Subgroups — #12087262

Order 1280.

710 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#500	C₂	1	1	yes	maximal, characteristic
2	#10614953	C₂	1	2	yes	maximal
3	#10614938	C₂	1	2	yes	maximal
4	#10610206	C₂	1	1	yes	maximal, characteristic
5	#10602161	C₂	1	4	yes	maximal, complemented
6	#10602155	C₂	1	2	yes	maximal, complemented
7	#10602153	C₂	1	2	yes	maximal, complemented
8	C₈×#681	C₂	1	1	yes	maximal, characteristic
9	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
10	#65804	C₂²	1	2	yes	—
11	C₅×#500	—	2	2	no	—
12	C₅×#500	C₂²	1	2	yes	—
13	C₅×#500	C₂²	1	2	yes	—
14	C₂×C₅×#107	—	2	2	no	—
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	#65517	—	2	4	no	—
17	#65405	—	2	2	no	—
18	#65381	C₂²	1	4	yes	—
19	#65379	—	2	4	no	—
20	#65370	C₂²	1	4	yes	—
21	#65369	—	2	4	no	—
22	#65368	C₂²	1	4	yes	—
23	#65367	C₂²	1	4	yes	—
24	#65253	—	2	4	no	—
25	#65214	C₂²	1	1	yes	characteristic
26	C₄×#681	C₂²	1	1	yes	characteristic
27	#64988	C₂²	1	1	yes	characteristic
28	C₄×Dic₂₀	—	2	2	no	—
29	#64971	C₂²	1	1	yes	characteristic
30	#64970	C₂²	1	2	yes	—
31	#64674	C₂²	1	2	yes	—
32	C₈×#168	C₂²	1	2	yes	—
33	#19633	—	5	1	no	maximal, Sylow
34	C₂²×C₅×Q₈	—	2	2	no	—
35	C₂²×C₅×C₈	D₄	1	2	yes	—
36	C₅×C₄⋊Q₈	—	2	2	no	—
37	C₅×C₄⋊Q₈	—	2	2	no	—
38	C₅×C₄⋊Q₈	—	2	2	no	—
39	C₅×C₄⋊Q₈	C₂³	1	4	yes	—
40	C₂×C₅×C₄⋊C₄	—	2	2	no	—
41	C₂×C₅×C₄⋊C₄	—	2	2	no	—
42	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₂×#689	—	2	8	no	—
44	#4072	—	2	2	no	—
45	#4070	—	2	2	no	—
46	#4070	—	2	2	no	—
47	C₂×#681	D₄	1	2	yes	—
48	C₂×Dic₂₀	—	2	2	no	—
49	#4016	—	2	2	no	—
50	#4015	C₂³	1	2	yes	—
51	#4015	C₂³	1	2	yes	—
52	C₄×Dic₁₀	—	2	2	no	—
53	C₄×Dic₁₀	—	2	2	no	—
54	C₅×#107	—	2	4	no	—
55	C₅×#107	—	2	4	no	—
56	C₄×C₅×C₈	D₄	1	2	yes	—
57	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
58	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
59	#3967	—	2	8	no	—
60	C₂²⋊#168	D₄	1	2	yes	—
61	#3950	—	2	2	no	—
62	#3948	—	2	4	no	—
63	#3940	—	2	8	no	—
64	#3939	—	2	8	no	—
65	C₄⋊#168	D₄	1	2	yes	—
66	#3935	D₄	1	2	yes	—
67	C₄×#168	C₂³	1	2	yes	—
68	C₄×#168	C₂³	1	2	yes	—
69	#3209	—	5	2	no	—
70	#3170	—	5	1	no	—
71	#3165	—	5	2	no	—
72	C₂×#500	D₅	1	1	yes	characteristic
73	#1723	—	5	4	no	—
74	#1713	—	5	2	no	—
75	#1707	—	5	2	no	—
76	C₈×M₁₆	—	5	1	no	—
77	C₂×C₅×Q₈	—	2	4	no	—
78	C₂×C₅×Q₈	—	4	2	no	—
79	C₂×C₅×Q₈	—	8	4	no	—
80	C₂²×C₄×C₅	—	4	2	no	—
81	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂×Dic₁₀	—	2	2	no	—
85	C₂×Dic₁₀	—	2	2	no	—
86	C₂×C₅×C₈	—	2	4	no	—
87	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
88	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
89	C₅×C₄⋊C₄	—	2	4	no	—
90	C₅×C₄⋊C₄	—	2	4	no	—
91	C₅×C₄⋊C₄	—	4	2	no	—
92	C₅×C₄⋊C₄	—	4	2	no	—
