Subgroups — #12252382

Order 1280.

682 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	#10602756	C₂	1	2	yes	maximal
3	#10600538	C₂	1	4	yes	maximal, complemented
4	#10600537	C₂	1	4	yes	maximal, complemented
5	#10600536	C₂	1	2	yes	maximal, complemented
6	C₄×#3944	C₂	1	2	yes	maximal, complemented
7	C₂³×C₅×C₈	C₄	1	1	yes	characteristic
8	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₅	C₄	1	1	yes	characteristic
10	C₂×C₄×C₅×C₈	C₄	1	2	yes	—
11	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
12	C₂×C₄×C₅×C₈	C₄	1	2	yes	—
13	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
17	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
18	C₂×#3944	C₂²	1	4	yes	—
19	C₂²⋊#672	C₂²	1	8	yes	—
20	C₄⋊#672	C₂²	1	8	yes	complemented
21	#64681	C₂²	1	4	yes	complemented
22	C₄×#672	C₂²	1	4	yes	complemented
23	#33418	—	5	1	no	maximal, Sylow
24	C₂³×C₄×C₅	C₈	1	2	yes	—
25	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₅×C₈	—	2	4	no	—
27	C₂²×C₅×C₈	—	2	4	no	—
28	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
32	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
33	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	C₂³	1	2	yes	—
35	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₅	C₈	1	2	yes	—
37	C₂×C₄²×C₅	C₈	1	2	yes	—
38	C₂×C₄²×C₅	C₈	1	2	yes	—
39	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
43	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
44	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
45	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
46	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
47	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
48	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
49	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
50	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
51	C₄×C₅×C₈	C₂³	1	2	yes	—
52	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
53	#3944	—	2	8	no	—
54	C₂×#672	C₂³	1	8	yes	—
55	C₂²×C₄×C₈	D₅	1	1	yes	characteristic
56	#2163	—	5	4	no	—
57	#2162	—	5	4	no	—
58	#2150	—	5	2	no	—
59	#2112	—	5	2	no	—
60	C₄×M₃₂	—	5	2	no	—
61	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₅	—	2	8	no	—
63	C₂²×C₄×C₅	—	2	2	no	—
64	C₂²×C₄×C₅	—	2	2	no	—
65	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
67	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
68	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
69	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
70	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
71	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
72	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
73	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₅×C₈	—	2	2	no	—
76	C₂×C₅×C₈	—	2	2	no	—
77	C₂×C₅×C₈	—	2	4	no	—
78	C₂×C₅×C₈	—	2	4	no	—
79	C₂×C₅×C₈	—	2	4	no	—
80	C₂×C₅×C₈	—	2	4	no	—
81	C₂×C₅×C₈	M₁₆	1	4	yes	—
82	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
83	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
84	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₅×C₈	#41	1	4	yes	—
88	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
