Subgroups — #13098983

Order 1600.

375 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₅×#123	C₂	1	1	yes	maximal, characteristic, complemented
2	#11716500	C₂	1	1	yes	maximal, characteristic, complemented
3	#11716499	C₂	1	1	yes	maximal, characteristic, complemented
4	#11716492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₂²⋊D₂₅	C₂	1	1	yes	maximal, characteristic, complemented
6	#11716460	C₂	1	1	yes	maximal, characteristic, complemented
7	#11716459	C₂	1	1	yes	maximal, characteristic, complemented
8	#11716456	C₂	1	1	yes	maximal, characteristic, complemented
9	#11716455	C₂	1	1	yes	maximal, characteristic, complemented
10	#11716447	C₂	1	1	yes	maximal, characteristic, complemented
11	#11716447	C₂	1	1	yes	maximal, characteristic, complemented
12	#11716446	C₂	1	1	yes	maximal, characteristic, complemented
13	#11716444	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²⋊C₄×D₂₅	C₂	1	1	yes	maximal, characteristic, complemented
15	#11716441	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂³×D₂₅	C₂²	1	1	yes	characteristic
17	C₂×D₄×C₂₅	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₂₅	C₂²	1	1	yes	characteristic, complemented
19	C₂×C₂²⋊D₂₅	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊D₂₅	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊D₂₅	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊D₂₅	C₂²	1	1	yes	characteristic, complemented
23	C₂×D₁₀₀	C₂²	1	1	yes	characteristic
24	C₂×D₁₀₀	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, complemented
28	C₂×Dic₅₀	C₂²	1	1	yes	characteristic, complemented
29	C₄⋊C₄×C₂₅	C₂²	1	1	yes	characteristic, complemented
30	C₄⋊C₄×C₂₅	C₂²	1	1	yes	characteristic, complemented
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, complemented
34	C₂²⋊Dic₂₅	C₂²	1	1	yes	characteristic
35	C₂²⋊Dic₂₅	C₂²	1	1	yes	characteristic
36	C₂²⋊Dic₂₅	C₂²	1	1	yes	characteristic, complemented
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	#87720	C₂²	1	1	yes	characteristic
43	#87720	C₂²	1	1	yes	characteristic, complemented
44	#87720	C₂²	1	1	yes	characteristic, complemented
45	C₄⋊Dic₂₅	C₂²	1	1	yes	characteristic, complemented
46	C₄⋊Dic₂₅	C₂²	1	1	yes	characteristic, complemented
47	#87718	C₂²	1	1	yes	characteristic, complemented
48	#87718	C₂²	1	1	yes	characteristic, complemented
49	C₄×Dic₂₅	C₂²	1	1	yes	characteristic, complemented
