11716988	C₂	1	1	yes	maximal, characteristic, complemented
9	#11716967	C₂	1	1	yes	maximal, characteristic
10	#11716961	C₂	1	2	yes	maximal, complemented
11	#11716956	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
13	C₂×C₅×Dic₁₀	C₂²	1	2	yes	—
14	C₂×#6039	C₂²	1	2	yes	—
15	C₂×#6036	C₂²	1	1	yes	characteristic
16	C₂×#6036	C₂²	1	1	yes	characteristic
17	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	#87808	C₂²	1	2	yes	—
19	C₄⋊#1039	C₂²	1	1	yes	characteristic
20	C₄×#1039	C₂²	1	2	yes	—
21	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
22	C₅×C₄⋊Dic₅	C₂²	1	2	yes	—
23	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
24	C₅×#683	C₂²	1	2	yes	—
25	C₅×#683	C₂²	1	2	yes	—
26	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
27	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
28	#87781	C₂²	1	2	yes	—
29	#87780	C₂²	1	2	yes	—
30	#87780	C₂²	1	2	yes	—
31	#87779	C₂²	1	2	yes	—
32	Dic₅²	C₂²	1	1	yes	characteristic
33	Dic₅²	C₂²	1	2	yes	—
34	Dic₅²	C₂²	1	1	yes	characteristic
35	#66062	—	5	1	no	maximal
36	#66022	—	5	1	no	maximal
37	C₂²×C₅⋊D₅	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₅²	C₂³	1	2	yes	—
39	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
40	C₂×#1039	C₂³	1	2	yes	—
41	C₂×#1039	C₂³	1	1	yes	characteristic
42	C₄×C₅⋊D₅	Q₈	1	4	yes	complemented
43	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
44	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
45	C₂×C₅×Dic₅	C₂³	1	2	yes	—
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	C₅×Dic₁₀	—	2	4	no	—
50	#6039	—	2	4	no	—
51	#6036	—	2	2	no	—
52	#6036	—	2	2	no	—
53	C₅×C₄⋊Q₈	—	5	1	no	—
54	C₅×#125	—	5	1	no	—
55	#4092	—	5	2	no	—
56	Q₈×Dic₅	—	5	2	no	—
57	#4043	—	5	2	no	—
58	#4042	—	5	2	no	—
59	#4042	—	5	2	no	—
60	#4038	—	5	1	no	—
61	#4038	—	5	1	no	—
62	#4038	—	5	1	no	—
63	#4038	—	5	1	no	—
64	#4038	—	10	2	no	—
65	#4036	—	5	1	no	—
66	#4036	D₅	1	1	yes	characteristic, complemented
67	#4034	—	5	1	no	—
68	#4034	D₅	1	1	yes	characteristic, complemented
69	#4033	—	5	2	no	—
70	#4033	—	5	2	no	—
71	#4033	—	5	2	no	—
72	C₄²×D₅	—	5	1	no	—
73	C₄²×D₅	—	5	1	no	—
74	#4016	—	5	1	no	—
75	#4015	—	5	1	no	—
76	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
77	C₂×C₅⋊D₅	—	2	1	no	—
78	C₂×C₅⋊D₅	—	2	1	no	—
79	C₂×C₅⋊D₅	C₂×Q₈	1	2	yes	—
