Subgroups — #12068543

Order 1280.

610 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	#10611007	C₂	1	1	yes	maximal, characteristic
2	#10610771	C₂	1	1	yes	maximal, characteristic
3	#10610731	C₂	1	1	yes	maximal, characteristic
4	C₅×#1570	C₂	1	1	yes	maximal, characteristic, complemented
5	#10601542	C₂	1	1	yes	maximal, characteristic, complemented
6	#10601284	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#64699	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599326	C₂	1	2	yes	maximal, complemented
9	#10599326	C₂	1	2	yes	maximal, complemented
10	#10599323	C₂	1	2	yes	maximal, complemented
11	#10599320	C₂	1	2	yes	maximal, complemented
12	C₄×#709	C₂²	1	1	yes	characteristic
13	C₅×#431	C₂²	1	1	yes	characteristic
14	C₅×#423	C₂²	1	1	yes	characteristic
15	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	#65418	—	2	1	no	—
17	#65411	—	2	1	no	—
18	#65399	—	2	1	no	—
19	#65390	—	2	1	no	—
20	#65213	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
22	C₂×C₄×#168	C₂²	1	1	yes	characteristic
23	#65129	—	2	2	no	—
24	#65129	—	2	2	no	—
25	#65124	C₂²	1	2	yes	—
26	#65124	C₂²	1	2	yes	—
27	#65123	C₄	1	1	yes	characteristic
28	#65123	C₄	1	1	yes	characteristic
29	#65122	C₂²	1	2	yes	—
30	#65122	C₂²	1	2	yes	—
31	#65119	C₄	1	1	yes	characteristic
32	#65119	C₄	1	1	yes	characteristic
33	#65014	C₄	1	2	yes	—
34	#65014	C₄	1	2	yes	—
35	#64974	C₂²	1	2	yes	—
36	C₄×C₈×D₅	C₂²	1	2	yes	—
37	C₅×#334	C₂²	1	2	yes	—
38	C₅×#334	C₂²	1	2	yes	—
39	#64699	C₂²	1	2	yes	—
40	#64699	C₂²	1	2	yes	—
41	#64699	C₂²	1	1	yes	characteristic
42	#64699	C₂²	1	1	yes	characteristic
43	#64699	C₂²	1	1	yes	characteristic
44	#64699	C₂²	1	1	yes	characteristic
45	#64677	C₂²	1	2	yes	—
46	#64677	C₂²	1	2	yes	—
47	#17829	—	5	1	no	maximal, Sylow
48	C₂×#709	D₄	1	1	yes	characteristic
49	C₂×C₅×M₁₆	—	2	1	no	—
50	C₂²×C₅×C₈	—	2	1	no	—
51	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
52	C₄×C₂²⋊D₅	C₂×C₄	1	2	yes	—
53	C₄×C₂²⋊D₅	Q₈	1	2	yes	—
54	#4072	D₄	1	1	yes	characteristic
55	C₂×#681	—	2	1	no	—
56	C₂²×#168	—	2	1	no	—
57	C₂²×#168	C₂×C₄	1	1	yes	characteristic
58	C₂²×#168	C₂×C₄	1	1	yes	characteristic
59	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
60	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
61	C₄×D₂₀	C₂×C₄	1	1	yes	characteristic
62	C₄×D₂₀	Q₈	1	1	yes	characteristic
63	#4018	D₄	1	2	yes	—
64	C₄²×D₅	C₂³	1	2	yes	—
65	C₄×Dic₁₀	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₁₀	Q₈	1	1	yes	characteristic
67	C₅×C₄⋊C₈	—	2	1	no	—
68	C₅×C₄⋊C₈	—	2	1	no	—
