Subgroups — #13099032

Order 1600.

321 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	C₂₅×#126	C₂	1	1	yes	maximal, characteristic, complemented
2	#11716458	C₂	1	1	yes	maximal, characteristic, complemented
3	#11716454	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄⋊C₄×D₂₅	C₂	1	1	yes	maximal, characteristic, complemented
5	#11716452	C₂	1	1	yes	maximal, characteristic, complemented
6	#11716451	C₂	1	1	yes	maximal, characteristic, complemented
7	#11716451	C₂	1	1	yes	maximal, characteristic, complemented
8	#11716445	C₂	1	1	yes	maximal, characteristic
9	#11716445	C₂	1	1	yes	maximal, characteristic
10	#11716443	C₂	1	1	yes	maximal, characteristic
11	#11716441	C₂	1	1	yes	maximal, characteristic, complemented
12	#11716440	C₂	1	1	yes	maximal, characteristic, complemented
13	#11716439	C₂	1	1	yes	maximal, characteristic, complemented
14	#11716434	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×Dic₅₀	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₂²⋊D₂₅	C₂²	1	1	yes	characteristic
17	C₂²×Dic₂₅	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₂₅	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₂₅	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₂₅	C₂²	1	1	yes	characteristic
21	C₂×Dic₅₀	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×C₂₅	C₂²	1	1	yes	characteristic
23	C₄⋊C₄×C₂₅	C₂²	1	1	yes	characteristic
24	C₄⋊C₄×C₂₅	C₂²	1	1	yes	characteristic
25	C₂²⋊C₄×C₂₅	C₂²	1	1	yes	characteristic
26	C₂²⋊C₄×C₂₅	C₂²	1	1	yes	characteristic
27	C₂²⋊C₄×C₂₅	C₂²	1	1	yes	characteristic
28	C₄²×C₂₅	C₂²	1	1	yes	characteristic
29	C₂²⋊Dic₂₅	C₂²	1	1	yes	characteristic
30	C₂²⋊Dic₂₅	C₂²	1	1	yes	characteristic
31	#87720	C₂²	1	1	yes	characteristic
32	#87720	C₂²	1	1	yes	characteristic
33	#87720	C₂²	1	1	yes	characteristic
34	C₄⋊Dic₂₅	C₂²	1	1	yes	characteristic
35	C₄⋊Dic₂₅	C₂²	1	1	yes	characteristic
36	C₄⋊Dic₂₅	C₂²	1	1	yes	characteristic
37	C₄⋊Dic₂₅	C₂²	1	1	yes	characteristic
38	#87718	C₂²	1	1	yes	characteristic
39	#87718	C₂²	1	1	yes	characteristic
40	#87718	C₂²	1	1	yes	characteristic
41	#87718	C₂²	1	1	yes	characteristic
42	#87718	C₂²	1	1	yes	characteristic
43	#87718	C₂²	1	1	yes	characteristic
44	#87718	C₂²	1	1	yes	characteristic
45	#87718	C₂²	1	1	yes	characteristic
46	#87718	C₂²	1	1	yes	characteristic
47	C₄×Dic₂₅	C₂²	1	1	yes	characteristic
48	C₄×Dic₂₅	C₂²	1	1	yes	characteristic
49	C₄×Dic₂₅	C₂²	1	1	yes	characteristic
50	C₄×Dic₂₅	C₂²	1	1	yes	characteristic
51	#66042	—	5	1	no	maximal
52	C₂³×C₂₅	C₂³	1	1	yes	characteristic
53	C₂²×D₂₅	C₂³	1	1	yes	characteristic
54	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
55	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
56	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
57	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
58	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
59	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
60	C₂²⋊D₂₅	—	2	2	no	—
61	C₂×Dic₂₅	—	2	2	no	—
62	C₂×Dic₂₅	C₂³	1	1	yes	characteristic
63	C₂×Dic₂₅	C₂³	1	1	yes	characteristic
64	C₂×Dic₂₅	C₂³	1	1	yes	characteristic
65	C₂×Dic₂₅	C₂³	1	1	yes	characteristic
