Subgroups — #12083239

Order 1280.

608 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3160	C₂	1	1	yes	maximal, characteristic, complemented
2	#10615085	C₂	1	2	yes	maximal
3	C₈×#709	C₂	1	1	yes	maximal, characteristic
4	#10602625	C₂	1	2	yes	maximal, complemented
5	#10602171	C₂	1	1	yes	maximal, characteristic, complemented
6	#10602170	C₂	1	1	yes	maximal, characteristic, complemented
7	#10602165	C₂	1	1	yes	maximal, characteristic, complemented
8	#10602164	C₂	1	1	yes	maximal, characteristic, complemented
9	#10602141	C₂	1	2	yes	maximal, complemented
10	#10602137	C₂	1	2	yes	maximal, complemented
11	#10600525	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#533	C₂²	1	2	yes	—
13	C₄×#709	C₂²	1	1	yes	characteristic
14	C₅×#500	C₂²	1	1	yes	characteristic
15	C₅×#500	C₂²	1	1	yes	characteristic
16	C₅×#499	—	2	1	no	—
17	C₅×#499	—	2	1	no	—
18	C₅×#498	C₂²	1	2	yes	—
19	C₅×#427	—	2	1	no	—
20	C₅×#427	—	2	1	no	—
21	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₈×C₂²⋊D₅	—	2	1	no	—
23	C₈×C₂²⋊D₅	—	2	1	no	—
24	#65380	C₂²	1	2	yes	—
25	#65379	C₂²	1	2	yes	—
26	#65373	C₂²	1	2	yes	—
27	#65372	C₂²	1	2	yes	—
28	#65336	—	2	2	no	—
29	#65334	—	2	2	no	—
30	#65319	—	2	2	no	—
31	#65315	—	2	2	no	—
32	#65215	C₂²	1	2	yes	—
33	#65214	C₂²	1	1	yes	characteristic
34	C₈×D₂₀	C₂²	1	1	yes	characteristic
35	C₈×D₂₀	C₂²	1	1	yes	characteristic
36	#64974	—	2	1	no	—
37	C₄×C₈×D₅	—	2	1	no	—
38	C₈×Dic₁₀	C₂²	1	1	yes	characteristic
39	C₈×Dic₁₀	C₂²	1	1	yes	characteristic
40	#64767	C₂²	1	2	yes	—
41	#64767	C₂²	1	2	yes	—
42	#64766	C₂²	1	2	yes	—
43	#64766	C₂²	1	2	yes	—
44	#64678	C₂²	1	1	yes	characteristic
45	#64678	C₂²	1	1	yes	characteristic
46	#64677	C₂²	1	2	yes	—
47	#19273	—	5	1	no	maximal, Sylow
48	C₂×#709	D₄	1	1	yes	characteristic
49	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
50	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
51	C₅×C₄⋊Q₈	C₂³	1	2	yes	—
52	C₅×C₄⋊Q₈	C₂³	1	2	yes	—
53	C₅×#125	—	2	2	no	—
54	C₅×C₂²⋊Q₈	—	2	2	no	—
55	C₅×C₂²⋊Q₈	—	2	2	no	—
56	C₄×C₅×Q₈	—	2	2	no	—
57	C₄×C₅×Q₈	—	2	2	no	—
58	C₅×#117	—	2	2	no	—
59	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
60	C₂×#689	—	2	2	no	—
61	C₂×#688	—	2	2	no	—
62	C₄×C₂²⋊D₅	—	2	1	no	—
63	C₄×C₂²⋊D₅	—	2	1	no	—
64	#4072	D₄	1	1	yes	characteristic
65	C₂×C₈×D₅	—	2	1	no	—
66	C₂×C₈×D₅	—	2	1	no	—
67	C₄×D₂₀	D₄	1	1	yes	characteristic
68	C₄×D₂₀	C₂³	1	1	yes	characteristic
69	#4018	—	2	1	no	—
70	C₄²×D₅	—	2	1	no	—
71	C₄×Dic₁₀	D₄	1	1	yes	characteristic
72	C₄×Dic₁₀	C₂³	1	1	yes	characteristic
73	C₅×#107	—	2	1	no	—
74	C₅×#107	—	2	1	no	—
75	C₅×#107	—	2	1	no	—
76	C₅×#107	—	2	1	no	—
