Subgroups — #12005196

Order 1280.

982 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×#3949	C₂	1	2	yes	maximal
3	C₂×#64771	C₂	1	2	yes	maximal
4	#10602328	C₂	1	2	yes	maximal
5	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊Dic₅	C₂²	1	4	yes	—
9	C₂×C₄×C₅×C₈	—	2	4	no	—
10	C₂×C₄×C₅×C₈	C₄	1	4	yes	complemented
11	C₂×#3963	—	2	4	no	—
12	C₂×#3949	—	2	4	no	—
13	C₂×#3949	—	2	4	no	—
14	C₂×#3949	C₄	1	8	yes	complemented
15	#64771	—	2	4	no	—
16	#10861	—	5	1	no	maximal, Sylow
17	C₂³×C₄×C₅	D₄	1	2	yes	—
18	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
19	C₂³×Dic₅	—	2	4	no	—
20	C₂²×C₅×C₈	—	2	2	no	—
21	C₂²×C₅×C₈	—	2	2	no	—
22	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
23	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
24	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₅×C₈	D₄	1	2	yes	—
26	C₂²×C₅×C₈	D₄	1	2	yes	—
27	C₂²×C₅×C₈	D₄	1	4	yes	—
28	C₂²×C₅×C₈	Q₈	1	2	yes	—
29	C₂²×C₅×C₈	Q₈	1	2	yes	—
30	C₂×C₄²×C₅	—	2	2	no	—
31	C₂×C₄²×C₅	—	2	2	no	—
32	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅	Q₈	1	2	yes	—
34	C₂×C₄⋊Dic₅	—	2	4	no	—
35	C₂×C₄⋊Dic₅	—	2	4	no	—
36	C₂×C₄⋊Dic₅	—	2	4	no	—
37	C₂×C₄⋊Dic₅	—	2	4	no	—
38	C₂×C₄⋊Dic₅	—	2	4	no	—
39	C₂×C₄⋊Dic₅	C₂×C₄	1	8	yes	—
40	C₄×C₅×C₈	—	2	8	no	—
41	#3963	—	4	8	no	—
42	#3949	—	2	16	no	—
43	C₂²×#106	—	5	2	no	—
44	C₂²×C₄×C₈	D₅	1	1	yes	characteristic
45	C₂×#340	—	5	2	no	—
46	C₄⋊#95	—	5	2	no	—
47	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅	—	2	4	no	—
49	C₂²×C₄×C₅	—	2	4	no	—
50	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
51	C₂²×C₄×C₅	SD₁₆	1	8	yes	—
52	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₅	#41	1	2	yes	—
61	C₂²×C₄×C₅	#41	1	2	yes	—
62	C₂²×Dic₅	—	2	8	no	—
63	C₂²×Dic₅	—	4	4	no	—
64	C₂²×Dic₅	—	4	4	no	—
65	C₂²×Dic₅	—	4	8	no	—
66	C₂²×Dic₅	—	4	8	no	—
67	C₂×C₅×C₈	—	2	2	no	—
68	C₂×C₅×C₈	—	2	2	no	—
69	C₂×C₅×C₈	—	2	4	no	—
70	C₂×C₅×C₈	—	2	4	no	—
71	C₂×C₅×C₈	C₄²	1	4	yes	—
72	C₂×C₅×C₈	C₂²⋊C₄	1	8	yes	—
73	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
74	C₂×C₅×C₈	C₄⋊C₄	1	8	yes	—
75	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
76	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
77	C₄²×C₅	—	2	2	no	—
78	C₄²×C₅	—	2	2	no	—
79	C₄²×C₅	—	2	4	no	—
80	C₄⋊Dic₅	—	2	8	no	—
81	C₄⋊Dic₅	—	2	8	no	—
82	C₄⋊Dic₅	—	4	4	no	—
83	C₄⋊Dic₅	—	4	4	no	—
84	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
85	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
86	C₂²×C₄⋊C₄	—	5	4	no	—
87	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
88	C₂×#106	—	5	8	no	—
89	C₂×#106	—	10	4	no	—
90	C₂×#106	—	10	4	no	—
91	C₂×C₄×C₈	—	2	4	no	—
92	C₂×C₄×C₈	Dic₅	1	4	yes	complemented
93	C₂×#95	—	10	4	no	—
94	#340	—	10	4	no	—
95	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂³×C₅	C₄×D₄	1	2	yes	—
97	C₂³×C₅	#121	1	2	yes	—
98	C₂³×C₅	#127	1	1	yes	characteristic
99	C₂³×C₅	C₄⋊Q₈	1	1	yes	characteristic
100	C₂³×C₅	C₂×SD₁₆	1	4	yes	—
101	C₂³×C₅	#135	1	4	yes	—
102	C₂×C₄×C₅	—	2	4	no	—
103	C₂×C₄×C₅	—	2	4	no	—
104	C₂×C₄×C₅	—	2	16	no	—
