Subgroups — #11992389

Order 1280.

434 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×#446	C₂	1	1	yes	maximal, characteristic
2	#10602419	C₂	1	2	yes	maximal
3	#10600557	C₂	1	1	yes	maximal, characteristic
4	C₂×#64678	C₂	1	1	yes	maximal, characteristic
5	#10598193	C₂	1	2	yes	maximal
6	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
7	C₅×#446	—	2	2	no	—
8	C₅×#446	C₄	1	4	yes	complemented
9	C₂×C₅×C₄⋊C₈	—	2	1	no	—
10	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₅×#97	—	2	1	no	—
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₂×#3967	—	2	2	no	—
14	C₂×#3949	—	2	1	no	—
15	C₂×#3949	—	2	1	no	—
16	#65224	C₂²	1	2	yes	—
17	C₂×C₄⋊#168	C₂²	1	2	yes	—
18	#64749	—	2	2	no	—
19	#64678	—	2	2	no	—
20	#8651	—	5	1	no	maximal, Sylow
21	C₂²×C₅×Q₈	D₄	1	1	yes	characteristic
22	C₂²×C₅×C₈	—	2	1	no	—
23	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
25	C₄×C₅×Q₈	—	2	1	no	—
26	C₄×C₅×Q₈	—	2	1	no	—
27	C₄×C₅×Q₈	C₂×C₄	1	2	yes	—
28	C₄×C₅×Q₈	Q₈	1	2	yes	—
29	C₂×C₅×C₄⋊C₄	—	2	1	no	—
30	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₅	—	2	1	no	—
32	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
33	C₂×C₄⋊Dic₅	—	2	2	no	—
34	C₂×C₄⋊Dic₅	—	2	2	no	—
35	C₂×C₄⋊Dic₅	—	2	2	no	—
36	C₂²×#168	—	2	2	no	—
37	C₅×C₄⋊C₈	—	2	4	no	—
38	C₅×C₄⋊C₈	—	2	1	no	—
39	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
40	C₅×#97	—	2	4	no	—
41	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
42	C₄×C₅×C₈	D₄	1	2	yes	—
43	#3967	—	4	4	no	—
44	#3949	—	4	2	no	—
45	#3949	—	4	2	no	—
46	C₄⋊#168	—	2	2	no	—
47	C₄⋊#168	—	2	2	no	—
48	C₂×#446	D₅	1	1	yes	characteristic
49	#1844	—	5	1	no	—
50	#1803	—	5	2	no	—
51	C₂×#335	—	5	1	no	—
52	#1296	—	5	2	no	—
53	C₂×C₅×Q₈	—	2	1	no	—
54	C₂×C₅×Q₈	—	2	2	no	—
55	C₂×C₅×Q₈	—	2	2	no	—
56	C₂×C₅×Q₈	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₅	—	2	1	no	—
58	C₂²×C₄×C₅	—	2	1	no	—
59	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	#41	1	1	yes	characteristic
62	C₂²×Dic₅	—	4	2	no	—
63	C₂×C₅×C₈	—	2	1	no	—
64	C₂×C₅×C₈	—	2	1	no	—
65	C₂×C₅×C₈	—	2	2	no	—
66	C₂×C₅×C₈	—	4	1	no	—
67	C₂×C₅×C₈	—	4	1	no	—
68	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
69	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
70	C₂×C₅×C₈	SD₁₆	1	2	yes	—
71	C₂×C₅×C₈	Q₁₆	1	2	yes	—
72	C₅×C₄⋊C₄	—	2	2	no	—
73	C₅×C₄⋊C₄	—	2	1	no	—
74	C₅×C₄⋊C₄	—	2	2	no	—
75	C₅×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
76	C₄²×C₅	—	2	2	no	—
77	C₄²×C₅	—	2	2	no	—
78	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
81	C₄²×C₅	#41	1	1	yes	characteristic
82	C₄⋊Dic₅	—	4	2	no	—
83	C₄⋊Dic₅	—	4	2	no	—
84	C₄⋊Dic₅	—	4	2	no	—
85	C₄⋊Dic₅	—	4	2	no	—
86	C₄⋊Dic₅	—	8	2	no	—
87	C₂×#168	—	4	2	no	—
88	C₂×#168	—	4	2	no	—
89	C₂×#168	—	4	2	no	—
90	C₂×C₄×Q₈	C₂×D₅	1	1	yes	characteristic
91	#446	—	2	2	no	—