77	C₅×#106	—	2	2	no	—
78	C₅×#106	—	2	2	no	—
79	C₄×C₅×C₈	D₄	1	1	yes	characteristic
80	C₄×C₅×C₈	D₄	1	1	yes	characteristic
81	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
82	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
83	#3967	—	2	2	no	—
84	#3967	—	2	2	no	—
85	C₂²⋊#168	—	2	2	no	—
86	C₂²⋊#168	—	2	1	no	—
87	#3952	—	2	1	no	—
88	#3952	—	2	1	no	—
89	#3946	—	2	1	no	—
90	#3946	—	2	1	no	—
91	C₈×Dic₅	—	2	1	no	—
92	C₈×Dic₅	—	2	1	no	—
93	#3942	—	2	2	no	—
94	#3941	—	2	2	no	—
95	#3940	—	2	2	no	—
96	#3939	—	2	2	no	—
97	C₄⋊#168	C₂³	1	2	yes	—
98	C₄⋊#168	C₂³	1	2	yes	—
99	C₄⋊#168	C₂³	1	1	yes	characteristic
100	C₄⋊#168	C₂³	1	1	yes	characteristic
101	#3935	—	2	2	no	—
102	C₄×#168	—	2	1	no	—
103	#3160	D₅	1	1	yes	characteristic, complemented
104	#3116	—	5	2	no	—
105	C₈×#41	—	5	1	no	—
106	#1676	—	5	1	no	—
107	#1675	—	5	1	no	—
108	#1674	—	5	1	no	—
109	#1673	—	5	1	no	—
110	#1672	—	5	2	no	—
111	#1671	—	5	2	no	—
112	#1568	—	5	1	no	—
113	#1554	—	5	2	no	—
114	C₂×C₅×Q₈	—	2	2	no	—
115	C₂×C₅×Q₈	—	2	2	no	—
116	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₅	#41	1	1	yes	characteristic
119	C₂×C₂²⋊D₅	—	2	1	no	—
120	#709	—	2	2	no	—
121	#709	—	2	2	no	—
122	C₂×D₂₀	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×D₅	—	2	1	no	—
124	C₂×C₄×D₅	—	2	1	no	—
125	C₂×C₄×D₅	—	2	1	no	—
126	C₂×Dic₁₀	C₂×D₄	1	1	yes	characteristic
127	C₂×C₅×C₈	—	2	2	no	—
128	C₂×C₅×C₈	—	2	2	no	—
129	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
130	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
131	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
132	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
133	C₅×C₄⋊C₄	—	2	2	no	—
134	C₅×C₄⋊C₄	—	2	2	no	—
135	C₅×C₄⋊C₄	—	2	2	no	—
136	C₅×C₄⋊C₄	—	2	2	no	—
137	C₅×C₄⋊C₄	—	4	2	no	—
138	C₅×C₄⋊C₄	—	4	2	no	—
139	C₅×C₂²⋊C₄	—	4	2	no	—
140	C₄²×C₅	—	4	2	no	—
141	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
142	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
143	C₄²×C₅	#41	1	1	yes	characteristic
144	C₄²×C₅	C₂⁴	1	1	yes	characteristic
145	C₂²⋊Dic₅	—	2	1	no	—
146	#689	—	4	4	no	—
147	#688	—	4	4	no	—
148	#685	—	2	1	no	—
149	#685	—	2	1	no	—
150	C₄⋊Dic₅	C₂×D₄	1	1	yes	characteristic
151	C₄⋊Dic₅	C₂×D₄	1	1	yes	characteristic
152	#683	—	2	1	no	—
153	#683	—	2	1	no	—
154	C₄×Dic₅	—	2	1	no	—
155	C₄×Dic₅	—	2	1	no	—
156	C₄×Dic₅	—	2	1	no	—
157	C₂×#168	—	2	1	no	—
158	C₂×#168	—	2	1	no	—
159	C₂×#168	—	2	2	no	—
160	C₂×#168	—	2	2	no	—
161	C₈×D₅	—	4	2	no	—
162	C₈×D₅	—	4	2	no	—
163	#533	C₂×D₅	1	2	yes	—
164	C₄×#41	—	5	1	no	—
165	#500	C₂×D₅	1	1	yes	characteristic
166	#500	C₂×D₅	1	1	yes	characteristic
