Subgroups — #11992350

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×C₈×Dic₁₀	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#334	C₂	1	1	yes	maximal, characteristic
3	#10601278	C₂	1	2	yes	maximal
4	#10600557	C₂	1	1	yes	maximal, characteristic
5	#10598193	C₂	1	2	yes	maximal
6	C₂×C₄×Dic₁₀	C₂²	1	1	yes	characteristic
7	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×#3949	—	2	1	no	—
10	C₂×#3949	—	2	1	no	—
11	C₂×#3948	—	2	2	no	—
12	C₂×#3946	—	2	1	no	—
13	C₂×C₈×Dic₅	—	2	1	no	—
14	#65224	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	C₅×#334	—	2	2	no	—
19	#64771	—	2	2	no	—
20	#8643	—	5	1	no	maximal, Sylow
21	C₂²×Dic₁₀	D₄	1	1	yes	characteristic
22	C₂²×C₅×C₈	—	2	2	no	—
23	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
26	C₂×C₄⋊Dic₅	—	2	2	no	—
27	C₂×C₄⋊Dic₅	—	2	2	no	—
28	C₂×C₄⋊Dic₅	—	2	2	no	—
29	C₂×C₄⋊Dic₅	D₄	1	1	yes	characteristic
30	C₂×#683	—	2	1	no	—
31	C₂×C₄×Dic₅	—	2	1	no	—
32	C₂²×#168	—	2	1	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×C₄⋊C₈	—	2	2	no	—
38	C₅×C₄⋊C₈	—	2	2	no	—
39	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
40	C₄×C₅×C₈	D₄	1	2	yes	—
41	#3949	—	4	2	no	—
42	#3949	—	4	2	no	—
43	#3948	—	4	4	no	—
44	#3946	—	2	4	no	—
45	C₈×Dic₅	—	2	4	no	—
46	C₄⋊#168	—	2	1	no	—
47	C₄⋊#168	C₂×C₄	1	2	yes	—
48	C₂×C₈×Q₈	—	5	1	no	—
49	#1844	—	5	1	no	—
50	#1803	—	5	2	no	—
51	C₂×#334	D₅	1	1	yes	characteristic
52	#1296	—	5	2	no	—
53	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅	#41	1	1	yes	characteristic
56	C₂²×Dic₅	—	2	1	no	—
57	C₂²×Dic₅	—	2	1	no	—
58	C₂²×Dic₅	—	4	2	no	—
59	C₂×Dic₁₀	—	2	1	no	—
60	C₂×Dic₁₀	—	2	2	no	—
61	C₂×Dic₁₀	C₄⋊C₄	1	2	yes	—
62	C₂×Dic₁₀	SD₁₆	1	4	yes	—
63	C₂×C₅×C₈	—	2	1	no	—
64	C₂×C₅×C₈	—	2	1	no	—
65	C₂×C₅×C₈	—	4	2	no	—
66	C₂×C₅×C₈	—	4	2	no	—
67	C₂×C₅×C₈	—	4	2	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	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
75	C₄²×C₅	#41	1	1	yes	characteristic
76	C₄⋊Dic₅	—	2	1	no	—
77	C₄⋊Dic₅	—	4	2	no	—
78	C₄⋊Dic₅	—	4	2	no	—
79	C₄⋊Dic₅	—	4	2	no	—
80	C₄⋊Dic₅	—	4	2	no	—
81	C₄⋊Dic₅	—	8	2	no	—
82	C₄⋊Dic₅	C₄⋊C₄	1	2	yes	—
83	#683	—	2	2	no	—
84	#683	—	2	2	no	—
85	C₄×Dic₅	—	2	2	no	—
86	C₄×Dic₅	—	2	2	no	—
87	C₂×#168	—	2	2	no	—
88	C₂×#168	—	4	1	no	—
89	C₂×#168	—	4	1	no	—
90	C₂×C₄×Q₈	—	5	1	no	—
91	C₈×Q₈	—	5	4	no	—
92	C₈×Q₈	—	10	2	no	—
93	C₂×#106	—	10	1	no	—
94	C₂×#106	—	10	1	no	—
95	C₂×C₄⋊C₈	—	5	1	no	—
96	C₂×C₄⋊C₈	—	10	1	no	—
97	C₂×C₄⋊C₈	C₂×D₅	1	2	yes	—
98	C₂×#103	—	10	2	no	—
99	C₂×C₄×C₈	—	10	1	no	—
100	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
101	#384	—	5	2	no	—
