Subgroups — #12099479

Order 1280.

772 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2873	C₂	1	1	yes	maximal, characteristic
2	C₂×#65414	C₂	1	1	yes	maximal, characteristic
3	#10615483	C₂	1	1	yes	maximal, characteristic
4	C₂×#64776	C₂	1	2	yes	maximal
5	C₂×#64751	C₂	1	2	yes	maximal
6	#10600689	C₂	1	8	yes	maximal, complemented
7	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₅×#117	C₂²	1	1	yes	characteristic
10	C₂×#4072	C₂²	1	2	yes	—
11	C₂²×#681	C₂²	1	2	yes	—
12	C₅×#405	C₄	1	4	yes	complemented
13	C₅×#405	C₂²	1	4	yes	—
14	C₄×C₅×M₁₆	C₄	1	4	yes	complemented
15	#65414	—	2	4	no	—
16	#65410	—	2	4	no	—
17	C₂×#3951	C₂²	1	4	yes	—
18	#65395	C₂²	1	4	yes	—
19	#64776	C₂²	1	8	yes	—
20	#64751	C₂²	1	8	yes	—
21	#20757	—	5	1	no	maximal, Sylow
22	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₂×C₅×M₁₆	—	2	2	no	—
24	C₂×C₅×M₁₆	—	2	2	no	—
25	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
26	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₅×M₁₆	D₄	1	4	yes	—
29	C₂×C₅×M₁₆	Q₈	1	4	yes	—
30	C₂×C₅×M₁₆	C₂³	1	2	yes	—
31	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
32	C₂²×C₅×C₈	C₂³	1	2	yes	—
33	C₅×#117	—	2	1	no	—
34	C₅×#117	—	2	1	no	—
35	C₅×#117	C₂×C₄	1	2	yes	—
36	C₅×#117	C₂³	1	2	yes	—
37	C₂×C₅×C₄⋊C₄	—	2	1	no	—
38	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
39	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
40	C₂×C₂²⋊Dic₅	—	2	2	no	—
41	#4072	—	2	2	no	—
42	#4072	—	2	2	no	—
43	#4072	—	2	2	no	—
44	#4072	C₂³	1	4	yes	—
45	C₂×C₄⋊Dic₅	—	2	2	no	—
46	C₂×C₄×Dic₅	—	2	2	no	—
47	C₂×#681	—	2	2	no	—
48	C₂×#681	—	2	2	no	—
49	C₂×#681	—	2	2	no	—
50	C₂×#681	—	2	2	no	—
51	C₂×#681	—	2	2	no	—
52	C₂×#681	C₂³	1	4	yes	—
53	C₂²×#168	—	2	2	no	—
54	C₅×#97	C₂×C₄	1	8	yes	complemented
55	C₄×C₅×C₈	C₂×C₄	1	8	yes	complemented
56	#3951	—	2	8	no	—
57	#3950	—	2	4	no	—
58	#3949	—	2	4	no	—
59	C₂×#430	—	5	1	no	—
60	#2911	—	5	1	no	—
61	#2873	D₅	1	1	yes	characteristic
62	#1833	—	5	8	no	—
63	C₂×#344	—	5	2	no	—
64	C₂×H(3,ℤ/4)	—	5	2	no	—
65	C₂⁴×C₅	#41	1	1	yes	characteristic
66	C₂²×C₄×C₅	—	2	2	no	—
67	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₅	#41	1	1	yes	characteristic
72	C₂²×C₄×C₅	#41	1	2	yes	—
73	C₂²×C₄×C₅	#41	1	2	yes	—
74	C₂²×C₄×C₅	#41	1	1	yes	characteristic
75	C₂²×C₄×C₅	#41	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
77	C₂²×Dic₅	—	4	2	no	—
78	C₅×M₁₆	—	2	4	no	—
79	C₅×M₁₆	—	2	2	no	—
80	C₅×M₁₆	—	2	2	no	—
81	C₅×M₁₆	C₄⋊C₄	1	8	yes	—
82	C₅×M₁₆	C₄⋊C₄	1	8	yes	—
83	C₂×C₅×C₈	—	2	4	no	—
