Subgroups — #11999080

Order 1280.

578 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#65377	C₂	1	1	yes	maximal, characteristic
2	C₅×#1990	C₂	1	1	yes	maximal, characteristic, complemented
3	#10602578	C₂	1	1	yes	maximal, characteristic, complemented
4	#10600565	C₂	1	1	yes	maximal, characteristic
5	C₂×#64675	C₂	1	1	yes	maximal, characteristic, complemented
6	#10600373	C₂	1	1	yes	maximal, characteristic
7	#10600369	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×#4020	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₅×#384	C₂²	1	1	yes	characteristic
12	C₂×#3967	—	2	2	no	—
13	C₂×#3953	—	2	2	no	—
14	#65377	—	2	2	no	—
15	#65377	C₄	1	4	yes	complemented
16	C₂×#3941	—	2	2	no	—
17	C₂×#3941	—	2	2	no	—
18	C₂×#3940	—	2	2	no	—
19	#65224	C₂²	1	1	yes	characteristic
20	C₂×C₄×#168	C₂²	1	1	yes	characteristic
21	C₂×C₄×#168	C₂²	1	1	yes	characteristic
22	C₅×#340	—	2	2	no	—
23	#64675	—	2	2	no	—
24	#9827	—	5	1	no	maximal, Sylow
25	C₂²×C₅×Q₈	—	2	1	no	—
26	C₂²×D₂₀	—	2	1	no	—
27	C₂²×D₂₀	—	2	1	no	—
28	C₂²×D₂₀	—	2	1	no	—
29	C₂²×C₅×C₈	D₄	1	2	yes	—
30	C₅×C₄⋊Q₈	—	2	1	no	—
31	C₅×C₄⋊Q₈	—	2	1	no	—
32	C₅×C₄⋊Q₈	—	2	1	no	—
33	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
34	C₂×C₅×C₄⋊C₄	—	2	1	no	—
35	C₂×C₅×C₄⋊C₄	—	2	1	no	—
36	C₂×C₅×C₄⋊C₄	—	2	1	no	—
37	C₂×C₅×C₄⋊C₄	—	2	1	no	—
38	C₂×C₅×C₄⋊C₄	—	2	1	no	—
39	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₂×C₄⋊Dic₅	—	2	1	no	—
41	C₂×C₄⋊Dic₅	—	2	1	no	—
42	C₂×C₄⋊Dic₅	—	2	1	no	—
43	C₂²×#168	D₄	1	2	yes	—
44	C₂²×#168	D₄	1	2	yes	—
45	#4020	—	2	1	no	—
46	#4020	—	2	1	no	—
47	#4020	—	2	1	no	—
48	#4020	C₂×C₄	1	2	yes	—
49	C₄×C₅×C₈	—	2	1	no	—
50	C₄×C₅×C₈	—	2	1	no	—
51	#3967	—	4	4	no	—
52	#3953	—	4	4	no	—
53	#3941	—	2	8	no	—
54	#3941	—	4	4	no	—
55	#3940	—	4	4	no	—
56	C₄×#168	—	2	1	no	—
57	C₄×#168	—	2	1	no	—
58	C₄×#168	C₂×C₄	1	2	yes	—
59	C₄×#168	D₄	1	2	yes	—
60	C₂×#486	—	5	1	no	—
61	#1990	D₅	1	1	yes	characteristic, complemented
62	#1957	—	5	1	no	—
63	#1957	—	5	1	no	—
64	#1956	—	5	1	no	—
65	#1844	—	5	1	no	—
66	C₂×#322	—	5	1	no	—
67	C₂³×D₅	—	4	1	no	—
68	C₂×C₅×Q₈	—	2	2	no	—
69	C₂×C₅×Q₈	—	4	1	no	—
70	C₂×C₅×Q₈	—	4	2	no	—
71	C₂×C₅×Q₈	—	4	2	no	—
72	C₂²×C₄×C₅	—	4	1	no	—
73	C₂²×C₄×C₅	—	4	1	no	—
74	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₂²×Dic₅	—	4	1	no	—
78	C₂×D₂₀	—	2	2	no	—
79	C₂×D₂₀	—	2	2	no	—
80	C₂×D₂₀	—	2	2	no	—
81	C₂×D₂₀	—	4	2	no	—
82	C₂×D₂₀	—	4	2	no	—
83	C₂×D₂₀	—	4	2	no	—
84	C₂×D₂₀	—	4	2	no	—
85	C₂×D₂₀	—	4	2	no	—
86	C₂×D₂₀	—	4	2	no	—
87	C₂×D₂₀	—	4	1	no	—
88	C₂×D₂₀	—	4	1	no	—
89	C₂×D₂₀	—	4	1	no	—
90	C₂×C₅×C₈	—	2	2	no	—
