Subgroups — #11990367

Order 1280.

482 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10610189	C₂	1	1	yes	maximal, characteristic
2	C₅×#1450	C₂	1	1	yes	maximal, characteristic, complemented
3	#10601582	C₂	1	1	yes	maximal, characteristic
4	#10601565	C₂	1	1	yes	maximal, characteristic
5	C₈×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
6	#10598237	C₂	1	1	yes	maximal, characteristic, complemented
7	#10598236	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#4047	—	2	1	no	—
9	#65811	C₂²	1	1	yes	characteristic
10	C₅×#431	C₂²	1	1	yes	characteristic
11	C₅×#431	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	#65399	—	2	1	no	—
14	C₂×C₈×Dic₅	—	2	1	no	—
15	C₄×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
16	#65213	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
18	#64977	—	2	1	no	—
19	#64977	C₄	1	2	yes	complemented
20	C₈×D₂₀	—	2	1	no	—
21	C₈×D₂₀	C₄	1	2	yes	complemented
22	C₅×#322	—	2	1	no	—
23	C₅×C₈²	—	2	1	no	—
24	#64770	—	2	1	no	—
25	#64706	C₄	1	2	yes	complemented
26	#64705	C₄	1	2	yes	complemented
27	#64690	C₄	1	4	yes	—
28	#8251	—	5	1	no	maximal, Sylow
29	C₂×#709	—	2	1	no	—
30	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
31	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
33	#4076	—	2	1	no	—
34	#4076	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊Dic₅	—	2	1	no	—
36	C₂×C₄×Dic₅	—	2	1	no	—
37	C₂×#681	—	2	1	no	—
38	C₂²×#168	—	2	1	no	—
39	#4047	—	4	2	no	—
40	#4047	—	4	2	no	—
41	C₂×#676	—	2	2	no	—
42	C₂×C₈×D₅	—	2	2	no	—
43	#4021	—	2	1	no	—
44	#4020	C₂×C₄	1	1	yes	characteristic
45	C₄×D₂₀	—	2	1	no	—
46	C₄×D₂₀	C₂×C₄	1	2	yes	—
47	#4015	C₂×C₄	1	1	yes	characteristic
48	C₅×C₄⋊C₈	—	2	1	no	—
49	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
50	C₅×C₂²⋊C₈	—	2	1	no	—
51	C₅×C₂²⋊C₈	C₂×C₄	1	2	yes	—
52	C₄×C₅×C₈	—	2	1	no	—
53	C₄×C₅×C₈	—	2	1	no	—
54	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
55	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
56	C₄×C₅×C₈	D₄	1	1	yes	characteristic
57	C₄×C₅×C₈	D₄	1	1	yes	characteristic
58	#3963	C₂×C₄	1	2	yes	—
59	#3952	—	2	2	no	—
60	#3952	—	2	2	no	—
61	C₈×Dic₅	—	4	2	no	—
62	C₄⋊#168	—	2	1	no	—
63	C₄⋊#168	—	2	1	no	—
64	C₄⋊#168	C₂×C₄	1	2	yes	—
65	C₄⋊#168	C₂×C₄	1	2	yes	—
66	#2931	—	5	1	no	—
67	C₈×C₄⋊C₄	—	5	1	no	—
68	#1540	—	5	1	no	—
69	#1502	—	5	1	no	—
70	#1450	D₅	1	1	yes	characteristic, complemented
71	#1303	—	5	1	no	—
72	#1301	—	5	1	no	—
73	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	#41	1	1	yes	characteristic
76	C₂×C₂²⋊D₅	—	2	2	no	—
