Subgroups — #11990003

Order 1280.

420 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10610186	C₂	1	1	yes	maximal, characteristic
2	C₅×#1450	C₂	1	1	yes	maximal, characteristic, complemented
3	#10601547	C₂	1	1	yes	maximal, characteristic
4	#10601547	C₂	1	1	yes	maximal, characteristic
5	C₂×#64674	C₂	1	1	yes	maximal, characteristic, complemented
6	#10598230	C₂	1	1	yes	maximal, characteristic, complemented
7	#10598230	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#709	C₂²	1	1	yes	characteristic
9	C₅×#431	C₂²	1	1	yes	characteristic
10	C₅×#431	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	#65399	C₄	1	2	yes	—
13	#65391	C₄	1	2	yes	—
14	C₄×#681	C₂²	1	1	yes	characteristic
15	C₂×C₄×#168	C₂²	1	1	yes	characteristic
16	C₂×C₄×#168	C₂²	1	1	yes	characteristic
17	#64979	—	2	2	no	—
18	#64974	—	2	1	no	—
19	#64974	—	2	1	no	—
20	C₅×#322	—	2	1	no	—
21	C₅×C₈²	—	2	1	no	—
22	#64704	C₄	1	2	yes	—
23	#64704	C₄	1	2	yes	—
24	#64703	C₄	1	2	yes	—
25	#64703	C₄	1	2	yes	—
26	#64674	—	2	2	no	—
27	#8189	—	5	1	no	maximal, Sylow
28	C₂×#709	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
31	C₄×C₂²⋊D₅	—	2	1	no	—
32	C₄×C₂²⋊D₅	—	2	1	no	—
33	#4072	C₂×C₄	1	1	yes	characteristic
34	C₂×#681	C₂×C₄	1	2	yes	—
35	C₂²×#168	—	2	1	no	—
36	C₂²×#168	—	2	1	no	—
37	C₄×D₂₀	C₂×C₄	1	1	yes	characteristic
38	C₄×D₂₀	C₂×C₄	1	1	yes	characteristic
39	#4018	—	2	1	no	—
40	C₄²×D₅	—	2	1	no	—
41	C₄×Dic₁₀	C₂×C₄	1	1	yes	characteristic
42	C₄×Dic₁₀	C₂×C₄	1	1	yes	characteristic
43	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
44	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
45	C₅×C₂²⋊C₈	—	2	1	no	—
46	C₅×C₂²⋊C₈	—	2	1	no	—
47	C₄×C₅×C₈	—	2	1	no	—
48	C₄×C₅×C₈	—	2	1	no	—
49	C₄×C₅×C₈	D₄	1	2	yes	—
50	C₄×C₅×C₈	D₄	1	1	yes	characteristic
51	C₄×C₅×C₈	D₄	1	1	yes	characteristic
52	#3952	—	2	2	no	—
53	#3952	—	2	2	no	—
54	#3946	—	2	2	no	—
55	#3946	—	2	2	no	—
56	#3935	—	2	1	no	—
57	C₄×#168	—	2	1	no	—
58	C₄×#168	—	2	1	no	—
59	C₄×#168	—	2	1	no	—
60	C₄×#168	C₂×C₄	1	2	yes	—
61	C₄×#168	C₂×C₄	1	2	yes	—
62	#2922	—	5	1	no	—
63	#1476	—	5	1	no	—
64	#1476	—	5	1	no	—
65	#1450	D₅	1	1	yes	characteristic, complemented
66	C₂×#322	—	5	1	no	—
67	#1278	—	5	1	no	—
68	#1278	—	5	1	no	—
69	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	#41	1	1	yes	characteristic
71	C₂²×C₄×C₅	#41	1	1	yes	characteristic
72	C₂×C₂²⋊D₅	—	2	1	no	—
73	#709	—	2	4	no	—
74	C₂×D₂₀	C₄²	1	1	yes	characteristic
75	C₂×C₄×D₅	—	2	1	no	—
76	C₂×C₄×D₅	—	2	1	no	—
77	C₂×C₄×D₅	—	2	1	no	—
78	C₂×Dic₁₀	C₄²	1	1	yes	characteristic
79	C₂×C₅×C₈	—	2	4	no	—
80	C₂×C₅×C₈	—	2	2	no	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
83	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
84	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂²⋊Dic₅	—	2	1	no	—
89	#685	—	2	1	no	—
90	#685	—	2	1	no	—
91	C₄⋊Dic₅	C₄²	1	1	yes	characteristic
