Subgroups — #12008506

Order 1280.

320 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	C₂×C₅×#433	C₂	1	1	yes	maximal, characteristic
2	#10600538	C₂	1	1	yes	maximal, characteristic, complemented
3	#10600537	C₂	1	1	yes	maximal, characteristic, complemented
4	#10598200	C₂	1	2	yes	maximal, complemented
5	#10598164	C₂	1	2	yes	maximal
6	C₂×C₅×#117	C₄	1	1	yes	characteristic
7	C₅×#433	—	2	1	no	—
8	C₅×#433	—	2	1	no	—
9	C₅×#433	—	2	1	no	—
10	C₅×#433	—	2	1	no	—
11	C₂×C₅×C₄⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₅×C₂²⋊C₈	C₄	1	1	yes	characteristic
14	C₂×C₅×C₂²⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₅×#97	C₄	1	1	yes	characteristic
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₂²⋊#672	C₂²	1	2	yes	—
18	C₂²⋊#672	C₂²	1	2	yes	—
19	C₄⋊#672	—	2	1	no	—
20	C₄⋊#672	—	2	1	no	—
21	#64681	—	2	1	no	—
22	C₄×#672	—	2	1	no	—
23	#11461	—	5	1	no	maximal, Sylow
24	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
29	C₅×#117	—	2	1	no	—
30	C₅×#117	—	2	1	no	—
31	C₅×#117	—	2	1	no	—
32	C₅×#117	—	2	1	no	—
33	C₂×C₅×C₄⋊C₄	C₈	1	1	yes	characteristic
34	C₂×C₅×C₄⋊C₄	C₈	1	1	yes	characteristic
35	C₂×C₅×C₂²⋊C₄	C₈	1	1	yes	characteristic
36	C₂×C₅×C₂²⋊C₄	C₈	1	1	yes	characteristic
37	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
39	C₅×C₄⋊C₈	—	2	1	no	—
40	C₅×C₄⋊C₈	—	2	1	no	—
41	C₅×C₄⋊C₈	—	2	1	no	—
42	C₅×C₄⋊C₈	—	2	1	no	—
43	C₅×C₂²⋊C₈	—	2	1	no	—
44	C₅×C₂²⋊C₈	—	2	1	no	—
45	C₅×C₂²⋊C₈	—	2	1	no	—
46	C₅×C₂²⋊C₈	—	2	1	no	—
47	C₅×#97	—	2	1	no	—
48	C₅×#97	—	2	1	no	—
49	C₄×C₅×C₈	D₄	1	1	yes	characteristic
50	C₄×C₅×C₈	D₄	1	1	yes	characteristic
51	C₄×C₅×C₈	D₄	1	1	yes	characteristic
52	C₄×C₅×C₈	D₄	1	1	yes	characteristic
53	C₂×#672	—	2	2	no	—
54	C₂×#672	—	2	2	no	—
55	C₂×#433	D₅	1	1	yes	characteristic
56	#2163	—	5	1	no	—
57	#2162	—	5	1	no	—
58	#1322	—	5	2	no	—
59	#1314	—	5	2	no	—
60	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₅	—	2	1	no	—
62	C₂²×C₄×C₅	—	2	1	no	—
63	C₂²×C₄×C₅	—	4	1	no	—
64	C₂²×C₄×C₅	—	4	1	no	—
65	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₅×C₈	—	2	1	no	—
72	C₂×C₅×C₈	—	2	1	no	—
73	C₂×C₅×C₈	—	4	1	no	—
74	C₂×C₅×C₈	—	4	1	no	—
75	C₂×C₅×C₈	—	4	1	no	—
76	C₂×C₅×C₈	—	4	1	no	—
77	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
80	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
81	C₅×C₄⋊C₄	—	2	1	no	—
82	C₅×C₄⋊C₄	—	2	1	no	—
83	C₅×C₄⋊C₄	—	2	1	no	—
84	C₅×C₄⋊C₄	—	2	1	no	—
85	C₅×C₂²⋊C₄	—	2	1	no	—
86	C₅×C₂²⋊C₄	—	2	1	no	—
87	C₅×C₂²⋊C₄	—	2	1	no	—
88	C₅×C₂²⋊C₄	—	2	1	no	—
89	C₄²×C₅	—	2	1	no	—
90	C₄²×C₅	—	2	1	no	—
91	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
92	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
93	C₄²×C₅	M₁₆	1	1	yes	characteristic
94	C₄²×C₅	M₁₆	1	1	yes	characteristic
95	#672	—	4	2	no	—
96	#672	—	4	2	no	—
97	C₂×#117	Dic₅	1	1	yes	characteristic
98	#433	—	2	1	no	—
99	#433	—	2	1	no	—
100	#433	—	2	1	no	—
101	#433	—	2	1	no	—