69	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
70	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
71	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
72	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
73	C₅×C₄⋊C₈	C₂³	1	2	yes	—
74	C₅×C₂²⋊C₈	C₂×C₄	1	2	yes	—
75	C₄×C₅×C₈	C₂³	1	2	yes	—
76	#3952	—	2	2	no	—
77	#3952	—	2	2	no	—
78	#3952	—	2	2	no	—
79	#3952	—	2	2	no	—
80	#3952	C₂×C₄	1	2	yes	—
81	#3952	C₂×C₄	1	2	yes	—
82	#3947	—	2	2	no	—
83	#3947	—	2	2	no	—
84	#3946	C₂×C₄	1	2	yes	—
85	#3946	C₂×C₄	1	2	yes	—
86	C₈×Dic₅	—	2	2	no	—
87	C₈×Dic₅	—	2	2	no	—
88	C₈×Dic₅	C₂×C₄	1	2	yes	—
89	C₈×Dic₅	C₂×C₄	1	2	yes	—
90	C₄⋊#168	—	2	1	no	—
91	C₄⋊#168	—	2	1	no	—
92	C₄⋊#168	C₂³	1	1	yes	characteristic
93	C₄⋊#168	C₂³	1	2	yes	—
94	C₄⋊#168	C₂³	1	1	yes	characteristic
95	C₄×#168	C₂×C₄	1	1	yes	characteristic
96	C₄×#168	C₂×C₄	1	1	yes	characteristic
97	C₄×#168	C₂³	1	1	yes	characteristic
98	C₄×#168	C₂³	1	1	yes	characteristic
99	C₈×#41	—	5	1	no	—
100	#2925	—	5	1	no	—
101	#2925	—	5	1	no	—
102	#1572	—	5	2	no	—
103	#1572	—	5	2	no	—
104	#1570	—	5	1	no	—
105	#1570	D₅	1	1	yes	characteristic, complemented
106	#1568	—	5	2	no	—
107	#1567	—	5	1	no	—
108	#1564	—	5	2	no	—
109	C₂×#335	—	5	1	no	—
110	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
113	C₂×C₂²⋊D₅	C₄⋊C₄	1	2	yes	—
114	#709	—	2	2	no	—
115	#709	—	2	2	no	—
116	C₂×D₂₀	C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
118	C₂×C₄×D₅	C₂×D₄	1	2	yes	—
119	C₂×C₄×D₅	C₂×Q₈	1	2	yes	—
120	C₂×Dic₁₀	C₄⋊C₄	1	1	yes	characteristic
121	C₅×M₁₆	—	4	2	no	—
122	C₂×C₅×C₈	—	2	1	no	—
123	C₂×C₅×C₈	—	2	1	no	—
124	C₂×C₅×C₈	—	2	2	no	—
125	C₂×C₅×C₈	—	4	2	no	—
126	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
127	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
128	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
129	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
131	C₄²×C₅	C₂⁴	1	1	yes	characteristic
132	C₂²⋊Dic₅	C₄⋊C₄	1	2	yes	—
133	#685	C₄⋊C₄	1	2	yes	—
134	#685	C₄⋊C₄	1	2	yes	—
135	C₄⋊Dic₅	C₄⋊C₄	1	1	yes	characteristic
136	C₄⋊Dic₅	C₄⋊C₄	1	1	yes	characteristic
137	#683	C₄⋊C₄	1	2	yes	—
138	#683	C₄⋊C₄	1	2	yes	—
139	C₄×Dic₅	C₂²×C₄	1	2	yes	—
140	C₄×Dic₅	C₂×D₄	1	2	yes	—
141	C₄×Dic₅	C₂×Q₈	1	2	yes	—
142	#681	—	4	2	no	—
143	C₂×#168	—	2	2	no	—
144	C₂×#168	—	2	1	no	—
145	C₂×#168	—	2	2	no	—
146	C₂×#168	—	2	1	no	—
147	C₂×#168	—	4	1	no	—
148	C₂×#168	—	4	2	no	—
149	C₂×#168	C₂²×C₄	1	1	yes	characteristic
150	C₂×#168	C₂²×C₄	1	1	yes	characteristic
151	C₂×#168	C₂²×C₄	1	1	yes	characteristic
152	C₂×#168	C₂²×C₄	1	1	yes	characteristic
153	C₈×D₅	—	2	4	no	—
