Subgroups — #11989404

Order 1280.

790 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×#65224	C₂	1	1	yes	maximal, characteristic
3	C₂×#65213	C₂	1	1	yes	maximal, characteristic
4	#10598193	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
6	C₂²×C₄⋊Dic₅	—	2	1	no	—
7	C₂²×C₄⋊Dic₅	—	2	1	no	—
8	C₂²×C₄⋊Dic₅	C₄	1	2	yes	—
9	C₂²×#681	—	2	1	no	—
10	C₅×#431	—	2	2	no	—
11	C₅×#431	C₄	1	4	yes	complemented
12	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₅×C₂²⋊C₈	—	2	1	no	—
14	C₂×C₄×C₅×C₈	—	2	1	no	—
15	#65224	—	2	1	no	—
16	#65224	—	2	1	no	—
17	#65224	C₄	1	2	yes	—
18	#65224	C₂²	1	2	yes	—
19	#65213	—	2	2	no	—
20	#65213	C₄	1	4	yes	complemented
21	C₂×C₄⋊#168	—	2	1	no	—
22	C₂×C₄⋊#168	C₂²	1	2	yes	—
23	#8043	—	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	C₂³×Dic₅	—	2	2	no	—
28	C₂²×C₅×C₈	—	2	2	no	—
29	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₅	D₄	1	2	yes	—
35	C₂×C₄²×C₅	Q₈	1	2	yes	—
36	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
37	C₂×C₄⋊Dic₅	—	2	2	no	—
38	C₂×C₄⋊Dic₅	—	2	2	no	—
39	C₂×C₄⋊Dic₅	—	2	2	no	—
40	C₂×C₄⋊Dic₅	—	2	2	no	—
41	C₂×C₄⋊Dic₅	—	2	2	no	—
42	C₂×C₄⋊Dic₅	—	4	4	no	—
43	C₂×C₄⋊Dic₅	—	4	2	no	—
44	C₂×C₄⋊Dic₅	—	4	2	no	—
45	C₂×C₄⋊Dic₅	—	4	4	no	—
46	C₂×C₄⋊Dic₅	C₂×C₄	1	4	yes	—
47	C₂×#681	—	2	4	no	—
48	C₂×#681	—	4	2	no	—
49	C₂×#681	—	4	2	no	—
50	C₂²×#168	—	2	2	no	—
51	C₅×C₄⋊C₈	—	2	2	no	—
52	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
53	C₅×C₂²⋊C₈	—	2	4	no	—
54	C₄×C₅×C₈	—	2	4	no	—
55	C₄⋊#168	—	2	4	no	—
56	C₄⋊#168	—	2	2	no	—
57	C₄⋊#168	C₂×C₄	1	4	yes	complemented
58	C₂×#431	D₅	1	1	yes	characteristic
59	C₂×#423	—	5	1	no	—
60	C₂×#384	—	5	1	no	—
61	#1296	—	5	4	no	—
62	C₂⁴×C₅	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	—	2	2	no	—
64	C₂²×C₄×C₅	—	2	2	no	—
65	C₂²×C₄×C₅	—	2	2	no	—
66	C₂²×C₄×C₅	—	2	2	no	—
67	C₂²×C₄×C₅	C₄²	1	2	yes	—
68	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
72	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
73	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
74	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
75	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
76	C₂²×C₄×C₅	D₈	1	2	yes	—
77	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
78	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
79	C₂²×C₄×C₅	Q₁₆	1	2	yes	—
80	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂²×Dic₅	—	2	4	no	—
84	C₂²×Dic₅	—	4	4	no	—
85	C₂²×Dic₅	—	4	4	no	—
86	C₂²×Dic₅	—	4	4	no	—
87	C₂×C₅×C₈	—	2	4	no	—
88	C₂×C₅×C₈	—	4	2	no	—
89	C₂×C₅×C₈	—	4	2	no	—
90	C₄²×C₅	C₄²	1	2	yes	—
91	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
92	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
93	C₄²×C₅	C₄⋊C₄	1	2	yes	—
94	C₄²×C₅	C₄⋊C₄	1	2	yes	—
95	C₄²×C₅	C₄⋊C₄	1	2	yes	—
96	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
100	C₄⋊Dic₅	—	4	4	no	—
