Subgroups — #12068509

Order 1280.

572 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2922	C₂	1	1	yes	maximal, characteristic
2	#10614252	C₂	1	2	yes	maximal
3	#10601546	C₂	1	2	yes	maximal, complemented
4	#10601545	C₂	1	2	yes	maximal, complemented
5	#10601544	C₂	1	2	yes	maximal, complemented
6	#10601542	C₂	1	2	yes	maximal, complemented
7	#10600644	C₂	1	2	yes	maximal, complemented
8	#10600525	C₂	1	1	yes	maximal, characteristic, complemented
9	#10600523	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₅×#41	C₂²	1	1	yes	characteristic
11	C₅×#433	C₄	1	4	yes	—
12	C₅×#431	C₂²	1	2	yes	—
13	C₅×#431	C₂²	1	2	yes	—
14	C₅×#424	C₄	1	2	yes	—
15	C₅×#408	C₄	1	2	yes	—
16	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	#65313	—	2	2	no	—
19	#65301	—	2	2	no	—
20	#65300	—	2	2	no	—
21	#65300	—	2	2	no	—
22	#65215	C₂²	1	2	yes	—
23	#65215	C₂²	1	2	yes	—
24	#65214	C₂²	1	2	yes	—
25	#65213	C₂²	1	2	yes	—
26	#65213	C₂²	1	2	yes	—
27	C₂×C₄⋊#168	C₂²	1	2	yes	—
28	#64678	C₂²	1	2	yes	—
29	#64678	C₂²	1	2	yes	—
30	#64678	C₂²	1	2	yes	—
31	#64677	C₂²	1	1	yes	characteristic
32	#64677	C₂²	1	2	yes	—
33	#64677	C₂²	1	2	yes	—
34	#64677	C₂²	1	2	yes	—
35	#64677	C₂²	1	1	yes	characteristic
36	#64677	C₂²	1	1	yes	characteristic
37	#64677	C₂²	1	1	yes	characteristic
38	#17824	—	5	1	no	maximal, Sylow
39	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
40	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
41	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
42	C₄×C₅×Q₈	D₄	1	1	yes	characteristic
43	C₄×C₅×Q₈	Q₈	1	1	yes	characteristic
44	C₄×C₅×D₄	D₄	1	1	yes	characteristic
45	C₄×C₅×D₄	D₄	1	2	yes	—
46	C₄×C₅×D₄	Q₈	1	1	yes	characteristic
47	C₄×C₅×D₄	Q₈	1	2	yes	—
48	C₅×#117	C₂×C₄	1	1	yes	characteristic
49	C₅×#117	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₅	C₂³	1	2	yes	—
52	C₂×#681	—	2	2	no	—
53	C₂×#681	—	2	2	no	—
54	C₂²×#168	—	2	2	no	—
55	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	—
56	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	—
57	C₅×C₂²⋊C₈	C₂×C₄	1	4	yes	—
58	C₅×C₂²⋊C₈	C₂×C₄	1	4	yes	—
59	C₅×#97	C₂×C₄	1	2	yes	—
60	C₄×C₅×C₈	C₂×C₄	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	C₂²⋊#168	—	2	2	no	—
65	C₂²⋊#168	—	2	2	no	—
66	C₂²⋊#168	—	2	2	no	—
67	C₄⋊#168	—	2	2	no	—
68	C₄⋊#168	—	2	2	no	—
69	C₄⋊#168	—	2	2	no	—
70	C₄⋊#168	C₂³	1	2	yes	—
71	C₄⋊#168	C₂³	1	2	yes	—
72	C₄⋊#168	C₂³	1	2	yes	—
73	C₄⋊#168	C₂³	1	2	yes	—
74	#3935	—	2	2	no	—
75	#3935	—	2	2	no	—
76	C₄×#168	—	2	2	no	—
77	#2989	—	5	2	no	—
78	#2922	D₅	1	1	yes	characteristic
79	#1574	—	5	2	no	—
80	#1573	—	5	2	no	—
81	#1572	—	5	2	no	—
82	#1570	—	5	2	no	—
83	#1569	—	5	2	no	—
84	#1568	—	5	1	no	—
85	#1566	—	5	1	no	—
86	C₅×#41	—	4	2	no	—
87	C₂×C₅×Q₈	C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₅×D₄	C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₅×D₄	C₄⋊C₄	1	2	yes	—
90	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
95	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
96	C₅×M₁₆	—	4	2	no	—
97	C₂×C₅×C₈	—	4	2	no	—
98	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
99	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
100	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
101	C₅×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
102	C₅×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
103	C₅×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
104	C₅×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
105	C₅×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