91	C₂×C₅×C₈	—	2	2	no	—
92	C₂×C₅×C₈	SD₁₆	1	4	yes	—
93	C₅×C₄⋊C₄	—	2	2	no	—
94	C₅×C₄⋊C₄	—	2	2	no	—
95	C₅×C₄⋊C₄	—	4	1	no	—
96	C₅×C₄⋊C₄	—	4	1	no	—
97	C₅×C₄⋊C₄	—	4	2	no	—
98	C₅×C₄⋊C₄	—	4	2	no	—
99	C₅×C₄⋊C₄	—	4	2	no	—
100	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₅	#41	1	1	yes	characteristic
103	C₄²×C₅	#41	1	1	yes	characteristic
104	C₄⋊Dic₅	—	4	2	no	—
105	C₄⋊Dic₅	—	4	2	no	—
106	C₄⋊Dic₅	—	4	2	no	—
107	C₂×#168	—	2	2	no	—
108	C₂×#168	—	2	2	no	—
109	C₂×#168	—	2	2	no	—
110	C₂×#168	—	2	2	no	—
111	C₂×#168	—	2	2	no	—
112	C₂×#168	C₂²⋊C₄	1	4	yes	—
113	C₂×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
114	C₂×#127	—	5	1	no	—
115	#486	—	5	4	no	—
116	#486	—	10	2	no	—
117	C₂×#106	—	10	2	no	—
118	C₂×#103	—	10	2	no	—
119	C₂×#102	—	10	2	no	—
120	C₂×#102	—	10	2	no	—
121	C₂×#102	—	10	2	no	—
122	C₂×C₄×C₈	—	5	1	no	—
123	C₂×C₄×C₈	—	5	1	no	—
124	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
125	#384	—	5	1	no	—
126	#384	C₂×D₅	1	1	yes	characteristic
127	#340	—	2	2	no	—
128	#322	—	10	2	no	—
129	C₂³×C₅	#127	1	1	yes	characteristic
130	C₂²×D₅	—	4	2	no	—
131	C₂²×D₅	—	8	1	no	—
132	C₂²×D₅	—	8	1	no	—
133	C₂²×D₅	—	8	1	no	—
134	C₂²×D₅	—	8	1	no	—
135	C₂²×D₅	—	8	1	no	—
136	C₂²×D₅	—	8	1	no	—
137	C₅×Q₈	—	4	4	no	—
138	C₅×Q₈	—	8	2	no	—
139	C₂×C₄×C₅	—	4	2	no	—
140	C₂×C₄×C₅	—	8	1	no	—
141	C₂×C₄×C₅	—	8	1	no	—
142	C₂×C₄×C₅	—	8	1	no	—
143	C₂×C₄×C₅	—	8	1	no	—
144	C₂×C₄×C₅	—	8	1	no	—
145	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₅	C₄×D₄	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	#124	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
152	C₂×C₄×C₅	#135	1	2	yes	—
153	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
154	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
155	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
156	C₂×C₄×C₅	#137	1	2	yes	—
157	C₂×Dic₅	—	8	1	no	—
158	C₂×Dic₅	—	8	1	no	—
159	C₂×Dic₅	—	8	1	no	—
160	D₂₀	—	4	4	no	—
161	D₂₀	—	4	4	no	—
162	D₂₀	—	4	4	no	—
163	D₂₀	—	8	2	no	—
164	D₂₀	—	8	2	no	—
165	D₂₀	—	8	2	no	—
166	C₅×C₈	—	2	4	no	—
167	#168	—	2	4	no	—
168	#168	—	4	2	no	—
169	C₂²×Q₈	—	2	1	no	—
170	C₂²×D₄	—	10	1	no	—
171	C₂²×D₄	—	10	1	no	—
172	C₂²×D₄	—	10	1	no	—
173	C₂²×C₈	—	5	2	no	—
174	C₂²×C₈	—	5	2	no	—
175	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
176	C₄⋊Q₈	—	2	1	no	—
177	C₄⋊Q₈	—	2	1	no	—
178	C₄⋊Q₈	—	2	1	no	—
179	C₄⋊Q₈	C₄×D₅	1	2	yes	—
180	#127	—	5	2	no	—
181	#127	—	10	1	no	—
182	#127	—	10	1	no	—
183	#127	—	10	1	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₄⋊C₄	—	2	1	no	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	—	10	1	no	—
