Subgroups — C₂×#10602245 #12252526

Order 1280.

662 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#504	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#65433	C₂	1	1	yes	maximal, characteristic
3	C₂×#65190	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#64781	C₂	1	1	yes	maximal, characteristic, complemented
5	#10602245	C₂	1	8	yes	maximal, direct product factor
6	C₂×#64735	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#64731	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₆×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅×#131	C₂²	1	1	yes	characteristic
10	C₂×#4096	C₄	1	1	yes	characteristic
11	C₂×#4047	C₂²	1	1	yes	characteristic
12	C₅×#504	—	2	2	no	—
13	C₅×#504	C₂²	1	4	yes	complemented
14	C₂×C₅×M₃₂	—	2	1	no	—
15	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
16	C₂²×C₅×C₁₆	—	2	1	no	—
17	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
18	C₂×#3969	C₄	1	1	yes	characteristic
19	#65433	C₄	1	4	yes	—
20	#65433	C₂²	1	4	yes	—
21	C₂×#3957	C₄	1	1	yes	characteristic
22	C₂×#3954	C₄	1	1	yes	characteristic
23	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
24	C₂×#3944	C₂²	1	1	yes	characteristic
25	C₂²×#672	C₂²	1	1	yes	characteristic
26	#65190	—	2	2	no	—
27	#65190	C₂²	1	4	yes	complemented
28	C₂×#3930	—	2	1	no	—
29	C₂×D₅×C₁₆	—	2	1	no	—
30	#64781	C₂²	1	4	yes	complemented
31	#64735	C₂²	1	4	yes	complemented
32	#64731	C₂²	1	4	yes	complemented
33	C₁₆×Dic₅	C₂²	1	4	yes	complemented
34	C₂×#2170	—	5	1	no	maximal, Sylow
35	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
36	C₂×#709	C₂×C₄	1	1	yes	characteristic
37	C₅×#131	—	2	1	no	—
38	C₅×#131	—	2	1	no	—
39	C₅×#131	C₂×C₄	1	2	yes	—
40	C₅×#131	C₂³	1	2	yes	—
41	C₂×C₅×M₁₆	—	2	1	no	—
42	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₅×C₈	—	2	1	no	—
44	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
45	#4096	C₈	1	4	yes	—
46	#4096	C₂×C₄	1	4	yes	—
47	C₂×#689	C₈	1	1	yes	characteristic
48	C₂×#688	C₈	1	1	yes	characteristic
49	C₂×#687	C₈	1	1	yes	characteristic
50	C₂×#686	C₈	1	1	yes	characteristic
51	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
52	C₂×#681	C₂×C₄	1	1	yes	characteristic
53	C₂²×#168	C₂×C₄	1	1	yes	characteristic
54	#4047	—	2	1	no	—
55	#4047	—	2	1	no	—
56	#4047	C₂×C₄	1	2	yes	—
57	#4047	C₂³	1	2	yes	—
58	C₂×#676	—	2	1	no	—
59	C₂×C₈×D₅	—	2	1	no	—
60	C₅×M₃₂	—	2	1	no	—
61	C₅×M₃₂	—	2	4	no	—
62	C₅×M₃₂	C₂³	1	2	yes	—
63	C₂×C₅×C₁₆	—	2	1	no	—
64	C₂×C₅×C₁₆	—	2	4	no	—
65	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
