Subgroups — #12252397

Order 1280.

582 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	#10610280	C₂	1	1	yes	maximal, characteristic
2	C₅×#2164	C₂	1	1	yes	maximal, characteristic, complemented
3	#10600676	C₂	1	1	yes	maximal, characteristic, complemented
4	#10600078	C₂	1	2	yes	maximal, complemented
5	#10600068	C₂	1	2	yes	maximal, complemented
6	#10599900	C₂	1	2	yes	maximal
7	#10599886	C₂	1	2	yes	maximal, complemented
8	#10599867	C₂	1	1	yes	maximal, characteristic, complemented
9	D₅×#346	C₂	1	1	yes	maximal, characteristic, complemented
10	#10599846	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#3930	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#4053	C₄	1	1	yes	characteristic
13	C₂×#4047	C₂²	1	1	yes	characteristic
14	#65856	C₄	1	1	yes	characteristic
15	C₅×#405	C₂²	1	1	yes	characteristic
16	M₁₆×Dic₅	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
18	#65286	C₄	1	1	yes	characteristic
19	C₂×#3930	C₂²	1	2	yes	—
20	C₂×D₅×C₁₆	C₂²	1	2	yes	—
21	#64997	C₄	1	1	yes	characteristic
22	#64997	C₄	1	1	yes	characteristic
23	#64996	C₄	1	1	yes	characteristic
24	#64996	C₄	1	1	yes	characteristic
25	#64978	C₂²	1	1	yes	characteristic
26	#64978	C₂²	1	1	yes	characteristic
27	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
28	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
29	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
31	C₅×#346	C₂²	1	1	yes	characteristic
32	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
33	C₂²⋊#672	C₂²	1	2	yes	—
34	#64727	C₂²	1	2	yes	—
35	#64727	C₂²	1	2	yes	—
36	#64722	C₂²	1	2	yes	—
37	#64722	C₂²	1	2	yes	—
38	#64721	C₂²	1	2	yes	—
39	#64721	C₂²	1	2	yes	—
40	C₄⋊#672	C₂²	1	2	yes	—
41	#64681	C₂²	1	1	yes	characteristic
42	C₄×#672	C₂²	1	1	yes	characteristic
43	#33421	—	5	1	no	maximal, Sylow
44	C₂×#709	C₂×C₄	1	1	yes	characteristic
45	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
47	C₅×#117	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
49	C₂×#681	C₂×C₄	1	1	yes	characteristic
50	C₂²×#168	C₂×C₄	1	1	yes	characteristic
51	#4053	—	2	4	no	—
52	#4047	—	2	4	no	—
53	C₂×#676	C₂×C₄	1	1	yes	characteristic
54	C₂×#676	C₂×C₄	1	1	yes	characteristic
55	C₂×#676	C₂×C₄	1	1	yes	characteristic
56	C₂×#676	C₂×C₄	1	1	yes	characteristic
57	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
58	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
59	C₂×C₈×D₅	C₂³	1	1	yes	characteristic
60	C₂×C₈×D₅	C₂³	1	1	yes	characteristic
61	#4039	C₈	1	2	yes	—
62	#4033	C₈	1	2	yes	—
63	#4027	C₈	1	2	yes	—
64	#4027	C₈	1	2	yes	—
65	#4018	C₂×C₄	1	1	yes	characteristic
66	#4018	C₂×C₄	1	1	yes	characteristic
67	C₄²×D₅	C₂×C₄	1	1	yes	characteristic
68	C₄²×D₅	C₂×C₄	1	1	yes	characteristic
69	C₂×C₅×C₁₆	C₂³	1	2	yes	—
70	C₂×C₅×C₁₆	C₂³	1	2	yes	—
71	C₅×#97	C₂×C₄	1	1	yes	characteristic
72	C₅×#97	C₂×C₄	1	1	yes	characteristic
73	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
74	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
75	#3947	C₂×C₄	1	1	yes	characteristic
76	#3947	C₂×C₄	1	1	yes	characteristic
77	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
78	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