105	C₂×C₄×C₅	#106	1	8	yes	complemented
106	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
109	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₅	#117	1	1	yes	characteristic
113	C₂×C₄×C₅	#117	1	1	yes	characteristic
114	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
116	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
117	C₂×C₄×C₅	#121	1	2	yes	—
118	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₅	#124	1	2	yes	—
121	C₂×C₄×C₅	#125	1	2	yes	—
122	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
124	C₂×Dic₅	—	4	8	no	—
125	C₂×Dic₅	—	4	16	no	—
126	C₂×Dic₅	—	8	8	no	—
127	C₂×Dic₅	—	8	4	no	—
128	C₂×Dic₅	—	8	4	no	—
129	C₅×C₈	—	2	4	no	—
130	C₅×C₈	#95	1	8	yes	complemented
131	C₂³×C₄	—	10	4	no	—
132	C₂³×C₄	D₂₀	1	2	yes	—
133	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
134	C₂²×C₈	—	2	2	no	—
135	C₂²×C₈	—	2	2	no	—
136	C₂²×C₈	Dic₁₀	1	2	yes	—
137	C₂²×C₈	Dic₁₀	1	2	yes	—
138	C₂²×C₈	C₄×D₅	1	4	yes	—
139	C₂²×C₈	D₂₀	1	2	yes	—
140	C₂²×C₈	D₂₀	1	2	yes	—
141	C₂²×C₈	C₂×Dic₅	1	2	yes	—
142	C₂²×C₈	C₂×Dic₅	1	2	yes	—
143	C₂²×C₈	C₂²⋊D₅	1	4	yes	—
144	C₂×C₄⋊C₄	—	5	8	no	—
145	C₂×C₄⋊C₄	—	10	4	no	—
146	C₂×C₄⋊C₄	—	10	4	no	—
147	C₂×C₄⋊C₄	—	10	4	no	—
148	C₂×C₄⋊C₄	—	10	4	no	—
149	C₂×C₄⋊C₄	—	10	4	no	—
150	C₂×C₄²	—	2	2	no	—
151	C₂×C₄²	—	2	2	no	—
152	C₂×C₄²	Dic₁₀	1	2	yes	—
153	C₂×C₄²	C₂×Dic₅	1	2	yes	—
154	#106	—	10	16	no	—
155	C₄×C₈	—	2	8	no	—
156	#95	—	20	8	no	—
157	C₂²×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
158	C₂²×C₅	#380	1	1	yes	characteristic
159	C₂²×C₅	#384	1	1	yes	characteristic
160	C₂²×C₅	#389	1	2	yes	—
161	C₂²×C₅	#390	1	2	yes	—
162	C₂²×C₅	C₂×#106	1	2	yes	—
163	C₂²×C₅	#427	1	2	yes	—
164	C₂²×C₅	C₄×SD₁₆	1	8	yes	—
165	C₂²×C₅	#465	1	8	yes	—
166	C₂²×C₅	#492	1	2	yes	—
167	C₂²×C₅	#495	1	2	yes	—
168	C₂²×C₅	#498	1	2	yes	—
169	C₂²×C₅	#499	1	2	yes	—
170	C₄×C₅	—	2	8	no	—
171	C₄×C₅	C₂×#95	1	1	yes	characteristic
172	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
173	C₄×C₅	#380	1	1	yes	characteristic
174	C₄×C₅	#382	1	1	yes	characteristic
175	C₄×C₅	#389	1	2	yes	—
176	C₄×C₅	#391	1	2	yes	—
177	Dic₅	—	8	8	no	—
178	C₂⁴	C₂×D₂₀	1	1	yes	characteristic
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	10	8	no	—
182	C₂²×C₄	—	20	4	no	—
183	C₂²×C₄	—	20	8	no	—
184	C₂²×C₄	—	20	8	no	—
185	C₂²×C₄	—	20	4	no	—
186	C₂²×C₄	#677	1	8	yes	—
187	C₂²×C₄	C₄⋊Dic₅	1	4	yes	—
188	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
189	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
190	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
192	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
194	C₂²×C₄	#709	1	2	yes	—
195	C₂²×C₄	#709	1	2	yes	—
196	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₂²⋊D₅	1	2	yes	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	C₄×Dic₅	1	4	yes	—
203	C₂×C₈	#683	1	8	yes	—
204	C₂×C₈	C₄⋊Dic₅	1	4	yes	—
205	C₂×C₈	C₄⋊Dic₅	1	4	yes	—
206	C₂×C₈	#685	1	8	yes	—
207	C₂×C₈	C₂²⋊Dic₅	1	4	yes	—
208	C₄⋊C₄	—	10	8	no	—
209	C₄⋊C₄	—	10	8	no	—
210	C₄⋊C₄	—	20	4	no	—