167	#499	—	2	1	no	—
168	#499	—	2	1	no	—
169	#498	C₂×D₅	1	2	yes	—
170	#485	—	10	2	no	—
171	#477	—	5	2	no	—
172	#476	—	5	2	no	—
173	#462	—	5	2	no	—
174	#461	—	5	2	no	—
175	#452	—	10	2	no	—
176	C₈×Q₈	—	5	1	no	—
177	C₈×Q₈	—	5	1	no	—
178	#441	—	10	2	no	—
179	#440	—	10	2	no	—
180	C₈×D₄	—	5	1	no	—
181	C₈×D₄	—	5	1	no	—
182	C₈×D₄	—	10	1	no	—
183	C₈×D₄	—	10	1	no	—
184	#432	—	5	2	no	—
185	#431	—	5	1	no	—
186	#431	—	10	1	no	—
187	#427	—	2	1	no	—
188	#427	—	2	1	no	—
189	C₂×C₄×C₈	—	10	1	no	—
190	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
191	#335	—	5	1	no	—
192	#335	—	5	1	no	—
193	#334	—	5	2	no	—
194	#333	—	5	2	no	—
195	#333	—	5	2	no	—
196	#332	—	5	2	no	—
197	#332	—	5	2	no	—
198	C₂³×C₅	#121	1	1	yes	characteristic
199	C₂²×D₅	—	2	1	no	—
200	C₅×Q₈	—	4	2	no	—
201	C₅×Q₈	—	4	2	no	—
202	C₅×Q₈	—	8	2	no	—
203	C₂×C₄×C₅	—	2	1	no	—
204	C₂×C₄×C₅	—	2	1	no	—
205	C₂×C₄×C₅	—	2	1	no	—
206	C₂×C₄×C₅	—	2	1	no	—
207	C₂×C₄×C₅	—	4	1	no	—
208	C₂×C₄×C₅	—	4	2	no	—
209	C₂×C₄×C₅	—	4	2	no	—
210	C₂×C₄×C₅	#121	1	1	yes	characteristic
211	C₂×C₄×C₅	#121	1	1	yes	characteristic
212	C₂×C₄×C₅	#121	1	1	yes	characteristic
213	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
215	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
216	C₂²⋊D₅	—	2	4	no	—
217	C₂×Dic₅	—	2	1	no	—
218	C₂×Dic₅	—	2	1	no	—
219	C₂×Dic₅	—	2	1	no	—
220	D₂₀	—	2	1	no	—
221	D₂₀	#135	1	2	yes	—
222	C₄×D₅	—	2	4	no	—
223	C₄×D₅	—	4	2	no	—
224	C₄×D₅	—	4	1	no	—
225	C₄×D₅	—	4	1	no	—
226	Dic₁₀	—	2	1	no	—
227	Dic₁₀	#135	1	2	yes	—
228	C₅×C₈	—	2	1	no	—
229	C₅×C₈	—	2	1	no	—
230	C₅×C₈	#137	1	2	yes	—
231	C₅×C₈	#137	1	2	yes	—
232	#168	—	4	2	no	—
233	#168	—	4	2	no	—
234	#168	—	4	1	no	—
235	#168	—	4	1	no	—
236	C₂×#41	—	5	1	no	—
237	C₂×Q₁₆	—	10	2	no	—
238	C₂×SD₁₆	—	10	2	no	—
239	C₂²×C₈	—	10	1	no	—
240	C₂²×C₈	—	10	1	no	—
241	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
242	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
243	C₄⋊Q₈	C₂²×D₅	1	2	yes	—
244	C₄⋊Q₈	C₂²×D₅	1	2	yes	—
245	#125	—	2	2	no	—
246	C₂²⋊Q₈	—	2	2	no	—
247	C₂²⋊Q₈	—	2	2	no	—
248	C₄×Q₈	—	2	2	no	—
249	C₄×Q₈	—	2	2	no	—
250	C₄×Q₈	—	5	1	no	—
251	C₄×Q₈	—	5	1	no	—
252	C₄×D₄	—	5	1	no	—
253	C₄×D₄	—	5	1	no	—
254	C₄×D₄	—	10	1	no	—
255	C₄×D₄	—	10	1	no	—
256	#117	—	2	2	no	—
257	#117	—	5	1	no	—
258	#117	—	10	1	no	—
259	C₂×C₄²	—	10	1	no	—
260	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
261	#107	—	2	1	no	—
262	#107	—	2	1	no	—
263	#107	—	2	1	no	—
264	#107	—	2	1	no	—
265	#107	—	10	2	no	—
266	#106	—	2	2	no	—
267	#106	—	2	2	no	—
268	#106	—	10	2	no	—
269	C₄⋊C₈	—	5	1	no	—
270	C₄⋊C₈	—	5	1	no	—
271	C₄⋊C₈	—	5	2	no	—
