Subgroups — #12097881

Order 1280.

1014 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#65183	C₂	1	1	yes	maximal, characteristic
2	C₂×#65184	C₂	1	1	yes	maximal, characteristic
3	#10610247	C₂	1	2	yes	maximal, complemented
4	#10610225	C₂	1	1	yes	maximal, characteristic
5	C₅×#1778	C₂	1	1	yes	maximal, characteristic, complemented
6	#10601496	C₂	1	1	yes	maximal, characteristic, complemented
7	#10601495	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#3957	C₂	1	2	yes	maximal, complemented
9	Dic₅×#97	C₂	1	1	yes	maximal, characteristic, complemented
10	#10599720	C₂	1	2	yes	maximal, complemented
11	#10599719	C₂	1	2	yes	maximal, complemented
12	C₂×#4053	C₂²	1	2	yes	—
13	C₂×#4053	C₂²	1	2	yes	—
14	C₂×#4050	C₂²	1	1	yes	characteristic
15	C₄×#709	C₂²	1	2	yes	—
16	C₂×C₅×#108	C₂²	1	1	yes	characteristic
17	C₂×C₅×#108	C₂²	1	1	yes	characteristic
18	C₄×C₅×M₁₆	C₂²	1	2	yes	—
19	C₂×C₅×#97	C₂²	1	1	yes	characteristic
20	C₂×#3957	C₂²	1	2	yes	—
21	C₂×#3957	C₂²	1	2	yes	—
22	#65411	C₂²	1	2	yes	—
23	#65411	C₂²	1	2	yes	—
24	#65402	C₂²	1	2	yes	—
25	#65395	C₂²	1	1	yes	characteristic
26	C₂×#3947	C₂²	1	1	yes	characteristic
27	C₂×#3947	C₂²	1	1	yes	characteristic
28	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
29	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
30	C₄²×Dic₅	C₂²	1	1	yes	characteristic
31	C₂×#3935	C₂²	1	1	yes	characteristic
32	#65184	C₄	1	8	yes	complemented
33	#65183	C₄	1	8	yes	complemented
34	#65005	C₄	1	2	yes	complemented
35	#65000	C₄	1	2	yes	complemented
36	#64999	C₄	1	4	yes	complemented
37	#64996	—	2	2	no	—
38	D₅×#97	—	2	2	no	—
39	#64978	—	2	2	no	—
40	#64978	—	2	2	no	—
41	C₅×#341	C₂²	1	2	yes	—
42	#64775	C₂²	1	2	yes	—
43	#64775	C₂²	1	2	yes	—
44	#20512	—	5	1	no	maximal, Sylow
45	C₂×#709	C₂³	1	2	yes	—
46	C₂×C₅×M₁₆	C₂³	1	2	yes	—
47	C₂×C₅×M₁₆	C₂³	1	2	yes	—
48	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
49	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
51	C₄×C₂²⋊D₅	—	2	4	no	—
52	#4072	C₂³	1	2	yes	—
53	C₂×C₄×Dic₅	—	2	2	no	—
54	C₂×C₄×Dic₅	C₂³	1	1	yes	characteristic
55	C₂×C₄×Dic₅	C₂³	1	1	yes	characteristic
56	C₂²×#168	C₂³	1	1	yes	characteristic
57	C₂²×#168	C₂³	1	1	yes	characteristic
58	#4053	—	2	4	no	—
59	#4053	—	2	4	no	—
60	#4053	C₂×C₄	1	8	yes	complemented
61	#4053	C₂×C₄	1	8	yes	complemented
62	C₂×Dic₂₀	C₂×C₄	1	1	yes	characteristic
63	#4050	C₂×C₄	1	4	yes	—
64	#4050	C₂×C₄	1	4	yes	—
65	C₂×D₄₀	C₂×C₄	1	1	yes	characteristic
66	C₂×#677	C₂×C₄	1	2	yes	—
67	C₂×#676	—	2	2	no	—
68	C₂×#676	—	2	2	no	—
69	C₂×C₈×D₅	—	2	2	no	—
70	C₂×C₈×D₅	—	2	2	no	—
71	C₄×D₂₀	C₂×C₄	1	4	yes	—
72	#4018	—	2	2	no	—
73	C₄²×D₅	—	2	2	no	—
