Subgroups — #11992543

Order 1280.

430 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10610772	C₂	1	1	yes	maximal, characteristic
2	#10601311	C₂	1	1	yes	maximal, characteristic, complemented
3	#10601296	C₂	1	1	yes	maximal, characteristic
4	#10601225	C₂	1	1	yes	maximal, characteristic
5	C₂×#64700	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1292	C₂	1	1	yes	maximal, characteristic, complemented
7	#10598202	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#4053	—	2	1	no	—
9	#65811	C₂²	1	1	yes	characteristic
10	C₅×#423	C₂²	1	1	yes	characteristic
11	C₅×#423	C₂²	1	1	yes	characteristic
12	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	#65418	—	2	1	no	—
14	C₂×#3954	—	2	1	no	—
15	C₂×#3954	—	2	1	no	—
16	#65213	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
18	C₂×C₄×#168	C₂²	1	1	yes	characteristic
19	#65127	—	2	1	no	—
20	#65127	C₄	1	2	yes	—
21	#65123	—	2	1	no	—
22	#65123	C₄	1	2	yes	—
23	#64706	—	2	1	no	—
24	#64705	—	2	1	no	—
25	#64705	—	2	1	no	—
26	#64702	—	2	1	no	—
27	#64700	—	2	1	no	—
28	#64700	—	2	1	no	—
29	#8682	—	5	1	no	maximal, Sylow
30	C₂×#709	—	2	1	no	—
31	C₂×C₅×M₁₆	—	2	1	no	—
32	C₂×C₅×M₁₆	—	2	1	no	—
33	C₂²×C₅×C₈	—	2	1	no	—
34	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
35	#4076	—	2	1	no	—
36	#4076	Q₈	1	2	yes	—
37	C₂×#681	—	2	1	no	—
38	C₂²×#168	—	2	1	no	—
39	C₂²×#168	D₄	1	1	yes	characteristic
40	C₂²×#168	D₄	1	1	yes	characteristic
41	#4053	—	4	2	no	—
42	#4053	—	4	2	no	—
43	C₂×#676	—	2	2	no	—
44	C₂×C₈×D₅	—	2	2	no	—
45	#4021	—	2	1	no	—
46	#4020	D₄	1	1	yes	characteristic
47	C₄×D₂₀	—	2	1	no	—
48	C₄×D₂₀	C₂×C₄	1	2	yes	—
49	#4015	D₄	1	1	yes	characteristic
50	C₅×C₄⋊C₈	—	2	1	no	—
51	C₅×C₄⋊C₈	—	2	1	no	—
52	C₅×C₄⋊C₈	—	2	1	no	—
53	C₅×C₄⋊C₈	—	2	1	no	—
54	C₅×C₄⋊C₈	—	2	1	no	—
55	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
56	#3954	—	4	2	no	—
57	#3954	—	4	2	no	—
58	#3952	—	2	2	no	—
59	#3952	—	2	2	no	—
60	C₄⋊#168	—	2	1	no	—
61	C₄⋊#168	—	2	1	no	—
62	C₄⋊#168	—	2	1	no	—
63	C₄⋊#168	—	2	1	no	—
64	C₄×#168	C₂×C₄	1	1	yes	characteristic
65	C₄×#168	C₂×C₄	1	1	yes	characteristic
66	C₄×#168	D₄	1	1	yes	characteristic
67	C₄×#168	D₄	1	1	yes	characteristic
68	#2931	—	5	1	no	—
69	#1848	—	5	1	no	—
70	C₂×#334	—	5	1	no	—
71	#1547	—	5	1	no	—
72	#1547	—	5	1	no	—
73	#1292	—	5	1	no	—
74	#1292	D₅	1	1	yes	characteristic, complemented
75	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
77	C₂²×C₄×C₅	#41	1	1	yes	characteristic
78	C₂×C₂²⋊D₅	—	2	2	no	—
79	#709	—	4	1	no	—
80	#709	—	4	1	no	—
