Subgroups — #11992356

Order 1280.

454 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₀	C₂	1	1	yes	maximal, characteristic
2	C₅×#1845	C₂	1	1	yes	maximal, characteristic
3	C₂×#64678	C₂	1	1	yes	maximal, characteristic
4	#10600381	C₂	1	2	yes	maximal
5	#10598125	C₂	1	2	yes	maximal
6	C₂×C₄×Dic₁₀	C₂²	1	1	yes	characteristic
7	C₂×C₅×#107	—	2	1	no	—
8	C₂×C₅×#107	—	2	1	no	—
9	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₅×#384	C₂²	1	2	yes	—
11	C₂×#3946	—	2	1	no	—
12	C₂×C₈×Dic₅	—	2	1	no	—
13	C₂×#3942	—	2	2	no	—
14	C₂×C₄⋊#168	C₂²	1	2	yes	—
15	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
16	C₈×Dic₁₀	—	2	2	no	—
17	C₈×Dic₁₀	C₄	1	4	yes	complemented
18	#64749	—	2	2	no	—
19	#64678	—	2	2	no	—
20	#8645	—	5	1	no	maximal, Sylow
21	C₂²×Dic₁₀	D₄	1	1	yes	characteristic
22	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
24	C₂×C₅×C₄⋊C₄	—	2	2	no	—
25	C₂×C₅×C₄⋊C₄	—	2	2	no	—
26	C₂×C₅×C₄⋊C₄	—	2	2	no	—
27	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊Dic₅	D₄	1	1	yes	characteristic
29	C₂×#683	—	2	1	no	—
30	C₂×C₄×Dic₅	—	2	1	no	—
31	C₂²×#168	—	2	1	no	—
32	C₂²×#168	—	2	2	no	—
33	C₄×Dic₁₀	—	2	1	no	—
34	C₄×Dic₁₀	—	2	1	no	—
35	C₄×Dic₁₀	C₂×C₄	1	2	yes	—
36	C₄×Dic₁₀	Q₈	1	2	yes	—
37	C₅×#107	—	4	2	no	—
38	C₅×#107	—	4	2	no	—
39	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
40	C₄×C₅×C₈	D₄	1	2	yes	—
41	#3946	—	2	4	no	—
42	C₈×Dic₅	—	2	4	no	—
43	#3942	—	4	4	no	—
44	C₄⋊#168	—	2	1	no	—
45	C₄⋊#168	—	2	2	no	—
46	C₄⋊#168	—	2	2	no	—
47	C₄⋊#168	C₂×C₄	1	2	yes	—
48	C₂×C₈×Q₈	—	5	1	no	—
49	#1845	D₅	1	1	yes	characteristic
50	#1803	—	5	2	no	—
51	C₂×#335	—	5	1	no	—
52	#1296	—	5	2	no	—
53	C₂²×C₄×C₅	—	4	2	no	—
54	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅	#41	1	1	yes	characteristic
57	C₂²×Dic₅	—	2	1	no	—
58	C₂²×Dic₅	—	2	1	no	—
59	C₂×Dic₁₀	—	2	1	no	—
60	C₂×Dic₁₀	—	2	1	no	—
61	C₂×Dic₁₀	—	2	1	no	—
62	C₂×Dic₁₀	C₄⋊C₄	1	2	yes	—
63	C₂×Dic₁₀	D₈	1	2	yes	—
64	C₂×Dic₁₀	Q₁₆	1	2	yes	—
65	C₂×C₅×C₈	—	2	1	no	—
66	C₂×C₅×C₈	—	2	1	no	—
67	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
68	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
69	C₂×C₅×C₈	SD₁₆	1	2	yes	—
70	C₂×C₅×C₈	Q₁₆	1	2	yes	—
71	C₅×C₄⋊C₄	—	4	2	no	—
72	C₅×C₄⋊C₄	—	4	2	no	—
73	C₅×C₄⋊C₄	—	4	2	no	—
74	C₅×C₄⋊C₄	—	4	2	no	—
75	C₅×C₄⋊C₄	—	8	2	no	—
76	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
79	C₄²×C₅	#41	1	1	yes	characteristic
80	C₄⋊Dic₅	—	2	1	no	—
81	C₄⋊Dic₅	C₄⋊C₄	1	2	yes	—
82	#683	—	2	2	no	—
83	#683	—	2	2	no	—
84	C₄×Dic₅	—	2	2	no	—
85	C₄×Dic₅	—	2	2	no	—
86	C₂×#168	—	2	2	no	—
87	C₂×#168	—	4	1	no	—
88	C₂×#168	—	4	1	no	—
89	C₂×#168	—	4	2	no	—
90	C₂×#168	—	4	2	no	—
91	C₂×#168	—	4	2	no	—
92	C₂×C₄×Q₈	—	5	1	no	—
93	C₈×Q₈	—	5	4	no	—
94	C₈×Q₈	—	10	2	no	—
95	C₂×#107	—	2	1	no	—
96	C₂×#107	—	2	1	no	—
97	C₂×C₄⋊C₈	—	5	1	no	—
98	C₂×C₄⋊C₈	—	5	2	no	—
99	C₂×C₄⋊C₈	—	10	1	no	—
100	C₂×#103	—	10	2	no	—
101	C₂×C₄×C₈	—	10	1	no	—
102	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
103	#384	C₂×D₅	1	2	yes	—
104	#340	—	10	2	no	—
105	#335	—	10	2	no	—
106	C₂³×C₅	#121	1	1	yes	characteristic