74	C₄×Dic₁₀	C₂×C₄	1	4	yes	—
75	C₅×#108	C₂×C₄	1	4	yes	complemented
76	C₅×#108	C₂×C₄	1	4	yes	complemented
77	C₅×#97	—	2	2	no	—
78	C₅×#97	C₂×C₄	1	2	yes	—
79	C₅×#97	C₂×C₄	1	2	yes	—
80	C₄×C₅×C₈	—	2	2	no	—
81	#3957	C₂×C₄	1	8	yes	complemented
82	#3957	C₂×C₄	1	8	yes	complemented
83	#3953	C₂×C₄	1	4	yes	—
84	#3950	C₂×C₄	1	2	yes	—
85	#3949	C₂×C₄	1	2	yes	—
86	#3948	C₂×C₄	1	4	yes	—
87	#3947	—	2	2	no	—
88	#3947	—	2	2	no	—
89	#3947	—	2	2	no	—
90	#3947	C₂×C₄	1	4	yes	complemented
91	C₈×Dic₅	—	2	2	no	—
92	C₈×Dic₅	—	2	2	no	—
93	C₈×Dic₅	—	2	2	no	—
94	C₈×Dic₅	C₂×C₄	1	4	yes	complemented
95	#3935	D₄	1	2	yes	—
96	#3935	D₄	1	2	yes	—
97	C₂×#444	—	5	1	no	—
98	C₂×#443	—	5	1	no	—
99	#2943	—	5	1	no	—
100	#2875	—	5	2	no	—
101	#1778	D₅	1	1	yes	characteristic, complemented
102	#1773	—	5	1	no	—
103	#1772	—	5	1	no	—
104	#1765	—	5	2	no	—
105	#1764	—	5	2	no	—
106	C₄×#104	—	5	2	no	—
107	C₄×#97	—	5	1	no	—
108	C₂²×C₄×C₅	#41	1	2	yes	—
109	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
110	C₂×C₂²⋊D₅	—	2	2	no	—
111	C₂²×Dic₅	—	2	1	no	—
112	C₂²×Dic₅	—	2	1	no	—
113	#709	—	2	2	no	—
114	#709	—	2	2	no	—
115	#709	C₂²×C₄	1	4	yes	—
116	#709	C₂²×C₄	1	4	yes	—
117	C₂×D₂₀	C₂²×C₄	1	2	yes	—
118	C₂×C₄×D₅	—	2	2	no	—
119	C₂×C₄×D₅	—	2	4	no	—
120	C₂×Dic₁₀	C₂²×C₄	1	2	yes	—
121	C₅×M₁₆	—	2	2	no	—
122	C₅×M₁₆	—	2	2	no	—
123	C₅×M₁₆	C₂²×C₄	1	4	yes	—
124	C₅×M₁₆	C₂²×C₄	1	4	yes	—
125	C₂×C₅×C₈	—	2	2	no	—
126	C₂×C₅×C₈	—	2	2	no	—
127	C₂×C₅×C₈	—	2	1	no	—
128	C₂×C₅×C₈	—	2	1	no	—
129	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
130	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
131	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
132	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
134	C₄²×C₅	C₂²×C₄	1	2	yes	—
135	C₄²×C₅	C₂×D₄	1	2	yes	—
136	C₂²⋊Dic₅	—	2	2	no	—
137	#685	—	2	4	no	—
138	C₄⋊Dic₅	C₂²×C₄	1	4	yes	—
139	#683	—	2	4	no	—
140	C₄×Dic₅	—	2	4	no	—
141	C₄×Dic₅	—	2	2	no	—
142	C₄×Dic₅	—	2	1	no	—
143	C₄×Dic₅	—	2	1	no	—
144	C₄×Dic₅	—	4	4	no	—
145	C₄×Dic₅	C₂²×C₄	1	2	yes	—
146	C₄×Dic₅	C₂²×C₄	1	2	yes	—
147	C₂×#168	C₂²×C₄	1	2	yes	—
148	C₂×#168	C₂²×C₄	1	2	yes	—
149	C₂×#168	C₂×D₄	1	1	yes	characteristic
150	C₂×#168	C₂×D₄	1	2	yes	—
151	C₂×#168	C₂×D₄	1	1	yes	characteristic
152	C₂×#168	#41	1	2	yes	—
153	C₂×#168	#41	1	2	yes	—
154	Dic₂₀	C₄²	1	4	yes	complemented
155	D₄₀	C₄²	1	4	yes	complemented
156	#677	C₄²	1	8	yes	complemented
157	#676	—	2	8	no	—
158	#676	—	2	8	no	—
159	C₈×D₅	—	2	8	no	—
160	C₈×D₅	—	2	8	no	—
161	C₂×#135	—	5	1	no	—
162	C₂×#131	—	5	2	no	—
163	C₂×#131	—	5	2	no	—
164	C₄×#41	—	5	2	no	—
165	#444	—	5	8	no	—
166	#443	—	5	8	no	—
167	#442	—	5	2	no	—
168	#441	—	5	2	no	—
169	#440	—	5	4	no	—
