Subgroups — #12252380

Order 1280.

682 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#10602224	C₂	1	2	yes	maximal
4	#10602213	C₂	1	2	yes	maximal, complemented
5	C₄×#3944	C₂	1	1	yes	maximal, characteristic, complemented
6	#10599866	C₂	1	4	yes	maximal, complemented
7	#10599845	C₂	1	4	yes	maximal, complemented
8	C₂²×C₈×D₅	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
10	C₂×C₄²×D₅	C₄	1	1	yes	characteristic
11	C₂×C₅×M₃₂	C₂²	1	2	yes	—
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₅	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
15	C₂×#3944	C₂²	1	2	yes	—
16	C₂×C₄×#168	C₄	1	1	yes	characteristic
17	C₄×C₈×D₅	C₄	1	2	yes	—
18	C₄×C₈×D₅	C₄	1	2	yes	—
19	C₄×C₈×D₅	C₂²	1	2	yes	—
20	C₄×C₈×D₅	C₂²	1	2	yes	—
21	C₅×#346	C₂²	1	2	yes	complemented
22	C₄×C₅×C₁₆	C₂²	1	2	yes	complemented
23	#64727	C₂²	1	8	yes	—
24	#64721	C₂²	1	8	yes	complemented
25	#64681	C₂²	1	2	yes	complemented
26	C₄×#672	C₂²	1	2	yes	complemented
27	#33418	—	5	1	no	maximal, Sylow
28	C₂²×C₄×D₅	C₈	1	2	yes	—
29	C₂²×C₄×D₅	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₅	C₈	1	2	yes	—
34	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
35	C₂²×#168	C₂×C₄	1	1	yes	characteristic
36	C₂²×#168	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×D₅	—	2	4	no	—
38	C₂×C₈×D₅	—	2	4	no	—
39	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₅	C₂³	1	2	yes	—
43	C₄²×D₅	C₈	1	2	yes	—
44	C₄²×D₅	C₈	1	2	yes	—
45	C₄²×D₅	C₂×C₄	1	2	yes	—
46	C₄²×D₅	C₂×C₄	1	2	yes	—
47	C₅×M₃₂	—	2	4	no	—
48	C₂×C₅×C₁₆	C₂³	1	4	yes	—
49	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
50	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
51	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
52	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
53	C₈×Dic₅	C₂×C₄	1	2	yes	—
54	C₈×Dic₅	C₂×C₄	1	2	yes	—
55	C₈×Dic₅	C₂×C₄	1	2	yes	—
56	C₈×Dic₅	C₂³	1	2	yes	—
57	#3944	—	2	4	no	—
58	C₂×#672	C₂³	1	4	yes	—
59	C₄×#168	C₂×C₄	1	2	yes	—
60	C₄×#168	C₂×C₄	1	2	yes	—
61	C₂²×C₄×C₈	—	5	1	no	—
62	#2163	—	5	4	no	—
63	#2162	—	5	4	no	—
64	#2150	—	5	2	no	—
65	#2112	—	5	2	no	—
66	C₄×M₃₂	—	5	1	no	—
67	C₄×M₃₂	D₅	1	1	yes	characteristic, complemented
68	C₂³×D₅	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
70	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂²×Dic₅	C₂×C₈	1	2	yes	—
72	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
73	C₂×C₄×D₅	—	2	2	no	—
74	C₂×C₄×D₅	—	2	2	no	—
75	C₂×C₄×D₅	—	2	8	no	—
76	C₂×C₄×D₅	C₂×C₈	1	2	yes	—
77	C₂×C₄×D₅	C₂×C₈	1	2	yes	—
78	C₂×C₄×D₅	C₂×C₈	1	2	yes	—
79	C₂×C₄×D₅	C₂×C₈	1	2	yes	—
80	C₂×C₄×D₅	C₂×C₈	1	2	yes	—
81	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
82	C₂×C₅×C₈	M₁₆	1	4	yes	—
83	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₅×C₈	#41	1	4	yes	—
87	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
88	C₄²×C₅	C₂×C₈	1	2	yes	—
89	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
91	C₄×Dic₅	C₂×C₈	1	1	yes	characteristic
92	C₄×Dic₅	C₂×C₈	1	1	yes	characteristic
93	C₄×Dic₅	C₂×C₈	1	2	yes	—
94	C₄×Dic₅	C₂×C₈	1	2	yes	—
95	C₄×Dic₅	C₂×C₈	1	2	yes	—
96	C₄×Dic₅	C₂×C₈	1	2	yes	—
97	C₄×Dic₅	C₂²×C₄	1	2	yes	—
98	C₂×#168	—	2	2	no	—
99	C₂×#168	—	2	2	no	—
100	C₂×#168	C₂²×C₄	1	2	yes	—
101	C₂×#168	C₂²×C₄	1	2	yes	—
