Subgroups — #13098873

Order 1600.

495 conjugacy classes of subgroups.

The subgroup lattice has 369 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#87744	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×D₂₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₄×C₂₅	C₂	1	1	yes	maximal, characteristic, complemented
4	#11716501	C₂	1	1	yes	maximal, characteristic, complemented
5	#11716499	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×C₂²⋊D₂₅	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₂²⋊D₂₅	C₂	1	1	yes	maximal, characteristic, complemented
8	#11716458	C₂	1	1	yes	maximal, characteristic, complemented
9	#11716456	C₂	1	1	yes	maximal, characteristic, complemented
10	#11716446	C₂	1	1	yes	maximal, characteristic, complemented
11	#11716445	C₂	1	1	yes	maximal, characteristic, complemented
12	#11716444	C₂	1	1	yes	maximal, characteristic, complemented
13	#11716440	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	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
19	C₂²×C₄×C₂₅	C₂²	1	1	yes	characteristic, complemented
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
23	C₂×C₂²⋊D₂₅	C₂²	1	1	yes	characteristic, complemented
24	C₂²×Dic₂₅	C₂²	1	1	yes	characteristic
25	#87744	—	2	4	no	—
26	C₂×D₁₀₀	C₂²	1	1	yes	characteristic
27	C₂×D₁₀₀	C₂²	1	1	yes	characteristic
28	C₂×C₄×D₂₅	—	2	4	no	—
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×C₄×D₂₅	C₂²	1	1	yes	characteristic
33	C₂×C₄×D₂₅	C₂²	1	1	yes	characteristic, complemented
34	C₂×Dic₅₀	C₂²	1	1	yes	characteristic, complemented
35	C₄⋊C₄×C₂₅	C₂²	1	1	yes	characteristic, complemented
36	C₂²⋊C₄×C₂₅	C₂²	1	1	yes	characteristic
37	C₂²⋊C₄×C₂₅	C₂²	1	1	yes	characteristic, complemented
38	C₄²×C₂₅	C₂²	1	1	yes	characteristic, complemented
39	C₂²⋊Dic₂₅	C₂²	1	1	yes	characteristic
40	C₂²⋊Dic₂₅	C₂²	1	1	yes	characteristic, complemented
41	#87720	C₂²	1	1	yes	characteristic
42	#87720	C₂²	1	1	yes	characteristic
43	#87720	C₂²	1	1	yes	characteristic
44	#87720	C₂²	1	1	yes	characteristic
45	#87720	C₂²	1	1	yes	characteristic, complemented
46	#87720	C₂²	1	1	yes	characteristic, complemented
47	C₄⋊Dic₂₅	C₂²	1	1	yes	characteristic, complemented
48	#87718	C₂²	1	1	yes	characteristic, complemented
49	#87718	C₂²	1	1	yes	characteristic, complemented
50	#87718	C₂²	1	1	yes	characteristic, complemented
51	#87718	C₂²	1	1	yes	characteristic, complemented
52	C₄×Dic₂₅	C₂²	1	1	yes	characteristic, complemented
53	#65881	—	5	1	no	maximal
54	C₂³×C₂₅	C₂³	1	1	yes	characteristic
