Subgroups — #12014155

Order 1280.

586 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#65390	C₂	1	1	yes	maximal, characteristic
3	C₂²×#3944	C₂	1	1	yes	maximal, characteristic
4	#10598301	C₂	1	4	yes	maximal, complemented
5	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₂×#4072	C₄	1	1	yes	characteristic
7	C₂²×#681	C₄	1	1	yes	characteristic
8	C₂×C₅×M₃₂	—	2	1	no	—
9	C₂×C₅×M₃₂	—	2	2	no	—
10	C₂×C₅×M₃₂	C₄	1	4	yes	complemented
11	C₂×C₅×M₃₂	C₂²	1	2	yes	—
12	C₂²×C₅×C₁₆	—	2	1	no	—
13	#65390	—	2	2	no	—
14	#65390	—	2	2	no	—
15	C₂×#3947	C₄	1	2	yes	—
16	C₂×C₈×Dic₅	C₂²	1	2	yes	—
17	C₂×#3944	—	2	2	no	—
18	C₂×#3944	—	2	1	no	—
19	C₂×#3944	C₄	1	4	yes	complemented
20	C₂×#3944	C₂²	1	2	yes	—
21	C₂²×#672	—	2	1	no	—
22	#12549	—	5	1	no	maximal, Sylow
23	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
27	C₂²×C₅×C₈	D₄	1	2	yes	—
28	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
30	C₂²×C₅×C₈	D₄	1	2	yes	—
31	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
32	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
33	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊Dic₅	—	2	1	no	—
35	#4072	—	2	2	no	—
36	#4072	C₈	1	4	yes	—
37	C₂×C₄⋊Dic₅	—	2	1	no	—
38	C₂×C₄×Dic₅	C₂×C₄	1	2	yes	—
39	C₂×#681	—	2	2	no	—
40	C₂×#681	—	2	2	no	—
41	C₂×#681	C₈	1	2	yes	—
42	C₂×#681	C₈	1	2	yes	—
43	C₂²×#168	C₂×C₄	1	2	yes	—
44	C₅×M₃₂	—	2	4	no	—
45	C₅×M₃₂	—	2	2	no	—
46	C₅×M₃₂	C₂×C₄	1	4	yes	—
47	C₂×C₅×C₁₆	—	2	4	no	—
48	C₂×C₅×C₁₆	—	2	2	no	—
49	#3947	—	2	4	no	—
50	C₈×Dic₅	—	2	2	no	—
51	C₈×Dic₅	—	2	2	no	—
52	#3944	—	2	4	no	—
53	#3944	—	2	2	no	—
54	#3944	C₂×C₄	1	4	yes	—
55	C₂×#672	—	2	4	no	—
56	C₂×#672	—	2	2	no	—
57	C₂²×M₃₂	—	5	1	no	—
58	C₂²×M₃₂	D₅	1	1	yes	characteristic
59	C₂×#405	—	5	1	no	—
60	#1376	—	5	4	no	—
61	C₂⁴×C₅	M₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₄²	1	2	yes	—
63	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
69	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
70	C₂²×C₄×C₅	M₁₆	1	2	yes	—
71	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂²×Dic₅	—	2	2	no	—
73	C₂×C₅×C₈	C₄²	1	2	yes	—
74	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
75	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
76	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
77	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
78	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
79	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
82	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
83	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
84	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
85	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
88	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
89	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
91	C₂²⋊Dic₅	—	2	4	no	—
92	C₄⋊Dic₅	—	2	4	no	—
93	C₄×Dic₅	—	2	2	no	—
94	C₄×Dic₅	C₂×C₈	1	4	yes	—
95	#681	—	2	4	no	—
96	#681	—	2	4	no	—
97	C₂×#168	—	2	2	no	—
98	C₂×#168	—	2	2	no	—
99	C₂×#168	C₂×C₈	1	4	yes	—
100	C₅×C₁₆	—	2	4	no	—
101	#672	—	2	4	no	—
102	C₂²×M₁₆	—	5	1	no	—
103	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
104	C₂×#117	—	5	1	no	—
105	C₂×M₃₂	—	2	2	no	—
106	C₂×M₃₂	—	2	1	no	—
107	C₂×M₃₂	—	5	4	no	—
108	C₂×M₃₂	—	5	2	no	—
109	C₂×M₃₂	—	10	2	no	—
110	C₂×M₃₂	—	10	1	no	—
111	C₂×M₃₂	Dic₅	1	4	yes	complemented
112	C₂×M₃₂	C₂×D₅	1	2	yes	—
