Subgroups — #12071062

Order 1280.

654 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10615074	C₂	1	1	yes	maximal, characteristic
2	#10615061	C₂	1	1	yes	maximal, characteristic, complemented
3	#10610735	C₂	1	1	yes	maximal, characteristic
4	C₅×#1481	C₂	1	1	yes	maximal, characteristic, complemented
5	#10601551	C₂	1	1	yes	maximal, characteristic
6	#10601532	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#64700	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599017	C₂	1	2	yes	maximal, complemented
9	#10599010	C₂	1	2	yes	maximal, complemented
10	#10599009	C₂	1	2	yes	maximal, complemented
11	#10599005	C₂	1	2	yes	maximal, complemented
12	C₅×#533	C₂²	1	1	yes	characteristic
13	C₂×#4047	—	2	1	no	—
14	#65811	C₂²	1	1	yes	characteristic
15	C₅×#431	C₂²	1	1	yes	characteristic
16	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	#65399	—	2	1	no	—
18	#65377	C₄	1	1	yes	characteristic
19	#65374	C₄	1	1	yes	characteristic
20	#65370	C₄	1	1	yes	characteristic
21	#65368	C₄	1	1	yes	characteristic
22	#65337	C₄	1	2	yes	—
23	#65337	C₄	1	2	yes	—
24	C₄×#688	C₂²	1	2	yes	—
25	C₄×#688	C₂²	1	2	yes	—
26	#65313	—	2	2	no	—
27	#65313	C₂²	1	2	yes	—
28	Q₈×#168	C₂²	1	2	yes	—
29	#65288	—	2	1	no	—
30	#65286	—	2	1	no	—
31	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
32	C₄×#681	C₂²	1	1	yes	characteristic
33	C₂×C₄×#168	C₂²	1	1	yes	characteristic
34	#65127	C₂²	1	2	yes	—
35	#65125	—	2	2	no	—
36	#65123	C₂²	1	2	yes	—
37	C₅×Q₈⋊C₈	C₂²	1	2	yes	—
38	C₅×Q₈⋊C₈	C₂²	1	2	yes	—
39	#64700	C₂²	1	1	yes	characteristic
40	#64700	C₂²	1	1	yes	characteristic
41	#64700	C₂²	1	1	yes	characteristic
42	#64700	C₂²	1	1	yes	characteristic
43	#64679	C₂²	1	2	yes	—
44	#64679	C₂²	1	2	yes	—
45	#64674	C₂²	1	2	yes	—
46	C₈×#168	C₂²	1	2	yes	—
47	#18067	—	5	1	no	maximal, Sylow
48	C₂×#709	—	2	1	no	—
49	C₂²×C₅×C₈	—	2	1	no	—
50	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
51	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
52	C₅×#125	—	2	1	no	—
53	C₅×C₂²⋊Q₈	—	2	1	no	—
54	C₅×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
55	C₄×C₅×Q₈	—	2	1	no	—
56	C₄×C₅×Q₈	C₂³	1	2	yes	—
57	C₅×#117	—	2	1	no	—
58	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
59	C₂×#688	C₂×C₄	1	2	yes	—
60	C₂×#688	C₂×C₄	1	2	yes	—
61	#4076	—	2	1	no	—
62	#4076	C₂×C₄	1	2	yes	—
63	C₂×#681	D₄	1	1	yes	characteristic
64	C₂×#681	D₄	1	1	yes	characteristic
65	C₂²×#168	—	2	1	no	—
66	C₂²×#168	C₂×C₄	1	1	yes	characteristic
67	C₂²×#168	C₂×C₄	1	1	yes	characteristic
68	#4047	—	4	4	no	—
69	C₂×#676	—	2	2	no	—
70	C₂×C₈×D₅	—	2	2	no	—
71	#4021	—	2	1	no	—
72	#4020	C₂×C₄	1	1	yes	characteristic
