Subgroups — #12071060

Order 1280.

722 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10610253	C₂	1	2	yes	maximal, complemented
2	#10610201	C₂	1	1	yes	maximal, characteristic
3	C₅×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#10601551	C₂	1	2	yes	maximal, complemented
5	C₂×#64678	C₂	1	1	yes	maximal, characteristic, complemented
6	#10598871	C₂	1	4	yes	maximal, complemented
7	#10598845	C₂	1	2	yes	maximal, complemented
8	C₈×D₄₀	C₂	1	2	yes	maximal, complemented
9	C₂×#4053	—	2	2	no	—
10	#65811	C₂²	1	2	yes	—
11	C₅×#431	C₂²	1	1	yes	characteristic
12	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	#65418	—	2	2	no	—
15	#65404	C₄	1	2	yes	—
16	#65404	C₄	1	2	yes	—
17	C₂×C₄⋊#168	C₂²	1	2	yes	—
18	#64995	—	2	4	no	—
19	#64984	C₄	1	1	yes	characteristic
20	#64983	C₄	1	2	yes	—
21	C₄×D₄₀	C₂²	1	2	yes	—
22	C₄×D₄₀	C₂²	1	2	yes	—
23	#64977	C₂²	1	4	yes	—
24	C₈×D₂₀	C₂²	1	4	yes	—
25	#64971	C₄	1	1	yes	characteristic
26	C₅×#322	C₂²	1	2	yes	—
27	C₅×C₈²	C₂²	1	2	yes	—
28	#64679	C₂²	1	4	yes	—
29	#64679	C₂²	1	4	yes	—
30	#64678	C₂²	1	1	yes	characteristic
31	#64678	C₂²	1	1	yes	characteristic
32	#64678	C₂²	1	2	yes	—
33	#18066	—	5	1	no	maximal, Sylow
34	C₂×#709	—	2	2	no	—
35	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
36	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
37	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
40	#4076	—	2	2	no	—
41	#4076	C₂×C₄	1	4	yes	—
42	C₂²×#168	—	2	2	no	—
43	#4053	—	4	8	no	—
44	C₂×D₄₀	C₂×C₄	1	2	yes	—
45	C₂×D₄₀	C₂×C₄	1	2	yes	—
46	C₂×#676	—	2	4	no	—
47	C₂×C₈×D₅	—	2	4	no	—
48	#4021	—	2	2	no	—
49	#4020	C₂×C₄	1	2	yes	—
50	C₄×D₂₀	—	2	2	no	—
51	C₄×D₂₀	C₂³	1	4	yes	—
52	#4015	C₂×C₄	1	2	yes	—
53	C₅×C₄⋊C₈	—	2	1	no	—
54	C₅×C₂²⋊C₈	—	2	1	no	—
55	C₅×#97	D₄	1	2	yes	—
56	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
57	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
58	C₄×C₅×C₈	C₂³	1	2	yes	—
59	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
60	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
61	C₄×C₅×C₈	C₂³	1	2	yes	—
62	#3953	C₂×C₄	1	4	yes	—
63	#3953	C₂×C₄	1	4	yes	—
64	#3952	—	2	4	no	—
65	#3952	—	2	4	no	—
66	#3950	C₂×C₄	1	2	yes	—
67	#3950	C₂×C₄	1	2	yes	—
68	C₄⋊#168	—	2	2	no	—
69	C₄⋊#168	C₂³	1	2	yes	—
70	C₄⋊#168	C₂³	1	2	yes	—
71	#3169	—	5	1	no	—
72	#2935	—	5	2	no	—
73	#1589	—	5	2	no	—
74	#1581	—	5	4	no	—
75	C₈×D₈	—	5	2	no	—