79	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
80	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₅	C₂³	1	1	yes	characteristic
82	C₈×Dic₅	C₂³	1	1	yes	characteristic
83	C₂×#672	C₂³	1	2	yes	—
84	C₂×#672	C₂³	1	2	yes	—
85	#3935	C₂×C₄	1	1	yes	characteristic
86	#3935	C₂×C₄	1	1	yes	characteristic
87	C₄×#168	C₂×C₄	1	1	yes	characteristic
88	C₄×#168	C₂×C₄	1	1	yes	characteristic
89	#3930	—	2	4	no	—
90	D₅×C₁₆	—	2	4	no	—
91	C₄×#131	—	5	1	no	—
92	#2164	—	5	1	no	—
93	#2164	—	5	1	no	—
94	#2164	—	5	1	no	—
95	#2164	D₅	1	1	yes	characteristic, complemented
96	#2163	—	5	2	no	—
97	#2163	—	5	2	no	—
98	#2151	—	5	2	no	—
99	#2113	—	5	2	no	—
100	C₄×M₃₂	—	5	1	no	—
101	C₂×#346	—	5	1	no	—
102	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₂²⋊D₅	C₂×C₈	1	1	yes	characteristic
104	C₂×C₂²⋊D₅	C₂×C₈	1	1	yes	characteristic
105	C₂²×Dic₅	C₂×C₈	1	2	yes	—
106	#709	—	2	2	no	—
107	#709	—	2	2	no	—
108	C₂×D₂₀	C₂×C₈	1	1	yes	characteristic
109	C₂×C₄×D₅	C₂×C₈	1	2	yes	—
110	C₂×C₄×D₅	C₂×C₈	1	2	yes	—
111	C₂×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
113	C₂×Dic₁₀	C₂×C₈	1	1	yes	characteristic
114	C₅×M₁₆	—	2	2	no	—
115	C₂×C₅×C₈	—	2	2	no	—
116	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
120	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
121	C₅×C₂²⋊C₄	C₂×C₈	1	2	yes	—
122	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
123	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
124	#685	C₂×C₈	1	2	yes	—
125	#685	C₂×C₈	1	2	yes	—
126	#683	C₂×C₈	1	2	yes	—
127	#683	C₂×C₈	1	2	yes	—
128	C₄×Dic₅	C₂×C₈	1	2	yes	—
129	C₄×Dic₅	C₂×C₈	1	2	yes	—
130	C₄×Dic₅	C₂×C₈	1	2	yes	—
131	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
132	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
133	#681	—	2	2	no	—
134	C₂×#168	—	2	2	no	—
135	C₂×#168	C₂²×C₄	1	1	yes	characteristic
136	C₂×#168	C₂²×C₄	1	1	yes	characteristic
137	C₂×#168	C₂²×C₄	1	1	yes	characteristic
138	C₂×#168	C₂²×C₄	1	1	yes	characteristic
139	#676	—	2	2	no	—
140	#676	—	2	2	no	—
141	#676	—	2	2	no	—
142	#676	—	2	2	no	—
143	C₈×D₅	—	2	2	no	—
144	C₈×D₅	—	2	2	no	—
145	C₈×D₅	M₁₆	1	4	yes	—
146	C₈×D₅	#41	1	4	yes	—
147	C₅×C₁₆	—	2	2	no	—
148	C₅×C₁₆	—	2	2	no	—
149	#672	—	2	2	no	—
150	#672	—	2	2	no	—
151	C₂×#131	—	5	1	no	—
152	C₂×#131	—	5	1	no	—
153	C₄×#41	—	5	1	no	—
154	C₂×M₃₂	—	5	2	no	—
155	C₂²×C₁₆	—	5	2	no	—
156	#405	—	5	1	no	—
157	#405	—	5	1	no	—
158	#405	—	5	1	no	—
159	#405	—	5	1	no	—
160	#405	—	5	1	no	—
161	#405	C₂×D₅	1	1	yes	characteristic
162	C₄×M₁₆	—	5	1	no	—
163	C₄×M₁₆	—	5	1	no	—
164	C₄×M₁₆	—	5	1	no	—
165	C₂×C₄×C₈	—	5	1	no	—
166	C₂×C₄×C₈	—	5	1	no	—
167	C₂×C₄×C₈	—	5	1	no	—
168	C₄⋊C₁₆	—	5	2	no	—
169	C₄⋊C₁₆	—	5	2	no	—
170	C₄⋊C₁₆	—	5	2	no	—
171	C₄⋊C₁₆	C₂×D₅	1	2	yes	—
172	C₂²⋊C₁₆	—	5	2	no	—
173	C₂²⋊C₁₆	—	5	2	no	—
174	C₂²⋊C₁₆	—	5	2	no	—
175	C₂²⋊C₁₆	C₂×D₅	1	2	yes	—
176	#346	—	5	2	no	—
177	#346	—	5	2	no	—
178	#346	—	5	1	no	—
179	#346	C₂×D₅	1	1	yes	characteristic
180	C₄×C₁₆	—	5	1	no	—
181	C₄×C₁₆	C₂×D₅	1	1	yes	characteristic
182	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
183	C₂²×D₅	C₂²×C₈	1	1	yes	characteristic
184	C₂²×D₅	C₂²×C₈	1	1	yes	characteristic
185	C₂×C₄×C₅	—	2	1	no	—
