Subgroups — #12003180

Order 1280.

490 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#65376	C₂	1	1	yes	maximal, characteristic
2	C₅×#1956	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#64767	C₂	1	1	yes	maximal, characteristic, complemented
4	#10600580	C₂	1	1	yes	maximal, characteristic
5	C₂×#64678	C₂	1	1	yes	maximal, characteristic, complemented
6	#10600377	C₂	1	1	yes	maximal, characteristic
7	#10598125	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₂₀	C₂²	1	1	yes	characteristic
10	C₂×C₅×#103	—	2	1	no	—
11	C₂×C₅×#103	—	2	1	no	—
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₅×#384	C₂²	1	1	yes	characteristic
14	C₂×#3952	—	2	1	no	—
15	C₂×#3952	—	2	1	no	—
16	#65376	—	2	2	no	—
17	#65376	—	2	2	no	—
18	C₂×#3941	—	2	1	no	—
19	C₂×#3941	—	2	1	no	—
20	C₂×#3939	—	2	1	no	—
21	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
24	#64767	—	2	2	no	—
25	#64749	—	2	1	no	—
26	#64678	—	2	2	no	—
27	#10485	—	5	1	no	maximal, Sylow
28	C₂²×C₅×Q₈	—	2	1	no	—
29	C₂²×D₂₀	D₄	1	1	yes	characteristic
30	C₂²×C₄×D₅	—	2	1	no	—
31	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
33	C₅×C₄⋊Q₈	—	2	1	no	—
34	C₅×C₄⋊Q₈	—	2	1	no	—
35	C₅×C₄⋊Q₈	—	2	1	no	—
36	C₅×C₄⋊Q₈	D₄	1	2	yes	—
37	C₂×C₅×C₄⋊C₄	—	2	1	no	—
38	C₂×C₅×C₄⋊C₄	—	2	1	no	—
39	C₂×C₅×C₄⋊C₄	—	2	1	no	—
40	C₂×C₅×C₄⋊C₄	—	2	1	no	—
41	C₂×C₅×C₄⋊C₄	—	2	1	no	—
42	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₂×#685	—	2	1	no	—
44	C₂×C₄⋊Dic₅	D₄	1	1	yes	characteristic
45	C₂²×#168	—	2	1	no	—
46	C₂²×#168	—	2	1	no	—
47	C₂²×#168	—	2	1	no	—
48	C₄×D₂₀	—	2	1	no	—
49	C₄×D₂₀	—	2	1	no	—
50	C₄×D₂₀	—	2	1	no	—
51	C₄×D₂₀	—	2	1	no	—
52	C₅×#103	—	4	2	no	—
53	C₅×#103	—	4	2	no	—
54	C₄×C₅×C₈	—	2	1	no	—
55	C₄×C₅×C₈	—	2	1	no	—
56	#3952	—	4	2	no	—
57	#3952	—	4	2	no	—
58	#3941	—	2	4	no	—
59	#3941	—	4	2	no	—
60	#3939	—	2	4	no	—
61	C₄⋊#168	—	2	1	no	—
62	C₄⋊#168	—	2	1	no	—
63	C₄⋊#168	—	2	1	no	—
64	C₄⋊#168	—	2	1	no	—
65	C₄⋊#168	—	2	1	no	—
66	C₄⋊#168	D₄	1	2	yes	—
67	C₂×#474	—	5	1	no	—
68	#1956	D₅	1	1	yes	characteristic, complemented
69	#1954	—	5	1	no	—
70	#1802	—	5	1	no	—
71	C₂×#335	—	5	1	no	—
72	C₂×#332	—	5	1	no	—
73	#1296	—	5	1	no	—
74	C₂³×D₅	—	2	1	no	—
75	C₂×C₅×Q₈	—	2	2	no	—
76	C₂×C₅×Q₈	—	4	1	no	—
77	C₂×C₅×Q₈	—	4	1	no	—
78	C₂×C₅×Q₈	—	4	1	no	—
79	C₂×C₅×Q₈	—	4	1	no	—
80	C₂×C₅×Q₈	—	4	1	no	—
81	C₂²×C₄×C₅	—	4	1	no	—
82	C₂²×C₄×C₅	—	4	1	no	—
83	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	#41	1	1	yes	characteristic