84	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
85	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
86	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
87	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
88	C₅×C₄⋊C₄	—	2	2	no	—
89	C₅×C₄⋊C₄	—	2	2	no	—
90	C₅×C₂²⋊C₄	—	2	2	no	—
91	C₅×C₂²⋊C₄	—	2	2	no	—
92	C₄²×C₅	—	2	2	no	—
93	C₄²×C₅	C₂²×C₄	1	4	yes	—
94	C₂²⋊Dic₅	—	2	4	no	—
95	C₂²⋊Dic₅	—	4	2	no	—
96	C₄⋊Dic₅	—	2	4	no	—
97	C₄⋊Dic₅	—	4	2	no	—
98	C₄×Dic₅	—	2	4	no	—
99	C₄×Dic₅	—	4	2	no	—
100	#681	—	2	4	no	—
101	#681	—	2	4	no	—
102	#681	—	4	2	no	—
103	#681	—	4	2	no	—
104	C₂×#168	—	2	4	no	—
105	C₂×#168	—	4	2	no	—
106	C₂×#168	—	4	2	no	—
107	C₂²×M₁₆	—	5	2	no	—
108	C₂²×M₁₆	C₂×D₅	1	1	yes	characteristic
109	C₂²×M₁₆	C₂×D₅	1	1	yes	characteristic
110	C₂×#117	—	5	2	no	—
111	C₂×#117	C₂×D₅	1	1	yes	characteristic
112	#430	—	10	4	no	—
113	C₂×#108	—	5	4	no	—
114	#428	—	10	4	no	—
115	#427	—	5	4	no	—
116	#405	Dic₅	1	4	yes	complemented
117	#405	C₂×D₅	1	4	yes	—
118	C₄×M₁₆	Dic₅	1	4	yes	complemented
119	#344	—	5	8	no	—
120	H(3,ℤ/4)	—	5	8	no	—
121	C₂³×C₅	—	8	1	no	—
122	C₂³×C₅	#117	1	1	yes	characteristic
123	C₂³×C₅	#117	1	1	yes	characteristic
124	C₂³×C₅	#124	1	2	yes	—
125	C₂³×C₅	#125	1	2	yes	—
126	C₂³×C₅	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₅	—	2	4	no	—
128	C₂×C₄×C₅	—	8	2	no	—
129	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×C₄×C₅	#117	1	2	yes	—
132	C₂×C₄×C₅	#117	1	2	yes	—
133	C₂×C₄×C₅	#117	1	1	yes	characteristic
134	C₂×C₄×C₅	#117	1	1	yes	characteristic
135	C₂×C₄×C₅	#124	1	2	yes	—
136	C₂×C₄×C₅	#124	1	2	yes	—
137	C₂×C₄×C₅	#124	1	2	yes	—
138	C₂×C₄×C₅	#125	1	2	yes	—
139	C₂×C₄×C₅	#125	1	2	yes	—
140	C₂×C₄×C₅	#125	1	2	yes	—
141	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
147	C₂×Dic₅	—	4	4	no	—
148	C₂×Dic₅	—	8	4	no	—
149	C₅×C₈	—	2	4	no	—
150	C₅×C₈	C₂×C₄⋊C₄	1	8	yes	—
151	#168	—	4	4	no	—
152	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
153	C₂×M₁₆	—	2	2	no	—
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	—	5	4	no	—
156	C₂×M₁₆	—	10	2	no	—
157	C₂×M₁₆	—	10	2	no	—
158	C₂×M₁₆	—	10	2	no	—
159	C₂×M₁₆	—	10	2	no	—
160	C₂×M₁₆	—	10	2	no	—
161	C₂×M₁₆	Dic₁₀	1	4	yes	—
162	C₂×M₁₆	D₂₀	1	4	yes	—
163	C₂×M₁₆	C₂×Dic₅	1	2	yes	—
164	C₂×M₁₆	C₂×Dic₅	1	2	yes	—
165	C₂×M₁₆	C₂×Dic₅	1	2	yes	—
166	C₂×M₁₆	C₂²×D₅	1	2	yes	—
167	C₂²×C₈	—	10	2	no	—
168	C₂²×C₈	C₂×Dic₅	1	2	yes	—
169	C₂²×C₈	C₂²×D₅	1	2	yes	—
170	#117	—	2	1	no	—
171	#117	—	2	1	no	—
172	#117	—	5	4	no	—
