Subgroups — #12003552

Order 1280.

422 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#64971	C₂	1	1	yes	maximal, characteristic
2	#10601252	C₂	1	2	yes	maximal
3	C₂×#64701	C₂	1	2	yes	maximal
4	#10600579	C₂	1	1	yes	maximal, characteristic
5	C₅×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×#4015	C₂²	1	2	yes	—
7	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×#3950	—	2	1	no	—
10	C₂×#3950	—	2	1	no	—
11	C₂×#3946	—	2	1	no	—
12	C₄×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
14	#64971	—	2	4	no	—
15	#64771	—	2	2	no	—
16	#64771	—	2	2	no	—
17	#64770	—	2	1	no	—
18	#64701	—	2	4	no	—
19	#10543	—	5	1	no	maximal, Sylow
20	C₂²×Dic₁₀	—	2	2	no	—
21	C₂²×C₅×C₈	—	2	2	no	—
22	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
25	C₂×C₄⋊Dic₅	—	2	2	no	—
26	C₂×C₄⋊Dic₅	—	2	2	no	—
27	C₂×C₄⋊Dic₅	D₄	1	1	yes	characteristic
28	C₂×C₄⋊Dic₅	D₄	1	1	yes	characteristic
29	C₂×C₄×Dic₅	—	2	1	no	—
30	C₂²×#168	—	2	1	no	—
31	#4015	—	2	2	no	—
32	#4015	—	2	2	no	—
33	#4015	—	2	2	no	—
34	#4015	D₄	1	4	yes	—
35	C₅×C₄⋊C₈	—	2	2	no	—
36	C₅×C₄⋊C₈	—	2	2	no	—
37	C₄×C₅×C₈	—	2	2	no	—
38	#3963	—	2	1	no	—
39	#3950	—	2	4	no	—
40	#3950	—	2	4	no	—
41	#3946	—	4	2	no	—
42	C₄⋊#168	—	2	2	no	—
43	C₂×#500	—	5	1	no	—
44	#1987	—	5	2	no	—
45	#1939	—	5	1	no	—
46	C₂×#333	—	5	2	no	—
47	#1277	D₅	1	1	yes	characteristic
48	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₅	#41	1	1	yes	characteristic
51	C₂²×Dic₅	—	2	1	no	—
52	C₂²×Dic₅	—	2	1	no	—
53	C₂²×Dic₅	—	4	2	no	—
54	C₂×Dic₁₀	—	2	4	no	—
55	C₂×Dic₁₀	—	4	2	no	—
56	C₂×Dic₁₀	—	8	2	no	—
57	C₂×Dic₁₀	—	8	2	no	—
58	C₂×C₅×C₈	—	2	1	no	—
59	C₂×C₅×C₈	—	2	1	no	—
60	C₂×C₅×C₈	—	2	1	no	—
61	C₂×C₅×C₈	—	2	1	no	—
62	C₂×C₅×C₈	—	4	2	no	—
63	C₂×C₅×C₈	—	4	2	no	—
64	C₂×C₅×C₈	—	4	2	no	—
65	C₂×C₅×C₈	SD₁₆	1	2	yes	—
66	C₂×C₅×C₈	Q₁₆	1	2	yes	—
67	C₄²×C₅	C₂×D₄	1	2	yes	—
68	C₄²×C₅	#41	1	2	yes	—
69	C₄⋊Dic₅	—	2	1	no	—
70	C₄⋊Dic₅	—	2	4	no	—
71	C₄⋊Dic₅	—	2	4	no	—
72	C₄⋊Dic₅	—	2	2	no	—
73	C₄⋊Dic₅	—	4	2	no	—
74	C₄⋊Dic₅	—	4	2	no	—
75	C₄⋊Dic₅	Q₁₆	1	2	yes	—
76	C₄×Dic₅	—	4	1	no	—
77	C₄×Dic₅	—	4	1	no	—
78	C₂×#168	—	4	1	no	—
79	C₂×#168	—	4	1	no	—
80	C₂×#168	—	4	1	no	—
81	C₂×C₄⋊Q₈	—	5	2	no	—
82	#500	—	10	4	no	—
83	C₂×#107	—	10	1	no	—
84	C₂×#107	—	10	1	no	—
85	C₂×C₄⋊C₈	—	5	1	no	—
86	C₂×C₄⋊C₈	—	10	1	no	—
87	C₂×C₄⋊C₈	C₂×D₅	1	2	yes	—
88	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
89	C₄×C₄⋊C₄	—	5	1	no	—
90	#340	—	10	2	no	—
91	#340	—	10	2	no	—
92	#336	—	10	1	no	—
93	#333	—	10	4	no	—
94	C₂³×C₅	#121	1	1	yes	characteristic
95	C₂×C₄×C₅	—	2	1	no	—
96	C₂×C₄×C₅	—	2	1	no	—
97	C₂×C₄×C₅	#120	1	1	yes	characteristic
