Subgroups — #11998656

Order 1280.

394 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10610173	C₂	1	1	yes	maximal, characteristic
2	C₅×#1564	C₂	1	1	yes	maximal, characteristic
3	#10601585	C₂	1	2	yes	maximal
4	#10601561	C₂	1	2	yes	maximal
5	#10600550	C₂	1	1	yes	maximal, characteristic
6	#65804	C₂²	1	2	yes	—
7	C₅×#431	C₂²	1	2	yes	—
8	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	#65395	—	2	1	no	—
10	#65395	—	2	1	no	—
11	C₂×#3947	—	2	1	no	—
12	C₄×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
14	#64972	—	2	1	no	—
15	#64972	—	2	1	no	—
16	#64971	C₄	1	1	yes	characteristic, complemented
17	#64971	C₄	1	1	yes	characteristic, complemented
18	#64969	C₄	1	2	yes	complemented
19	C₅×#335	—	2	1	no	—
20	C₅×#334	—	2	1	no	—
21	#64770	—	2	1	no	—
22	#64687	—	2	4	no	—
23	#64676	—	2	4	no	—
24	#9752	—	5	1	no	maximal, Sylow
25	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
26	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
28	#4072	—	2	2	no	—
29	C₂×C₄⋊Dic₅	—	2	1	no	—
30	#4070	—	2	2	no	—
31	#4070	—	2	2	no	—
32	C₂×C₄×Dic₅	—	2	1	no	—
33	C₂²×#168	—	2	1	no	—
34	#4016	—	2	2	no	—
35	#4015	C₂×C₄	1	2	yes	—
36	#4015	C₂×C₄	1	2	yes	—
37	C₄×Dic₁₀	—	2	2	no	—
38	C₄×Dic₁₀	—	2	2	no	—
39	C₅×C₄⋊C₈	—	2	2	no	—
40	C₅×C₂²⋊C₈	—	2	2	no	—
41	C₄×C₅×C₈	—	2	2	no	—
42	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
43	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
44	C₄×C₅×C₈	D₄	1	1	yes	characteristic
45	C₄×C₅×C₈	D₄	1	1	yes	characteristic
46	#3963	D₄	1	2	yes	—
47	#3950	—	2	1	no	—
48	#3950	—	2	1	no	—
49	#3950	—	2	1	no	—
50	#3950	—	2	1	no	—
51	#3949	—	2	2	no	—
52	#3949	—	2	2	no	—
53	#3947	—	4	2	no	—
54	C₄⋊#168	—	2	1	no	—
55	C₄⋊#168	D₄	1	2	yes	—
56	#3160	—	5	1	no	—
57	#1770	—	5	1	no	—
58	#1564	D₅	1	1	yes	characteristic
59	#1503	—	5	2	no	—
60	#1481	—	5	2	no	—
61	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	#41	1	1	yes	characteristic
64	C₂²×Dic₅	—	2	2	no	—
65	C₂×Dic₁₀	—	2	2	no	—
66	C₂×Dic₁₀	—	2	2	no	—
67	C₂×C₅×C₈	—	2	1	no	—
68	C₂×C₅×C₈	—	2	1	no	—
69	C₂×C₅×C₈	—	4	2	no	—
70	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₅×C₈	SD₁₆	1	2	yes	—
75	C₂×C₅×C₈	Q₁₆	1	2	yes	—
76	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₅	#41	1	1	yes	characteristic
80	C₂²⋊Dic₅	—	4	2	no	—
81	C₄⋊Dic₅	—	2	2	no	—
82	C₄⋊Dic₅	—	2	2	no	—
83	C₄⋊Dic₅	—	2	2	no	—
84	C₄⋊Dic₅	—	2	2	no	—
85	C₄⋊Dic₅	—	4	2	no	—
86	#683	—	4	2	no	—
87	#683	—	4	2	no	—
88	C₄×Dic₅	—	4	2	no	—
89	C₄×Dic₅	—	4	1	no	—
90	C₄×Dic₅	—	4	1	no	—
91	C₂×#168	—	2	2	no	—
92	C₂×#168	—	4	1	no	—
93	C₂×#168	—	4	1	no	—
94	#533	—	5	2	no	—
95	#500	—	5	1	no	—
96	#500	—	5	1	no	—
97	#499	—	10	1	no	—
98	#499	—	10	1	no	—
99	#498	—	5	2	no	—
100	#431	C₂×D₅	1	2	yes	—