102	#340	—	10	2	no	—
103	#334	—	2	2	no	—
104	C₂³×C₅	#121	1	1	yes	characteristic
105	C₂×C₄×C₅	—	2	2	no	—
106	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₅	#117	1	1	yes	characteristic
109	C₂×C₄×C₅	#121	1	1	yes	characteristic
110	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
114	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
115	C₂×C₄×C₅	#135	1	2	yes	—
116	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₂×C₄×C₅	#137	1	1	yes	characteristic
118	C₂×Dic₅	—	2	2	no	—
119	C₂×Dic₅	—	2	4	no	—
120	C₂×Dic₅	—	2	2	no	—
121	C₂×Dic₅	—	4	2	no	—
122	C₂×Dic₅	—	8	2	no	—
123	C₂×Dic₅	—	8	2	no	—
124	C₂×Dic₅	—	8	2	no	—
125	Dic₁₀	—	2	2	no	—
126	Dic₁₀	—	2	4	no	—
127	Dic₁₀	#106	1	4	yes	complemented
128	C₅×C₈	—	2	2	no	—
129	C₅×C₈	—	8	2	no	—
130	C₅×C₈	#103	1	4	yes	complemented
131	#168	—	4	2	no	—
132	C₂²×Q₈	—	5	1	no	—
133	C₂²×C₈	—	2	2	no	—
134	C₂²×C₈	—	10	1	no	—
135	C₂²×C₈	D₂₀	1	1	yes	characteristic
136	C₂²×C₈	D₂₀	1	1	yes	characteristic
137	C₄×Q₈	—	5	2	no	—
138	C₄×Q₈	—	5	2	no	—
139	C₄×Q₈	—	10	1	no	—
140	C₄×Q₈	—	10	1	no	—
141	C₂×C₄⋊C₄	—	5	1	no	—
142	C₂×C₄⋊C₄	—	10	1	no	—
143	C₂×C₄⋊C₄	—	10	2	no	—
144	C₂×C₄⋊C₄	—	10	2	no	—
145	C₂×C₄⋊C₄	—	10	2	no	—
146	C₂×C₄²	—	10	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	2	no	—
151	C₄⋊C₈	—	2	2	no	—
152	C₄⋊C₈	—	5	2	no	—
153	C₄⋊C₈	—	10	4	no	—
154	C₄⋊C₈	—	10	1	no	—
155	#103	—	20	4	no	—
156	C₄×C₈	—	10	4	no	—
157	C₄×C₈	C₄×D₅	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	#465	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	C₂×#103	1	1	yes	characteristic
167	C₄×C₅	#417	1	1	yes	characteristic
168	C₄×C₅	C₂×#106	1	1	yes	characteristic
169	C₄×C₅	#427	1	1	yes	characteristic
170	C₄×C₅	#465	1	2	yes	—
171	C₄×C₅	#475	1	1	yes	characteristic
172	C₄×C₅	#477	1	1	yes	characteristic
173	Dic₅	—	2	4	no	—
174	Dic₅	—	4	2	no	—
175	Dic₅	—	16	2	no	—
176	C₂×Q₈	—	5	2	no	—
177	C₂×Q₈	—	5	4	no	—
178	C₂×Q₈	—	10	2	no	—
179	C₂×Q₈	—	10	1	no	—
180	C₂²×C₄	—	10	1	no	—
181	C₂²×C₄	—	10	1	no	—
182	C₂²×C₄	—	20	2	no	—
183	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
184	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
185	C₂²×C₄	#713	1	1	yes	characteristic
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	—	10	2	no	—
192	C₂×C₈	—	20	1	no	—
193	C₂×C₈	—	20	1	no	—
194	C₂×C₈	#677	1	2	yes	—
195	C₂×C₈	Dic₂₀	1	2	yes	—
196	C₂×C₈	#685	1	2	yes	—
197	C₂×C₈	#685	1	2	yes	—
198	C₄⋊C₄	—	5	2	no	—
199	C₄⋊C₄	—	10	2	no	—
200	C₄⋊C₄	—	10	1	no	—
201	C₄⋊C₄	—	10	2	no	—
202	C₄⋊C₄	—	20	2	no	—
203	C₄⋊C₄	—	20	2	no	—
204	C₄⋊C₄	—	20	2	no	—