107	C₂×C₄×C₅	—	2	1	no	—
108	C₂×C₄×C₅	—	2	1	no	—
109	C₂×C₄×C₅	—	8	2	no	—
110	C₂×C₄×C₅	—	8	2	no	—
111	C₂×C₄×C₅	—	8	2	no	—
112	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅	#117	1	1	yes	characteristic
115	C₂×C₄×C₅	#121	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂×D₈	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	C₂×Q₁₆	1	1	yes	characteristic
122	C₂×C₄×C₅	#135	1	1	yes	characteristic
123	C₂×C₄×C₅	#135	1	1	yes	characteristic
124	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×C₄×C₅	#137	1	1	yes	characteristic
126	C₂×Dic₅	—	2	2	no	—
127	C₂×Dic₅	—	2	2	no	—
128	C₂×Dic₅	—	2	2	no	—
129	C₂×Dic₅	—	2	2	no	—
130	C₂×Dic₅	—	4	1	no	—
131	C₂×Dic₅	—	4	1	no	—
132	Dic₁₀	—	2	2	no	—
133	Dic₁₀	—	2	2	no	—
134	Dic₁₀	—	2	2	no	—
135	Dic₁₀	#107	1	4	yes	complemented
136	C₅×C₈	—	2	2	no	—
137	C₅×C₈	#103	1	4	yes	complemented
138	#168	—	4	2	no	—
139	#168	—	8	2	no	—
140	C₂²×Q₈	—	5	1	no	—
141	C₂²×C₈	—	10	1	no	—
142	C₂²×C₈	—	10	2	no	—
143	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
144	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
145	C₄×Q₈	—	5	2	no	—
146	C₄×Q₈	—	5	2	no	—
147	C₄×Q₈	—	10	1	no	—
148	C₄×Q₈	—	10	1	no	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄⋊C₄	—	2	2	no	—
151	C₂×C₄⋊C₄	—	2	2	no	—
152	C₂×C₄⋊C₄	—	5	1	no	—
153	C₂×C₄⋊C₄	—	10	1	no	—
154	C₂×C₄²	—	10	1	no	—
155	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
156	#107	—	4	2	no	—
157	#107	—	4	2	no	—
158	C₄⋊C₈	—	5	2	no	—
159	C₄⋊C₈	—	10	4	no	—
160	C₄⋊C₈	—	10	1	no	—
161	C₄⋊C₈	—	10	2	no	—
162	C₄⋊C₈	—	10	2	no	—
163	#103	—	20	4	no	—
164	C₄×C₈	—	10	4	no	—
165	C₄×C₈	C₄×D₅	1	2	yes	—
166	C₄×C₈	D₂₀	1	2	yes	—
167	C₂²×C₅	#380	1	1	yes	characteristic
168	C₂²×C₅	#460	1	1	yes	characteristic
169	C₂²×C₅	#462	1	1	yes	characteristic
170	C₂²×C₅	#466	1	1	yes	characteristic
171	C₂²×C₅	#467	1	1	yes	characteristic
172	C₂²×C₅	#472	1	1	yes	characteristic
173	C₂²×C₅	#473	1	1	yes	characteristic
174	C₄×C₅	—	2	1	no	—
175	C₄×C₅	—	2	1	no	—
176	C₄×C₅	—	16	2	no	—
177	C₄×C₅	C₂×#103	1	1	yes	characteristic
178	C₄×C₅	#417	1	1	yes	characteristic
179	C₄×C₅	C₂×#107	1	1	yes	characteristic
180	C₄×C₅	#427	1	1	yes	characteristic
181	C₄×C₅	#466	1	1	yes	characteristic
182	C₄×C₅	#467	1	1	yes	characteristic
183	C₄×C₅	#475	1	1	yes	characteristic
184	C₄×C₅	#477	1	1	yes	characteristic
185	Dic₅	—	2	4	no	—
186	Dic₅	—	4	2	no	—
187	C₂×Q₈	—	5	2	no	—
188	C₂×Q₈	—	5	2	no	—
189	C₂×Q₈	—	5	2	no	—
190	C₂×Q₈	—	10	1	no	—
191	C₂×Q₈	—	10	1	no	—
192	C₂×Q₈	—	10	1	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	—	10	1	no	—
195	C₂²×C₄	—	10	1	no	—
196	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
197	C₂²×C₄	#711	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	10	2	no	—
202	C₂×C₈	—	20	1	no	—
203	C₂×C₈	—	20	1	no	—
204	C₂×C₈	—	20	2	no	—
205	C₂×C₈	—	20	2	no	—
206	C₂×C₈	—	20	2	no	—
207	C₂×C₈	#685	1	2	yes	—
208	C₂×C₈	#685	1	2	yes	—
209	C₂×C₈	#687	1	2	yes	—
210	C₂×C₈	#689	1	2	yes	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	5	2	no	—
216	C₄⋊C₄	—	8	2	no	—
217	C₄⋊C₄	—	10	2	no	—
218	C₄⋊C₄	—	10	1	no	—
