Subgroups — #13107222

Order 1600.

856 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#11717218	C₂	1	1	yes	maximal, characteristic, complemented
3	#11717218	C₂	1	1	yes	maximal, characteristic, complemented
4	#11717216	C₂	1	1	yes	maximal, characteristic, complemented
5	#11717216	C₂	1	1	yes	maximal, characteristic, complemented
6	#11717214	C₂	1	1	yes	maximal, characteristic
7	#11717214	C₂	1	1	yes	maximal, characteristic
8	#11717213	C₂	1	1	yes	maximal, characteristic
9	#11717212	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄⋊C₄×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
11	#11717209	C₂	1	1	yes	maximal, characteristic, complemented
12	#11717209	C₂	1	1	yes	maximal, characteristic, complemented
13	#11717207	C₂	1	1	yes	maximal, characteristic, complemented
14	#11717192	C₂	1	1	yes	maximal, characteristic, complemented
15	#11717192	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
21	C₂×#6045	C₂²	1	1	yes	characteristic
22	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
26	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
28	C₄²×C₅²	C₂²	1	1	yes	characteristic
29	#87808	C₂²	1	1	yes	characteristic
30	#87808	C₂²	1	1	yes	characteristic
31	#87808	C₂²	1	1	yes	characteristic
32	#87808	C₂²	1	1	yes	characteristic
33	#87808	C₂²	1	1	yes	characteristic
34	#87808	C₂²	1	1	yes	characteristic
35	#87808	C₂²	1	1	yes	characteristic
36	#87808	C₂²	1	1	yes	characteristic
37	C₄⋊#1039	C₂²	1	1	yes	characteristic
38	C₄⋊#1039	C₂²	1	1	yes	characteristic
39	#87806	C₂²	1	1	yes	characteristic
40	#87806	C₂²	1	1	yes	characteristic
41	#87806	C₂²	1	1	yes	characteristic
42	#87806	C₂²	1	1	yes	characteristic
43	#87806	C₂²	1	1	yes	characteristic
44	#87806	C₂²	1	1	yes	characteristic
45	#87806	C₂²	1	1	yes	characteristic
46	#87806	C₂²	1	1	yes	characteristic
47	C₄×#1039	C₂²	1	1	yes	characteristic
48	C₄×#1039	C₂²	1	1	yes	characteristic
49	C₄×#1039	C₂²	1	1	yes	characteristic
50	C₄×#1039	C₂²	1	1	yes	characteristic
51	#66027	—	5	6	no	maximal
52	C₂²×C₅⋊D₅	C₂³	1	1	yes	characteristic
53	C₂²×C₅⋊D₅	C₂³	1	1	yes	characteristic
54	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
55	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
56	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
57	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
58	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
59	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
60	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
61	C₂×#1039	C₂³	1	1	yes	characteristic
62	C₂×#1039	C₂³	1	1	yes	characteristic
63	C₂×#1039	C₂³	1	1	yes	characteristic
64	C₂×#1039	C₂³	1	1	yes	characteristic
65	C₂×#1039	C₂³	1	1	yes	characteristic
66	C₂×#1039	C₂³	1	1	yes	characteristic
67	C₄⋊(C₅⋊D₅)	—	2	2	no	—
68	C₄×C₅⋊D₅	—	2	2	no	—
69	C₄×C₅⋊D₅	—	2	2	no	—
70	C₄×C₅⋊D₅	—	2	2	no	—
71	C₄×C₅⋊D₅	—	2	2	no	—
72	#6045	—	2	2	no	—
73	C₅×#125	D₅	1	6	yes	complemented
74	#4044	—	5	6	no	—
75	#4044	—	5	6	no	—
76	#4042	—	5	6	no	—
77	#4042	—	5	6	no	—
78	#4040	—	5	6	no	—
79	#4040	—	5	6	no	—
80	#4039	—	5	6	no	—
81	#4038	—	5	6	no	—
82	D₅×C₄⋊C₄	—	5	6	no	—
83	#4035	—	5	6	no	—
84	#4035	—	5	6	no	—
85	#4033	—	5	6	no	—
86	#4018	—	5	6	no	—
87	#4018	—	5	6	no	—
