Subgroups — #12003506

Order 1280.

622 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#64984	C₂	1	1	yes	maximal, characteristic
2	C₅×#1939	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#64705	C₂	1	2	yes	maximal, complemented
4	#10600280	C₂	1	2	yes	maximal, complemented
5	#10598145	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄₀	—	2	1	no	—
7	C₂²×D₄₀	—	2	1	no	—
8	C₂×#4020	C₂²	1	2	yes	—
9	C₂×C₅×C₄⋊C₈	—	2	1	no	—
10	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×#3941	—	2	2	no	—
14	C₂×#3941	—	2	2	no	—
15	C₂×C₄⋊#168	C₂²	1	2	yes	—
16	#64984	—	2	4	no	—
17	C₅×#336	—	2	1	no	—
18	#64705	—	2	4	no	—
19	#10536	—	5	1	no	maximal, Sylow
20	C₂²×D₂₀	—	2	2	no	—
21	C₂²×D₂₀	—	2	2	no	—
22	C₂²×D₂₀	—	2	2	no	—
23	C₂²×C₅×C₈	—	2	1	no	—
24	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
26	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
27	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₅	—	2	1	no	—
29	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
30	C₂²×#168	—	2	2	no	—
31	C₂×D₄₀	—	2	4	no	—
32	C₂×D₄₀	—	2	4	no	—
33	C₂×D₄₀	—	4	2	no	—
34	C₂×D₄₀	—	4	2	no	—
35	C₂×D₄₀	—	4	2	no	—
36	C₂×D₄₀	—	4	2	no	—
37	#4020	—	2	2	no	—
38	#4020	—	2	2	no	—
39	#4020	—	2	2	no	—
40	#4020	D₄	1	4	yes	—
41	C₅×C₄⋊C₈	—	2	2	no	—
42	C₅×C₄⋊C₈	—	4	2	no	—
43	C₄×C₅×C₈	—	2	2	no	—
44	C₅×#95	—	2	1	no	—
45	#3941	—	4	4	no	—
46	#3941	—	4	4	no	—
47	C₄⋊#168	—	2	2	no	—
48	C₄⋊#168	—	2	2	no	—
49	C₂×#493	—	5	1	no	—
50	#1982	—	5	2	no	—
51	#1939	D₅	1	1	yes	characteristic, complemented
52	C₂×#331	—	5	2	no	—
53	#1277	—	5	1	no	—
54	C₂³×D₅	—	4	2	no	—
55	C₂²×C₄×C₅	—	2	1	no	—
56	C₂²×C₄×C₅	—	2	1	no	—
57	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅	#41	1	1	yes	characteristic
60	C₂×D₂₀	—	2	4	no	—
61	C₂×D₂₀	—	2	4	no	—
62	C₂×D₂₀	—	2	4	no	—
63	C₂×D₂₀	—	4	2	no	—
64	C₂×D₂₀	—	4	2	no	—
65	C₂×D₂₀	—	4	2	no	—
66	C₂×D₂₀	—	4	2	no	—
67	C₂×D₂₀	—	4	2	no	—
68	C₂×D₂₀	—	4	2	no	—
69	C₂×D₂₀	—	4	2	no	—
70	C₂×D₂₀	—	4	2	no	—
71	C₂×D₂₀	—	4	2	no	—
72	C₂×D₂₀	—	4	2	no	—
73	C₂×D₂₀	—	4	2	no	—
74	C₂×D₂₀	—	8	2	no	—
75	C₂×D₂₀	—	8	2	no	—
76	C₂×C₅×C₈	—	2	1	no	—
77	C₂×C₅×C₈	—	2	1	no	—
78	C₂×C₅×C₈	—	2	1	no	—
79	C₂×C₅×C₈	—	2	1	no	—
80	C₂×C₅×C₈	—	4	1	no	—
81	C₂×C₅×C₈	—	4	1	no	—
82	C₂×C₅×C₈	—	4	1	no	—
83	C₂×C₅×C₈	D₈	1	2	yes	—
84	C₂×C₅×C₈	SD₁₆	1	2	yes	—
85	C₅×C₄⋊C₄	—	2	1	no	—
86	C₅×C₄⋊C₄	—	2	2	no	—