81	#709	—	4	1	no	—
82	#709	—	4	1	no	—
83	C₂×D₂₀	—	2	1	no	—
84	C₂×D₂₀	—	2	1	no	—
85	C₂×D₂₀	C₄⋊C₄	1	2	yes	—
86	C₂×C₄×D₅	—	2	2	no	—
87	C₂×Dic₁₀	—	2	1	no	—
88	C₅×M₁₆	—	4	1	no	—
89	C₅×M₁₆	—	4	1	no	—
90	C₅×M₁₆	—	4	1	no	—
91	C₅×M₁₆	—	4	1	no	—
92	C₂×C₅×C₈	—	2	2	no	—
93	C₂×C₅×C₈	—	4	1	no	—
94	C₂×C₅×C₈	—	4	1	no	—
95	C₂×C₅×C₈	—	8	1	no	—
96	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₅	#41	1	1	yes	characteristic
100	#685	—	2	2	no	—
101	#685	—	2	2	no	—
102	C₄⋊Dic₅	—	2	1	no	—
103	C₄⋊Dic₅	C₄⋊C₄	1	2	yes	—
104	#681	—	4	1	no	—
105	#681	—	4	1	no	—
106	C₂×#168	—	2	1	no	—
107	C₂×#168	—	2	1	no	—
108	C₂×#168	—	4	1	no	—
109	C₂×#168	—	4	1	no	—
110	C₂×#168	—	8	1	no	—
111	C₂×#168	C₂²⋊C₄	1	1	yes	characteristic
112	C₂×#168	C₂²⋊C₄	1	1	yes	characteristic
113	C₂×#168	C₂²⋊C₄	1	1	yes	characteristic
114	C₂×#168	C₂²⋊C₄	1	1	yes	characteristic
115	C₂×#168	D₈	1	2	yes	—
116	C₂×#168	SD₁₆	1	2	yes	—
117	#676	—	4	4	no	—
118	C₈×D₅	—	4	4	no	—
119	C₂×#131	—	10	1	no	—
120	#532	—	5	1	no	—
121	#436	—	5	2	no	—
122	#436	—	10	1	no	—
123	C₈×D₄	—	5	2	no	—
124	C₈×D₄	—	10	1	no	—
125	C₂×#108	—	10	1	no	—
126	C₂×#108	—	10	1	no	—
127	#423	—	5	1	no	—
128	#423	C₂×D₅	1	1	yes	characteristic
129	#423	C₂×D₅	1	1	yes	characteristic
130	C₂×C₄⋊C₈	—	5	1	no	—
131	C₂×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
132	#408	—	10	1	no	—
133	C₂×C₄×C₈	—	5	1	no	—
134	#334	—	10	1	no	—
135	#334	—	10	1	no	—
136	#332	—	10	1	no	—
137	#332	—	10	1	no	—
138	#331	—	10	1	no	—
139	#331	—	10	1	no	—
140	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
141	C₂²×D₅	—	2	2	no	—
142	C₂×C₄×C₅	—	2	1	no	—
143	C₂×C₄×C₅	—	2	1	no	—
144	C₂×C₄×C₅	—	4	1	no	—
145	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₅	#117	1	1	yes	characteristic
148	C₂×C₄×C₅	#121	1	1	yes	characteristic
149	C₂×C₄×C₅	#121	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
153	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
154	C₂×C₄×C₅	#137	1	1	yes	characteristic
155	C₂²⋊D₅	—	4	2	no	—
156	C₂²⋊D₅	—	4	2	no	—
157	C₂×Dic₅	—	2	2	no	—
158	D₂₀	—	4	2	no	—
159	D₂₀	—	4	1	no	—
160	D₂₀	—	4	1	no	—
161	D₂₀	—	4	1	no	—
162	D₂₀	—	4	1	no	—
163	C₄×D₅	—	4	2	no	—
164	C₄×D₅	—	4	2	no	—
165	Dic₁₀	—	4	1	no	—
166	Dic₁₀	—	4	1	no	—
167	C₅×C₈	—	4	2	no	—
168	C₅×C₈	—	8	1	no	—
169	C₅×C₈	—	8	1	no	—
170	#168	—	2	1	no	—
171	#168	—	2	1	no	—
172	#168	—	8	1	no	—
173	#168	—	8	1	no	—
174	#168	#102	1	2	yes	—
175	#168	#102	1	2	yes	—
176	C₂×#41	—	10	1	no	—
177	#131	—	20	2	no	—
178	#131	—	20	2	no	—
179	C₂×M₁₆	—	2	1	no	—
180	C₂×M₁₆	—	2	1	no	—