73	C₄×D₂₀	—	2	1	no	—
74	C₄×D₂₀	C₂³	1	2	yes	—
75	#4015	C₂×C₄	1	1	yes	characteristic
76	C₅×C₄⋊C₈	—	2	1	no	—
77	C₅×C₄⋊C₈	—	2	1	no	—
78	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
79	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
80	C₅×C₂²⋊C₈	—	2	1	no	—
81	C₄×C₅×C₈	C₂³	1	2	yes	—
82	#3967	C₂×C₄	1	2	yes	—
83	#3967	C₂×C₄	1	2	yes	—
84	#3952	—	2	2	no	—
85	#3952	—	2	2	no	—
86	#3941	C₂×C₄	1	2	yes	—
87	#3941	C₂×C₄	1	2	yes	—
88	#3940	C₂×C₄	1	2	yes	—
89	#3940	C₂×C₄	1	2	yes	—
90	C₄⋊#168	—	2	1	no	—
91	C₄⋊#168	—	2	2	no	—
92	C₄⋊#168	—	2	2	no	—
93	C₄⋊#168	C₂³	1	1	yes	characteristic
94	C₄⋊#168	C₂³	1	1	yes	characteristic
95	#3935	—	2	2	no	—
96	#3935	D₄	1	2	yes	—
97	C₄×#168	—	2	2	no	—
98	C₄×#168	C₂×C₄	1	1	yes	characteristic
99	C₄×#168	C₂×C₄	1	1	yes	characteristic
100	C₄×#168	C₂³	1	1	yes	characteristic
101	C₄×#168	C₂³	1	1	yes	characteristic
102	C₄×#168	C₂³	1	2	yes	—
103	#3167	—	5	1	no	—
104	#2962	—	5	1	no	—
105	#2935	—	5	1	no	—
106	#1590	—	5	2	no	—
107	#1583	—	5	2	no	—
108	#1582	—	5	2	no	—
109	C₈×SD₁₆	—	5	2	no	—
110	C₂×#334	—	5	1	no	—
111	#1481	D₅	1	1	yes	characteristic, complemented
112	#1480	—	5	1	no	—
113	C₈×M₁₆	—	5	1	no	—
114	C₂×C₅×Q₈	—	2	1	no	—
115	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
116	C₂²×C₄×C₅	C₂²×C₄	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	2	no	—
120	#709	—	4	2	no	—
121	#709	—	4	2	no	—
122	C₂×D₂₀	—	2	1	no	—
123	C₂×D₂₀	—	2	1	no	—
124	C₂×D₂₀	C₂²×C₄	1	2	yes	—
125	C₂×C₄×D₅	—	2	2	no	—
126	C₂×Dic₁₀	—	2	1	no	—
127	C₂×C₅×C₈	—	2	1	no	—
128	C₂×C₅×C₈	—	2	1	no	—
129	C₂×C₅×C₈	—	2	2	no	—
130	C₂×C₅×C₈	—	4	2	no	—
131	C₅×C₄⋊C₄	—	2	1	no	—
132	C₅×C₄⋊C₄	—	2	1	no	—
133	C₅×C₄⋊C₄	—	2	1	no	—
134	C₅×C₄⋊C₄	—	2	2	no	—
135	C₅×C₄⋊C₄	—	2	2	no	—
136	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
137	C₅×C₂²⋊C₄	—	2	2	no	—
138	C₄²×C₅	—	2	2	no	—
139	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
140	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
141	C₄²×C₅	#41	1	1	yes	characteristic
142	C₄²×C₅	C₂⁴	1	1	yes	characteristic
143	#688	—	2	4	no	—
144	#688	—	4	2	no	—
145	#685	—	2	2	no	—
146	#685	—	2	2	no	—
147	C₄⋊Dic₅	—	2	1	no	—
148	C₄⋊Dic₅	C₂²×C₄	1	2	yes	—
149	#681	—	2	2	no	—
150	#681	SD₁₆	1	4	yes	complemented
151	C₂×#168	—	2	2	no	—
152	C₂×#168	—	2	1	no	—
153	C₂×#168	—	2	1	no	—
154	C₂×#168	—	4	1	no	—
155	C₂×#168	—	4	2	no	—
156	C₂×#168	C₂²×C₄	1	1	yes	characteristic
157	C₂×#168	C₂²×C₄	1	1	yes	characteristic
158	C₂×#168	C₂²×C₄	1	1	yes	characteristic
159	C₂×#168	C₂²×C₄	1	1	yes	characteristic
160	C₂×#168	C₂×D₄	1	2	yes	—
161	C₂×#168	C₂×D₄	1	2	yes	—
162	#676	—	4	4	no	—
163	C₈×D₅	—	4	4	no	—
