Subgroups — #11989413

Order 1280.

730 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#423	C₂	1	1	yes	maximal, characteristic
2	#10613793	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊#168	C₂	1	1	yes	maximal, characteristic
4	#10598193	C₂	1	4	yes	maximal, complemented
5	C₂²×C₅×M₁₆	—	2	1	no	—
6	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₂×#4072	—	2	1	no	—
8	C₂³×#168	—	2	1	no	—
9	C₅×#423	—	2	2	no	—
10	C₅×#423	C₄	1	4	yes	complemented
11	C₂×C₅×C₄⋊C₈	—	2	1	no	—
12	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
13	#65501	—	2	1	no	—
14	#65501	C₄	1	2	yes	—
15	#65225	—	2	1	no	—
16	#65224	C₂²	1	2	yes	—
17	C₄×C₄⋊Dic₅	—	2	1	no	—
18	C₄×C₄⋊Dic₅	C₄	1	2	yes	—
19	C₂×C₄⋊#168	—	2	1	no	—
20	C₂×C₄⋊#168	—	2	2	no	—
21	C₂×C₄⋊#168	C₄	1	4	yes	complemented
22	C₂×C₄⋊#168	C₂²	1	2	yes	—
23	#8045	—	5	1	no	maximal, Sylow
24	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
27	C₂×C₅×M₁₆	—	2	4	no	—
28	C₂×C₅×M₁₆	—	4	2	no	—
29	C₂×C₅×M₁₆	—	4	2	no	—
30	C₂²×C₅×C₈	—	2	2	no	—
31	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	D₄	1	2	yes	—
37	C₂×C₄²×C₅	Q₈	1	2	yes	—
38	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
39	C₂×C₂²⋊Dic₅	—	2	2	no	—
40	#4072	—	4	2	no	—
41	#4072	—	4	2	no	—
42	C₂×C₄⋊Dic₅	—	2	2	no	—
43	C₂×C₄⋊Dic₅	—	2	2	no	—
44	C₂×C₄⋊Dic₅	C₂×C₄	1	4	yes	—
45	C₂×C₄×Dic₅	—	2	2	no	—
46	C₂²×#168	—	2	4	no	—
47	C₂²×#168	—	2	2	no	—
48	C₂²×#168	—	4	4	no	—
49	C₅×C₄⋊C₈	—	2	4	no	—
50	C₅×C₄⋊C₈	—	2	2	no	—
51	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	—
52	#3963	—	2	2	no	—
53	#3963	—	2	2	no	—
54	C₄⋊#168	—	2	4	no	—
55	C₄⋊#168	—	2	2	no	—
56	C₄⋊#168	C₂×C₄	1	4	yes	—
57	C₂×#423	D₅	1	1	yes	characteristic
58	C₂²×C₄⋊C₈	—	5	1	no	—
59	#2285	—	5	1	no	—
60	#1296	—	5	4	no	—
61	C₂⁴×C₅	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	—	2	2	no	—
63	C₂²×C₄×C₅	—	2	2	no	—
64	C₂²×C₄×C₅	—	4	2	no	—
65	C₂²×C₄×C₅	C₄²	1	2	yes	—
66	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
74	C₂²×C₄×C₅	D₈	1	2	yes	—
75	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
76	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
77	C₂²×C₄×C₅	Q₁₆	1	2	yes	—
78	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂²×Dic₅	—	2	4	no	—
82	C₅×M₁₆	—	4	8	no	—
83	C₂×C₅×C₈	—	2	4	no	—
84	C₂×C₅×C₈	—	4	2	no	—
85	C₂×C₅×C₈	—	4	2	no	—
86	C₄²×C₅	C₄²	1	2	yes	—
87	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
88	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
89	C₄²×C₅	C₄⋊C₄	1	2	yes	—
90	C₄²×C₅	C₄⋊C₄	1	2	yes	—
91	C₄²×C₅	C₄⋊C₄	1	2	yes	—