181	C₂×M₁₆	—	10	2	no	—
182	C₂×M₁₆	—	10	1	no	—
183	C₂²×C₈	—	2	1	no	—
184	C₂²×C₈	—	5	1	no	—
185	C₂²×C₈	—	5	1	no	—
186	C₂²×C₈	—	10	2	no	—
187	C₂²×C₈	—	10	1	no	—
188	C₄⋊Q₈	—	5	1	no	—
189	#127	—	5	1	no	—
190	#124	—	10	1	no	—
191	#121	—	5	2	no	—
192	#121	—	10	1	no	—
193	C₄×D₄	—	5	2	no	—
194	C₄×D₄	—	10	1	no	—
195	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
196	#108	—	20	2	no	—
197	#108	—	20	2	no	—
198	C₄⋊C₈	—	2	1	no	—
199	C₄⋊C₈	—	2	1	no	—
200	C₄⋊C₈	—	2	1	no	—
201	C₄⋊C₈	—	2	1	no	—
202	C₄⋊C₈	—	2	1	no	—
203	C₄⋊C₈	—	10	1	no	—
204	C₄⋊C₈	—	10	1	no	—
205	C₄⋊C₈	—	10	1	no	—
206	C₄⋊C₈	—	10	1	no	—
207	C₄⋊C₈	C₄×D₅	1	2	yes	—
208	C₂²⋊C₈	—	10	2	no	—
209	C₂²⋊C₈	—	10	2	no	—
210	C₄×C₈	—	5	1	no	—
211	C₄×C₈	—	5	1	no	—
212	C₄×C₈	—	5	1	no	—
213	C₄×C₈	—	5	1	no	—
214	C₂²×C₅	—	2	1	no	—
215	C₂²×C₅	—	2	1	no	—
216	C₂²×C₅	#380	1	1	yes	characteristic
217	C₂²×C₅	#474	1	1	yes	characteristic
218	C₂²×C₅	#476	1	1	yes	characteristic
219	C₂×D₅	—	4	2	no	—
220	C₂×D₅	—	4	2	no	—
221	C₂×D₅	—	4	2	no	—
222	C₄×C₅	—	2	1	no	—
223	C₄×C₅	—	2	1	no	—
224	C₄×C₅	—	2	1	no	—
225	C₄×C₅	C₂×#102	1	1	yes	characteristic
226	C₄×C₅	#417	1	1	yes	characteristic
227	C₄×C₅	#459	1	1	yes	characteristic
228	C₄×C₅	#461	1	1	yes	characteristic
229	C₄×C₅	#471	1	1	yes	characteristic
230	C₄×C₅	#471	1	1	yes	characteristic
231	Dic₅	—	4	2	no	—
232	#41	—	20	1	no	—
233	#41	—	20	1	no	—
234	#41	—	20	1	no	—
235	#41	—	20	1	no	—
236	C₂×Q₈	—	10	1	no	—
237	C₂×D₄	—	5	2	no	—
238	C₂×D₄	—	10	2	no	—
239	C₂×D₄	—	10	1	no	—
240	C₂×D₄	—	10	1	no	—
241	C₂²×C₄	—	10	2	no	—
242	C₂²×C₄	#709	1	1	yes	characteristic
243	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
244	C₂²×C₄	Q₈×D₅	1	1	yes	characteristic
245	M₁₆	—	4	1	no	—
246	M₁₆	—	4	1	no	—
247	M₁₆	—	4	1	no	—
248	M₁₆	—	4	1	no	—
249	M₁₆	—	20	4	no	—
250	M₁₆	—	20	1	no	—
251	M₁₆	—	20	1	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	5	1	no	—
256	C₂×C₈	—	5	1	no	—
257	C₂×C₈	—	5	2	no	—
258	C₂×C₈	—	5	2	no	—
259	C₂×C₈	—	5	1	no	—
260	C₂×C₈	—	5	1	no	—
261	C₂×C₈	—	8	1	no	—
262	C₂×C₈	—	10	1	no	—
263	C₂×C₈	—	10	1	no	—
264	C₂×C₈	—	20	4	no	—
265	C₂×C₈	—	20	1	no	—
266	C₂×C₈	—	20	1	no	—
267	C₂×C₈	—	40	1	no	—
268	C₄⋊C₄	—	5	2	no	—
269	C₄⋊C₄	—	10	1	no	—
270	C₂²⋊C₄	—	10	2	no	—
271	C₂²⋊C₄	—	10	2	no	—
272	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
273	C₄²	#709	1	1	yes	characteristic
274	C₄²	D₄×D₅	1	1	yes	characteristic
275	C₄²	D₄×D₅	1	1	yes	characteristic
276	C₂×C₅	—	2	1	no	—
277	C₂×C₅	—	2	1	no	—
278	C₂×C₅	#1802	1	1	yes	characteristic
