Subgroups — #12071079

Order 1280.

586 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10610239	C₂	1	2	yes	maximal
2	#10610206	C₂	1	1	yes	maximal, characteristic
3	C₅×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
4	#10601561	C₂	1	2	yes	maximal
5	C₂×#64678	C₂	1	1	yes	maximal, characteristic
6	#10598879	C₂	1	4	yes	maximal, complemented
7	#10598849	C₂	1	2	yes	maximal, complemented
8	C₈×Dic₂₀	C₂	1	2	yes	maximal, complemented
9	#65804	C₂²	1	2	yes	—
10	C₅×#431	C₂²	1	1	yes	characteristic
11	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	#65411	—	2	2	no	—
14	#65408	—	2	2	no	—
15	#65405	C₄	1	2	yes	—
16	#65405	C₄	1	2	yes	—
17	C₂×C₄⋊#168	C₂²	1	2	yes	—
18	#64990	—	2	4	no	—
19	#64988	C₄	1	1	yes	characteristic
20	C₄×Dic₂₀	C₂²	1	2	yes	—
21	C₄×Dic₂₀	C₂²	1	2	yes	—
22	#64971	C₄	1	1	yes	characteristic
23	#64970	C₄	1	2	yes	—
24	#64968	C₂²	1	4	yes	—
25	C₈×Dic₁₀	C₂²	1	4	yes	—
26	C₅×#322	C₂²	1	2	yes	—
27	C₅×C₈²	C₂²	1	2	yes	—
28	#64678	C₂²	1	1	yes	characteristic
29	#64678	C₂²	1	1	yes	characteristic
30	#64678	C₂²	1	2	yes	—
31	#64676	C₂²	1	4	yes	—
32	#64676	C₂²	1	4	yes	—
33	#18068	—	5	1	no	maximal, Sylow
34	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
35	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
36	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
39	#4072	—	2	2	no	—
40	#4070	—	2	2	no	—
41	#4070	C₂×C₄	1	4	yes	—
42	C₂²×#168	—	2	2	no	—
43	C₂×Dic₂₀	C₂×C₄	1	2	yes	—
44	C₂×Dic₂₀	C₂×C₄	1	2	yes	—
45	#4016	—	2	2	no	—
46	#4015	C₂×C₄	1	2	yes	—
47	#4015	C₂×C₄	1	2	yes	—
48	C₄×Dic₁₀	—	2	2	no	—
49	C₄×Dic₁₀	C₂³	1	4	yes	—
50	C₅×C₄⋊C₈	—	2	1	no	—
51	C₅×C₂²⋊C₈	—	2	1	no	—
52	C₅×#97	D₄	1	2	yes	—
53	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
54	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
55	C₄×C₅×C₈	C₂³	1	2	yes	—
56	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
57	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
58	C₄×C₅×C₈	C₂³	1	2	yes	—
59	#3950	C₂×C₄	1	2	yes	—
60	#3950	C₂×C₄	1	2	yes	—
61	#3948	C₂×C₄	1	4	yes	—
62	#3948	C₂×C₄	1	4	yes	—
63	#3947	—	2	4	no	—
64	#3946	—	2	4	no	—
65	#3946	—	2	4	no	—
66	C₈×Dic₅	—	2	4	no	—
67	C₄⋊#168	—	2	2	no	—
68	C₄⋊#168	C₂³	1	2	yes	—
69	C₄⋊#168	C₂³	1	2	yes	—
70	#3170	—	5	1	no	—
71	#2962	—	5	2	no	—
72	#1591	—	5	2	no	—
73	#1584	—	5	4	no	—
74	C₈×Q₁₆	—	5	2	no	—
75	C₂×#335	—	5	1	no	—
76	#1481	—	5	2	no	—
77	C₈×M₁₆	D₅	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₅	#41	1	1	yes	characteristic
81	C₂×Dic₁₀	—	2	2	no	—
82	C₂×Dic₁₀	C₂²×C₄	1	4	yes	—
83	C₅×M₁₆	—	2	2	no	—
84	C₅×M₁₆	Q₁₆	1	4	yes	complemented
85	C₂×C₅×C₈	—	2	2	no	—
86	C₂×C₅×C₈	—	2	2	no	—
87	C₂×C₅×C₈	—	4	1	no	—
88	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
93	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
94	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₅	#41	1	1	yes	characteristic
97	C₄²×C₅	C₂⁴	1	1	yes	characteristic
98	C₂²⋊Dic₅	—	2	4	no	—
99	C₄⋊Dic₅	—	2	2	no	—
100	C₄⋊Dic₅	—	2	2	no	—
101	C₄⋊Dic₅	—	2	2	no	—
102	C₄⋊Dic₅	C₂²×C₄	1	4	yes	—
