Subgroups — #12014385

Order 1280.

486 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₅×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#346	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#64727	C₂	1	1	yes	maximal, characteristic
4	C₂×#64722	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×#685	C₂	1	1	yes	maximal, characteristic
6	#10598295	C₂	1	1	yes	maximal, characteristic, complemented
7	#10598148	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
9	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂×#3952	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
14	C₂²×#672	C₂²	1	1	yes	characteristic
15	C₂²×#672	C₂²	1	1	yes	characteristic
16	C₄×#685	C₄	1	1	yes	characteristic
17	C₂×D₅×C₁₆	—	2	2	no	—
18	C₂×D₅×C₁₆	—	2	2	no	—
19	C₂×D₅×C₁₆	C₄	1	4	yes	complemented
20	C₅×#346	—	2	2	no	—
21	#64770	C₄	1	1	yes	characteristic
22	#64744	C₄	1	1	yes	characteristic
23	#64727	C₄	1	4	yes	complemented
24	#64722	—	2	2	no	—
25	#12611	—	5	1	no	maximal, Sylow
26	C₂²×C₄×D₅	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	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
30	C₂×#685	C₈	1	2	yes	—
31	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
32	C₂²×#168	C₂×C₄	1	1	yes	characteristic
33	C₂²×#168	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×D₅	—	2	1	no	—
35	C₂×C₈×D₅	—	2	1	no	—
36	C₂×C₈×D₅	—	2	1	no	—
37	C₂×C₈×D₅	—	2	1	no	—
38	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
40	C₂×C₅×C₁₆	—	2	1	no	—
41	C₂×C₅×C₁₆	—	2	1	no	—
42	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
43	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
44	C₂×C₅×C₁₆	D₄	1	2	yes	—
45	C₂×C₅×C₁₆	D₄	1	2	yes	—
46	C₄×C₅×C₈	—	2	1	no	—
47	C₄×C₅×C₈	—	2	1	no	—
48	#3963	C₈	1	2	yes	—
49	#3952	C₈	1	4	yes	—
50	C₈×Dic₅	—	2	1	no	—
51	C₈×Dic₅	—	2	1	no	—
52	C₂×#672	—	2	1	no	—
53	C₂×#672	—	2	1	no	—
54	C₂×#672	—	2	1	no	—
55	C₂×#672	—	2	1	no	—
56	C₂×#672	C₂×C₄	1	2	yes	—
57	C₂×#672	C₂×C₄	1	2	yes	—
58	D₅×C₁₆	—	2	8	no	—
59	C₂³×C₁₆	—	5	1	no	—
60	C₂×C₂²⋊C₁₆	—	5	1	no	—
61	C₂×#346	—	5	1	no	—
62	C₂×#346	D₅	1	1	yes	characteristic, complemented
63	C₈×C₂²⋊C₄	—	5	1	no	—
64	#1374	—	5	1	no	—
65	#1374	—	5	1	no	—
66	C₂³×D₅	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
68	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂²×Dic₅	C₂×C₈	1	2	yes	—
70	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
71	C₂×C₄×D₅	—	2	1	no	—
72	C₂×C₄×D₅	—	2	1	no	—
73	C₂×C₄×D₅	—	2	1	no	—
74	C₂×C₄×D₅	—	2	1	no	—
75	C₂×C₄×D₅	C₄²	1	2	yes	—
76	C₂×C₄×D₅	C₂×C₈	1	2	yes	—
77	C₂×C₅×C₈	—	2	1	no	—
78	C₂×C₅×C₈	—	2	1	no	—
79	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
80	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
83	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
84	C₂×C₅×C₈	#41	1	1	yes	characteristic
85	C₂×C₅×C₈	#41	1	1	yes	characteristic
86	C₄²×C₅	—	2	1	no	—
87	C₄²×C₅	—	2	1	no	—
88	#685	—	2	4	no	—
89	C₄×Dic₅	—	2	1	no	—
90	C₄×Dic₅	—	2	1	no	—
91	C₂×#168	—	2	1	no	—
92	C₂×#168	—	2	1	no	—
93	C₂×#168	—	2	1	no	—
94	C₂×#168	—	2	1	no	—
95	C₂×#168	C₄²	1	2	yes	—
96	C₂×#168	C₂×C₈	1	2	yes	—
97	C₈×D₅	—	2	2	no	—
98	C₈×D₅	—	2	2	no	—
99	C₈×D₅	—	2	2	no	—
100	C₈×D₅	—	2	2	no	—
101	C₅×C₁₆	—	2	2	no	—
102	C₅×C₁₆	C₂²⋊C₄	1	4	yes	complemented