190	C₂×C₄⋊C₄	—	10	1	no	—
191	C₂×C₄⋊C₄	—	10	1	no	—
192	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
193	#106	—	20	4	no	—
194	#103	—	20	4	no	—
195	#102	—	10	8	no	—
196	#102	—	20	4	no	—
197	#102	—	20	4	no	—
198	C₄×C₈	—	2	1	no	—
199	C₄×C₈	—	2	1	no	—
200	C₄×C₈	—	5	2	no	—
201	C₄×C₈	—	5	2	no	—
202	C₄×C₈	—	10	1	no	—
203	C₄×C₈	—	10	1	no	—
204	C₂²×C₅	#390	1	1	yes	characteristic
205	C₂²×C₅	#492	1	1	yes	characteristic
206	C₂²×C₅	#495	1	1	yes	characteristic
207	C₂²×C₅	#496	1	1	yes	characteristic
208	C₂²×C₅	#496	1	1	yes	characteristic
209	C₂²×C₅	#496	1	1	yes	characteristic
210	C₂²×C₅	#497	1	1	yes	characteristic
211	C₂×D₅	—	8	2	no	—
212	C₂×D₅	—	8	2	no	—
213	C₂×D₅	—	8	2	no	—
214	C₂×D₅	—	16	1	no	—
215	C₂×D₅	—	16	1	no	—
216	C₂×D₅	—	16	1	no	—
217	C₂×D₅	—	16	1	no	—
218	C₄×C₅	—	8	2	no	—
219	C₄×C₅	—	16	1	no	—
220	C₄×C₅	#417	1	2	yes	—
221	C₄×C₅	C₄×SD₁₆	1	2	yes	—
222	C₄×C₅	#442	1	2	yes	—
223	C₄×C₅	#465	1	2	yes	—
224	C₄×C₅	#469	1	2	yes	—
225	C₄×C₅	#489	1	2	yes	—
226	Dic₅	—	16	1	no	—
227	C₂⁴	—	20	1	no	—
228	C₂×Q₈	—	2	2	no	—
229	C₂×Q₈	—	4	1	no	—
230	C₂×Q₈	—	4	2	no	—
231	C₂×Q₈	—	4	2	no	—
232	C₂×D₄	—	10	2	no	—
233	C₂×D₄	—	10	2	no	—
234	C₂×D₄	—	10	2	no	—
235	C₂×D₄	—	20	2	no	—
236	C₂×D₄	—	20	1	no	—
237	C₂×D₄	—	20	1	no	—
238	C₂×D₄	—	20	2	no	—
239	C₂×D₄	—	20	2	no	—
240	C₂×D₄	—	20	2	no	—
241	C₂×D₄	—	20	2	no	—
242	C₂×D₄	—	20	1	no	—
243	C₂×D₄	—	20	2	no	—
244	C₂²×C₄	—	4	1	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	—	20	1	no	—
247	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
248	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	5	4	no	—
253	C₂×C₈	—	10	2	no	—
254	C₂×C₈	—	10	2	no	—
255	C₂×C₈	—	10	2	no	—
256	C₂×C₈	—	10	2	no	—
257	C₂×C₈	—	10	2	no	—
258	C₂×C₈	#688	1	4	yes	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	20	2	no	—
267	C₄⋊C₄	—	20	2	no	—
268	C₄⋊C₄	—	20	2	no	—
269	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
270	C₄²	#709	1	1	yes	characteristic
271	C₄²	D₄×D₅	1	1	yes	characteristic
272	C₄²	#711	1	1	yes	characteristic
273	C₂×C₅	#1714	1	1	yes	characteristic
274	C₂×C₅	#1715	1	1	yes	characteristic
275	C₂×C₅	#1717	1	1	yes	characteristic
276	C₂×C₅	#1720	1	1	yes	characteristic
277	C₂×C₅	#1959	1	1	yes	characteristic
278	C₂×C₅	#1968	1	1	yes	characteristic
279	C₂×C₅	#1972	1	1	yes	characteristic
280	D₅	—	16	2	no	—
281	C₂³	—	20	2	no	—
282	C₂³	—	40	1	no	—
283	C₂³	—	40	1	no	—
284	C₂³	—	40	1	no	—
285	C₂³	—	40	1	no	—
286	C₂³	—	40	1	no	—
287	C₂³	—	40	1	no	—
288	C₂³	#4086	1	1	yes	characteristic
289	Q₈	—	4	4	no	—
290	Q₈	—	8	2	no	—