101	#427	—	10	1	no	—
102	#427	—	10	1	no	—
103	C₂×C₄⋊C₈	—	5	1	no	—
104	C₂×#97	—	10	1	no	—
105	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
106	C₄×C₄⋊C₄	—	5	1	no	—
107	#336	—	10	1	no	—
108	#335	—	2	1	no	—
109	#334	—	2	1	no	—
110	#328	—	10	4	no	—
111	Q₈⋊C₈	—	10	4	no	—
112	C₂³×C₅	#121	1	1	yes	characteristic
113	C₂×C₄×C₅	—	2	2	no	—
114	C₂×C₄×C₅	—	4	1	no	—
115	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₅	#121	1	1	yes	characteristic
119	C₂×C₄×C₅	#124	1	1	yes	characteristic
120	C₂×C₄×C₅	#127	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
123	C₂×C₄×C₅	#137	1	1	yes	characteristic
124	C₂×C₄×C₅	#137	1	1	yes	characteristic
125	C₂×Dic₅	—	4	2	no	—
126	C₂×Dic₅	—	4	2	no	—
127	C₂×Dic₅	—	4	2	no	—
128	C₂×Dic₅	—	4	2	no	—
129	C₂×Dic₅	—	4	2	no	—
130	Dic₁₀	—	4	4	no	—
131	Dic₁₀	—	8	2	no	—
132	C₅×C₈	—	2	1	no	—
133	C₅×C₈	—	2	1	no	—
134	C₅×C₈	—	8	2	no	—
135	C₅×C₈	#103	1	2	yes	—
136	C₅×C₈	#103	1	2	yes	—
137	#168	—	4	2	no	—
138	C₂²×C₈	—	10	1	no	—
139	C₂²×C₈	D₂₀	1	1	yes	characteristic
140	C₂²×C₈	D₂₀	1	1	yes	characteristic
141	C₄⋊Q₈	—	5	2	no	—
142	C₄⋊Q₈	—	5	2	no	—
143	#125	—	10	2	no	—
144	C₂²⋊Q₈	—	10	2	no	—
145	C₂²⋊Q₈	—	10	2	no	—
146	C₄×Q₈	—	10	2	no	—
147	C₄×Q₈	—	10	2	no	—
148	#117	—	10	2	no	—
149	C₂×C₄⋊C₄	—	10	1	no	—
150	C₂×C₄²	—	10	1	no	—
151	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
152	#107	—	10	1	no	—
153	#107	—	10	1	no	—
154	#107	—	10	1	no	—
155	#107	—	10	1	no	—
156	#106	—	10	2	no	—
157	#106	—	10	2	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	5	2	no	—
160	C₄⋊C₈	—	10	1	no	—
161	C₂²⋊C₈	—	2	2	no	—
162	#97	—	20	2	no	—
163	C₄×C₈	—	2	2	no	—
164	C₄×C₈	C₄×D₅	1	1	yes	characteristic
165	C₄×C₈	C₄×D₅	1	1	yes	characteristic
166	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
167	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
168	#95	—	5	2	no	—
169	C₂²×C₅	—	2	1	no	—
170	C₂²×C₅	—	2	1	no	—
171	C₂²×C₅	#390	1	1	yes	characteristic
172	C₂²×C₅	#461	1	1	yes	characteristic
173	C₂²×C₅	#462	1	1	yes	characteristic
174	C₄×C₅	—	2	1	no	—
175	C₄×C₅	—	2	1	no	—
176	C₄×C₅	—	2	1	no	—
177	C₄×C₅	C₂×#103	1	1	yes	characteristic
178	C₄×C₅	#419	1	1	yes	characteristic
179	C₄×C₅	#444	1	2	yes	—
180	C₄×C₅	#461	1	1	yes	characteristic
181	C₄×C₅	#462	1	1	yes	characteristic
182	Dic₅	—	8	2	no	—
183	Dic₅	—	8	2	no	—
184	Dic₅	—	8	2	no	—
185	C₂×Q₈	—	10	2	no	—
186	C₂×Q₈	—	10	2	no	—
187	C₂²×C₄	—	10	2	no	—
188	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
189	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
190	C₂²×C₄	#713	1	1	yes	characteristic
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	—	10	2	no	—
195	C₂×C₈	—	20	1	no	—
196	C₂×C₈	—	20	1	no	—
197	C₂×C₈	#677	1	2	yes	—