66	C₂×C₅×C₁₆	D₄	1	2	yes	—
67	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
68	C₂×C₅×C₁₆	C₂³	1	2	yes	—
69	#3969	C₂×C₄	1	4	yes	—
70	#3957	C₂×C₄	1	4	yes	—
71	#3954	C₂×C₄	1	4	yes	—
72	C₈×Dic₅	C₂×C₄	1	2	yes	—
73	C₈×Dic₅	C₂³	1	2	yes	—
74	#3944	—	2	1	no	—
75	#3944	C₂³	1	2	yes	—
76	C₂×#672	—	2	1	no	—
77	C₂×#672	—	2	1	no	—
78	C₂×#672	—	2	1	no	—
79	C₂×#672	C₂³	1	2	yes	—
80	#3930	—	2	4	no	—
81	D₅×C₁₆	—	2	4	no	—
82	C₂×#504	—	5	1	no	—
83	C₂×#504	D₅	1	1	yes	characteristic, complemented
84	C₂×#443	—	5	1	no	—
85	#2170	—	5	8	no	—
86	C₂×#364	—	5	1	no	—
87	C₂×#350	—	5	1	no	—
88	C₂×#350	—	5	1	no	—
89	C₂×C₄×C₁₆	—	5	1	no	—
90	C₅×#41	—	2	1	no	—
91	C₅×#41	—	2	1	no	—
92	C₅×#41	C₂×C₈	1	2	yes	—
93	C₅×#41	C₂²×C₄	1	2	yes	—
94	C₂×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
95	C₂×C₅×D₄	—	2	1	no	—
96	C₂×C₅×D₄	C₂×C₈	1	1	yes	characteristic
97	C₂²×C₄×C₅	—	2	1	no	—
98	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₂²⋊D₅	—	2	1	no	—
100	C₂²×Dic₅	—	2	1	no	—
101	#709	—	2	1	no	—
102	#709	—	2	1	no	—
103	#709	C₂×C₈	1	2	yes	—
104	#709	C₂²×C₄	1	2	yes	—
105	C₂×D₂₀	C₂×C₈	1	1	yes	characteristic
106	C₂×C₄×D₅	—	2	1	no	—
107	C₂×Dic₁₀	C₂×C₈	1	1	yes	characteristic
108	C₅×M₁₆	—	2	1	no	—
109	C₅×M₁₆	—	2	2	no	—
110	C₅×M₁₆	—	2	2	no	—
111	C₅×M₁₆	C₂²×C₄	1	2	yes	—
112	C₂×C₅×C₈	—	2	1	no	—
113	C₂×C₅×C₈	—	2	2	no	—
114	C₂×C₅×C₈	—	2	2	no	—
115	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
117	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
118	C₂×C₅×C₈	#41	1	1	yes	characteristic
119	C₂×C₅×C₈	#41	1	1	yes	characteristic
120	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
121	#689	C₂×C₈	1	4	yes	—
122	#688	C₂×C₈	1	4	yes	—
123	#687	C₂×C₈	1	4	yes	—
124	#686	C₂×C₈	1	4	yes	—
125	C₄×Dic₅	—	2	1	no	—
126	C₄×Dic₅	C₂²×C₄	1	2	yes	—
127	#681	—	2	1	no	—
128	#681	C₂²×C₄	1	2	yes	—
129	C₂×#168	—	2	1	no	—
130	C₂×#168	C₂×C₈	1	2	yes	—
131	C₂×#168	C₂×C₈	1	1	yes	characteristic
132	C₂×#168	C₂×C₈	1	1	yes	characteristic
133	C₂×#168	C₂²×C₄	1	2	yes	—
134	#676	—	2	2	no	—
135	#676	—	2	2	no	—
136	C₈×D₅	—	2	2	no	—
137	C₈×D₅	—	2	2	no	—
138	C₅×C₁₆	—	2	2	no	—
139	C₅×C₁₆	C₂×D₄	1	2	yes	—
140	C₅×C₁₆	C₂×D₄	1	2	yes	—
141	#672	—	2	2	no	—
142	#672	—	2	2	no	—
143	C₂×#135	—	5	1	no	—
144	C₂×#131	—	5	1	no	—
145	C₂×#131	C₂×D₅	1	1	yes	characteristic
146	#504	—	2	2	no	—
147	#504	—	5	4	no	—
148	#504	—	10	2	no	—
149	#504	C₂×D₅	1	4	yes	complemented
150	C₂×M₃₂	—	2	1	no	—
151	C₂×M₃₂	—	5	1	no	—
152	C₂×M₃₂	—	10	1	no	—
153	C₂×M₃₂	C₂×D₅	1	1	yes	characteristic
154	C₂²×C₁₆	—	2	1	no	—
