Subgroups — #12071052

Order 1280.

652 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	#10611024	C₂	1	2	yes	maximal, complemented
2	C₄×#4050	C₂	1	1	yes	maximal, characteristic
3	C₅×#1451	C₂	1	1	yes	maximal, characteristic, complemented
4	#10600647	C₂	1	2	yes	maximal, complemented
5	#10600523	C₂	1	1	yes	maximal, characteristic, complemented
6	#10598882	C₂	1	2	yes	maximal, complemented
7	#10598872	C₂	1	4	yes	maximal, complemented
8	#10598870	C₂	1	2	yes	maximal, complemented
9	C₂×#4050	C₄	1	1	yes	characteristic
10	C₄×#709	C₂²	1	2	yes	—
11	C₅×#431	C₂²	1	1	yes	characteristic
12	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	#65402	C₄	1	2	yes	—
15	#65395	C₄	1	1	yes	characteristic
16	#65213	C₂²	1	2	yes	—
17	#65124	—	2	2	no	—
18	D₅×C₄⋊C₈	—	2	2	no	—
19	#65017	—	2	4	no	—
20	C₄×Dic₂₀	C₄	1	1	yes	characteristic
21	C₄×Dic₂₀	C₂²	1	1	yes	characteristic
22	C₄×D₄₀	C₄	1	1	yes	characteristic
23	C₄×D₄₀	C₂²	1	1	yes	characteristic
24	C₄×#677	C₄	1	2	yes	—
25	C₄×#677	C₂²	1	2	yes	—
26	C₈×D₂₀	C₂²	1	4	yes	—
27	#64968	C₂²	1	4	yes	—
28	C₅×#322	C₂²	1	2	yes	—
29	C₅×#322	C₂²	1	2	yes	—
30	#64679	C₂²	1	4	yes	—
31	#64677	C₂²	1	2	yes	—
32	#64677	C₂²	1	2	yes	—
33	#64676	C₂²	1	4	yes	—
34	#18065	—	5	1	no	maximal, Sylow
35	C₂×#709	C₂×C₄	1	2	yes	—
36	C₂×C₅×M₁₆	—	2	1	no	—
37	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₄×C₂²⋊D₅	—	2	2	no	—
41	C₄×C₂²⋊D₅	—	2	2	no	—
42	#4072	C₂×C₄	1	2	yes	—
43	C₂×#681	—	2	2	no	—
44	C₂×Dic₂₀	C₂×C₄	1	1	yes	characteristic
45	#4050	—	2	4	no	—
46	C₂×D₄₀	C₂×C₄	1	1	yes	characteristic
47	C₂×#677	C₂×C₄	1	2	yes	—
48	C₂×C₈×D₅	—	2	4	no	—
49	C₄×D₂₀	C₂×C₄	1	2	yes	—
50	C₄×D₂₀	C₂³	1	2	yes	—
51	#4018	—	2	2	no	—
52	C₄²×D₅	—	2	2	no	—
53	C₄×Dic₁₀	C₂×C₄	1	2	yes	—
54	C₄×Dic₁₀	C₂³	1	2	yes	—
55	C₅×C₄⋊C₈	D₄	1	2	yes	—
56	C₅×C₂²⋊C₈	D₄	1	2	yes	—
57	C₅×#97	—	2	1	no	—
58	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
59	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
60	C₄×C₅×C₈	C₂³	1	2	yes	—
61	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
62	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
63	C₄×C₅×C₈	C₂³	1	2	yes	—
64	#3953	C₂×C₄	1	2	yes	—
65	#3953	C₂×C₄	1	2	yes	—
66	#3952	—	2	4	no	—
67	#3950	C₂×C₄	1	1	yes	characteristic
68	#3950	C₂×C₄	1	1	yes	characteristic
69	#3949	C₂×C₄	1	2	yes	—
70	#3948	C₂×C₄	1	2	yes	—
71	#3948	C₂×C₄	1	2	yes	—
72	#3947	—	2	4	no	—
73	#3946	—	2	4	no	—
74	C₄⋊#168	—	2	2	no	—
75	C₄⋊#168	C₂³	1	4	yes	—
76	#2990	—	5	2	no	—
77	C₄×#135	—	5	1	no	—
78	#1593	—	5	2	no	—
79	#1582	—	5	4	no	—