87	C₅×C₄⋊C₄	D₈	1	2	yes	complemented
88	C₄²×C₅	—	4	1	no	—
89	C₄²×C₅	—	4	1	no	—
90	C₄²×C₅	C₂×D₄	1	2	yes	—
91	C₄²×C₅	#41	1	2	yes	—
92	C₂×#168	—	4	2	no	—
93	C₂×#168	—	4	2	no	—
94	C₂×#168	—	4	2	no	—
95	D₄₀	—	4	8	no	—
96	D₄₀	—	4	8	no	—
97	C₂²×D₈	—	10	1	no	—
98	C₂²×D₈	—	10	1	no	—
99	C₂×#127	—	5	2	no	—
100	#493	—	10	4	no	—
101	C₂×C₄⋊C₈	—	2	1	no	—
102	C₂×C₄⋊C₈	—	5	2	no	—
103	C₂×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
104	C₂×#102	—	10	2	no	—
105	C₂×#102	—	10	2	no	—
106	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
107	C₄×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
108	#336	—	2	1	no	—
109	#331	—	10	4	no	—
110	C₂³×C₅	#121	1	1	yes	characteristic
111	C₂²×D₅	—	4	4	no	—
112	C₂²×D₅	—	8	2	no	—
113	C₂²×D₅	—	8	2	no	—
114	C₂²×D₅	—	8	2	no	—
115	C₂²×D₅	—	8	2	no	—
116	C₂²×D₅	—	8	2	no	—
117	C₂²×D₅	—	8	2	no	—
118	C₂×C₄×C₅	—	2	1	no	—
119	C₂×C₄×C₅	—	2	1	no	—
120	C₂×C₄×C₅	—	2	2	no	—
121	C₂×C₄×C₅	—	4	1	no	—
122	C₂×C₄×C₅	—	4	1	no	—
123	C₂×C₄×C₅	—	4	1	no	—
124	C₂×C₄×C₅	—	4	1	no	—
125	C₂×C₄×C₅	—	4	1	no	—
126	C₂×C₄×C₅	#120	1	1	yes	characteristic
127	C₂×C₄×C₅	#121	1	2	yes	—
128	C₂×C₄×C₅	#123	1	2	yes	—
129	C₂×C₄×C₅	#124	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
133	C₂×C₄×C₅	#135	1	1	yes	characteristic
134	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
135	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
137	D₂₀	—	4	4	no	—
138	D₂₀	—	4	4	no	—
139	D₂₀	—	4	4	no	—
140	D₂₀	—	4	4	no	—
141	D₂₀	—	8	4	no	—
142	D₂₀	—	8	4	no	—
143	D₂₀	—	8	2	no	—
144	D₂₀	—	8	2	no	—
145	D₂₀	—	8	2	no	—
146	D₂₀	—	8	2	no	—
147	C₅×C₈	—	2	2	no	—
148	C₅×C₈	—	2	2	no	—
149	C₅×C₈	—	8	1	no	—
150	#168	—	8	2	no	—
151	C₂²×D₄	—	10	2	no	—
152	C₂²×D₄	—	10	2	no	—
153	C₂²×D₄	—	10	2	no	—
154	C₂×D₈	—	10	4	no	—
155	C₂×D₈	—	10	4	no	—
156	C₂×D₈	—	20	2	no	—
157	C₂×D₈	—	20	2	no	—
158	C₂×D₈	—	20	2	no	—
159	C₂×D₈	—	20	2	no	—
160	C₂²×C₈	—	2	1	no	—
161	C₂²×C₈	—	10	2	no	—
162	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
163	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
164	#127	—	5	4	no	—
165	#127	—	10	2	no	—
166	#127	—	10	2	no	—
167	#127	—	10	2	no	—
168	C₂×C₄⋊C₄	D₂₀	1	1	yes	characteristic
169	C₂×C₄⋊C₄	D₂₀	1	1	yes	characteristic
170	C₂×C₄²	—	2	1	no	—
171	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
172	C₄⋊C₈	—	2	2	no	—
173	C₄⋊C₈	—	4	2	no	—
174	C₄⋊C₈	—	10	2	no	—
175	C₄⋊C₈	—	10	2	no	—