50	C₄×Dic₂₅	C₂²	1	1	yes	characteristic, complemented
51	#65991	—	5	1	no	maximal
52	C₂³×C₂₅	C₂³	1	1	yes	characteristic
53	C₂³×C₂₅	C₂³	1	1	yes	characteristic
54	C₂²×D₂₅	—	2	2	no	—
55	C₂²×D₂₅	—	4	1	no	—
56	C₂²×D₂₅	—	4	1	no	—
57	C₂²×D₂₅	C₂³	1	1	yes	characteristic
58	C₂²×D₂₅	C₂³	1	1	yes	characteristic
59	C₂²×D₂₅	C₂³	1	1	yes	characteristic
60	D₄×C₂₅	—	4	1	no	—
61	C₂×C₄×C₂₅	—	4	1	no	—
62	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
63	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
64	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
65	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
66	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
67	C₂²⋊D₂₅	—	2	2	no	—
68	C₂²⋊D₂₅	—	4	1	no	—
69	C₂²⋊D₂₅	—	4	1	no	—
70	C₂²⋊D₂₅	—	4	1	no	—
71	C₂×Dic₂₅	C₂³	1	1	yes	characteristic
72	C₂×Dic₂₅	C₂³	1	1	yes	characteristic
73	C₂×Dic₂₅	C₂³	1	1	yes	characteristic
74	C₂×Dic₂₅	C₂³	1	1	yes	characteristic
75	C₂×Dic₂₅	C₂³	1	1	yes	characteristic
76	D₁₀₀	—	2	2	no	—
77	D₁₀₀	—	4	1	no	—
78	C₄×D₂₅	—	4	1	no	—
79	C₄×D₂₅	—	4	1	no	—
80	C₄×D₂₅	—	4	1	no	—
81	Dic₅₀	—	4	1	no	—
82	C₅×#123	D₅	1	1	yes	characteristic
83	#4084	—	5	1	no	—
84	#4083	—	5	1	no	—
85	#4076	—	5	1	no	—
86	C₄×C₂²⋊D₅	—	5	1	no	—
87	#4044	—	5	1	no	—
88	#4043	—	5	1	no	—
89	#4040	—	5	1	no	—
90	#4039	—	5	1	no	—
91	#4031	—	5	1	no	—
92	#4031	—	5	1	no	—
93	#4030	—	5	1	no	—
94	#4028	—	5	1	no	—
95	D₅×C₂²⋊C₄	—	5	1	no	—
96	#4025	—	5	1	no	—
97	C₂²×C₂₅	—	2	1	no	—
98	C₂²×C₂₅	—	2	1	no	—
99	C₂²×C₂₅	—	2	1	no	—
100	C₂²×C₂₅	—	2	1	no	—
101	C₂²×C₂₅	—	2	1	no	—
102	C₂²×C₂₅	—	2	1	no	—
103	C₂²×C₂₅	C₂⁴	1	1	yes	characteristic
104	C₂×D₂₅	—	2	1	no	—
105	C₂×D₂₅	—	2	1	no	—
106	C₂×D₂₅	—	2	1	no	—
107	C₂×D₂₅	—	2	1	no	—
108	C₂×D₂₅	—	2	1	no	—
109	C₂×D₂₅	—	2	1	no	—
110	C₂×D₂₅	—	2	1	no	—
111	C₂×D₂₅	—	2	1	no	—
112	C₂×D₂₅	—	4	4	no	—
113	C₂×D₂₅	#41	1	2	yes	—
114	C₄×C₂₅	—	2	1	no	—
115	C₄×C₂₅	—	2	1	no	—
116	C₄×C₂₅	—	2	1	no	—
117	C₄×C₂₅	—	2	1	no	—
118	C₄×C₂₅	—	2	1	no	—
119	Dic₂₅	—	2	1	no	—
120	Dic₂₅	—	2	1	no	—
121	Dic₂₅	—	2	1	no	—
122	Dic₂₅	—	2	1	no	—
123	Dic₂₅	—	2	1	no	—
124	C₂³×D₅	—	5	1	no	—
125	C₂×C₅×D₄	C₂×D₅	1	1	yes	characteristic
126	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
127	C₂×C₂²⋊D₅	—	5	1	no	—
128	C₂×C₂²⋊D₅	—	5	1	no	—
129	C₂×C₂²⋊D₅	—	5	1	no	—
130	C₂×C₂²⋊D₅	—	5	1	no	—
131	C₂×D₂₀	—	5	1	no	—
132	C₂×D₂₀	—	5	1	no	—
133	C₂×C₄×D₅	—	5	1	no	—
134	C₂×C₄×D₅	—	5	1	no	—