80	#1580	—	5	2	no	—
81	#1566	—	5	1	no	—
82	#1489	—	5	2	no	—
83	#1451	D₅	1	1	yes	characteristic, complemented
84	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₅	#41	1	1	yes	characteristic
87	C₂×C₂²⋊D₅	—	2	2	no	—
88	#709	—	2	4	no	—
89	#709	—	4	2	no	—
90	C₂×D₂₀	C₂²×C₄	1	2	yes	—
91	C₂×C₄×D₅	—	2	2	no	—
92	C₂×C₄×D₅	—	2	2	no	—
93	C₂×C₄×D₅	—	2	2	no	—
94	C₂×Dic₁₀	C₂²×C₄	1	2	yes	—
95	C₅×M₁₆	—	4	2	no	—
96	C₂×C₅×C₈	—	2	2	no	—
97	C₂×C₅×C₈	—	2	2	no	—
98	C₂×C₅×C₈	—	4	1	no	—
99	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
104	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₅	#41	1	1	yes	characteristic
107	C₄²×C₅	C₂⁴	1	1	yes	characteristic
108	C₂²⋊Dic₅	—	2	2	no	—
109	#685	—	2	2	no	—
110	#685	—	2	2	no	—
111	C₄⋊Dic₅	C₂²×C₄	1	2	yes	—
112	C₄⋊Dic₅	C₂²×C₄	1	2	yes	—
113	#683	—	2	2	no	—
114	#683	—	2	2	no	—
115	C₄×Dic₅	—	2	2	no	—
116	C₄×Dic₅	—	2	2	no	—
117	C₄×Dic₅	—	2	2	no	—
118	#681	—	8	2	no	—
119	C₂×#168	—	2	4	no	—
120	Dic₂₀	—	2	2	no	—
121	D₄₀	M₁₆	1	4	yes	complemented
122	#677	—	2	4	no	—
123	C₈×D₅	—	8	4	no	—
124	C₂×#135	—	5	1	no	—
125	C₄×#41	—	5	2	no	—
126	#446	—	5	4	no	—
127	C₄×Q₁₆	—	5	1	no	—
128	C₄×Q₁₆	—	5	1	no	—
129	C₄×SD₁₆	—	5	2	no	—
130	C₄×SD₁₆	—	5	2	no	—
131	C₄×D₈	—	5	1	no	—
132	C₄×D₈	—	5	1	no	—
133	#435	—	10	4	no	—
134	C₈×D₄	—	5	4	no	—
135	#432	—	10	2	no	—
136	#431	C₂×D₅	1	1	yes	characteristic
137	#427	—	5	1	no	—
138	#423	—	5	2	no	—
139	C₂×C₄⋊C₈	—	10	2	no	—
140	#416	—	5	2	no	—
141	C₄×M₁₆	C₂×D₅	1	1	yes	characteristic
142	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
143	#334	—	5	2	no	—
144	#334	—	5	2	no	—
145	Q₈⋊C₈	—	5	4	no	—
146	D₄⋊C₈	—	5	4	no	—
147	#322	C₂×D₅	1	2	yes	—
148	#322	C₂×D₅	1	2	yes	—
149	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
150	C₂²×D₅	—	2	2	no	—
151	C₂×C₄×C₅	—	2	1	no	—
152	C₂×C₄×C₅	—	2	1	no	—
153	C₂×C₄×C₅	—	2	2	no	—
154	C₂×C₄×C₅	—	2	2	no	—
155	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
156	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
157	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
161	C₂²⋊D₅	—	4	4	no	—
162	C₂×Dic₅	—	2	2	no	—
163	C₂×Dic₅	—	2	2	no	—
164	C₂×Dic₅	—	2	2	no	—
165	D₂₀	—	2	2	no	—
166	D₂₀	C₂×M₁₆	1	4	yes	—
167	C₄×D₅	—	4	4	no	—
168	C₄×D₅	—	4	2	no	—
169	C₄×D₅	—	4	2	no	—
170	C₄×D₅	—	4	4	no	—
171	Dic₁₀	—	2	2	no	—
172	Dic₁₀	—	2	2	no	—
173	C₅×C₈	—	2	1	no	—
174	C₅×C₈	—	2	1	no	—
175	C₅×C₈	—	2	1	no	—
176	C₅×C₈	—	2	4	no	—
177	C₅×C₈	—	4	2	no	—
178	C₅×C₈	C₂×M₁₆	1	2	yes	—
179	#168	—	4	4	no	—
180	C₂×#41	—	5	2	no	—
181	#135	—	10	4	no	—
