Subgroups — #12003518

Order 1280.

486 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#64988	C₂	1	1	yes	maximal, characteristic
2	C₅×#1939	C₂	1	1	yes	maximal, characteristic
3	C₂×#64702	C₂	1	2	yes	maximal
4	#10600283	C₂	1	2	yes	maximal
5	#10598145	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₀	—	2	1	no	—
7	C₂²×Dic₂₀	—	2	1	no	—
8	C₂×#4015	C₂²	1	2	yes	—
9	C₂×C₅×C₄⋊C₈	—	2	1	no	—
10	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#168	C₂²	1	2	yes	—
14	#64988	—	2	4	no	—
15	C₅×#336	—	2	1	no	—
16	#64749	—	2	2	no	—
17	#64749	—	2	2	no	—
18	#64702	—	2	4	no	—
19	#10538	—	5	1	no	maximal, Sylow
20	C₂²×Dic₁₀	—	2	2	no	—
21	C₂²×C₅×C₈	—	2	1	no	—
22	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
24	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
25	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₅	—	2	1	no	—
27	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊Dic₅	—	2	2	no	—
29	C₂×C₄⋊Dic₅	—	2	2	no	—
30	C₂²×#168	—	2	2	no	—
31	C₂×Dic₂₀	—	2	4	no	—
32	C₂×Dic₂₀	—	2	4	no	—
33	C₂×Dic₂₀	—	4	2	no	—
34	C₂×Dic₂₀	—	4	2	no	—
35	C₂×Dic₂₀	—	4	2	no	—
36	C₂×Dic₂₀	—	4	2	no	—
37	#4015	—	2	2	no	—
38	#4015	—	2	2	no	—
39	#4015	—	2	2	no	—
40	#4015	D₄	1	4	yes	—
41	C₅×C₄⋊C₈	—	2	2	no	—
42	C₅×C₄⋊C₈	—	4	2	no	—
43	C₄×C₅×C₈	—	2	2	no	—
44	C₅×#95	—	2	1	no	—
45	C₄⋊#168	—	2	2	no	—
46	C₄⋊#168	—	2	2	no	—
47	C₂×#494	—	5	1	no	—
48	#1988	—	5	2	no	—
49	#1939	D₅	1	1	yes	characteristic
50	C₂×#332	—	5	2	no	—
51	#1277	—	5	1	no	—
52	C₂²×C₄×C₅	—	2	1	no	—
53	C₂²×C₄×C₅	—	2	1	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	—	4	2	no	—
58	C₂×Dic₁₀	—	2	4	no	—
59	C₂×Dic₁₀	—	4	2	no	—
60	C₂×Dic₁₀	—	4	2	no	—
61	C₂×Dic₁₀	—	4	2	no	—
62	C₂×Dic₁₀	—	4	2	no	—
63	C₂×Dic₁₀	—	4	2	no	—
64	C₂×C₅×C₈	—	2	1	no	—
65	C₂×C₅×C₈	—	2	1	no	—
66	C₂×C₅×C₈	—	2	1	no	—
67	C₂×C₅×C₈	—	2	1	no	—
68	C₂×C₅×C₈	—	4	1	no	—
69	C₂×C₅×C₈	—	4	1	no	—
70	C₂×C₅×C₈	—	4	1	no	—
71	C₂×C₅×C₈	D₈	1	2	yes	—
72	C₂×C₅×C₈	SD₁₆	1	2	yes	—
73	C₅×C₄⋊C₄	—	2	1	no	—
74	C₅×C₄⋊C₄	—	2	2	no	—
75	C₅×C₄⋊C₄	Q₁₆	1	2	yes	complemented
76	C₄²×C₅	—	4	1	no	—
77	C₄²×C₅	—	4	1	no	—
78	C₄²×C₅	C₂×D₄	1	2	yes	—
79	C₄²×C₅	#41	1	2	yes	—
80	C₄⋊Dic₅	—	2	4	no	—
81	C₄⋊Dic₅	—	2	4	no	—
82	C₄⋊Dic₅	—	4	2	no	—
83	C₄⋊Dic₅	—	4	2	no	—
84	C₂×#168	—	4	2	no	—
85	C₂×#168	—	4	2	no	—
86	C₂×#168	—	4	2	no	—
87	Dic₂₀	—	4	8	no	—
88	Dic₂₀	—	4	8	no	—
89	C₂²×Q₁₆	—	10	1	no	—
90	C₂²×Q₁₆	—	10	1	no	—
91	C₂×C₄⋊Q₈	—	5	2	no	—
92	#494	—	10	4	no	—
93	C₂×C₄⋊C₈	—	2	1	no	—
94	C₂×C₄⋊C₈	—	5	2	no	—
95	C₂×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
96	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
97	C₄×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
98	#340	—	10	2	no	—
99	#340	—	10	2	no	—
100	#336	—	2	1	no	—
101	#332	—	10	4	no	—
