Subgroups — #13098982

Order 1600.

438 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#87746	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#87744	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₅×#123	C₂	1	1	yes	maximal, characteristic, complemented
4	#11716501	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#87718	C₂	1	1	yes	maximal, characteristic, complemented
6	#11716458	C₂	1	1	yes	maximal, characteristic, complemented
7	#11716456	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄⋊C₄×D₂₅	C₂	1	1	yes	maximal, characteristic, complemented
9	#11716450	C₂	1	1	yes	maximal, characteristic, complemented
10	#11716448	C₂	1	1	yes	maximal, characteristic, complemented
11	#11716446	C₂	1	1	yes	maximal, characteristic, complemented
12	#11716445	C₂	1	1	yes	maximal, characteristic, complemented
13	#11716443	C₂	1	1	yes	maximal, characteristic, complemented
14	#11716443	C₂	1	1	yes	maximal, characteristic, complemented
15	#11716440	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₄×C₂₅	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₂₅	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊D₂₅	C₂²	1	1	yes	characteristic
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	C₂²×Dic₂₅	C₂²	1	1	yes	characteristic
23	C₂²×Dic₂₅	C₂²	1	1	yes	characteristic
24	C₂²×Dic₂₅	C₂²	1	1	yes	characteristic
25	#87746	—	2	4	no	—
26	#87744	—	2	4	no	—
27	C₂×D₁₀₀	C₂²	1	1	yes	characteristic
28	C₂×C₄×D₂₅	C₂²	1	1	yes	characteristic
29	C₂×C₄×D₂₅	C₂²	1	1	yes	characteristic
30	C₂×C₄×D₂₅	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₂₅	C₂²	1	1	yes	characteristic
32	C₂×Dic₅₀	C₂²	1	1	yes	characteristic
33	C₂×Dic₅₀	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×C₂₅	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×C₂₅	C₂²	1	1	yes	characteristic
36	C₂²⋊C₄×C₂₅	C₂²	1	1	yes	characteristic
37	C₂²⋊C₄×C₂₅	C₂²	1	1	yes	characteristic
38	C₂²⋊C₄×C₂₅	C₂²	1	1	yes	characteristic
39	C₂²⋊Dic₂₅	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
44	C₄⋊Dic₂₅	C₂²	1	1	yes	characteristic
45	C₄⋊Dic₂₅	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	#87718	C₂²	1	1	yes	characteristic
50	#87718	C₂²	1	1	yes	characteristic
51	#87718	C₂²	1	1	yes	characteristic
52	C₄×Dic₂₅	C₂²	1	1	yes	characteristic
53	#65989	—	5	1	no	maximal
54	C₂³×C₂₅	C₂³	1	1	yes	characteristic
55	C₂³×C₂₅	C₂³	1	1	yes	characteristic
56	C₂²×D₂₅	C₂³	1	1	yes	characteristic
57	C₂²×D₂₅	C₂³	1	1	yes	characteristic
58	D₄×C₂₅	—	2	2	no	—
59	C₂×C₄×C₂₅	—	2	2	no	—
60	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