164	C₂×#131	—	10	1	no	—
165	#533	C₂×D₅	1	1	yes	characteristic
166	#532	—	5	1	no	—
167	#498	—	5	1	no	—
168	#492	—	5	1	no	—
169	#487	—	5	1	no	—
170	#486	—	5	1	no	—
171	#465	—	5	2	no	—
172	#465	—	5	2	no	—
173	#446	—	5	2	no	—
174	#446	—	10	2	no	—
175	C₈×Q₈	—	5	2	no	—
176	C₄×SD₁₆	—	5	2	no	—
177	C₄×SD₁₆	—	5	2	no	—
178	#436	—	5	2	no	—
179	#435	—	10	2	no	—
180	C₈×D₄	—	5	2	no	—
181	#431	C₂×D₅	1	1	yes	characteristic
182	#424	—	10	1	no	—
183	C₂×C₄⋊C₈	—	5	1	no	—
184	C₂×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
185	#408	—	10	1	no	—
186	#405	—	10	1	no	—
187	C₄×M₁₆	—	5	1	no	—
188	C₂×C₄×C₈	—	5	1	no	—
189	#334	—	5	1	no	—
190	#334	—	5	1	no	—
191	#334	—	5	1	no	—
192	#334	—	5	1	no	—
193	Q₈⋊C₈	C₂×D₅	1	2	yes	—
194	Q₈⋊C₈	C₂×D₅	1	2	yes	—
195	D₄⋊C₈	—	5	2	no	—
196	D₄⋊C₈	—	5	2	no	—
197	#322	—	5	2	no	—
198	C₈²	—	5	2	no	—
199	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
200	C₂²×D₅	—	2	2	no	—
201	C₅×Q₈	—	2	2	no	—
202	C₅×Q₈	—	2	2	no	—
203	C₅×Q₈	—	8	1	no	—
204	C₂×C₄×C₅	—	2	1	no	—
205	C₂×C₄×C₅	—	2	1	no	—
206	C₂×C₄×C₅	—	2	2	no	—
207	C₂×C₄×C₅	—	2	2	no	—
208	C₂×C₄×C₅	—	4	1	no	—
209	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
210	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
211	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
212	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
213	C₂×C₄×C₅	#135	1	2	yes	—
214	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
215	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
216	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
217	C₂²⋊D₅	—	4	4	no	—
218	C₂×Dic₅	—	2	2	no	—
219	D₂₀	—	2	2	no	—
220	D₂₀	—	2	2	no	—
221	D₂₀	—	4	2	no	—
222	D₂₀	—	8	1	no	—
223	C₄×D₅	—	4	2	no	—
224	C₄×D₅	—	4	2	no	—
225	Dic₁₀	—	4	2	no	—
226	C₅×C₈	—	4	2	no	—
227	C₅×C₈	—	4	1	no	—
228	C₅×C₈	—	4	1	no	—
229	#168	—	2	1	no	—
230	#168	—	2	1	no	—
231	#168	—	2	1	no	—
232	#168	—	2	2	no	—
233	#168	—	4	1	no	—
234	#168	—	4	1	no	—
235	#168	#131	1	2	yes	complemented
236	#168	C₂×SD₁₆	1	4	yes	—
237	C₂×#41	—	10	1	no	—
238	C₂×SD₁₆	—	5	2	no	—
239	C₂×SD₁₆	—	5	2	no	—
240	#131	—	20	4	no	—
241	C₂×M₁₆	—	5	1	no	—
242	C₂×M₁₆	—	5	1	no	—
243	C₂×M₁₆	—	10	2	no	—
244	C₂²×C₈	—	2	1	no	—
245	C₂²×C₈	—	5	1	no	—
246	C₂²×C₈	—	5	1	no	—
247	C₂²×C₈	—	10	2	no	—
248	C₂²×C₈	—	10	1	no	—
249	C₄⋊Q₈	—	5	1	no	—
250	C₄⋊Q₈	C₄×D₅	1	1	yes	characteristic
251	C₄⋊Q₈	C₄×D₅	1	1	yes	characteristic
252	#127	—	5	1	no	—
253	#125	—	2	1	no	—
254	#124	—	10	1	no	—
255	C₂²⋊Q₈	—	2	1	no	—
256	C₂²⋊Q₈	C₄×D₅	1	2	yes	—
257	#121	—	5	2	no	—
258	#121	—	10	1	no	—
259	C₄×Q₈	—	2	1	no	—
260	C₄×Q₈	C₂²×D₅	1	2	yes	—
261	C₄×D₄	—	5	2	no	—
