Subgroups — #11992574

Order 1280.

414 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2931	C₂	1	1	yes	maximal, characteristic
2	#10602599	C₂	1	2	yes	maximal
3	#10600584	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#64678	C₂	1	1	yes	maximal, characteristic, complemented
5	#10598202	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#131	—	2	1	no	—
7	C₅×#532	C₂²	1	1	yes	characteristic
8	C₅×#436	—	2	1	no	—
9	C₅×#436	C₄	1	2	yes	—
10	C₅×C₈×D₄	—	2	1	no	—
11	C₅×C₈×D₄	C₄	1	2	yes	—
12	C₅×#423	C₂²	1	1	yes	characteristic
13	C₅×#408	—	2	1	no	—
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	#65213	C₂²	1	2	yes	—
16	C₂×C₄⋊#168	C₂²	1	2	yes	—
17	#64772	—	2	1	no	—
18	#64772	—	2	1	no	—
19	#64767	—	2	2	no	—
20	#64764	—	2	2	no	—
21	#64678	—	2	1	no	—
22	#64678	—	2	1	no	—
23	#8688	—	5	1	no	maximal, Sylow
24	C₂×C₅×#41	—	2	1	no	—
25	C₅×#131	—	4	4	no	—
26	C₂×C₅×M₁₆	—	2	1	no	—
27	C₂×C₅×M₁₆	—	2	2	no	—
28	C₂²×C₅×C₈	—	2	2	no	—
29	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
30	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
31	C₅×C₄⋊Q₈	D₄	1	1	yes	characteristic
32	C₅×#127	D₄	1	1	yes	characteristic
33	C₅×#124	—	2	1	no	—
34	C₅×#121	—	2	1	no	—
35	C₅×#121	Q₈	1	2	yes	—
36	C₄×C₅×D₄	—	2	1	no	—
37	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
39	C₂×#681	—	2	2	no	—
40	C₂²×#168	—	2	2	no	—
41	C₅×C₄⋊C₈	—	2	1	no	—
42	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
43	C₅×C₂²⋊C₈	—	2	2	no	—
44	C₅×C₂²⋊C₈	—	2	2	no	—
45	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
46	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
47	C₄×C₅×C₈	D₄	1	1	yes	characteristic
48	C₄×C₅×C₈	D₄	1	1	yes	characteristic
49	C₄⋊#168	—	2	2	no	—
50	C₄⋊#168	—	2	2	no	—
51	C₄⋊#168	—	2	2	no	—
52	C₄⋊#168	—	2	2	no	—
53	#2931	D₅	1	1	yes	characteristic
54	#1837	—	5	1	no	—
55	C₂×#335	—	5	1	no	—
56	#1547	—	5	2	no	—
57	#1292	—	5	2	no	—
58	C₅×#41	—	4	2	no	—
59	C₅×#41	—	4	2	no	—
60	C₂×C₅×Q₈	—	2	1	no	—
61	C₂×C₅×D₄	—	2	2	no	—
62	C₂×C₅×D₄	—	2	1	no	—
63	C₂×C₅×D₄	—	2	1	no	—
64	C₂×C₅×D₄	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₅	—	2	2	no	—
66	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₅	#41	1	1	yes	characteristic
69	C₅×M₁₆	—	4	4	no	—
70	C₅×M₁₆	—	4	2	no	—
71	C₂×C₅×C₈	—	2	2	no	—
72	C₂×C₅×C₈	—	2	2	no	—
73	C₂×C₅×C₈	—	2	2	no	—
74	C₂×C₅×C₈	—	4	4	no	—
75	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
79	C₅×C₄⋊C₄	—	2	1	no	—
80	C₅×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
81	C₅×C₂²⋊C₄	—	2	2	no	—
82	C₅×C₂²⋊C₄	—	2	2	no	—
83	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₅	#41	1	1	yes	characteristic
87	#681	—	4	2	no	—
88	#681	—	4	2	no	—
89	C₂×#168	—	4	2	no	—
90	C₂×#168	—	4	2	no	—
91	C₂×#168	—	8	2	no	—
92	C₂×#131	—	2	1	no	—
93	#532	C₂×D₅	1	1	yes	characteristic
94	#436	—	2	1	no	—
95	#436	Dic₅	1	2	yes	—
96	C₈×D₄	—	2	1	no	—
97	C₈×D₄	Dic₅	1	2	yes	—
98	#423	—	5	2	no	—
99	#423	C₂×D₅	1	1	yes	characteristic
100	C₂×C₄⋊C₈	—	5	2	no	—
101	#408	—	2	1	no	—
102	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
103	#341	—	10	1	no	—
104	#341	—	10	1	no	—
105	#335	—	10	1	no	—
106	#335	—	10	1	no	—
107	#332	—	10	2	no	—
108	#331	—	10	2	no	—
109	C₂³×C₅	—	2	2	no	—
110	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
111	C₅×Q₈	—	4	2	no	—
112	C₅×D₄	—	4	2	no	—
113	C₅×D₄	—	4	4	no	—
114	C₅×D₄	—	4	2	no	—
115	C₅×D₄	—	4	1	no	—
116	C₅×D₄	—	4	1	no	—
117	C₂×C₄×C₅	—	2	1	no	—
118	C₂×C₄×C₅	—	2	1	no	—
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	—	4	1	no	—
121	C₂×C₄×C₅	—	4	2	no	—
122	C₂×C₄×C₅	—	4	2	no	—
123	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₅	#117	1	1	yes	characteristic
126	C₂×C₄×C₅	#121	1	1	yes	characteristic
127	C₂×C₄×C₅	#121	1	1	yes	characteristic
128	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
129	C₂×C₄×C₅	#135	1	2	yes	—
130	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
131	C₅×C₈	—	2	1	no	—
132	C₅×C₈	—	2	1	no	—
133	C₅×C₈	—	2	1	no	—
134	C₅×C₈	—	2	1	no	—
135	C₅×C₈	—	4	2	no	—
136	#168	—	8	2	no	—
137	#168	—	8	2	no	—
138	C₂×#41	—	2	1	no	—
139	#131	—	4	4	no	—
140	C₂×M₁₆	—	2	1	no	—
141	C₂×M₁₆	—	2	2	no	—
142	C₂×M₁₆	—	10	2	no	—
143	C₂²×C₈	—	2	2	no	—
144	C₂²×C₈	—	10	2	no	—