77	C₂²×Dic₅	—	2	2	no	—
78	#709	—	4	1	no	—
79	#709	—	4	1	no	—
80	#709	—	4	1	no	—
81	#709	—	4	1	no	—
82	C₂×D₂₀	—	2	1	no	—
83	C₂×D₂₀	—	2	1	no	—
84	C₂×D₂₀	C₄²	1	2	yes	—
85	C₂×C₄×D₅	—	2	2	no	—
86	C₂×Dic₁₀	—	2	1	no	—
87	C₂×C₅×C₈	—	2	1	no	—
88	C₂×C₅×C₈	—	2	1	no	—
89	C₂×C₅×C₈	—	2	2	no	—
90	C₂×C₅×C₈	—	2	2	no	—
91	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₅×C₈	D₈	1	2	yes	—
96	C₂×C₅×C₈	SD₁₆	1	2	yes	—
97	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₅	#41	1	1	yes	characteristic
101	#685	—	2	2	no	—
102	#685	—	2	2	no	—
103	C₄⋊Dic₅	—	2	4	no	—
104	C₄⋊Dic₅	—	2	1	no	—
105	C₄⋊Dic₅	C₄²	1	2	yes	—
106	C₄×Dic₅	—	4	1	no	—
107	C₄×Dic₅	—	4	1	no	—
108	#681	—	4	1	no	—
109	#681	—	4	1	no	—
110	C₂×#168	—	2	2	no	—
111	C₂×#168	—	2	2	no	—
112	C₂×#168	—	4	1	no	—
113	C₂×#168	—	4	1	no	—
114	#676	—	4	4	no	—
115	C₈×D₅	—	4	4	no	—
116	C₂×#131	—	10	1	no	—
117	#532	—	5	1	no	—
118	#436	—	5	2	no	—
119	#436	—	10	1	no	—
120	C₈×D₄	—	5	2	no	—
121	C₈×D₄	—	10	1	no	—
122	#431	C₂×D₅	1	1	yes	characteristic
123	#431	C₂×D₅	1	1	yes	characteristic
124	#423	—	5	1	no	—
125	C₂×C₄⋊C₈	—	5	1	no	—
126	#408	—	10	1	no	—
127	C₂×C₄×C₈	—	10	1	no	—
128	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
129	C₄×C₄⋊C₄	—	5	1	no	—
130	#336	—	10	1	no	—
131	#332	—	5	2	no	—
132	#331	—	5	2	no	—
133	#327	—	5	4	no	—
134	#322	—	2	1	no	—
135	C₈²	—	2	1	no	—
136	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
137	C₂²×D₅	—	2	2	no	—
138	C₂×C₄×C₅	—	2	2	no	—
139	C₂×C₄×C₅	—	4	1	no	—
140	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₅	#117	1	1	yes	characteristic
143	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
148	C₂×C₄×C₅	#135	1	1	yes	characteristic
149	C₂×C₄×C₅	#135	1	1	yes	characteristic
150	C₂²⋊D₅	—	4	2	no	—
151	C₂²⋊D₅	—	4	2	no	—
152	C₂×Dic₅	—	2	4	no	—
153	C₂×Dic₅	—	2	2	no	—
154	C₂×Dic₅	—	4	2	no	—
155	C₂×Dic₅	—	4	2	no	—
156	D₂₀	—	4	2	no	—
157	D₂₀	—	4	1	no	—
158	D₂₀	—	4	1	no	—
159	D₂₀	—	4	2	no	—
160	C₄×D₅	—	4	2	no	—
161	C₄×D₅	—	4	2	no	—
162	Dic₁₀	—	4	1	no	—
163	Dic₁₀	—	4	1	no	—
164	C₅×C₈	—	2	1	no	—
165	C₅×C₈	—	2	1	no	—
166	C₅×C₈	—	4	2	no	—
167	C₅×C₈	—	4	2	no	—
168	C₅×C₈	#102	1	2	yes	—
169	C₅×C₈	#102	1	2	yes	—
170	#168	—	4	2	no	—
171	#168	—	4	2	no	—
172	C₂×#41	—	10	1	no	—
173	#131	—	20	2	no	—
174	#131	—	20	2	no	—
175	C₂×M₁₆	—	10	1	no	—
176	C₂×M₁₆	—	10	2	no	—
177	C₂²×C₈	—	10	1	no	—
178	C₂²×C₈	—	10	2	no	—
179	C₂²×C₈	D₂₀	1	1	yes	characteristic
180	C₂²×C₈	D₂₀	1	1	yes	characteristic
181	C₄⋊Q₈	—	5	1	no	—
182	#127	—	5	1	no	—
183	#124	—	10	1	no	—
184	#121	—	5	2	no	—