103	#672	—	2	2	no	—
104	#672	—	2	2	no	—
105	C₂³×C₈	—	5	1	no	—
106	C₂²×C₁₆	—	5	1	no	—
107	C₂²×C₁₆	—	5	1	no	—
108	C₂²×C₁₆	—	5	4	no	—
109	C₂²×C₁₆	—	10	2	no	—
110	C₂²×C₁₆	—	10	2	no	—
111	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
112	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
113	C₂×C₂²⋊C₈	—	5	1	no	—
114	C₂×C₄×C₈	—	5	1	no	—
115	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
116	C₄×C₂²⋊C₄	—	5	1	no	—
117	C₂²⋊C₁₆	—	5	4	no	—
118	#346	—	2	2	no	—
119	#346	—	10	2	no	—
120	#336	—	5	1	no	—
121	#336	—	5	1	no	—
122	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
123	C₂²×D₅	—	2	1	no	—
124	C₂²×D₅	—	2	1	no	—
125	C₂²×D₅	—	2	2	no	—
126	C₂²×D₅	C₄×C₈	1	2	yes	—
127	C₂²×D₅	M₃₂	1	4	yes	—
128	C₂×C₄×C₅	—	2	2	no	—
129	C₂×C₄×C₅	—	2	2	no	—
130	C₂×C₄×C₅	—	2	2	no	—
131	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
132	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₄×C₅	#131	1	1	yes	characteristic
136	C₂×C₄×C₅	#131	1	1	yes	characteristic
137	C₂×Dic₅	—	2	1	no	—
138	C₂×Dic₅	—	2	1	no	—
139	C₂×Dic₅	—	2	2	no	—
140	C₂×Dic₅	—	2	2	no	—
141	C₂×Dic₅	—	2	2	no	—
142	C₂×Dic₅	C₄×C₈	1	2	yes	—
143	C₄×D₅	—	2	2	no	—
144	C₄×D₅	—	2	2	no	—
145	C₄×D₅	—	2	2	no	—
146	C₄×D₅	—	2	2	no	—
147	C₅×C₈	—	2	2	no	—
148	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₅×C₈	#117	1	1	yes	characteristic
150	C₅×C₈	C₄×D₄	1	1	yes	characteristic
151	C₅×C₈	C₄×D₄	1	1	yes	characteristic
152	#168	—	2	2	no	—
153	#168	—	2	2	no	—
154	C₂³×C₄	—	5	1	no	—
155	C₂²×C₈	—	5	1	no	—
156	C₂²×C₈	—	5	1	no	—
157	C₂²×C₈	—	5	2	no	—
158	C₂²×C₈	—	5	2	no	—
159	C₂²×C₈	—	10	1	no	—
160	C₂²×C₈	—	10	1	no	—
161	C₂²×C₈	—	10	1	no	—
162	C₂²×C₈	—	10	1	no	—
163	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
164	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
165	C₂×C₂²⋊C₄	—	5	2	no	—
166	C₂×C₄²	—	5	1	no	—
167	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
168	C₂×C₁₆	—	2	1	no	—
169	C₂×C₁₆	—	2	1	no	—
170	C₂×C₁₆	—	5	2	no	—
171	C₂×C₁₆	—	5	2	no	—
172	C₂×C₁₆	—	10	1	no	—
173	C₂×C₁₆	—	10	1	no	—
174	C₂×C₁₆	—	10	1	no	—
175	C₂×C₁₆	—	10	1	no	—
176	C₂×C₁₆	—	10	8	no	—
177	C₂×C₁₆	C₄×D₅	1	2	yes	—
178	C₂×C₁₆	C₄×D₅	1	2	yes	—
179	C₂×C₁₆	D₂₀	1	2	yes	—
180	C₂×C₁₆	C₂²⋊D₅	1	2	yes	—
181	C₂²⋊C₈	—	5	4	no	—
182	C₄×C₈	—	2	1	no	—
183	C₄×C₈	—	2	1	no	—
184	C₄×C₈	—	10	1	no	—
185	C₄×C₈	—	10	1	no	—
186	#95	—	5	2	no	—
187	C₂²×C₅	C₂×C₄×C₈	1	1	yes	characteristic
188	C₂²×C₅	C₈×D₄	1	1	yes	characteristic
189	C₂²×C₅	C₈×D₄	1	1	yes	characteristic
190	C₂²×C₅	C₂×M₃₂	1	2	yes	—
191	C₂²×C₅	#504	1	2	yes	—
192	C₂×D₅	—	2	2	no	—
193	C₂×D₅	—	2	4	no	—
194	C₂×D₅	—	2	4	no	—
195	C₂×D₅	—	2	2	no	—
196	C₂×D₅	#346	1	4	yes	complemented
197	C₄×C₅	—	4	2	no	—
198	C₄×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
199	C₄×C₅	#405	1	1	yes	characteristic
200	C₄×C₅	C₈×D₄	1	1	yes	characteristic
201	C₄×C₅	C₈×D₄	1	1	yes	characteristic
202	Dic₅	—	2	2	no	—
203	Dic₅	—	4	2	no	—
204	C₂⁴	—	5	1	no	—
205	C₂²×C₄	—	5	1	no	—
206	C₂²×C₄	—	5	2	no	—
207	C₂²×C₄	—	5	2	no	—
208	C₂²×C₄	—	5	2	no	—
209	C₂²×C₄	—	10	1	no	—
210	C₂²×C₄	—	10	1	no	—
211	C₂²×C₄	—	10	1	no	—
212	C₂²×C₄	—	10	1	no	—
213	C₂²×C₄	C₈×D₅	1	2	yes	—
214	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	5	2	no	—
218	C₂×C₈	—	5	2	no	—
219	C₂×C₈	—	10	1	no	—
220	C₂×C₈	—	10	1	no	—