102	C₂×C₄⋊C₈	Dic₅	1	1	yes	characteristic
103	C₂×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
104	C₂×C₂²⋊C₈	Dic₅	1	1	yes	characteristic
105	C₂×C₂²⋊C₈	C₂×D₅	1	1	yes	characteristic
106	C₂×#97	Dic₅	1	1	yes	characteristic
107	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
108	C₄⋊C₁₆	—	10	1	no	—
109	C₄⋊C₁₆	—	10	1	no	—
110	C₂²⋊C₁₆	—	5	2	no	—
111	C₂²⋊C₁₆	—	5	2	no	—
112	#346	—	10	1	no	—
113	C₄×C₁₆	—	10	1	no	—
114	C₂³×C₅	—	2	1	no	—
115	C₂³×C₅	—	2	2	no	—
116	C₂³×C₅	—	4	1	no	—
117	C₂³×C₅	—	4	1	no	—
118	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	—	2	1	no	—
121	C₂×C₄×C₅	—	2	1	no	—
122	C₂×C₄×C₅	—	2	1	no	—
123	C₂×C₄×C₅	—	2	1	no	—
124	C₂×C₄×C₅	—	2	1	no	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	—	4	1	no	—
127	C₂×C₄×C₅	—	4	2	no	—
128	C₂×C₄×C₅	—	4	1	no	—
129	C₂×C₄×C₅	—	4	1	no	—
130	C₂×C₄×C₅	—	4	1	no	—
131	C₂×C₄×C₅	—	4	1	no	—
132	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
138	C₅×C₈	—	2	1	no	—
139	C₅×C₈	—	2	1	no	—
140	C₅×C₈	—	2	1	no	—
141	C₅×C₈	—	2	1	no	—
142	C₅×C₈	—	4	1	no	—
143	C₅×C₈	—	4	1	no	—
144	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
145	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
146	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
147	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
148	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
149	#117	—	2	1	no	—
150	#117	—	2	1	no	—
151	#117	—	2	1	no	—
152	#117	—	2	1	no	—
153	C₂×C₄⋊C₄	#168	1	1	yes	characteristic
154	C₂×C₄⋊C₄	#168	1	1	yes	characteristic
155	C₂×C₂²⋊C₄	#168	1	1	yes	characteristic
156	C₂×C₂²⋊C₄	#168	1	1	yes	characteristic
157	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
158	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
159	C₂×C₁₆	—	10	2	no	—
160	C₂×C₁₆	—	10	2	no	—
161	C₄⋊C₈	—	2	1	no	—
162	C₄⋊C₈	—	2	1	no	—
163	C₄⋊C₈	—	2	1	no	—
164	C₄⋊C₈	—	2	1	no	—
165	C₂²⋊C₈	—	2	1	no	—
166	C₂²⋊C₈	—	2	1	no	—
167	C₂²⋊C₈	—	2	1	no	—
168	C₂²⋊C₈	—	2	1	no	—
169	#97	—	2	1	no	—
170	#97	—	2	1	no	—
171	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
172	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
173	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
174	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
175	C₂²×C₅	—	2	1	no	—
176	C₂²×C₅	—	2	1	no	—
177	C₂²×C₅	—	4	1	no	—
178	C₂²×C₅	—	4	1	no	—
179	C₂²×C₅	—	4	1	no	—
180	C₂²×C₅	—	8	2	no	—
181	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
182	C₂²×C₅	#504	1	2	yes	—
183	C₄×C₅	—	2	1	no	—
184	C₄×C₅	—	2	1	no	—
185	C₄×C₅	—	2	1	no	—
186	C₄×C₅	—	2	1	no	—
187	C₄×C₅	—	2	1	no	—
188	C₄×C₅	—	4	1	no	—
189	C₄×C₅	—	4	1	no	—
190	C₄×C₅	—	4	1	no	—
191	C₄×C₅	#410	1	1	yes	characteristic
192	C₄×C₅	#413	1	1	yes	characteristic
193	C₂⁴	C₂×#168	1	1	yes	characteristic
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	4	1	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	C₂×#168	1	1	yes	characteristic