93	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₅	C₂⁴	1	1	yes	characteristic
97	C₂²⋊Dic₅	—	4	2	no	—
98	#689	—	8	8	no	—
99	C₄⋊Dic₅	—	2	2	no	—
100	C₄⋊Dic₅	—	2	2	no	—
101	C₄⋊Dic₅	—	2	2	no	—
102	C₄⋊Dic₅	—	2	2	no	—
103	#683	—	4	2	no	—
104	#683	—	4	2	no	—
105	C₄×Dic₅	—	4	2	no	—
106	#681	Q₁₆	1	8	yes	complemented
107	C₂×#168	C₂×D₄	1	4	yes	—
108	C₂×#168	C₂×D₄	1	4	yes	—
109	Dic₂₀	—	4	4	no	—
110	#533	—	5	2	no	—
111	C₂×C₄⋊Q₈	C₂×D₅	1	2	yes	—
112	#501	—	10	4	no	—
113	#500	—	2	2	no	—
114	#500	—	5	4	no	—
115	#500	—	5	1	no	—
116	#500	C₂×D₅	1	2	yes	—
117	#500	C₂×D₅	1	2	yes	—
118	#498	—	5	4	no	—
119	#494	—	5	4	no	—
120	#494	—	5	1	no	—
121	#487	—	5	4	no	—
122	#487	—	5	2	no	—
123	#467	—	10	2	no	—
124	#462	—	10	4	no	—
125	#451	—	10	4	no	—
126	C₄×Q₁₆	—	10	2	no	—
127	#431	—	5	1	no	—
128	#428	—	10	4	no	—
129	C₂×#107	—	2	2	no	—
130	C₄×M₁₆	—	5	1	no	—
131	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
132	#322	—	5	2	no	—
133	C₈²	—	5	2	no	—
134	C₂³×C₅	#127	1	1	yes	characteristic
135	C₅×Q₈	—	4	8	no	—
136	C₅×Q₈	—	8	4	no	—
137	C₂×C₄×C₅	—	2	2	no	—
138	C₂×C₄×C₅	—	2	2	no	—
139	C₂×C₄×C₅	—	4	4	no	—
140	C₂×C₄×C₅	—	8	4	no	—
141	C₂×C₄×C₅	#127	1	1	yes	characteristic
142	C₂×C₄×C₅	#127	1	1	yes	characteristic
143	C₂×C₄×C₅	#127	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂×Q₁₆	1	4	yes	—
145	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
148	C₂×Dic₅	—	4	2	no	—
149	C₂×Dic₅	—	4	2	no	—
150	Dic₁₀	—	4	4	no	—
151	Dic₁₀	—	4	4	no	—
152	C₅×C₈	—	2	2	no	—
153	C₅×C₈	#137	1	4	yes	—
154	#168	—	2	4	no	—
155	#168	C₂×Q₁₆	1	8	yes	—
156	C₂²×Q₈	—	2	2	no	—
157	C₂×Q₁₆	—	10	8	no	—
158	C₂×Q₁₆	—	10	2	no	—
159	C₂×M₁₆	—	5	2	no	—
160	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
161	C₄⋊Q₈	—	2	2	no	—
162	C₄⋊Q₈	—	2	2	no	—
163	C₄⋊Q₈	—	2	2	no	—
164	C₄⋊Q₈	—	5	2	no	—
165	C₄⋊Q₈	—	5	2	no	—
166	C₄⋊Q₈	C₂²×D₅	1	4	yes	—
167	#125	—	10	2	no	—
168	C₂²⋊Q₈	—	10	2	no	—
169	C₂²⋊Q₈	—	10	2	no	—
170	C₄×Q₈	—	10	2	no	—
171	C₄×Q₈	—	10	2	no	—
172	#117	—	10	2	no	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
176	#107	—	2	4	no	—
177	#107	—	2	4	no	—
178	#107	—	10	8	no	—
179	#107	—	10	2	no	—
180	#106	—	10	8	no	—
181	C₄⋊C₈	—	5	2	no	—
182	#103	—	10	8	no	—
183	#103	—	10	4	no	—
184	C₂²⋊C₈	—	5	2	no	—
185	#97	—	5	2	no	—
186	C₄×C₈	—	5	2	no	—
187	C₄×C₈	—	5	2	no	—
188	C₄×C₈	C₂²⋊D₅	1	2	yes	—
189	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
190	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
191	C₂²×C₅	—	2	1	no	—
192	C₂²×C₅	—	2	1	no	—
193	C₂²×C₅	#494	1	2	yes	—
194	C₂²×C₅	C₂×#127	1	1	yes	characteristic
195	C₄×C₅	—	2	1	no	—
196	C₄×C₅	—	2	1	no	—
197	C₄×C₅	—	8	4	no	—
198	C₄×C₅	#494	1	2	yes	—
199	C₄×C₅	C₂²×Q₁₆	1	2	yes	—
200	C₄×C₅	C₂×#137	1	2	yes	—
201	C₄×C₅	#576	1	2	yes	—