86	C₂²×Dic₅	—	2	1	no	—
87	C₂×D₂₀	—	2	1	no	—
88	C₂×D₂₀	—	2	1	no	—
89	C₂×D₂₀	—	2	1	no	—
90	C₂×D₂₀	—	2	1	no	—
91	C₂×D₂₀	—	2	1	no	—
92	C₂×D₂₀	Q₁₆	1	2	yes	—
93	C₂×C₄×D₅	—	4	1	no	—
94	C₂×C₄×D₅	—	4	1	no	—
95	C₂×C₄×D₅	—	4	1	no	—
96	C₂×C₄×D₅	—	4	1	no	—
97	C₂×C₄×D₅	—	4	1	no	—
98	C₂×C₄×D₅	—	4	1	no	—
99	C₂×C₅×C₈	—	2	1	no	—
100	C₂×C₅×C₈	—	2	1	no	—
101	C₂×C₅×C₈	—	2	1	no	—
102	C₂×C₅×C₈	—	2	1	no	—
103	C₂×C₅×C₈	—	2	1	no	—
104	C₂×C₅×C₈	D₈	1	2	yes	—
105	C₅×C₄⋊C₄	—	2	2	no	—
106	C₅×C₄⋊C₄	—	2	2	no	—
107	C₅×C₄⋊C₄	—	4	1	no	—
108	C₅×C₄⋊C₄	—	4	1	no	—
109	C₅×C₄⋊C₄	—	4	1	no	—
110	C₅×C₄⋊C₄	—	4	1	no	—
111	C₅×C₄⋊C₄	—	4	1	no	—
112	C₅×C₄⋊C₄	—	4	1	no	—
113	C₅×C₄⋊C₄	—	8	1	no	—
114	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₅	#41	1	1	yes	characteristic
117	C₄²×C₅	#41	1	1	yes	characteristic
118	#685	—	4	1	no	—
119	#685	—	4	1	no	—
120	C₄⋊Dic₅	—	2	1	no	—
121	C₄⋊Dic₅	—	2	1	no	—
122	C₂×#168	—	2	2	no	—
123	C₂×#168	—	4	1	no	—
124	C₂×#168	—	4	1	no	—
125	C₂×#168	—	4	1	no	—
126	C₂×#168	—	4	1	no	—
127	C₂×#168	—	4	1	no	—
128	C₂×#168	—	4	1	no	—
129	C₂×#168	—	4	1	no	—
130	C₂×#168	—	4	1	no	—
131	C₂×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
132	C₂×C₄×D₄	—	5	1	no	—
133	#474	—	10	2	no	—
134	#474	—	10	2	no	—
135	C₂×#107	—	10	1	no	—
136	C₂×C₄⋊C₈	—	5	1	no	—
137	C₂×C₄⋊C₈	—	5	1	no	—
138	C₂×C₄⋊C₈	—	5	1	no	—
139	C₂×#103	—	2	1	no	—
140	C₂×#103	—	2	1	no	—
141	C₂×#102	—	10	1	no	—
142	C₂×#102	—	10	1	no	—
143	C₂×C₂²⋊C₈	—	10	1	no	—
144	C₂×C₂²⋊C₈	—	10	1	no	—
145	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
146	#384	C₂×D₅	1	1	yes	characteristic
147	#340	—	10	1	no	—
148	#335	—	10	2	no	—
149	#332	—	10	2	no	—
150	C₂³×C₅	#121	1	1	yes	characteristic
151	C₂²×D₅	—	2	2	no	—
152	C₂²×D₅	—	4	1	no	—
153	C₂²×D₅	—	4	1	no	—
154	C₂²×D₅	—	4	1	no	—
155	C₂²×D₅	—	4	1	no	—
156	C₂²×D₅	—	4	1	no	—
157	C₂²×D₅	—	4	1	no	—
158	C₅×Q₈	—	4	2	no	—
159	C₅×Q₈	—	4	2	no	—
160	C₅×Q₈	—	8	1	no	—
161	C₅×Q₈	—	8	1	no	—
162	C₂×C₄×C₅	—	2	1	no	—
163	C₂×C₄×C₅	—	2	1	no	—
164	C₂×C₄×C₅	—	4	2	no	—
165	C₂×C₄×C₅	—	8	1	no	—
166	C₂×C₄×C₅	—	8	1	no	—
167	C₂×C₄×C₅	—	8	1	no	—
168	C₂×C₄×C₅	—	8	1	no	—
169	C₂×C₄×C₅	—	8	1	no	—
170	C₂×C₄×C₅	#120	1	1	yes	characteristic
171	C₂×C₄×C₅	#121	1	1	yes	characteristic
172	C₂×C₄×C₅	#121	1	1	yes	characteristic
173	C₂×C₄×C₅	#123	1	1	yes	characteristic
174	C₂×C₄×C₅	#123	1	1	yes	characteristic
175	C₂×C₄×C₅	#124	1	1	yes	characteristic
176	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