55	C₂³×C₂₅	C₂³	1	1	yes	characteristic
56	C₂²×D₂₅	—	2	2	no	—
57	C₂²×D₂₅	—	2	2	no	—
58	C₂²×D₂₅	—	2	2	no	—
59	C₂²×D₂₅	C₂³	1	1	yes	characteristic
60	C₂²×D₂₅	C₂³	1	1	yes	characteristic
61	C₂²×D₂₅	C₂³	1	1	yes	characteristic
62	D₄×C₂₅	—	2	2	no	—
63	C₂×C₄×C₂₅	—	2	2	no	—
64	C₂×C₄×C₂₅	—	2	2	no	—
65	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
66	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
67	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
68	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
69	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
70	C₂²⋊D₂₅	—	2	2	no	—
71	C₂²⋊D₂₅	—	2	2	no	—
72	C₂²⋊D₂₅	—	2	2	no	—
73	C₂²⋊D₂₅	—	2	2	no	—
74	C₂×Dic₂₅	—	2	2	no	—
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	C₂×Dic₂₅	C₂³	1	1	yes	characteristic
79	C₂×Dic₂₅	C₂³	1	1	yes	characteristic
80	D₁₀₀	—	2	2	no	—
81	D₁₀₀	—	2	2	no	—
82	C₄×D₂₅	—	2	2	no	—
83	C₄×D₂₅	—	2	2	no	—
84	C₄×D₂₅	—	2	2	no	—
85	C₄×D₂₅	—	2	2	no	—
86	C₄×D₂₅	D₄	1	4	yes	complemented
87	Dic₅₀	—	2	2	no	—
88	C₂×#709	—	5	1	no	—
89	C₂²×C₄×D₅	—	5	1	no	—
90	C₄×C₅×D₄	D₅	1	1	yes	characteristic
91	#4085	—	5	1	no	—
92	#4083	—	5	1	no	—
93	C₄×C₂²⋊D₅	—	5	1	no	—
94	C₄×C₂²⋊D₅	—	5	1	no	—
95	#4042	—	5	1	no	—
96	#4040	—	5	1	no	—
97	#4030	—	5	1	no	—
98	#4029	—	5	1	no	—
99	#4028	—	5	1	no	—
100	#4024	—	5	1	no	—
101	C₄×D₂₀	—	5	1	no	—
102	#4018	—	5	1	no	—
103	C₂²×C₂₅	—	2	1	no	—
104	C₂²×C₂₅	—	2	1	no	—
105	C₂²×C₂₅	—	2	1	no	—
106	C₂²×C₂₅	—	2	1	no	—
107	C₂²×C₂₅	—	2	1	no	—
108	C₂²×C₂₅	#41	1	2	yes	complemented
109	C₂²×C₂₅	C₂⁴	1	1	yes	characteristic
110	C₂×D₂₅	—	2	8	no	—
111	C₂×D₂₅	—	2	1	no	—
112	C₂×D₂₅	—	2	1	no	—
113	C₂×D₂₅	—	2	1	no	—
114	C₂×D₂₅	—	2	1	no	—
115	C₂×D₂₅	—	2	1	no	—
116	C₂×D₂₅	—	2	1	no	—
117	C₂×D₂₅	—	2	1	no	—
118	C₂×D₂₅	C₂×D₄	1	2	yes	—
119	C₂×D₂₅	#41	1	2	yes	complemented
120	C₄×C₂₅	—	2	1	no	—
121	C₄×C₂₅	—	2	1	no	—
122	C₄×C₂₅	—	2	1	no	—
123	C₄×C₂₅	—	2	1	no	—
124	C₄×C₂₅	C₂×D₄	1	2	yes	—
125	Dic₂₅	—	2	1	no	—
126	Dic₂₅	—	2	1	no	—
127	Dic₂₅	—	2	1	no	—
128	Dic₂₅	—	2	1	no	—
129	Dic₂₅	C₂×D₄	1	2	yes	—
130	C₂³×D₅	—	5	1	no	—
131	C₂×C₅×D₄	C₂×D₅	1	1	yes	characteristic
132	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
133	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
134	C₂×C₂²⋊D₅	—	5	1	no	—
135	C₂×C₂²⋊D₅	—	5	1	no	—
136	C₂×C₂²⋊D₅	—	5	1	no	—
137	C₂×C₂²⋊D₅	—	5	1	no	—
138	C₂²×Dic₅	—	5	1	no	—
139	#709	—	10	4	no	—
140	C₂×D₂₀	—	5	1	no	—
141	C₂×D₂₀	—	5	1	no	—
142	C₂×C₄×D₅	—	5	1	no	—
143	C₂×C₄×D₅	—	5	1	no	—
144	C₂×C₄×D₅	—	5	1	no	—