279	C₂×C₅	#1813	1	1	yes	characteristic
280	C₂×C₅	#1815	1	1	yes	characteristic
281	D₅	—	8	2	no	—
282	C₂³	—	10	2	no	—
283	C₂³	#4042	1	1	yes	characteristic
284	Q₈	—	20	1	no	—
285	Q₈	—	20	1	no	—
286	D₄	—	20	2	no	—
287	D₄	—	20	2	no	—
288	D₄	—	20	2	no	—
289	D₄	—	20	1	no	—
290	D₄	—	20	1	no	—
291	D₄	—	20	1	no	—
292	D₄	—	20	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	10	2	no	—
297	C₂×C₄	—	20	2	no	—
298	C₂×C₄	—	20	2	no	—
299	C₂×C₄	#4018	1	1	yes	characteristic
300	C₂×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
301	C₂×C₄	#4030	1	1	yes	characteristic
302	C₂×C₄	#4030	1	1	yes	characteristic
303	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
304	C₂×C₄	#4042	1	1	yes	characteristic
305	C₂×C₄	D₅×D₈	1	1	yes	characteristic
306	C₂×C₄	D₅×SD₁₆	1	1	yes	characteristic
307	C₂×C₄	#4060	1	1	yes	characteristic
308	C₂×C₄	#4064	1	1	yes	characteristic
309	C₈	—	4	2	no	—
310	C₈	—	5	2	no	—
311	C₈	—	5	2	no	—
312	C₈	—	8	1	no	—
313	C₈	—	8	1	no	—
314	C₈	—	10	1	no	—
315	C₈	—	10	1	no	—
316	C₈	—	40	1	no	—
317	C₈	—	40	1	no	—
318	C₅	#8682	1	1	yes	characteristic, complemented, Sylow
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	20	2	no	—
322	C₂²	—	20	2	no	—
323	C₂²	—	20	2	no	—
324	C₂²	#64956	1	1	yes	characteristic
325	C₂²	#65159	1	1	yes	characteristic
326	C₂²	#65179	1	1	yes	characteristic
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	20	2	no	—
331	C₄	#65037	1	1	yes	characteristic
332	C₄	#65043	1	1	yes	characteristic
333	C₄	D₅×#102	1	1	yes	characteristic
334	C₄	#65075	1	1	yes	characteristic
335	C₄	#65092	1	1	yes	characteristic
336	C₄	#65107	1	1	yes	characteristic
337	C₂	—	2	1	no	—
338	C₂	—	2	1	no	—
339	C₂	—	40	2	no	—
340	C₂	#10599517	1	1	yes	characteristic
341	C₂	#10599685	1	1	yes	characteristic
342	C₂	#10599793	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#64956
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1292	C₂
socle	C₂²×C₅	#380
prime radical	#65811	C₂²
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1292	D₅
hypocenter	C₅	#8682
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1292

Complements

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

N	H ≅ G/N	type
2. #10601311	C₂	semidirect
5. C₂×#64700	C₂	semidirect
6. C₅×#1292	C₂	semidirect
7. #10598202	C₂	semidirect
74. #1292	D₅	semidirect
318. C₅	#8682	semidirect

Maximal subgroups

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

subgroup	index
#10610772	2
#10601311	2
#10601296	2
#10601225	2
C₂×#64700	2
C₅×#1292	2
#10598202	2
#8682	5