Subgroups — #12015220

Order 1280.

566 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×#64727	C₂	1	2	yes	maximal
3	C₂×#64721	C₂	1	2	yes	maximal, complemented
4	C₅×#1374	C₂	1	1	yes	maximal, characteristic, complemented
5	#10598148	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₅	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
8	C₂×C₄²×D₅	C₄	1	1	yes	characteristic
9	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₅	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
13	C₂²×#672	C₂²	1	2	yes	—
14	C₂×C₄×#168	C₄	1	1	yes	characteristic
15	C₄×C₈×D₅	—	2	4	no	—
16	#64727	—	2	4	no	—
17	#64721	—	2	4	no	—
18	#12783	—	5	1	no	maximal, Sylow
19	C₂²×C₄×D₅	C₈	1	1	yes	characteristic
20	C₂²×C₄×D₅	C₈	1	1	yes	characteristic
21	C₂²×C₄×D₅	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₅	C₈	1	1	yes	characteristic
26	C₂×C₄×Dic₅	C₈	1	1	yes	characteristic
27	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
28	C₂²×#168	C₂×C₄	1	1	yes	characteristic
29	C₂²×#168	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×D₅	—	2	2	no	—
31	C₂×C₈×D₅	—	2	2	no	—
32	C₂×C₈×D₅	—	2	2	no	—
33	C₂×C₈×D₅	—	2	2	no	—
34	C₂×C₈×D₅	—	2	2	no	—
35	C₂×C₈×D₅	D₄	1	2	yes	—
36	C₂×C₈×D₅	Q₈	1	2	yes	—
37	C₄²×D₅	—	2	2	no	—
38	C₄²×D₅	—	2	2	no	—
39	C₂×C₅×C₁₆	—	2	2	no	—
40	C₂×C₅×C₁₆	—	2	2	no	—
41	C₂×C₅×C₁₆	—	2	2	no	—
42	C₄×C₅×C₈	—	2	2	no	—
43	C₈×Dic₅	—	2	2	no	—
44	C₈×Dic₅	D₄	1	4	yes	—
45	C₂×#672	—	2	2	no	—
46	C₂×#672	—	2	2	no	—
47	C₂×#672	—	2	2	no	—
48	C₄×#168	—	2	2	no	—
49	C₂²×C₄×C₈	—	5	1	no	—
50	C₂×C₄⋊C₁₆	—	5	2	no	—
51	C₂×C₂²⋊C₁₆	—	5	2	no	—
52	#1374	—	5	1	no	—
53	#1374	D₅	1	1	yes	characteristic, complemented
54	C₂³×D₅	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
59	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
60	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
61	C₂×C₄×D₅	—	2	2	no	—
62	C₂×C₄×D₅	—	2	2	no	—
63	C₂×C₄×D₅	—	2	1	no	—
64	C₂×C₄×D₅	—	2	1	no	—
65	C₂×C₄×D₅	—	2	1	no	—
66	C₂×C₄×D₅	—	2	1	no	—
67	C₂×C₄×D₅	—	2	2	no	—
68	C₂×C₄×D₅	—	2	2	no	—
69	C₂×C₄×D₅	—	2	2	no	—
70	C₂×C₄×D₅	C₁₆	1	4	yes	complemented
71	C₂×C₄×D₅	C₄⋊C₄	1	2	yes	—
72	C₂×C₄×D₅	M₁₆	1	2	yes	—
73	C₂×C₅×C₈	—	2	1	no	—
74	C₂×C₅×C₈	—	2	1	no	—
75	C₂×C₅×C₈	M₁₆	1	2	yes	—
76	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
77	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
78	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
79	C₂×C₅×C₈	#41	1	2	yes	—
80	C₄²×C₅	—	2	1	no	—