154	C₈×D₅	—	4	2	no	—
155	C₄×#41	—	5	1	no	—
156	C₈×Q₈	—	5	1	no	—
157	C₈×Q₈	—	5	1	no	—
158	C₈×D₄	—	5	1	no	—
159	C₈×D₄	—	5	1	no	—
160	C₈×D₄	—	5	2	no	—
161	C₈×D₄	—	5	2	no	—
162	#433	—	10	2	no	—
163	#433	—	10	2	no	—
164	#432	—	5	2	no	—
165	#432	—	5	2	no	—
166	#431	—	5	2	no	—
167	#431	—	5	2	no	—
168	#431	—	5	2	no	—
169	#431	C₂×D₅	1	1	yes	characteristic
170	#423	—	5	1	no	—
171	#423	C₂×D₅	1	1	yes	characteristic
172	C₂×C₄⋊C₈	—	5	1	no	—
173	C₂×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
174	#408	—	10	1	no	—
175	#408	—	10	1	no	—
176	#405	—	10	1	no	—
177	#405	—	10	1	no	—
178	C₂×C₄×C₈	—	5	1	no	—
179	C₂×C₄×C₈	—	5	2	no	—
180	#335	—	5	1	no	—
181	#335	—	5	1	no	—
182	#335	—	5	1	no	—
183	#335	—	5	1	no	—
184	#335	—	5	2	no	—
185	#335	—	5	2	no	—
186	#334	—	5	2	no	—
187	#334	—	5	2	no	—
188	#334	C₂×D₅	1	2	yes	—
189	#334	C₂×D₅	1	2	yes	—
190	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
191	C₂²×D₅	C₂×C₄⋊C₄	1	2	yes	—
192	C₂×C₄×C₅	—	2	1	no	—
193	C₂×C₄×C₅	—	2	1	no	—
194	C₂×C₄×C₅	—	4	1	no	—
195	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
196	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
197	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
198	C₂×C₄×C₅	#135	1	2	yes	—
199	C₂×C₄×C₅	#135	1	2	yes	—
200	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
201	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
202	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
203	C₂²⋊D₅	—	2	4	no	—
204	C₂×Dic₅	C₂×C₄⋊C₄	1	2	yes	—
205	C₂×Dic₅	C₂×C₄⋊C₄	1	2	yes	—
206	C₂×Dic₅	C₂×C₄⋊C₄	1	2	yes	—
207	D₂₀	—	2	2	no	—
208	C₄×D₅	—	2	2	no	—
209	C₄×D₅	—	2	2	no	—
210	C₄×D₅	—	2	2	no	—
211	C₄×D₅	—	2	2	no	—
212	C₄×D₅	—	4	2	no	—
213	Dic₁₀	—	2	2	no	—
214	C₅×C₈	—	2	2	no	—
215	C₅×C₈	—	2	2	no	—
216	C₅×C₈	—	4	1	no	—
217	C₅×C₈	—	4	1	no	—
218	C₅×C₈	—	4	2	no	—
219	#168	—	2	1	no	—
220	#168	—	2	1	no	—
221	#168	—	2	1	no	—
222	#168	—	4	1	no	—
223	#168	—	4	1	no	—
224	#168	—	4	2	no	—
225	#168	#131	1	2	yes	—
226	C₂×#41	—	5	1	no	—
227	C₂×M₁₆	—	2	1	no	—
228	C₂×M₁₆	—	10	1	no	—
229	C₂²×C₈	—	2	1	no	—
230	C₂²×C₈	—	5	1	no	—
231	C₂²×C₈	—	5	1	no	—
232	C₂²×C₈	—	5	2	no	—
233	C₂²×C₈	—	5	2	no	—
234	C₂²×C₈	—	10	1	no	—
235	C₄×Q₈	—	5	1	no	—
236	C₄×Q₈	—	5	1	no	—
237	C₄×D₄	—	5	2	no	—
238	C₄×D₄	—	5	2	no	—
239	C₄×D₄	—	5	1	no	—
240	C₄×D₄	—	5	1	no	—
241	#117	—	5	1	no	—
242	#117	—	5	2	no	—
243	C₂×C₄²	—	5	2	no	—
244	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
245	C₄⋊C₈	—	2	1	no	—
246	C₄⋊C₈	—	2	1	no	—
247	C₄⋊C₈	—	5	2	no	—
248	C₄⋊C₈	—	5	2	no	—
249	C₄⋊C₈	—	5	1	no	—
250	C₄⋊C₈	—	5	1	no	—