103	#683	—	2	4	no	—
104	#683	—	2	4	no	—
105	C₄×Dic₅	—	2	4	no	—
106	C₂×#168	—	2	2	no	—
107	C₂×#168	—	2	2	no	—
108	C₂×#168	—	4	4	no	—
109	Dic₂₀	—	2	4	no	—
110	Dic₂₀	—	4	2	no	—
111	#533	—	5	2	no	—
112	#500	—	5	1	no	—
113	#494	—	5	1	no	—
114	#487	—	5	2	no	—
115	#467	—	5	2	no	—
116	#467	—	5	2	no	—
117	#446	—	5	4	no	—
118	#446	—	10	4	no	—
119	C₈×Q₈	—	5	4	no	—
120	C₄×Q₁₆	—	5	2	no	—
121	C₄×Q₁₆	—	5	2	no	—
122	#431	C₂×D₅	1	1	yes	characteristic
123	#424	—	10	2	no	—
124	C₂×C₄⋊C₈	—	5	2	no	—
125	#405	—	10	2	no	—
126	C₄×M₁₆	C₂×D₅	1	1	yes	characteristic
127	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
128	#335	—	5	1	no	—
129	#335	—	5	2	no	—
130	#335	—	5	1	no	—
131	Q₈⋊C₈	—	5	4	no	—
132	Q₈⋊C₈	—	5	4	no	—
133	#322	C₂×D₅	1	2	yes	—
134	C₈²	C₂×D₅	1	2	yes	—
135	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅	—	2	1	no	—
137	C₂×C₄×C₅	—	2	1	no	—
138	C₂×C₄×C₅	—	4	1	no	—
139	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
143	C₂×C₄×C₅	#135	1	2	yes	—
144	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
147	C₂×Dic₅	—	2	4	no	—
148	C₂×Dic₅	—	2	4	no	—
149	Dic₁₀	—	2	4	no	—
150	Dic₁₀	—	2	4	no	—
151	Dic₁₀	—	8	2	no	—
152	C₅×C₈	—	2	1	no	—
153	C₅×C₈	—	2	1	no	—
154	C₅×C₈	—	2	1	no	—
155	C₅×C₈	—	2	2	no	—
156	C₅×C₈	—	4	2	no	—
157	C₅×C₈	#131	1	2	yes	—
158	C₅×C₈	C₂×Q₁₆	1	4	yes	—
159	#168	—	4	2	no	—
160	#168	—	4	2	no	—
161	C₂×Q₁₆	—	5	2	no	—
162	C₂×Q₁₆	—	5	2	no	—
163	C₂×M₁₆	D₂₀	1	1	yes	characteristic
164	C₂×M₁₆	D₂₀	1	1	yes	characteristic
165	C₂²×C₈	—	10	2	no	—
166	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
167	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
168	C₄⋊Q₈	—	5	2	no	—
169	C₄⋊Q₈	—	5	2	no	—
170	#125	—	10	2	no	—
171	C₂²⋊Q₈	—	5	4	no	—
172	C₂²⋊Q₈	—	10	2	no	—
173	C₄×Q₈	—	5	4	no	—
174	C₄×Q₈	—	10	2	no	—
175	#117	—	10	2	no	—
176	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
177	#107	—	5	2	no	—
178	#107	—	5	2	no	—
179	C₄⋊C₈	—	2	1	no	—
180	C₄⋊C₈	—	5	2	no	—
181	C₄⋊C₈	—	5	2	no	—
182	C₄⋊C₈	—	10	4	no	—
183	C₄⋊C₈	—	10	4	no	—
184	C₄⋊C₈	—	10	2	no	—
185	#103	—	5	4	no	—
186	#103	—	5	4	no	—
187	C₂²⋊C₈	—	2	1	no	—
188	#97	—	10	4	no	—
189	#97	D₂₀	1	2	yes	—
190	C₄×C₈	—	10	4	no	—
191	C₄×C₈	C₄×D₅	1	1	yes	characteristic
192	C₄×C₈	C₄×D₅	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₅	1	2	yes	—
194	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
195	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
196	C₄×C₈	C₂²×D₅	1	2	yes	—
197	C₂²×C₅	—	2	1	no	—
198	C₂²×C₅	—	2	1	no	—
199	C₂²×C₅	C₄×Q₁₆	1	2	yes	—
200	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
201	C₄×C₅	—	2	1	no	—
202	C₄×C₅	—	2	1	no	—
203	C₄×C₅	—	2	1	no	—
204	C₄×C₅	C₄×Q₁₆	1	2	yes	—
205	C₄×C₅	C₂×#131	1	1	yes	characteristic
206	C₄×C₅	#568	1	1	yes	characteristic
207	C₄×C₅	C₂²×Q₁₆	1	1	yes	characteristic
208	C₄×C₅	C₂×#135	1	1	yes	characteristic
209	Dic₅	—	4	4	no	—
210	Dic₅	—	4	4	no	—
211	C₂×Q₈	—	5	4	no	—
212	C₂×Q₈	—	10	2	no	—
213	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
215	C₂²×C₄	#709	1	1	yes	characteristic