186	C₂×C₄×C₅	—	2	1	no	—
187	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
188	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
189	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
190	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
191	C₂²⋊D₅	—	2	2	no	—
192	C₂²⋊D₅	—	2	2	no	—
193	C₂×Dic₅	—	4	2	no	—
194	C₂×Dic₅	C₂²×C₈	1	2	yes	—
195	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
196	C₂×Dic₅	C₂²×C₈	1	2	yes	—
197	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
198	D₂₀	—	2	2	no	—
199	C₄×D₅	—	2	1	no	—
200	C₄×D₅	—	2	1	no	—
201	C₄×D₅	—	4	2	no	—
202	C₄×D₅	—	4	2	no	—
203	C₄×D₅	#117	1	2	yes	—
204	C₄×D₅	C₂×M₁₆	1	2	yes	—
205	Dic₁₀	—	2	2	no	—
206	C₅×C₈	—	2	1	no	—
207	C₅×C₈	—	2	1	no	—
208	C₅×C₈	C₂×M₁₆	1	2	yes	—
209	C₅×C₈	C₂×#41	1	2	yes	—
210	#168	—	2	1	no	—
211	#168	—	2	1	no	—
212	#168	#117	1	2	yes	—
213	#168	C₂×M₁₆	1	2	yes	—
214	C₂×#41	—	5	1	no	—
215	#131	—	10	4	no	—
216	#131	—	10	4	no	—
217	C₂×M₁₆	—	5	1	no	—
218	C₂×M₁₆	—	5	1	no	—
219	C₂×M₁₆	—	5	1	no	—
220	C₂×M₁₆	—	5	1	no	—
221	C₂×M₁₆	—	5	1	no	—
222	C₂×M₁₆	C₄×D₅	1	1	yes	characteristic
223	C₂²×C₈	—	5	1	no	—
224	C₂²×C₈	—	5	1	no	—
225	C₂²×C₈	—	5	1	no	—
226	C₂²×C₈	—	5	1	no	—
227	C₂²×C₈	—	5	1	no	—
228	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
229	C₄×Q₈	—	5	2	no	—
230	C₄×D₄	—	5	2	no	—
231	C₄×D₄	—	5	2	no	—
232	C₄×D₄	—	5	2	no	—
233	#117	—	5	1	no	—
234	#117	—	5	1	no	—
235	#117	C₄×D₅	1	1	yes	characteristic
236	C₂×C₄²	—	5	1	no	—
237	C₂×C₄²	—	5	1	no	—
238	C₂×C₄²	—	5	1	no	—
239	M₃₂	—	10	4	no	—
240	C₂×C₁₆	—	5	2	no	—
241	C₂×C₁₆	—	5	2	no	—
242	C₂×C₁₆	—	10	4	no	—
243	C₂×C₁₆	C₂²×D₅	1	2	yes	—
244	C₂×C₁₆	C₂²×D₅	1	2	yes	—
245	#97	—	5	1	no	—
246	#97	—	5	1	no	—
247	#97	—	5	1	no	—
248	#97	—	5	1	no	—
249	#97	C₄×D₅	1	1	yes	characteristic
250	#97	C₄×D₅	1	1	yes	characteristic
251	C₄×C₈	—	5	1	no	—
252	C₄×C₈	—	5	1	no	—
253	C₄×C₈	—	5	1	no	—
254	C₄×C₈	—	5	1	no	—
255	C₄×C₈	—	5	1	no	—
256	C₄×C₈	—	5	1	no	—
257	C₄×C₈	—	5	1	no	—
258	C₄×C₈	—	5	1	no	—
259	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
260	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
261	C₂²×C₅	—	2	2	no	—
262	C₂²×C₅	—	2	1	no	—
263	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
264	C₂×D₅	—	2	2	no	—
265	C₂×D₅	—	2	2	no	—
266	C₂×D₅	—	2	1	no	—
267	C₂×D₅	#431	1	2	yes	—
268	C₄×C₅	—	2	2	no	—
269	C₄×C₅	—	2	2	no	—
270	C₄×C₅	—	2	1	no	—
271	C₄×C₅	C₂×#117	1	1	yes	characteristic
272	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
273	Dic₅	—	2	2	no	—
274	Dic₅	—	2	2	no	—
275	Dic₅	—	2	1	no	—
276	Dic₅	#431	1	2	yes	—
277	#41	—	10	2	no	—
278	#41	—	10	2	no	—
279	C₂×Q₈	—	5	1	no	—
280	C₂×D₄	—	5	1	no	—
281	C₂×D₄	—	5	1	no	—
282	C₂×D₄	—	5	1	no	—
283	C₂²×C₄	—	5	1	no	—
284	C₂²×C₄	—	5	2	no	—
285	C₂²×C₄	—	5	1	no	—
286	C₂²×C₄	—	5	2	no	—
287	C₂²×C₄	—	5	2	no	—
288	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
289	M₁₆	—	2	2	no	—
290	M₁₆	—	10	2	no	—
291	M₁₆	—	10	2	no	—
292	M₁₆	—	10	2	no	—
293	M₁₆	—	10	2	no	—
294	M₁₆	—	10	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	5	1	no	—
297	C₂×C₈	—	5	1	no	—
298	C₂×C₈	—	5	4	no	—
299	C₂×C₈	—	5	4	no	—
300	C₂×C₈	—	5	1	no	—
301	C₂×C₈	—	5	1	no	—
