Subgroups — #12003560

Order 1280.

454 conjugacy classes of subgroups.

The subgroup lattice has 270 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	#10602571	C₂	1	2	yes	maximal
3	C₂×#64767	C₂	1	2	yes	maximal
4	#10600579	C₂	1	1	yes	maximal, characteristic
5	#10598145	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₅×#500	—	2	4	no	—
8	C₂×C₅×#107	—	2	1	no	—
9	C₂×C₅×#107	—	2	1	no	—
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×#3967	—	2	2	no	—
12	C₂×#3967	—	2	2	no	—
13	C₂×#3946	—	2	1	no	—
14	C₄×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#168	C₂²	1	2	yes	—
16	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
17	#64770	—	2	1	no	—
18	#64767	—	2	4	no	—
19	#10545	—	5	1	no	maximal, Sylow
20	C₂²×C₅×Q₈	—	2	2	no	—
21	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
22	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
23	C₅×C₄⋊Q₈	—	2	2	no	—
24	C₅×C₄⋊Q₈	—	2	2	no	—
25	C₅×C₄⋊Q₈	—	2	2	no	—
26	C₅×C₄⋊Q₈	D₄	1	4	yes	—
27	C₂×C₅×C₄⋊C₄	—	2	2	no	—
28	C₂×C₅×C₄⋊C₄	—	2	2	no	—
29	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊Dic₅	D₄	1	1	yes	characteristic
31	C₂×C₄⋊Dic₅	D₄	1	1	yes	characteristic
32	C₂×C₄×Dic₅	—	2	1	no	—
33	C₂²×#168	—	2	1	no	—
34	C₂²×#168	—	2	2	no	—
35	C₅×#107	—	2	4	no	—
36	C₅×#107	—	2	4	no	—
37	C₄×C₅×C₈	—	2	2	no	—
38	#3967	—	4	4	no	—
39	#3967	—	4	4	no	—
40	#3963	—	2	1	no	—
41	#3946	—	4	2	no	—
42	C₄⋊#168	—	2	2	no	—
43	C₄⋊#168	—	2	2	no	—
44	C₄⋊#168	—	2	2	no	—
45	C₂×#500	D₅	1	1	yes	characteristic
46	#1981	—	5	2	no	—
47	#1939	—	5	1	no	—
48	C₂×#332	—	5	2	no	—
49	#1277	—	5	1	no	—
50	C₂×C₅×Q₈	—	2	4	no	—
51	C₂×C₅×Q₈	—	4	2	no	—
52	C₂×C₅×Q₈	—	4	2	no	—
53	C₂×C₅×Q₈	—	4	2	no	—
54	C₂×C₅×Q₈	—	4	2	no	—
55	C₂×C₅×Q₈	—	4	2	no	—
56	C₂²×C₄×C₅	—	4	2	no	—
57	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅	#41	1	1	yes	characteristic
60	C₂²×Dic₅	—	2	1	no	—
61	C₂²×Dic₅	—	2	1	no	—
62	C₂×C₅×C₈	—	2	1	no	—
63	C₂×C₅×C₈	—	2	1	no	—
64	C₂×C₅×C₈	—	2	1	no	—
65	C₂×C₅×C₈	—	2	1	no	—
66	C₂×C₅×C₈	SD₁₆	1	2	yes	—
67	C₂×C₅×C₈	Q₁₆	1	2	yes	—
68	C₅×C₄⋊C₄	—	2	4	no	—
69	C₅×C₄⋊C₄	—	2	4	no	—
70	C₅×C₄⋊C₄	—	4	2	no	—
71	C₅×C₄⋊C₄	—	4	2	no	—
72	C₄²×C₅	C₂×D₄	1	2	yes	—
73	C₄²×C₅	#41	1	2	yes	—
74	C₄⋊Dic₅	—	2	1	no	—
75	C₄⋊Dic₅	—	2	2	no	—
76	C₄⋊Dic₅	D₈	1	2	yes	—
77	C₄×Dic₅	—	4	1	no	—
78	C₄×Dic₅	—	4	1	no	—
79	C₂×#168	—	4	1	no	—
80	C₂×#168	—	4	2	no	—
81	C₂×#168	—	4	2	no	—
82	C₂×#168	—	4	1	no	—
83	C₂×#168	—	4	2	no	—
84	C₂×#168	—	4	1	no	—
85	C₂×C₄⋊Q₈	C₂×D₅	1	2	yes	—
86	#500	—	2	4	no	—
87	C₂×#107	—	2	1	no	—
88	C₂×#107	—	2	1	no	—
89	C₂×C₄⋊C₈	—	5	1	no	—
90	C₂×C₄⋊C₈	—	5	2	no	—
91	C₂×C₄⋊C₈	—	10	1	no	—
92	C₂×#103	—	10	2	no	—
93	C₂×#103	—	10	2	no	—
94	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