101	C₄⋊Dic₅	—	4	4	no	—
102	C₄⋊Dic₅	—	4	4	no	—
103	C₄⋊Dic₅	—	4	2	no	—
104	C₄⋊Dic₅	—	4	2	no	—
105	C₄⋊Dic₅	—	4	4	no	—
106	C₄⋊Dic₅	—	4	4	no	—
107	#681	—	4	8	no	—
108	C₂×#168	—	2	4	no	—
109	C₂×#168	—	4	2	no	—
110	C₂×#168	—	4	2	no	—
111	C₂²×M₁₆	—	10	1	no	—
112	C₂²×C₄⋊C₄	—	5	2	no	—
113	C₂²×C₄⋊C₄	—	10	1	no	—
114	C₂²×C₄⋊C₄	—	10	1	no	—
115	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
116	#431	—	2	2	no	—
117	#431	Dic₅	1	4	yes	complemented
118	#423	—	5	4	no	—
119	#423	—	10	2	no	—
120	C₂×C₄⋊C₈	—	5	2	no	—
121	C₂×C₄⋊C₈	—	10	1	no	—
122	C₂×C₄⋊C₈	C₂×D₅	1	2	yes	—
123	C₂×C₂²⋊C₈	—	2	1	no	—
124	C₂×C₄×C₈	—	2	1	no	—
125	#384	—	5	2	no	—
126	#384	—	5	2	no	—
127	#384	—	10	1	no	—
128	#384	—	10	1	no	—
129	C₂³×C₅	—	2	1	no	—
130	C₂³×C₅	—	2	1	no	—
131	C₂³×C₅	—	2	1	no	—
132	C₂³×C₅	—	2	1	no	—
133	C₂³×C₅	#95	1	2	yes	—
134	C₂³×C₅	#102	1	2	yes	—
135	C₂³×C₅	#103	1	2	yes	—
136	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₅	—	2	1	no	—
138	C₂×C₄×C₅	—	2	1	no	—
139	C₂×C₄×C₅	—	2	1	no	—
140	C₂×C₄×C₅	—	2	1	no	—
141	C₂×C₄×C₅	—	2	4	no	—
142	C₂×C₄×C₅	—	2	4	no	—
143	C₂×C₄×C₅	—	2	4	no	—
144	C₂×C₄×C₅	—	2	4	no	—
145	C₂×C₄×C₅	#95	1	2	yes	—
146	C₂×C₄×C₅	#95	1	2	yes	—
147	C₂×C₄×C₅	#95	1	2	yes	—
148	C₂×C₄×C₅	#100	1	2	yes	—
149	C₂×C₄×C₅	#101	1	2	yes	—
150	C₂×C₄×C₅	#102	1	2	yes	—
151	C₂×C₄×C₅	#102	1	4	yes	—
152	C₂×C₄×C₅	#103	1	2	yes	—
153	C₂×C₄×C₅	#103	1	4	yes	—
154	C₂×C₄×C₅	#106	1	4	yes	—
155	C₂×C₄×C₅	#107	1	4	yes	—
156	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
158	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
161	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
164	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
165	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
166	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
167	C₂×C₄×C₅	#137	1	2	yes	—
168	C₂×Dic₅	—	4	4	no	—
169	C₂×Dic₅	—	4	4	no	—
170	C₂×Dic₅	—	4	4	no	—
171	C₂×Dic₅	—	8	8	no	—
172	C₅×C₈	—	4	4	no	—
173	#168	—	4	4	no	—
174	C₂³×C₄	—	10	2	no	—
175	C₂³×C₄	C₄×D₅	1	1	yes	characteristic
176	C₂³×C₄	D₂₀	1	1	yes	characteristic
177	C₂³×C₄	C₂²⋊D₅	1	1	yes	characteristic
178	C₂×M₁₆	—	10	4	no	—
179	C₂×M₁₆	—	20	2	no	—
180	C₂×M₁₆	—	20	2	no	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	10	2	no	—
183	C₂×C₄⋊C₄	—	5	4	no	—
184	C₂×C₄⋊C₄	—	10	2	no	—
185	C₂×C₄⋊C₄	—	10	2	no	—
186	C₂×C₄⋊C₄	—	10	2	no	—
187	C₂×C₄⋊C₄	—	10	2	no	—
188	C₂×C₄⋊C₄	—	10	2	no	—
189	C₂×C₄⋊C₄	—	20	4	no	—
190	C₂×C₄⋊C₄	—	20	2	no	—
191	C₂×C₄⋊C₄	—	20	2	no	—
192	C₂×C₄⋊C₄	—	20	4	no	—
193	C₂×C₄²	Dic₁₀	1	2	yes	—
194	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
195	C₂×C₄²	C₄×D₅	1	2	yes	—
196	C₂×C₄²	D₂₀	1	1	yes	characteristic
197	C₂×C₄²	C₂×Dic₅	1	2	yes	—
198	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
199	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