106	C₅×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
107	C₅×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
108	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₅	C₂²×C₄	1	2	yes	—
110	C₄²×C₅	C₂×D₄	1	2	yes	—
111	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
113	C₄²×C₅	C₂×Q₈	1	2	yes	—
114	C₄²×C₅	C₂⁴	1	1	yes	characteristic
115	#681	—	8	2	no	—
116	#681	—	8	2	no	—
117	C₂×#168	—	2	2	no	—
118	C₂×#168	—	2	2	no	—
119	C₂×#168	—	2	2	no	—
120	C₂×#168	—	2	2	no	—
121	C₂×#168	—	8	2	no	—
122	C₄×#41	C₂×D₅	1	1	yes	characteristic
123	#446	—	10	2	no	—
124	#436	—	10	2	no	—
125	#435	—	10	2	no	—
126	#435	—	10	2	no	—
127	#433	Dic₅	1	4	yes	—
128	#432	—	5	2	no	—
129	#432	—	5	2	no	—
130	#431	—	5	2	no	—
131	#431	C₂×D₅	1	2	yes	—
132	#431	C₂×D₅	1	2	yes	—
133	#424	Dic₅	1	2	yes	—
134	#423	—	5	2	no	—
135	#423	—	5	2	no	—
136	C₂×C₄⋊C₈	—	5	2	no	—
137	#408	Dic₅	1	2	yes	—
138	C₄×M₁₆	C₂×D₅	1	1	yes	characteristic
139	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
140	#335	—	5	2	no	—
141	#335	—	5	2	no	—
142	#335	—	5	2	no	—
143	#334	—	5	2	no	—
144	#334	—	5	1	no	—
145	#334	—	5	1	no	—
146	#334	—	5	1	no	—
147	#334	—	5	1	no	—
148	#334	—	5	2	no	—
149	#334	—	5	2	no	—
150	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
152	C₅×Q₈	—	4	1	no	—
153	C₅×D₄	—	4	1	no	—
154	C₅×D₄	—	4	2	no	—
155	C₂×C₄×C₅	—	2	2	no	—
156	C₂×C₄×C₅	—	2	4	no	—
157	C₂×C₄×C₅	—	2	2	no	—
158	C₂×C₄×C₅	—	2	2	no	—
159	C₂×C₄×C₅	—	2	2	no	—
160	C₂×C₄×C₅	—	4	1	no	—
161	C₂×C₄×C₅	—	4	2	no	—
162	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
164	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
165	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
166	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
167	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
169	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
170	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
171	C₅×C₈	—	2	2	no	—
172	C₅×C₈	—	2	2	no	—
173	C₅×C₈	—	2	4	no	—
174	#168	—	4	2	no	—
175	#168	—	4	2	no	—
176	#168	—	4	2	no	—
177	#168	—	4	2	no	—
178	C₂×#41	C₂×Dic₅	1	1	yes	characteristic
179	C₂×M₁₆	—	10	2	no	—
180	C₂×M₁₆	—	10	2	no	—
181	C₂×M₁₆	C₂×Dic₅	1	2	yes	—
182	C₂²×C₈	—	10	2	no	—
183	C₂²×C₈	C₂×Dic₅	1	2	yes	—
184	C₄×Q₈	Dic₁₀	1	1	yes	characteristic
185	C₄×Q₈	D₂₀	1	1	yes	characteristic
186	C₄×D₄	Dic₁₀	1	1	yes	characteristic
187	C₄×D₄	Dic₁₀	1	2	yes	—
188	C₄×D₄	D₂₀	1	1	yes	characteristic
189	C₄×D₄	D₂₀	1	2	yes	—
190	#117	C₂×Dic₅	1	1	yes	characteristic
191	#117	C₂×Dic₅	1	2	yes	—
192	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
193	C₂×C₄²	C₂²×D₅	1	2	yes	—
194	C₄⋊C₈	—	5	2	no	—
195	C₄⋊C₈	—	5	2	no	—
196	C₄⋊C₈	—	5	2	no	—
197	C₄⋊C₈	—	5	2	no	—
198	C₄⋊C₈	—	10	2	no	—
199	C₄⋊C₈	—	10	2	no	—
200	C₄⋊C₈	—	10	2	no	—
201	C₄⋊C₈	C₂×Dic₅	1	4	yes	—
202	C₄⋊C₈	C₂×Dic₅	1	4	yes	—
203	C₂²⋊C₈	—	10	2	no	—
204	C₂²⋊C₈	—	10	2	no	—
205	C₂²⋊C₈	—	10	2	no	—
206	C₂²⋊C₈	C₂×Dic₅	1	4	yes	—
207	C₂²⋊C₈	C₂×Dic₅	1	4	yes	—
208	#97	—	10	2	no	—
209	#97	—	10	2	no	—
210	#97	C₂×Dic₅	1	2	yes	—
211	C₄×C₈	—	10	2	no	—
212	C₄×C₈	C₂×Dic₅	1	2	yes	—
213	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
214	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
215	C₄×C₈	C₂²×D₅	1	2	yes	—
216	C₂²×C₅	—	2	1	no	—
217	C₂²×C₅	—	2	2	no	—
218	C₂²×C₅	—	2	2	no	—
219	C₂²×C₅	—	2	1	no	—
220	C₂²×C₅	—	2	1	no	—
221	C₂²×C₅	—	2	2	no	—
222	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
223	C₄×C₅	—	2	1	no	—
224	C₄×C₅	—	2	2	no	—
225	C₄×C₅	—	2	2	no	—
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	—	2	1	no	—
