Subgroups — #12068579

Order 1280.

588 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10611016	C₂	1	2	yes	maximal
2	#10610178	C₂	1	1	yes	maximal, characteristic
3	C₅×#1564	C₂	1	1	yes	maximal, characteristic, complemented
4	#10601540	C₂	1	2	yes	maximal, complemented
5	#10600525	C₂	1	1	yes	maximal, characteristic, complemented
6	#10599364	C₂	1	2	yes	maximal, complemented
7	#10599361	C₂	1	2	yes	maximal, complemented
8	#10599328	C₂	1	2	yes	maximal, complemented
9	#10599327	C₂	1	2	yes	maximal, complemented
10	C₄×#709	C₂²	1	1	yes	characteristic
11	C₅×#431	C₂²	1	2	yes	—
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	#65400	C₄	1	2	yes	—
14	#65400	C₄	1	2	yes	—
15	#65215	C₂²	1	2	yes	—
16	#65214	C₂²	1	1	yes	characteristic
17	#65126	C₂²	1	2	yes	—
18	#65124	C₂²	1	2	yes	—
19	#65122	C₂²	1	2	yes	—
20	D₅×C₄⋊C₈	C₂²	1	2	yes	—
21	#65016	—	2	2	no	—
22	#65009	—	2	2	no	—
23	#64979	—	2	2	no	—
24	#64978	—	2	2	no	—
25	#64977	C₄	1	1	yes	characteristic
26	#64977	C₄	1	1	yes	characteristic
27	C₄×#676	C₂²	1	2	yes	—
28	#64974	C₂²	1	2	yes	—
29	#64968	C₄	1	1	yes	characteristic
30	#64968	C₄	1	1	yes	characteristic
31	C₅×#335	C₂²	1	1	yes	characteristic
32	C₅×#335	C₂²	1	1	yes	characteristic
33	C₅×#334	C₂²	1	2	yes	—
34	#64700	C₂²	1	2	yes	—
35	#64700	C₂²	1	2	yes	—
36	#64699	C₂²	1	2	yes	—
37	#64699	C₂²	1	2	yes	—
38	#64678	C₂²	1	1	yes	characteristic
39	#64678	C₂²	1	1	yes	characteristic
40	#64677	C₂²	1	2	yes	—
41	#17833	—	5	1	no	maximal, Sylow
42	C₂×#709	D₄	1	1	yes	characteristic
43	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₄×C₂²⋊D₅	C₂×C₄	1	2	yes	—
47	C₄×C₂²⋊D₅	Q₈	1	2	yes	—
48	#4072	D₄	1	1	yes	characteristic
49	C₂×#676	—	2	2	no	—
50	C₂×#676	C₂×C₄	1	2	yes	—
51	C₂×#676	C₂×C₄	1	2	yes	—
52	C₂×C₈×D₅	—	2	2	no	—
53	C₄×D₂₀	C₂×C₄	1	1	yes	characteristic
54	C₄×D₂₀	Q₈	1	1	yes	characteristic
55	#4018	D₄	1	2	yes	—
56	C₄²×D₅	C₂³	1	2	yes	—
57	C₄×Dic₁₀	C₂×C₄	1	1	yes	characteristic
58	C₄×Dic₁₀	Q₈	1	1	yes	characteristic
59	C₅×C₄⋊C₈	C₂³	1	2	yes	—
60	C₅×C₄⋊C₈	C₂³	1	2	yes	—
61	C₅×C₂²⋊C₈	—	2	2	no	—
62	C₄×C₅×C₈	—	2	2	no	—
63	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
64	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
65	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
66	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
67	C₂²⋊#168	—	2	2	no	—
68	C₂²⋊#168	C₂×C₄	1	2	yes	—
69	#3952	—	2	2	no	—
70	#3952	—	2	2	no	—
71	#3952	C₂×C₄	1	2	yes	—
72	#3952	C₂×C₄	1	2	yes	—
73	#3947	—	2	2	no	—
74	#3947	C₂×C₄	1	2	yes	—
75	#3947	C₂×C₄	1	2	yes	—
76	#3946	—	2	2	no	—
77	#3946	—	2	2	no	—
78	#3946	C₂×C₄	1	2	yes	—
79	#3946	C₂×C₄	1	2	yes	—
80	C₈×Dic₅	—	2	2	no	—
81	C₄⋊#168	C₂×C₄	1	1	yes	characteristic
82	C₄⋊#168	C₂×C₄	1	1	yes	characteristic
83	C₄⋊#168	C₂³	1	2	yes	—