155	C₂²×C₁₆	—	5	1	no	—
156	C₂²×C₁₆	—	10	1	no	—
157	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
158	#443	—	5	4	no	—
159	#443	—	5	4	no	—
160	C₂×#108	—	5	1	no	—
161	C₂×#104	—	5	1	no	—
162	C₂×#104	—	5	1	no	—
163	C₂×C₄×C₈	—	5	1	no	—
164	#364	—	5	4	no	—
165	#350	—	5	4	no	—
166	#350	—	5	4	no	—
167	C₄×C₁₆	—	5	4	no	—
168	C₂³×C₅	—	2	1	no	—
169	C₂³×C₅	#131	1	1	yes	characteristic
170	C₂²×D₅	—	2	1	no	—
171	C₅×Q₈	—	2	1	no	—
172	C₅×Q₈	C₂²×C₈	1	2	yes	—
173	C₅×D₄	—	2	2	no	—
174	C₅×D₄	—	2	2	no	—
175	C₅×D₄	—	2	1	no	—
176	C₅×D₄	C₂²×C₈	1	2	yes	—
177	C₂×C₄×C₅	—	2	1	no	—
178	C₂×C₄×C₅	—	2	2	no	—
179	C₂×C₄×C₅	—	2	2	no	—
180	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
181	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
182	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
183	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
184	C₂×C₄×C₅	#131	1	1	yes	characteristic
185	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
186	C₂²⋊D₅	—	2	2	no	—
187	C₂²⋊D₅	—	2	2	no	—
188	C₂×Dic₅	—	2	4	no	—
189	C₂×Dic₅	—	2	2	no	—
190	C₂×Dic₅	—	2	1	no	—
191	D₂₀	—	2	1	no	—
192	D₂₀	C₂²×C₈	1	2	yes	—
193	C₄×D₅	—	2	2	no	—
194	C₄×D₅	—	2	2	no	—
195	Dic₁₀	—	2	1	no	—
196	Dic₁₀	C₂²×C₈	1	2	yes	—
197	C₅×C₈	—	2	2	no	—
198	C₅×C₈	C₄×D₄	1	1	yes	characteristic
199	C₅×C₈	C₄×D₄	1	1	yes	characteristic
200	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
201	C₅×C₈	C₂×#41	1	1	yes	characteristic
202	#168	—	2	2	no	—
203	#168	C₂²×C₈	1	2	yes	—
204	#168	C₂²×C₈	1	2	yes	—
205	C₂×#41	—	5	1	no	—
206	C₂×#41	C₄×D₅	1	1	yes	characteristic
207	#135	—	5	4	no	—
208	#135	—	5	4	no	—
209	C₂×Q₁₆	—	5	1	no	—
210	C₂×SD₁₆	—	5	1	no	—
211	C₂×SD₁₆	—	5	1	no	—
212	C₂×D₈	—	5	1	no	—
213	#131	—	2	1	no	—
214	#131	—	2	1	no	—
215	#131	—	5	2	no	—
216	#131	—	5	2	no	—
217	#131	—	10	1	no	—
218	#131	—	10	1	no	—
219	#131	C₄×D₅	1	2	yes	—
220	#131	C₂²×D₅	1	2	yes	—
221	C₂×M₁₆	—	2	1	no	—
222	C₂×M₁₆	—	5	1	no	—
223	C₂×M₁₆	—	10	1	no	—
224	C₂×M₁₆	C₄×D₅	1	1	yes	characteristic
225	C₂²×C₈	—	2	1	no	—
226	C₂²×C₈	—	5	1	no	—
227	C₂²×C₈	—	10	1	no	—
228	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
229	C₂×C₄²	—	5	1	no	—
230	M₃₂	—	2	1	no	—
231	M₃₂	—	2	4	no	—
232	M₃₂	—	5	2	no	—
233	M₃₂	—	10	1	no	—
234	M₃₂	—	10	4	no	—
235	M₃₂	C₂²×D₅	1	2	yes	—
236	C₂×C₁₆	—	2	1	no	—
237	C₂×C₁₆	—	2	4	no	—
238	C₂×C₁₆	—	5	2	no	—
239	C₂×C₁₆	—	10	1	no	—
240	C₂×C₁₆	—	10	1	no	—
241	C₂×C₁₆	—	10	1	no	—
242	C₂×C₁₆	—	10	4	no	—
243	C₂×C₁₆	C₂²⋊D₅	1	1	yes	characteristic
244	C₂×C₁₆	C₂²⋊D₅	1	2	yes	—
245	C₂×C₁₆	C₂²⋊D₅	1	1	yes	characteristic