135	C₂×C₄×D₅	—	5	1	no	—
136	C₂×Dic₁₀	—	5	1	no	—
137	C₅×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
138	C₅×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
139	C₅×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
140	C₅×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
141	C₅×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
142	C₂²⋊Dic₅	—	5	1	no	—
143	C₂²⋊Dic₅	—	5	1	no	—
144	C₂²⋊Dic₅	—	5	1	no	—
145	#685	—	5	1	no	—
146	#685	—	5	1	no	—
147	#685	—	5	1	no	—
148	#685	—	5	1	no	—
149	#685	—	5	1	no	—
150	#685	—	5	1	no	—
151	#685	—	5	1	no	—
152	#685	—	5	1	no	—
153	C₄⋊Dic₅	—	5	1	no	—
154	C₄⋊Dic₅	—	5	1	no	—
155	#683	—	5	1	no	—
156	#683	—	5	1	no	—
157	C₄×Dic₅	—	5	1	no	—
158	C₄×Dic₅	—	5	1	no	—
159	#538	—	25	1	no	Sylow
160	C₂×C₂₅	—	4	1	no	—
161	C₂×C₂₅	—	4	1	no	—
162	C₂×C₂₅	C₂×#41	1	1	yes	characteristic
163	C₂×C₂₅	H(5,𝔽₂)	1	1	yes	characteristic
164	C₂×C₂₅	H(5,𝔽₂)	1	1	yes	characteristic
165	D₂₅	—	2	2	no	—
166	D₂₅	—	4	1	no	—
167	D₂₅	—	4	1	no	—
168	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
169	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
170	C₂²×D₅	—	5	1	no	—
171	C₂²×D₅	—	5	1	no	—
172	C₂²×D₅	—	5	1	no	—
173	C₂²×D₅	—	10	2	no	—
174	C₂²×D₅	—	20	1	no	—
175	C₂²×D₅	—	20	1	no	—
176	C₅×D₄	—	4	1	no	—
177	C₂×C₄×C₅	—	4	1	no	—
178	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
179	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
182	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
183	C₂²⋊D₅	—	10	2	no	—
184	C₂²⋊D₅	—	20	1	no	—
185	C₂²⋊D₅	—	20	1	no	—
186	C₂²⋊D₅	—	20	1	no	—
187	C₂×Dic₅	—	5	1	no	—
188	C₂×Dic₅	—	5	1	no	—
189	C₂×Dic₅	—	5	1	no	—
190	C₂×Dic₅	—	5	1	no	—
191	C₂×Dic₅	—	5	1	no	—
192	D₂₀	—	10	2	no	—
193	D₂₀	—	20	1	no	—
194	C₄×D₅	—	20	1	no	—
195	C₄×D₅	—	20	1	no	—
196	C₄×D₅	—	20	1	no	—
197	Dic₁₀	—	20	1	no	—
198	#126	—	25	1	no	—
199	#126	—	25	1	no	—
200	#124	—	25	1	no	—
201	#124	—	25	1	no	—
202	#123	—	25	1	no	—
203	#123	D₂₅	1	1	yes	characteristic, complemented
204	C₂²⋊Q₈	—	25	1	no	—
205	#121	—	25	1	no	—
206	#121	—	25	1	no	—
207	#121	—	25	1	no	—
208	#120	—	25	1	no	—
209	#120	—	25	1	no	—
210	C₄×D₄	—	25	1	no	—
211	C₄×D₄	—	25	1	no	—
212	C₂×C₂²⋊C₄	—	25	1	no	—
213	C₂₅	#538	1	1	yes	characteristic, complemented, Sylow
214	C₂²×C₅	—	2	1	no	—
215	C₂²×C₅	—	2	1	no	—
216	C₂²×C₅	—	2	1	no	—
217	C₂²×C₅	—	2	1	no	—
218	C₂²×C₅	—	2	1	no	—
219	C₂²×C₅	—	2	1	no	—