92	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
96	C₂²⋊Dic₅	—	4	4	no	—
97	C₄⋊Dic₅	—	4	2	no	—
98	C₄⋊Dic₅	—	4	2	no	—
99	C₄⋊Dic₅	—	4	2	no	—
100	C₄⋊Dic₅	—	4	2	no	—
101	C₄⋊Dic₅	—	4	4	no	—
102	C₄×Dic₅	—	4	4	no	—
103	C₂×#168	—	2	4	no	—
104	C₂×#168	—	4	8	no	—
105	C₂×#168	—	4	4	no	—
106	C₂²×M₁₆	—	2	1	no	—
107	C₂³×C₈	—	10	1	no	—
108	C₂×#117	—	10	1	no	—
109	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
110	#423	—	2	2	no	—
111	#423	Dic₅	1	4	yes	complemented
112	C₂×C₄⋊C₈	—	2	1	no	—
113	C₂×C₄⋊C₈	—	5	4	no	—
114	C₂×C₄⋊C₈	—	5	2	no	—
115	C₂×C₄⋊C₈	—	10	1	no	—
116	C₂×C₄⋊C₈	—	10	2	no	—
117	C₂×C₄⋊C₈	C₂×D₅	1	2	yes	—
118	#384	—	5	2	no	—
119	#382	—	10	1	no	—
120	#380	—	5	2	no	—
121	#380	—	10	1	no	—
122	C₄×C₄⋊C₄	—	5	2	no	—
123	C₄×C₄⋊C₄	—	10	1	no	—
124	C₂³×C₅	—	2	2	no	—
125	C₂³×C₅	—	2	2	no	—
126	C₂³×C₅	#95	1	2	yes	—
127	C₂³×C₅	#102	1	2	yes	—
128	C₂³×C₅	#103	1	2	yes	—
129	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₅	—	2	2	no	—
131	C₂×C₄×C₅	—	2	2	no	—
132	C₂×C₄×C₅	—	2	2	no	—
133	C₂×C₄×C₅	—	2	2	no	—
134	C₂×C₄×C₅	—	2	2	no	—
135	C₂×C₄×C₅	—	2	4	no	—
136	C₂×C₄×C₅	—	4	2	no	—
137	C₂×C₄×C₅	—	4	2	no	—
138	C₂×C₄×C₅	#95	1	2	yes	—
139	C₂×C₄×C₅	#95	1	2	yes	—
140	C₂×C₄×C₅	#95	1	2	yes	—
141	C₂×C₄×C₅	#102	1	2	yes	—
142	C₂×C₄×C₅	#102	1	4	yes	—
143	C₂×C₄×C₅	#103	1	2	yes	—
144	C₂×C₄×C₅	#103	1	4	yes	—
145	C₂×C₄×C₅	#106	1	4	yes	—
146	C₂×C₄×C₅	#107	1	4	yes	—
147	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
155	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
156	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
157	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
158	C₂×C₄×C₅	#137	1	2	yes	—
159	C₂×Dic₅	—	4	4	no	—
160	C₂×Dic₅	—	4	4	no	—
161	C₂×Dic₅	—	4	4	no	—
162	C₅×C₈	—	4	4	no	—
163	#168	—	4	4	no	—
164	C₂³×C₄	C₄×D₅	1	1	yes	characteristic
165	C₂³×C₄	D₂₀	1	1	yes	characteristic
166	C₂³×C₄	C₂²⋊D₅	1	1	yes	characteristic
167	C₂×M₁₆	—	2	4	no	—
168	C₂×M₁₆	—	4	2	no	—
169	C₂×M₁₆	—	4	2	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	10	4	no	—
172	C₂²×C₈	—	10	2	no	—
173	C₂²×C₈	—	20	4	no	—
174	#117	—	20	2	no	—
175	#117	—	20	2	no	—
176	C₂×C₄⋊C₄	—	5	4	no	—
177	C₂×C₄⋊C₄	—	10	2	no	—
178	C₂×C₄⋊C₄	—	10	2	no	—
179	C₂×C₂²⋊C₄	—	10	2	no	—
180	C₂×C₄²	—	10	2	no	—
181	C₂×C₄²	Dic₁₀	1	2	yes	—
182	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
183	C₂×C₄²	C₄×D₅	1	2	yes	—
184	C₂×C₄²	D₂₀	1	1	yes	characteristic
185	C₂×C₄²	C₂×Dic₅	1	2	yes	—
186	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