102	C₂³×C₅	#121	1	1	yes	characteristic
103	C₂×C₄×C₅	—	2	1	no	—
104	C₂×C₄×C₅	—	2	1	no	—
105	C₂×C₄×C₅	—	2	2	no	—
106	C₂×C₄×C₅	—	4	1	no	—
107	C₂×C₄×C₅	—	4	1	no	—
108	C₂×C₄×C₅	—	4	1	no	—
109	C₂×C₄×C₅	—	4	1	no	—
110	C₂×C₄×C₅	—	4	1	no	—
111	C₂×C₄×C₅	#120	1	1	yes	characteristic
112	C₂×C₄×C₅	#121	1	2	yes	—
113	C₂×C₄×C₅	#123	1	2	yes	—
114	C₂×C₄×C₅	#124	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
118	C₂×C₄×C₅	#135	1	1	yes	characteristic
119	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂×C₄×C₅	#137	1	2	yes	—
122	C₂×Dic₅	—	4	4	no	—
123	C₂×Dic₅	—	8	2	no	—
124	C₂×Dic₅	—	8	2	no	—
125	Dic₁₀	—	4	4	no	—
126	Dic₁₀	—	4	4	no	—
127	Dic₁₀	—	8	2	no	—
128	Dic₁₀	—	8	2	no	—
129	C₅×C₈	—	2	2	no	—
130	C₅×C₈	—	2	2	no	—
131	C₅×C₈	—	8	1	no	—
132	#168	—	8	2	no	—
133	C₂²×Q₈	—	10	2	no	—
134	C₂×Q₁₆	—	10	4	no	—
135	C₂×Q₁₆	—	10	4	no	—
136	C₂×Q₁₆	—	20	2	no	—
137	C₂×Q₁₆	—	20	2	no	—
138	C₂×Q₁₆	—	20	2	no	—
139	C₂×Q₁₆	—	20	2	no	—
140	C₂²×C₈	—	2	1	no	—
141	C₂²×C₈	—	10	2	no	—
142	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
143	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
144	C₄⋊Q₈	—	5	4	no	—
145	C₄⋊Q₈	—	10	2	no	—
146	C₄⋊Q₈	—	10	2	no	—
147	C₄⋊Q₈	—	10	2	no	—
148	C₂×C₄⋊C₄	—	10	2	no	—
149	C₂×C₄⋊C₄	—	10	2	no	—
150	C₂×C₄⋊C₄	D₂₀	1	1	yes	characteristic
151	C₂×C₄⋊C₄	D₂₀	1	1	yes	characteristic
152	C₂×C₄²	—	2	1	no	—
153	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
154	C₄⋊C₈	—	2	2	no	—
155	C₄⋊C₈	—	4	2	no	—
156	C₄⋊C₈	—	10	2	no	—
157	C₄⋊C₈	—	10	2	no	—
158	C₄×C₈	—	2	2	no	—
159	#95	—	2	1	no	—
160	C₂²×C₅	#394	1	1	yes	characteristic
161	C₂²×C₅	#459	1	1	yes	characteristic
162	C₂²×C₅	#460	1	1	yes	characteristic
163	C₂²×C₅	#467	1	1	yes	characteristic
164	C₂²×C₅	#470	1	1	yes	characteristic
165	C₂²×C₅	#473	1	1	yes	characteristic
166	C₂²×C₅	#473	1	1	yes	characteristic
167	C₄×C₅	—	2	2	no	—
168	C₄×C₅	—	4	2	no	—
169	C₄×C₅	—	8	1	no	—
170	C₄×C₅	#447	1	1	yes	characteristic
171	C₄×C₅	#450	1	1	yes	characteristic
172	C₄×C₅	#462	1	2	yes	—
173	C₄×C₅	#464	1	2	yes	—
174	C₄×C₅	#486	1	1	yes	characteristic
175	C₄×C₅	#490	1	1	yes	characteristic
176	Dic₅	—	8	4	no	—
177	C₂×Q₈	—	10	4	no	—
178	C₂×Q₈	—	20	2	no	—
179	C₂×Q₈	—	20	2	no	—
180	C₂×Q₈	—	20	2	no	—
181	C₂×Q₈	—	20	2	no	—
182	C₂×Q₈	—	20	2	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	20	2	no	—
186	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
187	C₂²×C₄	#709	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
189	Q₁₆	—	20	8	no	—
190	Q₁₆	—	20	8	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	20	2	no	—
199	C₂×C₈	—	20	2	no	—
200	C₂×C₈	—	20	2	no	—
201	C₂×C₈	#686	1	2	yes	—
202	C₂×C₈	#688	1	2	yes	—
203	C₄⋊C₄	—	2	1	no	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	10	4	no	—
206	C₄⋊C₄	—	10	4	no	—