220	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
221	C₂×D₅	—	5	2	no	—
222	C₂×D₅	—	10	1	no	—
223	C₂×D₅	—	10	1	no	—
224	C₂×D₅	—	10	1	no	—
225	C₂×D₅	—	10	1	no	—
226	C₂×D₅	—	10	1	no	—
227	C₂×D₅	—	10	1	no	—
228	C₂×D₅	—	10	1	no	—
229	C₂×D₅	—	10	1	no	—
230	C₂×D₅	—	20	4	no	—
231	C₄×C₅	—	2	1	no	—
232	C₄×C₅	—	2	1	no	—
233	C₄×C₅	—	2	1	no	—
234	C₄×C₅	—	2	1	no	—
235	C₄×C₅	—	2	1	no	—
236	Dic₅	—	10	1	no	—
237	Dic₅	—	10	1	no	—
238	Dic₅	—	10	1	no	—
239	Dic₅	—	10	1	no	—
240	Dic₅	—	10	1	no	—
241	C₂⁴	—	25	1	no	—
242	C₂×Q₈	—	25	1	no	—
243	C₂×D₄	—	25	1	no	—
244	C₂×D₄	—	25	1	no	—
245	C₂×D₄	—	25	1	no	—
246	C₂×D₄	—	25	1	no	—
247	C₂×D₄	—	25	1	no	—
248	C₂×D₄	—	25	1	no	—
249	C₂×D₄	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	C₂×D₂₅	1	1	yes	characteristic, complemented
254	C₄⋊C₄	—	25	1	no	—
255	C₄⋊C₄	—	25	1	no	—
256	C₄⋊C₄	—	25	1	no	—
257	C₄⋊C₄	—	25	1	no	—
258	C₄⋊C₄	C₂×D₂₅	1	1	yes	characteristic, complemented
259	C₄⋊C₄	C₂×D₂₅	1	1	yes	characteristic, complemented
260	C₂²⋊C₄	—	25	1	no	—
261	C₂²⋊C₄	—	25	1	no	—
262	C₂²⋊C₄	—	25	1	no	—
263	C₂²⋊C₄	—	25	1	no	—
264	C₂²⋊C₄	—	25	1	no	—
265	C₂²⋊C₄	—	25	1	no	—
266	C₂²⋊C₄	—	25	1	no	—
267	C₂²⋊C₄	—	25	1	no	—
268	C₂²⋊C₄	—	25	1	no	—
269	C₂²⋊C₄	—	25	1	no	—
270	C₂²⋊C₄	—	25	1	no	—
271	C₂²⋊C₄	C₂×D₂₅	1	1	yes	characteristic
272	C₂²⋊C₄	C₂×D₂₅	1	1	yes	characteristic
273	C₂²⋊C₄	C₂×D₂₅	1	1	yes	characteristic, complemented
274	C₄²	—	25	1	no	—
275	C₄²	—	25	1	no	—
276	C₂×C₅	—	4	1	no	—
277	C₂×C₅	—	4	1	no	—
278	C₂×C₅	#4144	1	1	yes	characteristic
279	C₂×C₅	D₅×#41	1	1	yes	characteristic
280	C₂×C₅	#4149	1	1	yes	characteristic
281	D₅	—	10	2	no	—
282	D₅	—	20	1	no	—
283	D₅	—	20	1	no	—
284	C₂³	—	25	1	no	—
285	C₂³	—	25	1	no	—
286	C₂³	—	25	1	no	—
287	C₂³	—	50	2	no	—
288	C₂³	—	100	1	no	—
289	C₂³	—	100	1	no	—
290	C₂³	C₂²×D₂₅	1	1	yes	characteristic
291	C₂³	C₂²×D₂₅	1	1	yes	characteristic
292	Q₈	—	100	1	no	—
293	D₄	—	4	1	no	—
294	D₄	—	50	2	no	—
295	D₄	—	50	2	no	—
296	D₄	—	100	1	no	—
297	D₄	—	100	1	no	—
298	D₄	—	100	1	no	—
299	D₄	—	100	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	25	1	no	—
302	C₂×C₄	—	25	1	no	—
303	C₂×C₄	—	25	1	no	—
304	C₂×C₄	—	25	1	no	—
305	C₂×C₄	—	25	1	no	—
306	C₂×C₄	—	100	1	no	—
307	C₂×C₄	—	100	1	no	—
308	C₂×C₄	—	100	1	no	—
309	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
310	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
