Subgroups — #12069817

Order 1280.

750 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10610201	C₂	1	1	yes	maximal, characteristic
2	#10610189	C₂	1	2	yes	maximal, complemented
3	C₂×C₅×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
4	#10600528	C₂	1	2	yes	maximal, complemented
5	#10600524	C₂	1	1	yes	maximal, characteristic, complemented
6	#10598845	C₂	1	4	yes	maximal, complemented
7	C₈×D₄₀	C₂	1	4	yes	maximal, complemented
8	C₂×#4047	—	2	2	no	—
9	#65811	C₂²	1	2	yes	—
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	#65404	C₄	1	4	yes	—
14	#65399	—	2	2	no	—
15	#65213	C₂²	1	2	yes	—
16	#64984	C₄	1	1	yes	characteristic
17	#64983	C₄	1	2	yes	—
18	C₄×D₄₀	C₂²	1	4	yes	—
19	#64977	—	2	2	no	—
20	#64977	C₂²	1	4	yes	—
21	C₈×D₂₀	—	2	2	no	—
22	C₈×D₂₀	C₂²	1	4	yes	—
23	#64971	C₄	1	1	yes	characteristic
24	C₅×C₈²	C₂²	1	2	yes	—
25	C₅×C₈²	C₂²	1	2	yes	—
26	#64679	C₂²	1	8	yes	—
27	#64678	C₂²	1	4	yes	—
28	#17956	—	5	1	no	maximal, Sylow
29	C₂×#709	—	2	2	no	—
30	C₂²×C₅×C₈	—	2	1	no	—
31	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
32	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
33	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
35	#4076	—	2	2	no	—
36	#4076	C₂×C₄	1	4	yes	—
37	C₂×#681	—	2	2	no	—
38	C₂×D₄₀	C₂×C₄	1	4	yes	—
39	#4047	—	4	8	no	—
40	C₂×#676	—	2	4	no	—
41	C₂×C₈×D₅	—	2	4	no	—
42	#4021	—	2	2	no	—
43	#4020	C₂×C₄	1	2	yes	—
44	C₄×D₂₀	—	2	2	no	—
45	C₄×D₂₀	C₂³	1	4	yes	—
46	#4015	C₂×C₄	1	2	yes	—
47	C₄×C₅×C₈	—	2	1	no	—
48	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
49	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
50	C₄×C₅×C₈	D₄	1	1	yes	characteristic
51	C₄×C₅×C₈	D₄	1	1	yes	characteristic
52	C₄×C₅×C₈	C₂³	1	2	yes	—
53	C₄×C₅×C₈	C₂³	1	2	yes	—
54	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
55	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
56	#3953	C₂×C₄	1	8	yes	—
57	#3952	—	2	4	no	—
58	#3952	—	2	4	no	—
59	#3950	C₂×C₄	1	4	yes	—
60	C₄⋊#168	—	2	2	no	—
61	C₄⋊#168	C₂³	1	4	yes	—
62	#3169	—	5	1	no	—
63	#2931	—	5	2	no	—
64	#1589	—	5	4	no	—
65	C₈×D₈	—	5	4	no	—
66	#1567	—	5	1	no	—
67	#1492	—	5	2	no	—
68	C₂×C₈²	D₅	1	1	yes	characteristic, complemented
69	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	#41	1	1	yes	characteristic
72	C₂×C₂²⋊D₅	—	2	4	no	—
73	#709	—	4	4	no	—
74	#709	—	4	4	no	—
75	C₂×D₂₀	—	2	2	no	—
76	C₂×D₂₀	—	2	2	no	—
77	C₂×D₂₀	C₂²×C₄	1	4	yes	—
78	C₂×C₄×D₅	—	2	4	no	—
79	C₂×Dic₁₀	—	2	2	no	—
80	C₂×C₅×C₈	—	2	4	no	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₂×C₅×C₈	—	2	2	no	—
83	C₂×C₅×C₈	—	2	4	no	—
84	C₂×C₅×C₈	D₈	1	4	yes	complemented
85	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
86	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
87	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
88	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
89	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₅	#41	1	1	yes	characteristic
94	C₄²×C₅	C₂⁴	1	1	yes	characteristic
95	#685	—	2	4	no	—
96	#685	—	2	4	no	—
97	C₄⋊Dic₅	—	2	2	no	—
98	C₄⋊Dic₅	C₂²×C₄	1	4	yes	—