145	C₂×C₄×D₅	—	5	1	no	—
146	C₂×C₄×D₅	—	5	1	no	—
147	C₂×C₄×D₅	—	10	4	no	—
148	C₂×Dic₁₀	—	5	1	no	—
149	C₅×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
150	C₅×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
151	C₅×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
152	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
153	C₂²⋊Dic₅	—	5	1	no	—
154	C₂²⋊Dic₅	—	5	1	no	—
155	#685	—	5	1	no	—
156	#685	—	5	1	no	—
157	#685	—	5	1	no	—
158	#685	—	5	1	no	—
159	#685	—	5	1	no	—
160	#685	—	5	1	no	—
161	C₄⋊Dic₅	—	5	1	no	—
162	#683	—	5	1	no	—
163	#683	—	5	1	no	—
164	#683	—	5	1	no	—
165	#683	—	5	1	no	—
166	C₄×Dic₅	—	5	1	no	—
167	#525	—	25	1	no	Sylow
168	C₂×C₂₅	—	2	2	no	—
169	C₂×C₂₅	—	4	1	no	—
170	C₂×C₂₅	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₂₅	C₂×#41	1	1	yes	characteristic
172	C₂×C₂₅	C₂×#41	1	1	yes	characteristic
173	D₂₅	—	2	2	no	—
174	D₂₅	—	2	2	no	—
175	D₂₅	—	4	1	no	—
176	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
177	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
178	C₂²×D₅	—	5	1	no	—
179	C₂²×D₅	—	5	1	no	—
180	C₂²×D₅	—	5	1	no	—
181	C₂²×D₅	—	10	2	no	—
182	C₂²×D₅	—	10	2	no	—
183	C₂²×D₅	—	10	2	no	—
184	C₅×D₄	—	2	2	no	—
185	C₂×C₄×C₅	—	2	2	no	—
186	C₂×C₄×C₅	—	2	2	no	—
187	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
188	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
189	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
190	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
191	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
192	C₂²⋊D₅	—	10	2	no	—
193	C₂²⋊D₅	—	10	2	no	—
194	C₂²⋊D₅	—	10	2	no	—
195	C₂²⋊D₅	—	10	2	no	—
196	C₂×Dic₅	—	5	1	no	—
197	C₂×Dic₅	—	5	1	no	—
198	C₂×Dic₅	—	5	1	no	—
199	C₂×Dic₅	—	5	1	no	—
200	C₂×Dic₅	—	5	1	no	—
201	C₂×Dic₅	—	10	2	no	—
202	D₂₀	—	10	2	no	—
203	D₂₀	—	10	2	no	—
204	C₄×D₅	—	5	4	no	—
205	C₄×D₅	—	10	2	no	—
206	C₄×D₅	—	10	2	no	—
207	C₄×D₅	—	10	2	no	—
208	C₄×D₅	—	10	2	no	—
209	Dic₁₀	—	10	2	no	—
210	C₂×#41	—	25	1	no	—
211	C₂³×C₄	—	25	1	no	—
212	#123	—	25	1	no	—
213	#123	—	25	1	no	—
214	C₂²⋊Q₈	—	25	1	no	—
215	C₂²⋊Q₈	—	25	1	no	—
216	#121	—	25	1	no	—
217	#121	—	25	1	no	—
218	#120	—	25	1	no	—
219	#120	—	25	1	no	—
220	C₄×D₄	—	25	1	no	—
221	C₄×D₄	—	25	1	no	—
222	C₄×D₄	—	25	1	no	—
223	C₄×D₄	D₂₅	1	1	yes	characteristic, complemented
224	#117	—	25	1	no	—
225	C₂₅	#525	1	1	yes	characteristic, complemented, Sylow
226	C₂²×C₅	—	2	1	no	—
227	C₂²×C₅	—	2	1	no	—
228	C₂²×C₅	—	2	1	no	—
229	C₂²×C₅	—	2	1	no	—
230	C₂²×C₅	—	2	1	no	—
231	C₂²×C₅	#709	1	2	yes	—