92	C₄⋊Dic₅	C₄²	1	1	yes	characteristic
93	#683	—	2	1	no	—
94	#683	—	2	1	no	—
95	C₄×Dic₅	—	2	1	no	—
96	C₄×Dic₅	—	2	1	no	—
97	C₄×Dic₅	—	2	1	no	—
98	#681	—	2	4	no	—
99	C₂×#168	—	2	4	no	—
100	C₂×#168	—	2	2	no	—
101	C₂×#168	—	2	2	no	—
102	C₂×#168	—	2	2	no	—
103	C₂×#168	—	4	2	no	—
104	C₄×#41	—	5	1	no	—
105	#433	—	10	2	no	—
106	#431	—	10	1	no	—
107	#431	—	10	1	no	—
108	#431	C₂×D₅	1	1	yes	characteristic
109	#431	C₂×D₅	1	1	yes	characteristic
110	#424	—	5	2	no	—
111	#408	—	5	2	no	—
112	C₄×M₁₆	—	5	1	no	—
113	C₂×C₄×C₈	—	5	1	no	—
114	C₂×C₄×C₈	—	5	1	no	—
115	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
116	Q₈⋊C₈	—	5	2	no	—
117	Q₈⋊C₈	—	5	2	no	—
118	D₄⋊C₈	—	5	2	no	—
119	D₄⋊C₈	—	5	2	no	—
120	#322	—	2	1	no	—
121	#322	—	10	2	no	—
122	C₈²	—	2	1	no	—
123	C₂³×C₅	#117	1	1	yes	characteristic
124	C₂²×D₅	—	2	1	no	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
128	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₅	#131	1	4	yes	—
134	C₂²⋊D₅	—	4	2	no	—
135	C₂×Dic₅	—	2	1	no	—
136	C₂×Dic₅	—	2	1	no	—
137	C₂×Dic₅	—	2	1	no	—
138	D₂₀	—	2	2	no	—
139	C₄×D₅	—	4	2	no	—
140	C₄×D₅	—	4	2	no	—
141	C₄×D₅	—	4	2	no	—
142	Dic₁₀	—	2	2	no	—
143	C₅×C₈	—	2	2	no	—
144	C₅×C₈	—	2	2	no	—
145	C₅×C₈	—	4	2	no	—
146	C₅×C₈	—	4	2	no	—
147	#168	—	2	4	no	—
148	#168	—	4	2	no	—
149	#168	—	4	2	no	—
150	C₂×#41	—	5	1	no	—
151	C₂×M₁₆	—	5	2	no	—
152	C₂²×C₈	—	10	1	no	—
153	C₂²×C₈	—	10	1	no	—
154	C₂²×C₈	C₄×D₅	1	2	yes	—
155	C₄×Q₈	—	5	1	no	—
156	C₄×Q₈	—	5	1	no	—
157	C₄×D₄	—	5	1	no	—
158	C₄×D₄	—	5	1	no	—
159	C₄×D₄	—	10	1	no	—
160	C₄×D₄	—	10	1	no	—
161	#117	—	5	1	no	—
162	#117	—	10	1	no	—
163	C₂×C₄²	—	10	1	no	—
164	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
165	C₄⋊C₈	—	10	2	no	—
166	C₄⋊C₈	—	10	2	no	—
167	C₄⋊C₈	C₄×D₅	1	2	yes	—
168	C₄⋊C₈	C₄×D₅	1	2	yes	—
169	C₂²⋊C₈	—	2	1	no	—
170	C₂²⋊C₈	—	2	1	no	—
171	C₂²⋊C₈	—	10	2	no	—
172	C₂²⋊C₈	—	10	2	no	—
173	#97	—	10	1	no	—
174	C₄×C₈	—	2	1	no	—
175	C₄×C₈	—	2	1	no	—
176	C₄×C₈	—	5	2	no	—
177	C₄×C₈	—	5	2	no	—
178	C₄×C₈	—	10	1	no	—
179	C₄×C₈	—	10	1	no	—
180	C₄×C₈	—	10	1	no	—
181	C₄×C₈	D₂₀	1	2	yes	—
182	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
183	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
184	C₂²×C₅	—	2	1	no	—
185	C₂²×C₅	—	2	1	no	—
186	C₂²×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
187	C₂²×C₅	#433	1	2	yes	—
188	C₂×D₅	—	4	1	no	—
189	C₂×D₅	—	4	1	no	—
190	C₂×D₅	—	4	1	no	—
191	C₄×C₅	—	2	1	no	—
192	C₄×C₅	—	2	1	no	—
193	C₄×C₅	#405	1	2	yes	—
194	C₄×C₅	#408	1	2	yes	—
195	C₄×C₅	#443	1	2	yes	—
196	C₄×C₅	#444	1	2	yes	—
197	Dic₅	—	4	1	no	—
198	Dic₅	—	4	1	no	—
199	Dic₅	—	4	1	no	—
200	#41	—	10	4	no	—
201	C₂×Q₈	—	5	1	no	—
202	C₂×D₄	—	5	1	no	—
203	C₂×D₄	—	10	1	no	—
204	C₂²×C₄	—	10	1	no	—
205	C₂²×C₄	—	10	1	no	—
206	C₂²×C₄	—	10	1	no	—
207	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
208	C₂²×C₄	#709	1	1	yes	characteristic