113	C₂²×C₁₆	—	2	1	no	—
114	C₂²×C₁₆	—	10	1	no	—
115	#405	—	10	2	no	—
116	#405	—	10	2	no	—
117	C₂×#97	—	5	2	no	—
118	C₂×C₄×C₈	—	5	2	no	—
119	C₂³×C₅	—	4	2	no	—
120	C₂³×C₅	#97	1	2	yes	—
121	C₂³×C₅	C₂²⋊C₈	1	2	yes	—
122	C₂³×C₅	C₄⋊C₈	1	1	yes	characteristic
123	C₂³×C₅	C₄⋊C₈	1	1	yes	characteristic
124	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
125	C₂×C₄×C₅	#95	1	2	yes	—
126	C₂×C₄×C₅	#95	1	2	yes	—
127	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
128	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
129	C₂×C₄×C₅	#97	1	2	yes	—
130	C₂×C₄×C₅	C₂²⋊C₈	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	C₄⋊C₈	1	1	yes	characteristic
134	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
135	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
136	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
137	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
143	C₂×Dic₅	—	2	4	no	—
144	C₂×Dic₅	—	4	4	no	—
145	C₅×C₈	#95	1	2	yes	—
146	C₅×C₈	#95	1	2	yes	—
147	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
151	#168	—	2	4	no	—
152	C₂³×C₄	C₄×D₅	1	1	yes	characteristic
153	C₂×M₁₆	—	5	2	no	—
154	C₂×M₁₆	—	5	2	no	—
155	C₂×M₁₆	—	10	2	no	—
156	C₂×M₁₆	—	10	2	no	—
157	C₂²×C₈	—	5	2	no	—
158	C₂²×C₈	Dic₁₀	1	1	yes	characteristic
159	C₂²×C₈	Dic₁₀	1	1	yes	characteristic
160	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
161	C₂²×C₈	C₄×D₅	1	2	yes	—
162	C₂²×C₈	D₂₀	1	2	yes	—
163	C₂²×C₈	C₂×Dic₅	1	2	yes	—
164	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
165	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
166	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
167	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
168	#117	—	5	4	no	—
169	#117	—	10	2	no	—
170	C₂×C₄⋊C₄	—	10	1	no	—
171	C₂×C₂²⋊C₄	—	10	1	no	—
172	C₂×C₄²	—	5	2	no	—
173	M₃₂	—	2	4	no	—
174	M₃₂	—	2	2	no	—
175	M₃₂	—	5	4	no	—
176	M₃₂	—	10	4	no	—
177	M₃₂	—	10	2	no	—
178	M₃₂	C₂×Dic₅	1	4	yes	—
179	C₂×C₁₆	—	2	4	no	—
180	C₂×C₁₆	—	2	2	no	—
181	C₂×C₁₆	—	10	4	no	—
182	C₂×C₁₆	—	10	2	no	—
183	#97	—	10	4	no	—
184	C₄×C₈	—	10	2	no	—
185	C₄×C₈	—	10	2	no	—
186	C₂²×C₅	—	2	2	no	—
187	C₂²×C₅	—	2	2	no	—
188	C₂²×C₅	—	2	2	no	—
189	C₂²×C₅	—	4	2	no	—
190	C₂²×C₅	—	4	2	no	—
191	C₂²×C₅	#336	1	2	yes	—
192	C₂²×C₅	#336	1	2	yes	—
193	C₂²×C₅	C₂×#97	1	1	yes	characteristic
194	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
195	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
196	C₄×C₅	#336	1	2	yes	—
197	C₄×C₅	#336	1	2	yes	—
198	C₄×C₅	C₂×#95	1	1	yes	characteristic
199	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
200	C₄×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
201	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
202	Dic₅	—	4	4	no	—
203	C₂⁴	#676	1	1	yes	characteristic
204	C₂²×C₄	—	10	2	no	—
205	C₂²×C₄	C₈×D₅	1	2	yes	—
206	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
207	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
208	C₂²×C₄	#676	1	1	yes	characteristic
209	C₂²×C₄	#676	1	2	yes	—
210	C₂²×C₄	C₄×Dic₅	1	2	yes	—
211	C₂²×C₄	#683	1	1	yes	characteristic
212	C₂²×C₄	#683	1	1	yes	characteristic
213	C₂²×C₄	#685	1	2	yes	—
214	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
215	M₁₆	—	10	4	no	—
216	M₁₆	—	10	4	no	—
217	C₂×C₈	—	5	4	no	—
218	C₂×C₈	—	10	2	no	—
219	C₂×C₈	—	10	2	no	—
220	C₂×C₈	C₄×Dic₅	1	2	yes	—
221	C₂×C₈	#683	1	1	yes	characteristic