84	C₄⋊#168	C₂³	1	2	yes	—
85	#3935	—	2	2	no	—
86	C₄×#168	C₂³	1	2	yes	—
87	#2965	—	5	1	no	—
88	#2925	—	5	2	no	—
89	#1574	—	5	2	no	—
90	#1573	—	5	2	no	—
91	#1572	—	5	2	no	—
92	#1569	—	5	2	no	—
93	#1568	—	5	1	no	—
94	#1565	—	5	2	no	—
95	#1564	D₅	1	1	yes	characteristic, complemented
96	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
99	C₂×C₂²⋊D₅	C₄⋊C₄	1	2	yes	—
100	#709	—	4	1	no	—
101	#709	—	4	1	no	—
102	C₂×D₂₀	C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
104	C₂×C₄×D₅	C₂×D₄	1	2	yes	—
105	C₂×C₄×D₅	C₂×Q₈	1	2	yes	—
106	C₂×Dic₁₀	C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₅×C₈	—	2	2	no	—
108	C₂×C₅×C₈	—	2	2	no	—
109	C₂×C₅×C₈	—	2	2	no	—
110	C₂×C₅×C₈	—	4	1	no	—
111	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
115	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
116	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
118	C₄²×C₅	C₂⁴	1	1	yes	characteristic
119	C₂²⋊Dic₅	C₄⋊C₄	1	2	yes	—
120	#685	C₄⋊C₄	1	2	yes	—
121	#685	C₄⋊C₄	1	2	yes	—
122	C₄⋊Dic₅	C₄⋊C₄	1	1	yes	characteristic
123	C₄⋊Dic₅	C₄⋊C₄	1	1	yes	characteristic
124	#683	C₄⋊C₄	1	2	yes	—
125	#683	C₄⋊C₄	1	2	yes	—
126	C₄×Dic₅	C₂²×C₄	1	2	yes	—
127	C₄×Dic₅	C₂×D₄	1	2	yes	—
128	C₄×Dic₅	C₂×Q₈	1	2	yes	—
129	C₂×#168	—	2	2	no	—
130	C₂×#168	—	2	2	no	—
131	C₂×#168	C₂²×C₄	1	2	yes	—
132	C₂×#168	C₂²×C₄	1	2	yes	—
133	#676	—	2	4	no	—
134	#676	—	4	2	no	—
135	#676	—	4	4	no	—
136	C₈×D₅	—	4	4	no	—
137	C₄×#41	—	5	1	no	—
138	#446	—	5	1	no	—
139	#446	—	5	1	no	—
140	#436	—	5	1	no	—
141	#436	—	5	1	no	—
142	#435	—	5	2	no	—
143	#435	—	5	2	no	—
144	#433	—	10	2	no	—
145	#433	—	10	2	no	—
146	#432	—	5	2	no	—
147	#432	—	5	2	no	—
148	#431	—	5	1	no	—
149	#431	—	5	2	no	—
150	#431	—	5	2	no	—
151	#431	C₂×D₅	1	2	yes	—
152	#423	—	5	2	no	—
153	C₂×C₄⋊C₈	—	5	2	no	—
154	#408	—	10	2	no	—
155	#405	—	10	2	no	—
156	C₄×M₁₆	—	5	2	no	—
157	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
158	#335	—	5	2	no	—
159	#335	—	5	2	no	—
160	#335	—	5	1	no	—
161	#335	—	5	1	no	—
162	#335	C₂×D₅	1	1	yes	characteristic
163	#335	C₂×D₅	1	1	yes	characteristic
164	#334	—	5	2	no	—
165	#334	—	5	2	no	—
166	#334	—	5	2	no	—
167	#334	C₂×D₅	1	2	yes	—
168	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
169	C₂²×D₅	C₂×C₄⋊C₄	1	2	yes	—
170	C₂×C₄×C₅	—	2	1	no	—
171	C₂×C₄×C₅	—	2	1	no	—
172	C₂×C₄×C₅	—	2	1	no	—
173	C₂×C₄×C₅	—	2	1	no	—
174	C₂×C₄×C₅	—	4	1	no	—
175	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
176	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
178	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
179	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
181	C₂²⋊D₅	—	4	2	no	—
182	C₂×Dic₅	C₂×C₄⋊C₄	1	2	yes	—
183	C₂×Dic₅	C₂×C₄⋊C₄	1	2	yes	—
184	C₂×Dic₅	C₂×C₄⋊C₄	1	2	yes	—
185	D₂₀	—	4	1	no	—