102	C₈×D₅	—	2	4	no	—
103	C₈×D₅	—	2	4	no	—
104	C₈×D₅	—	2	4	no	—
105	C₈×D₅	—	2	4	no	—
106	C₅×C₁₆	—	2	4	no	—
107	#672	—	2	4	no	—
108	C₂³×C₈	—	5	1	no	—
109	C₂³×C₈	—	5	1	no	—
110	C₂²×C₄²	—	5	1	no	—
111	C₂×M₃₂	—	5	2	no	—
112	C₂×M₃₂	C₂×D₅	1	2	yes	—
113	C₂×C₄×C₈	—	5	1	no	—
114	C₂×C₄×C₈	—	5	1	no	—
115	C₂×C₄×C₈	—	5	1	no	—
116	C₂×C₄×C₈	—	5	2	no	—
117	C₂×C₄×C₈	—	5	2	no	—
118	C₂×C₄×C₈	—	5	2	no	—
119	C₂×C₄×C₈	—	5	2	no	—
120	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
121	C₄⋊C₁₆	—	5	8	no	—
122	C₂²⋊C₁₆	—	5	8	no	—
123	#346	—	5	2	no	—
124	#346	C₂×D₅	1	2	yes	complemented
125	C₄×C₁₆	—	5	2	no	—
126	C₄×C₁₆	C₂×D₅	1	2	yes	complemented
127	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
128	C₂²×D₅	—	2	2	no	—
129	C₂²×D₅	—	2	4	no	—
130	C₂²×D₅	C₂²×C₈	1	2	yes	—
131	C₂×C₄×C₅	—	2	4	no	—
132	C₂×C₄×C₅	#117	1	2	yes	—
133	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
134	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
136	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
137	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
138	C₂×Dic₅	—	2	2	no	—
139	C₂×Dic₅	—	2	4	no	—
140	C₂×Dic₅	C₂²×C₈	1	2	yes	—
141	C₂×Dic₅	C₂²×C₈	1	2	yes	—
142	C₂×Dic₅	C₂²×C₈	1	2	yes	—
143	C₄×D₅	—	2	4	no	—
144	C₄×D₅	—	2	2	no	—
145	C₄×D₅	—	2	2	no	—
146	C₄×D₅	—	2	4	no	—
147	C₄×D₅	—	2	4	no	—
148	C₄×D₅	—	2	4	no	—
149	C₄×D₅	M₃₂	1	8	yes	complemented
150	C₅×C₈	#117	1	2	yes	—
151	C₅×C₈	C₂×M₁₆	1	2	yes	—
152	C₅×C₈	C₂×M₁₆	1	2	yes	—
153	C₅×C₈	C₂×#41	1	2	yes	—
154	#168	—	2	2	no	—
155	#168	—	2	2	no	—
156	C₂³×C₄	—	5	2	no	—
157	C₂³×C₄	—	5	1	no	—
158	C₂²×C₈	—	5	1	no	—
159	C₂²×C₈	—	5	1	no	—
160	C₂²×C₈	—	5	2	no	—
161	C₂²×C₈	—	5	2	no	—
162	C₂²×C₈	—	5	2	no	—
163	C₂²×C₈	—	5	2	no	—
164	C₂²×C₈	—	10	4	no	—
165	C₂²×C₈	—	10	4	no	—
166	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
167	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
168	C₂×C₄²	—	5	2	no	—
169	C₂×C₄²	—	5	1	no	—
170	C₂×C₄²	—	5	2	no	—
171	C₂×C₄²	—	5	2	no	—
172	C₂×C₄²	—	5	2	no	—
173	C₂×C₄²	—	5	2	no	—
174	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
175	M₃₂	—	2	4	no	—
176	M₃₂	—	10	4	no	—
177	C₂×C₁₆	—	5	4	no	—
178	C₂×C₁₆	C₂²×D₅	1	4	yes	—
179	C₄×C₈	—	5	2	no	—
180	C₄×C₈	—	5	2	no	—
181	C₄×C₈	—	5	2	no	—
182	C₄×C₈	—	5	2	no	—
183	C₄×C₈	—	5	2	no	—
184	C₄×C₈	—	5	2	no	—
185	C₄×C₈	C₄×D₅	1	1	yes	characteristic
186	C₄×C₈	C₄×D₅	1	1	yes	characteristic
187	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
188	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
189	C₂²×C₅	—	2	2	no	—
190	C₂²×C₅	#431	1	2	yes	—
191	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
192	C₂×D₅	—	2	8	no	—
193	C₂×D₅	—	2	2	no	—
194	C₂×D₅	—	2	2	no	—
195	C₂×D₅	C₂×M₃₂	1	4	yes	—
196	C₄×C₅	—	2	2	no	—
197	C₄×C₅	#431	1	2	yes	—
198	C₄×C₅	C₂×M₃₂	1	4	yes	—
199	C₄×C₅	C₂×#117	1	1	yes	characteristic
200	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
201	Dic₅	—	2	2	no	—
202	Dic₅	—	2	2	no	—
203	Dic₅	C₂×M₃₂	1	4	yes	—
204	C₂⁴	—	5	1	no	—
205	C₂²×C₄	—	5	2	no	—
206	C₂²×C₄	—	5	2	no	—
207	C₂²×C₄	—	5	1	no	—
208	C₂²×C₄	—	5	2	no	—
209	C₂²×C₄	—	5	2	no	—
210	C₂²×C₄	—	5	2	no	—
211	C₂²×C₄	—	5	2	no	—
212	C₂²×C₄	—	5	2	no	—
213	C₂²×C₄	—	10	8	no	—
214	C₂²×C₄	—	10	2	no	—
215	C₂²×C₄	—	10	2	no	—