170	C₂×#108	C₂×D₅	1	1	yes	characteristic
171	C₂×#108	C₂×D₅	1	1	yes	characteristic
172	#427	—	5	1	no	—
173	C₂×#104	—	5	2	no	—
174	C₂×#104	—	5	2	no	—
175	#416	—	5	2	no	—
176	#405	—	5	2	no	—
177	#405	—	5	2	no	—
178	#405	—	10	2	no	—
179	#405	—	10	2	no	—
180	C₄×M₁₆	—	10	2	no	—
181	C₄×M₁₆	C₂×D₅	1	2	yes	—
182	C₂×#97	—	5	1	no	—
183	C₂×#97	—	5	1	no	—
184	C₂×#97	—	5	1	no	—
185	C₂×#97	—	10	2	no	—
186	C₂×#97	C₂×D₅	1	1	yes	characteristic
187	C₂×C₄×C₈	—	5	1	no	—
188	C₂×C₄×C₈	—	5	1	no	—
189	C₄³	—	5	1	no	—
190	#341	C₂×D₅	1	2	yes	—
191	#339	—	5	2	no	—
192	#339	—	5	2	no	—
193	C₂³×C₅	C₂×#41	1	1	yes	characteristic
194	C₂²×D₅	—	2	2	no	—
195	C₂×C₄×C₅	—	4	2	no	—
196	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
197	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
198	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
199	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
200	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
201	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
202	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
203	C₂²⋊D₅	—	2	4	no	—
204	C₂²⋊D₅	—	2	4	no	—
205	C₂×Dic₅	—	2	4	no	—
206	C₂×Dic₅	—	2	2	no	—
207	C₂×Dic₅	—	2	2	no	—
208	C₂×Dic₅	—	2	4	no	—
209	C₂×Dic₅	—	2	2	no	—
210	C₂×Dic₅	—	4	2	no	—
211	D₂₀	—	2	2	no	—
212	D₂₀	C₂×C₄²	1	4	yes	—
213	C₄×D₅	—	2	4	no	—
214	C₄×D₅	—	2	4	no	—
215	C₄×D₅	—	8	4	no	—
216	Dic₁₀	—	2	2	no	—
217	Dic₁₀	C₂×C₄²	1	4	yes	—
218	C₅×C₈	—	2	4	no	—
219	C₅×C₈	—	2	4	no	—
220	C₅×C₈	—	2	2	no	—
221	C₅×C₈	C₂×C₄²	1	2	yes	—
222	C₅×C₈	C₂×C₄²	1	2	yes	—
223	#168	C₄×D₄	1	2	yes	—
224	#168	C₄×D₄	1	4	yes	—
225	#168	C₄×D₄	1	2	yes	—
226	C₂×#41	—	5	2	no	—
227	#135	—	5	4	no	—
228	#135	—	5	4	no	—
229	C₂×Q₁₆	—	5	1	no	—
230	C₂×SD₁₆	—	5	2	no	—
231	C₂×D₈	—	5	1	no	—
232	#131	—	5	8	no	—
233	#131	—	5	8	no	—
234	#131	—	10	4	no	—
235	#131	—	10	4	no	—
236	C₂×M₁₆	—	10	2	no	—
237	C₂×M₁₆	—	10	2	no	—
238	C₂×M₁₆	C₂²×D₅	1	2	yes	—
239	C₂×M₁₆	C₂²×D₅	1	2	yes	—
240	C₂²×C₈	—	5	1	no	—
241	C₂²×C₈	—	5	1	no	—
242	C₂²×C₈	—	10	2	no	—
243	C₂²×C₈	—	10	2	no	—
244	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
245	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
246	C₄×Q₈	—	5	4	no	—
247	C₄×D₄	—	5	4	no	—
248	C₄×D₄	—	10	4	no	—
249	#117	—	5	2	no	—
250	#117	—	10	2	no	—
251	C₂×C₄²	—	5	1	no	—
252	C₂×C₄²	—	5	1	no	—
253	C₂×C₄²	—	10	2	no	—
254	C₂×C₄²	—	10	2	no	—
255	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
256	#108	C₄×D₅	1	4	yes	complemented
257	#108	C₄×D₅	1	4	yes	complemented
258	#107	—	5	2	no	—
259	#106	—	5	2	no	—
260	#104	—	5	8	no	—
261	#104	—	5	8	no	—
262	#103	—	5	4	no	—
263	#102	—	5	4	no	—
264	#97	—	2	2	no	—