186	C₄×D₅	—	2	2	no	—
187	C₄×D₅	—	2	2	no	—
188	C₄×D₅	—	2	2	no	—
189	C₄×D₅	—	2	2	no	—
190	C₄×D₅	—	4	2	no	—
191	Dic₁₀	—	4	1	no	—
192	C₅×C₈	—	2	1	no	—
193	C₅×C₈	—	2	1	no	—
194	C₅×C₈	—	2	1	no	—
195	C₅×C₈	—	4	2	no	—
196	C₅×C₈	—	4	2	no	—
197	C₅×C₈	#131	1	2	yes	—
198	#168	—	2	2	no	—
199	#168	—	2	2	no	—
200	#168	—	4	2	no	—
201	#168	—	4	2	no	—
202	C₂×#41	—	5	1	no	—
203	C₂×M₁₆	—	5	2	no	—
204	C₂×M₁₆	—	5	2	no	—
205	C₂×M₁₆	—	10	2	no	—
206	C₂²×C₈	—	10	2	no	—
207	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
208	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
209	C₄×Q₈	—	5	1	no	—
210	C₄×Q₈	—	5	1	no	—
211	C₄×D₄	—	5	2	no	—
212	C₄×D₄	—	5	2	no	—
213	C₄×D₄	—	5	1	no	—
214	C₄×D₄	—	5	1	no	—
215	#117	—	5	1	no	—
216	#117	—	5	2	no	—
217	C₂×C₄²	—	5	2	no	—
218	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
219	C₄⋊C₈	—	5	2	no	—
220	C₄⋊C₈	—	5	2	no	—
221	C₄⋊C₈	—	5	1	no	—
222	C₄⋊C₈	—	5	1	no	—
223	C₄⋊C₈	—	5	2	no	—
224	C₄⋊C₈	—	5	2	no	—
225	C₄⋊C₈	—	10	2	no	—
226	C₄⋊C₈	—	10	2	no	—
227	C₄⋊C₈	C₂²×D₅	1	2	yes	—
228	C₄⋊C₈	C₂²×D₅	1	2	yes	—
229	C₂²⋊C₈	—	2	2	no	—
230	C₂²⋊C₈	—	5	2	no	—
231	C₂²⋊C₈	—	5	2	no	—
232	C₂²⋊C₈	—	5	2	no	—
233	C₂²⋊C₈	—	10	2	no	—
234	C₂²⋊C₈	—	10	2	no	—
235	C₂²⋊C₈	—	10	2	no	—
236	#97	—	5	2	no	—
237	#97	—	5	2	no	—
238	#97	—	10	2	no	—
239	#97	—	10	2	no	—
240	C₄×C₈	—	2	2	no	—
241	C₄×C₈	—	5	2	no	—
242	C₄×C₈	—	10	2	no	—
243	C₄×C₈	C₄×D₅	1	1	yes	characteristic
244	C₄×C₈	C₄×D₅	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
246	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
247	C₂²×C₅	—	2	1	no	—
248	C₂²×C₅	—	2	1	no	—
249	C₂²×C₅	—	2	1	no	—
250	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
251	C₂×D₅	—	2	2	no	—
252	C₂×D₅	—	2	2	no	—
253	C₂×D₅	—	2	2	no	—
254	C₄×C₅	—	2	1	no	—
255	C₄×C₅	—	2	1	no	—
256	C₄×C₅	—	2	1	no	—
257	C₄×C₅	—	2	1	no	—
258	C₄×C₅	C₂×#131	1	1	yes	characteristic
259	C₄×C₅	#568	1	1	yes	characteristic
260	C₄×C₅	#575	1	1	yes	characteristic
261	C₄×C₅	#575	1	1	yes	characteristic
262	Dic₅	—	2	2	no	—
263	Dic₅	—	2	2	no	—
264	Dic₅	—	2	2	no	—
265	#41	—	20	1	no	—
266	#41	—	20	1	no	—
267	C₂×Q₈	—	5	1	no	—
268	C₂×D₄	—	5	2	no	—
269	C₂×D₄	—	5	1	no	—
270	C₂²×C₄	—	5	2	no	—
271	C₂²×C₄	—	5	2	no	—
272	C₂²×C₄	—	5	2	no	—
273	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
274	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
275	C₂²×C₄	Q₈×D₅	1	1	yes	characteristic
276	M₁₆	—	10	4	no	—
277	M₁₆	—	20	2	no	—
278	M₁₆	—	20	4	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	4	1	no	—
283	C₂×C₈	—	5	2	no	—
284	C₂×C₈	—	5	2	no	—
285	C₂×C₈	—	10	2	no	—
286	C₂×C₈	—	10	2	no	—
287	C₂×C₈	—	20	4	no	—
288	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
289	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