145	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
146	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
147	C₄⋊Q₈	D₂₀	1	1	yes	characteristic
148	#127	D₂₀	1	1	yes	characteristic
149	#124	—	2	1	no	—
150	#121	—	2	1	no	—
151	#121	Dic₁₀	1	2	yes	—
152	C₄×D₄	—	2	1	no	—
153	C₄×D₄	C₂×Dic₅	1	2	yes	—
154	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
155	C₄⋊C₈	—	2	1	no	—
156	C₄⋊C₈	—	10	2	no	—
157	C₄⋊C₈	—	10	2	no	—
158	C₄⋊C₈	—	10	2	no	—
159	C₄⋊C₈	—	10	2	no	—
160	C₄⋊C₈	C₂×Dic₅	1	2	yes	—
161	C₂²⋊C₈	—	2	2	no	—
162	C₂²⋊C₈	—	2	2	no	—
163	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
164	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
165	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
166	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
167	C₂²×C₅	—	2	1	no	—
168	C₂²×C₅	—	2	1	no	—
169	C₂²×C₅	—	4	2	no	—
170	C₂²×C₅	—	4	2	no	—
171	C₂²×C₅	—	4	2	no	—
172	C₂²×C₅	#380	1	1	yes	characteristic
173	C₂²×C₅	#478	1	2	yes	—
174	C₄×C₅	—	2	1	no	—
175	C₄×C₅	—	2	1	no	—
176	C₄×C₅	—	2	1	no	—
177	C₄×C₅	—	4	2	no	—
178	C₄×C₅	#416	1	1	yes	characteristic
179	C₄×C₅	#418	1	1	yes	characteristic
180	C₄×C₅	#463	1	2	yes	—
181	C₄×C₅	#472	1	1	yes	characteristic
182	C₄×C₅	#473	1	1	yes	characteristic
183	#41	—	4	2	no	—
184	#41	—	4	2	no	—
185	C₂×Q₈	—	2	1	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	—	2	1	no	—
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	C₄⋊Dic₅	1	2	yes	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
192	C₂²×C₄	#709	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
194	M₁₆	—	4	4	no	—
195	M₁₆	—	4	2	no	—
196	M₁₆	—	20	2	no	—
197	M₁₆	—	20	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	4	4	no	—
202	C₂×C₈	—	20	2	no	—
203	C₂×C₈	—	20	2	no	—
204	C₂×C₈	—	40	2	no	—
205	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
206	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
207	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
208	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	C₄⋊Dic₅	1	2	yes	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₄²	C₂×D₂₀	1	1	yes	characteristic
214	C₄²	#709	1	1	yes	characteristic
215	C₄²	C₂²×Dic₅	1	1	yes	characteristic
216	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
217	C₂×C₅	—	2	2	no	—
218	C₂×C₅	—	8	2	no	—
219	C₂×C₅	#1804	1	1	yes	characteristic
220	C₂×C₅	#1813	1	1	yes	characteristic
221	C₂×C₅	#1816	1	1	yes	characteristic
222	C₂³	—	2	2	no	—
223	C₂³	#4070	1	1	yes	characteristic
224	Q₈	—	4	2	no	—
225	D₄	—	4	2	no	—
226	D₄	—	4	4	no	—
227	D₄	—	4	2	no	—
228	D₄	—	4	1	no	—
229	D₄	—	4	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	#4070	1	1	yes	characteristic
237	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
238	C₂×C₄	#4072	1	1	yes	characteristic
239	C₂×C₄	#4076	1	1	yes	characteristic
240	C₂×C₄	#4076	1	1	yes	characteristic
241	C₂×C₄	#4078	1	2	yes	—
242	C₂×C₄	#4096	1	2	yes	—
243	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	—	2	1	no	—
248	C₈	—	4	2	no	—
249	C₈	—	40	2	no	—
250	C₈	—	40	2	no	—
251	C₅	#8688	1	1	yes	characteristic, complemented, Sylow
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	4	2	no	—
257	C₂²	#65298	1	2	yes	—
258	C₂²	#65501	1	1	yes	characteristic
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	4	2	no	—
263	C₄	#65305	1	2	yes	—
264	C₄	#65431	1	1	yes	characteristic
265	C₄	#65432	1	1	yes	characteristic
266	C₄	#65504	1	1	yes	characteristic
267	C₄	#65522	1	1	yes	characteristic
268	C₂	—	2	2	no	—
269	C₂	—	8	2	no	—
270	C₂	#10602398	1	1	yes	characteristic
271	C₂	#10602807	1	1	yes	characteristic
272	C₂	#10602809	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#65501
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#2931	C₂
socle	C₂²×C₅	#380
prime radical	C₅×#532	C₂²
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#2931	D₅
hypocenter	C₅	#8688
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931

Complements

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

N	H ≅ G/N	type
3. #10600584	C₂	semidirect
4. C₂×#64678	C₂	semidirect
5. #10598202	C₂	semidirect
251. C₅	#8688	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2931	2
#10602599	2
#10600584	2
C₂×#64678	2
#10598202	2
#8688	5