265	#97	—	5	4	no	—
266	#97	—	5	2	no	—
267	#97	—	5	2	no	—
268	#97	—	10	2	no	—
269	#97	—	10	2	no	—
270	#97	—	10	2	no	—
271	#97	C₄×D₅	1	2	yes	—
272	#97	C₄×D₅	1	2	yes	—
273	C₄×C₈	—	2	2	no	—
274	C₄×C₈	—	5	4	no	—
275	C₄×C₈	—	10	2	no	—
276	C₄×C₈	—	10	2	no	—
277	C₄×C₈	—	10	2	no	—
278	C₂²×C₅	—	2	2	no	—
279	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
280	C₂²×C₅	C₄×#41	1	1	yes	characteristic
281	C₂²×C₅	C₄×#41	1	1	yes	characteristic
282	C₂×D₅	—	2	4	no	—
283	C₂×D₅	—	4	4	no	—
284	C₄×C₅	—	2	4	no	—
285	C₄×C₅	C₂²×C₄²	1	1	yes	characteristic
286	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
287	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
288	C₄×C₅	C₄×#41	1	1	yes	characteristic
289	Dic₅	—	2	4	no	—
290	Dic₅	—	2	4	no	—
291	Dic₅	—	4	2	no	—
292	Dic₅	—	4	4	no	—
293	#41	—	5	4	no	—
294	#41	—	5	4	no	—
295	#41	—	10	2	no	—
296	#41	—	10	2	no	—
297	C₂×Q₈	—	5	2	no	—
298	C₂×D₄	—	5	2	no	—
299	C₂×D₄	—	10	2	no	—
300	C₂²×C₄	—	10	1	no	—
301	C₂²×C₄	—	10	2	no	—
302	C₂²×C₄	—	10	1	no	—
303	C₂²×C₄	—	10	4	no	—
304	C₂²×C₄	#713	1	2	yes	—
305	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
306	Q₁₆	—	5	4	no	—
307	SD₁₆	—	5	8	no	—
308	D₈	—	5	4	no	—
309	M₁₆	—	2	2	no	—
310	M₁₆	—	2	2	no	—
311	M₁₆	—	10	8	no	—
312	M₁₆	—	10	8	no	—
313	M₁₆	C₂×C₄×D₅	1	4	yes	—
314	M₁₆	C₂×C₄×D₅	1	4	yes	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	2	1	no	—
318	C₂×C₈	—	2	1	no	—
319	C₂×C₈	—	5	2	no	—
320	C₂×C₈	—	5	2	no	—
321	C₂×C₈	—	5	1	no	—
322	C₂×C₈	—	5	2	no	—
323	C₂×C₈	—	5	1	no	—
324	C₂×C₈	—	5	2	no	—
325	C₂×C₈	—	5	2	no	—
326	C₂×C₈	—	10	8	no	—
327	C₂×C₈	—	10	8	no	—
328	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
329	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
330	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
331	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
332	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
333	C₄⋊C₄	—	5	4	no	—
334	C₄⋊C₄	—	10	4	no	—
335	C₂²⋊C₄	—	10	2	no	—
336	C₂²⋊C₄	—	10	4	no	—
337	C₄²	—	5	2	no	—
338	C₄²	—	5	2	no	—
339	C₄²	—	10	2	no	—
340	C₄²	—	10	1	no	—
341	C₄²	—	10	1	no	—
342	C₄²	—	10	4	no	—
343	C₄²	—	20	4	no	—
344	C₄²	C₂×C₄×D₅	1	2	yes	—
345	C₄²	D₄×D₅	1	2	yes	—
346	C₂×C₅	—	2	1	no	—
347	C₂×C₅	—	2	1	no	—
348	C₂×C₅	C₄²×D₄	1	1	yes	characteristic
349	C₂×C₅	#2956	1	2	yes	—
350	D₅	—	4	4	no	—
351	C₂³	—	10	2	no	—
352	C₂³	C₂×#713	1	1	yes	characteristic
353	Q₈	—	5	4	no	—
354	Q₈	—	10	2	no	—
355	D₄	—	5	4	no	—
356	D₄	—	10	4	no	—
357	D₄	—	10	4	no	—
358	D₄	—	10	2	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	—	10	4	no	—
361	C₂×C₄	—	10	2	no	—
362	C₂×C₄	—	10	2	no	—
363	C₂×C₄	—	10	4	no	—
364	C₂×C₄	—	10	4	no	—