290	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
291	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
292	C₄⋊C₄	—	5	2	no	—
293	C₄⋊C₄	—	5	2	no	—
294	C₄⋊C₄	—	5	1	no	—
295	C₄⋊C₄	—	5	1	no	—
296	C₂²⋊C₄	—	5	2	no	—
297	C₂²⋊C₄	—	5	2	no	—
298	C₂²⋊C₄	—	5	2	no	—
299	C₄²	—	5	2	no	—
300	C₄²	—	5	2	no	—
301	C₄²	—	5	2	no	—
302	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
303	C₄²	D₄×D₅	1	1	yes	characteristic
304	C₄²	Q₈×D₅	1	1	yes	characteristic
305	C₄²	C₂³×D₅	1	1	yes	characteristic
306	C₂×C₅	—	4	1	no	—
307	C₂×C₅	#2906	1	1	yes	characteristic
308	C₂×C₅	#2911	1	1	yes	characteristic
309	C₂×C₅	#2916	1	1	yes	characteristic
310	D₅	—	4	2	no	—
311	C₂³	—	5	2	no	—
312	C₂³	D₅×C₄⋊C₄	1	1	yes	characteristic
313	Q₈	—	20	1	no	—
314	D₄	—	20	1	no	—
315	D₄	—	20	2	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	5	2	no	—
322	C₂×C₄	—	5	2	no	—
323	C₂×C₄	—	5	2	no	—
324	C₂×C₄	—	10	2	no	—
325	C₂×C₄	—	10	2	no	—
326	C₂×C₄	—	10	2	no	—
327	C₂×C₄	—	10	2	no	—
328	C₂×C₄	—	20	2	no	—
329	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
330	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
331	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
332	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
334	C₂×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₈	—	2	1	no	—
338	C₈	—	4	2	no	—
339	C₈	—	4	2	no	—
340	C₈	—	10	2	no	—
341	C₈	—	10	2	no	—
342	C₈	—	20	2	no	—
343	C₈	—	20	2	no	—
344	C₈	#4047	1	2	yes	—
345	C₅	#17833	1	1	yes	characteristic, complemented, Sylow
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	10	2	no	—
350	C₂²	—	10	2	no	—
351	C₂²	—	10	2	no	—
352	C₂²	C₂×D₅×C₄⋊C₄	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	2	1	no	—
357	C₄	—	10	2	no	—
358	C₄	—	10	2	no	—
359	C₄	—	10	2	no	—
360	C₄	C₂×#4047	1	1	yes	characteristic
361	C₄	#66079	1	1	yes	characteristic
362	C₄	#66102	1	1	yes	characteristic
363	C₄	#66102	1	1	yes	characteristic
364	C₂	—	4	1	no	—
365	C₂	—	20	2	no	—
366	C₂	#10610743	1	1	yes	characteristic
367	C₂	#10610908	1	1	yes	characteristic
368	C₂	#10610983	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×#1564	C₂
socle	C₂²×C₅	C₂²×C₄⋊C₄
prime radical	C₂×#709	D₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	#1564	D₅
hypocenter	C₅	#17833
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
3. C₅×#1564	C₂	semidirect
4. #10601540	C₂	semidirect
5. #10600525	C₂	semidirect
6. #10599364	C₂	semidirect
7. #10599361	C₂	semidirect
8. #10599328	C₂	semidirect
9. #10599327	C₂	semidirect
95. #1564	D₅	semidirect
345. C₅	#17833	semidirect

Maximal subgroups

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

subgroup	index
#10611016	2
#10610178	2
C₅×#1564	2
#10601540	2
#10600525	2
#10599364	2
#10599361	2
#10599328	2
#10599327	2
#17833	5