98	C₂×C₄×C₅	#121	1	2	yes	—
99	C₂×C₄×C₅	#123	1	2	yes	—
100	C₂×C₄×C₅	#124	1	1	yes	characteristic
101	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
102	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
103	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
104	C₂×C₄×C₅	#135	1	1	yes	characteristic
105	C₂×C₄×C₅	#137	1	1	yes	characteristic
106	C₂×C₄×C₅	#137	1	1	yes	characteristic
107	C₂×C₄×C₅	#137	1	2	yes	—
108	C₂×Dic₅	—	2	2	no	—
109	C₂×Dic₅	—	4	1	no	—
110	C₂×Dic₅	—	4	1	no	—
111	C₂×Dic₅	—	4	4	no	—
112	C₂×Dic₅	—	4	1	no	—
113	C₂×Dic₅	—	4	1	no	—
114	C₂×Dic₅	—	4	1	no	—
115	C₂×Dic₅	—	8	2	no	—
116	C₂×Dic₅	—	8	2	no	—
117	Dic₁₀	—	8	4	no	—
118	Dic₁₀	—	8	4	no	—
119	C₅×C₈	—	2	2	no	—
120	C₅×C₈	—	2	2	no	—
121	C₅×C₈	—	8	2	no	—
122	#168	—	8	1	no	—
123	C₂²×Q₈	—	10	2	no	—
124	C₂²×C₈	—	2	2	no	—
125	C₂²×C₈	—	10	1	no	—
126	C₂²×C₈	D₂₀	1	1	yes	characteristic
127	C₂²×C₈	D₂₀	1	1	yes	characteristic
128	C₄⋊Q₈	—	5	4	no	—
129	C₄⋊Q₈	—	10	2	no	—
130	C₄⋊Q₈	—	10	2	no	—
131	C₄⋊Q₈	—	10	2	no	—
132	C₂×C₄⋊C₄	—	5	1	no	—
133	C₂×C₄⋊C₄	—	5	1	no	—
134	C₂×C₄⋊C₄	—	10	2	no	—
135	C₂×C₄⋊C₄	—	10	2	no	—
136	C₂×C₄²	—	10	1	no	—
137	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
138	#107	—	10	4	no	—
139	#107	—	10	4	no	—
140	C₄⋊C₈	—	2	2	no	—
141	C₄⋊C₈	—	2	2	no	—
142	C₄⋊C₈	—	10	2	no	—
143	C₄⋊C₈	—	20	2	no	—
144	C₄×C₈	—	2	2	no	—
145	#95	—	10	1	no	—
146	C₂²×C₅	#394	1	1	yes	characteristic
147	C₂²×C₅	#461	1	1	yes	characteristic
148	C₂²×C₅	#462	1	1	yes	characteristic
149	C₂²×C₅	#467	1	1	yes	characteristic
150	C₂²×C₅	#470	1	1	yes	characteristic
151	C₂²×C₅	#472	1	1	yes	characteristic
152	C₂²×C₅	#472	1	1	yes	characteristic
153	C₄×C₅	—	2	2	no	—
154	C₄×C₅	#448	1	1	yes	characteristic
155	C₄×C₅	#451	1	1	yes	characteristic
156	C₄×C₅	#462	1	2	yes	—
157	C₄×C₅	#464	1	2	yes	—
158	C₄×C₅	#487	1	1	yes	characteristic
159	C₄×C₅	#491	1	1	yes	characteristic
160	Dic₅	—	4	2	no	—
161	Dic₅	—	8	1	no	—
162	Dic₅	—	8	4	no	—
163	C₂×Q₈	—	10	4	no	—
164	C₂×Q₈	—	20	2	no	—
165	C₂×Q₈	—	40	2	no	—
166	C₂×Q₈	—	40	2	no	—
167	C₂²×C₄	—	10	1	no	—
168	C₂²×C₄	—	10	1	no	—
169	C₂²×C₄	—	20	2	no	—
170	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
171	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
172	C₂²×C₄	#713	1	1	yes	characteristic
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	—	4	2	no	—
179	C₂×C₈	—	4	2	no	—
180	C₂×C₈	—	20	1	no	—
181	C₂×C₈	—	20	1	no	—
182	C₂×C₈	—	20	1	no	—
183	C₂×C₈	#677	1	2	yes	—
184	C₂×C₈	Dic₂₀	1	2	yes	—
185	C₄⋊C₄	—	5	2	no	—
186	C₄⋊C₄	—	10	1	no	—
187	C₄⋊C₄	—	10	4	no	—
188	C₄⋊C₄	—	10	4	no	—
189	C₄⋊C₄	—	10	2	no	—