80	C₄×C₅²	—	2	2	no	—
81	C₄×C₅²	C₂×Q₈	1	2	yes	—
82	#1039	—	2	2	no	—
83	#1039	C₂×Q₈	1	2	yes	—
84	C₅×Dic₅	—	2	1	no	—
85	C₅×Dic₅	—	2	1	no	—
86	C₅×Dic₅	—	2	2	no	—
87	C₅×Dic₅	—	2	2	no	—
88	C₅×Dic₅	#41	1	2	yes	—
89	C₅×Dic₅	#41	1	2	yes	—
90	C₂×C₅×Q₈	—	5	2	no	—
91	C₂×C₄×D₅	—	5	1	no	—
92	C₂×C₄×D₅	—	5	1	no	—
93	C₂×C₄×D₅	—	5	1	no	—
94	C₂×C₄×D₅	—	5	1	no	—
95	C₂×C₄×D₅	—	5	1	no	—
96	C₂×C₄×D₅	—	5	1	no	—
97	C₂×C₄×D₅	—	10	2	no	—
98	C₂×Dic₁₀	—	5	2	no	—
99	C₂×Dic₁₀	—	5	2	no	—
100	C₂×Dic₁₀	C₂×D₅	1	2	yes	—
101	C₅×C₄⋊C₄	—	2	2	no	—
102	C₅×C₄⋊C₄	—	5	2	no	—
103	C₅×C₄⋊C₄	—	5	2	no	—
104	C₅×C₄⋊C₄	—	5	1	no	—
105	C₅×C₄⋊C₄	—	5	2	no	—
106	C₅×C₄⋊C₄	—	5	1	no	—
107	C₅×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
108	C₅×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
109	C₄²×C₅	—	5	1	no	—
110	C₄²×C₅	—	5	1	no	—
111	#685	—	5	2	no	—
112	#685	—	5	2	no	—
113	#685	—	5	2	no	—
114	#685	—	5	2	no	—
115	#685	—	10	4	no	—
116	C₄⋊Dic₅	—	5	1	no	—
117	C₄⋊Dic₅	—	5	1	no	—
118	C₄⋊Dic₅	—	5	2	no	—
119	C₄⋊Dic₅	—	5	2	no	—
120	C₄⋊Dic₅	—	10	2	no	—
121	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
122	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
123	C₄⋊Dic₅	C₂×D₅	1	2	yes	—
124	#683	—	5	2	no	—
125	#683	—	5	2	no	—
126	#683	—	5	2	no	—
127	#683	—	5	2	no	—
128	#683	C₂×D₅	1	2	yes	—
129	#683	C₂×D₅	1	2	yes	—
130	C₄×Dic₅	—	5	2	no	—
131	C₄×Dic₅	—	5	2	no	—
132	C₄×Dic₅	—	5	1	no	—
133	C₄×Dic₅	—	5	1	no	—
134	C₄×Dic₅	—	5	1	no	—
135	C₄×Dic₅	—	5	1	no	—
136	C₄×Dic₅	—	5	2	no	—
137	C₄×Dic₅	—	5	2	no	—
138	C₄×Dic₅	—	10	4	no	—
139	C₄×Dic₅	C₂×D₅	1	1	yes	characteristic
140	C₄×Dic₅	C₂×D₅	1	1	yes	characteristic
141	#533	—	25	1	no	Sylow
142	C₂×C₅²	C₂²×Q₈	1	1	yes	characteristic
143	C₂×C₅²	C₂×#41	1	1	yes	characteristic
144	C₂×C₅²	C₂×#41	1	1	yes	characteristic
145	C₅⋊D₅	—	2	2	no	—
146	C₂²×D₅	—	5	1	no	—
147	C₂²×D₅	—	5	1	no	—
148	C₂²×D₅	—	10	2	no	—
149	C₅×Q₈	—	10	4	no	—
150	C₂×C₄×C₅	—	2	2	no	—
151	C₂×C₄×C₅	—	2	4	no	—
152	C₂×C₄×C₅	—	5	1	no	—
153	C₂×C₄×C₅	—	5	1	no	—
154	C₂×C₄×C₅	—	5	1	no	—
155	C₂×C₄×C₅	—	5	1	no	—
156	C₂×C₄×C₅	—	5	2	no	—
157	C₂×C₄×C₅	—	5	2	no	—
158	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
159	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
160	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