66	C₂×Dic₂₅	C₂³	1	1	yes	characteristic
67	C₂×Dic₂₅	C₂³	1	1	yes	characteristic
68	C₂×Dic₂₅	C₂³	1	1	yes	characteristic
69	C₄×D₂₅	—	2	2	no	—
70	C₄×D₂₅	—	2	2	no	—
71	C₄×D₂₅	—	2	2	no	—
72	Dic₅₀	—	2	2	no	—
73	C₅×#126	D₅	1	1	yes	characteristic
74	#4042	—	5	1	no	—
75	#4038	—	5	1	no	—
76	D₅×C₄⋊C₄	—	5	1	no	—
77	#4036	—	5	1	no	—
78	#4035	—	5	1	no	—
79	#4035	—	5	1	no	—
80	#4029	—	5	1	no	—
81	#4029	—	5	1	no	—
82	#4027	—	5	1	no	—
83	#4025	—	5	1	no	—
84	#4024	—	5	1	no	—
85	#4023	—	5	1	no	—
86	#4018	—	5	1	no	—
87	C₄×Dic₁₀	—	5	1	no	—
88	C₂²×C₂₅	—	2	1	no	—
89	C₂²×C₂₅	—	2	1	no	—
90	C₂²×C₂₅	—	2	1	no	—
91	C₂²×C₂₅	C₂⁴	1	1	yes	characteristic
92	C₂×D₂₅	—	2	1	no	—
93	C₂×D₂₅	—	2	1	no	—
94	C₂×D₂₅	#41	1	2	yes	—
95	C₄×C₂₅	—	2	1	no	—
96	C₄×C₂₅	—	2	1	no	—
97	C₄×C₂₅	—	2	1	no	—
98	C₄×C₂₅	—	2	1	no	—
99	C₄×C₂₅	—	2	1	no	—
100	C₄×C₂₅	—	2	1	no	—
101	Dic₂₅	—	2	1	no	—
102	Dic₂₅	—	2	1	no	—
103	Dic₂₅	—	2	1	no	—
104	Dic₂₅	—	2	1	no	—
105	Dic₂₅	—	2	1	no	—
106	Dic₂₅	—	2	1	no	—
107	Dic₂₅	#41	1	2	yes	—
108	C₂×C₂²⋊D₅	—	5	1	no	—
109	C₂²×Dic₅	—	5	1	no	—
110	C₂×C₄×D₅	—	5	1	no	—
111	C₂×C₄×D₅	—	5	1	no	—
112	C₂×C₄×D₅	—	5	1	no	—
113	C₂×Dic₁₀	—	5	1	no	—
114	C₅×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
115	C₅×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
116	C₅×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
117	C₅×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
118	C₅×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
119	C₅×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
120	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
121	C₂²⋊Dic₅	—	5	1	no	—
122	C₂²⋊Dic₅	—	5	1	no	—
123	#685	—	5	1	no	—
124	#685	—	5	1	no	—
125	#685	—	5	1	no	—
126	C₄⋊Dic₅	—	5	1	no	—
127	C₄⋊Dic₅	—	5	1	no	—
128	C₄⋊Dic₅	—	5	1	no	—
129	C₄⋊Dic₅	—	5	1	no	—
130	#683	—	5	1	no	—
131	#683	—	5	1	no	—
132	#683	—	5	1	no	—
133	#683	—	5	1	no	—
134	#683	—	5	1	no	—
135	#683	—	5	1	no	—
136	#683	—	5	1	no	—
137	#683	—	5	1	no	—
138	#683	—	5	1	no	—
139	C₄×Dic₅	—	5	1	no	—
140	C₄×Dic₅	—	5	1	no	—
141	C₄×Dic₅	—	5	1	no	—
142	C₄×Dic₅	—	5	1	no	—
143	#552	—	25	1	no	Sylow
144	C₂×C₂₅	—	4	1	no	—
145	C₂×C₂₅	C₂×#41	1	1	yes	characteristic
146	C₂×C₂₅	C₂×#41	1	1	yes	characteristic
147	C₂×C₂₅	2₋¹⁺⁴	1	1	yes	characteristic
148	D₂₅	—	2	2	no	—
149	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
150	C₂²×D₅	—	5	1	no	—
151	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
155	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
156	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
157	C₂²⋊D₅	—	10	2	no	—
158	C₂×Dic₅	—	5	1	no	—
159	C₂×Dic₅	—	5	1	no	—
160	C₂×Dic₅	—	5	1	no	—
161	C₂×Dic₅	—	5	1	no	—
162	C₂×Dic₅	—	5	1	no	—
163	C₂×Dic₅	—	5	1	no	—
164	C₂×Dic₅	—	5	1	no	—
165	C₂×Dic₅	—	10	2	no	—
166	C₄×D₅	—	10	2	no	—
167	C₄×D₅	—	10	2	no	—
168	C₄×D₅	—	10	2	no	—
169	Dic₁₀	—	10	2	no	—
170	#126	—	25	1	no	—
171	#126	D₂₅	1	1	yes	characteristic, complemented
172	#125	—	25	1	no	—
173	#125	—	25	1	no	—
174	#125	—	25	1	no	—
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	C₂₅	#552	1	1	yes	characteristic, complemented, Sylow
186	C₂²×C₅	—	2	1	no	—
187	C₂²×C₅	—	2	1	no	—
