Subgroups — #12097390

Order 1280.

1126 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#405	C₂	1	1	yes	maximal, characteristic
2	C₂×#65395	C₂	1	1	yes	maximal, characteristic
3	C₂²×#3950	C₂	1	1	yes	maximal, characteristic
4	#10613954	C₂	1	2	yes	maximal
5	#10602778	C₂	1	2	yes	maximal
6	#10600607	C₂	1	4	yes	maximal, complemented
7	C₄×#3950	C₂	1	4	yes	maximal, complemented
8	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
9	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₅×#117	C₂²	1	1	yes	characteristic
11	C₂×#4072	C₂²	1	2	yes	—
12	C₂²×C₄⋊Dic₅	C₂²	1	2	yes	—
13	C₅×#405	C₄	1	8	yes	complemented
14	C₂×C₅×#97	C₂²	1	2	yes	—
15	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
16	#65395	C₄	1	8	yes	complemented
17	C₂×#3950	C₄	1	8	yes	complemented
18	C₂×#3950	C₂²	1	2	yes	—
19	C₂×#3950	C₂²	1	2	yes	—
20	C₂×#3950	C₂²	1	2	yes	—
21	C₂×#3949	C₂²	1	2	yes	—
22	C₄⋊C₄×Dic₅	—	2	4	no	—
23	C₂²⋊C₄×Dic₅	—	2	4	no	—
24	C₄×C₄⋊Dic₅	C₂²	1	8	yes	—
25	#64771	C₂²	1	8	yes	—
26	#20455	—	5	1	no	maximal, Sylow
27	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂³×Dic₅	—	2	2	no	—
29	C₂×C₅×M₁₆	—	2	2	no	—
30	C₂×C₅×M₁₆	—	2	2	no	—
31	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
32	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
33	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
34	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
35	C₂²×C₅×C₈	C₂³	1	2	yes	—
36	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
37	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
38	C₅×#117	C₂×C₄	1	4	yes	—
39	C₅×#117	Q₈	1	4	yes	—
40	C₂×C₅×C₄⋊C₄	D₄	1	2	yes	—
41	C₂×C₅×C₂²⋊C₄	D₄	1	2	yes	—
42	C₂×C₄²×C₅	C₂³	1	2	yes	—
43	C₂×C₂²⋊Dic₅	—	2	2	no	—
44	#4072	—	2	4	no	—
45	#4072	C₂×C₄	1	8	yes	—
46	C₂×C₄⋊Dic₅	—	2	2	no	—
47	C₂×C₄⋊Dic₅	—	2	4	no	—
48	C₂×C₄⋊Dic₅	C₂×C₄	1	8	yes	—
49	C₂×C₄⋊Dic₅	C₂³	1	4	yes	—
50	C₂×C₄⋊Dic₅	C₂³	1	4	yes	—
51	C₂×C₄×Dic₅	—	2	2	no	—
52	C₂×C₄×Dic₅	—	2	8	no	—
53	C₅×#97	C₂×C₄	1	8	yes	complemented
54	C₄×C₅×C₈	C₂×C₄	1	8	yes	complemented
55	#3963	—	2	8	no	—
56	#3950	C₂×C₄	1	8	yes	—
57	#3950	C₂×C₄	1	8	yes	—
58	#3950	C₂×C₄	1	8	yes	complemented
59	#3949	C₂×C₄	1	8	yes	complemented
60	C₂×#427	—	5	1	no	—
61	C₂²×#107	—	5	1	no	—
62	C₂×#405	D₅	1	1	yes	characteristic
63	#2274	—	5	2	no	—
64	#1776	—	5	4	no	—
65	C₄×#107	—	5	4	no	—
66	#1736	—	5	2	no	—
67	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
72	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
74	C₂²×C₄×C₅	#41	1	2	yes	—
75	C₂²×C₄×C₅	#41	1	2	yes	—
76	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
77	C₂²×Dic₅	—	2	4	no	—
78	C₂²×Dic₅	—	2	4	no	—
79	C₂²×Dic₅	—	2	8	no	—
80	C₂²×Dic₅	—	8	4	no	—
81	C₅×M₁₆	—	2	8	no	—
82	C₂×C₅×C₈	—	2	2	no	—