216	C₂²×C₄	C₈×D₅	1	2	yes	—
217	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
218	C₂×C₈	—	5	2	no	—
219	C₂×C₈	—	5	2	no	—
220	C₂×C₈	—	10	2	no	—
221	C₂×C₈	—	10	2	no	—
222	C₂×C₈	—	10	4	no	—
223	C₂×C₈	—	10	4	no	—
224	C₂×C₈	—	10	4	no	—
225	C₂×C₈	—	10	4	no	—
226	C₂×C₈	#676	1	4	yes	—
227	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
228	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
229	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
230	C₂×C₈	#709	1	4	yes	—
231	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
232	C₄²	—	5	2	no	—
233	C₄²	—	5	1	no	—
234	C₄²	—	5	2	no	—
235	C₄²	—	5	2	no	—
236	C₄²	—	5	2	no	—
237	C₄²	—	5	2	no	—
238	C₄²	—	5	1	no	—
239	C₄²	C₈×D₅	1	2	yes	—
240	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
241	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
242	C₁₆	—	2	4	no	—
243	C₁₆	—	10	4	no	—
244	C₂×C₅	—	2	2	no	—
245	C₂×C₅	C₂×#431	1	1	yes	characteristic
246	C₂×C₅	C₂²×M₃₂	1	2	yes	—
247	D₅	—	2	4	no	—
248	C₂³	—	5	2	no	—
249	C₂³	—	10	4	no	—
250	C₂³	—	10	2	no	—
251	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	5	2	no	—
254	C₂×C₄	—	5	8	no	—
255	C₂×C₄	—	5	2	no	—
256	C₂×C₄	—	5	2	no	—
257	C₂×C₄	—	10	4	no	—
258	C₂×C₄	—	10	4	no	—
259	C₂×C₄	—	10	4	no	—
260	C₂×C₄	—	10	2	no	—
261	C₂×C₄	—	10	2	no	—
262	C₂×C₄	—	10	2	no	—
263	C₂×C₄	—	10	4	no	—
264	C₂×C₄	—	10	4	no	—
265	C₂×C₄	#4018	1	2	yes	—
266	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
267	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
268	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
269	C₂×C₄	C₂×#676	1	2	yes	—
270	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
271	C₈	—	10	2	no	—
272	C₈	—	10	2	no	—
273	C₈	#4018	1	2	yes	—
274	C₈	C₂×#676	1	2	yes	—
275	C₈	C₂×#676	1	2	yes	—
276	C₈	C₂×#709	1	2	yes	—
277	C₅	#33418	1	1	yes	characteristic, complemented, Sylow
278	C₂²	—	2	2	no	—
279	C₂²	—	5	4	no	—
280	C₂²	—	10	8	no	—
281	C₂²	—	10	2	no	—
282	C₂²	—	10	2	no	—
283	C₂²	#64974	1	2	yes	—
284	C₂²	C₂²×C₈×D₅	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	5	4	no	—
287	C₄	—	10	2	no	—
288	C₄	—	10	2	no	—
289	C₄	#64974	1	2	yes	—
290	C₄	D₅×M₃₂	1	4	yes	—
291	C₄	C₂×#4018	1	1	yes	characteristic
292	C₄	C₂²×#676	1	1	yes	characteristic
293	C₂	—	2	2	no	—
294	C₂	—	10	4	no	—
295	C₂	C₂×#64974	1	1	yes	characteristic
296	C₂	C₂×D₅×M₃₂	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²×D₅
commutator, socle	C₂²×C₅	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₄×C₅×M₃₂	C₂
prime radical	C₂³×D₅	C₂×C₈
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	C₄×M₃₂	D₅
hypocenter	C₅	#33418
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
2. C₄×C₅×M₃₂	C₂	semidirect
4. #10602213	C₂	semidirect
5. C₄×#3944	C₂	semidirect
6. #10599866	C₂	semidirect
7. #10599845	C₂	semidirect
21. C₅×#346	C₂²	semidirect
22. C₄×C₅×C₁₆	C₂²	semidirect
24. #64721	C₂²	semidirect
25. #64681	C₂²	semidirect
26. C₄×#672	C₂²	semidirect
67. C₄×M₃₂	D₅	semidirect
124. #346	C₂×D₅	semidirect
126. C₄×C₁₆	C₂×D₅	semidirect
149. C₄×D₅	M₃₂	semidirect
277. C₅	#33418	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₈×D₅	2
C₄×C₅×M₃₂	2
#10602224	2
#10602213	2
C₄×#3944	2
#10599866	2
#10599845	2
#33418	5