207	C₄⋊C₄	—	20	2	no	—
208	C₄⋊C₄	—	20	2	no	—
209	C₄⋊C₄	Dic₂₀	1	2	yes	complemented
210	C₄²	—	4	1	no	—
211	C₄²	—	4	1	no	—
212	C₄²	D₄×D₅	1	2	yes	—
213	C₄²	#713	1	2	yes	—
214	C₂×C₅	#1669	1	1	yes	characteristic
215	C₂×C₅	#1670	1	1	yes	characteristic
216	C₂×C₅	#1689	1	1	yes	characteristic
217	C₂×C₅	#1690	1	1	yes	characteristic
218	C₂×C₅	#2041	1	1	yes	characteristic
219	C₂×C₅	#2047	1	1	yes	characteristic
220	C₂×C₅	#2056	1	1	yes	characteristic
221	C₂³	#4076	1	1	yes	characteristic
222	Q₈	—	20	4	no	—
223	Q₈	—	20	4	no	—
224	Q₈	—	40	2	no	—
225	Q₈	—	40	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	4	1	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	20	4	no	—
235	C₂×C₄	—	40	2	no	—
236	C₂×C₄	—	40	2	no	—
237	C₂×C₄	#4040	1	2	yes	—
238	C₂×C₄	#4041	1	2	yes	—
239	C₂×C₄	#4050	1	1	yes	characteristic
240	C₂×C₄	C₂×Dic₂₀	1	1	yes	characteristic
241	C₂×C₄	#4055	1	2	yes	—
242	C₂×C₄	C₂×#686	1	1	yes	characteristic
243	C₂×C₄	#4083	1	1	yes	characteristic
244	C₂×C₄	C₂×#688	1	1	yes	characteristic
245	C₂×C₄	#4093	1	1	yes	characteristic
246	C₂×C₄	#4095	1	1	yes	characteristic
247	C₂×C₄	#4095	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	—	2	2	no	—
250	C₈	—	8	1	no	—
251	C₈	—	40	2	no	—
252	C₅	#10538	1	1	yes	characteristic, complemented, Sylow
253	C₂²	#65280	1	1	yes	characteristic
254	C₂²	#65304	1	1	yes	characteristic
255	C₂²	#65316	1	1	yes	characteristic
256	C₂²	#65405	1	1	yes	characteristic
257	C₂²	#65426	1	1	yes	characteristic
258	C₂²	#65432	1	1	yes	characteristic
259	C₂²	#65432	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	4	2	no	—
262	C₄	—	8	1	no	—
263	C₄	—	40	4	no	—
264	C₄	#65134	1	2	yes	—
265	C₄	#65138	1	2	yes	—
266	C₄	#65325	1	1	yes	characteristic
267	C₄	#65335	1	1	yes	characteristic
268	C₄	#65356	1	1	yes	characteristic
269	C₄	#65377	1	1	yes	characteristic
270	C₂	#10600420	1	1	yes	characteristic
271	C₂	#10601280	1	1	yes	characteristic
272	C₂	#10601336	1	1	yes	characteristic
273	C₂	#10601643	1	1	yes	characteristic
274	C₂	#10601664	1	1	yes	characteristic
275	C₂	#10601735	1	1	yes	characteristic
276	C₂	#10601755	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#4076
commutator, cosocle	C₂×C₄²×C₅	C₂³
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1939	C₂
socle, prime radical	C₂³×C₅	#121
hypercenter	#1939	D₅
hypocenter	C₅	#10538
LCS	G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1939

Complements

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

N	H ≅ G/N	type
75. C₅×C₄⋊C₄	Q₁₆	semidirect
209. C₄⋊C₄	Dic₂₀	semidirect
252. C₅	#10538	semidirect

Maximal subgroups

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

subgroup	index
C₂×#64988	2
C₅×#1939	2
C₂×#64702	2
#10600283	2
#10598145	2
#10538	5