83	C₂×C₅×C₈	—	2	2	no	—
84	C₂×C₅×C₈	C₄²	1	16	yes	complemented
85	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
86	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
87	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
88	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
89	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
90	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
91	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
92	C₅×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
93	C₅×C₂²⋊C₄	C₄⋊C₄	1	8	yes	—
94	C₄²×C₅	C₂²×C₄	1	4	yes	—
95	C₄²×C₅	C₂×Q₈	1	4	yes	—
96	C₂²⋊Dic₅	—	2	8	no	—
97	C₄⋊Dic₅	—	2	8	no	—
98	C₄⋊Dic₅	—	2	4	no	—
99	C₄⋊Dic₅	—	2	4	no	—
100	C₄⋊Dic₅	C₂²×C₄	1	8	yes	—
101	C₄⋊Dic₅	C₂²×C₄	1	8	yes	—
102	C₄×Dic₅	—	2	8	no	—
103	C₄×Dic₅	—	8	8	no	—
104	C₂²×M₁₆	C₂×D₅	1	1	yes	characteristic
105	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
106	C₂×#117	—	5	2	no	—
107	C₂×#117	C₂×D₅	1	1	yes	characteristic
108	C₂²×C₄⋊C₄	—	5	2	no	—
109	#427	—	5	8	no	—
110	C₂×#107	—	5	8	no	—
111	C₂×#107	—	5	2	no	—
112	C₂×#107	—	5	2	no	—
113	C₂×#107	—	5	2	no	—
114	C₂×#106	—	5	2	no	—
115	#405	Dic₅	1	8	yes	complemented
116	C₂×#97	C₂×D₅	1	2	yes	—
117	C₂×C₄×C₈	C₂×D₅	1	2	yes	—
118	C₄×C₄⋊C₄	—	5	8	no	—
119	C₄×C₄⋊C₄	—	10	4	no	—
120	C₄×C₂²⋊C₄	—	10	4	no	—
121	#340	—	5	8	no	—
122	C₂³×C₅	—	2	2	no	—
123	C₂³×C₅	—	2	2	no	—
124	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
125	C₂³×C₅	C₄×D₄	1	2	yes	—
126	C₂³×C₅	C₄×D₄	1	2	yes	—
127	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₅	—	2	4	no	—
129	C₂×C₄×C₅	—	2	4	no	—
130	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
131	C₂×C₄×C₅	C₂×C₄⋊C₄	1	8	yes	—
132	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
133	C₂×C₄×C₅	#117	1	4	yes	—
134	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
135	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
136	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
137	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
138	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂³×C₄	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×Dic₅	—	2	8	no	—
145	C₂×Dic₅	—	2	8	no	—
146	C₂×Dic₅	—	4	4	no	—
147	C₂×Dic₅	—	4	8	no	—
148	C₂×Dic₅	—	4	4	no	—
149	C₂×Dic₅	—	8	8	no	—
150	C₅×C₈	—	2	4	no	—
151	C₅×C₈	C₂×C₄²	1	4	yes	—
152	C₅×C₈	C₂×C₄²	1	4	yes	—
153	C₂³×C₄	—	10	2	no	—
154	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
155	C₂×M₁₆	—	2	2	no	—
156	C₂×M₁₆	—	2	2	no	—
157	C₂×M₁₆	C₂×Dic₅	1	4	yes	—
158	C₂²×C₈	C₄×D₅	1	4	yes	—
159	C₂²×C₈	C₄×D₅	1	4	yes	—
160	C₂²×C₈	C₂×Dic₅	1	4	yes	—
161	C₂²×C₈	C₂²×D₅	1	2	yes	—
162	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