61	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
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	C₂²⋊D₂₅	—	2	2	no	—
66	C₂²⋊D₂₅	—	2	2	no	—
67	C₂²⋊D₂₅	—	2	2	no	—
68	C₂²⋊D₂₅	D₄	1	4	yes	complemented
69	C₂×Dic₂₅	—	2	2	no	—
70	C₂×Dic₂₅	—	2	2	no	—
71	C₂×Dic₂₅	—	2	2	no	—
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	C₂×Dic₂₅	C₂³	1	1	yes	characteristic
77	C₂×Dic₂₅	C₂³	1	1	yes	characteristic
78	D₁₀₀	—	2	2	no	—
79	C₄×D₂₅	—	2	2	no	—
80	C₄×D₂₅	—	2	2	no	—
81	C₄×D₂₅	—	2	2	no	—
82	C₄×D₂₅	—	2	2	no	—
83	Dic₅₀	—	2	2	no	—
84	Dic₅₀	—	2	2	no	—
85	C₂×#711	—	5	1	no	—
86	C₂×#709	—	5	1	no	—
87	C₅×#123	D₅	1	1	yes	characteristic
88	#4085	—	5	1	no	—
89	C₂×#683	—	5	1	no	—
90	#4042	—	5	1	no	—
91	#4040	—	5	1	no	—
92	D₅×C₄⋊C₄	—	5	1	no	—
93	#4034	—	5	1	no	—
94	#4032	—	5	1	no	—
95	#4030	—	5	1	no	—
96	#4029	—	5	1	no	—
97	#4027	—	5	1	no	—
98	#4027	—	5	1	no	—
99	#4024	—	5	1	no	—
100	C₂²×C₂₅	—	2	1	no	—
101	C₂²×C₂₅	—	2	1	no	—
102	C₂²×C₂₅	—	2	1	no	—
103	C₂²×C₂₅	—	2	1	no	—
104	C₂²×C₂₅	—	2	1	no	—
105	C₂²×C₂₅	C₂×D₄	1	2	yes	—
106	C₂²×C₂₅	C₂⁴	1	1	yes	characteristic
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	C₂×D₄	1	2	yes	—
113	C₄×C₂₅	—	2	1	no	—
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	Dic₂₅	—	2	1	no	—
119	Dic₂₅	—	2	1	no	—
120	Dic₂₅	—	2	1	no	—
121	Dic₂₅	—	2	1	no	—
122	Dic₂₅	C₂×D₄	1	2	yes	—
123	Dic₂₅	#41	1	2	yes	complemented
124	C₂×C₅×D₄	C₂×D₅	1	1	yes	characteristic
125	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
126	C₂×C₂²⋊D₅	—	5	1	no	—
127	C₂×C₂²⋊D₅	—	5	1	no	—
128	C₂×C₂²⋊D₅	—	5	1	no	—
129	C₂×C₂²⋊D₅	—	5	1	no	—
130	C₂²×Dic₅	—	5	1	no	—
131	C₂²×Dic₅	—	5	1	no	—
132	C₂²×Dic₅	—	5	1	no	—
133	#711	—	10	4	no	—
134	#709	—	10	4	no	—
135	C₂×D₂₀	—	5	1	no	—
136	C₂×C₄×D₅	—	5	1	no	—
137	C₂×C₄×D₅	—	5	1	no	—
138	C₂×C₄×D₅	—	5	1	no	—
139	C₂×C₄×D₅	—	5	1	no	—
140	C₂×Dic₁₀	—	5	1	no	—
141	C₂×Dic₁₀	—	5	1	no	—
142	C₅×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
143	C₅×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
144	C₅×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
145	C₅×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
146	C₅×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
147	C₂²⋊Dic₅	—	5	1	no	—
148	#685	—	5	1	no	—
149	#685	—	5	1	no	—
150	#685	—	5	1	no	—
151	#685	—	5	1	no	—
152	C₄⋊Dic₅	—	5	1	no	—
153	C₄⋊Dic₅	—	5	1	no	—
154	#683	—	5	1	no	—
155	#683	—	5	1	no	—
156	#683	—	5	1	no	—
157	#683	—	5	1	no	—
158	#683	—	5	1	no	—
159	#683	—	5	1	no	—
160	C₄×Dic₅	—	5	1	no	—
161	#547	—	25	1	no	Sylow