178	C₂×C₄×C₅	#135	1	1	yes	characteristic
179	C₂×C₄×C₅	#135	1	1	yes	characteristic
180	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
181	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
183	C₂×C₄×C₅	#137	1	1	yes	characteristic
184	C₂×Dic₅	—	4	1	no	—
185	C₂×Dic₅	—	4	1	no	—
186	C₂×Dic₅	—	4	1	no	—
187	D₂₀	—	2	2	no	—
188	D₂₀	—	2	2	no	—
189	D₂₀	—	4	1	no	—
190	D₂₀	—	4	1	no	—
191	C₄×D₅	—	8	1	no	—
192	C₄×D₅	—	8	1	no	—
193	C₄×D₅	—	8	1	no	—
194	C₄×D₅	—	8	1	no	—
195	C₅×C₈	—	2	2	no	—
196	C₅×C₈	—	4	1	no	—
197	#168	—	4	2	no	—
198	#168	—	8	1	no	—
199	#168	—	8	1	no	—
200	C₂²×Q₈	—	2	1	no	—
201	C₂²×D₄	—	5	1	no	—
202	C₂³×C₄	—	10	1	no	—
203	C₂²×C₈	—	10	1	no	—
204	C₂²×C₈	—	10	1	no	—
205	C₂²×C₈	—	10	1	no	—
206	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
207	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
208	C₄⋊Q₈	—	2	1	no	—
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	D₂₀	1	2	yes	—
212	C₄×D₄	—	10	1	no	—
213	C₄×D₄	—	10	1	no	—
214	C₄×D₄	—	10	1	no	—
215	C₄×D₄	—	10	1	no	—
216	C₂×C₄⋊C₄	—	2	1	no	—
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	5	1	no	—
222	C₂×C₂²⋊C₄	—	10	1	no	—
223	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
224	#107	—	10	4	no	—
225	C₄⋊C₈	—	5	2	no	—
226	C₄⋊C₈	—	10	1	no	—
227	C₄⋊C₈	—	10	1	no	—
228	C₄⋊C₈	—	10	1	no	—
229	C₄⋊C₈	—	10	1	no	—
230	C₄⋊C₈	—	10	1	no	—
231	#103	—	4	2	no	—
232	#103	—	4	2	no	—
233	#102	—	10	4	no	—
234	#102	—	20	2	no	—
235	C₂²⋊C₈	—	20	2	no	—
236	C₂²⋊C₈	—	20	2	no	—
237	C₄×C₈	—	2	1	no	—
238	C₄×C₈	—	2	1	no	—
239	C₂²×C₅	#394	1	1	yes	characteristic
240	C₂²×C₅	#459	1	1	yes	characteristic
241	C₂²×C₅	#463	1	1	yes	characteristic
242	C₂²×C₅	#467	1	1	yes	characteristic
243	C₂²×C₅	#468	1	1	yes	characteristic
244	C₂²×C₅	#471	1	1	yes	characteristic
245	C₂²×C₅	#473	1	1	yes	characteristic
246	C₂×D₅	—	4	2	no	—
247	C₂×D₅	—	4	2	no	—
248	C₂×D₅	—	4	2	no	—
249	C₂×D₅	—	8	1	no	—
250	C₂×D₅	—	8	1	no	—
251	C₂×D₅	—	8	1	no	—
252	C₂×D₅	—	8	1	no	—
253	C₄×C₅	—	2	1	no	—
254	C₄×C₅	—	2	1	no	—
255	C₄×C₅	—	8	2	no	—
256	C₄×C₅	—	16	1	no	—
257	C₄×C₅	#449	1	1	yes	characteristic
258	C₄×C₅	#451	1	1	yes	characteristic
259	C₄×C₅	#459	1	1	yes	characteristic
260	C₄×C₅	#463	1	1	yes	characteristic
261	C₄×C₅	#466	1	1	yes	characteristic
262	C₄×C₅	#469	1	1	yes	characteristic
263	C₄×C₅	#484	1	1	yes	characteristic
264	C₄×C₅	#485	1	1	yes	characteristic
265	Dic₅	—	8	1	no	—