291	D₄	—	20	4	no	—
292	D₄	—	20	4	no	—
293	D₄	—	20	4	no	—
294	D₄	—	40	2	no	—
295	D₄	—	40	2	no	—
296	D₄	—	40	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	8	1	no	—
299	C₂×C₄	—	8	1	no	—
300	C₂×C₄	—	8	1	no	—
301	C₂×C₄	—	8	1	no	—
302	C₂×C₄	—	8	1	no	—
303	C₂×C₄	—	40	1	no	—
304	C₂×C₄	—	40	1	no	—
305	C₂×C₄	—	40	1	no	—
306	C₂×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
307	C₂×C₄	#4027	1	1	yes	characteristic
308	C₂×C₄	#4030	1	1	yes	characteristic
309	C₂×C₄	#4031	1	1	yes	characteristic
310	C₂×C₄	#4057	1	2	yes	—
311	C₂×C₄	#4060	1	2	yes	—
312	C₂×C₄	#4060	1	2	yes	—
313	C₂×C₄	#4064	1	2	yes	—
314	C₂×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
315	C₂×C₄	#4085	1	1	yes	characteristic
316	C₂×C₄	C₂×#688	1	2	yes	—
317	C₂×C₄	#4096	1	2	yes	—
318	C₈	—	2	4	no	—
319	C₈	—	10	4	no	—
320	C₈	—	20	2	no	—
321	C₅	#9827	1	1	yes	characteristic, complemented, Sylow
322	C₂²	—	40	2	no	—
323	C₂²	—	40	2	no	—
324	C₂²	—	40	2	no	—
325	C₂²	—	80	1	no	—
326	C₂²	—	80	1	no	—
327	C₂²	—	80	1	no	—
328	C₂²	—	80	1	no	—
329	C₂²	#65246	1	1	yes	characteristic
330	C₂²	#65348	1	1	yes	characteristic
331	C₂²	#65374	1	1	yes	characteristic
332	C₂²	#65443	1	1	yes	characteristic
333	C₂²	#65465	1	1	yes	characteristic
334	C₂²	#65465	1	1	yes	characteristic
335	C₂²	#65479	1	1	yes	characteristic
336	C₄	—	8	2	no	—
337	C₄	—	16	1	no	—
338	C₄	—	80	1	no	—
339	C₄	#65071	1	2	yes	—
340	C₄	#65074	1	2	yes	—
341	C₄	#65084	1	2	yes	—
342	C₄	#65092	1	2	yes	—
343	C₄	C₄×#688	1	2	yes	—
344	C₄	#65337	1	2	yes	—
345	C₂	—	80	2	no	—
346	C₂	#10600469	1	1	yes	characteristic
347	C₂	#10600984	1	1	yes	characteristic
348	C₂	#10601020	1	1	yes	characteristic
349	C₂	#10601980	1	1	yes	characteristic
350	C₂	#10602018	1	1	yes	characteristic
351	C₂	#10602129	1	1	yes	characteristic
352	C₂	#10602156	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#4086
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1990	C₂
socle	C₂³×C₅	#127
prime radical	C₂×#4020	C₂²
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1990	D₅
hypocenter	C₅	#9827
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1990

Complements

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

N	H ≅ G/N	type
2. C₅×#1990	C₂	semidirect
3. #10602578	C₂	semidirect
5. C₂×#64675	C₂	semidirect
7. #10600369	C₂	semidirect
15. #65377	C₄	semidirect
61. #1990	D₅	semidirect
321. C₅	#9827	semidirect

Maximal subgroups

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

subgroup	index
C₂×#65377	2
C₅×#1990	2
#10602578	2
#10600565	2
C₂×#64675	2
#10600373	2
#10600369	2
#9827	5