92	#446	Dic₅	1	4	yes	complemented
93	C₂×#106	—	10	1	no	—
94	C₂×#106	—	10	1	no	—
95	C₂×C₄⋊C₈	—	2	1	no	—
96	C₂×C₄⋊C₈	—	5	2	no	—
97	C₂×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
98	C₂×#103	—	10	2	no	—
99	C₂×#97	—	2	1	no	—
100	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
101	#384	—	5	2	no	—
102	#340	—	10	2	no	—
103	#335	—	10	2	no	—
104	C₂³×C₅	#121	1	1	yes	characteristic
105	C₅×Q₈	—	2	4	no	—
106	C₅×Q₈	—	4	1	no	—
107	C₅×Q₈	—	4	1	no	—
108	C₂×C₄×C₅	—	2	2	no	—
109	C₂×C₄×C₅	—	2	2	no	—
110	C₂×C₄×C₅	—	2	2	no	—
111	C₂×C₄×C₅	—	4	2	no	—
112	C₂×C₄×C₅	—	4	2	no	—
113	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₅	#117	1	1	yes	characteristic
116	C₂×C₄×C₅	#121	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
121	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
122	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₂×C₄×C₅	#137	1	1	yes	characteristic
124	C₂×C₄×C₅	#137	1	2	yes	—
125	C₂×Dic₅	—	8	2	no	—
126	C₂×Dic₅	—	8	2	no	—
127	C₂×Dic₅	—	8	2	no	—
128	C₅×C₈	—	2	2	no	—
129	C₅×C₈	—	4	2	no	—
130	C₅×C₈	#103	1	4	yes	complemented
131	#168	—	8	2	no	—
132	C₂²×Q₈	D₂₀	1	1	yes	characteristic
133	C₂²×C₈	—	2	1	no	—
134	C₂²×C₈	—	10	2	no	—
135	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
136	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
137	C₄×Q₈	—	2	1	no	—
138	C₄×Q₈	—	2	1	no	—
139	C₄×Q₈	Dic₁₀	1	2	yes	—
140	C₄×Q₈	C₂×Dic₅	1	2	yes	—
141	C₂×C₄⋊C₄	—	2	1	no	—
142	C₂×C₄⋊C₄	—	10	2	no	—
143	C₂×C₄⋊C₄	—	10	2	no	—
144	C₂×C₄⋊C₄	—	10	2	no	—
145	C₂×C₄⋊C₄	D₂₀	1	1	yes	characteristic
146	C₂×C₄²	—	2	1	no	—
147	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
148	#106	—	20	2	no	—
149	#106	—	20	2	no	—
150	C₄⋊C₈	—	2	4	no	—
151	C₄⋊C₈	—	2	1	no	—
152	C₄⋊C₈	—	10	2	no	—
153	C₄⋊C₈	—	10	2	no	—
154	C₄⋊C₈	C₂×Dic₅	1	2	yes	—
155	#103	—	20	4	no	—
156	#97	—	2	4	no	—
157	C₄×C₈	C₂×Dic₅	1	2	yes	—
158	C₄×C₈	C₂²⋊D₅	1	2	yes	—
159	C₂²×C₅	#380	1	1	yes	characteristic
160	C₂²×C₅	#460	1	1	yes	characteristic
161	C₂²×C₅	#462	1	1	yes	characteristic
162	C₂²×C₅	#468	1	2	yes	—
163	C₂²×C₅	#471	1	2	yes	—
164	C₄×C₅	—	2	1	no	—
165	C₄×C₅	—	2	1	no	—
166	C₄×C₅	—	4	2	no	—
167	C₄×C₅	—	4	2	no	—
168	C₄×C₅	C₂×#103	1	1	yes	characteristic
169	C₄×C₅	#417	1	1	yes	characteristic
170	C₄×C₅	#428	1	2	yes	—
171	C₄×C₅	#469	1	1	yes	characteristic
172	C₄×C₅	#470	1	1	yes	characteristic
173	C₄×C₅	#475	1	1	yes	characteristic
174	C₄×C₅	#477	1	1	yes	characteristic
175	Dic₅	—	16	2	no	—
176	C₂×Q₈	—	2	1	no	—
177	C₂×Q₈	—	2	2	no	—
178	C₂×Q₈	—	2	2	no	—
179	C₂×Q₈	C₄⋊Dic₅	1	2	yes	—
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	20	2	no	—
183	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
184	C₂²×C₄	#709	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	4	1	no	—
190	C₂×C₈	—	4	1	no	—
191	C₂×C₈	—	20	2	no	—
192	C₂×C₈	—	20	2	no	—