162	C₂×C₂₅	—	2	2	no	—
163	C₂×C₂₅	—	4	1	no	—
164	C₂×C₂₅	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₂₅	C₂×#41	1	1	yes	characteristic
166	C₂×C₂₅	2₋¹⁺⁴	1	1	yes	characteristic
167	D₂₅	—	2	2	no	—
168	D₂₅	—	4	1	no	—
169	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
170	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
171	C₂²×D₅	—	5	1	no	—
172	C₂²×D₅	—	5	1	no	—
173	C₅×D₄	—	2	2	no	—
174	C₂×C₄×C₅	—	2	2	no	—
175	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
176	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
178	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
179	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
180	C₂²⋊D₅	—	5	4	no	—
181	C₂²⋊D₅	—	10	2	no	—
182	C₂²⋊D₅	—	10	2	no	—
183	C₂²⋊D₅	—	10	2	no	—
184	C₂×Dic₅	—	5	1	no	—
185	C₂×Dic₅	—	5	1	no	—
186	C₂×Dic₅	—	5	1	no	—
187	C₂×Dic₅	—	5	1	no	—
188	C₂×Dic₅	—	5	1	no	—
189	C₂×Dic₅	—	5	1	no	—
190	C₂×Dic₅	—	10	2	no	—
191	C₂×Dic₅	—	10	2	no	—
192	C₂×Dic₅	—	10	2	no	—
193	D₂₀	—	10	2	no	—
194	C₄×D₅	—	10	2	no	—
195	C₄×D₅	—	10	2	no	—
196	C₄×D₅	—	10	2	no	—
197	C₄×D₅	—	10	2	no	—
198	Dic₁₀	—	10	2	no	—
199	Dic₁₀	—	10	2	no	—
200	C₂×#41	—	25	1	no	—
201	C₂×#41	—	25	1	no	—
202	C₄⋊Q₈	—	25	1	no	—
203	#123	—	25	1	no	—
204	#123	—	25	1	no	—
205	#123	—	25	1	no	—
206	#123	D₂₅	1	1	yes	characteristic, complemented
207	C₂²⋊Q₈	—	25	1	no	—
208	C₂²⋊Q₈	—	25	1	no	—
209	#121	—	25	1	no	—
210	#121	—	25	1	no	—
211	C₄×D₄	—	25	1	no	—
212	C₄×D₄	—	25	1	no	—
213	C₂×C₄⋊C₄	—	25	1	no	—
214	C₂×C₄⋊C₄	—	25	1	no	—
215	C₂₅	#547	1	1	yes	characteristic, complemented, Sylow
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	—	2	1	no	—
221	C₂²×C₅	D₄×D₅	1	2	yes	—
222	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
223	C₂×D₅	—	5	2	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	C₄×C₅	—	2	1	no	—
230	C₄×C₅	—	2	1	no	—
231	C₄×C₅	—	2	1	no	—
232	C₄×C₅	—	2	1	no	—
233	C₄×C₅	—	2	1	no	—
234	Dic₅	—	5	2	no	—
235	Dic₅	—	5	2	no	—
236	Dic₅	—	10	1	no	—
237	Dic₅	—	10	1	no	—
238	Dic₅	—	10	1	no	—
239	Dic₅	—	10	1	no	—
240	#41	—	50	4	no	—
241	#41	—	50	4	no	—
242	C₂×Q₈	—	25	1	no	—
243	C₂×Q₈	—	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	—	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	C₄⋊C₄	—	25	1	no	—
259	C₄⋊C₄	—	25	1	no	—
260	C₄⋊C₄	—	25	1	no	—
261	C₄⋊C₄	—	25	1	no	—
262	C₄⋊C₄	—	25	1	no	—
263	C₄⋊C₄	—	25	1	no	—
264	C₄⋊C₄	—	25	1	no	—
265	C₄⋊C₄	—	25	1	no	—
266	C₄⋊C₄	C₂×D₂₅	1	1	yes	characteristic
267	C₄⋊C₄	C₂×D₂₅	1	1	yes	characteristic
268	C₂²⋊C₄	—	25	1	no	—
269	C₂²⋊C₄	—	25	1	no	—
270	C₂²⋊C₄	—	25	1	no	—