246	C₂×C₁₆	C₂²×D₅	1	2	yes	—
247	#108	—	5	4	no	—
248	#104	—	5	4	no	—
249	#104	—	5	4	no	—
250	C₄×C₈	—	5	2	no	—
251	C₄×C₈	—	5	2	no	—
252	C₂²×C₅	—	2	2	no	—
253	C₂²×C₅	—	2	2	no	—
254	C₂²×C₅	—	4	2	no	—
255	C₂²×C₅	C₈×D₄	1	1	yes	characteristic
256	C₂²×C₅	C₈×D₄	1	1	yes	characteristic
257	C₂²×C₅	C₂×#131	1	1	yes	characteristic
258	C₂×D₅	—	2	2	no	—
259	C₂×D₅	—	4	2	no	—
260	C₄×C₅	—	2	2	no	—
261	C₄×C₅	C₈×D₄	1	1	yes	characteristic
262	C₄×C₅	C₈×D₄	1	1	yes	characteristic
263	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
264	C₄×C₅	C₂³×C₈	1	1	yes	characteristic
265	Dic₅	—	2	4	no	—
266	Dic₅	—	2	2	no	—
267	#41	—	2	1	no	—
268	#41	—	2	1	no	—
269	#41	—	5	2	no	—
270	#41	—	5	2	no	—
271	#41	—	10	1	no	—
272	#41	—	10	1	no	—
273	#41	C₈×D₅	1	2	yes	—
274	#41	C₂×C₄×D₅	1	2	yes	—
275	C₂×Q₈	—	5	1	no	—
276	C₂×Q₈	C₈×D₅	1	1	yes	characteristic
277	C₂×D₄	—	2	1	no	—
278	C₂×D₄	—	5	1	no	—
279	C₂×D₄	—	10	1	no	—
280	C₂×D₄	C₈×D₅	1	1	yes	characteristic
281	C₂²×C₄	—	2	1	no	—
282	C₂²×C₄	—	10	1	no	—
283	C₂²×C₄	—	10	1	no	—
284	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
285	Q₁₆	—	5	4	no	—
286	SD₁₆	—	5	4	no	—
287	SD₁₆	—	5	4	no	—
288	D₈	—	5	4	no	—
289	M₁₆	—	2	1	no	—
290	M₁₆	—	2	2	no	—
291	M₁₆	—	2	2	no	—
292	M₁₆	—	5	2	no	—
293	M₁₆	—	10	1	no	—
294	M₁₆	—	10	2	no	—
295	M₁₆	—	10	2	no	—
296	M₁₆	C₂×C₄×D₅	1	2	yes	—
297	C₂×C₈	—	2	1	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	5	2	no	—
301	C₂×C₈	—	5	1	no	—
302	C₂×C₈	—	5	1	no	—
303	C₂×C₈	—	5	2	no	—
304	C₂×C₈	—	10	1	no	—
305	C₂×C₈	—	10	2	no	—
306	C₂×C₈	—	10	2	no	—
307	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
308	C₂×C₈	#709	1	1	yes	characteristic
309	C₂×C₈	#709	1	1	yes	characteristic
310	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
311	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
312	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
313	C₄²	—	5	2	no	—
314	C₄²	—	10	1	no	—
315	C₁₆	—	2	2	no	—
316	C₁₆	—	10	2	no	—
317	C₁₆	—	10	2	no	—
318	C₁₆	C₂×C₂²⋊D₅	1	2	yes	—
319	C₁₆	C₂×C₂²⋊D₅	1	2	yes	—
320	C₂×C₅	—	2	1	no	—
321	C₂×C₅	—	2	1	no	—
322	C₂×C₅	—	4	2	no	—
323	C₂×C₅	#2170	1	2	yes	complemented
324	C₂×C₅	C₂×C₈×D₄	1	1	yes	characteristic
325	D₅	—	4	2	no	—
326	C₂³	—	2	1	no	—
327	C₂³	—	10	1	no	—
328	C₂³	#4047	1	1	yes	characteristic
329	Q₈	—	2	1	no	—
330	Q₈	—	5	2	no	—
331	Q₈	—	10	1	no	—
332	Q₈	C₂×C₈×D₅	1	2	yes	—
333	D₄	—	2	2	no	—
334	D₄	—	2	2	no	—
335	D₄	—	2	1	no	—
336	D₄	—	5	2	no	—
337	D₄	—	10	2	no	—
338	D₄	—	10	2	no	—
339	D₄	—	10	1	no	—