182	C₂×Q₁₆	—	5	1	no	—
183	C₂×SD₁₆	—	5	2	no	—
184	C₂×D₈	—	5	1	no	—
185	C₂×M₁₆	—	2	1	no	—
186	C₂×M₁₆	—	10	2	no	—
187	C₂²×C₈	—	10	4	no	—
188	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
189	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
190	C₄×Q₈	—	5	2	no	—
191	C₄×Q₈	—	5	2	no	—
192	C₄×D₄	—	5	2	no	—
193	C₄×D₄	—	5	2	no	—
194	C₄×D₄	—	10	2	no	—
195	C₄×D₄	—	10	2	no	—
196	#117	—	5	2	no	—
197	#117	—	10	2	no	—
198	C₂×C₄²	—	10	2	no	—
199	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
200	#107	—	5	1	no	—
201	#107	—	5	1	no	—
202	#106	—	5	2	no	—
203	C₄⋊C₈	—	5	4	no	—
204	C₄⋊C₈	—	10	4	no	—
205	C₄⋊C₈	—	10	2	no	—
206	C₄⋊C₈	D₂₀	1	2	yes	—
207	#103	—	5	2	no	—
208	#103	—	5	2	no	—
209	#102	—	5	2	no	—
210	#102	—	5	2	no	—
211	C₂²⋊C₈	—	10	4	no	—
212	C₂²⋊C₈	D₂₀	1	2	yes	—
213	#97	—	2	1	no	—
214	#97	—	10	4	no	—
215	C₄×C₈	C₄×D₅	1	1	yes	characteristic
216	C₄×C₈	C₄×D₅	1	1	yes	characteristic
217	C₄×C₈	C₂²×D₅	1	2	yes	—
218	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
219	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
220	C₄×C₈	C₂²×D₅	1	2	yes	—
221	C₂²×C₅	—	2	1	no	—
222	C₂²×C₅	—	2	1	no	—
223	C₂²×C₅	—	2	1	no	—
224	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
225	C₂×D₅	—	2	4	no	—
226	C₂×D₅	—	4	2	no	—
227	C₂×D₅	—	4	2	no	—
228	C₄×C₅	—	2	1	no	—
229	C₄×C₅	—	2	1	no	—
230	C₄×C₅	—	2	1	no	—
231	C₄×C₅	#444	1	2	yes	—
232	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
233	C₄×C₅	#568	1	1	yes	characteristic
234	C₄×C₅	#577	1	2	yes	—
235	Dic₅	—	4	2	no	—
236	Dic₅	—	4	2	no	—
237	Dic₅	—	4	2	no	—
238	#41	—	10	4	no	—
239	#41	—	20	2	no	—
240	C₂×Q₈	—	5	2	no	—
241	C₂×D₄	—	5	2	no	—
242	C₂×D₄	—	10	2	no	—
243	C₂²×C₄	—	10	2	no	—
244	C₂²×C₄	—	10	2	no	—
245	C₂²×C₄	—	10	2	no	—
246	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
248	C₂²×C₄	#709	1	1	yes	characteristic
249	Q₁₆	—	10	2	no	—
250	SD₁₆	—	10	4	no	—
251	D₈	—	5	4	no	—
252	M₁₆	—	4	2	no	—
253	M₁₆	—	40	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	10	4	no	—
258	C₂×C₈	—	40	4	no	—
259	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
260	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
261	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
262	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
263	C₂×C₈	C₂×D₂₀	1	4	yes	—
264	C₄⋊C₄	—	5	2	no	—
265	C₄⋊C₄	—	5	2	no	—
266	C₄⋊C₄	—	10	2	no	—
267	C₄⋊C₄	—	10	2	no	—
268	C₂²⋊C₄	—	10	2	no	—
269	C₂²⋊C₄	—	10	2	no	—
270	C₂²⋊C₄	—	10	2	no	—
271	C₄²	—	10	2	no	—
272	C₄²	—	10	2	no	—
273	C₄²	—	10	2	no	—