232	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
233	C₂×D₅	—	5	2	no	—
234	C₂×D₅	—	5	2	no	—
235	C₂×D₅	—	10	8	no	—
236	C₂×D₅	—	10	1	no	—
237	C₂×D₅	—	10	1	no	—
238	C₂×D₅	—	10	1	no	—
239	C₂×D₅	—	10	1	no	—
240	C₂×D₅	—	10	1	no	—
241	C₂×D₅	—	10	1	no	—
242	C₂×D₅	—	10	1	no	—
243	C₄×C₅	—	2	1	no	—
244	C₄×C₅	—	2	1	no	—
245	C₄×C₅	—	2	1	no	—
246	C₄×C₅	—	2	1	no	—
247	C₄×C₅	D₄×D₅	1	2	yes	—
248	Dic₅	—	5	2	no	—
249	Dic₅	—	10	1	no	—
250	Dic₅	—	10	1	no	—
251	Dic₅	—	10	1	no	—
252	Dic₅	—	10	1	no	—
253	C₂⁴	—	25	1	no	—
254	#41	—	50	4	no	—
255	C₂×Q₈	—	25	1	no	—
256	C₂×D₄	—	25	1	no	—
257	C₂×D₄	—	25	1	no	—
258	C₂×D₄	—	25	1	no	—
259	C₂×D₄	—	25	1	no	—
260	C₂×D₄	—	25	1	no	—
261	C₂×D₄	—	25	1	no	—
262	C₂×D₄	C₂×D₂₅	1	1	yes	characteristic
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	—	50	4	no	—
270	C₂²×C₄	C₂×D₂₅	1	1	yes	characteristic
271	C₂²×C₄	C₂×D₂₅	1	1	yes	characteristic, complemented
272	C₄⋊C₄	—	25	1	no	—
273	C₄⋊C₄	—	25	1	no	—
274	C₄⋊C₄	—	25	1	no	—
275	C₄⋊C₄	—	25	1	no	—
276	C₄⋊C₄	—	25	1	no	—
277	C₄⋊C₄	C₂×D₂₅	1	1	yes	characteristic, complemented
278	C₂²⋊C₄	—	25	1	no	—
279	C₂²⋊C₄	—	25	1	no	—
280	C₂²⋊C₄	—	25	1	no	—
281	C₂²⋊C₄	—	25	1	no	—
282	C₂²⋊C₄	—	25	1	no	—
283	C₂²⋊C₄	—	25	1	no	—
284	C₂²⋊C₄	—	25	1	no	—
285	C₂²⋊C₄	—	25	1	no	—
286	C₂²⋊C₄	C₂×D₂₅	1	1	yes	characteristic
287	C₂²⋊C₄	C₂×D₂₅	1	1	yes	characteristic, complemented
288	C₄²	—	25	1	no	—
289	C₄²	C₂×D₂₅	1	1	yes	characteristic, complemented
290	C₂×C₅	—	2	2	no	—
291	C₂×C₅	—	4	1	no	—
292	C₂×C₅	C₂×#709	1	1	yes	characteristic
293	C₂×C₅	C₂×D₄×D₅	1	1	yes	characteristic
294	C₂×C₅	D₅×#41	1	1	yes	characteristic
295	D₅	—	10	2	no	—
296	D₅	—	10	2	no	—
297	D₅	—	20	1	no	—
298	C₂³	—	25	1	no	—
299	C₂³	—	25	1	no	—
300	C₂³	—	25	1	no	—
301	C₂³	—	50	2	no	—
302	C₂³	—	50	2	no	—
303	C₂³	—	50	2	no	—
304	C₂³	C₂²×D₂₅	1	1	yes	characteristic
305	C₂³	C₂²×D₂₅	1	1	yes	characteristic
306	Q₈	—	50	2	no	—
307	D₄	—	2	2	no	—
308	D₄	—	50	2	no	—
309	D₄	—	50	2	no	—
310	D₄	—	50	2	no	—
311	D₄	—	50	2	no	—
312	D₄	—	50	2	no	—
313	D₄	—	50	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	25	1	no	—
317	C₂×C₄	—	25	4	no	—
318	C₂×C₄	—	25	1	no	—
319	C₂×C₄	—	25	1	no	—
320	C₂×C₄	—	25	1	no	—
321	C₂×C₄	—	25	1	no	—
322	C₂×C₄	—	50	2	no	—
323	C₂×C₄	—	50	2	no	—
324	C₂×C₄	—	50	2	no	—
325	C₂×C₄	—	50	2	no	—
326	C₂×C₄	—	50	2	no	—
327	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
328	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
329	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