198	C₂×C₈	Dic₂₀	1	2	yes	—
199	C₂×C₈	#685	1	1	yes	characteristic
200	C₂×C₈	#685	1	1	yes	characteristic
201	C₂×C₈	#685	1	1	yes	characteristic
202	C₂×C₈	#685	1	1	yes	characteristic
203	C₄⋊C₄	—	10	2	no	—
204	C₄⋊C₄	—	10	2	no	—
205	C₄⋊C₄	—	10	2	no	—
206	C₄⋊C₄	—	10	2	no	—
207	C₄⋊C₄	—	20	2	no	—
208	C₄⋊C₄	—	20	2	no	—
209	C₄⋊C₄	—	20	2	no	—
210	C₂²⋊C₄	—	20	2	no	—
211	C₄²	—	20	2	no	—
212	C₄²	—	20	1	no	—
213	C₄²	—	20	1	no	—
214	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
215	C₄²	D₄×D₅	1	1	yes	characteristic
216	C₄²	#713	1	1	yes	characteristic
217	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
218	C₂×C₅	—	2	1	no	—
219	C₂×C₅	—	2	1	no	—
220	C₂×C₅	#1975	1	1	yes	characteristic
221	C₂×C₅	#1976	1	1	yes	characteristic
222	C₂×C₅	#1981	1	1	yes	characteristic
223	C₂³	#4041	1	1	yes	characteristic
224	Q₈	—	20	4	no	—
225	Q₈	—	40	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	—	20	2	no	—
229	C₂×C₄	—	20	2	no	—
230	C₂×C₄	—	20	2	no	—
231	C₂×C₄	—	20	2	no	—
232	C₂×C₄	—	20	2	no	—
233	C₂×C₄	#4039	1	1	yes	characteristic
234	C₂×C₄	#4039	1	1	yes	characteristic
235	C₂×C₄	#4041	1	1	yes	characteristic
236	C₂×C₄	C₂×#677	1	1	yes	characteristic
237	C₂×C₄	C₂×Dic₂₀	1	1	yes	characteristic
238	C₂×C₄	#4055	1	1	yes	characteristic
239	C₂×C₄	#4055	1	1	yes	characteristic
240	C₂×C₄	C₂×#685	1	1	yes	characteristic
241	C₂×C₄	#4086	1	1	yes	characteristic
242	C₂×C₄	#4093	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	8	2	no	—
246	C₈	—	20	2	no	—
247	C₈	#3948	1	2	yes	—
248	C₈	#3948	1	2	yes	—
249	C₅	#9752	1	1	yes	characteristic, complemented, Sylow
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	#65130	1	1	yes	characteristic
253	C₂²	#65138	1	1	yes	characteristic
254	C₂²	#65277	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	40	2	no	—
259	C₄	—	40	2	no	—
260	C₄	—	40	2	no	—
261	C₄	#65130	1	1	yes	characteristic
262	C₄	#65138	1	1	yes	characteristic
263	C₄	#65183	1	2	yes	—
264	C₄	C₂×#3948	1	1	yes	characteristic
265	C₄	#65409	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	#10601249	1	1	yes	characteristic
269	C₂	#10601451	1	1	yes	characteristic
270	C₂	#10601490	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#4041
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1564	C₂
socle	C₂²×C₅	#390
prime radical	C₂³×C₅	#121
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1564	D₅
hypocenter	C₅	#9752
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
16. #64971	C₄	semidirect
17. #64971	C₄	semidirect
18. #64969	C₄	semidirect
249. C₅	#9752	semidirect

Maximal subgroups

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

subgroup	index
#10610173	2
C₅×#1564	2
#10601585	2
#10601561	2
#10600550	2
#9752	5