Subgroups — #12097519

Order 1280.

1078 conjugacy classes of subgroups.

The subgroup lattice has 278 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	C₄×#4072	C₂	1	1	yes	maximal, characteristic
3	C₂×#65214	C₂	1	1	yes	maximal, characteristic
4	#10602760	C₂	1	4	yes	maximal
5	#10600550	C₂	1	4	yes	maximal, complemented
6	#10600543	C₂	1	2	yes	maximal, complemented
7	C₄×#3947	C₂	1	2	yes	maximal, complemented
8	C₂³×C₅×C₈	C₂²	1	2	yes	—
9	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂×#4072	C₄	1	2	yes	—
11	C₂×C₄×C₅×C₈	C₄	1	8	yes	complemented
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
15	C₄×C₂²⋊Dic₅	C₄	1	2	yes	—
16	C₄×C₂²⋊Dic₅	C₂²	1	2	yes	—
17	C₂×C₂²⋊#168	C₂²	1	2	yes	—
18	C₂×#3947	C₂²	1	8	yes	—
19	C₄×C₄⋊Dic₅	C₄	1	2	yes	—
20	C₄×C₄⋊Dic₅	C₂²	1	2	yes	—
21	#65221	C₄	1	2	yes	—
22	C₄²×Dic₅	C₂²	1	2	yes	—
23	#65214	C₄	1	8	yes	complemented
24	C₂×C₄⋊#168	C₂²	1	2	yes	—
25	C₂×C₄×#168	C₂²	1	2	yes	—
26	#64770	C₂²	1	8	yes	—
27	#20473	—	5	1	no	maximal, Sylow
28	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂²×C₅×C₈	—	2	8	no	—
32	C₂²×C₅×C₈	C₂×C₄	1	8	yes	—
33	C₂²×C₅×C₈	C₂³	1	2	yes	—
34	C₂²×C₅×C₈	C₂³	1	2	yes	—
35	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
41	C₂×C₂²⋊Dic₅	C₂×C₄	1	4	yes	—
42	#4072	—	2	8	no	—
43	C₂×C₄⋊Dic₅	C₂×C₄	1	4	yes	—
44	C₂×C₄×Dic₅	C₂×C₄	1	4	yes	—
45	C₂×C₄×Dic₅	C₂×C₄	1	4	yes	—
46	C₂×C₄×Dic₅	C₂³	1	4	yes	—
47	C₂²×#168	C₂³	1	4	yes	—
48	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
49	C₄×C₅×C₈	C₂×C₄	1	8	yes	—
50	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
51	#3963	C₂×C₄	1	4	yes	—
52	#3963	C₂×C₄	1	4	yes	—
53	C₂²⋊#168	C₂×C₄	1	8	yes	—
54	#3947	—	2	16	no	—
55	C₄⋊#168	C₂×C₄	1	8	yes	—
56	C₄×#168	C₂×C₄	1	8	yes	—
57	C₂×#431	—	5	1	no	—
58	C₂²×C₄×C₈	D₅	1	1	yes	characteristic
59	C₄×#117	—	5	1	no	—
60	#1770	—	5	4	no	—
61	#1752	—	5	4	no	—
62	#1744	—	5	2	no	—
63	C₄×#97	—	5	2	no	—
64	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	—	2	4	no	—
66	C₂²×C₄×C₅	—	2	4	no	—
67	C₂²×C₄×C₅	C₄²	1	4	yes	—
68	C₂²×C₄×C₅	C₄²	1	4	yes	—
69	C₂²×C₄×C₅	M₁₆	1	8	yes	—
70	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
78	C₂²×Dic₅	C₂²×C₄	1	8	yes	—
79	C₂×C₅×C₈	—	2	4	no	—
80	C₂×C₅×C₈	—	2	16	no	—
81	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
82	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
83	C₂×C₅×C₈	#41	1	8	yes	—
84	C₄²×C₅	C₄²	1	4	yes	—
85	C₄²×C₅	C₄²	1	4	yes	—
86	C₄²×C₅	C₂²×C₄	1	2	yes	—
87	C₄²×C₅	C₂²×C₄	1	2	yes	—
88	C₄²×C₅	C₂²×C₄	1	2	yes	—
89	C₄²×C₅	C₂²×C₄	1	2	yes	—
90	C₂²⋊Dic₅	—	2	8	no	—
91	C₄⋊Dic₅	—	2	8	no	—
92	C₄×Dic₅	—	2	8	no	—
93	C₄×Dic₅	—	2	8	no	—
94	C₄×Dic₅	—	2	8	no	—
95	C₂×#168	—	2	8	no	—
96	C₂×#168	C₂²×C₄	1	8	yes	—
97	C₂³×C₈	C₂×D₅	1	2	yes	—
98	C₂×#117	—	5	2	no	—
99	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
100	#431	—	5	8	no	—
101	C₂×C₄⋊C₈	—	5	2	no	—
102	C₂×C₂²⋊C₈	—	5	2	no	—
103	C₂×#97	—	5	8	no	—
104	C₂×C₄×C₈	—	5	2	no	—
105	C₂×C₄×C₈	Dic₅	1	8	yes	complemented
106	C₂×C₄×C₈	C₂×D₅	1	2	yes	—