311	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
312	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
313	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
314	C₅	#65991	1	1	yes	characteristic
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	25	2	no	—
322	C₂²	—	50	1	no	—
323	C₂²	—	50	1	no	—
324	C₂²	—	50	1	no	—
325	C₂²	—	50	1	no	—
326	C₂²	—	50	1	no	—
327	C₂²	—	50	1	no	—
328	C₂²	—	50	1	no	—
329	C₂²	—	50	1	no	—
330	C₂²	—	100	4	no	—
331	C₂²	C₂³×D₂₅	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	50	1	no	—
338	C₄	—	50	1	no	—
339	C₄	—	50	1	no	—
340	C₄	—	50	1	no	—
341	C₄	—	50	1	no	—
342	C₂	—	4	1	no	—
343	C₂	—	4	1	no	—
344	C₂	—	50	2	no	—
345	C₂	—	100	1	no	—
346	C₂	—	100	1	no	—
347	C₂	#11716563	1	1	yes	characteristic
348	C₂	#41×D₂₅	1	1	yes	characteristic
349	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×#123	C₂
hypercenter	#123	D₂₅
hypocenter	C₂₅	#538
LCS	G ▷ C₂²×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂² ◁ #123

Complements

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

N	H ≅ G/N	type
1. C₂₅×#123	C₂	semidirect
2. #11716500	C₂	semidirect
3. #11716499	C₂	semidirect
4. #11716492	C₂	semidirect
5. C₄×C₂²⋊D₂₅	C₂	semidirect
6. #11716460	C₂	semidirect
7. #11716459	C₂	semidirect
8. #11716456	C₂	semidirect
9. #11716455	C₂	semidirect
10. #11716447	C₂	semidirect
11. #11716447	C₂	semidirect
12. #11716446	C₂	semidirect
13. #11716444	C₂	semidirect
14. C₂²⋊C₄×D₂₅	C₂	semidirect
15. #11716441	C₂	semidirect
18. C₂²×C₄×C₂₅	C₂²	semidirect
22. C₂×C₂²⋊D₂₅	C₂²	semidirect
27. C₂×C₄×D₂₅	C₂²	semidirect
28. C₂×Dic₅₀	C₂²	semidirect
29. C₄⋊C₄×C₂₅	C₂²	semidirect
30. C₄⋊C₄×C₂₅	C₂²	semidirect
33. C₂²⋊C₄×C₂₅	C₂²	semidirect
36. C₂²⋊Dic₂₅	C₂²	semidirect
43. #87720	C₂²	semidirect
44. #87720	C₂²	semidirect
45. C₄⋊Dic₂₅	C₂²	semidirect
46. C₄⋊Dic₂₅	C₂²	semidirect
47. #87718	C₂²	semidirect
48. #87718	C₂²	semidirect
49. C₄×Dic₂₅	C₂²	semidirect
50. C₄×Dic₂₅	C₂²	semidirect
203. #123	D₂₅	semidirect
213. C₂₅	#538	semidirect
253. C₂²×C₄	C₂×D₂₅	semidirect
258. C₄⋊C₄	C₂×D₂₅	semidirect
259. C₄⋊C₄	C₂×D₂₅	semidirect
273. C₂²⋊C₄	C₂×D₂₅	semidirect

Maximal subgroups

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

subgroup	index
C₂₅×#123	2
#11716500	2
#11716499	2
#11716492	2
C₄×C₂²⋊D₂₅	2
#11716460	2
#11716459	2
#11716456	2
#11716455	2
#11716447	2
#11716446	2
#11716444	2
C₂²⋊C₄×D₂₅	2
#11716441	2
#65991	5