176	#102	—	20	4	no	—
177	#102	—	20	4	no	—
178	C₄×C₈	—	2	2	no	—
179	#95	—	2	1	no	—
180	C₂²×C₅	#394	1	1	yes	characteristic
181	C₂²×C₅	#459	1	1	yes	characteristic
182	C₂²×C₅	#460	1	1	yes	characteristic
183	C₂²×C₅	#466	1	1	yes	characteristic
184	C₂²×C₅	#469	1	1	yes	characteristic
185	C₂²×C₅	#472	1	1	yes	characteristic
186	C₂²×C₅	#472	1	1	yes	characteristic
187	C₂×D₅	—	8	4	no	—
188	C₂×D₅	—	8	4	no	—
189	C₂×D₅	—	8	4	no	—
190	C₂×D₅	—	16	2	no	—
191	C₂×D₅	—	16	2	no	—
192	C₂×D₅	—	16	2	no	—
193	C₂×D₅	—	16	2	no	—
194	C₄×C₅	—	2	2	no	—
195	C₄×C₅	—	4	2	no	—
196	C₄×C₅	—	8	1	no	—
197	C₄×C₅	#447	1	1	yes	characteristic
198	C₄×C₅	#450	1	1	yes	characteristic
199	C₄×C₅	#459	1	2	yes	—
200	C₄×C₅	#463	1	2	yes	—
201	C₄×C₅	#486	1	1	yes	characteristic
202	C₄×C₅	#490	1	1	yes	characteristic
203	C₂⁴	—	20	2	no	—
204	C₂×D₄	—	10	4	no	—
205	C₂×D₄	—	10	4	no	—
206	C₂×D₄	—	10	4	no	—
207	C₂×D₄	—	20	2	no	—
208	C₂×D₄	—	20	2	no	—
209	C₂×D₄	—	20	2	no	—
210	C₂×D₄	—	20	2	no	—
211	C₂×D₄	—	20	2	no	—
212	C₂×D₄	—	20	2	no	—
213	C₂×D₄	—	20	2	no	—
214	C₂×D₄	—	20	2	no	—
215	C₂×D₄	—	20	2	no	—
216	C₂×D₄	—	20	2	no	—
217	C₂×D₄	—	20	2	no	—
218	C₂×D₄	—	40	2	no	—
219	C₂×D₄	—	40	2	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
223	C₂²×C₄	#709	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
225	D₈	—	20	8	no	—
226	D₈	—	20	8	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	4	1	no	—
234	C₂×C₈	—	20	2	no	—
235	C₂×C₈	—	20	2	no	—
236	C₂×C₈	—	20	2	no	—
237	C₂×C₈	#686	1	2	yes	—
238	C₂×C₈	#688	1	2	yes	—
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	D₄₀	1	2	yes	complemented
242	C₄²	—	4	1	no	—
243	C₄²	—	4	1	no	—
244	C₄²	D₄×D₅	1	2	yes	—
245	C₄²	#713	1	2	yes	—
246	C₂×C₅	#1667	1	1	yes	characteristic
247	C₂×C₅	#1668	1	1	yes	characteristic
248	C₂×C₅	#1687	1	1	yes	characteristic
249	C₂×C₅	#1688	1	1	yes	characteristic
250	C₂×C₅	#2041	1	1	yes	characteristic
251	C₂×C₅	#2045	1	1	yes	characteristic
252	C₂×C₅	#2054	1	1	yes	characteristic
253	D₅	—	16	4	no	—
254	C₂³	—	20	4	no	—
255	C₂³	—	40	2	no	—
256	C₂³	—	40	2	no	—
257	C₂³	—	40	2	no	—
258	C₂³	—	40	2	no	—
259	C₂³	—	40	2	no	—
260	C₂³	—	40	2	no	—
261	C₂³	#4076	1	1	yes	characteristic
262	D₄	—	20	4	no	—
263	D₄	—	20	4	no	—
264	D₄	—	20	4	no	—
265	D₄	—	20	4	no	—
266	D₄	—	40	4	no	—
267	D₄	—	40	4	no	—
268	D₄	—	40	2	no	—
269	D₄	—	40	2	no	—
270	D₄	—	40	2	no	—