266	C₂⁴	—	10	1	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	—	4	1	no	—
269	C₂×Q₈	—	4	1	no	—
270	C₂×Q₈	—	4	1	no	—
271	C₂×Q₈	—	4	1	no	—
272	C₂×Q₈	—	4	1	no	—
273	C₂×D₄	—	5	2	no	—
274	C₂×D₄	—	10	1	no	—
275	C₂×D₄	—	10	1	no	—
276	C₂×D₄	—	10	1	no	—
277	C₂×D₄	—	10	1	no	—
278	C₂×D₄	—	10	1	no	—
279	C₂²×C₄	—	4	1	no	—
280	C₂²×C₄	—	4	1	no	—
281	C₂²×C₄	—	10	1	no	—
282	C₂²×C₄	—	20	1	no	—
283	C₂²×C₄	—	20	1	no	—
284	C₂²×C₄	—	20	1	no	—
285	C₂²×C₄	—	20	1	no	—
286	C₂²×C₄	—	20	1	no	—
287	C₂²×C₄	—	20	1	no	—
288	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
289	C₂²×C₄	#711	1	1	yes	characteristic
290	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	2	1	no	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	10	2	no	—
297	C₂×C₈	—	20	1	no	—
298	C₂×C₈	—	20	1	no	—
299	C₂×C₈	—	20	1	no	—
300	C₂×C₈	—	20	1	no	—
301	C₂×C₈	—	20	1	no	—
302	C₂×C₈	—	20	1	no	—
303	C₂×C₈	—	20	1	no	—
304	C₂×C₈	—	20	1	no	—
305	C₂×C₈	#686	1	2	yes	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	4	1	no	—
309	C₄⋊C₄	—	4	1	no	—
310	C₄⋊C₄	—	4	1	no	—
311	C₄⋊C₄	—	4	1	no	—
312	C₄⋊C₄	—	4	1	no	—
313	C₄⋊C₄	—	4	1	no	—
314	C₄⋊C₄	—	8	1	no	—
315	C₄⋊C₄	—	10	1	no	—
316	C₄⋊C₄	—	10	1	no	—
317	C₂²⋊C₄	—	20	1	no	—
318	C₂²⋊C₄	—	20	1	no	—
319	C₄²	C₂×D₂₀	1	1	yes	characteristic
320	C₄²	#709	1	1	yes	characteristic
321	C₄²	D₄×D₅	1	1	yes	characteristic
322	C₄²	#711	1	1	yes	characteristic
323	C₂×C₅	#1669	1	1	yes	characteristic
324	C₂×C₅	#1683	1	1	yes	characteristic
325	C₂×C₅	#1685	1	1	yes	characteristic
326	C₂×C₅	#1699	1	1	yes	characteristic
327	C₂×C₅	#2036	1	1	yes	characteristic
328	C₂×C₅	#2052	1	1	yes	characteristic
329	C₂×C₅	#2054	1	1	yes	characteristic
330	D₅	—	8	2	no	—
331	C₂³	—	10	2	no	—
332	C₂³	—	20	1	no	—
333	C₂³	—	20	1	no	—
334	C₂³	—	20	1	no	—
335	C₂³	—	20	1	no	—
336	C₂³	—	20	1	no	—
337	C₂³	—	20	1	no	—
338	C₂³	#4084	1	1	yes	characteristic
339	Q₈	—	4	2	no	—
340	Q₈	—	4	2	no	—
341	Q₈	—	8	1	no	—
342	Q₈	—	8	1	no	—
343	D₄	—	10	2	no	—
344	D₄	—	10	2	no	—
345	D₄	—	20	1	no	—
346	D₄	—	20	1	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	8	1	no	—
351	C₂×C₄	—	8	1	no	—
352	C₂×C₄	—	8	1	no	—
353	C₂×C₄	—	8	1	no	—
354	C₂×C₄	—	8	1	no	—
355	C₂×C₄	—	20	1	no	—
356	C₂×C₄	—	20	1	no	—
357	C₂×C₄	—	20	1	no	—
358	C₂×C₄	—	40	1	no	—
359	C₂×C₄	—	40	1	no	—
360	C₂×C₄	—	40	1	no	—
361	C₂×C₄	—	40	1	no	—
362	C₂×C₄	#4028	1	1	yes	characteristic
363	C₂×C₄	#4029	1	1	yes	characteristic
364	C₂×C₄	#4031	1	1	yes	characteristic