81	C₄²×C₅	—	2	1	no	—
82	C₄×Dic₅	—	2	2	no	—
83	C₄×Dic₅	—	2	2	no	—
84	C₄×Dic₅	C₂²⋊C₄	1	2	yes	—
85	C₄×Dic₅	M₁₆	1	2	yes	—
86	C₂×#168	—	2	1	no	—
87	C₂×#168	—	2	1	no	—
88	C₂×#168	—	2	1	no	—
89	C₂×#168	—	2	1	no	—
90	C₂×#168	C₂²⋊C₄	1	2	yes	—
91	C₂×#168	C₄⋊C₄	1	2	yes	—
92	C₈×D₅	—	2	8	no	—
93	C₈×D₅	—	2	4	no	—
94	C₈×D₅	—	2	4	no	—
95	C₅×C₁₆	—	4	2	no	—
96	#672	—	4	2	no	—
97	C₂³×C₈	—	5	1	no	—
98	C₂³×C₈	—	5	1	no	—
99	C₂²×C₄²	—	5	1	no	—
100	C₂²×C₁₆	—	5	2	no	—
101	C₂²×C₁₆	C₂×D₅	1	2	yes	—
102	C₂×C₄×C₈	—	5	1	no	—
103	C₂×C₄×C₈	—	5	1	no	—
104	C₂×C₄×C₈	—	5	1	no	—
105	C₂×C₄×C₈	—	10	4	no	—
106	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
107	C₄⋊C₁₆	—	10	4	no	—
108	C₂²⋊C₁₆	—	10	4	no	—
109	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
110	C₂²×D₅	—	2	1	no	—
111	C₂²×D₅	—	2	1	no	—
112	C₂²×D₅	—	2	1	no	—
113	C₂²×D₅	—	2	1	no	—
114	C₂²×D₅	C₄⋊C₈	1	2	yes	—
115	C₂²×D₅	C₂×C₁₆	1	2	yes	—
116	C₂²×D₅	M₃₂	1	2	yes	—
117	C₂×C₄×C₅	—	2	1	no	—
118	C₂×C₄×C₅	—	2	1	no	—
119	C₂×C₄×C₅	—	2	1	no	—
120	C₂×C₄×C₅	—	2	1	no	—
121	C₂×C₄×C₅	C₂×C₁₆	1	2	yes	—
122	C₂×C₄×C₅	M₃₂	1	2	yes	—
123	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₅	#117	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
128	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
129	C₂×Dic₅	—	2	1	no	—
130	C₂×Dic₅	—	2	1	no	—
131	C₂×Dic₅	—	2	1	no	—
132	C₂×Dic₅	—	2	1	no	—
133	C₂×Dic₅	C₂²⋊C₈	1	2	yes	—
134	C₂×Dic₅	C₂²⋊C₈	1	2	yes	—
135	C₂×Dic₅	C₄⋊C₈	1	2	yes	—
136	C₂×Dic₅	C₂×C₁₆	1	2	yes	—
137	C₂×Dic₅	M₃₂	1	2	yes	—
138	C₄×D₅	—	2	8	no	—
139	C₄×D₅	—	2	2	no	—
140	C₄×D₅	—	2	2	no	—
141	C₄×D₅	—	2	2	no	—
142	C₄×D₅	—	2	2	no	—
143	C₄×D₅	—	2	8	no	—
144	C₅×C₈	—	2	2	no	—
145	C₅×C₈	#121	1	2	yes	—
146	C₅×C₈	C₂²⋊Q₈	1	2	yes	—
147	#168	—	2	2	no	—
148	#168	—	2	2	no	—
149	C₂³×C₄	—	5	1	no	—
150	C₂³×C₄	—	5	1	no	—
151	C₂³×C₄	—	5	1	no	—
152	C₂²×C₈	—	5	1	no	—
153	C₂²×C₈	—	5	1	no	—
154	C₂²×C₈	—	5	2	no	—
155	C₂²×C₈	—	5	2	no	—
156	C₂²×C₈	—	10	2	no	—
157	C₂²×C₈	—	10	2	no	—
158	C₂²×C₈	—	10	2	no	—
159	C₂²×C₈	—	10	2	no	—
160	C₂²×C₈	—	10	2	no	—
161	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
162	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
163	C₂×C₄²	—	5	1	no	—
164	C₂×C₄²	—	5	1	no	—
165	C₂×C₄²	—	5	1	no	—
166	C₂×C₄²	—	10	2	no	—
167	C₂×C₄²	—	10	2	no	—
168	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
169	C₂×C₁₆	—	2	2	no	—
170	C₂×C₁₆	—	2	2	no	—
171	C₂×C₁₆	—	2	2	no	—
172	C₂×C₁₆	—	10	2	no	—
173	C₂×C₁₆	—	10	2	no	—