89	C₄²×C₅	C₂×C₈	1	2	yes	—
90	C₄²×C₅	C₂×C₈	1	2	yes	—
91	C₄²×C₅	C₂×C₈	1	2	yes	—
92	C₄²×C₅	C₂×C₈	1	2	yes	—
93	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
94	C₄²×C₅	C₂×C₈	1	2	yes	—
95	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
96	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₅	C₂²×C₄	1	2	yes	—
98	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
99	#672	—	2	8	no	—
100	C₂³×C₈	Dic₅	1	1	yes	characteristic
101	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
102	C₂²×C₄²	Dic₅	1	1	yes	characteristic
103	C₂×M₃₂	—	5	4	no	—
104	C₂×C₄×C₈	Dic₅	1	1	yes	characteristic
105	C₂×C₄×C₈	Dic₅	1	2	yes	—
106	C₂×C₄×C₈	Dic₅	1	2	yes	—
107	C₂×C₄×C₈	Dic₅	1	1	yes	characteristic
108	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
109	C₂×C₄×C₈	C₂×D₅	1	2	yes	—
110	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
111	C₂×C₄×C₈	C₂×D₅	1	2	yes	—
112	C₄⋊C₁₆	—	5	8	no	—
113	C₂²⋊C₁₆	—	5	8	no	—
114	#346	—	5	4	no	—
115	C₄×C₁₆	—	5	4	no	—
116	C₂³×C₅	—	2	2	no	—
117	C₂³×C₅	—	2	4	no	—
118	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
119	C₂³×C₅	C₂²×C₈	1	2	yes	—
120	C₂×C₄×C₅	—	2	2	no	—
121	C₂×C₄×C₅	—	2	4	no	—
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	2	4	no	—
125	C₂×C₄×C₅	—	2	4	no	—
126	C₂×C₄×C₅	—	2	4	no	—
127	C₂×C₄×C₅	—	2	4	no	—
128	C₂×C₄×C₅	—	2	4	no	—
129	C₂×C₄×C₅	M₃₂	1	8	yes	complemented
130	C₂×C₄×C₅	#117	1	2	yes	—
131	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
132	C₂×C₄×C₅	C₂²×C₈	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	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	C₅×C₈	—	2	2	no	—
140	C₅×C₈	—	2	2	no	—
141	C₅×C₈	#117	1	2	yes	—
142	C₅×C₈	C₂×M₁₆	1	2	yes	—
143	C₅×C₈	C₂×M₁₆	1	2	yes	—
144	C₅×C₈	C₂×#41	1	2	yes	—
145	C₂³×C₄	#168	1	2	yes	—
146	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
147	C₂²×C₈	—	2	4	no	—
148	C₂²×C₈	—	2	4	no	—
149	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
150	C₂²×C₈	C₂×Dic₅	1	2	yes	—
151	C₂²×C₈	C₂×Dic₅	1	2	yes	—
152	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
153	C₂²×C₈	C₂×Dic₅	1	2	yes	—
154	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
155	C₂²×C₈	C₂²×D₅	1	2	yes	—
156	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
157	C₂×C₄²	#168	1	2	yes	—
158	C₂×C₄²	#168	1	2	yes	—
159	C₂×C₄²	#168	1	2	yes	—
160	C₂×C₄²	C₂×Dic₅	1	2	yes	—
161	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
162	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
163	C₂×C₄²	C₂×Dic₅	1	2	yes	—
164	M₃₂	—	10	8	no	—
165	C₂×C₁₆	—	5	8	no	—
166	C₄×C₈	C₂×Dic₅	1	2	yes	—
167	C₄×C₈	C₂×Dic₅	1	2	yes	—
168	C₄×C₈	C₂×Dic₅	1	2	yes	—
169	C₄×C₈	C₂×Dic₅	1	2	yes	—
170	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
171	C₄×C₈	C₂×Dic₅	1	2	yes	—
172	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
173	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
174	C₄×C₈	C₂²×D₅	1	2	yes	—
175	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
176	C₂²×C₅	—	2	8	no	—
177	C₂²×C₅	—	2	2	no	—
178	C₂²×C₅	—	2	2	no	—
179	C₂²×C₅	—	2	2	no	—
180	C₂²×C₅	#431	1	2	yes	—
181	C₂²×C₅	C₂×M₃₂	1	4	yes	—