185	#121	—	10	1	no	—
186	C₄×D₄	—	5	2	no	—
187	C₄×D₄	—	10	1	no	—
188	C₂×C₄⋊C₄	—	10	1	no	—
189	C₂×C₄²	—	10	1	no	—
190	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
191	C₄⋊C₈	—	2	1	no	—
192	C₄⋊C₈	—	5	2	no	—
193	C₄⋊C₈	—	5	2	no	—
194	C₄⋊C₈	—	10	1	no	—
195	C₄⋊C₈	—	10	1	no	—
196	C₄⋊C₈	C₄×D₅	1	2	yes	—
197	C₂²⋊C₈	—	2	1	no	—
198	C₂²⋊C₈	—	10	2	no	—
199	C₂²⋊C₈	—	10	2	no	—
200	C₂²⋊C₈	C₄×D₅	1	2	yes	—
201	C₄×C₈	—	2	1	no	—
202	C₄×C₈	—	2	1	no	—
203	C₄×C₈	—	20	2	no	—
204	C₄×C₈	C₄×D₅	1	1	yes	characteristic
205	C₄×C₈	C₄×D₅	1	1	yes	characteristic
206	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
207	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
208	#95	—	5	2	no	—
209	C₂²×C₅	—	2	1	no	—
210	C₂²×C₅	—	2	1	no	—
211	C₂²×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
212	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
213	C₂²×C₅	C₄×SD₁₆	1	1	yes	characteristic
214	C₂×D₅	—	4	2	no	—
215	C₂×D₅	—	4	2	no	—
216	C₂×D₅	—	4	2	no	—
217	C₄×C₅	—	2	1	no	—
218	C₄×C₅	—	2	1	no	—
219	C₄×C₅	—	2	1	no	—
220	C₄×C₅	C₂×#102	1	1	yes	characteristic
221	C₄×C₅	#416	1	1	yes	characteristic
222	C₄×C₅	C₄×D₈	1	1	yes	characteristic
223	C₄×C₅	C₄×SD₁₆	1	1	yes	characteristic
224	C₄×C₅	#443	1	2	yes	—
225	Dic₅	—	4	4	no	—
226	Dic₅	—	4	2	no	—
227	#41	—	20	1	no	—
228	#41	—	20	1	no	—
229	#41	—	20	1	no	—
230	#41	—	20	1	no	—
231	C₂×Q₈	—	10	1	no	—
232	C₂×D₄	—	5	2	no	—
233	C₂×D₄	—	10	2	no	—
234	C₂×D₄	—	10	1	no	—
235	C₂×D₄	—	10	1	no	—
236	C₂²×C₄	—	10	2	no	—
237	C₂²×C₄	—	10	2	no	—
238	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
240	C₂²×C₄	#709	1	1	yes	characteristic
241	M₁₆	—	20	1	no	—
242	M₁₆	—	20	1	no	—
243	M₁₆	—	20	4	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	10	2	no	—
249	C₂×C₈	—	10	2	no	—
250	C₂×C₈	—	20	1	no	—
251	C₂×C₈	—	20	1	no	—
252	C₂×C₈	—	20	4	no	—
253	C₂×C₈	#677	1	2	yes	—
254	C₂×C₈	D₄₀	1	2	yes	—
255	C₂×C₈	#685	1	1	yes	characteristic
256	C₂×C₈	#685	1	1	yes	characteristic
257	C₂×C₈	#685	1	1	yes	characteristic
258	C₂×C₈	#685	1	1	yes	characteristic
259	C₄⋊C₄	—	5	2	no	—
260	C₄⋊C₄	—	10	4	no	—
261	C₄⋊C₄	—	10	1	no	—
262	C₂²⋊C₄	—	10	2	no	—
263	C₂²⋊C₄	—	10	2	no	—
264	C₄²	—	20	1	no	—
265	C₄²	—	20	1	no	—
266	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
267	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
268	C₄²	#709	1	1	yes	characteristic
269	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
270	C₂×C₅	—	2	1	no	—
271	C₂×C₅	—	2	1	no	—
272	C₂×C₅	#1757	1	1	yes	characteristic
273	C₂×C₅	C₄×#102	1	1	yes	characteristic