271	C₂²⋊C₄	—	25	1	no	—
272	C₂²⋊C₄	—	25	1	no	—
273	C₂²⋊C₄	C₂×D₂₅	1	1	yes	characteristic
274	C₂²⋊C₄	C₂×D₂₅	1	1	yes	characteristic
275	C₂²⋊C₄	C₂×D₂₅	1	1	yes	characteristic
276	C₄²	—	25	1	no	—
277	C₂×C₅	—	2	2	no	—
278	C₂×C₅	—	4	1	no	—
279	C₂×C₅	C₂×D₄×D₅	1	1	yes	characteristic
280	C₂×C₅	D₅×#41	1	1	yes	characteristic
281	C₂×C₅	#4150	1	1	yes	characteristic
282	D₅	—	10	2	no	—
283	D₅	—	20	1	no	—
284	C₂³	—	25	1	no	—
285	C₂³	—	25	1	no	—
286	C₂³	C₂²×D₂₅	1	1	yes	characteristic
287	C₂³	C₂²×D₂₅	1	1	yes	characteristic
288	Q₈	—	50	2	no	—
289	Q₈	—	50	2	no	—
290	D₄	—	2	2	no	—
291	D₄	—	25	4	no	—
292	D₄	—	50	2	no	—
293	D₄	—	50	2	no	—
294	D₄	—	50	2	no	—
295	D₄	—	50	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	25	1	no	—
298	C₂×C₄	—	25	1	no	—
299	C₂×C₄	—	25	1	no	—
300	C₂×C₄	—	25	1	no	—
301	C₂×C₄	—	25	1	no	—
302	C₂×C₄	—	25	1	no	—
303	C₂×C₄	—	50	2	no	—
304	C₂×C₄	—	50	2	no	—
305	C₂×C₄	—	50	2	no	—
306	C₂×C₄	—	50	2	no	—
307	C₂×C₄	—	50	2	no	—
308	C₂×C₄	—	50	2	no	—
309	C₂×C₄	—	50	2	no	—
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	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
315	C₅	#65989	1	1	yes	characteristic
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	D₄×D₂₅	1	2	yes	—
328	C₂²	C₂³×D₂₅	1	1	yes	characteristic
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	25	2	no	—
335	C₄	—	25	2	no	—
336	C₄	—	50	1	no	—
337	C₄	—	50	1	no	—
338	C₄	—	50	1	no	—
339	C₄	—	50	1	no	—
340	C₂	—	2	2	no	—
341	C₂	—	4	1	no	—
342	C₂	—	50	2	no	—
343	C₂	—	100	1	no	—
344	C₂	C₂×D₄×D₂₅	1	1	yes	characteristic
345	C₂	#41×D₂₅	1	1	yes	characteristic
346	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×#123	C₂
hypercenter	#123	D₂₅
hypocenter	C₂₅	#547
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. C₂×#87746	C₂	semidirect
2. C₂×#87744	C₂	semidirect
3. C₂₅×#123	C₂	semidirect
4. #11716501	C₂	semidirect
5. C₂×#87718	C₂	semidirect
6. #11716458	C₂	semidirect
7. #11716456	C₂	semidirect
8. C₄⋊C₄×D₂₅	C₂	semidirect
9. #11716450	C₂	semidirect
10. #11716448	C₂	semidirect
11. #11716446	C₂	semidirect
12. #11716445	C₂	semidirect
13. #11716443	C₂	semidirect
14. #11716443	C₂	semidirect
15. #11716440	C₂	semidirect
68. C₂²⋊D₂₅	D₄	semidirect
123. Dic₂₅	#41	semidirect
206. #123	D₂₅	semidirect
215. C₂₅	#547	semidirect

Maximal subgroups

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

subgroup	index
C₂×#87746	2
C₂×#87744	2
C₂₅×#123	2
#11716501	2
C₂×#87718	2
#11716458	2
#11716456	2
C₄⋊C₄×D₂₅	2
#11716450	2
#11716448	2
#11716446	2
#11716445	2
#11716443	2
#11716440	2
#65989	5