193	C₂×C₈	—	20	2	no	—
194	C₂×C₈	#688	1	2	yes	—
195	C₂×C₈	#689	1	2	yes	—
196	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
197	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	2	1	no	—
201	C₄⋊C₄	—	20	2	no	—
202	C₄⋊C₄	—	20	2	no	—
203	C₄⋊C₄	—	20	2	no	—
204	C₄⋊C₄	—	20	2	no	—
205	C₄⋊C₄	—	40	2	no	—
206	C₄⋊C₄	C₄⋊Dic₅	1	2	yes	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	2	no	—
209	C₄²	C₂×Dic₁₀	1	1	yes	characteristic
210	C₄²	#709	1	1	yes	characteristic
211	C₄²	C₂²×Dic₅	1	1	yes	characteristic
212	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
213	C₂×C₅	#1686	1	2	yes	—
214	C₂×C₅	#1692	1	2	yes	—
215	C₂×C₅	#1803	1	1	yes	characteristic
216	C₂×C₅	#1809	1	1	yes	characteristic
217	C₂×C₅	#1815	1	1	yes	characteristic
218	C₂³	#4076	1	1	yes	characteristic
219	Q₈	—	2	4	no	—
220	Q₈	—	4	1	no	—
221	Q₈	—	4	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	40	2	no	—
228	C₂×C₄	—	40	2	no	—
229	C₂×C₄	—	40	2	no	—
230	C₂×C₄	#4054	1	2	yes	—
231	C₂×C₄	#4055	1	2	yes	—
232	C₂×C₄	#4070	1	1	yes	characteristic
233	C₂×C₄	#4070	1	1	yes	characteristic
234	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
235	C₂×C₄	#4072	1	1	yes	characteristic
236	C₂×C₄	#4076	1	1	yes	characteristic
237	C₂×C₄	C₂×#688	1	1	yes	characteristic
238	C₂×C₄	C₂×#689	1	1	yes	characteristic
239	C₂×C₄	#4095	1	1	yes	characteristic
240	C₂×C₄	#4097	1	1	yes	characteristic
241	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
242	C₈	—	2	2	no	—
243	C₈	—	4	2	no	—
244	C₈	—	40	2	no	—
245	C₈	#3967	1	4	yes	complemented
246	C₅	#8651	1	1	yes	characteristic, complemented, Sylow
247	C₂²	#65316	1	1	yes	characteristic
248	C₂²	#65320	1	1	yes	characteristic
249	C₂²	#65423	1	2	yes	—
250	C₂²	#65430	1	2	yes	—
251	C₂²	#65501	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	4	2	no	—
255	C₄	—	4	2	no	—
256	C₄	—	80	2	no	—
257	C₄	#65309	1	1	yes	characteristic
258	C₄	#65310	1	1	yes	characteristic
259	C₄	#65410	1	2	yes	—
260	C₄	#65424	1	1	yes	characteristic
261	C₄	#65426	1	1	yes	characteristic
262	C₄	C₂×#3967	1	1	yes	characteristic
263	C₄	#65521	1	1	yes	characteristic
264	C₂	#10601751	1	2	yes	—
265	C₂	#10601756	1	2	yes	—
266	C₂	#10602419	1	1	yes	characteristic
267	C₂	#10602781	1	1	yes	characteristic
268	C₂	#10602808	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#4076
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₂×C₅×#446	C₂
socle, prime radical	C₂³×C₅	#121
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂×#446	D₅
hypocenter	C₅	#8651
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
8. C₅×#446	C₄	semidirect
92. #446	Dic₅	semidirect
130. C₅×C₈	#103	semidirect
245. C₈	#3967	semidirect
246. C₅	#8651	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#446	2
#10602419	2
#10600557	2
C₂×#64678	2
#10598193	2
#8651	5