161	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
162	C₂×Dic₅	—	5	1	no	—
163	C₂×Dic₅	—	5	1	no	—
164	C₂×Dic₅	—	5	2	no	—
165	C₂×Dic₅	—	5	2	no	—
166	C₂×Dic₅	—	10	2	no	—
167	C₂×Dic₅	—	10	4	no	—
168	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
169	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
170	C₂×Dic₅	C₂²×D₅	1	2	yes	—
171	C₂×Dic₅	C₂²×D₅	1	2	yes	—
172	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
173	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
174	C₄×D₅	—	5	4	no	—
175	C₄×D₅	—	5	4	no	—
176	C₄×D₅	—	10	8	no	—
177	C₄×D₅	—	10	2	no	—
178	C₄×D₅	—	10	2	no	—
179	C₄×D₅	—	10	2	no	—
180	C₄×D₅	—	10	2	no	—
181	Dic₁₀	—	2	4	no	—
182	Dic₁₀	—	10	4	no	—
183	Dic₁₀	—	10	4	no	—
184	C₄⋊Q₈	—	5	1	no	—
185	C₄⋊Q₈	—	25	1	no	—
186	#125	—	5	1	no	—
187	#125	—	25	1	no	—
188	C₂²⋊Q₈	—	25	2	no	—
189	C₂²⋊Q₈	—	25	2	no	—
190	C₄×Q₈	—	25	2	no	—
191	C₄×Q₈	—	25	2	no	—
192	#117	—	25	1	no	—
193	#117	—	25	1	no	—
194	C₂×C₄²	—	25	1	no	—
195	C₅²	#533	1	1	yes	characteristic, complemented, Sylow
196	C₂²×C₅	—	2	2	no	—
197	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
198	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
199	C₂×D₅	—	5	2	no	—
200	C₂×D₅	—	5	2	no	—
201	C₂×D₅	—	10	4	no	—
202	C₂×D₅	—	10	1	no	—
203	C₂×D₅	—	10	1	no	—
204	C₂×D₅	—	10	1	no	—
205	C₂×D₅	—	10	1	no	—
206	C₂×D₅	—	20	2	no	—
207	C₂×D₅	—	20	2	no	—
208	C₄×C₅	—	2	2	no	—
209	C₄×C₅	—	2	2	no	—
210	C₄×C₅	—	2	4	no	—
211	C₄×C₅	—	4	4	no	—
212	C₄×C₅	—	5	2	no	—
213	C₄×C₅	—	5	2	no	—
214	C₄×C₅	—	10	1	no	—
215	C₄×C₅	—	10	2	no	—
216	C₄×C₅	—	10	1	no	—
217	C₄×C₅	—	10	2	no	—
218	C₄×C₅	Q₈×D₅	1	2	yes	—
219	C₄×C₅	Q₈×D₅	1	2	yes	—
220	Dic₅	—	2	2	no	—
221	Dic₅	—	2	2	no	—
222	Dic₅	—	2	1	no	—
223	Dic₅	—	2	1	no	—
224	Dic₅	—	5	2	no	—
225	Dic₅	—	5	2	no	—
226	Dic₅	—	10	4	no	—
227	Dic₅	—	10	2	no	—
228	Dic₅	—	10	2	no	—
229	Dic₅	—	20	4	no	—
230	Dic₅	#711	1	2	yes	—
231	Dic₅	#713	1	2	yes	—
232	C₂×Q₈	—	5	2	no	—
233	C₂×Q₈	—	25	2	no	—
234	C₂²×C₄	—	25	1	no	—
235	C₂²×C₄	—	25	1	no	—
236	C₂²×C₄	—	25	1	no	—
237	C₄⋊C₄	—	5	1	no	—
238	C₄⋊C₄	—	5	1	no	—
239	C₄⋊C₄	—	5	2	no	—
240	C₄⋊C₄	—	5	2	no	—
241	C₄⋊C₄	—	5	2	no	—
242	C₄⋊C₄	—	25	2	no	—
243	C₄⋊C₄	—	25	2	no	—
244	C₄⋊C₄	—	25	2	no	—
245	C₄⋊C₄	—	25	1	no	—
246	C₄⋊C₄	D₅²	1	1	yes	characteristic