209	C₂²×C₄	#709	1	1	yes	characteristic
210	M₁₆	—	10	4	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	10	4	no	—
215	C₂×C₈	—	10	2	no	—
216	C₂×C₈	—	10	2	no	—
217	C₂×C₈	—	10	2	no	—
218	C₂×C₈	—	20	2	no	—
219	C₂×C₈	#685	1	2	yes	—
220	C₂×C₈	#685	1	2	yes	—
221	C₄⋊C₄	—	5	1	no	—
222	C₄⋊C₄	—	5	1	no	—
223	C₄⋊C₄	—	10	1	no	—
224	C₄⋊C₄	—	10	1	no	—
225	C₂²⋊C₄	—	10	1	no	—
226	C₂²⋊C₄	—	10	1	no	—
227	C₂²⋊C₄	—	10	1	no	—
228	C₄²	—	10	1	no	—
229	C₄²	—	10	1	no	—
230	C₄²	—	10	1	no	—
231	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
232	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
233	C₄²	C₂×D₂₀	1	1	yes	characteristic
234	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
235	C₂×C₅	—	2	2	no	—
236	C₂×C₅	#1750	1	1	yes	characteristic
237	C₂×C₅	#1764	1	1	yes	characteristic
238	C₂×C₅	#1765	1	1	yes	characteristic
239	D₅	—	8	1	no	—
240	C₂³	—	10	1	no	—
241	C₂³	#4018	1	1	yes	characteristic
242	Q₈	—	10	2	no	—
243	D₄	—	10	2	no	—
244	D₄	—	20	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	10	1	no	—
248	C₂×C₄	—	10	1	no	—
249	C₂×C₄	—	10	1	no	—
250	C₂×C₄	—	20	2	no	—
251	C₂×C₄	—	20	2	no	—
252	C₂×C₄	—	20	2	no	—
253	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
254	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
255	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
256	C₂×C₄	#4047	1	4	yes	—
257	C₂×C₄	C₂×#685	1	1	yes	characteristic
258	C₂×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
259	C₂×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
260	C₈	—	2	2	no	—
261	C₈	—	2	2	no	—
262	C₈	—	4	2	no	—
263	C₈	—	4	2	no	—
264	C₈	—	10	4	no	—
265	C₈	—	20	2	no	—
266	C₈	—	20	2	no	—
267	C₅	#8189	1	1	yes	characteristic, complemented, Sylow
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	20	1	no	—
271	C₂²	—	20	1	no	—
272	C₂²	—	20	1	no	—
273	C₂²	#64979	1	2	yes	—
274	C₂²	C₄×#685	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	20	1	no	—
278	C₄	—	20	1	no	—
279	C₄	—	20	1	no	—
280	C₄	#64978	1	2	yes	—
281	C₄	#65399	1	2	yes	—
282	C₄	#65433	1	2	yes	—
283	C₄	#65434	1	2	yes	—
284	C₂	—	2	2	no	—
285	C₂	—	40	1	no	—
286	C₂	#10600546	1	1	yes	characteristic
287	C₂	#10600680	1	1	yes	characteristic
288	C₂	#10600681	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×C₂²⋊D₅
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1450	C₂
socle	C₂²×C₅	C₄×C₂²⋊C₄
prime radical	C₂×#709	C₂×C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1450	D₅
hypocenter	C₅	#8189
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
2. C₅×#1450	C₂	semidirect
5. C₂×#64674	C₂	semidirect
6. #10598230	C₂	semidirect
7. #10598230	C₂	semidirect
65. #1450	D₅	semidirect
267. C₅	#8189	semidirect

Maximal subgroups

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

subgroup	index
#10610186	2
C₅×#1450	2
#10601547	2
C₂×#64674	2
#10598230	2
#8189	5