95	C₄×C₄⋊C₄	—	5	1	no	—
96	#336	—	10	1	no	—
97	#332	—	10	4	no	—
98	C₂³×C₅	#121	1	1	yes	characteristic
99	C₅×Q₈	—	4	4	no	—
100	C₅×Q₈	—	4	4	no	—
101	C₅×Q₈	—	8	2	no	—
102	C₅×Q₈	—	8	2	no	—
103	C₂×C₄×C₅	—	2	1	no	—
104	C₂×C₄×C₅	—	2	1	no	—
105	C₂×C₄×C₅	—	4	4	no	—
106	C₂×C₄×C₅	—	8	2	no	—
107	C₂×C₄×C₅	—	8	2	no	—
108	C₂×C₄×C₅	#120	1	1	yes	characteristic
109	C₂×C₄×C₅	#121	1	2	yes	—
110	C₂×C₄×C₅	#123	1	2	yes	—
111	C₂×C₄×C₅	#124	1	1	yes	characteristic
112	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
115	C₂×C₄×C₅	#135	1	1	yes	characteristic
116	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
117	C₂×C₄×C₅	#137	1	1	yes	characteristic
118	C₂×C₄×C₅	#137	1	1	yes	characteristic
119	C₂×Dic₅	—	2	2	no	—
120	C₂×Dic₅	—	4	1	no	—
121	C₂×Dic₅	—	4	1	no	—
122	C₂×Dic₅	—	4	1	no	—
123	C₂×Dic₅	—	4	1	no	—
124	C₂×Dic₅	—	4	1	no	—
125	C₅×C₈	—	2	2	no	—
126	C₅×C₈	—	2	2	no	—
127	#168	—	8	1	no	—
128	#168	—	8	2	no	—
129	C₂²×Q₈	—	2	2	no	—
130	C₂²×C₈	—	10	2	no	—
131	C₂²×C₈	—	10	1	no	—
132	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
133	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
134	C₄⋊Q₈	—	2	2	no	—
135	C₄⋊Q₈	—	2	2	no	—
136	C₄⋊Q₈	—	2	2	no	—
137	C₄⋊Q₈	C₂²⋊D₅	1	4	yes	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	—	5	1	no	—
141	C₂×C₄⋊C₄	—	5	1	no	—
142	C₂×C₄²	—	10	1	no	—
143	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
144	#107	—	2	4	no	—
145	#107	—	2	4	no	—
146	C₄⋊C₈	—	10	2	no	—
147	C₄⋊C₈	—	10	2	no	—
148	C₄⋊C₈	—	10	2	no	—
149	C₄⋊C₈	—	20	2	no	—
150	#103	—	20	4	no	—
151	#103	—	20	4	no	—
152	C₄×C₈	—	2	2	no	—
153	#95	—	10	1	no	—
154	C₂²×C₅	#394	1	1	yes	characteristic
155	C₂²×C₅	#461	1	1	yes	characteristic
156	C₂²×C₅	#462	1	1	yes	characteristic
157	C₂²×C₅	#466	1	1	yes	characteristic
158	C₂²×C₅	#469	1	1	yes	characteristic
159	C₂²×C₅	#473	1	1	yes	characteristic
160	C₂²×C₅	#473	1	1	yes	characteristic
161	C₄×C₅	—	2	2	no	—
162	C₄×C₅	—	8	4	no	—
163	C₄×C₅	#448	1	1	yes	characteristic
164	C₄×C₅	#451	1	1	yes	characteristic
165	C₄×C₅	#459	1	2	yes	—
166	C₄×C₅	#463	1	2	yes	—
167	C₄×C₅	#487	1	1	yes	characteristic
168	C₄×C₅	#491	1	1	yes	characteristic
169	Dic₅	—	4	2	no	—
170	Dic₅	—	8	1	no	—
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×Q₈	—	4	2	no	—
175	C₂×Q₈	—	4	2	no	—
176	C₂×Q₈	—	4	2	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	—	10	1	no	—
179	C₂²×C₄	—	10	1	no	—
180	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
181	C₂²×C₄	#711	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	20	1	no	—
188	C₂×C₈	—	20	2	no	—
189	C₂×C₈	—	20	2	no	—
190	C₂×C₈	—	20	2	no	—
191	C₂×C₈	—	20	1	no	—
192	C₂×C₈	—	20	1	no	—
193	C₂×C₈	#688	1	2	yes	—
194	C₂×C₈	#689	1	2	yes	—