107	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
108	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
109	C₄×C₄⋊C₄	—	5	2	no	—
110	C₄×C₄⋊C₄	—	5	2	no	—
111	C₄×C₂²⋊C₄	—	5	2	no	—
112	C₄×C₂²⋊C₄	—	5	2	no	—
113	#376	—	5	2	no	—
114	C₄³	—	5	2	no	—
115	#336	—	5	8	no	—
116	C₂³×C₅	—	2	2	no	—
117	C₂³×C₅	—	2	2	no	—
118	C₂³×C₅	C₂×C₄²	1	2	yes	—
119	C₂³×C₅	C₂×M₁₆	1	4	yes	—
120	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₅	—	2	2	no	—
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	—	2	4	no	—
124	C₂×C₄×C₅	—	2	4	no	—
125	C₂×C₄×C₅	—	2	8	no	—
126	C₂×C₄×C₅	#97	1	16	yes	complemented
127	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
128	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
129	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
130	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
131	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
132	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
133	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
134	C₂×C₄×C₅	#131	1	4	yes	—
135	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂×#41	1	4	yes	—
138	C₂×Dic₅	—	2	8	no	—
139	C₂×Dic₅	—	2	8	no	—
140	C₂×Dic₅	—	2	8	no	—
141	C₅×C₈	—	2	4	no	—
142	C₅×C₈	#117	1	8	yes	complemented
143	#168	—	2	8	no	—
144	C₂³×C₄	C₄×D₅	1	1	yes	characteristic
145	C₂³×C₄	C₄×D₅	1	1	yes	characteristic
146	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
147	C₂²×C₈	—	2	8	no	—
148	C₂²×C₈	—	5	4	no	—
149	C₂²×C₈	C₂×Dic₅	1	8	yes	—
150	C₂²×C₈	C₂²×D₅	1	2	yes	—
151	C₂²×C₈	C₂²×D₅	1	2	yes	—
152	#117	—	10	8	no	—
153	C₂×C₄⋊C₄	—	5	4	no	—
154	C₂×C₂²⋊C₄	—	5	4	no	—
155	C₂×C₄²	—	5	4	no	—
156	C₂×C₄²	—	5	4	no	—
157	C₂×C₄²	—	5	4	no	—
158	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
159	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
160	C₂×C₄²	C₄×D₅	1	4	yes	—
161	C₂×C₄²	C₂×Dic₅	1	4	yes	—
162	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
163	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
164	C₄⋊C₈	—	5	8	no	—
165	C₂²⋊C₈	—	5	8	no	—
166	#97	—	10	16	no	—
167	C₄×C₈	—	5	8	no	—
168	C₄×C₈	C₂×Dic₅	1	8	yes	—
169	C₄×C₈	C₂×Dic₅	1	4	yes	—
170	C₄×C₈	C₂×Dic₅	1	4	yes	—
171	#95	—	5	4	no	—
172	#95	—	5	4	no	—
173	C₂²×C₅	—	2	2	no	—
174	C₂²×C₅	—	2	8	no	—
175	C₂²×C₅	—	2	2	no	—
176	C₂²×C₅	C₂×#97	1	4	yes	—
177	C₂²×C₅	C₂²×C₄²	1	1	yes	characteristic
178	C₂²×C₅	C₄×#41	1	2	yes	—
179	C₂²×C₅	C₂²×M₁₆	1	2	yes	—
180	C₂²×C₅	C₂×#131	1	2	yes	—
181	C₄×C₅	—	2	2	no	—
182	C₄×C₅	—	2	2	no	—
183	C₄×C₅	C₂×#97	1	4	yes	—
184	C₄×C₅	C₂×#97	1	4	yes	—
185	C₄×C₅	#405	1	4	yes	—
186	C₄×C₅	C₂×#117	1	2	yes	—
187	C₄×C₅	C₄×#41	1	2	yes	—
188	Dic₅	—	4	8	no	—
189	C₂⁴	C₂×C₄×D₅	1	1	yes	characteristic
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	5	8	no	—
193	C₂²×C₄	#676	1	8	yes	—
194	C₂²×C₄	C₄×Dic₅	1	4	yes	—
195	C₂²×C₄	C₄×Dic₅	1	4	yes	—
196	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
197	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