188	C₂²×C₅	—	2	1	no	—
189	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
190	C₂×D₅	—	5	2	no	—
191	C₂×D₅	—	10	1	no	—
192	C₂×D₅	—	10	1	no	—
193	C₄×C₅	—	2	1	no	—
194	C₄×C₅	—	2	1	no	—
195	C₄×C₅	—	2	1	no	—
196	C₄×C₅	—	2	1	no	—
197	C₄×C₅	—	2	1	no	—
198	C₄×C₅	—	2	1	no	—
199	Dic₅	—	5	2	no	—
200	Dic₅	—	10	1	no	—
201	Dic₅	—	10	1	no	—
202	Dic₅	—	10	1	no	—
203	Dic₅	—	10	1	no	—
204	Dic₅	—	10	1	no	—
205	Dic₅	—	10	1	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	—	25	1	no	—
223	C₄⋊C₄	—	25	1	no	—
224	C₄⋊C₄	—	25	1	no	—
225	C₄⋊C₄	C₂×D₂₅	1	1	yes	characteristic
226	C₄⋊C₄	C₂×D₂₅	1	1	yes	characteristic
227	C₄⋊C₄	C₂×D₂₅	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	C₂×D₂₅	1	1	yes	characteristic
234	C₂²⋊C₄	C₂×D₂₅	1	1	yes	characteristic
235	C₂²⋊C₄	C₂×D₂₅	1	1	yes	characteristic
236	C₄²	—	25	1	no	—
237	C₄²	—	25	1	no	—
238	C₄²	—	25	1	no	—
239	C₄²	—	25	1	no	—
240	C₄²	C₂×D₂₅	1	1	yes	characteristic
241	C₂×C₅	—	4	1	no	—
242	C₂×C₅	D₅×#41	1	1	yes	characteristic
243	C₂×C₅	D₅×#41	1	1	yes	characteristic
244	C₂×C₅	#4150	1	1	yes	characteristic
245	D₅	—	10	2	no	—
246	C₂³	—	25	1	no	—
247	C₂³	C₂²×D₂₅	1	1	yes	characteristic
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	—	25	1	no	—
257	C₂×C₄	—	50	2	no	—
258	C₂×C₄	—	50	2	no	—
259	C₂×C₄	—	50	2	no	—
260	C₂×C₄	—	50	2	no	—
261	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
262	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
263	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
264	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
265	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
266	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
267	C₅	#66042	1	1	yes	characteristic
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	25	2	no	—
272	C₂²	—	50	1	no	—
273	C₂²	—	50	1	no	—
274	C₂²	C₂³×D₂₅	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	—	25	2	no	—
282	C₄	—	50	1	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	—	4	1	no	—
289	C₂	—	50	2	no	—
290	C₂	#41×D₂₅	1	1	yes	characteristic
291	C₂	#41×D₂₅	1	1	yes	characteristic
292	C₂	#11716569	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×D₂₅
commutator, cosocle	C₂²×C₂₅	C₂⁴
Frattini, socle	C₂²×C₅	C₂³×D₅
Fitting	C₂₅×#126	C₂
prime radical	C₂×C₂²⋊D₂₅	C₂²
hypercenter	#126	D₂₅
hypocenter	C₂₅	#552
LCS	G ▷ C₂²×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂² ◁ #126

Complements

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

N	H ≅ G/N	type
1. C₂₅×#126	C₂	semidirect
2. #11716458	C₂	semidirect
3. #11716454	C₂	semidirect
4. C₄⋊C₄×D₂₅	C₂	semidirect
5. #11716452	C₂	semidirect
6. #11716451	C₂	semidirect
7. #11716451	C₂	semidirect
11. #11716441	C₂	semidirect
12. #11716440	C₂	semidirect
13. #11716439	C₂	semidirect
14. #11716434	C₂	semidirect
15. C₄×Dic₅₀	C₂	semidirect
171. #126	D₂₅	semidirect
185. C₂₅	#552	semidirect

Maximal subgroups

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

subgroup	index
C₂₅×#126	2
#11716458	2
#11716454	2
C₄⋊C₄×D₂₅	2
#11716452	2
#11716451	2
#11716445	2
#11716443	2
#11716441	2
#11716440	2
#11716439	2
#11716434	2
C₄×Dic₅₀	2
#66042	5