173	#117	—	10	2	no	—
174	#117	—	10	2	no	—
175	#117	—	10	2	no	—
176	#117	C₂×Dic₅	1	2	yes	—
177	#117	C₂²×D₅	1	2	yes	—
178	C₂×C₄⋊C₄	—	2	1	no	—
179	C₂×C₄⋊C₄	—	10	2	no	—
180	C₂×C₂²⋊C₄	—	2	1	no	—
181	C₂×C₂²⋊C₄	—	10	2	no	—
182	C₂×C₄²	—	10	2	no	—
183	C₂×C₄²	C₂×Dic₅	1	2	yes	—
184	#108	—	10	8	no	—
185	#107	—	10	4	no	—
186	#106	—	10	4	no	—
187	#97	C₂×Dic₅	1	8	yes	complemented
188	C₄×C₈	C₂×Dic₅	1	8	yes	complemented
189	C₂²×C₅	—	4	2	no	—
190	C₂²×C₅	—	4	1	no	—
191	C₂²×C₅	—	4	1	no	—
192	C₂²×C₅	—	4	2	no	—
193	C₂²×C₅	—	4	1	no	—
194	C₂²×C₅	#382	1	2	yes	—
195	C₂²×C₅	#382	1	2	yes	—
196	C₂²×C₅	C₂×#117	1	1	yes	characteristic
197	C₂²×C₅	C₂×#124	1	1	yes	characteristic
198	C₂²×C₅	C₂×#125	1	1	yes	characteristic
199	C₄×C₅	—	4	4	no	—
200	C₄×C₅	#382	1	2	yes	—
201	C₄×C₅	#382	1	2	yes	—
202	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
203	C₄×C₅	C₂×#117	1	1	yes	characteristic
204	C₄×C₅	C₂×#124	1	1	yes	characteristic
205	C₄×C₅	C₂×#125	1	1	yes	characteristic
206	Dic₅	—	8	4	no	—
207	C₂⁴	#709	1	1	yes	characteristic
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	20	2	no	—
210	C₂²×C₄	C₂×Dic₁₀	1	2	yes	—
211	C₂²×C₄	C₂×D₂₀	1	2	yes	—
212	C₂²×C₄	#709	1	1	yes	characteristic
213	C₂²×C₄	#709	1	1	yes	characteristic
214	C₂²×C₄	#709	1	2	yes	—
215	C₂²×C₄	#709	1	2	yes	—
216	C₂²×C₄	#709	1	1	yes	characteristic
217	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
218	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
219	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
220	M₁₆	—	2	2	no	—
221	M₁₆	—	2	2	no	—
222	M₁₆	—	2	4	no	—
223	M₁₆	—	10	4	no	—
224	M₁₆	—	10	4	no	—
225	M₁₆	—	20	2	no	—
226	M₁₆	—	20	2	no	—
227	M₁₆	C₄⋊Dic₅	1	8	yes	—
228	M₁₆	C₄⋊Dic₅	1	8	yes	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	10	4	no	—
231	C₂×C₈	—	20	2	no	—
232	C₂×C₈	—	20	2	no	—
233	C₂×C₈	C₂×Dic₁₀	1	4	yes	—
234	C₂×C₈	C₂×D₂₀	1	4	yes	—
235	C₂×C₈	C₂²×Dic₅	1	4	yes	—
236	C₂×C₈	C₂²×Dic₅	1	4	yes	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	10	4	no	—
240	C₄⋊C₄	—	20	2	no	—
241	C₂²⋊C₄	—	2	2	no	—
242	C₂²⋊C₄	—	2	2	no	—
243	C₂²⋊C₄	—	10	4	no	—
244	C₂²⋊C₄	—	20	2	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	10	4	no	—
247	C₄²	—	20	2	no	—
248	C₄²	C₂²×Dic₅	1	4	yes	—
249	C₂×C₅	—	2	1	no	—
250	C₂×C₅	—	2	1	no	—
251	C₂×C₅	—	2	1	no	—
252	C₂×C₅	—	2	2	no	—
253	C₂×C₅	—	2	2	no	—
254	C₂×C₅	C₂×#382	1	1	yes	characteristic
255	C₂³	—	8	1	no	—
256	C₂³	#4016	1	2	yes	—
257	C₂³	#4021	1	2	yes	—
258	C₂³	#4072	1	1	yes	characteristic
