Subgroups — #12099480

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	C₂×C₅×#430	C₂	1	1	yes	maximal, characteristic
2	#10615483	C₂	1	1	yes	maximal, characteristic
3	#10615478	C₂	1	1	yes	maximal, characteristic
4	C₂×#64779	C₂	1	2	yes	maximal
5	C₂×#64777	C₂	1	2	yes	maximal
6	#10601491	C₂	1	8	yes	maximal, complemented
7	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
9	C₂×#4072	C₂²	1	2	yes	—
10	C₂×#4072	C₂²	1	1	yes	characteristic
11	C₂²×#681	C₂²	1	1	yes	characteristic
12	C₅×#430	—	2	4	no	—
13	C₂×C₅×#108	C₂²	1	4	yes	—
14	#65411	C₄	1	4	yes	complemented
15	#65410	—	2	4	no	—
16	M₁₆×Dic₅	C₄	1	4	yes	complemented
17	#65395	C₂²	1	4	yes	—
18	#65390	C₂²	1	4	yes	—
19	#64779	C₂²	1	8	yes	—
20	#64777	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	—	2	2	no	—
26	C₂×C₅×M₁₆	—	2	2	no	—
27	C₂×C₅×M₁₆	—	2	2	no	—
28	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
29	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
30	C₂×C₅×M₁₆	D₄	1	4	yes	—
31	C₂×C₅×M₁₆	Q₈	1	4	yes	—
32	C₂×C₅×M₁₆	C₂³	1	4	yes	—
33	C₂²×C₅×C₈	—	2	2	no	—
34	C₂²×C₅×C₈	C₂³	1	2	yes	—
35	C₂×C₂²⋊Dic₅	—	2	1	no	—
36	C₂×C₂²⋊Dic₅	—	2	2	no	—
37	#4072	—	2	1	no	—
38	#4072	—	2	1	no	—
39	#4072	—	2	2	no	—
40	#4072	—	2	2	no	—
41	#4072	—	2	2	no	—
42	#4072	C₂×C₄	1	2	yes	—
43	#4072	C₂³	1	4	yes	—
44	#4072	C₂³	1	2	yes	—
45	C₂×C₄⋊Dic₅	—	2	1	no	—
46	C₂×C₄⋊Dic₅	—	2	2	no	—
47	C₂×C₄×Dic₅	—	2	2	no	—
48	C₂×C₄×Dic₅	C₂×C₄	1	2	yes	—
49	C₂×#681	—	2	2	no	—
50	C₂×#681	—	2	2	no	—
51	C₂×#681	C₂×C₄	1	2	yes	—
52	C₂×#681	C₂³	1	2	yes	—
53	C₂²×#168	C₂×C₄	1	2	yes	—
54	C₅×#108	—	2	8	no	—
55	#3950	—	2	4	no	—
56	#3949	—	2	4	no	—
57	#3947	C₂×C₄	1	8	yes	complemented
58	C₈×Dic₅	C₂×C₄	1	8	yes	complemented
59	C₂×#430	D₅	1	1	yes	characteristic
60	#2911	—	5	1	no	—
61	#2873	—	5	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₅	#41	1	2	yes	—
71	C₂²×C₄×C₅	#41	1	1	yes	characteristic
72	C₂²×C₄×C₅	#41	1	1	yes	characteristic
73	C₂²×C₄×C₅	#41	1	1	yes	characteristic
74	C₂²×C₄×C₅	#41	1	2	yes	—
75	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
76	C₂²×Dic₅	—	2	2	no	—
77	C₂²×Dic₅	—	4	2	no	—
78	C₅×M₁₆	—	2	4	no	—
79	C₅×M₁₆	—	2	4	no	—
80	C₅×M₁₆	—	4	2	no	—
81	C₅×M₁₆	—	4	2	no	—
82	C₅×M₁₆	C₄⋊C₄	1	8	yes	—
83	C₅×M₁₆	C₄⋊C₄	1	8	yes	—
84	C₂×C₅×C₈	—	2	4	no	—
85	C₂×C₅×C₈	—	4	2	no	—
86	C₂×C₅×C₈	—	4	2	no	—
87	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
88	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
89	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
90	C₂²⋊Dic₅	—	2	2	no	—
91	C₂²⋊Dic₅	—	2	2	no	—
92	C₂²⋊Dic₅	—	2	4	no	—
93	C₂²⋊Dic₅	—	4	2	no	—
94	C₄⋊Dic₅	—	2	2	no	—
95	C₄⋊Dic₅	—	2	2	no	—
96	C₄⋊Dic₅	—	2	4	no	—
97	C₄⋊Dic₅	—	4	2	no	—
98	C₄×Dic₅	—	2	2	no	—
99	C₄×Dic₅	—	2	4	no	—
100	C₄×Dic₅	—	4	2	no	—
101	C₄×Dic₅	C₂²×C₄	1	4	yes	—
102	#681	—	2	4	no	—