330	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
331	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
332	C₅	#65881	1	1	yes	characteristic
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	25	2	no	—
339	C₂²	—	25	2	no	—
340	C₂²	—	50	8	no	—
341	C₂²	—	50	1	no	—
342	C₂²	—	50	1	no	—
343	C₂²	—	50	1	no	—
344	C₂²	—	50	1	no	—
345	C₂²	—	50	1	no	—
346	C₂²	—	50	1	no	—
347	C₂²	—	50	1	no	—
348	C₂²	#87744	1	2	yes	complemented
349	C₂²	C₂³×D₂₅	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	25	2	no	—
355	C₄	—	50	1	no	—
356	C₄	—	50	1	no	—
357	C₄	—	50	1	no	—
358	C₄	—	50	1	no	—
359	C₄	D₄×D₂₅	1	2	yes	—
360	C₂	—	2	2	no	—
361	C₂	—	4	1	no	—
362	C₂	—	50	2	no	—
363	C₂	—	50	2	no	—
364	C₂	—	100	1	no	—
365	C₂	C₂×#87744	1	1	yes	characteristic
366	C₂	C₂×D₄×D₂₅	1	1	yes	characteristic
367	C₂	#41×D₂₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₅
commutator, cosocle	C₂²×C₂₅	C₂⁴
Frattini, socle	C₂²×C₅	C₂³×D₅
Fitting	C₄×D₄×C₂₅	C₂
hypercenter	C₄×D₄	D₂₅
hypocenter	C₂₅	#525
LCS	G ▷ C₂²×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#87744	C₂	semidirect
2. C₂²×C₄×D₂₅	C₂	semidirect
3. C₄×D₄×C₂₅	C₂	semidirect
4. #11716501	C₂	semidirect
5. #11716499	C₂	semidirect
6. C₄×C₂²⋊D₂₅	C₂	semidirect
7. C₄×C₂²⋊D₂₅	C₂	semidirect
8. #11716458	C₂	semidirect
9. #11716456	C₂	semidirect
10. #11716446	C₂	semidirect
11. #11716445	C₂	semidirect
12. #11716444	C₂	semidirect
13. #11716440	C₂	semidirect
14. C₄×D₁₀₀	C₂	semidirect
15. #11716434	C₂	semidirect
19. C₂²×C₄×C₂₅	C₂²	semidirect
23. C₂×C₂²⋊D₂₅	C₂²	semidirect
33. C₂×C₄×D₂₅	C₂²	semidirect
34. C₂×Dic₅₀	C₂²	semidirect
35. C₄⋊C₄×C₂₅	C₂²	semidirect
37. C₂²⋊C₄×C₂₅	C₂²	semidirect
38. C₄²×C₂₅	C₂²	semidirect
40. C₂²⋊Dic₂₅	C₂²	semidirect
45. #87720	C₂²	semidirect
46. #87720	C₂²	semidirect
47. C₄⋊Dic₂₅	C₂²	semidirect
48. #87718	C₂²	semidirect
49. #87718	C₂²	semidirect
50. #87718	C₂²	semidirect
51. #87718	C₂²	semidirect
52. C₄×Dic₂₅	C₂²	semidirect
86. C₄×D₂₅	D₄	semidirect
108. C₂²×C₂₅	#41	semidirect
119. C₂×D₂₅	#41	semidirect
223. C₄×D₄	D₂₅	semidirect
225. C₂₅	#525	semidirect
271. C₂²×C₄	C₂×D₂₅	semidirect
277. C₄⋊C₄	C₂×D₂₅	semidirect
287. C₂²⋊C₄	C₂×D₂₅	semidirect
289. C₄²	C₂×D₂₅	semidirect
348. C₂²	#87744	semidirect

Maximal subgroups

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

subgroup	index
C₂×#87744	2
C₂²×C₄×D₂₅	2
C₄×D₄×C₂₅	2
#11716501	2
#11716499	2
C₄×C₂²⋊D₂₅	2
#11716458	2
#11716456	2
#11716446	2
#11716445	2
#11716444	2
#11716440	2
C₄×D₁₀₀	2
#11716434	2
#65881	5