247	C₂²⋊C₄	—	25	2	no	—
248	C₂²⋊C₄	—	25	2	no	—
249	C₄²	—	5	1	no	—
250	C₄²	—	5	1	no	—
251	C₄²	—	25	1	no	—
252	C₄²	—	25	1	no	—
253	C₄²	—	25	2	no	—
254	C₄²	—	25	2	no	—
255	C₂×C₅	—	2	2	no	—
256	C₂×C₅	—	2	2	no	—
257	C₂×C₅	—	2	2	no	—
258	C₂×C₅	C₂×#711	1	1	yes	characteristic
259	C₂×C₅	C₂×Q₈×D₅	1	1	yes	characteristic
260	C₂×C₅	C₂×Q₈×D₅	1	1	yes	characteristic
261	C₂×C₅	C₂×#713	1	1	yes	characteristic
262	C₂×C₅	D₅×#41	1	1	yes	characteristic
263	C₂×C₅	D₅×#41	1	1	yes	characteristic
264	D₅	—	10	2	no	—
265	D₅	—	10	2	no	—
266	D₅	—	20	4	no	—
267	C₂³	—	25	1	no	—
268	Q₈	—	10	4	no	—
269	Q₈	—	50	4	no	—
270	C₂×C₄	—	5	1	no	—
271	C₂×C₄	—	5	1	no	—
272	C₂×C₄	—	5	1	no	—
273	C₂×C₄	—	5	1	no	—
274	C₂×C₄	—	5	2	no	—
275	C₂×C₄	—	5	2	no	—
276	C₂×C₄	—	25	1	no	—
277	C₂×C₄	—	25	4	no	—
278	C₂×C₄	—	25	2	no	—
279	C₂×C₄	—	50	2	no	—
280	C₂×C₄	—	50	2	no	—
281	C₂×C₄	C₂×D₅²	1	2	yes	—
282	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
283	C₅	—	2	2	no	—
284	C₅	#66022	1	1	yes	characteristic, complemented
285	C₅	#66062	1	1	yes	characteristic, complemented
286	C₂²	—	25	2	no	—
287	C₂²	—	50	1	no	—
288	C₂²	—	50	1	no	—
289	C₂²	C₂²×D₅²	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	5	2	no	—
292	C₄	—	5	2	no	—
293	C₄	—	10	1	no	—
294	C₄	—	10	2	no	—
295	C₄	—	10	1	no	—
296	C₄	—	10	2	no	—
297	C₄	—	25	2	no	—
298	C₄	—	50	2	no	—
299	C₄	#87872	1	2	yes	—
300	C₂	—	50	2	no	—
301	C₂	C₂×#87872	1	1	yes	characteristic
302	C₂	D₅×#711	1	1	yes	characteristic
303	C₂	D₅×#713	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅²
commutator, socle, cosocle	C₂²×C₅²	C₂⁴
Fitting	C₅²×C₄⋊C₄	C₂²
prime radical	C₂²×C₅⋊D₅	C₂³
hypercenter	C₄⋊C₄	D₅²
hypocenter	C₅²	#533
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₅×#4036	C₂	semidirect
3. C₅×#4034	C₂	semidirect
4. #11717036	C₂	semidirect
8. #11716988	C₂	semidirect
10. #11716961	C₂	semidirect
11. #11716956	C₂	semidirect
42. C₄×C₅⋊D₅	Q₈	semidirect
66. #4036	D₅	semidirect
68. #4034	D₅	semidirect
195. C₅²	#533	semidirect
284. C₅	#66022	semidirect
285. C₅	#66062	semidirect

Maximal subgroups

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

subgroup	index
#11717212	2
C₅×#4036	2
C₅×#4034	2
#11717036	2
#11717009	2
#11717008	2
C₄×#6036	2
#11716988	2
#11716967	2
#11716961	2
#11716956	2
#66062	5
#66022	5