228	C₄×C₅	—	2	1	no	—
229	C₄×C₅	—	2	2	no	—
230	C₄×C₅	#568	1	2	yes	—
231	C₄×C₅	#577	1	2	yes	—
232	#41	—	4	2	no	—
233	C₂×Q₈	C₄⋊Dic₅	1	1	yes	characteristic
234	C₂×D₄	C₄⋊Dic₅	1	1	yes	characteristic
235	C₂×D₄	C₄⋊Dic₅	1	2	yes	—
236	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
237	C₂²×C₄	C₂×Dic₁₀	1	2	yes	—
238	C₂²×C₄	C₂×D₂₀	1	2	yes	—
239	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
240	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
241	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
242	M₁₆	—	4	2	no	—
243	M₁₆	—	40	2	no	—
244	M₁₆	—	40	2	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	10	2	no	—
247	C₂×C₈	—	10	2	no	—
248	C₂×C₈	—	10	2	no	—
249	C₂×C₈	—	10	2	no	—
250	C₂×C₈	—	40	2	no	—
251	C₂×C₈	C₂²×Dic₅	1	2	yes	—
252	C₂×C₈	C₂²×Dic₅	1	2	yes	—
253	C₂×C₈	C₂²×Dic₅	1	4	yes	—
254	C₄⋊C₄	C₄⋊Dic₅	1	1	yes	characteristic
255	C₄⋊C₄	C₄⋊Dic₅	1	2	yes	—
256	C₄⋊C₄	C₄⋊Dic₅	1	1	yes	characteristic
257	C₄⋊C₄	C₄⋊Dic₅	1	2	yes	—
258	C₂²⋊C₄	C₄⋊Dic₅	1	2	yes	—
259	C₂²⋊C₄	C₄⋊Dic₅	1	2	yes	—
260	C₂²⋊C₄	C₄⋊Dic₅	1	2	yes	—
261	C₄²	C₂×Dic₁₀	1	1	yes	characteristic
262	C₄²	C₂×Dic₁₀	1	2	yes	—
263	C₄²	C₂×D₂₀	1	2	yes	—
264	C₄²	C₂×D₂₀	1	1	yes	characteristic
265	C₄²	C₂²×Dic₅	1	1	yes	characteristic
266	C₄²	C₂²×Dic₅	1	2	yes	—
267	C₄²	C₂³×D₅	1	1	yes	characteristic
268	C₂×C₅	—	4	1	no	—
269	C₂×C₅	—	4	2	no	—
270	C₂×C₅	#2905	1	1	yes	characteristic
271	C₂×C₅	#2912	1	1	yes	characteristic
272	C₂×C₅	#2916	1	1	yes	characteristic
273	C₂³	C₂×C₄⋊Dic₅	1	2	yes	—
274	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
275	Q₈	—	4	1	no	—
276	D₄	—	4	1	no	—
277	D₄	—	4	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
286	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
287	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
288	C₂×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
289	C₂×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
290	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
291	C₂×C₄	C₂²×Dic₁₀	1	1	yes	characteristic
292	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
293	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
294	C₈	—	2	2	no	—
295	C₈	—	2	2	no	—
296	C₈	—	2	4	no	—
297	C₈	—	20	2	no	—
298	C₈	—	20	2	no	—
299	C₈	—	20	2	no	—
300	C₈	—	20	2	no	—
301	C₅	#17824	1	1	yes	characteristic, complemented, Sylow
302	C₂²	—	2	1	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	1	no	—
308	C₂²	C₂²×C₄⋊Dic₅	1	1	yes	characteristic
309	C₄	—	2	1	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	1	no	—
316	C₄	#66085	1	2	yes	—
317	C₄	#66153	1	2	yes	—
318	C₂	—	4	1	no	—
319	C₂	—	4	2	no	—
320	C₂	#10613935	1	1	yes	characteristic
321	C₂	#10615515	1	1	yes	characteristic
322	C₂	#10615520	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₀
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₄²	C₂³×D₅
Fitting	C₅×#2922	C₂
socle	C₂²×C₅	C₂²×C₄⋊C₄
prime radical	C₂×C₅×#41	C₂×C₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	#2922	D₅
hypocenter	C₅	#17824
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ #2922

Complements

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

N	H ≅ G/N	type
3. #10601546	C₂	semidirect
4. #10601545	C₂	semidirect
5. #10601544	C₂	semidirect
6. #10601542	C₂	semidirect
7. #10600644	C₂	semidirect
8. #10600525	C₂	semidirect
9. #10600523	C₂	semidirect
301. C₅	#17824	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2922	2
#10614252	2
#10601546	2
#10601545	2
#10601544	2
#10601542	2
#10600644	2
#10600525	2
#10600523	2
#17824	5