Subgroups — #13099033

Order 1600.

357 conjugacy classes of subgroups.

The subgroup lattice has 330 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	#11716460	C₂	1	1	yes	maximal, characteristic, complemented
3	#11716457	C₂	1	1	yes	maximal, characteristic, complemented
4	#11716456	C₂	1	1	yes	maximal, characteristic, complemented
5	#11716455	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄⋊C₄×D₂₅	C₂	1	1	yes	maximal, characteristic, complemented
7	#11716451	C₂	1	1	yes	maximal, characteristic, complemented
8	#11716448	C₂	1	1	yes	maximal, characteristic, complemented
9	#11716446	C₂	1	1	yes	maximal, characteristic, complemented
10	#11716446	C₂	1	1	yes	maximal, characteristic, complemented
11	#11716446	C₂	1	1	yes	maximal, characteristic, complemented
12	#11716443	C₂	1	1	yes	maximal, characteristic, complemented
13	#11716443	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×D₁₀₀	C₂	1	1	yes	maximal, characteristic, complemented
15	#11716434	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₂²⋊D₂₅	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊D₂₅	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊D₂₅	C₂²	1	1	yes	characteristic
19	C₂²×Dic₂₅	C₂²	1	1	yes	characteristic
20	C₂×D₁₀₀	C₂²	1	1	yes	characteristic
21	C₂×D₁₀₀	C₂²	1	1	yes	characteristic
22	C₂×D₁₀₀	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₂₅	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₂₅	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₂₅	C₂²	1	1	yes	characteristic
26	C₂×C₄×D₂₅	C₂²	1	1	yes	characteristic
27	C₂×C₄×D₂₅	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×C₂₅	C₂²	1	1	yes	characteristic
29	C₄⋊C₄×C₂₅	C₂²	1	1	yes	characteristic
30	C₄⋊C₄×C₂₅	C₂²	1	1	yes	characteristic
31	C₂²⋊C₄×C₂₅	C₂²	1	1	yes	characteristic
32	C₂²⋊C₄×C₂₅	C₂²	1	1	yes	characteristic
33	C₂²⋊C₄×C₂₅	C₂²	1	1	yes	characteristic
34	C₄²×C₂₅	C₂²	1	1	yes	characteristic
35	#87720	C₂²	1	1	yes	characteristic
36	#87720	C₂²	1	1	yes	characteristic
37	#87720	C₂²	1	1	yes	characteristic
38	#87720	C₂²	1	1	yes	characteristic
39	#87720	C₂²	1	1	yes	characteristic
40	#87720	C₂²	1	1	yes	characteristic
41	#87720	C₂²	1	1	yes	characteristic
42	C₄⋊Dic₂₅	C₂²	1	1	yes	characteristic
43	C₄⋊Dic₂₅	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	#87718	C₂²	1	1	yes	characteristic
48	#87718	C₂²	1	1	yes	characteristic
49	C₄×Dic₂₅	C₂²	1	1	yes	characteristic
50	C₄×Dic₂₅	C₂²	1	1	yes	characteristic
51	#66043	—	5	1	no	maximal
52	C₂³×C₂₅	C₂³	1	1	yes	characteristic
53	C₂²×D₂₅	C₂³	1	1	yes	characteristic
54	C₂²×D₂₅	C₂³	1	1	yes	characteristic