199	C₂²×C₄	C₂×#168	1	1	yes	characteristic
200	C₂²×C₄	C₂×#168	1	1	yes	characteristic
201	C₂²×C₄	C₂×#168	1	1	yes	characteristic
202	C₂²×C₄	C₂×#168	1	1	yes	characteristic
203	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	4	1	no	—
207	C₂×C₈	—	4	1	no	—
208	C₂×C₈	—	4	1	no	—
209	C₂×C₈	—	4	1	no	—
210	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
211	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
212	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
213	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
214	C₄⋊C₄	—	2	1	no	—
215	C₄⋊C₄	—	2	1	no	—
216	C₄⋊C₄	—	2	1	no	—
217	C₄⋊C₄	—	2	1	no	—
218	C₂²⋊C₄	—	2	1	no	—
219	C₂²⋊C₄	—	2	1	no	—
220	C₂²⋊C₄	—	2	1	no	—
221	C₂²⋊C₄	—	2	1	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	2	1	no	—
224	C₄²	#681	1	1	yes	characteristic
225	C₄²	#681	1	1	yes	characteristic
226	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
227	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
228	C₁₆	—	20	2	no	—
229	C₁₆	—	20	2	no	—
230	C₂×C₅	—	2	2	no	—
231	C₂×C₅	—	8	1	no	—
232	C₂×C₅	#1458	1	1	yes	characteristic
233	C₂×C₅	#2113	1	1	yes	characteristic
234	C₂×C₅	#2115	1	1	yes	characteristic
235	C₂³	—	2	1	no	—
236	C₂³	—	2	2	no	—
237	C₂³	—	4	1	no	—
238	C₂³	—	4	1	no	—
239	C₂³	C₂²×#168	1	1	yes	characteristic
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
254	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
255	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
256	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
257	C₂×C₄	C₂×#681	1	1	yes	characteristic
258	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	2	1	no	—
262	C₈	—	2	1	no	—
263	C₈	—	4	1	no	—
264	C₈	—	4	1	no	—
265	C₅	#11461	1	1	yes	characteristic, complemented, Sylow
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	4	1	no	—
269	C₂²	—	4	1	no	—
270	C₂²	—	4	1	no	—
271	C₂²	—	8	2	no	—
272	C₂²	#65429	1	2	yes	—
273	C₂²	C₂×C₂²⋊#168	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	4	1	no	—
280	C₄	—	4	1	no	—
281	C₄	—	4	1	no	—
282	C₄	#65526	1	1	yes	characteristic
283	C₄	#65529	1	1	yes	characteristic
284	C₂	—	2	2	no	—
285	C₂	—	8	1	no	—
286	C₂	#10602364	1	1	yes	characteristic
287	C₂	#10602797	1	1	yes	characteristic
288	C₂	#10602800	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₅
commutator	C₂³×C₅	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂×C₅×#433	C₂
socle	C₂²×C₅	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₅	C₂×C₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×#433	D₅
hypocenter	C₅	#11461
LCS	G ▷ C₂³×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#433

Complements

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

N	H ≅ G/N	type
2. #10600538	C₂	semidirect
3. #10600537	C₂	semidirect
4. #10598200	C₂	semidirect
265. C₅	#11461	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#433	2
#10600538	2
#10600537	2
#10598200	2
#10598164	2
#11461	5