216	Q₁₆	—	10	4	no	—
217	Q₁₆	—	20	2	no	—
218	M₁₆	—	2	2	no	—
219	M₁₆	Dic₂₀	1	4	yes	complemented
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	10	2	no	—
224	C₂×C₈	—	10	2	no	—
225	C₂×C₈	—	20	4	no	—
226	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
227	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
228	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
229	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
230	C₂×C₈	C₂×D₂₀	1	2	yes	—
231	C₂×C₈	C₂×D₂₀	1	2	yes	—
232	C₄⋊C₄	—	5	4	no	—
233	C₄⋊C₄	—	10	4	no	—
234	C₄⋊C₄	—	10	2	no	—
235	C₄⋊C₄	—	10	2	no	—
236	C₄⋊C₄	—	10	4	no	—
237	C₄⋊C₄	—	10	2	no	—
238	C₂²⋊C₄	—	10	4	no	—
239	C₄²	—	10	4	no	—
240	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
241	C₄²	C₂×D₂₀	1	1	yes	characteristic
242	C₄²	#709	1	1	yes	characteristic
243	C₄²	C₂³×D₅	1	1	yes	characteristic
244	C₂×C₅	—	2	2	no	—
245	C₂×C₅	#2935	1	1	yes	characteristic
246	C₂×C₅	C₂×C₄×Q₁₆	1	1	yes	characteristic
247	C₂×C₅	#2957	1	1	yes	characteristic
248	C₂³	C₄×D₂₀	1	1	yes	characteristic
249	Q₈	—	10	4	no	—
250	Q₈	—	10	4	no	—
251	Q₈	—	40	2	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	10	4	no	—
256	C₂×C₄	—	10	4	no	—
257	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
258	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
259	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
260	C₂×C₄	#4050	1	2	yes	—
261	C₂×C₄	C₂×Dic₂₀	1	2	yes	—
262	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
263	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
264	C₂×C₄	C₂×#709	1	1	yes	characteristic
265	C₈	—	2	2	no	—
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	4	2	no	—
270	C₈	—	20	2	no	—
271	C₈	—	20	2	no	—
272	C₈	C₂×Dic₂₀	1	4	yes	—
273	C₈	#4053	1	2	yes	—
274	C₅	#18068	1	1	yes	characteristic, complemented, Sylow
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	C₄×Dic₂₀	1	2	yes	—
278	C₂²	C₂×C₄×D₂₀	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	20	4	no	—
283	C₄	—	20	4	no	—
284	C₄	C₄×Dic₂₀	1	2	yes	—
285	C₄	C₂×#4050	1	1	yes	characteristic
286	C₄	C₂²×Dic₂₀	1	1	yes	characteristic
287	C₄	C₂×#4053	1	1	yes	characteristic
288	C₄	#66079	1	1	yes	characteristic
289	C₂	—	2	2	no	—
290	C₂	C₂×C₄×Dic₂₀	1	1	yes	characteristic
291	C₂	#10610253	1	1	yes	characteristic
292	C₂	#10610311	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₀
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₄²	C₂³×D₅
Fitting	C₅×C₈×M₁₆	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂³×C₅	C₄×D₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	C₈×M₁₆	D₅
hypocenter	C₅	#18068
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
6. #10598879	C₂	semidirect
7. #10598849	C₂	semidirect
8. C₈×Dic₂₀	C₂	semidirect
84. C₅×M₁₆	Q₁₆	semidirect
219. M₁₆	Dic₂₀	semidirect
274. C₅	#18068	semidirect

Maximal subgroups

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

subgroup	index
#10610239	2
#10610206	2
C₅×C₈×M₁₆	2
#10601561	2
C₂×#64678	2
#10598879	2
#10598849	2
C₈×Dic₂₀	2
#18068	5