200	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
201	C₄⋊C₈	—	2	2	no	—
202	C₄⋊C₈	—	5	4	no	—
203	C₄⋊C₈	—	10	4	no	—
204	C₄⋊C₈	—	10	2	no	—
205	C₄⋊C₈	C₂×Dic₅	1	4	yes	complemented
206	C₂²⋊C₈	—	2	4	no	—
207	C₄×C₈	—	2	4	no	—
208	C₂²×C₅	—	2	2	no	—
209	C₂²×C₅	—	2	2	no	—
210	C₂²×C₅	—	2	2	no	—
211	C₂²×C₅	—	2	2	no	—
212	C₂²×C₅	—	2	2	no	—
213	C₂²×C₅	—	2	2	no	—
214	C₂²×C₅	#340	1	4	yes	—
215	C₂²×C₅	C₂×#95	1	1	yes	characteristic
216	C₂²×C₅	C₂×#100	1	1	yes	characteristic
217	C₂²×C₅	C₂×#101	1	1	yes	characteristic
218	C₂²×C₅	C₂×#102	1	1	yes	characteristic
219	C₂²×C₅	C₂×#103	1	1	yes	characteristic
220	C₂²×C₅	#418	1	1	yes	characteristic
221	C₂²×C₅	#419	1	1	yes	characteristic
222	C₄×C₅	—	2	2	no	—
223	C₄×C₅	—	2	1	no	—
224	C₄×C₅	—	2	1	no	—
225	C₄×C₅	#340	1	4	yes	—
226	C₄×C₅	#343	1	4	yes	—
227	C₄×C₅	C₂×#102	1	1	yes	characteristic
228	C₄×C₅	C₂×#103	1	1	yes	characteristic
229	C₄×C₅	#417	1	2	yes	—
230	C₄×C₅	C₂×#106	1	1	yes	characteristic
231	C₄×C₅	C₂×#107	1	1	yes	characteristic
232	C₄×C₅	#428	1	2	yes	—
233	Dic₅	—	8	4	no	—
234	C₂⁴	#685	1	1	yes	characteristic
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	10	4	no	—
240	C₂²×C₄	—	20	4	no	—
241	C₂²×C₄	—	20	4	no	—
242	C₂²×C₄	—	20	4	no	—
243	C₂²×C₄	#677	1	2	yes	—
244	C₂²×C₄	#677	1	2	yes	—
245	C₂²×C₄	D₄₀	1	2	yes	—
246	C₂²×C₄	Dic₂₀	1	2	yes	—
247	C₂²×C₄	C₄×Dic₅	1	2	yes	—
248	C₂²×C₄	#683	1	2	yes	—
249	C₂²×C₄	#683	1	2	yes	—
250	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
251	C₂²×C₄	#685	1	1	yes	characteristic
252	C₂²×C₄	#685	1	1	yes	characteristic
253	C₂²×C₄	#685	1	2	yes	—
254	C₂²×C₄	#685	1	1	yes	characteristic
255	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
256	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
258	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
259	M₁₆	—	20	8	no	—
260	C₂×C₈	—	2	4	no	—
261	C₂×C₈	—	4	2	no	—
262	C₂×C₈	—	4	2	no	—
263	C₂×C₈	—	10	4	no	—
264	C₂×C₈	—	20	2	no	—
265	C₂×C₈	—	20	2	no	—
266	C₄⋊C₄	—	20	4	no	—
267	C₄⋊C₄	—	20	4	no	—
268	C₄⋊C₄	—	20	4	no	—
269	C₄⋊C₄	—	20	4	no	—
270	C₄⋊C₄	—	20	2	no	—
271	C₄⋊C₄	—	20	2	no	—
272	C₄⋊C₄	—	20	4	no	—
273	C₄²	C₄×Dic₅	1	2	yes	—
274	C₄²	#683	1	2	yes	—
275	C₄²	#683	1	2	yes	—
276	C₄²	C₄⋊Dic₅	1	2	yes	—
277	C₄²	#685	1	2	yes	—
278	C₄²	C₂²⋊Dic₅	1	2	yes	—
279	C₄²	C₂×Dic₁₀	1	1	yes	characteristic
280	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
281	C₄²	C₂²×Dic₅	1	1	yes	characteristic
282	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
283	C₂×C₅	—	2	4	no	—
284	C₂×C₅	#1545	1	1	yes	characteristic
285	C₂×C₅	#1546	1	1	yes	characteristic
286	C₂×C₅	#1548	1	1	yes	characteristic
287	C₂×C₅	#1549	1	1	yes	characteristic
288	C₂×C₅	C₂×#340	1	1	yes	characteristic
289	C₂×C₅	#1736	1	1	yes	characteristic
290	C₂×C₅	C₂×#343	1	1	yes	characteristic
291	C₂³	—	2	1	no	—
292	C₂³	—	2	1	no	—
293	C₂³	—	2	1	no	—
294	C₂³	—	2	1	no	—