222	C₂×C₈	#683	1	1	yes	characteristic
223	C₂×C₈	#683	1	2	yes	—
224	C₂×C₈	#683	1	2	yes	—
225	C₂×C₈	C₄⋊Dic₅	1	2	yes	—
226	C₂×C₈	C₄⋊Dic₅	1	2	yes	—
227	C₂×C₈	#685	1	2	yes	—
228	C₂×C₈	#685	1	2	yes	—
229	C₂×C₈	#685	1	2	yes	—
230	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
231	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
232	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
233	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
234	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
235	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
236	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
237	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
238	C₄⋊C₄	—	10	4	no	—
239	C₂²⋊C₄	—	10	4	no	—
240	C₄²	—	5	4	no	—
241	C₄²	—	10	2	no	—
242	C₁₆	—	2	4	no	—
243	C₁₆	—	10	4	no	—
244	C₂×C₅	—	2	1	no	—
245	C₂×C₅	—	2	2	no	—
246	C₂×C₅	—	2	2	no	—
247	C₂×C₅	—	2	1	no	—
248	C₂×C₅	C₂×#336	1	1	yes	characteristic
249	C₂×C₅	#2153	1	2	yes	—
250	C₂³	—	4	2	no	—
251	C₂³	#3946	1	1	yes	characteristic
252	C₂³	#3946	1	1	yes	characteristic
253	C₂³	#3947	1	2	yes	—
254	C₂³	#3952	1	2	yes	—
255	C₂³	C₂×#676	1	1	yes	characteristic
256	C₂×C₄	—	10	4	no	—
257	C₂×C₄	—	20	4	no	—
258	C₂×C₄	C₈×Dic₅	1	2	yes	—
259	C₂×C₄	C₈×Dic₅	1	2	yes	—
260	C₂×C₄	#3946	1	1	yes	characteristic
261	C₂×C₄	#3946	1	1	yes	characteristic
262	C₂×C₄	#3946	1	2	yes	—
263	C₂×C₄	#3946	1	2	yes	—
264	C₂×C₄	#3947	1	2	yes	—
265	C₂×C₄	#3952	1	2	yes	—
266	C₂×C₄	#3952	1	2	yes	—
267	C₂×C₄	#3952	1	2	yes	—
268	C₂×C₄	#3963	1	2	yes	—
269	C₂×C₄	#3963	1	2	yes	—
270	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
271	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
272	C₂×C₄	C₂×#676	1	1	yes	characteristic
273	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
274	C₂×C₄	C₂×#683	1	1	yes	characteristic
275	C₂×C₄	C₂×#685	1	1	yes	characteristic
276	C₈	—	10	4	no	—
277	C₈	#3963	1	2	yes	—
278	C₈	#3963	1	2	yes	—
279	C₈	C₂×#683	1	1	yes	characteristic
280	C₈	C₂×C₄⋊Dic₅	1	1	yes	characteristic
281	C₈	C₂×#685	1	1	yes	characteristic
282	C₈	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
283	C₅	#12549	1	1	yes	characteristic, complemented, Sylow
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	2	no	—
289	C₂²	#64770	1	2	yes	—
290	C₂²	#64770	1	2	yes	—
291	C₂²	C₂×#3946	1	1	yes	characteristic
292	C₂²	C₂×#3947	1	1	yes	characteristic
293	C₂²	C₂×#3952	1	1	yes	characteristic
294	C₄	—	20	4	no	—
295	C₄	#64770	1	2	yes	—
296	C₄	#64770	1	2	yes	—
297	C₄	C₂×C₈×Dic₅	1	1	yes	characteristic
298	C₄	C₂×#3946	1	1	yes	characteristic
299	C₄	C₂×#3952	1	1	yes	characteristic
300	C₄	C₂×#3963	1	1	yes	characteristic
301	C₂	—	2	1	no	—
302	C₂	—	2	2	no	—
303	C₂	—	2	2	no	—
304	C₂	—	2	1	no	—
305	C₂	#10602216	1	2	yes	—
306	C₂	C₂×#64770	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅
commutator	C₄×C₅	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂²×C₅×M₃₂	C₂
socle	C₂²×C₅	C₂×C₄⋊C₈
prime radical	C₂⁴×C₅	M₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂²×M₃₂	D₅
hypocenter	C₅	#12549
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
4. #10598301	C₂	semidirect
10. C₂×C₅×M₃₂	C₄	semidirect
19. C₂×#3944	C₄	semidirect
111. C₂×M₃₂	Dic₅	semidirect
283. C₅	#12549	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅×M₃₂	2
C₂×#65390	2
C₂²×#3944	2
#10598301	2
#12549	5