274	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
275	C₄²	C₂×D₂₀	1	1	yes	characteristic
276	C₄²	#709	1	1	yes	characteristic
277	C₄²	C₂³×D₅	1	1	yes	characteristic
278	C₂×C₅	—	4	1	no	—
279	C₂×C₅	D₄×M₁₆	1	1	yes	characteristic
280	C₂×C₅	#2952	1	1	yes	characteristic
281	C₂×C₅	C₂×#444	1	1	yes	characteristic
282	D₅	—	4	4	no	—
283	C₂³	—	10	2	no	—
284	C₂³	C₄×D₂₀	1	1	yes	characteristic
285	Q₈	—	10	2	no	—
286	Q₈	—	10	2	no	—
287	D₄	—	5	4	no	—
288	D₄	—	10	2	no	—
289	D₄	—	20	4	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	10	2	no	—
295	C₂×C₄	—	10	2	no	—
296	C₂×C₄	—	10	2	no	—
297	C₂×C₄	—	20	4	no	—
298	C₂×C₄	—	20	2	no	—
299	C₂×C₄	—	20	2	no	—
300	C₂×C₄	—	20	4	no	—
301	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
302	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
303	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
305	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
306	C₂×C₄	C₂×#709	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	4	no	—
311	C₈	—	4	2	no	—
312	C₈	—	20	4	no	—
313	C₈	D₅×M₁₆	1	2	yes	—
314	C₅	#18065	1	1	yes	characteristic, complemented, Sylow
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	10	4	no	—
319	C₂²	—	20	2	no	—
320	C₂²	—	20	2	no	—
321	C₂²	C₂×C₄×D₂₀	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	20	2	no	—
326	C₄	—	20	2	no	—
327	C₄	—	20	2	no	—
328	C₄	#65006	1	2	yes	—
329	C₄	C₂×D₅×M₁₆	1	1	yes	characteristic
330	C₄	#66079	1	1	yes	characteristic
331	C₄	#66085	1	2	yes	—
332	C₂	—	4	1	no	—
333	C₂	—	20	4	no	—
334	C₂	C₂×#65006	1	1	yes	characteristic
335	C₂	M₁₆×D₂₀	1	1	yes	characteristic
336	C₂	#10610305	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₀
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₄²	C₂³×D₅
Fitting	C₅×#1451	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂×#4050	C₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	#1451	D₅
hypocenter	C₅	#18065
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1451

Complements

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

N	H ≅ G/N	type
1. #10611024	C₂	semidirect
3. C₅×#1451	C₂	semidirect
4. #10600647	C₂	semidirect
5. #10600523	C₂	semidirect
6. #10598882	C₂	semidirect
7. #10598872	C₂	semidirect
8. #10598870	C₂	semidirect
83. #1451	D₅	semidirect
121. D₄₀	M₁₆	semidirect
314. C₅	#18065	semidirect

Maximal subgroups

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

subgroup	index
#10611024	2
C₄×#4050	2
C₅×#1451	2
#10600647	2
#10600523	2
#10598882	2
#10598872	2
#10598870	2
#18065	5