259	C₂³	#4072	1	1	yes	characteristic
260	C₂³	C₂×#709	1	1	yes	characteristic
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	8	2	no	—
263	C₂×C₄	—	20	4	no	—
264	C₂×C₄	—	40	4	no	—
265	C₂×C₄	#4016	1	2	yes	—
266	C₂×C₄	#4016	1	2	yes	—
267	C₂×C₄	#4016	1	2	yes	—
268	C₂×C₄	#4021	1	2	yes	—
269	C₂×C₄	#4021	1	2	yes	—
270	C₂×C₄	#4021	1	2	yes	—
271	C₂×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
272	C₂×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
273	C₂×C₄	#4072	1	2	yes	—
274	C₂×C₄	#4072	1	2	yes	—
275	C₂×C₄	#4072	1	1	yes	characteristic
276	C₂×C₄	#4072	1	1	yes	characteristic
277	C₂×C₄	C₂²×Dic₁₀	1	1	yes	characteristic
278	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
279	C₂×C₄	C₂×#709	1	1	yes	characteristic
280	C₂×C₄	C₂×#709	1	1	yes	characteristic
281	C₂×C₄	C₂×#709	1	1	yes	characteristic
282	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
283	C₈	—	2	4	no	—
284	C₈	—	20	4	no	—
285	C₈	C₂×C₄⋊Dic₅	1	8	yes	—
286	C₅	#20757	1	1	yes	characteristic, complemented, Sylow
287	C₂²	—	4	1	no	—
288	C₂²	—	4	2	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	2	no	—
291	C₂²	—	4	1	no	—
292	C₂²	#65225	1	2	yes	—
293	C₂²	#65225	1	2	yes	—
294	C₂²	C₂×#4016	1	1	yes	characteristic
295	C₂²	C₂×#4021	1	1	yes	characteristic
296	C₂²	C₂×#4072	1	1	yes	characteristic
297	C₄	—	4	4	no	—
298	C₄	—	40	4	no	—
299	C₄	#65225	1	2	yes	—
300	C₄	#65225	1	2	yes	—
301	C₄	C₂×#4016	1	1	yes	characteristic
302	C₄	C₂×#4021	1	1	yes	characteristic
303	C₄	C₂²×C₄⋊Dic₅	1	1	yes	characteristic
304	C₄	C₂×#4072	1	1	yes	characteristic
305	C₂	—	2	1	no	—
306	C₂	—	2	1	no	—
307	C₂	—	2	1	no	—
308	C₂	—	2	2	no	—
309	C₂	—	2	2	no	—
310	C₂	C₂×#65225	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#4072
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₅×#2873	C₂
socle	C₂×C₅	C₂×#382
prime radical	C₂⁴×C₅	#41
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	#2873	D₅
hypocenter	C₅	#20757
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×#117 ◁ #2873

Complements

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

N	H ≅ G/N	type
6. #10600689	C₂	semidirect
12. C₅×#405	C₄	semidirect
14. C₄×C₅×M₁₆	C₄	semidirect
54. C₅×#97	C₂×C₄	semidirect
55. C₄×C₅×C₈	C₂×C₄	semidirect
116. #405	Dic₅	semidirect
118. C₄×M₁₆	Dic₅	semidirect
187. #97	C₂×Dic₅	semidirect
188. C₄×C₈	C₂×Dic₅	semidirect
286. C₅	#20757	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2873	2
C₂×#65414	2
#10615483	2
C₂×#64776	2
C₂×#64751	2
#10600689	2
#20757	5