274	C₂×C₅	#1764	1	1	yes	characteristic
275	D₅	—	8	2	no	—
276	C₂³	—	10	2	no	—
277	C₂³	C₄×D₂₀	1	1	yes	characteristic
278	Q₈	—	20	1	no	—
279	Q₈	—	20	1	no	—
280	D₄	—	20	2	no	—
281	D₄	—	20	2	no	—
282	D₄	—	20	2	no	—
283	D₄	—	20	2	no	—
284	D₄	—	20	1	no	—
285	D₄	—	20	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	10	4	no	—
289	C₂×C₄	—	10	2	no	—
290	C₂×C₄	—	20	2	no	—
291	C₂×C₄	—	20	2	no	—
292	C₂×C₄	—	20	2	no	—
293	C₂×C₄	—	20	2	no	—
294	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
295	C₂×C₄	#4018	1	1	yes	characteristic
296	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
297	C₂×C₄	C₂×#677	1	1	yes	characteristic
298	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
299	C₂×C₄	#4050	1	1	yes	characteristic
300	C₂×C₄	#4050	1	1	yes	characteristic
301	C₂×C₄	C₂×#685	1	1	yes	characteristic
302	C₂×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
303	C₂×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	4	2	no	—
307	C₈	—	4	2	no	—
308	C₈	—	20	2	no	—
309	C₈	—	20	2	no	—
310	C₈	#3953	1	2	yes	—
311	C₈	#3953	1	2	yes	—
312	C₅	#8251	1	1	yes	characteristic, complemented, Sylow
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	20	2	no	—
316	C₂²	—	20	2	no	—
317	C₂²	—	20	2	no	—
318	C₂²	C₄×#677	1	1	yes	characteristic
319	C₂²	C₄×D₄₀	1	1	yes	characteristic
320	C₂²	C₄×#685	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	20	4	no	—
325	C₄	—	20	2	no	—
326	C₄	C₄×#677	1	1	yes	characteristic
327	C₄	C₄×D₄₀	1	1	yes	characteristic
328	C₄	C₂×#3953	1	1	yes	characteristic
329	C₄	#65402	1	1	yes	characteristic
330	C₄	#65433	1	2	yes	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	40	2	no	—
334	C₂	C₄×#3953	1	1	yes	characteristic
335	C₂	#10600680	1	1	yes	characteristic
336	C₂	#10600691	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#685
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1450	C₂
socle	C₂²×C₅	C₄×C₂²⋊C₄
prime radical	#65811	C₂²
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1450	D₅
hypocenter	C₅	#8251
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
2. C₅×#1450	C₂	semidirect
5. C₈×C₄⋊Dic₅	C₂	semidirect
6. #10598237	C₂	semidirect
7. #10598236	C₂	semidirect
19. #64977	C₄	semidirect
21. C₈×D₂₀	C₄	semidirect
25. #64706	C₄	semidirect
26. #64705	C₄	semidirect
70. #1450	D₅	semidirect
312. C₅	#8251	semidirect

Maximal subgroups

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

subgroup	index
#10610189	2
C₅×#1450	2
#10601582	2
#10601565	2
C₈×C₄⋊Dic₅	2
#10598237	2
#10598236	2
#8251	5