365	C₂×C₄	—	10	4	no	—
366	C₂×C₄	—	10	2	no	—
367	C₂×C₄	—	20	2	no	—
368	C₂×C₄	—	40	4	no	—
369	C₂×C₄	#4039	1	4	yes	—
370	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
371	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
372	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
373	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
374	C₂×C₄	D₅×#41	1	1	yes	characteristic
375	C₂×C₄	D₅×#41	1	1	yes	characteristic
376	C₈	—	2	2	no	—
377	C₈	—	2	4	no	—
378	C₈	—	2	4	no	—
379	C₈	—	5	2	no	—
380	C₈	—	5	4	no	—
381	C₈	—	5	2	no	—
382	C₈	C₄²×D₅	1	2	yes	—
383	C₈	C₄²×D₅	1	2	yes	—
384	C₅	#20512	1	1	yes	characteristic, complemented, Sylow
385	C₂²	—	2	2	no	—
386	C₂²	—	10	4	no	—
387	C₂²	—	20	4	no	—
388	C₂²	C₂×#4039	1	1	yes	characteristic
389	C₂²	C₄×#713	1	1	yes	characteristic
390	C₂²	C₄×#713	1	1	yes	characteristic
391	C₄	—	2	4	no	—
392	C₄	—	10	4	no	—
393	C₄	—	10	4	no	—
394	C₄	—	20	2	no	—
395	C₄	—	20	4	no	—
396	C₄	C₂×C₄²×D₅	1	1	yes	characteristic
397	C₄	#65856	1	1	yes	characteristic
398	C₄	C₄×D₄×D₅	1	1	yes	characteristic
399	C₄	C₄×D₄×D₅	1	1	yes	characteristic
400	C₂	—	2	1	no	—
401	C₂	—	2	1	no	—
402	C₂	—	20	4	no	—
403	C₂	C₄×#4039	1	1	yes	characteristic
404	C₂	#10610989	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅
commutator	C₄×C₅	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₅×#1778	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂×#4050	C₂²
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	#1778	D₅
hypocenter	C₅	#20512
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1778

Complements

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

N	H ≅ G/N	type
3. #10610247	C₂	semidirect
5. C₅×#1778	C₂	semidirect
6. #10601496	C₂	semidirect
7. #10601495	C₂	semidirect
8. C₄×#3957	C₂	semidirect
9. Dic₅×#97	C₂	semidirect
10. #10599720	C₂	semidirect
11. #10599719	C₂	semidirect
32. #65184	C₄	semidirect
33. #65183	C₄	semidirect
34. #65005	C₄	semidirect
35. #65000	C₄	semidirect
36. #64999	C₄	semidirect
60. #4053	C₂×C₄	semidirect
61. #4053	C₂×C₄	semidirect
75. C₅×#108	C₂×C₄	semidirect
76. C₅×#108	C₂×C₄	semidirect
81. #3957	C₂×C₄	semidirect
82. #3957	C₂×C₄	semidirect
90. #3947	C₂×C₄	semidirect
94. C₈×Dic₅	C₂×C₄	semidirect
101. #1778	D₅	semidirect
154. Dic₂₀	C₄²	semidirect
155. D₄₀	C₄²	semidirect
156. #677	C₄²	semidirect
256. #108	C₄×D₅	semidirect
257. #108	C₄×D₅	semidirect
384. C₅	#20512	semidirect

Maximal subgroups

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

subgroup	index
C₂×#65183	2
C₂×#65184	2
#10610247	2
#10610225	2
C₅×#1778	2
#10601496	2
#10601495	2
C₄×#3957	2
Dic₅×#97	2
#10599720	2
#10599719	2
#20512	5