163	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
164	#117	—	5	8	no	—
165	#117	—	10	4	no	—
166	#117	Dic₁₀	1	4	yes	—
167	#117	C₂×Dic₅	1	4	yes	—
168	C₂×C₄⋊C₄	—	5	8	no	—
169	C₂×C₄⋊C₄	—	5	4	no	—
170	C₂×C₄⋊C₄	—	5	4	no	—
171	C₂×C₄⋊C₄	—	10	2	no	—
172	C₂×C₄⋊C₄	—	10	4	no	—
173	C₂×C₄⋊C₄	D₂₀	1	2	yes	—
174	C₂×C₂²⋊C₄	—	10	2	no	—
175	C₂×C₂²⋊C₄	D₂₀	1	2	yes	—
176	C₂×C₄²	—	10	2	no	—
177	C₂×C₄²	—	10	8	no	—
178	C₂×C₄²	C₂²×D₅	1	2	yes	—
179	#107	—	5	8	no	—
180	#107	—	5	8	no	—
181	#107	—	5	8	no	—
182	#106	—	5	8	no	—
183	#97	C₂×Dic₅	1	8	yes	complemented
184	C₄×C₈	C₂×Dic₅	1	8	yes	complemented
185	#95	—	10	8	no	—
186	C₂²×C₅	—	2	4	no	—
187	C₂²×C₅	—	2	2	no	—
188	C₂²×C₅	—	2	2	no	—
189	C₂²×C₅	—	2	4	no	—
190	C₂²×C₅	C₄×C₄⋊C₄	1	4	yes	—
191	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
192	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
193	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
194	C₂²×C₅	#576	1	2	yes	—
195	C₂²×C₅	#578	1	2	yes	—
196	C₄×C₅	—	2	8	no	—
197	C₄×C₅	C₄×C₄⋊C₄	1	4	yes	—
198	C₄×C₅	C₂²×C₄²	1	1	yes	characteristic
199	C₄×C₅	C₂×#117	1	1	yes	characteristic
200	C₄×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
201	C₄×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
202	Dic₅	—	4	8	no	—
203	Dic₅	—	4	8	no	—
204	C₂⁴	C₂×D₂₀	1	1	yes	characteristic
205	C₂²×C₄	—	10	4	no	—
206	C₂²×C₄	—	10	4	no	—
207	C₂²×C₄	—	10	8	no	—
208	C₂²×C₄	—	40	4	no	—
209	C₂²×C₄	C₂×Dic₁₀	1	2	yes	—
210	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
211	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
212	C₂²×C₄	C₂×D₂₀	1	4	yes	—
213	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
214	C₂²×C₄	#709	1	2	yes	—
215	C₂²×C₄	#709	1	2	yes	—
216	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
217	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
218	M₁₆	—	2	8	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	C₄×Dic₅	1	16	yes	complemented
222	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
223	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
224	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
225	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
226	C₂×C₈	C₂²×Dic₅	1	4	yes	—
227	C₂×C₈	C₂²×Dic₅	1	2	yes	—
228	C₂×C₈	C₂²×Dic₅	1	2	yes	—
229	C₄⋊C₄	—	5	8	no	—
230	C₄⋊C₄	—	5	8	no	—
231	C₄⋊C₄	—	10	4	no	—
232	C₄⋊C₄	—	10	4	no	—
233	C₄⋊C₄	—	10	8	no	—
234	C₄⋊C₄	C₄⋊Dic₅	1	8	yes	—
235	C₂²⋊C₄	—	10	8	no	—
236	C₂²⋊C₄	C₄⋊Dic₅	1	8	yes	—
237	C₄²	—	10	8	no	—
238	C₄²	—	40	8	no	—
239	C₄²	C₂×Dic₁₀	1	4	yes	—
240	C₄²	C₂²×Dic₅	1	4	yes	—
241	C₂×C₅	—	2	4	no	—
242	C₂×C₅	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
243	C₂×C₅	#2913	1	2	yes	—
244	C₂×C₅	#2948	1	1	yes	characteristic
245	C₂×C₅	#2950	1	1	yes	characteristic