262	C₄×D₄	—	10	1	no	—
263	#117	—	2	1	no	—
264	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
265	#106	—	5	2	no	—
266	#106	—	5	2	no	—
267	C₄⋊C₈	—	2	1	no	—
268	C₄⋊C₈	—	2	1	no	—
269	C₄⋊C₈	—	5	1	no	—
270	C₄⋊C₈	—	5	1	no	—
271	C₄⋊C₈	—	10	2	no	—
272	C₄⋊C₈	—	10	2	no	—
273	C₄⋊C₈	—	10	1	no	—
274	C₄⋊C₈	C₂²×D₅	1	1	yes	characteristic
275	C₄⋊C₈	C₂²×D₅	1	1	yes	characteristic
276	#103	—	5	2	no	—
277	#103	—	5	2	no	—
278	#102	—	5	2	no	—
279	#102	—	5	2	no	—
280	C₂²⋊C₈	—	2	1	no	—
281	C₂²⋊C₈	—	10	2	no	—
282	C₂²⋊C₈	—	10	2	no	—
283	#97	—	5	2	no	—
284	#97	—	10	2	no	—
285	C₄×C₈	—	5	1	no	—
286	C₄×C₈	—	5	1	no	—
287	C₄×C₈	—	5	1	no	—
288	C₄×C₈	—	5	2	no	—
289	C₄×C₈	—	5	1	no	—
290	C₄×C₈	—	10	2	no	—
291	C₄×C₈	C₂²×D₅	1	2	yes	—
292	C₂²×C₅	—	2	1	no	—
293	C₂²×C₅	—	2	1	no	—
294	C₂²×C₅	C₄×SD₁₆	1	2	yes	—
295	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
296	C₂×D₅	—	4	2	no	—
297	C₂×D₅	—	4	2	no	—
298	C₂×D₅	—	4	2	no	—
299	C₄×C₅	—	2	1	no	—
300	C₄×C₅	—	2	1	no	—
301	C₄×C₅	—	2	1	no	—
302	C₄×C₅	—	4	2	no	—
303	C₄×C₅	—	4	2	no	—
304	C₄×C₅	C₄×SD₁₆	1	2	yes	—
305	C₄×C₅	C₂×#131	1	1	yes	characteristic
306	C₄×C₅	#568	1	1	yes	characteristic
307	C₄×C₅	C₂²×SD₁₆	1	1	yes	characteristic
308	C₄×C₅	C₂×#135	1	1	yes	characteristic
309	Dic₅	—	4	2	no	—
310	#41	—	20	2	no	—
311	#41	—	20	2	no	—
312	C₂×Q₈	—	2	1	no	—
313	C₂×Q₈	—	10	1	no	—
314	C₂×Q₈	C₂×C₄×D₅	1	2	yes	—
315	C₂×D₄	—	5	2	no	—
316	C₂×D₄	—	10	2	no	—
317	C₂×D₄	—	10	1	no	—
318	C₂×D₄	—	10	1	no	—
319	C₂²×C₄	—	10	2	no	—
320	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
321	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
322	C₂²×C₄	#711	1	1	yes	characteristic
323	SD₁₆	—	10	4	no	—
324	SD₁₆	—	20	2	no	—
325	M₁₆	—	5	4	no	—
326	M₁₆	—	10	2	no	—
327	M₁₆	—	20	4	no	—
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	2	2	no	—
331	C₂×C₈	—	4	2	no	—
332	C₂×C₈	—	5	1	no	—
333	C₂×C₈	—	5	2	no	—
334	C₂×C₈	—	5	1	no	—
335	C₂×C₈	—	5	1	no	—
336	C₂×C₈	—	5	1	no	—
337	C₂×C₈	—	5	2	no	—
338	C₂×C₈	—	10	2	no	—
339	C₂×C₈	—	10	1	no	—
340	C₂×C₈	—	10	1	no	—
341	C₂×C₈	—	20	4	no	—
342	C₂×C₈	—	20	1	no	—
343	C₂×C₈	—	20	2	no	—
344	C₄⋊C₄	—	2	2	no	—
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	2	1	no	—
347	C₄⋊C₄	—	2	1	no	—
348	C₄⋊C₄	—	2	2	no	—
349	C₄⋊C₄	—	5	2	no	—
350	C₄⋊C₄	—	10	1	no	—
351	C₄⋊C₄	C₂×C₄×D₅	1	2	yes	—
352	C₂²⋊C₄	—	2	2	no	—
353	C₂²⋊C₄	—	10	2	no	—
354	C₂²⋊C₄	—	10	2	no	—
355	C₄²	—	2	2	no	—
356	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
357	C₄²	D₄×D₅	1	1	yes	characteristic
358	C₄²	#711	1	1	yes	characteristic
359	C₄²	C₂³×D₅	1	1	yes	characteristic
360	C₂×C₅	—	2	2	no	—
361	C₂×C₅	#2935	1	1	yes	characteristic