272	C₄⋊C₈	—	5	2	no	—
273	C₄⋊C₈	—	10	1	no	—
274	C₄⋊C₈	—	10	1	no	—
275	#103	—	10	2	no	—
276	#103	—	10	2	no	—
277	#103	—	10	2	no	—
278	#102	—	10	2	no	—
279	C₂²⋊C₈	—	10	2	no	—
280	C₂²⋊C₈	—	10	1	no	—
281	C₂²⋊C₈	—	10	1	no	—
282	C₂²⋊C₈	—	10	1	no	—
283	#97	—	10	2	no	—
284	C₄×C₈	—	10	1	no	—
285	C₄×C₈	—	10	1	no	—
286	C₄×C₈	—	10	1	no	—
287	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
288	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
289	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
290	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
291	C₂²×C₅	—	2	1	no	—
292	C₂²×C₅	—	2	1	no	—
293	C₂²×C₅	—	2	1	no	—
294	C₂²×C₅	C₂×#121	1	1	yes	characteristic
295	C₂×D₅	—	2	2	no	—
296	C₂×D₅	—	4	1	no	—
297	C₂×D₅	—	4	1	no	—
298	C₄×C₅	—	2	1	no	—
299	C₄×C₅	—	2	1	no	—
300	C₄×C₅	—	2	1	no	—
301	C₄×C₅	—	2	1	no	—
302	C₄×C₅	—	8	2	no	—
303	C₄×C₅	—	8	2	no	—
304	C₄×C₅	C₂×#135	1	1	yes	characteristic
305	C₄×C₅	C₂×#135	1	1	yes	characteristic
306	C₄×C₅	C₂×#137	1	1	yes	characteristic
307	C₄×C₅	C₂×#137	1	1	yes	characteristic
308	Dic₅	—	2	2	no	—
309	Dic₅	—	4	1	no	—
310	Dic₅	—	4	1	no	—
311	#41	—	10	2	no	—
312	#41	—	10	2	no	—
313	C₂×Q₈	—	2	2	no	—
314	C₂×Q₈	—	2	2	no	—
315	C₂×Q₈	—	5	1	no	—
316	C₂×D₄	—	5	1	no	—
317	C₂×D₄	—	10	1	no	—
318	C₂²×C₄	—	10	1	no	—
319	C₂²×C₄	—	10	1	no	—
320	C₂²×C₄	—	10	1	no	—
321	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
322	C₂²×C₄	#711	1	1	yes	characteristic
323	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
324	Q₁₆	—	20	4	no	—
325	SD₁₆	—	20	4	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	10	2	no	—
329	C₂×C₈	—	10	2	no	—
330	C₂×C₈	—	10	1	no	—
331	C₂×C₈	—	10	1	no	—
332	C₂×C₈	—	20	2	no	—
333	C₂×C₈	—	20	2	no	—
334	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
335	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
336	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
337	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
338	C₄⋊C₄	—	2	2	no	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	2	2	no	—
342	C₄⋊C₄	—	4	2	no	—
343	C₄⋊C₄	—	4	2	no	—
344	C₄⋊C₄	—	5	1	no	—
345	C₄⋊C₄	—	5	1	no	—
346	C₄⋊C₄	—	10	1	no	—
347	C₄⋊C₄	—	10	1	no	—
348	C₂²⋊C₄	—	4	2	no	—
349	C₂²⋊C₄	—	10	1	no	—
350	C₂²⋊C₄	—	10	1	no	—
351	C₂²⋊C₄	—	10	1	no	—
352	C₄²	—	4	2	no	—
353	C₄²	—	10	1	no	—
354	C₄²	—	10	1	no	—
355	C₄²	—	10	1	no	—
356	C₄²	D₄×D₅	1	1	yes	characteristic
357	C₄²	#711	1	1	yes	characteristic
358	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
359	C₄²	C₂³×D₅	1	1	yes	characteristic
360	C₂×C₅	—	4	1	no	—
361	C₂×C₅	#3039	1	1	yes	characteristic
362	C₂×C₅	#3050	1	1	yes	characteristic