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	2	4	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₄⋊C₄	—	5	2	no	—
200	C₄⋊C₄	—	10	1	no	—
201	C₄⋊C₄	—	10	2	no	—
202	C₄²	—	20	1	no	—
203	C₄²	—	20	1	no	—
204	C₄²	#711	1	2	yes	—
205	C₄²	C₂×C₂²⋊D₅	1	2	yes	—
206	C₂×C₅	#1673	1	1	yes	characteristic
207	C₂×C₅	#1674	1	1	yes	characteristic
208	C₂×C₅	#1693	1	1	yes	characteristic
209	C₂×C₅	#1694	1	1	yes	characteristic
210	C₂×C₅	#2043	1	1	yes	characteristic
211	C₂×C₅	#2047	1	1	yes	characteristic
212	C₂×C₅	#2054	1	1	yes	characteristic
213	C₂³	#4084	1	1	yes	characteristic
214	Q₈	—	4	4	no	—
215	Q₈	—	4	4	no	—
216	Q₈	—	8	2	no	—
217	Q₈	—	8	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	8	2	no	—
222	C₂×C₄	—	8	2	no	—
223	C₂×C₄	—	10	2	no	—
224	C₂×C₄	—	20	1	no	—
225	C₂×C₄	—	20	1	no	—
226	C₂×C₄	—	20	1	no	—
227	C₂×C₄	—	20	1	no	—
228	C₂×C₄	—	20	1	no	—
229	C₂×C₄	D₅×D₈	1	1	yes	characteristic
230	C₂×C₄	#4057	1	2	yes	—
231	C₂×C₄	#4058	1	1	yes	characteristic
232	C₂×C₄	#4081	1	2	yes	—
233	C₂×C₄	#4082	1	1	yes	characteristic
234	C₂×C₄	#4085	1	2	yes	—
235	C₂×C₄	C₂×#688	1	1	yes	characteristic
236	C₂×C₄	#4088	1	1	yes	characteristic
237	C₂×C₄	#4088	1	1	yes	characteristic
238	C₂×C₄	C₂×#689	1	1	yes	characteristic
239	C₂×C₄	#4099	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	—	2	2	no	—
242	C₈	—	40	2	no	—
243	C₈	—	40	1	no	—
244	C₅	#10545	1	1	yes	characteristic, complemented, Sylow
245	C₂²	#65372	1	1	yes	characteristic
246	C₂²	#65379	1	1	yes	characteristic
247	C₂²	#65446	1	1	yes	characteristic
248	C₂²	#65448	1	1	yes	characteristic
249	C₂²	#65481	1	1	yes	characteristic
250	C₂²	#65481	1	1	yes	characteristic
251	C₂²	#65511	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	8	4	no	—
254	C₄	—	20	2	no	—
255	C₄	—	40	1	no	—
256	C₄	#65370	1	1	yes	characteristic
257	C₄	#65371	1	1	yes	characteristic
258	C₄	#65439	1	2	yes	—
259	C₄	#65442	1	2	yes	—
260	C₄	#65516	1	1	yes	characteristic
261	C₄	#65517	1	1	yes	characteristic
262	C₂	#10602164	1	1	yes	characteristic
263	C₂	#10602166	1	1	yes	characteristic
264	C₂	#10602170	1	1	yes	characteristic
265	C₂	#10602172	1	1	yes	characteristic
266	C₂	#10602574	1	1	yes	characteristic
267	C₂	#10602833	1	1	yes	characteristic
268	C₂	#10602863	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#4084
commutator, cosocle	C₂×C₄²×C₅	C₂³
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₂×C₅×#500	C₂
socle, prime radical	C₂³×C₅	#121
hypercenter	C₂×#500	D₅
hypocenter	C₅	#10545
LCS	G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
244. C₅	#10545	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
#10602571	2
C₂×#64767	2
#10600579	2
#10598145	2
#10545	5