205	C₄⋊C₄	—	20	2	no	—
206	C₄⋊C₄	—	40	2	no	—
207	C₄²	—	10	2	no	—
208	C₄²	—	10	2	no	—
209	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
210	C₄²	#711	1	1	yes	characteristic
211	C₄²	Q₈×D₅	1	1	yes	characteristic
212	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
213	C₂×C₅	#1666	1	2	yes	—
214	C₂×C₅	#1672	1	2	yes	—
215	C₂×C₅	#1803	1	1	yes	characteristic
216	C₂×C₅	#1807	1	1	yes	characteristic
217	C₂×C₅	#1815	1	1	yes	characteristic
218	C₂³	#4041	1	1	yes	characteristic
219	Q₈	—	5	4	no	—
220	Q₈	—	10	2	no	—
221	Q₈	—	10	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	10	4	no	—
224	C₂×C₄	—	10	2	no	—
225	C₂×C₄	—	10	2	no	—
226	C₂×C₄	—	20	2	no	—
227	C₂×C₄	—	40	2	no	—
228	C₂×C₄	—	40	2	no	—
229	C₂×C₄	—	40	2	no	—
230	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
231	C₂×C₄	#4038	1	1	yes	characteristic
232	C₂×C₄	#4043	1	1	yes	characteristic
233	C₂×C₄	C₂×#677	1	1	yes	characteristic
234	C₂×C₄	C₂×Dic₂₀	1	1	yes	characteristic
235	C₂×C₄	#4054	1	1	yes	characteristic
236	C₂×C₄	#4055	1	1	yes	characteristic
237	C₂×C₄	D₅×SD₁₆	1	2	yes	—
238	C₂×C₄	#4062	1	2	yes	—
239	C₂×C₄	C₂×#685	1	1	yes	characteristic
240	C₂×C₄	#4084	1	1	yes	characteristic
241	C₂×C₄	#4092	1	1	yes	characteristic
242	C₈	—	2	2	no	—
243	C₈	—	8	2	no	—
244	C₈	—	20	2	no	—
245	C₈	#3948	1	4	yes	complemented
246	C₅	#8643	1	1	yes	characteristic, complemented, Sylow
247	C₂²	#65137	1	1	yes	characteristic
248	C₂²	#65138	1	1	yes	characteristic
249	C₂²	#65153	1	2	yes	—
250	C₂²	#65187	1	2	yes	—
251	C₂²	#65276	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	10	4	no	—
255	C₄	—	20	2	no	—
256	C₄	—	80	2	no	—
257	C₄	#65139	1	1	yes	characteristic
258	C₄	#65140	1	1	yes	characteristic
259	C₄	D₅×#106	1	1	yes	characteristic
260	C₄	#65149	1	1	yes	characteristic
261	C₄	C₂×#3948	1	1	yes	characteristic
262	C₄	#65422	1	1	yes	characteristic
263	C₄	#65460	1	2	yes	—
264	C₂	#10599339	1	2	yes	—
265	C₂	#10599340	1	2	yes	—
266	C₂	#10601278	1	1	yes	characteristic
267	C₂	#10601370	1	1	yes	characteristic
268	C₂	#10601513	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#4041
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₂×C₅×#334	C₂
socle, prime radical	C₂³×C₅	#121
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂×#334	D₅
hypocenter	C₅	#8643
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
17. C₈×Dic₁₀	C₄	semidirect
127. Dic₁₀	#106	semidirect
130. C₅×C₈	#103	semidirect
245. C₈	#3948	semidirect
246. C₅	#8643	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
C₂×C₅×#334	2
#10601278	2
#10600557	2
#10598193	2
#8643	5