362	C₂×C₅	C₂×C₄×SD₁₆	1	1	yes	characteristic
363	C₂×C₅	#2957	1	1	yes	characteristic
364	D₅	—	8	2	no	—
365	C₂³	—	10	2	no	—
366	C₂³	#4027	1	1	yes	characteristic
367	Q₈	—	2	2	no	—
368	Q₈	—	2	2	no	—
369	Q₈	—	8	1	no	—
370	Q₈	—	20	2	no	—
371	D₄	—	10	2	no	—
372	D₄	—	10	2	no	—
373	D₄	—	20	4	no	—
374	D₄	—	20	2	no	—
375	D₄	—	40	1	no	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	4	1	no	—
381	C₂×C₄	—	10	2	no	—
382	C₂×C₄	—	20	2	no	—
383	C₂×C₄	—	20	2	no	—
384	C₂×C₄	#4027	1	1	yes	characteristic
385	C₂×C₄	#4027	1	1	yes	characteristic
386	C₂×C₄	#4027	1	1	yes	characteristic
387	C₂×C₄	D₅×SD₁₆	1	2	yes	—
388	C₂×C₄	#4065	1	2	yes	—
389	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
390	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
391	C₂×C₄	C₂×#711	1	1	yes	characteristic
392	C₈	—	4	2	no	—
393	C₈	—	4	1	no	—
394	C₈	—	4	1	no	—
395	C₈	—	5	2	no	—
396	C₈	—	5	4	no	—
397	C₈	—	10	1	no	—
398	C₈	—	10	1	no	—
399	C₈	—	10	1	no	—
400	C₈	—	10	2	no	—
401	C₈	—	20	1	no	—
402	C₈	—	20	1	no	—
403	C₅	#18067	1	1	yes	characteristic, complemented, Sylow
404	C₂²	—	2	1	no	—
405	C₂²	—	2	1	no	—
406	C₂²	—	20	2	no	—
407	C₂²	—	20	2	no	—
408	C₂²	—	20	2	no	—
409	C₂²	#65077	1	2	yes	—
410	C₂²	C₂×#4027	1	1	yes	characteristic
411	C₄	—	2	1	no	—
412	C₄	—	2	1	no	—
413	C₄	—	2	1	no	—
414	C₄	—	4	2	no	—
415	C₄	—	4	2	no	—
416	C₄	—	20	2	no	—
417	C₄	#65077	1	2	yes	—
418	C₄	D₅×#131	1	1	yes	characteristic
419	C₄	#66084	1	1	yes	characteristic
420	C₄	C₂×D₅×SD₁₆	1	1	yes	characteristic
421	C₄	C₂×#4065	1	1	yes	characteristic
422	C₂	—	2	2	no	—
423	C₂	—	40	2	no	—
424	C₂	#10610391	1	1	yes	characteristic
425	C₂	C₂×#65077	1	1	yes	characteristic
426	C₂	#10610721	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₄²	C₂³×D₅
Fitting	C₅×#1481	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	#65811	C₂²
cosocle	C₄²×C₅	C₂⁴
hypercenter	#1481	D₅
hypocenter	C₅	#18067
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1481

Complements

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

N	H ≅ G/N	type
2. #10615061	C₂	semidirect
4. C₅×#1481	C₂	semidirect
6. #10601532	C₂	semidirect
7. C₂×#64700	C₂	semidirect
8. #10599017	C₂	semidirect
9. #10599010	C₂	semidirect
10. #10599009	C₂	semidirect
11. #10599005	C₂	semidirect
111. #1481	D₅	semidirect
150. #681	SD₁₆	semidirect
235. #168	#131	semidirect
403. C₅	#18067	semidirect

Maximal subgroups

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

subgroup	index
#10615074	2
#10615061	2
#10610735	2
C₅×#1481	2
#10601551	2
#10601532	2
C₂×#64700	2
#10599017	2
#10599010	2
#10599009	2
#10599005	2
#18067	5