Subgroups — #12252381

Order 1280.

682 conjugacy classes of subgroups.

The subgroup lattice has 304 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	C₅×#2150	C₂	1	1	yes	maximal, characteristic, complemented
3	#10602224	C₂	1	2	yes	maximal
4	Dic₅×M₃₂	C₂	1	2	yes	maximal, complemented
5	#10600536	C₂	1	1	yes	maximal, characteristic, complemented
6	#10600068	C₂	1	4	yes	maximal, complemented
7	#10600067	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×C₄⋊C₁₆	C₂²	1	4	yes	complemented
22	#64727	C₂²	1	8	yes	—
23	#64722	C₂²	1	4	yes	complemented
24	C₁₆×Dic₅	C₂²	1	4	yes	complemented
25	C₄⋊#672	C₂²	1	4	yes	complemented
26	#33418	—	5	1	no	maximal, Sylow
27	C₂²×C₄×D₅	C₈	1	2	yes	—
28	C₂²×C₄×D₅	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₅	C₈	1	2	yes	—
33	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
34	C₂²×#168	C₂×C₄	1	1	yes	characteristic
35	C₂²×#168	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×D₅	—	2	2	no	—
37	C₂×C₈×D₅	—	2	2	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	2	yes	—
50	C₄×C₅×C₈	C₂³	1	2	yes	—
51	C₈×Dic₅	C₂×C₄	1	2	yes	—
52	C₈×Dic₅	C₂×C₄	1	2	yes	—
53	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
54	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
55	C₈×Dic₅	C₂³	1	1	yes	characteristic
56	C₈×Dic₅	C₂³	1	1	yes	characteristic
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	1	no	—
65	#2150	D₅	1	1	yes	characteristic, complemented
66	#2112	—	5	2	no	—
67	C₄×M₃₂	—	5	2	no	—
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	—	2	2	no	—
83	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
84	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₅×C₈	#41	1	2	yes	—
86	C₂×C₅×C₈	#41	1	2	yes	—
87	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
88	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
89	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
90	C₄²×C₅	C₂²×C₄	1	2	yes	—
91	C₄×Dic₅	C₂×C₈	1	2	yes	—
92	C₄×Dic₅	C₂×C₈	1	2	yes	—
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	C₂²×C₄	1	1	yes	characteristic
97	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
98	C₂×#168	—	2	2	no	—
99	C₂×#168	M₁₆	1	4	yes	—
100	C₂×#168	C₂²×C₄	1	2	yes	—
101	C₂×#168	C₂²×C₄	1	1	yes	characteristic
102	C₂×#168	C₂²×C₄	1	1	yes	characteristic
103	C₈×D₅	—	2	4	no	—
104	C₈×D₅	—	2	2	no	—
105	C₈×D₅	—	2	2	no	—
106	C₈×D₅	—	2	4	no	—
107	C₈×D₅	—	2	4	no	—
108	C₅×C₁₆	—	2	4	no	—
109	#672	—	2	4	no	—
110	C₂³×C₈	—	5	1	no	—
111	C₂³×C₈	—	5	1	no	—
112	C₂²×C₄²	—	5	1	no	—
113	C₂×M₃₂	—	5	2	no	—
114	C₂×M₃₂	C₂×D₅	1	2	yes	—
115	C₂×C₄×C₈	—	5	1	no	—
116	C₂×C₄×C₈	—	5	1	no	—
117	C₂×C₄×C₈	—	5	1	no	—
118	C₂×C₄×C₈	—	5	2	no	—
119	C₂×C₄×C₈	—	5	2	no	—
120	C₂×C₄×C₈	—	5	2	no	—
121	C₂×C₄×C₈	—	5	2	no	—
122	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
123	C₄⋊C₁₆	—	5	4	no	—