99	#681	—	4	4	no	—
100	C₂×#168	—	2	4	no	—
101	D₄₀	—	2	8	no	—
102	#676	—	4	8	no	—
103	C₈×D₅	—	4	8	no	—
104	C₂×#131	—	10	2	no	—
105	#532	—	5	2	no	—
106	#500	—	5	1	no	—
107	#493	—	5	1	no	—
108	#486	—	5	2	no	—
109	#466	—	5	4	no	—
110	C₄×D₈	—	5	4	no	—
111	#436	—	5	4	no	—
112	#436	—	10	2	no	—
113	C₈×D₄	—	5	4	no	—
114	C₈×D₄	—	10	2	no	—
115	#423	—	5	2	no	—
116	#408	—	10	2	no	—
117	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
119	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
120	#335	—	5	4	no	—
121	D₄⋊C₈	—	5	8	no	—
122	C₈²	C₂×D₅	1	2	yes	—
123	C₈²	C₂×D₅	1	2	yes	—
124	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
125	C₂²×D₅	—	2	4	no	—
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	—	2	2	no	—
128	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₅	#135	1	2	yes	—
133	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
136	C₂²⋊D₅	—	4	8	no	—
137	C₂×Dic₅	—	2	4	no	—
138	D₂₀	—	2	8	no	—
139	D₂₀	—	4	4	no	—
140	D₂₀	—	4	4	no	—
141	C₄×D₅	—	4	4	no	—
142	C₄×D₅	—	4	4	no	—
143	Dic₁₀	—	4	4	no	—
144	C₅×C₈	—	2	4	no	—
145	C₅×C₈	—	2	2	no	—
146	C₅×C₈	—	2	1	no	—
147	C₅×C₈	—	2	1	no	—
148	C₅×C₈	#131	1	4	yes	complemented
149	C₅×C₈	C₂×D₈	1	2	yes	—
150	C₅×C₈	C₂×D₈	1	2	yes	—
151	#168	—	4	4	no	—
152	C₂×#41	—	10	2	no	—
153	C₂×D₈	—	5	4	no	—
154	#131	—	20	8	no	—
155	C₂×M₁₆	—	10	2	no	—
156	C₂×M₁₆	—	10	4	no	—
157	C₂²×C₈	—	2	1	no	—
158	C₂²×C₈	—	10	4	no	—
159	C₂²×C₈	C₄×D₅	1	2	yes	—
160	C₂²×C₈	D₂₀	1	1	yes	characteristic
161	C₂²×C₈	D₂₀	1	1	yes	characteristic
162	C₄⋊Q₈	—	5	2	no	—
163	#127	—	5	2	no	—
164	#124	—	10	2	no	—
165	#121	—	5	4	no	—
166	#121	—	10	2	no	—
167	C₄×D₄	—	5	4	no	—
168	C₄×D₄	—	10	2	no	—
169	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
170	#107	—	5	4	no	—
171	C₄⋊C₈	—	5	4	no	—
172	C₄⋊C₈	—	10	2	no	—
173	#102	—	5	8	no	—
174	C₂²⋊C₈	—	10	4	no	—
175	C₂²⋊C₈	—	10	4	no	—
176	C₄×C₈	—	2	1	no	—
177	C₄×C₈	C₄×D₅	1	1	yes	characteristic
178	C₄×C₈	C₄×D₅	1	1	yes	characteristic
179	C₄×C₈	D₂₀	1	1	yes	characteristic
180	C₄×C₈	D₂₀	1	1	yes	characteristic
181	C₄×C₈	C₂²×D₅	1	2	yes	—
182	C₄×C₈	C₂²×D₅	1	2	yes	—
183	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
184	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
185	C₂²×C₅	—	2	1	no	—
186	C₂²×C₅	—	2	1	no	—
187	C₂²×C₅	C₄×D₈	1	2	yes	—
188	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
189	C₂×D₅	—	4	4	no	—
190	C₂×D₅	—	4	4	no	—
191	C₂×D₅	—	4	4	no	—
192	C₄×C₅	—	2	1	no	—
193	C₄×C₅	—	2	1	no	—
194	C₄×C₅	C₄×D₈	1	2	yes	—
195	C₄×C₅	#443	1	2	yes	—
196	C₄×C₅	C₂×#131	1	2	yes	—
197	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
198	C₄×C₅	C₂×#135	1	1	yes	characteristic
199	Dic₅	—	4	4	no	—
200	#41	—	20	4	no	—
201	#41	—	20	4	no	—
202	C₂×Q₈	—	10	2	no	—
203	C₂×D₄	—	5	4	no	—
204	C₂×D₄	—	10	4	no	—
205	C₂×D₄	—	10	2	no	—
206	C₂×D₄	—	10	2	no	—
207	C₂²×C₄	—	10	4	no	—
208	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
210	C₂²×C₄	#709	1	1	yes	characteristic
211	D₈	—	10	8	no	—
212	M₁₆	—	20	4	no	—
213	M₁₆	—	20	8	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	10	4	no	—