219	C₄⋊C₄	—	10	2	no	—
220	C₄²	—	10	2	no	—
221	C₄²	—	10	2	no	—
222	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
223	C₄²	C₂×D₂₀	1	1	yes	characteristic
224	C₄²	Q₈×D₅	1	1	yes	characteristic
225	C₄²	#713	1	1	yes	characteristic
226	C₂×C₅	#1668	1	1	yes	characteristic
227	C₂×C₅	#1670	1	1	yes	characteristic
228	C₂×C₅	#1674	1	1	yes	characteristic
229	C₂×C₅	#1676	1	1	yes	characteristic
230	C₂×C₅	#1803	1	1	yes	characteristic
231	C₂×C₅	#1808	1	1	yes	characteristic
232	C₂×C₅	#1816	1	1	yes	characteristic
233	C₂³	#4084	1	1	yes	characteristic
234	Q₈	—	5	4	no	—
235	Q₈	—	10	2	no	—
236	Q₈	—	10	2	no	—
237	Q₈	—	10	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	8	2	no	—
241	C₂×C₄	—	8	2	no	—
242	C₂×C₄	—	8	2	no	—
243	C₂×C₄	—	10	2	no	—
244	C₂×C₄	—	10	2	no	—
245	C₂×C₄	—	10	2	no	—
246	C₂×C₄	—	10	2	no	—
247	C₂×C₄	—	20	1	no	—
248	C₂×C₄	—	20	1	no	—
249	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
250	C₂×C₄	#4038	1	1	yes	characteristic
251	C₂×C₄	#4041	1	1	yes	characteristic
252	C₂×C₄	#4043	1	1	yes	characteristic
253	C₂×C₄	D₅×D₈	1	1	yes	characteristic
254	C₂×C₄	#4058	1	1	yes	characteristic
255	C₂×C₄	D₅×Q₁₆	1	1	yes	characteristic
256	C₂×C₄	#4065	1	1	yes	characteristic
257	C₂×C₄	C₂×#685	1	1	yes	characteristic
258	C₂×C₄	#4078	1	1	yes	characteristic
259	C₂×C₄	C₂×#687	1	1	yes	characteristic
260	C₂×C₄	#4088	1	1	yes	characteristic
261	C₂×C₄	C₂×#689	1	1	yes	characteristic
262	C₂×C₄	#4092	1	1	yes	characteristic
263	C₈	—	2	2	no	—
264	C₈	—	20	2	no	—
265	C₈	—	40	2	no	—
266	C₈	#3942	1	4	yes	complemented
267	C₅	#8645	1	1	yes	characteristic, complemented, Sylow
268	C₂²	#65276	1	1	yes	characteristic
269	C₂²	#65364	1	1	yes	characteristic
270	C₂²	#65379	1	1	yes	characteristic
271	C₂²	#65446	1	1	yes	characteristic
272	C₂²	#65449	1	1	yes	characteristic
273	C₂²	#65477	1	1	yes	characteristic
274	C₂²	#65481	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	10	4	no	—
278	C₄	—	16	2	no	—
279	C₄	—	20	2	no	—
280	C₄	D₅×#107	1	1	yes	characteristic
281	C₄	#65169	1	1	yes	characteristic
282	C₄	#65172	1	1	yes	characteristic
283	C₄	#65176	1	1	yes	characteristic
284	C₄	#65255	1	1	yes	characteristic
285	C₄	C₂×#3942	1	1	yes	characteristic
286	C₄	#65380	1	1	yes	characteristic
287	C₄	#65383	1	1	yes	characteristic
288	C₂	#10600381	1	1	yes	characteristic
289	C₂	#10601423	1	1	yes	characteristic
290	C₂	#10601439	1	1	yes	characteristic
291	C₂	#10602079	1	1	yes	characteristic
292	C₂	#10602101	1	1	yes	characteristic
293	C₂	#10602170	1	1	yes	characteristic
294	C₂	#10602171	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#4084
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1845	C₂
socle, prime radical	C₂³×C₅	#121
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1845	D₅
hypocenter	C₅	#8645
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
17. C₈×Dic₁₀	C₄	semidirect
135. Dic₁₀	#107	semidirect
137. C₅×C₈	#103	semidirect
266. C₈	#3942	semidirect
267. C₅	#8645	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₀	2
C₅×#1845	2
C₂×#64678	2
#10600381	2
#10598125	2
#8645	5