211	C₄⋊C₄	—	20	4	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	4	no	—
214	C₄²	—	2	2	no	—
215	C₂×C₅	C₄⋊#95	1	1	yes	characteristic
216	C₂×C₅	C₄×#106	1	2	yes	—
217	C₂×C₅	#1807	1	2	yes	—
218	C₂×C₅	#1844	1	1	yes	characteristic
219	C₂×C₅	#1846	1	1	yes	characteristic
220	C₂×C₅	#1941	1	4	yes	—
221	C₂×C₅	#1968	1	4	yes	—
222	C₂³	#4015	1	1	yes	characteristic
223	C₂³	C₄×D₂₀	1	2	yes	—
224	C₂³	#4020	1	1	yes	characteristic
225	C₂³	C₂×#677	1	4	yes	—
226	C₂³	#4050	1	4	yes	—
227	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
228	C₂³	#4076	1	2	yes	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	16	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	20	8	no	—
233	C₂×C₄	—	20	16	no	—
234	C₂×C₄	—	40	8	no	—
235	C₂×C₄	—	40	4	no	—
236	C₂×C₄	—	40	4	no	—
237	C₂×C₄	#3949	1	8	yes	complemented
238	C₂×C₄	C₄×Dic₁₀	1	2	yes	—
239	C₂×C₄	C₄×Dic₁₀	1	2	yes	—
240	C₂×C₄	#4015	1	1	yes	characteristic
241	C₂×C₄	#4015	1	1	yes	characteristic
242	C₂×C₄	#4016	1	2	yes	—
243	C₂×C₄	C₄×D₂₀	1	2	yes	—
244	C₂×C₄	#4021	1	2	yes	—
245	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
246	C₂×C₄	C₂×#683	1	2	yes	—
247	C₂×C₄	#4070	1	2	yes	—
248	C₂×C₄	#4070	1	2	yes	—
249	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
250	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
251	C₂×C₄	#4072	1	1	yes	characteristic
252	C₂×C₄	#4072	1	1	yes	characteristic
253	C₂×C₄	C₂×#685	1	2	yes	—
254	C₂×C₄	#4076	1	2	yes	—
255	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
256	C₈	—	2	4	no	—
257	C₈	#3963	1	8	yes	complemented
258	C₅	#10861	1	1	yes	characteristic, complemented, Sylow
259	C₂²	#64969	1	2	yes	—
260	C₂²	#64972	1	2	yes	—
261	C₂²	C₄×#677	1	8	yes	—
262	C₂²	#64982	1	2	yes	—
263	C₂²	#64985	1	2	yes	—
264	C₂²	#65217	1	2	yes	—
265	C₂²	C₄×C₄⋊Dic₅	1	1	yes	characteristic
266	C₂²	#65224	1	1	yes	characteristic
267	C₂²	#65228	1	2	yes	—
268	C₂²	C₂×#3949	1	2	yes	—
269	C₂²	#65395	1	2	yes	—
270	C₂²	#65403	1	8	yes	—
271	C₂²	#65501	1	1	yes	characteristic
272	C₄	—	2	8	no	—
273	C₄	—	40	8	no	—
274	C₄	#65217	1	2	yes	—
275	C₄	#65218	1	2	yes	—
276	C₄	C₄×C₄⋊Dic₅	1	1	yes	characteristic
277	C₄	#65225	1	1	yes	characteristic
278	C₄	C₂×#3963	1	1	yes	characteristic
279	C₄	#65501	1	1	yes	characteristic
280	C₂	C₄×#3949	1	2	yes	—
281	C₂	#10600557	1	1	yes	characteristic
282	C₂	#10600559	1	1	yes	characteristic
283	C₂	#10600562	1	4	yes	—
284	C₂	#10600567	1	4	yes	—
285	C₂	#10602328	1	1	yes	characteristic
286	C₂	#10602763	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅
Fitting	C₂²×C₄×C₅×C₈	C₂
socle, prime radical	C₂⁴×C₅	C₂×D₄
cosocle	C₂³×C₄×C₅	C₂³
hypercenter	C₂²×C₄×C₈	D₅
hypocenter	C₅	#10861
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
10. C₂×C₄×C₅×C₈	C₄	semidirect
14. C₂×#3949	C₄	semidirect
92. C₂×C₄×C₈	Dic₅	semidirect
105. C₂×C₄×C₅	#106	semidirect
130. C₅×C₈	#95	semidirect
237. C₂×C₄	#3949	semidirect
257. C₈	#3963	semidirect
258. C₅	#10861	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₅×C₈	2
C₂²×#3949	2
C₂×#64771	2
#10602328	2
#10861	5