219	C₂×C₈	—	20	8	no	—
220	C₂×C₈	D₄₀	1	4	yes	complemented
221	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
222	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
223	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
224	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
225	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
226	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
227	C₄⋊C₄	—	5	4	no	—
228	C₄⋊C₄	—	10	2	no	—
229	C₂²⋊C₄	—	10	4	no	—
230	C₂²⋊C₄	—	10	4	no	—
231	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
232	C₄²	C₂×D₂₀	1	1	yes	characteristic
233	C₄²	#709	1	1	yes	characteristic
234	C₄²	C₂³×D₅	1	1	yes	characteristic
235	C₂×C₅	—	2	2	no	—
236	C₂×C₅	#2931	1	1	yes	characteristic
237	C₂×C₅	C₂×C₄×D₈	1	1	yes	characteristic
238	C₂×C₅	C₂×#443	1	1	yes	characteristic
239	D₅	—	8	4	no	—
240	C₂³	—	10	4	no	—
241	C₂³	C₄×D₂₀	1	1	yes	characteristic
242	Q₈	—	20	4	no	—
243	D₄	—	10	8	no	—
244	D₄	—	20	8	no	—
245	D₄	—	20	4	no	—
246	D₄	—	20	4	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	10	4	no	—
250	C₂×C₄	—	20	4	no	—
251	C₂×C₄	—	20	4	no	—
252	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
253	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
254	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
255	C₂×C₄	C₂×D₄₀	1	2	yes	—
256	C₂×C₄	#4050	1	2	yes	—
257	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
258	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
259	C₂×C₄	C₂×#709	1	1	yes	characteristic
260	C₈	—	2	4	no	—
261	C₈	—	2	2	no	—
262	C₈	—	2	1	no	—
263	C₈	—	2	1	no	—
264	C₈	—	20	4	no	—
265	C₈	#4047	1	4	yes	complemented
266	C₈	C₂×D₄₀	1	2	yes	—
267	C₈	C₂×D₄₀	1	2	yes	—
268	C₅	#17956	1	1	yes	characteristic, complemented, Sylow
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	20	4	no	—
272	C₂²	—	20	4	no	—
273	C₂²	—	20	4	no	—
274	C₂²	C₄×D₄₀	1	2	yes	—
275	C₂²	C₂×C₄×D₂₀	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	20	4	no	—
279	C₄	C₄×D₄₀	1	2	yes	—
280	C₄	#64989	1	2	yes	—
281	C₄	C₂×#4047	1	2	yes	—
282	C₄	C₂²×D₄₀	1	1	yes	characteristic
283	C₄	C₂×#4050	1	1	yes	characteristic
284	C₂	—	2	2	no	—
285	C₂	—	40	4	no	—
286	C₂	#10610189	1	1	yes	characteristic
287	C₂	C₂×C₄×D₄₀	1	1	yes	characteristic
288	C₂	C₂×#64989	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₄²	C₂³×D₅
Fitting	C₂×C₅×C₈²	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	#10610201	C₂
cosocle	C₄²×C₅	C₂⁴
hypercenter	C₂×C₈²	D₅
hypocenter	C₅	#17956
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
2. #10610189	C₂	semidirect
3. C₂×C₅×C₈²	C₂	semidirect
4. #10600528	C₂	semidirect
5. #10600524	C₂	semidirect
6. #10598845	C₂	semidirect
7. C₈×D₄₀	C₂	semidirect
68. C₂×C₈²	D₅	semidirect
84. C₂×C₅×C₈	D₈	semidirect
148. C₅×C₈	#131	semidirect
220. C₂×C₈	D₄₀	semidirect
265. C₈	#4047	semidirect
268. C₅	#17956	semidirect

Maximal subgroups

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

subgroup	index
#10610201	2
#10610189	2
C₂×C₅×C₈²	2
#10600528	2
#10600524	2
#10598845	2
C₈×D₄₀	2
#17956	5