271	D₄	—	40	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	#4040	1	2	yes	—
281	C₂×C₄	#4041	1	2	yes	—
282	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
283	C₂×C₄	#4050	1	1	yes	characteristic
284	C₂×C₄	#4054	1	2	yes	—
285	C₂×C₄	C₂×#686	1	1	yes	characteristic
286	C₂×C₄	#4083	1	1	yes	characteristic
287	C₂×C₄	C₂×#688	1	1	yes	characteristic
288	C₂×C₄	#4093	1	1	yes	characteristic
289	C₂×C₄	#4095	1	1	yes	characteristic
290	C₂×C₄	#4095	1	1	yes	characteristic
291	C₈	—	2	2	no	—
292	C₈	—	2	2	no	—
293	C₈	—	8	1	no	—
294	C₈	—	40	2	no	—
295	C₅	#10536	1	1	yes	characteristic, complemented, Sylow
296	C₂²	—	40	4	no	—
297	C₂²	—	40	4	no	—
298	C₂²	—	40	4	no	—
299	C₂²	—	80	2	no	—
300	C₂²	—	80	2	no	—
301	C₂²	—	80	2	no	—
302	C₂²	—	80	2	no	—
303	C₂²	#65280	1	1	yes	characteristic
304	C₂²	#65304	1	1	yes	characteristic
305	C₂²	#65316	1	1	yes	characteristic
306	C₂²	#65404	1	1	yes	characteristic
307	C₂²	#65424	1	1	yes	characteristic
308	C₂²	#65431	1	1	yes	characteristic
309	C₂²	#65431	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	4	2	no	—
312	C₄	—	8	1	no	—
313	C₄	#65132	1	2	yes	—
314	C₄	#65133	1	2	yes	—
315	C₄	#65325	1	1	yes	characteristic
316	C₄	#65335	1	1	yes	characteristic
317	C₄	#65356	1	1	yes	characteristic
318	C₄	#65377	1	1	yes	characteristic
319	C₂	—	80	4	no	—
320	C₂	#10600420	1	1	yes	characteristic
321	C₂	#10601274	1	1	yes	characteristic
322	C₂	#10601333	1	1	yes	characteristic
323	C₂	#10601640	1	1	yes	characteristic
324	C₂	#10601661	1	1	yes	characteristic
325	C₂	#10601732	1	1	yes	characteristic
326	C₂	#10601752	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#4076
commutator, cosocle	C₂×C₄²×C₅	C₂³
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1939	C₂
socle	C₂³×C₅	#121
prime radical	C₂×#64984	C₂
hypercenter	#1939	D₅
hypocenter	C₅	#10536
LCS	G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1939

Complements

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

N	H ≅ G/N	type
2. C₅×#1939	C₂	semidirect
3. C₂×#64705	C₂	semidirect
4. #10600280	C₂	semidirect
5. #10598145	C₂	semidirect
51. #1939	D₅	semidirect
87. C₅×C₄⋊C₄	D₈	semidirect
241. C₄⋊C₄	D₄₀	semidirect
295. C₅	#10536	semidirect

Maximal subgroups

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

subgroup	index
C₂×#64984	2
C₅×#1939	2
C₂×#64705	2
#10600280	2
#10598145	2
#10536	5