124	C₄⋊C₁₆	C₂×D₅	1	4	yes	complemented
125	C₂²⋊C₁₆	—	5	8	no	—
126	#346	—	5	4	no	—
127	C₄×C₁₆	—	5	4	no	—
128	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
129	C₂²×D₅	—	2	2	no	—
130	C₂²×D₅	—	2	4	no	—
131	C₂²×D₅	C₂²×C₈	1	2	yes	—
132	C₂×C₄×C₅	—	2	4	no	—
133	C₂×C₄×C₅	#117	1	2	yes	—
134	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
135	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
137	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
138	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
139	C₂×Dic₅	—	2	2	no	—
140	C₂×Dic₅	—	2	4	no	—
141	C₂×Dic₅	C₂²×C₈	1	2	yes	—
142	C₂×Dic₅	C₂²×C₈	1	2	yes	—
143	C₂×Dic₅	C₂²×C₈	1	2	yes	—
144	C₄×D₅	—	2	4	no	—
145	C₄×D₅	—	2	2	no	—
146	C₄×D₅	—	2	2	no	—
147	C₄×D₅	—	2	4	no	—
148	C₄×D₅	—	2	4	no	—
149	C₄×D₅	—	2	4	no	—
150	C₄×D₅	M₃₂	1	8	yes	complemented
151	C₅×C₈	—	2	2	no	—
152	C₅×C₈	#117	1	2	yes	—
153	C₅×C₈	C₂×#41	1	1	yes	characteristic
154	C₅×C₈	C₂×#41	1	1	yes	characteristic
155	#168	—	2	2	no	—
156	#168	C₂×M₁₆	1	2	yes	—
157	#168	C₂×M₁₆	1	2	yes	—
158	C₂³×C₄	—	5	2	no	—
159	C₂³×C₄	—	5	1	no	—
160	C₂²×C₈	—	5	1	no	—
161	C₂²×C₈	—	5	1	no	—
162	C₂²×C₈	—	5	2	no	—
163	C₂²×C₈	—	5	2	no	—
164	C₂²×C₈	—	5	2	no	—
165	C₂²×C₈	—	5	2	no	—
166	C₂²×C₈	—	10	4	no	—
167	C₂²×C₈	—	10	2	no	—
168	C₂²×C₈	—	10	2	no	—
169	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
170	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
171	C₂×C₄²	—	5	2	no	—
172	C₂×C₄²	—	5	1	no	—
173	C₂×C₄²	—	5	2	no	—
174	C₂×C₄²	—	5	2	no	—
175	C₂×C₄²	—	5	2	no	—
176	C₂×C₄²	—	5	2	no	—
177	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
178	M₃₂	—	2	4	no	—
179	M₃₂	—	10	4	no	—
180	C₂×C₁₆	—	5	4	no	—
181	C₂×C₁₆	C₂²×D₅	1	4	yes	—
182	C₄×C₈	—	5	1	no	—
183	C₄×C₈	—	5	1	no	—
184	C₄×C₈	—	5	2	no	—
185	C₄×C₈	—	5	2	no	—
186	C₄×C₈	—	5	1	no	—
187	C₄×C₈	—	5	1	no	—
188	C₄×C₈	—	5	2	no	—
189	C₄×C₈	—	5	2	no	—
190	C₄×C₈	C₄×D₅	1	2	yes	—
191	C₄×C₈	C₂²×D₅	1	2	yes	—
192	C₂²×C₅	—	2	2	no	—
193	C₂²×C₅	#431	1	2	yes	—
194	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
195	C₂×D₅	—	2	8	no	—
196	C₂×D₅	—	2	2	no	—
197	C₂×D₅	—	2	2	no	—
198	C₂×D₅	C₂×M₃₂	1	4	yes	—
199	C₄×C₅	—	2	2	no	—
200	C₄×C₅	#431	1	2	yes	—
201	C₄×C₅	C₂×M₃₂	1	4	yes	—
202	C₄×C₅	C₂×#117	1	1	yes	characteristic
203	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
204	Dic₅	—	2	2	no	—
205	Dic₅	—	2	2	no	—
206	Dic₅	C₂×M₃₂	1	4	yes	—
207	C₂⁴	—	5	1	no	—
208	C₂²×C₄	—	5	2	no	—
209	C₂²×C₄	—	5	2	no	—
210	C₂²×C₄	—	5	1	no	—
211	C₂²×C₄	—	5	2	no	—
212	C₂²×C₄	—	5	2	no	—
213	C₂²×C₄	—	5	2	no	—
214	C₂²×C₄	—	5	2	no	—
215	C₂²×C₄	—	5	2	no	—
216	C₂²×C₄	—	10	8	no	—
217	C₂²×C₄	—	10	2	no	—
218	C₂²×C₄	—	10	2	no	—
219	C₂²×C₄	C₈×D₅	1	2	yes	—
220	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	5	1	no	—
223	C₂×C₈	—	5	1	no	—
224	C₂×C₈	—	5	4	no	—
225	C₂×C₈	—	5	2	no	—
226	C₂×C₈	—	10	4	no	—