55	C₂²×D₂₅	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	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
61	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
62	C₂²⋊D₂₅	—	2	2	no	—
63	C₂²⋊D₂₅	—	2	2	no	—
64	C₂²⋊D₂₅	—	4	1	no	—
65	C₂×Dic₂₅	—	4	1	no	—
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	C₂×Dic₂₅	C₂³	1	1	yes	characteristic
70	C₂×Dic₂₅	C₂³	1	1	yes	characteristic
71	D₁₀₀	—	2	2	no	—
72	D₁₀₀	—	2	2	no	—
73	D₁₀₀	—	4	1	no	—
74	C₄×D₂₅	—	2	2	no	—
75	C₄×D₂₅	—	2	2	no	—
76	C₄×D₂₅	—	4	1	no	—
77	C₄×D₂₅	—	4	1	no	—
78	C₄×D₂₅	—	4	1	no	—
79	C₅×#126	D₅	1	1	yes	characteristic
80	#4044	—	5	1	no	—
81	#4041	—	5	1	no	—
82	#4040	—	5	1	no	—
83	#4039	—	5	1	no	—
84	D₅×C₄⋊C₄	—	5	1	no	—
85	#4035	—	5	1	no	—
86	#4032	—	5	1	no	—
87	#4030	—	5	1	no	—
88	#4030	—	5	1	no	—
89	#4030	—	5	1	no	—
90	#4027	—	5	1	no	—
91	#4027	—	5	1	no	—
92	C₄×D₂₀	—	5	1	no	—
93	#4018	—	5	1	no	—
94	C₂²×C₂₅	—	2	1	no	—
95	C₂²×C₂₅	—	2	1	no	—
96	C₂²×C₂₅	—	2	1	no	—
97	C₂²×C₂₅	C₂⁴	1	1	yes	characteristic
98	C₂×D₂₅	—	2	1	no	—
99	C₂×D₂₅	—	2	1	no	—
100	C₂×D₂₅	—	2	1	no	—
101	C₂×D₂₅	—	2	1	no	—
102	C₂×D₂₅	—	2	1	no	—
103	C₂×D₂₅	—	2	1	no	—
104	C₂×D₂₅	—	2	1	no	—
105	C₂×D₂₅	—	2	1	no	—
106	C₂×D₂₅	#41	1	2	yes	—
107	C₄×C₂₅	—	2	1	no	—
108	C₄×C₂₅	—	2	1	no	—
109	C₄×C₂₅	—	2	1	no	—
110	C₄×C₂₅	—	2	1	no	—
111	C₄×C₂₅	—	2	1	no	—
112	C₄×C₂₅	—	2	1	no	—
113	Dic₂₅	—	2	1	no	—
114	Dic₂₅	—	2	1	no	—
115	Dic₂₅	—	2	1	no	—
116	Dic₂₅	—	2	1	no	—
117	Dic₂₅	#41	1	2	yes	—
118	C₂×C₂²⋊D₅	—	5	1	no	—
119	C₂×C₂²⋊D₅	—	5	1	no	—
120	C₂×C₂²⋊D₅	—	5	1	no	—
121	C₂²×Dic₅	—	5	1	no	—
122	C₂×D₂₀	—	5	1	no	—
123	C₂×D₂₀	—	5	1	no	—
124	C₂×D₂₀	—	5	1	no	—
125	C₂×C₄×D₅	—	5	1	no	—
126	C₂×C₄×D₅	—	5	1	no	—
127	C₂×C₄×D₅	—	5	1	no	—
128	C₂×C₄×D₅	—	5	1	no	—
129	C₂×C₄×D₅	—	5	1	no	—
130	C₅×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
131	C₅×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
132	C₅×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
133	C₅×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
134	C₅×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
135	C₅×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
136	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
137	#685	—	5	1	no	—
138	#685	—	5	1	no	—
139	#685	—	5	1	no	—
140	#685	—	5	1	no	—
141	#685	—	5	1	no	—
142	#685	—	5	1	no	—
143	#685	—	5	1	no	—
144	C₄⋊Dic₅	—	5	1	no	—
145	C₄⋊Dic₅	—	5	1	no	—
146	#683	—	5	1	no	—
147	#683	—	5	1	no	—
148	#683	—	5	1	no	—
149	#683	—	5	1	no	—
150	#683	—	5	1	no	—
151	C₄×Dic₅	—	5	1	no	—
152	C₄×Dic₅	—	5	1	no	—
153	#553	—	25	1	no	Sylow
154	C₂×C₂₅	—	4	1	no	—
155	C₂×C₂₅	C₂×#41	1	1	yes	characteristic