365	C₂×C₄	#4032	1	1	yes	characteristic
366	C₂×C₄	#4060	1	1	yes	characteristic
367	C₂×C₄	#4061	1	1	yes	characteristic
368	C₂×C₄	D₅×Q₁₆	1	1	yes	characteristic
369	C₂×C₄	#4065	1	1	yes	characteristic
370	C₂×C₄	#4076	1	1	yes	characteristic
371	C₂×C₄	C₂×#686	1	1	yes	characteristic
372	C₂×C₄	#4078	1	1	yes	characteristic
373	C₂×C₄	#4085	1	1	yes	characteristic
374	C₂×C₄	#4095	1	1	yes	characteristic
375	C₂×C₄	#4096	1	1	yes	characteristic
376	C₈	—	2	2	no	—
377	C₈	—	4	1	no	—
378	C₈	—	20	2	no	—
379	C₈	—	40	1	no	—
380	C₈	—	40	1	no	—
381	C₅	#10485	1	1	yes	characteristic, complemented, Sylow
382	C₂²	—	20	2	no	—
383	C₂²	—	20	2	no	—
384	C₂²	—	20	2	no	—
385	C₂²	—	40	1	no	—
386	C₂²	—	40	1	no	—
387	C₂²	—	40	1	no	—
388	C₂²	—	40	1	no	—
389	C₂²	#65250	1	1	yes	characteristic
390	C₂²	#65346	1	1	yes	characteristic
391	C₂²	#65361	1	1	yes	characteristic
392	C₂²	#65463	1	1	yes	characteristic
393	C₂²	#65466	1	1	yes	characteristic
394	C₂²	#65477	1	1	yes	characteristic
395	C₂²	#65481	1	1	yes	characteristic
396	C₄	—	2	1	no	—
397	C₄	—	2	1	no	—
398	C₄	—	8	2	no	—
399	C₄	—	16	1	no	—
400	C₄	—	40	1	no	—
401	C₄	#65097	1	1	yes	characteristic
402	C₄	#65098	1	1	yes	characteristic
403	C₄	#65100	1	1	yes	characteristic
404	C₄	#65103	1	1	yes	characteristic
405	C₄	#65304	1	1	yes	characteristic
406	C₄	#65305	1	1	yes	characteristic
407	C₄	#65327	1	1	yes	characteristic
408	C₄	#65328	1	1	yes	characteristic
409	C₂	—	40	2	no	—
410	C₂	#10600443	1	1	yes	characteristic
411	C₂	#10601037	1	1	yes	characteristic
412	C₂	#10601057	1	1	yes	characteristic
413	C₂	#10602026	1	1	yes	characteristic
414	C₂	#10602027	1	1	yes	characteristic
415	C₂	#10602061	1	1	yes	characteristic
416	C₂	#10602098	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#4084
commutator, cosocle	C₂×C₄²×C₅	C₂³
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1956	C₂
socle	C₂³×C₅	#121
prime radical	C₂²×D₂₀	D₄
hypercenter	#1956	D₅
hypocenter	C₅	#10485
LCS	G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

Complements

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

N	H ≅ G/N	type
2. C₅×#1956	C₂	semidirect
3. C₂×#64767	C₂	semidirect
5. C₂×#64678	C₂	semidirect
7. #10598125	C₂	semidirect
68. #1956	D₅	semidirect
381. C₅	#10485	semidirect

Maximal subgroups

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

subgroup	index
C₂×#65376	2
C₅×#1956	2
C₂×#64767	2
#10600580	2
C₂×#64678	2
#10600377	2
#10598125	2
#10485	5