103	#681	—	2	2	no	—
104	#681	—	2	2	no	—
105	C₂×#168	—	2	4	no	—
106	C₂×#168	C₂²×C₄	1	4	yes	—
107	C₂²×M₁₆	—	5	1	no	—
108	C₂²×M₁₆	C₂×D₅	1	2	yes	—
109	C₂²×M₁₆	C₂×D₅	1	1	yes	characteristic
110	C₂×#117	—	5	1	no	—
111	C₂×#117	—	5	2	no	—
112	#430	—	2	4	no	—
113	C₂×#108	C₂×D₅	1	4	yes	—
114	#428	—	10	4	no	—
115	#427	—	5	4	no	—
116	#405	—	5	4	no	—
117	#405	—	5	4	no	—
118	C₄×M₁₆	—	5	4	no	—
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	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₅	#117	1	2	yes	—
130	C₂×C₄×C₅	#117	1	2	yes	—
131	C₂×C₄×C₅	#117	1	1	yes	characteristic
132	C₂×C₄×C₅	#117	1	1	yes	characteristic
133	C₂×C₄×C₅	#124	1	2	yes	—
134	C₂×C₄×C₅	#124	1	2	yes	—
135	C₂×C₄×C₅	#124	1	2	yes	—
136	C₂×C₄×C₅	#125	1	2	yes	—
137	C₂×C₄×C₅	#125	1	2	yes	—
138	C₂×C₄×C₅	#125	1	2	yes	—
139	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
145	C₂×Dic₅	—	2	4	no	—
146	C₂×Dic₅	—	4	4	no	—
147	C₂×Dic₅	—	8	2	no	—
148	C₂×Dic₅	—	8	4	no	—
149	C₅×C₈	—	4	4	no	—
150	C₅×C₈	C₂×C₄⋊C₄	1	8	yes	—
151	#168	—	2	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	—	2	2	no	—
156	C₂×M₁₆	—	2	2	no	—
157	C₂×M₁₆	—	2	2	no	—
158	C₂×M₁₆	—	5	2	no	—
159	C₂×M₁₆	—	5	2	no	—
160	C₂×M₁₆	—	10	2	no	—
161	C₂×M₁₆	—	10	2	no	—
162	C₂×M₁₆	Dic₁₀	1	4	yes	—
163	C₂×M₁₆	C₄×D₅	1	2	yes	—
164	C₂×M₁₆	C₄×D₅	1	2	yes	—
165	C₂×M₁₆	C₂²⋊D₅	1	4	yes	—
166	C₂×M₁₆	C₂²×D₅	1	4	yes	—
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	—	5	2	no	—
169	C₂²×C₈	C₂²×D₅	1	2	yes	—
170	#117	—	5	2	no	—
171	#117	—	5	2	no	—
172	#117	—	5	4	no	—
173	#117	—	10	1	no	—
174	#117	—	10	1	no	—
175	#117	—	10	2	no	—
176	#117	—	10	2	no	—
177	#117	—	10	2	no	—
178	C₂×C₄⋊C₄	—	10	1	no	—
179	C₂×C₄⋊C₄	—	10	2	no	—
180	C₂×C₂²⋊C₄	—	10	1	no	—
181	C₂×C₂²⋊C₄	—	10	2	no	—
182	C₂×C₄²	—	5	2	no	—
183	C₂×C₄²	—	10	2	no	—
184	#108	—	2	8	no	—
185	#107	—	10	4	no	—
186	#106	—	10	4	no	—
187	#97	—	5	8	no	—
188	C₄×C₈	—	5	8	no	—
189	C₂²×C₅	—	4	1	no	—
190	C₂²×C₅	—	4	2	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	#382	1	2	yes	—
200	C₄×C₅	#382	1	2	yes	—
201	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
202	C₄×C₅	C₂×#117	1	1	yes	characteristic
203	C₄×C₅	C₂×#124	1	1	yes	characteristic
204	C₄×C₅	C₂×#125	1	1	yes	characteristic
205	Dic₅	—	4	4	no	—
206	Dic₅	—	8	4	no	—
207	C₂⁴	#713	1	1	yes	characteristic
208	C₂²×C₄	—	10	2	no	—
209	C₂²×C₄	—	20	2	no	—
210	C₂²×C₄	C₂×Dic₁₀	1	2	yes	—
211	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
213	C₂²×C₄	#711	1	1	yes	characteristic
214	C₂²×C₄	#711	1	1	yes	characteristic
215	C₂²×C₄	#711	1	2	yes	—
216	C₂²×C₄	#713	1	2	yes	—
217	C₂²×C₄	#713	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₂²⋊D₅	1	2	yes	—
219	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
220	M₁₆	—	2	4	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	—	4	2	no	—