227	C₂×C₈	—	10	4	no	—
228	C₂×C₈	—	10	2	no	—
229	C₂×C₈	—	10	2	no	—
230	C₂×C₈	—	10	2	no	—
231	C₂×C₈	—	10	4	no	—
232	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
233	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
234	C₂×C₈	#711	1	2	yes	—
235	C₂×C₈	#713	1	2	yes	—
236	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
237	C₄²	—	5	1	no	—
238	C₄²	—	5	1	no	—
239	C₄²	—	5	2	no	—
240	C₄²	—	5	2	no	—
241	C₄²	—	5	2	no	—
242	C₄²	—	5	2	no	—
243	C₄²	—	5	2	no	—
244	C₄²	C₈×D₅	1	1	yes	characteristic
245	C₄²	C₈×D₅	1	1	yes	characteristic
246	C₄²	C₂×C₄×D₅	1	2	yes	—
247	C₁₆	—	2	4	no	—
248	C₁₆	—	10	4	no	—
249	C₂×C₅	—	2	2	no	—
250	C₂×C₅	C₂×#431	1	1	yes	characteristic
251	C₂×C₅	C₂²×M₃₂	1	2	yes	—
252	D₅	—	2	4	no	—
253	C₂³	—	5	2	no	—
254	C₂³	—	10	4	no	—
255	C₂³	—	10	2	no	—
256	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	5	2	no	—
259	C₂×C₄	—	5	8	no	—
260	C₂×C₄	—	5	2	no	—
261	C₂×C₄	—	5	2	no	—
262	C₂×C₄	—	10	4	no	—
263	C₂×C₄	—	10	4	no	—
264	C₂×C₄	—	10	4	no	—
265	C₂×C₄	—	10	2	no	—
266	C₂×C₄	—	10	2	no	—
267	C₂×C₄	—	10	2	no	—
268	C₂×C₄	—	10	4	no	—
269	C₂×C₄	—	10	4	no	—
270	C₂×C₄	#4038	1	2	yes	—
271	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
272	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
273	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
274	C₂×C₄	D₅×M₁₆	1	2	yes	—
275	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	—	5	2	no	—
278	C₈	—	5	2	no	—
279	C₈	—	10	2	no	—
280	C₈	#4038	1	2	yes	—
281	C₈	C₂×#711	1	1	yes	characteristic
282	C₈	C₂×#713	1	1	yes	characteristic
283	C₅	#33418	1	1	yes	characteristic, complemented, Sylow
284	C₂²	—	2	2	no	—
285	C₂²	—	5	4	no	—
286	C₂²	—	10	8	no	—
287	C₂²	—	10	2	no	—
288	C₂²	—	10	2	no	—
289	C₂²	#65122	1	2	yes	—
290	C₂²	C₂²×C₈×D₅	1	1	yes	characteristic
291	C₄	—	2	2	no	—
292	C₄	—	5	4	no	—
293	C₄	—	10	2	no	—
294	C₄	—	10	2	no	—
295	C₄	#65122	1	2	yes	—
296	C₄	D₅×M₃₂	1	4	yes	—
297	C₄	C₂×#4038	1	1	yes	characteristic
298	C₄	C₂×D₅×M₁₆	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	10	4	no	—
301	C₂	C₂×#65122	1	1	yes	characteristic
302	C₂	C₂×D₅×M₃₂	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₈
Fitting	C₅×#2150	C₂
prime radical	C₂³×D₅	C₂×C₈
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	#2150	D₅
hypocenter	C₅	#33418
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
2. C₅×#2150	C₂	semidirect
4. Dic₅×M₃₂	C₂	semidirect
5. #10600536	C₂	semidirect
6. #10600068	C₂	semidirect
7. #10600067	C₂	semidirect
21. C₅×C₄⋊C₁₆	C₂²	semidirect
23. #64722	C₂²	semidirect
24. C₁₆×Dic₅	C₂²	semidirect
25. C₄⋊#672	C₂²	semidirect
65. #2150	D₅	semidirect
124. C₄⋊C₁₆	C₂×D₅	semidirect
150. C₄×D₅	M₃₂	semidirect
283. 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
C₅×#2150	2
#10602224	2
Dic₅×M₃₂	2
#10600536	2
#10600068	2
#10600067	2
#33418	5