363	C₂×C₅	#3067	1	1	yes	characteristic
364	D₅	—	4	2	no	—
365	C₂³	—	10	1	no	—
366	C₂³	#4084	1	1	yes	characteristic
367	Q₈	—	4	2	no	—
368	Q₈	—	4	2	no	—
369	Q₈	—	5	2	no	—
370	Q₈	—	8	2	no	—
371	Q₈	—	10	1	no	—
372	D₄	—	5	2	no	—
373	D₄	—	10	4	no	—
374	D₄	—	10	1	no	—
375	C₂×C₄	—	2	1	no	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	4	1	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	10	4	no	—
383	C₂×C₄	—	10	1	no	—
384	C₂×C₄	—	10	1	no	—
385	C₂×C₄	—	10	1	no	—
386	C₂×C₄	—	20	2	no	—
387	C₂×C₄	—	20	1	no	—
388	C₂×C₄	—	20	1	no	—
389	C₂×C₄	#4084	1	1	yes	characteristic
390	C₂×C₄	#4084	1	1	yes	characteristic
391	C₂×C₄	#4084	1	1	yes	characteristic
392	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
393	C₂×C₄	C₂×#711	1	1	yes	characteristic
394	C₂×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
395	C₈	—	2	1	no	—
396	C₈	—	2	1	no	—
397	C₈	—	20	2	no	—
398	C₈	—	20	2	no	—
399	C₈	—	20	1	no	—
400	C₈	—	20	1	no	—
401	C₈	#4088	1	2	yes	—
402	C₈	#4088	1	2	yes	—
403	C₅	#19273	1	1	yes	characteristic, complemented, Sylow
404	C₂²	—	2	1	no	—
405	C₂²	—	2	1	no	—
406	C₂²	—	2	1	no	—
407	C₂²	—	10	2	no	—
408	C₂²	—	20	1	no	—
409	C₂²	—	20	1	no	—
410	C₂²	C₂×#4084	1	1	yes	characteristic
411	C₄	—	2	1	no	—
412	C₄	—	2	1	no	—
413	C₄	—	2	1	no	—
414	C₄	—	2	1	no	—
415	C₄	—	8	2	no	—
416	C₄	—	8	2	no	—
417	C₄	—	10	2	no	—
418	C₄	—	20	1	no	—
419	C₄	—	20	1	no	—
420	C₄	D₅×#135	1	1	yes	characteristic
421	C₄	D₅×#135	1	1	yes	characteristic
422	C₄	C₂×#4088	1	1	yes	characteristic
423	C₄	C₂×#4088	1	1	yes	characteristic
424	C₂	—	4	1	no	—
425	C₂	—	20	2	no	—
426	C₂	#10614940	1	1	yes	characteristic
427	C₂	#10615645	1	1	yes	characteristic
428	C₂	#10615659	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#4084
commutator, cosocle	C₄²×C₅	C₂⁴
Frattini	C₄²	C₂³×D₅
Fitting	C₅×#3160	C₂
socle	C₂²×C₅	C₂×#121
prime radical	C₂×#709	D₄
hypercenter	#3160	D₅
hypocenter	C₅	#19273
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3160

Complements

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

N	H ≅ G/N	type
1. C₅×#3160	C₂	semidirect
4. #10602625	C₂	semidirect
5. #10602171	C₂	semidirect
6. #10602170	C₂	semidirect
7. #10602165	C₂	semidirect
8. #10602164	C₂	semidirect
9. #10602141	C₂	semidirect
10. #10602137	C₂	semidirect
11. #10600525	C₂	semidirect
103. #3160	D₅	semidirect
403. C₅	#19273	semidirect

Maximal subgroups

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

subgroup	index
C₅×#3160	2
#10615085	2
C₈×#709	2
#10602625	2
#10602171	2
#10602170	2
#10602165	2
#10602164	2
#10602141	2
#10602137	2
#10600525	2
#19273	5