182	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
183	C₄×C₅	—	2	2	no	—
184	C₄×C₅	—	2	2	no	—
185	C₄×C₅	—	2	2	no	—
186	C₄×C₅	#431	1	2	yes	—
187	C₄×C₅	C₂×M₃₂	1	4	yes	—
188	C₄×C₅	C₂×M₃₂	1	4	yes	—
189	C₄×C₅	C₂×#117	1	1	yes	characteristic
190	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
191	C₂⁴	C₂×#168	1	1	yes	characteristic
192	C₂²×C₄	—	2	8	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	C₂×#168	1	1	yes	characteristic
196	C₂²×C₄	C₂×#168	1	2	yes	—
197	C₂²×C₄	C₂×#168	1	2	yes	—
198	C₂²×C₄	C₂×#168	1	2	yes	—
199	C₂²×C₄	C₂×#168	1	2	yes	—
200	C₂²×C₄	C₂×#168	1	2	yes	—
201	C₂²×C₄	C₂×#168	1	2	yes	—
202	C₂²×C₄	C₂×#168	1	2	yes	—
203	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
204	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	#681	1	4	yes	—
212	C₂×C₈	#709	1	4	yes	—
213	C₂×C₈	C₂²×Dic₅	1	2	yes	—
214	C₂×C₈	C₂²×Dic₅	1	2	yes	—
215	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
216	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
217	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
218	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
219	C₄²	C₂×#168	1	2	yes	—
220	C₄²	C₂×#168	1	2	yes	—
221	C₄²	C₂×#168	1	2	yes	—
222	C₄²	C₂×#168	1	2	yes	—
223	C₄²	C₂×#168	1	1	yes	characteristic
224	C₄²	C₂×#168	1	2	yes	—
225	C₄²	C₂×#168	1	1	yes	characteristic
226	C₄²	C₂²×Dic₅	1	1	yes	characteristic
227	C₄²	C₂²×Dic₅	1	2	yes	—
228	C₄²	C₂²×Dic₅	1	1	yes	characteristic
229	C₁₆	—	10	8	no	—
230	C₂×C₅	—	2	2	no	—
231	C₂×C₅	—	2	4	no	—
232	C₂×C₅	C₂×#431	1	1	yes	characteristic
233	C₂×C₅	C₂²×M₃₂	1	2	yes	—
234	C₂³	—	2	4	no	—
235	C₂³	—	2	2	no	—
236	C₂³	C₂²×#168	1	2	yes	—
237	C₂³	C₂²×#168	1	1	yes	characteristic
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	#3944	1	8	yes	complemented
248	C₂×C₄	C₂²×#168	1	2	yes	—
249	C₂×C₄	C₂²×#168	1	2	yes	—
250	C₂×C₄	C₂²×#168	1	2	yes	—
251	C₂×C₄	C₂²×#168	1	2	yes	—
252	C₂×C₄	C₂²×#168	1	2	yes	—
253	C₂×C₄	C₂²×#168	1	1	yes	characteristic
254	C₂×C₄	C₂×#681	1	2	yes	—
255	C₂×C₄	#4072	1	2	yes	—
256	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	2	2	no	—
259	C₈	C₂×#681	1	2	yes	—
260	C₈	C₂×#681	1	2	yes	—
261	C₈	#4072	1	2	yes	—
262	C₈	C₂×#709	1	2	yes	—
263	C₅	#33418	1	1	yes	characteristic, complemented, Sylow
264	C₂²	—	2	8	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	#65214	1	2	yes	—
269	C₂²	C₂×#3944	1	4	yes	—
270	C₂²	C₂³×#168	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	#65214	1	2	yes	—
275	C₄	C₂×#3944	1	4	yes	—
276	C₄	C₂×#3944	1	4	yes	—
277	C₄	C₂²×#681	1	1	yes	characteristic
278	C₄	C₂×#4072	1	1	yes	characteristic
279	C₂	—	2	4	no	—
280	C₂	—	2	2	no	—
281	C₂	C₂×#65214	1	1	yes	characteristic
282	C₂	C₂²×#3944	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. #10600538	C₂	semidirect
4. #10600537	C₂	semidirect
5. #10600536	C₂	semidirect
6. C₄×#3944	C₂	semidirect
20. C₄⋊#672	C₂²	semidirect
21. #64681	C₂²	semidirect
22. C₄×#672	C₂²	semidirect
129. C₂×C₄×C₅	M₃₂	semidirect
247. C₂×C₄	#3944	semidirect
263. C₅	#33418	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₅×C₈	2
#10602756	2
#10600538	2
#10600537	2
#10600536	2
C₄×#3944	2
#33418	5