202	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
203	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	16	no	—
206	C₂×C₈	—	5	8	no	—
207	C₂×C₈	—	10	8	no	—
208	C₂×C₈	#709	1	8	yes	—
209	C₂×C₈	C₂²×Dic₅	1	4	yes	—
210	C₂×C₈	C₂²×Dic₅	1	4	yes	—
211	C₄⋊C₄	—	10	8	no	—
212	C₂²⋊C₄	—	10	8	no	—
213	C₄²	—	10	8	no	—
214	C₄²	—	10	8	no	—
215	C₄²	—	10	8	no	—
216	C₄²	C₄×Dic₅	1	4	yes	—
217	C₄²	C₄×Dic₅	1	4	yes	—
218	C₄²	C₂×C₄×D₅	1	2	yes	—
219	C₄²	C₂×C₄×D₅	1	2	yes	—
220	C₄²	C₂²×Dic₅	1	2	yes	—
221	C₄²	C₂²×Dic₅	1	2	yes	—
222	C₂×C₅	—	2	4	no	—
223	C₂×C₅	C₄×#117	1	1	yes	characteristic
224	C₂×C₅	C₂²×#97	1	1	yes	characteristic
225	C₂×C₅	C₂×#405	1	1	yes	characteristic
226	C₂×C₅	#2965	1	4	yes	—
227	C₂³	—	2	2	no	—
228	C₂³	—	2	2	no	—
229	C₂³	C₂×#676	1	4	yes	—
230	C₂³	C₂×C₄×Dic₅	1	2	yes	—
231	C₂³	C₂²×C₄×D₅	1	1	yes	characteristic
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	8	no	—
237	C₂×C₄	—	10	8	no	—
238	C₂×C₄	—	10	8	no	—
239	C₂×C₄	—	10	8	no	—
240	C₂×C₄	#3947	1	16	yes	complemented
241	C₂×C₄	C₂×#676	1	4	yes	—
242	C₂×C₄	C₂×#676	1	4	yes	—
243	C₂×C₄	#4047	1	4	yes	—
244	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
245	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
246	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
247	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
248	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
249	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
250	C₂×C₄	C₂×#709	1	4	yes	—
251	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
252	C₈	—	2	4	no	—
253	C₈	—	10	8	no	—
254	C₈	#4072	1	8	yes	complemented
255	C₅	#20473	1	1	yes	characteristic, complemented, Sylow
256	C₂²	—	2	8	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	C₂×#3947	1	4	yes	—
260	C₂²	C₄×#709	1	2	yes	—
261	C₂²	C₂²×#676	1	2	yes	—
262	C₂²	C₂×#4047	1	2	yes	—
263	C₂²	C₂²×C₄×Dic₅	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	20	8	no	—
267	C₄	C₂×#3947	1	4	yes	—
268	C₄	C₂×#3947	1	4	yes	—
269	C₄	#65390	1	4	yes	—
270	C₄	C₄×#709	1	2	yes	—
271	C₄	C₂×#4072	1	2	yes	—
272	C₂	—	2	4	no	—
273	C₂	#10610178	1	4	yes	—
274	C₂	C₄×#4072	1	1	yes	characteristic
275	C₂	C₂²×#3947	1	1	yes	characteristic
276	C₂	C₂×#65390	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂²×D₅
commutator	C₂²×C₅	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₂²×C₄×C₅×C₈	C₂
socle	C₂³×C₅	C₂³×C₄
prime radical	C₂⁴×C₅	C₂²×C₄
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	C₂²×C₄×C₈	D₅
hypocenter	C₅	#20473
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
5. #10600550	C₂	semidirect
6. #10600543	C₂	semidirect
7. C₄×#3947	C₂	semidirect
11. C₂×C₄×C₅×C₈	C₄	semidirect
23. #65214	C₄	semidirect
105. C₂×C₄×C₈	Dic₅	semidirect
126. C₂×C₄×C₅	#97	semidirect
142. C₅×C₈	#117	semidirect
240. C₂×C₄	#3947	semidirect
254. C₈	#4072	semidirect
255. C₅	#20473	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
C₄×#4072	2
C₂×#65214	2
#10602760	2
#10600550	2
#10600543	2
C₄×#3947	2
#20473	5