156	C₂×C₂₅	C₂×#41	1	1	yes	characteristic
157	C₂×C₂₅	H(5,𝔽₂)	1	1	yes	characteristic
158	D₂₅	—	2	2	no	—
159	D₂₅	—	4	1	no	—
160	D₂₅	—	4	1	no	—
161	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
162	C₂²×D₅	—	5	1	no	—
163	C₂²×D₅	—	5	1	no	—
164	C₂²×D₅	—	5	1	no	—
165	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
166	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
167	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
169	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
170	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
171	C₂²⋊D₅	—	10	2	no	—
172	C₂²⋊D₅	—	10	2	no	—
173	C₂²⋊D₅	—	20	1	no	—
174	C₂×Dic₅	—	5	1	no	—
175	C₂×Dic₅	—	5	1	no	—
176	C₂×Dic₅	—	5	1	no	—
177	C₂×Dic₅	—	5	1	no	—
178	C₂×Dic₅	—	5	1	no	—
179	C₂×Dic₅	—	20	1	no	—
180	D₂₀	—	10	2	no	—
181	D₂₀	—	10	2	no	—
182	D₂₀	—	20	1	no	—
183	C₄×D₅	—	10	2	no	—
184	C₄×D₅	—	10	2	no	—
185	C₄×D₅	—	20	1	no	—
186	C₄×D₅	—	20	1	no	—
187	C₄×D₅	—	20	1	no	—
188	#126	—	25	1	no	—
189	#126	D₂₅	1	1	yes	characteristic, complemented
190	#125	—	25	1	no	—
191	#123	—	25	1	no	—
192	#123	—	25	1	no	—
193	#121	—	25	1	no	—
194	#121	—	25	1	no	—
195	#121	—	25	1	no	—
196	#121	—	25	1	no	—
197	C₄×D₄	—	25	1	no	—
198	C₄×D₄	—	25	1	no	—
199	C₄×D₄	—	25	1	no	—
200	C₄×D₄	—	25	1	no	—
201	#117	—	25	1	no	—
202	C₂×C₄⋊C₄	—	25	1	no	—
203	C₂₅	#553	1	1	yes	characteristic, complemented, Sylow
204	C₂²×C₅	—	2	1	no	—
205	C₂²×C₅	—	2	1	no	—
206	C₂²×C₅	—	2	1	no	—
207	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
208	C₂×D₅	—	5	2	no	—
209	C₂×D₅	—	10	1	no	—
210	C₂×D₅	—	10	1	no	—
211	C₂×D₅	—	10	1	no	—
212	C₂×D₅	—	10	1	no	—
213	C₂×D₅	—	10	1	no	—
214	C₂×D₅	—	10	1	no	—
215	C₂×D₅	—	10	1	no	—
216	C₂×D₅	—	10	1	no	—
217	C₄×C₅	—	2	1	no	—
218	C₄×C₅	—	2	1	no	—
219	C₄×C₅	—	2	1	no	—
220	C₄×C₅	—	2	1	no	—
221	C₄×C₅	—	2	1	no	—
222	C₄×C₅	—	2	1	no	—
223	Dic₅	—	5	2	no	—
224	Dic₅	—	10	1	no	—
225	Dic₅	—	10	1	no	—
226	Dic₅	—	10	1	no	—
227	Dic₅	—	10	1	no	—
228	C₂×D₄	—	25	1	no	—
229	C₂×D₄	—	25	1	no	—
230	C₂×D₄	—	25	1	no	—
231	C₂×D₄	—	25	1	no	—
232	C₂×D₄	—	25	1	no	—
233	C₂×D₄	—	25	1	no	—
234	C₂²×C₄	—	25	1	no	—
235	C₂²×C₄	—	25	1	no	—
236	C₂²×C₄	—	25	1	no	—
237	C₂²×C₄	—	25	1	no	—
238	C₂²×C₄	—	25	1	no	—
239	C₂²×C₄	—	25	1	no	—
240	C₄⋊C₄	—	25	1	no	—
241	C₄⋊C₄	—	25	1	no	—
242	C₄⋊C₄	—	25	1	no	—
243	C₄⋊C₄	—	25	1	no	—
244	C₄⋊C₄	—	25	1	no	—
245	C₄⋊C₄	—	25	1	no	—
246	C₄⋊C₄	—	25	1	no	—
247	C₄⋊C₄	C₂×D₂₅	1	1	yes	characteristic
248	C₄⋊C₄	C₂×D₂₅	1	1	yes	characteristic
249	C₄⋊C₄	C₂×D₂₅	1	1	yes	characteristic
250	C₂²⋊C₄	—	25	1	no	—
251	C₂²⋊C₄	—	25	1	no	—
252	C₂²⋊C₄	—	25	1	no	—
253	C₂²⋊C₄	—	25	1	no	—