174	C₂×C₁₆	—	10	2	no	—
175	C₄×C₈	—	2	2	no	—
176	C₄×C₈	—	5	4	no	—
177	C₄×C₈	—	10	2	no	—
178	C₄×C₈	—	10	2	no	—
179	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
180	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
181	C₂²×C₅	#431	1	1	yes	characteristic
182	C₂²×C₅	C₂²×C₁₆	1	1	yes	characteristic
183	C₂²×C₅	C₂×M₃₂	1	1	yes	characteristic
184	C₂²×C₅	C₂×M₃₂	1	1	yes	characteristic
185	C₂²×C₅	C₂×M₃₂	1	1	yes	characteristic
186	C₂×D₅	—	2	2	no	—
187	C₂×D₅	—	2	2	no	—
188	C₂×D₅	—	2	2	no	—
189	C₂×D₅	—	2	2	no	—
190	C₂×D₅	—	2	2	no	—
191	C₂×D₅	—	2	2	no	—
192	C₂×D₅	C₄⋊C₁₆	1	4	yes	complemented
193	C₄×C₅	—	2	2	no	—
194	C₄×C₅	—	2	2	no	—
195	C₄×C₅	#380	1	1	yes	characteristic
196	C₄×C₅	#407	1	1	yes	characteristic
197	C₄×C₅	#423	1	1	yes	characteristic
198	C₄×C₅	#432	1	1	yes	characteristic
199	Dic₅	—	2	2	no	—
200	Dic₅	—	2	2	no	—
201	Dic₅	C₂²⋊C₁₆	1	4	yes	complemented
202	C₂⁴	—	5	1	no	—
203	C₂²×C₄	—	5	1	no	—
204	C₂²×C₄	—	5	1	no	—
205	C₂²×C₄	—	5	1	no	—
206	C₂²×C₄	—	5	4	no	—
207	C₂²×C₄	—	5	2	no	—
208	C₂²×C₄	—	5	2	no	—
209	C₂²×C₄	—	10	2	no	—
210	C₂²×C₄	—	10	2	no	—
211	C₂²×C₄	—	10	2	no	—
212	C₂²×C₄	—	10	2	no	—
213	C₂²×C₄	—	10	2	no	—
214	C₂²×C₄	—	10	1	no	—
215	C₂²×C₄	—	10	1	no	—
216	C₂²×C₄	—	10	1	no	—
217	C₂²×C₄	—	10	1	no	—
218	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
219	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	5	2	no	—
224	C₂×C₈	—	5	2	no	—
225	C₂×C₈	—	10	8	no	—
226	C₂×C₈	—	10	1	no	—
227	C₂×C₈	—	10	1	no	—
228	C₂×C₈	—	10	1	no	—
229	C₂×C₈	—	10	1	no	—
230	C₂×C₈	—	10	4	no	—
231	C₂×C₈	—	10	4	no	—
232	C₂×C₈	#676	1	2	yes	—
233	C₂×C₈	#709	1	2	yes	—
234	C₂×C₈	D₄×D₅	1	1	yes	characteristic
235	C₂×C₈	D₄×D₅	1	2	yes	—
236	C₂×C₈	Q₈×D₅	1	1	yes	characteristic
237	C₄²	—	2	1	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	5	2	no	—
240	C₄²	—	5	2	no	—
241	C₄²	—	10	2	no	—
242	C₄²	—	10	2	no	—
243	C₁₆	—	4	2	no	—
244	C₁₆	—	20	2	no	—
245	C₂×C₅	#1797	1	1	yes	characteristic
246	C₂×C₅	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
247	C₂×C₅	#2112	1	1	yes	characteristic
248	C₂×C₅	C₂×C₄⋊C₁₆	1	1	yes	characteristic
249	C₂×C₅	#2150	1	1	yes	characteristic
250	C₂×C₅	#2162	1	1	yes	characteristic
251	C₂×C₅	#2163	1	1	yes	characteristic
252	D₅	—	2	4	no	—
253	C₂³	—	5	2	no	—
254	C₂³	—	5	2	no	—
255	C₂³	—	5	2	no	—
256	C₂³	—	10	1	no	—
257	C₂³	—	10	1	no	—
258	C₂³	—	10	1	no	—
259	C₂³	—	10	1	no	—
260	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	5	2	no	—