88	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
89	C₂×C₅⋊D₅	—	2	1	no	—
90	C₂×C₅⋊D₅	—	2	1	no	—
91	C₂×C₅⋊D₅	—	2	1	no	—
92	C₂×C₅⋊D₅	—	2	1	no	—
93	C₂×C₅⋊D₅	—	2	1	no	—
94	C₂×C₅⋊D₅	#41	1	2	yes	—
95	C₄×C₅²	—	2	1	no	—
96	C₄×C₅²	—	2	1	no	—
97	C₄×C₅²	—	2	1	no	—
98	C₄×C₅²	—	2	1	no	—
99	C₄×C₅²	—	2	1	no	—
100	C₄×C₅²	—	2	1	no	—
101	C₄×C₅²	—	2	1	no	—
102	#1039	—	2	1	no	—
103	#1039	—	2	1	no	—
104	#1039	—	2	1	no	—
105	#1039	—	2	1	no	—
106	#1039	—	2	1	no	—
107	#1039	#41	1	2	yes	—
108	C₂×D₂₀	—	5	6	no	—
109	C₂×C₄×D₅	—	5	6	no	—
110	C₂×C₄×D₅	—	5	6	no	—
111	C₂×C₄×D₅	—	5	6	no	—
112	C₂×C₄×D₅	—	5	6	no	—
113	C₂×Dic₁₀	—	5	6	no	—
114	C₅×C₄⋊C₄	C₂×D₅	1	6	yes	—
115	C₅×C₄⋊C₄	C₂×D₅	1	6	yes	—
116	C₅×C₄⋊C₄	C₂×D₅	1	6	yes	—
117	C₅×C₄⋊C₄	C₂×D₅	1	6	yes	—
118	C₅×C₄⋊C₄	C₂×D₅	1	6	yes	—
119	C₅×C₄⋊C₄	C₂×D₅	1	6	yes	—
120	C₄²×C₅	C₂×D₅	1	6	yes	—
121	#685	—	5	6	no	—
122	#685	—	5	6	no	—
123	#685	—	5	6	no	—
124	#685	—	5	6	no	—
125	#685	—	5	6	no	—
126	#685	—	5	6	no	—
127	#685	—	5	6	no	—
128	#685	—	5	6	no	—
129	C₄⋊Dic₅	—	5	6	no	—
130	C₄⋊Dic₅	—	5	6	no	—
131	#683	—	5	6	no	—
132	#683	—	5	6	no	—
133	#683	—	5	6	no	—
134	#683	—	5	6	no	—
135	#683	—	5	6	no	—
136	#683	—	5	6	no	—
137	#683	—	5	6	no	—
138	#683	—	5	6	no	—
139	C₄×Dic₅	—	5	6	no	—
140	C₄×Dic₅	—	5	6	no	—
141	C₄×Dic₅	—	5	6	no	—
142	C₄×Dic₅	—	5	6	no	—
143	#552	—	25	1	no	Sylow
144	C₂×C₅²	C₂×#41	1	1	yes	characteristic
145	C₂×C₅²	C₂×#41	1	1	yes	characteristic
146	C₂×C₅²	2₋¹⁺⁴	1	1	yes	characteristic
147	C₅⋊D₅	—	2	2	no	—
148	C₅⋊D₅	—	4	1	no	—
149	C₂²×D₅	—	5	6	no	—
150	C₂²×D₅	—	5	6	no	—
151	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
152	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
153	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
154	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
155	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
156	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
157	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
158	C₂×Dic₅	—	5	6	no	—
159	C₂×Dic₅	—	5	6	no	—
160	C₂×Dic₅	—	5	6	no	—
161	C₂×Dic₅	—	5	6	no	—
162	C₂×Dic₅	—	5	6	no	—
163	C₂×Dic₅	—	5	6	no	—
164	D₂₀	—	10	12	no	—
165	C₄×D₅	—	10	12	no	—
166	C₄×D₅	—	10	12	no	—
167	C₄×D₅	—	10	12	no	—
168	C₄×D₅	—	10	12	no	—
169	Dic₁₀	—	10	12	no	—
170	#126	—	25	1	no	—
171	#126	—	25	1	no	—
172	#125	—	25	1	no	—
173	#125	—	25	1	no	—
174	#125	C₅⋊D₅	1	1	yes	characteristic, complemented
175	#123	—	25	1	no	—
176	#123	—	25	1	no	—
177	C₂²⋊Q₈	—	25	1	no	—
178	C₂²⋊Q₈	—	25	1	no	—
179	C₄×Q₈	—	25	1	no	—
180	C₄×D₄	—	25	1	no	—
181	#117	—	25	1	no	—
182	#117	—	25	1	no	—
183	#117	—	25	1	no	—
184	C₂×C₄⋊C₄	—	25	1	no	—
185	C₅²	#552	1	1	yes	characteristic, complemented, Sylow
186	C₂²×C₅	C₂³×D₅	1	6	yes	—
187	C₂×D₅	—	5	12	no	—
188	C₂×D₅	—	10	6	no	—
189	C₂×D₅	—	10	6	no	—
190	C₂×D₅	—	10	6	no	—
191	C₂×D₅	—	10	6	no	—
192	C₂×D₅	—	10	6	no	—
193	C₄×C₅	—	2	6	no	—
194	C₄×C₅	—	2	6	no	—
195	C₄×C₅	—	2	6	no	—
196	C₄×C₅	—	2	6	no	—
197	C₄×C₅	—	2	6	no	—
198	C₄×C₅	—	2	6	no	—
199	C₄×C₅	—	2	6	no	—
200	Dic₅	—	5	12	no	—
201	Dic₅	—	10	6	no	—
202	Dic₅	—	10	6	no	—
203	Dic₅	—	10	6	no	—
204	Dic₅	—	10	6	no	—
205	Dic₅	—	10	6	no	—
206	C₂×Q₈	—	25	1	no	—
207	C₂×D₄	—	25	1	no	—
208	C₂²×C₄	—	25	1	no	—
209	C₂²×C₄	—	25	1	no	—