221	C₂×C₈	—	10	1	no	—
222	C₂×C₈	—	10	1	no	—
223	C₂×C₈	—	10	2	no	—
224	C₂×C₈	—	10	2	no	—
225	C₂×C₈	—	10	2	no	—
226	C₂×C₈	—	10	2	no	—
227	C₂×C₈	C₈×D₅	1	2	yes	—
228	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
229	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
230	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
231	C₂×C₈	#709	1	1	yes	characteristic
232	C₂×C₈	#709	1	1	yes	characteristic
233	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
234	C₂²⋊C₄	—	10	4	no	—
235	C₄²	—	2	1	no	—
236	C₄²	—	2	1	no	—
237	C₄²	—	10	1	no	—
238	C₄²	—	10	1	no	—
239	C₁₆	—	2	2	no	—
240	C₁₆	—	10	2	no	—
241	C₁₆	—	10	2	no	—
242	C₁₆	#685	1	4	yes	complemented
243	C₂×C₅	C₈×C₂²⋊C₄	1	1	yes	characteristic
244	C₂×C₅	C₂×#346	1	1	yes	characteristic
245	C₂×C₅	#2108	1	1	yes	characteristic
246	C₂×C₅	#2168	1	2	yes	—
247	C₂×C₅	#2168	1	2	yes	—
248	D₅	—	2	4	no	—
249	C₂³	—	5	4	no	—
250	C₂³	—	5	2	no	—
251	C₂³	—	10	1	no	—
252	C₂³	—	10	2	no	—
253	C₂³	—	10	1	no	—
254	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	5	2	no	—
259	C₂×C₄	—	10	1	no	—
260	C₂×C₄	—	10	1	no	—
261	C₂×C₄	—	10	2	no	—
262	C₂×C₄	—	10	2	no	—
263	C₂×C₄	—	10	2	no	—
264	C₂×C₄	—	10	2	no	—
265	C₂×C₄	—	10	2	no	—
266	C₂×C₄	—	10	2	no	—
267	C₂×C₄	—	10	2	no	—
268	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
269	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
270	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
271	C₂×C₄	#4047	1	1	yes	characteristic
272	C₂×C₄	#4047	1	1	yes	characteristic
273	C₂×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
274	C₈	—	2	2	no	—
275	C₈	—	10	2	no	—
276	C₈	—	10	2	no	—
277	C₈	#4018	1	1	yes	characteristic
278	C₈	C₄×D₂₀	1	1	yes	characteristic
279	C₈	C₂×#685	1	1	yes	characteristic
280	C₈	C₄×C₂²⋊D₅	1	1	yes	characteristic
281	C₅	#12611	1	1	yes	characteristic, complemented, Sylow
282	C₂²	—	5	4	no	—
283	C₂²	—	10	2	no	—
284	C₂²	—	10	2	no	—
285	C₂²	—	10	4	no	—
286	C₂²	—	10	4	no	—
287	C₂²	C₄×C₈×D₅	1	1	yes	characteristic
288	C₂²	C₈×D₂₀	1	1	yes	characteristic
289	C₂²	D₅×M₃₂	1	2	yes	—
290	C₂²	#65196	1	2	yes	—
291	C₂²	C₈×C₂²⋊D₅	1	1	yes	characteristic
292	C₄	—	4	2	no	—
293	C₄	—	10	2	no	—
294	C₄	—	20	2	no	—
295	C₄	C₈×D₂₀	1	1	yes	characteristic
296	C₄	#64978	1	1	yes	characteristic
297	C₄	C₄×#685	1	1	yes	characteristic
298	C₄	C₈×C₂²⋊D₅	1	1	yes	characteristic
299	C₂	—	10	4	no	—
300	C₂	D₅×#346	1	1	yes	characteristic
301	C₂	#10599865	1	1	yes	characteristic
302	C₂	#10599868	1	2	yes	—
303	C₂	C₈×#685	1	1	yes	characteristic
304	C₂	#10602226	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅
commutator	C₂²×C₅	C₂×C₄×C₈
Fitting	C₂×C₅×#346	C₂
socle	C₂³×C₅	C₂²×C₈
prime radical	C₂³×D₅	C₂×C₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×#346	D₅
hypocenter	C₅	#12611
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#346	C₂	semidirect
4. C₂×#64722	C₂	semidirect
6. #10598295	C₂	semidirect
7. #10598148	C₂	semidirect
19. C₂×D₅×C₁₆	C₄	semidirect
23. #64727	C₄	semidirect
62. C₂×#346	D₅	semidirect
102. C₅×C₁₆	C₂²⋊C₄	semidirect
196. C₂×D₅	#346	semidirect
242. C₁₆	#685	semidirect
281. C₅	#12611	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₅×C₁₆	2
C₂×C₅×#346	2
C₂×#64727	2
C₂×#64722	2
C₈×#685	2
#10598295	2
#10598148	2
#12611	5