202	Dic₅	—	8	2	no	—
203	Dic₅	—	8	2	no	—
204	C₂×Q₈	—	2	4	no	—
205	C₂×Q₈	—	4	2	no	—
206	C₂×Q₈	—	8	4	no	—
207	C₂×Q₈	—	10	2	no	—
208	C₂×Q₈	—	10	2	no	—
209	C₂²×C₄	—	4	2	no	—
210	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
211	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
213	Q₁₆	—	20	4	no	—
214	Q₁₆	—	40	8	no	—
215	M₁₆	—	5	8	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	5	4	no	—
218	C₂×C₈	—	5	4	no	—
219	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
220	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₄⋊C₄	—	10	2	no	—
226	C₄⋊C₄	—	10	2	no	—
227	C₄⋊C₄	—	10	2	no	—
228	C₄⋊C₄	—	10	2	no	—
229	C₄⋊C₄	—	20	2	no	—
230	C₄⋊C₄	—	20	2	no	—
231	C₂²⋊C₄	—	20	2	no	—
232	C₄²	—	20	2	no	—
233	C₄²	D₄×D₅	1	1	yes	characteristic
234	C₄²	D₄×D₅	1	1	yes	characteristic
235	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
236	C₄²	C₂³×D₅	1	1	yes	characteristic
237	C₂×C₅	—	2	2	no	—
238	C₂×C₅	C₂×#494	1	1	yes	characteristic
239	C₂×C₅	#3152	1	1	yes	characteristic
240	C₂×C₅	#3155	1	1	yes	characteristic
241	C₂³	#4086	1	1	yes	characteristic
242	Q₈	—	4	8	no	—
243	Q₈	—	8	4	no	—
244	Q₈	—	20	4	no	—
245	Q₈	—	20	4	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	8	4	no	—
250	C₂×C₄	—	20	2	no	—
251	C₂×C₄	—	20	2	no	—
252	C₂×C₄	D₅×Q₁₆	1	4	yes	—
253	C₂×C₄	#4086	1	1	yes	characteristic
254	C₂×C₄	#4086	1	1	yes	characteristic
255	C₂×C₄	#4086	1	1	yes	characteristic
256	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
257	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
258	C₂×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
259	C₈	—	2	2	no	—
260	C₈	—	5	8	no	—
261	C₈	—	10	4	no	—
262	C₈	#4088	1	4	yes	—
263	C₅	#19633	1	1	yes	characteristic, complemented, Sylow
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	#65470	1	2	yes	—
267	C₂²	C₂×#4086	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	8	4	no	—
271	C₄	—	40	2	no	—
272	C₄	—	40	2	no	—
273	C₄	#65470	1	2	yes	—
274	C₄	#66091	1	2	yes	—
275	C₄	C₂×D₅×Q₁₆	1	2	yes	—
276	C₄	C₂×#4088	1	2	yes	—
277	C₂	—	2	2	no	—
278	C₂	#10614934	1	1	yes	characteristic
279	C₂	#10615561	1	1	yes	characteristic
280	C₂	C₂×#65470	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#4086
commutator, cosocle	C₄²×C₅	C₂⁴
Frattini	C₄²	C₂³×D₅
Fitting	C₂×C₅×#500	C₂
socle	C₂²×C₅	C₂×#127
prime radical	C₂³×C₅	#127
hypercenter	C₂×#500	D₅
hypocenter	C₅	#19633
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
5. #10602161	C₂	semidirect
6. #10602155	C₂	semidirect
7. #10602153	C₂	semidirect
106. #681	Q₁₆	semidirect
263. C₅	#19633	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#500	2
#10614953	2
#10614938	2
#10610206	2
#10602161	2
#10602155	2
#10602153	2
C₈×#681	2
#19633	5