295	C₂³	#3948	1	2	yes	—
296	C₂³	#3953	1	2	yes	—
297	C₂³	#3963	1	2	yes	—
298	C₂³	C₂×#685	1	1	yes	characteristic
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	20	4	no	—
308	C₂×C₄	—	20	4	no	—
309	C₂×C₄	—	20	4	no	—
310	C₂×C₄	—	40	8	no	—
311	C₂×C₄	#3948	1	2	yes	—
312	C₂×C₄	#3948	1	4	yes	—
313	C₂×C₄	#3949	1	4	yes	—
314	C₂×C₄	#3950	1	4	yes	—
315	C₂×C₄	#3953	1	2	yes	—
316	C₂×C₄	#3953	1	4	yes	—
317	C₂×C₄	#3955	1	2	yes	—
318	C₂×C₄	#3956	1	2	yes	—
319	C₂×C₄	#3963	1	2	yes	—
320	C₂×C₄	#3963	1	2	yes	—
321	C₂×C₄	#3963	1	2	yes	—
322	C₂×C₄	C₂×#677	1	1	yes	characteristic
323	C₂×C₄	C₂×#677	1	1	yes	characteristic
324	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
325	C₂×C₄	C₂×Dic₂₀	1	1	yes	characteristic
326	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
327	C₂×C₄	C₂×#683	1	1	yes	characteristic
328	C₂×C₄	C₂×#683	1	1	yes	characteristic
329	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
330	C₂×C₄	C₂×#685	1	1	yes	characteristic
331	C₂×C₄	#4095	1	2	yes	—
332	C₂×C₄	#4097	1	2	yes	—
333	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
334	C₈	—	4	4	no	—
335	C₈	—	20	4	no	—
336	C₅	#8043	1	1	yes	characteristic, complemented, Sylow
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	2	2	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	2	2	no	—
343	C₂²	#64771	1	4	yes	—
344	C₂²	#65290	1	1	yes	characteristic
345	C₂²	#65293	1	1	yes	characteristic
346	C₂²	C₂×#3948	1	1	yes	characteristic
347	C₂²	C₂×#3953	1	1	yes	characteristic
348	C₂²	C₂×#3955	1	1	yes	characteristic
349	C₂²	C₂×#3956	1	1	yes	characteristic
350	C₂²	C₂×#3963	1	1	yes	characteristic
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	2	no	—
354	C₄	—	40	4	no	—
355	C₄	#64771	1	4	yes	—
356	C₄	#64774	1	4	yes	—
357	C₄	#65284	1	2	yes	—
358	C₄	C₂×#3948	1	1	yes	characteristic
359	C₄	C₂×#3949	1	1	yes	characteristic
360	C₄	C₂×#3950	1	1	yes	characteristic
361	C₄	C₂×#3953	1	1	yes	characteristic
362	C₄	#65521	1	2	yes	—
363	C₂	—	2	4	no	—
364	C₂	#10601567	1	1	yes	characteristic
365	C₂	#10601571	1	1	yes	characteristic
366	C₂	#10601572	1	1	yes	characteristic
367	C₂	#10601573	1	1	yes	characteristic
368	C₂	#10602374	1	1	yes	characteristic
369	C₂	C₂×#64771	1	1	yes	characteristic
370	C₂	C₂×#64774	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×#431	C₂
socle	C₂³×C₅	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₅	C₂²⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂×#431	D₅
hypocenter	C₅	#8043
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
4. #10598193	C₂	semidirect
11. C₅×#431	C₄	semidirect
20. #65213	C₄	semidirect
52. C₅×C₄⋊C₈	C₂×C₄	semidirect
57. C₄⋊#168	C₂×C₄	semidirect
117. #431	Dic₅	semidirect
205. C₄⋊C₈	C₂×Dic₅	semidirect
336. C₅	#8043	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#431	2
C₂×#65224	2
C₂×#65213	2
#10598193	2
#8043	5