251	C₄⋊C₈	—	5	2	no	—
252	C₄⋊C₈	—	10	1	no	—
253	C₄⋊C₈	—	10	1	no	—
254	C₄⋊C₈	C₄×D₅	1	1	yes	characteristic
255	C₄⋊C₈	C₄×D₅	1	1	yes	characteristic
256	C₄⋊C₈	C₂²×D₅	1	1	yes	characteristic
257	C₄⋊C₈	C₂²×D₅	1	1	yes	characteristic
258	C₄⋊C₈	C₂²×D₅	1	2	yes	—
259	C₂²⋊C₈	—	5	2	no	—
260	C₂²⋊C₈	—	5	2	no	—
261	C₂²⋊C₈	—	10	2	no	—
262	C₂²⋊C₈	—	10	2	no	—
263	C₂²⋊C₈	—	10	2	no	—
264	C₂²⋊C₈	—	10	2	no	—
265	C₂²⋊C₈	C₄×D₅	1	2	yes	—
266	#97	—	10	2	no	—
267	#97	—	10	2	no	—
268	C₄×C₈	—	5	2	no	—
269	C₄×C₈	—	5	2	no	—
270	C₄×C₈	—	5	1	no	—
271	C₄×C₈	—	5	1	no	—
272	C₄×C₈	—	5	1	no	—
273	C₄×C₈	—	5	1	no	—
274	C₄×C₈	—	10	2	no	—
275	C₄×C₈	—	10	2	no	—
276	C₄×C₈	C₂²×D₅	1	2	yes	—
277	C₂²×C₅	—	2	1	no	—
278	C₂²×C₅	—	2	1	no	—
279	C₂²×C₅	#427	1	2	yes	—
280	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
281	C₂×D₅	—	2	2	no	—
282	C₂×D₅	—	2	2	no	—
283	C₂×D₅	—	2	2	no	—
284	C₄×C₅	—	2	1	no	—
285	C₄×C₅	—	2	1	no	—
286	C₄×C₅	—	2	1	no	—
287	C₄×C₅	#427	1	2	yes	—
288	C₄×C₅	C₂×#131	1	1	yes	characteristic
289	C₄×C₅	#568	1	1	yes	characteristic
290	C₄×C₅	C₂×#135	1	1	yes	characteristic
291	C₄×C₅	C₂×#135	1	1	yes	characteristic
292	Dic₅	—	2	2	no	—
293	Dic₅	—	2	2	no	—
294	Dic₅	—	2	2	no	—
295	#41	—	10	2	no	—
296	#41	—	10	2	no	—
297	C₂×Q₈	—	5	1	no	—
298	C₂×D₄	—	5	2	no	—
299	C₂×D₄	—	5	1	no	—
300	C₂²×C₄	—	5	2	no	—
301	C₂²×C₄	—	5	2	no	—
302	C₂²×C₄	—	5	2	no	—
303	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
304	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
305	C₂²×C₄	Q₈×D₅	1	1	yes	characteristic
306	M₁₆	—	4	2	no	—
307	M₁₆	—	20	2	no	—
308	C₂×C₈	—	2	1	no	—
309	C₂×C₈	—	2	2	no	—
310	C₂×C₈	—	2	1	no	—
311	C₂×C₈	—	4	2	no	—
312	C₂×C₈	—	5	1	no	—
313	C₂×C₈	—	5	1	no	—
314	C₂×C₈	—	5	1	no	—
315	C₂×C₈	—	5	1	no	—
316	C₂×C₈	—	10	4	no	—
317	C₂×C₈	—	10	2	no	—
318	C₂×C₈	—	10	1	no	—
319	C₂×C₈	—	10	2	no	—
320	C₂×C₈	—	10	1	no	—
321	C₂×C₈	—	20	2	no	—
322	C₂×C₈	—	20	1	no	—
323	C₂×C₈	—	20	2	no	—
324	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
325	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
326	C₄⋊C₄	—	5	2	no	—
327	C₄⋊C₄	—	5	2	no	—
328	C₄⋊C₄	—	5	1	no	—
329	C₄⋊C₄	—	5	1	no	—
330	C₂²⋊C₄	—	5	2	no	—
331	C₂²⋊C₄	—	5	2	no	—
332	C₂²⋊C₄	—	5	2	no	—
333	C₄²	—	5	2	no	—
334	C₄²	—	5	2	no	—
335	C₄²	—	5	2	no	—
336	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
337	C₄²	D₄×D₅	1	1	yes	characteristic
338	C₄²	Q₈×D₅	1	1	yes	characteristic
339	C₄²	C₂³×D₅	1	1	yes	characteristic
340	C₂×C₅	—	2	2	no	—
341	C₂×C₅	#2906	1	1	yes	characteristic
342	C₂×C₅	C₂×#427	1	1	yes	characteristic