340	D₄	C₂×C₈×D₅	1	2	yes	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	10	4	no	—
345	C₂×C₄	—	10	2	no	—
346	C₂×C₄	—	10	1	no	—
347	C₂×C₄	—	10	2	no	—
348	C₂×C₄	—	10	2	no	—
349	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
350	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
351	C₂×C₄	#4047	1	1	yes	characteristic
352	C₂×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
353	C₂×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
354	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
355	C₈	—	2	2	no	—
356	C₈	—	5	2	no	—
357	C₈	—	5	2	no	—
358	C₈	—	10	2	no	—
359	C₈	C₄×C₂²⋊D₅	1	1	yes	characteristic
360	C₈	C₄×C₂²⋊D₅	1	1	yes	characteristic
361	C₈	C₂×#709	1	1	yes	characteristic
362	C₈	C₂²×C₂²⋊D₅	1	1	yes	characteristic
363	C₅	C₂×#2170	1	1	yes	characteristic, complemented, Sylow
364	C₂²	—	2	2	no	—
365	C₂²	—	2	2	no	—
366	C₂²	—	4	2	no	—
367	C₂²	—	10	2	no	—
368	C₂²	—	20	2	no	—
369	C₂²	C₈×C₂²⋊D₅	1	1	yes	characteristic
370	C₂²	C₈×C₂²⋊D₅	1	1	yes	characteristic
371	C₂²	C₂×#4047	1	1	yes	characteristic
372	C₄	—	2	2	no	—
373	C₄	—	10	4	no	—
374	C₄	—	10	2	no	—
375	C₄	C₈×C₂²⋊D₅	1	1	yes	characteristic
376	C₄	C₈×C₂²⋊D₅	1	1	yes	characteristic
377	C₄	C₂²×C₈×D₅	1	1	yes	characteristic
378	C₄	C₂×C₄×C₂²⋊D₅	1	1	yes	characteristic
379	C₂	—	2	1	no	—
380	C₂	—	2	1	no	—
381	C₂	—	4	2	no	—
382	C₂	—	20	2	no	—
383	C₂	#10602245	1	2	yes	direct product factor
384	C₂	C₂×C₈×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂²⋊D₅
commutator	C₄×C₅	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₂×C₅×#504	C₂
socle	C₂²×C₅	C₈×D₄
prime radical	C₂×#4096	C₄
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	C₂×#504	D₅
hypocenter	C₅	C₂×#2170
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆ ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#504	C₂	semidirect
3. C₂×#65190	C₂	semidirect
4. C₂×#64781	C₂	semidirect
5. #10602245	C₂	direct (N × H)
6. C₂×#64735	C₂	semidirect
7. C₂×#64731	C₂	semidirect
8. C₂×C₁₆×Dic₅	C₂	semidirect
13. C₅×#504	C₂²	semidirect
27. #65190	C₂²	semidirect
30. #64781	C₂²	semidirect
31. #64735	C₂²	semidirect
32. #64731	C₂²	semidirect
33. C₁₆×Dic₅	C₂²	semidirect
83. C₂×#504	D₅	semidirect
149. #504	C₂×D₅	semidirect
323. C₂×C₅	#2170	semidirect
363. C₅	C₂×#2170	semidirect
383. C₂	#10602245	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#504	2
C₂×#65433	2
C₂×#65190	2
C₂×#64781	2
#10602245	2
C₂×#64735	2
C₂×#64731	2
C₂×C₁₆×Dic₅	2
C₂×#2170	5

Subgroups — C2×#10602245 #12252526

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10602245 #12252526