187	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
188	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
189	C₄⋊C₈	—	2	4	no	—
190	C₄⋊C₈	—	2	2	no	—
191	C₄⋊C₈	—	5	4	no	—
192	C₄⋊C₈	—	10	4	no	—
193	C₄⋊C₈	—	10	2	no	—
194	C₄⋊C₈	C₂×Dic₅	1	4	yes	—
195	#95	—	10	2	no	—
196	#95	—	10	2	no	—
197	C₂²×C₅	—	2	4	no	—
198	C₂²×C₅	—	2	4	no	—
199	C₂²×C₅	—	2	2	no	—
200	C₂²×C₅	—	2	2	no	—
201	C₂²×C₅	#340	1	4	yes	—
202	C₂²×C₅	C₂×#95	1	1	yes	characteristic
203	C₂²×C₅	#413	1	2	yes	—
204	C₂²×C₅	C₂×#102	1	1	yes	characteristic
205	C₂²×C₅	C₂×#103	1	1	yes	characteristic
206	C₂²×C₅	#417	1	2	yes	—
207	C₄×C₅	—	2	2	no	—
208	C₄×C₅	—	2	2	no	—
209	C₄×C₅	—	2	1	no	—
210	C₄×C₅	—	2	1	no	—
211	C₄×C₅	#340	1	4	yes	—
212	C₄×C₅	C₂×#102	1	1	yes	characteristic
213	C₄×C₅	C₂×#103	1	1	yes	characteristic
214	C₄×C₅	#418	1	1	yes	characteristic
215	C₄×C₅	#419	1	1	yes	characteristic
216	C₄×C₅	C₂×#106	1	1	yes	characteristic
217	C₄×C₅	C₂×#107	1	1	yes	characteristic
218	C₄×C₅	#428	1	2	yes	—
219	Dic₅	—	8	4	no	—
220	C₂⁴	#685	1	1	yes	characteristic
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	—	10	4	no	—
225	C₂²×C₄	C₄×Dic₅	1	2	yes	—
226	C₂²×C₄	#683	1	2	yes	—
227	C₂²×C₄	#683	1	2	yes	—
228	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
229	C₂²×C₄	#685	1	1	yes	characteristic
230	C₂²×C₄	#685	1	1	yes	characteristic
231	C₂²×C₄	#685	1	2	yes	—
232	C₂²×C₄	#685	1	1	yes	characteristic
233	C₂²×C₄	#686	1	2	yes	—
234	C₂²×C₄	#687	1	2	yes	—
235	C₂²×C₄	#688	1	2	yes	—
236	C₂²×C₄	#689	1	2	yes	—
237	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
238	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
241	M₁₆	—	4	8	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	4	2	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	10	4	no	—
246	C₂×C₈	—	20	8	no	—
247	C₂×C₈	—	20	4	no	—
248	C₄⋊C₄	—	20	4	no	—
249	C₄⋊C₄	—	20	2	no	—
250	C₄⋊C₄	—	20	2	no	—
251	C₄⋊C₄	—	20	2	no	—
252	C₄⋊C₄	—	20	2	no	—
253	C₂²⋊C₄	—	20	4	no	—
254	C₄²	—	20	4	no	—
255	C₄²	C₄×Dic₅	1	2	yes	—
256	C₄²	#683	1	2	yes	—
257	C₄²	#683	1	2	yes	—
258	C₄²	C₄⋊Dic₅	1	2	yes	—
259	C₄²	#685	1	2	yes	—
260	C₄²	C₂²⋊Dic₅	1	2	yes	—
261	C₄²	C₂×Dic₁₀	1	1	yes	characteristic
262	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
263	C₄²	C₂²×Dic₅	1	1	yes	characteristic
264	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
265	C₂×C₅	—	2	4	no	—
266	C₂×C₅	#1547	1	2	yes	—
267	C₂×C₅	#1550	1	2	yes	—
268	C₂×C₅	C₂×#340	1	1	yes	characteristic
269	C₂×C₅	#1736	1	1	yes	characteristic
270	C₂×C₅	#1742	1	1	yes	characteristic