190	C₄⋊C₄	—	20	2	no	—
191	C₄⋊C₄	—	20	2	no	—
192	C₄²	—	20	1	no	—
193	C₄²	—	20	1	no	—
194	C₄²	#709	1	2	yes	—
195	C₄²	D₄×D₅	1	2	yes	—
196	C₂×C₅	#1675	1	1	yes	characteristic
197	C₂×C₅	#1676	1	1	yes	characteristic
198	C₂×C₅	#1695	1	1	yes	characteristic
199	C₂×C₅	#1696	1	1	yes	characteristic
200	C₂×C₅	#2043	1	1	yes	characteristic
201	C₂×C₅	#2045	1	1	yes	characteristic
202	C₂×C₅	#2056	1	1	yes	characteristic
203	C₂³	#4041	1	1	yes	characteristic
204	Q₈	—	40	4	no	—
205	Q₈	—	40	4	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	10	2	no	—
209	C₂×C₄	—	20	1	no	—
210	C₂×C₄	—	20	1	no	—
211	C₂×C₄	—	20	1	no	—
212	C₂×C₄	—	20	1	no	—
213	C₂×C₄	—	20	1	no	—
214	C₂×C₄	—	20	4	no	—
215	C₂×C₄	—	40	2	no	—
216	C₂×C₄	—	40	2	no	—
217	C₂×C₄	#4021	1	1	yes	characteristic
218	C₂×C₄	#4028	1	1	yes	characteristic
219	C₂×C₄	#4030	1	2	yes	—
220	C₂×C₄	#4040	1	2	yes	—
221	C₂×C₄	C₂×#677	1	1	yes	characteristic
222	C₂×C₄	C₂×Dic₂₀	1	1	yes	characteristic
223	C₂×C₄	#4055	1	1	yes	characteristic
224	C₂×C₄	#4055	1	1	yes	characteristic
225	C₂×C₄	#4058	1	1	yes	characteristic
226	C₂×C₄	#4061	1	2	yes	—
227	C₂×C₄	D₅×Q₁₆	1	1	yes	characteristic
228	C₈	—	2	2	no	—
229	C₈	—	2	2	no	—
230	C₈	—	8	2	no	—
231	C₈	—	40	1	no	—
232	C₅	#10543	1	1	yes	characteristic, complemented, Sylow
233	C₂²	#64961	1	1	yes	characteristic
234	C₂²	#65130	1	1	yes	characteristic
235	C₂²	#65138	1	1	yes	characteristic
236	C₂²	#65157	1	1	yes	characteristic
237	C₂²	#65176	1	1	yes	characteristic
238	C₂²	#65186	1	1	yes	characteristic
239	C₂²	#65186	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	20	2	no	—
242	C₄	—	40	4	no	—
243	C₄	—	40	1	no	—
244	C₄	#64970	1	1	yes	characteristic
245	C₄	#64992	1	1	yes	characteristic
246	C₄	#65027	1	1	yes	characteristic
247	C₄	#65028	1	1	yes	characteristic
248	C₄	#65056	1	2	yes	—
249	C₄	#65082	1	2	yes	—
250	C₂	#10599336	1	1	yes	characteristic
251	C₂	#10599346	1	1	yes	characteristic
252	C₂	#10599372	1	1	yes	characteristic
253	C₂	#10599382	1	1	yes	characteristic
254	C₂	#10599424	1	1	yes	characteristic
255	C₂	#10599657	1	1	yes	characteristic
256	C₂	#10599725	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#4041
commutator, cosocle	C₂×C₄²×C₅	C₂³
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1277	C₂
socle, prime radical	C₂³×C₅	#121
hypercenter	#1277	D₅
hypocenter	C₅	#10543
LCS	G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
232. C₅	#10543	semidirect

Maximal subgroups

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

subgroup	index
C₂×#64971	2
#10601252	2
C₂×#64701	2
#10600579	2
C₅×#1277	2
#10543	5