246	C₂×C₅	#2951	1	2	yes	—
247	C₂³	—	2	2	no	—
248	C₂³	—	2	2	no	—
249	C₂³	C₄×D₂₀	1	2	yes	—
250	C₂³	C₄×D₂₀	1	2	yes	—
251	C₂³	C₂×C₄⋊Dic₅	1	2	yes	—
252	C₂³	C₂²×D₂₀	1	1	yes	characteristic
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	10	8	no	—
256	C₂×C₄	—	10	8	no	—
257	C₂×C₄	—	20	4	no	—
258	C₂×C₄	—	20	8	no	—
259	C₂×C₄	—	20	4	no	—
260	C₂×C₄	—	40	8	no	—
261	C₂×C₄	C₄×Dic₁₀	1	4	yes	—
262	C₂×C₄	C₄×Dic₁₀	1	4	yes	—
263	C₂×C₄	C₄×D₂₀	1	2	yes	—
264	C₂×C₄	C₄×D₂₀	1	2	yes	—
265	C₂×C₄	C₂×C₄×Dic₅	1	4	yes	—
266	C₂×C₄	C₂×C₄⋊Dic₅	1	8	yes	—
267	C₂×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
268	C₂×C₄	#4072	1	4	yes	—
269	C₂×C₄	C₂²×Dic₁₀	1	1	yes	characteristic
270	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
271	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
272	C₂×C₄	C₂×#709	1	1	yes	characteristic
273	C₂×C₄	C₂×#709	1	1	yes	characteristic
274	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
275	C₈	—	2	4	no	—
276	C₈	C₂×C₄×Dic₅	1	4	yes	—
277	C₈	C₂×C₄×Dic₅	1	4	yes	—
278	C₅	#20455	1	1	yes	characteristic, complemented, Sylow
279	C₂²	—	2	2	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	4	no	—
283	C₂²	C₄×C₄⋊Dic₅	1	4	yes	—
284	C₂²	C₂×C₄×D₂₀	1	1	yes	characteristic
285	C₂²	C₂×C₄×D₂₀	1	1	yes	characteristic
286	C₂²	#66086	1	2	yes	—
287	C₂²	#66087	1	2	yes	—
288	C₂²	C₂²×C₄⋊Dic₅	1	1	yes	characteristic
289	C₄	—	2	8	no	—
290	C₄	—	20	8	no	—
291	C₄	—	20	8	no	—
292	C₄	C₄×C₄⋊Dic₅	1	4	yes	—
293	C₄	C₂×C₄×Dic₁₀	1	1	yes	characteristic
294	C₄	C₂×C₄×Dic₁₀	1	1	yes	characteristic
295	C₄	C₂²×C₄×Dic₅	1	1	yes	characteristic
296	C₄	C₂×#4072	1	1	yes	characteristic
297	C₂	—	2	4	no	—
298	C₂	#10610292	1	1	yes	characteristic
299	C₂	#10610293	1	2	yes	—
300	C₂	#10610304	1	1	yes	characteristic
301	C₂	C₂×C₄×C₄⋊Dic₅	1	1	yes	characteristic
302	C₂	#10615516	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₂₀
commutator	C₄×C₅	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₂×C₅×#405	C₂
socle	C₂³×C₅	C₂²×D₄
prime radical	C₂⁴×C₅	C₂×D₄
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	C₂×#405	D₅
hypocenter	C₅	#20455
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
6. #10600607	C₂	semidirect
7. C₄×#3950	C₂	semidirect
13. C₅×#405	C₄	semidirect
16. #65395	C₄	semidirect
17. C₂×#3950	C₄	semidirect
53. C₅×#97	C₂×C₄	semidirect
54. C₄×C₅×C₈	C₂×C₄	semidirect
58. #3950	C₂×C₄	semidirect
59. #3949	C₂×C₄	semidirect
84. C₂×C₅×C₈	C₄²	semidirect
115. #405	Dic₅	semidirect
183. #97	C₂×Dic₅	semidirect
184. C₄×C₈	C₂×Dic₅	semidirect
221. C₂×C₈	C₄×Dic₅	semidirect
278. C₅	#20455	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#405	2
C₂×#65395	2
C₂²×#3950	2
#10613954	2
#10602778	2
#10600607	2
C₄×#3950	2
#20455	5