271	C₂³	—	2	2	no	—
272	C₂³	—	2	2	no	—
273	C₂³	#3941	1	2	yes	—
274	C₂³	#3942	1	2	yes	—
275	C₂³	#3963	1	2	yes	—
276	C₂³	C₂×#685	1	1	yes	characteristic
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	20	4	no	—
286	C₂×C₄	—	20	4	no	—
287	C₂×C₄	—	20	4	no	—
288	C₂×C₄	#3939	1	4	yes	—
289	C₂×C₄	#3940	1	4	yes	—
290	C₂×C₄	#3941	1	2	yes	—
291	C₂×C₄	#3942	1	2	yes	—
292	C₂×C₄	#3963	1	2	yes	—
293	C₂×C₄	#3963	1	2	yes	—
294	C₂×C₄	#3963	1	2	yes	—
295	C₂×C₄	#3964	1	4	yes	—
296	C₂×C₄	#3967	1	4	yes	—
297	C₂×C₄	#4054	1	2	yes	—
298	C₂×C₄	#4055	1	2	yes	—
299	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
300	C₂×C₄	C₂×#683	1	1	yes	characteristic
301	C₂×C₄	C₂×#683	1	1	yes	characteristic
302	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
303	C₂×C₄	C₂×#685	1	1	yes	characteristic
304	C₂×C₄	C₂×#686	1	1	yes	characteristic
305	C₂×C₄	C₂×#687	1	1	yes	characteristic
306	C₂×C₄	C₂×#688	1	1	yes	characteristic
307	C₂×C₄	C₂×#689	1	1	yes	characteristic
308	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
309	C₈	—	4	4	no	—
310	C₈	—	20	4	no	—
311	C₅	#8045	1	1	yes	characteristic, complemented, Sylow
312	C₂²	—	2	2	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	4	no	—
315	C₂²	—	2	4	no	—
316	C₂²	#64749	1	4	yes	—
317	C₂²	C₂×#3941	1	1	yes	characteristic
318	C₂²	C₂×#3942	1	1	yes	characteristic
319	C₂²	#65421	1	2	yes	—
320	C₂²	#65422	1	2	yes	—
321	C₂²	C₂×#3963	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	40	4	no	—
327	C₄	#64749	1	4	yes	—
328	C₄	C₂×#3939	1	1	yes	characteristic
329	C₄	C₂×#3940	1	1	yes	characteristic
330	C₄	#65409	1	1	yes	characteristic
331	C₄	#65410	1	2	yes	—
332	C₄	#65420	1	1	yes	characteristic
333	C₄	C₂×#3964	1	1	yes	characteristic
334	C₄	C₂×#3967	1	1	yes	characteristic
335	C₂	—	2	4	no	—
336	C₂	#10601296	1	2	yes	—
337	C₂	#10601302	1	2	yes	—
338	C₂	C₂×#64749	1	1	yes	characteristic
339	C₂	#10602778	1	1	yes	characteristic
340	C₂	#10602780	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#685
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₂×C₅×#423	C₂
socle	C₂³×C₅	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₅	C₂²⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂×#423	D₅
hypocenter	C₅	#8045
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
4. #10598193	C₂	semidirect
10. C₅×#423	C₄	semidirect
21. C₂×C₄⋊#168	C₄	semidirect
111. #423	Dic₅	semidirect
311. C₅	#8045	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#423	2
#10613793	2
C₂²×C₄⋊#168	2
#10598193	2
#8045	5