254	C₂²⋊C₄	—	25	1	no	—
255	C₂²⋊C₄	—	25	1	no	—
256	C₂²⋊C₄	—	25	1	no	—
257	C₂²⋊C₄	C₂×D₂₅	1	1	yes	characteristic
258	C₂²⋊C₄	C₂×D₂₅	1	1	yes	characteristic
259	C₂²⋊C₄	C₂×D₂₅	1	1	yes	characteristic
260	C₄²	—	25	1	no	—
261	C₄²	—	25	1	no	—
262	C₄²	C₂×D₂₅	1	1	yes	characteristic
263	C₂×C₅	—	4	1	no	—
264	C₂×C₅	D₅×#41	1	1	yes	characteristic
265	C₂×C₅	D₅×#41	1	1	yes	characteristic
266	C₂×C₅	#4149	1	1	yes	characteristic
267	D₅	—	10	2	no	—
268	D₅	—	20	1	no	—
269	D₅	—	20	1	no	—
270	C₂³	—	25	1	no	—
271	C₂³	—	25	1	no	—
272	C₂³	—	25	1	no	—
273	C₂³	C₂²×D₂₅	1	1	yes	characteristic
274	D₄	—	50	2	no	—
275	D₄	—	50	2	no	—
276	D₄	—	50	2	no	—
277	D₄	—	50	2	no	—
278	D₄	—	100	1	no	—
279	D₄	—	100	1	no	—
280	C₂×C₄	—	25	1	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	2	no	—
286	C₂×C₄	—	50	2	no	—
287	C₂×C₄	—	100	1	no	—
288	C₂×C₄	—	100	1	no	—
289	C₂×C₄	—	100	1	no	—
290	C₂×C₄	—	100	1	no	—
291	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
292	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
293	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
295	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
296	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
297	C₅	#66043	1	1	yes	characteristic
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	25	2	no	—
302	C₂²	—	50	1	no	—
303	C₂²	—	50	1	no	—
304	C₂²	—	50	1	no	—
305	C₂²	—	50	1	no	—
306	C₂²	—	50	1	no	—
307	C₂²	—	50	1	no	—
308	C₂²	—	50	1	no	—
309	C₂²	—	50	1	no	—
310	C₂²	C₂³×D₂₅	1	1	yes	characteristic
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	25	2	no	—
318	C₄	—	50	1	no	—
319	C₄	—	50	1	no	—
320	C₄	—	50	1	no	—
321	C₄	—	50	1	no	—
322	C₂	—	4	1	no	—
323	C₂	—	50	2	no	—
324	C₂	—	100	1	no	—
325	C₂	—	100	1	no	—
326	C₂	#41×D₂₅	1	1	yes	characteristic
327	C₂	#41×D₂₅	1	1	yes	characteristic
328	C₂	#11716568	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
hypercenter	#126	D₂₅
hypocenter	C₂₅	#553
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. #11716460	C₂	semidirect
3. #11716457	C₂	semidirect
4. #11716456	C₂	semidirect
5. #11716455	C₂	semidirect
6. C₄⋊C₄×D₂₅	C₂	semidirect
7. #11716451	C₂	semidirect
8. #11716448	C₂	semidirect
9. #11716446	C₂	semidirect
10. #11716446	C₂	semidirect
11. #11716446	C₂	semidirect
12. #11716443	C₂	semidirect
13. #11716443	C₂	semidirect
14. C₄×D₁₀₀	C₂	semidirect
15. #11716434	C₂	semidirect
189. #126	D₂₅	semidirect
203. C₂₅	#553	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
#11716460	2
#11716457	2
#11716456	2
#11716455	2
C₄⋊C₄×D₂₅	2
#11716451	2
#11716448	2
#11716446	2
#11716443	2
C₄×D₁₀₀	2
#11716434	2
#66043	5