343	C₂×C₅	#2916	1	1	yes	characteristic
344	D₅	—	4	2	no	—
345	C₂³	—	5	2	no	—
346	C₂³	D₅×C₄⋊C₄	1	1	yes	characteristic
347	Q₈	—	10	2	no	—
348	D₄	—	10	4	no	—
349	D₄	—	10	2	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	4	1	no	—
353	C₂×C₄	—	5	2	no	—
354	C₂×C₄	—	5	2	no	—
355	C₂×C₄	—	5	2	no	—
356	C₂×C₄	—	10	2	no	—
357	C₂×C₄	—	10	2	no	—
358	C₂×C₄	—	10	2	no	—
359	C₂×C₄	—	10	2	no	—
360	C₂×C₄	—	20	2	no	—
361	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
362	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
363	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
364	C₂×C₄	#4062	1	2	yes	—
365	C₂×C₄	#4062	1	2	yes	—
366	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
367	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
368	C₂×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
369	C₈	—	2	2	no	—
370	C₈	—	2	2	no	—
371	C₈	—	4	1	no	—
372	C₈	—	4	1	no	—
373	C₈	—	4	2	no	—
374	C₈	—	5	2	no	—
375	C₈	—	10	1	no	—
376	C₈	—	10	1	no	—
377	C₈	—	10	1	no	—
378	C₈	—	20	1	no	—
379	C₈	—	20	1	no	—
380	C₈	—	20	2	no	—
381	C₅	#17829	1	1	yes	characteristic, complemented, Sylow
382	C₂²	—	2	1	no	—
383	C₂²	—	2	1	no	—
384	C₂²	—	10	2	no	—
385	C₂²	—	10	2	no	—
386	C₂²	—	10	2	no	—
387	C₂²	#65149	1	2	yes	—
388	C₂²	C₂×D₅×C₄⋊C₄	1	1	yes	characteristic
389	C₄	—	2	1	no	—
390	C₄	—	2	1	no	—
391	C₄	—	2	1	no	—
392	C₄	—	10	2	no	—
393	C₄	—	10	2	no	—
394	C₄	—	10	2	no	—
395	C₄	#65149	1	2	yes	—
396	C₄	D₅×#131	1	1	yes	characteristic
397	C₄	#66084	1	1	yes	characteristic
398	C₄	C₂×#4062	1	1	yes	characteristic
399	C₄	C₂×#4062	1	1	yes	characteristic
400	C₂	—	2	2	no	—
401	C₂	—	20	2	no	—
402	C₂	#10610812	1	1	yes	characteristic
403	C₂	C₂×#65149	1	1	yes	characteristic
404	C₂	#10610991	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×#1570	C₂
socle	C₂²×C₅	C₂²×C₄⋊C₄
prime radical	C₂×#709	D₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	#1570	D₅
hypocenter	C₅	#17829
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1570

Complements

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

N	H ≅ G/N	type
4. C₅×#1570	C₂	semidirect
5. #10601542	C₂	semidirect
6. #10601284	C₂	semidirect
7. C₂×#64699	C₂	semidirect
8. #10599326	C₂	semidirect
9. #10599326	C₂	semidirect
10. #10599323	C₂	semidirect
11. #10599320	C₂	semidirect
105. #1570	D₅	semidirect
381. C₅	#17829	semidirect

Maximal subgroups

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

subgroup	index
#10611007	2
#10610771	2
#10610731	2
C₅×#1570	2
#10601542	2
#10601284	2
C₂×#64699	2
#10599326	2
#10599323	2
#10599320	2
#17829	5