223	M₁₆	—	4	2	no	—
224	M₁₆	—	10	2	no	—
225	M₁₆	—	10	2	no	—
226	M₁₆	—	10	4	no	—
227	M₁₆	#683	1	8	yes	—
228	M₁₆	#683	1	8	yes	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	5	4	no	—
233	C₂×C₈	—	10	4	no	—
234	C₂×C₈	C₂×Dic₁₀	1	4	yes	—
235	C₂×C₈	C₂×C₄×D₅	1	4	yes	—
236	C₂×C₈	C₂×C₂²⋊D₅	1	4	yes	—
237	C₄⋊C₄	—	10	2	no	—
238	C₄⋊C₄	—	10	2	no	—
239	C₄⋊C₄	—	10	4	no	—
240	C₄⋊C₄	—	20	2	no	—
241	C₂²⋊C₄	—	10	2	no	—
242	C₂²⋊C₄	—	10	2	no	—
243	C₂²⋊C₄	—	10	4	no	—
244	C₂²⋊C₄	—	20	2	no	—
245	C₄²	—	5	4	no	—
246	C₄²	—	10	2	no	—
247	C₄²	—	10	4	no	—
248	C₄²	—	20	2	no	—
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	#4036	1	2	yes	—
257	C₂³	#4038	1	1	yes	characteristic
258	C₂³	#4038	1	1	yes	characteristic
259	C₂³	#4093	1	2	yes	—
260	C₂³	C₂×#713	1	1	yes	characteristic
261	C₂×C₄	—	10	4	no	—
262	C₂×C₄	—	20	4	no	—
263	C₂×C₄	—	40	2	no	—
264	C₂×C₄	—	40	4	no	—
265	C₂×C₄	#4036	1	2	yes	—
266	C₂×C₄	#4036	1	2	yes	—
267	C₂×C₄	#4036	1	2	yes	—
268	C₂×C₄	#4038	1	2	yes	—
269	C₂×C₄	#4038	1	2	yes	—
270	C₂×C₄	#4038	1	1	yes	characteristic
271	C₂×C₄	#4038	1	1	yes	characteristic
272	C₂×C₄	C₂×#683	1	2	yes	—
273	C₂×C₄	C₂×#683	1	2	yes	—
274	C₂×C₄	#4082	1	2	yes	—
275	C₂×C₄	#4082	1	2	yes	—
276	C₂×C₄	#4093	1	2	yes	—
277	C₂×C₄	C₂²×Dic₁₀	1	1	yes	characteristic
278	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
279	C₂×C₄	C₂×#711	1	1	yes	characteristic
280	C₂×C₄	C₂×#711	1	1	yes	characteristic
281	C₂×C₄	C₂×#713	1	1	yes	characteristic
282	C₂×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
283	C₈	—	4	4	no	—
284	C₈	—	10	4	no	—
285	C₈	C₂×#683	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	#65266	1	2	yes	—
293	C₂²	#65266	1	2	yes	—
294	C₂²	C₂×#4036	1	1	yes	characteristic
295	C₂²	C₂×#4038	1	1	yes	characteristic
296	C₂²	C₂×#4093	1	1	yes	characteristic
297	C₄	—	20	4	no	—
298	C₄	—	40	4	no	—
299	C₄	#65266	1	2	yes	—
300	C₄	#65266	1	2	yes	—
301	C₄	C₂×#4036	1	1	yes	characteristic
302	C₄	C₂×#4038	1	1	yes	characteristic
303	C₄	C₂²×#683	1	1	yes	characteristic
304	C₄	C₂×#4082	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×#65266	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#4038
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₂×C₅×#430	C₂
socle	C₂×C₅	C₂×#382
prime radical	C₂⁴×C₅	#41
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	C₂×#430	D₅
hypocenter	C₅	#20757
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#430

Complements

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

N	H ≅ G/N	type
6. #10601491	C₂	semidirect
14. #65411	C₄	semidirect
16. M₁₆×Dic₅	C₄	semidirect
57. #3947	C₂×C₄	semidirect
58. C₈×Dic₅	C₂×C₄	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
C₂×C₅×#430	2
#10615483	2
#10615478	2
C₂×#64779	2
C₂×#64777	2
#10601491	2
#20757	5