302	C₂×C₈	—	10	2	no	—
303	C₂×C₈	—	10	2	no	—
304	C₂×C₈	—	10	2	no	—
305	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
306	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
307	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
308	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
309	C₄⋊C₄	—	5	2	no	—
310	C₄⋊C₄	—	5	2	no	—
311	C₄⋊C₄	C₈×D₅	1	2	yes	—
312	C₂²⋊C₄	—	5	2	no	—
313	C₂²⋊C₄	—	5	2	no	—
314	C₂²⋊C₄	C₈×D₅	1	2	yes	—
315	C₄²	—	5	1	no	—
316	C₄²	—	5	1	no	—
317	C₄²	—	5	2	no	—
318	C₄²	—	5	2	no	—
319	C₄²	—	5	2	no	—
320	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
321	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
322	C₁₆	—	2	2	no	—
323	C₁₆	—	2	2	no	—
324	C₁₆	—	10	2	no	—
325	C₁₆	—	10	2	no	—
326	C₂×C₅	—	4	1	no	—
327	C₂×C₅	C₂×#431	1	1	yes	characteristic
328	C₂×C₅	#3407	1	2	yes	—
329	D₅	—	2	2	no	—
330	D₅	—	4	1	no	—
331	C₂³	—	5	1	no	—
332	C₂³	—	5	1	no	—
333	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
334	Q₈	—	10	2	no	—
335	D₄	—	10	2	no	—
336	D₄	—	10	2	no	—
337	D₄	—	10	2	no	—
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	5	1	no	—
341	C₂×C₄	—	5	2	no	—
342	C₂×C₄	—	5	2	no	—
343	C₂×C₄	—	5	2	no	—
344	C₂×C₄	—	5	2	no	—
345	C₂×C₄	—	5	1	no	—
346	C₂×C₄	—	10	1	no	—
347	C₂×C₄	—	10	1	no	—
348	C₂×C₄	—	20	2	no	—
349	C₂×C₄	—	20	2	no	—
350	C₂×C₄	—	20	2	no	—
351	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
352	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
353	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
354	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
355	C₈	—	2	1	no	—
356	C₈	—	2	1	no	—
357	C₈	—	5	2	no	—
358	C₈	—	5	2	no	—
359	C₈	—	10	1	no	—
360	C₈	—	10	1	no	—
361	C₈	D₅×M₁₆	1	2	yes	—
362	C₈	D₅×#41	1	2	yes	—
363	C₅	#33421	1	1	yes	characteristic, complemented, Sylow
364	C₂²	—	2	1	no	—
365	C₂²	—	2	2	no	—
366	C₂²	—	5	2	no	—
367	C₂²	—	10	2	no	—
368	C₂²	—	10	1	no	—
369	C₂²	—	10	2	no	—
370	C₂²	C₂²×C₈×D₅	1	1	yes	characteristic
371	C₄	—	2	1	no	—
372	C₄	—	2	2	no	—
373	C₄	—	2	2	no	—
374	C₄	—	5	2	no	—
375	C₄	—	10	1	no	—
376	C₄	—	10	2	no	—
377	C₄	—	10	2	no	—
378	C₄	D₅×#117	1	1	yes	characteristic
379	C₄	C₂×D₅×M₁₆	1	1	yes	characteristic
380	C₂	—	4	1	no	—
381	C₂	—	10	2	no	—
382	C₂	—	20	1	no	—
383	C₂	D₅×#431	1	1	yes	characteristic
384	C₂	#10613743	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₈
Fitting	C₅×#2164	C₂
prime radical	C₂×#709	C₂×C₄
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	#2164	D₅
hypocenter	C₅	#33421
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ #2164

Complements

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

N	H ≅ G/N	type
2. C₅×#2164	C₂	semidirect
3. #10600676	C₂	semidirect
4. #10600078	C₂	semidirect
5. #10600068	C₂	semidirect
7. #10599886	C₂	semidirect
8. #10599867	C₂	semidirect
9. D₅×#346	C₂	semidirect
10. #10599846	C₂	semidirect
11. C₄×#3930	C₂	semidirect
95. #2164	D₅	semidirect
363. C₅	#33421	semidirect

Maximal subgroups

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

subgroup	index
#10610280	2
C₅×#2164	2
#10600676	2
#10600078	2
#10600068	2
#10599900	2
#10599886	2
#10599867	2
D₅×#346	2
#10599846	2
C₄×#3930	2
#33421	5