210	C₂²×C₄	—	25	1	no	—
211	C₂²×C₄	—	25	1	no	—
212	C₄⋊C₄	—	25	1	no	—
213	C₄⋊C₄	—	25	1	no	—
214	C₄⋊C₄	—	25	1	no	—
215	C₄⋊C₄	—	25	1	no	—
216	C₄⋊C₄	—	25	1	no	—
217	C₄⋊C₄	—	25	1	no	—
218	C₄⋊C₄	—	25	1	no	—
219	C₄⋊C₄	—	25	1	no	—
220	C₄⋊C₄	—	25	1	no	—
221	C₄⋊C₄	—	25	1	no	—
222	C₄⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
223	C₄⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
224	C₄⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
225	C₄⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
226	C₄⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
227	C₄⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
228	C₂²⋊C₄	—	25	1	no	—
229	C₂²⋊C₄	—	25	1	no	—
230	C₂²⋊C₄	—	25	1	no	—
231	C₂²⋊C₄	—	25	1	no	—
232	C₂²⋊C₄	—	25	1	no	—
233	C₂²⋊C₄	—	25	1	no	—
234	C₂²⋊C₄	—	25	1	no	—
235	C₂²⋊C₄	—	25	1	no	—
236	C₄²	—	25	1	no	—
237	C₄²	—	25	1	no	—
238	C₄²	—	25	1	no	—
239	C₄²	—	25	1	no	—
240	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
241	C₂×C₅	#4147	1	6	yes	—
242	C₂×C₅	D₅×#41	1	6	yes	—
243	C₂×C₅	D₅×#41	1	6	yes	—
244	D₅	—	10	12	no	—
245	D₅	—	20	6	no	—
246	C₂³	—	25	1	no	—
247	C₂³	—	25	1	no	—
248	Q₈	—	50	2	no	—
249	D₄	—	50	2	no	—
250	C₂×C₄	—	25	1	no	—
251	C₂×C₄	—	25	1	no	—
252	C₂×C₄	—	25	1	no	—
253	C₂×C₄	—	25	1	no	—
254	C₂×C₄	—	25	1	no	—
255	C₂×C₄	—	25	1	no	—
256	C₂×C₄	—	50	2	no	—
257	C₂×C₄	—	50	2	no	—
258	C₂×C₄	—	50	2	no	—
259	C₂×C₄	—	50	2	no	—
260	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
261	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
262	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
263	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
264	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
265	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
266	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
267	C₅	#66027	1	6	yes	complemented
268	C₂²	—	25	2	no	—
269	C₂²	—	50	1	no	—
270	C₂²	—	50	1	no	—
271	C₂²	—	50	1	no	—
272	C₂²	—	50	1	no	—
273	C₂²	—	50	1	no	—
274	C₂²	C₂³×C₅⋊D₅	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	25	2	no	—
283	C₄	—	50	1	no	—
284	C₄	—	50	1	no	—
285	C₄	—	50	1	no	—
286	C₄	—	50	1	no	—
287	C₄	—	50	1	no	—
288	C₂	—	50	2	no	—
289	C₂	—	100	1	no	—
290	C₂	#11717520	1	1	yes	characteristic
291	C₂	#41×C₅⋊D₅	1	1	yes	characteristic
292	C₂	#41×C₅⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₅⋊D₅
commutator, socle, cosocle	C₂²×C₅²	C₂⁴
Fitting	C₅²×#125	C₂
prime radical	C₂×C₄⋊(C₅⋊D₅)	C₂²
hypercenter	#125	C₅⋊D₅
hypocenter	C₅²	#552
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. C₅²×#125	C₂	semidirect
2. #11717218	C₂	semidirect
3. #11717218	C₂	semidirect
4. #11717216	C₂	semidirect
5. #11717216	C₂	semidirect
9. #11717212	C₂	semidirect
10. C₄⋊C₄×C₅⋊D₅	C₂	semidirect
11. #11717209	C₂	semidirect
12. #11717209	C₂	semidirect
13. #11717207	C₂	semidirect
14. #11717192	C₂	semidirect
15. #11717192	C₂	semidirect
73. C₅×#125	D₅	semidirect
174. #125	C₅⋊D₅	semidirect
185. C₅²	#552	semidirect
267. C₅	#66027	semidirect

Maximal subgroups

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

subgroup	index
C₅²×#125	2
#11717218	2
#11717216	2
#11717214	2
#11717213	2
#11717212	2
C₄⋊C₄×C₅⋊D₅	2
#11717209	2
#11717207	2
#11717192	2
#66027	5