266	C₂×C₄	—	5	2	no	—
267	C₂×C₄	—	5	2	no	—
268	C₂×C₄	—	5	2	no	—
269	C₂×C₄	—	5	2	no	—
270	C₂×C₄	—	10	8	no	—
271	C₂×C₄	—	10	8	no	—
272	C₂×C₄	—	10	1	no	—
273	C₂×C₄	—	10	1	no	—
274	C₂×C₄	—	10	1	no	—
275	C₂×C₄	—	10	1	no	—
276	C₂×C₄	—	10	2	no	—
277	C₂×C₄	—	10	2	no	—
278	C₂×C₄	—	10	2	no	—
279	C₂×C₄	—	10	2	no	—
280	C₂×C₄	D₅×C₁₆	1	2	yes	—
281	C₂×C₄	#3930	1	2	yes	—
282	C₂×C₄	#4018	1	1	yes	characteristic
283	C₂×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
284	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
285	C₂×C₄	C₂×#676	1	1	yes	characteristic
286	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
287	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
288	C₈	—	2	2	no	—
289	C₈	—	10	2	no	—
290	C₈	—	10	2	no	—
291	C₈	#4030	1	2	yes	—
292	C₈	#4042	1	2	yes	—
293	C₅	#12783	1	1	yes	characteristic, complemented, Sylow
294	C₂²	—	5	4	no	—
295	C₂²	—	10	2	no	—
296	C₂²	—	10	2	no	—
297	C₂²	—	10	2	no	—
298	C₂²	—	10	2	no	—
299	C₂²	—	10	2	no	—
300	C₂²	—	10	2	no	—
301	C₂²	#64974	1	1	yes	characteristic
302	C₂²	D₅×C₂²⋊C₈	1	1	yes	characteristic
303	C₂²	D₅×C₄⋊C₈	1	1	yes	characteristic
304	C₂²	C₂×D₅×C₁₆	1	1	yes	characteristic
305	C₂²	C₂×#3930	1	1	yes	characteristic
306	C₂²	D₅×M₃₂	1	1	yes	characteristic
307	C₂²	D₅×M₃₂	1	1	yes	characteristic
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	5	4	no	—
311	C₄	—	10	2	no	—
312	C₄	—	10	2	no	—
313	C₄	#64956	1	1	yes	characteristic
314	C₄	#64994	1	1	yes	characteristic
315	C₄	#65015	1	1	yes	characteristic
316	C₄	#65126	1	1	yes	characteristic
317	C₂	—	10	4	no	—
318	C₂	#10599455	1	1	yes	characteristic
319	C₂	#10599845	1	1	yes	characteristic
320	C₂	#10599866	1	1	yes	characteristic
321	C₂	D₅×C₂²⋊C₁₆	1	1	yes	characteristic
322	C₂	#10599899	1	1	yes	characteristic
323	C₂	D₅×C₄⋊C₁₆	1	1	yes	characteristic
324	C₂	#10600077	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅
commutator	C₂²×C₅	C₂²×C₁₆
Fitting	C₅×#1374	C₂
socle	C₂³×C₅	C₂²×C₈
prime radical	C₂³×D₅	C₂×C₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#1374	D₅
hypocenter	C₅	#12783
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
3. C₂×#64721	C₂	semidirect
4. C₅×#1374	C₂	semidirect
5. #10598148	C₂	semidirect
53. #1374	D₅	semidirect
70. C₂×C₄×D₅	C₁₆	semidirect
192. C₂×D₅	C₄⋊C₁₆	semidirect
201. Dic₅	C₂²⋊C₁₆	semidirect
293. C₅	#12783	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₈×D₅	2
C₂×#64727	2
C₂×#64721	2
C₅×#1374	2
#10598148	2
#12783	5