76	C₂×#335	—	5	1	no	—
77	#1480	—	5	2	no	—
78	C₈×M₁₆	D₅	1	1	yes	characteristic, complemented
79	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₅	#41	1	1	yes	characteristic
82	C₂×C₂²⋊D₅	—	2	4	no	—
83	#709	—	4	4	no	—
84	#709	—	4	4	no	—
85	C₂×D₂₀	—	2	2	no	—
86	C₂×D₂₀	—	2	2	no	—
87	C₂×D₂₀	C₂²×C₄	1	4	yes	—
88	C₂×C₄×D₅	—	2	4	no	—
89	C₂×Dic₁₀	—	2	2	no	—
90	C₅×M₁₆	—	2	2	no	—
91	C₅×M₁₆	D₈	1	4	yes	complemented
92	C₂×C₅×C₈	—	2	2	no	—
93	C₂×C₅×C₈	—	2	2	no	—
94	C₂×C₅×C₈	—	4	1	no	—
95	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
100	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
101	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₅	#41	1	1	yes	characteristic
104	C₄²×C₅	C₂⁴	1	1	yes	characteristic
105	#685	—	2	4	no	—
106	#685	—	2	4	no	—
107	C₄⋊Dic₅	—	2	2	no	—
108	C₄⋊Dic₅	C₂²×C₄	1	4	yes	—
109	C₂×#168	—	2	2	no	—
110	C₂×#168	—	2	2	no	—
111	C₂×#168	—	4	4	no	—
112	D₄₀	—	2	4	no	—
113	D₄₀	—	4	2	no	—
114	#676	—	4	8	no	—
115	C₈×D₅	—	4	8	no	—
116	C₂×#131	—	10	2	no	—
117	#532	—	5	2	no	—
118	#500	—	5	1	no	—
119	#493	—	5	1	no	—
120	#486	—	5	2	no	—
121	#466	—	5	2	no	—
122	#466	—	5	2	no	—
123	C₄×D₈	—	5	2	no	—
124	C₄×D₈	—	5	2	no	—
125	#436	—	5	4	no	—
126	#435	—	10	4	no	—
127	C₈×D₄	—	5	4	no	—
128	#431	C₂×D₅	1	1	yes	characteristic
129	C₂×C₄⋊C₈	—	5	2	no	—
130	#408	—	10	2	no	—
131	C₄×M₁₆	C₂×D₅	1	1	yes	characteristic
132	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
133	#335	—	5	1	no	—
134	#335	—	5	2	no	—
135	#335	—	5	1	no	—
136	D₄⋊C₈	—	5	4	no	—
137	D₄⋊C₈	—	5	4	no	—
138	#322	C₂×D₅	1	2	yes	—
139	C₈²	C₂×D₅	1	2	yes	—
140	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
141	C₂²×D₅	—	2	4	no	—
142	C₂×C₄×C₅	—	2	1	no	—
143	C₂×C₄×C₅	—	2	1	no	—
144	C₂×C₄×C₅	—	4	1	no	—
145	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
149	C₂×C₄×C₅	#135	1	2	yes	—
150	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
153	C₂²⋊D₅	—	4	8	no	—
154	C₂×Dic₅	—	2	4	no	—
155	D₂₀	—	2	4	no	—
156	D₂₀	—	2	4	no	—
157	D₂₀	—	4	4	no	—
158	D₂₀	—	8	2	no	—
159	C₄×D₅	—	4	4	no	—
160	C₄×D₅	—	4	4	no	—
161	Dic₁₀	—	4	4	no	—
162	C₅×C₈	—	2	1	no	—
163	C₅×C₈	—	2	1	no	—
164	C₅×C₈	—	2	1	no	—
165	C₅×C₈	—	2	2	no	—
166	C₅×C₈	—	4	2	no	—
167	C₅×C₈	#131	1	2	yes	complemented
168	C₅×C₈	C₂×D₈	1	4	yes	—
169	#168	—	4	2	no	—
170	#168	—	4	2	no	—
171	C₂×#41	—	10	2	no	—
172	C₂×D₈	—	5	2	no	—
173	C₂×D₈	—	5	2	no	—
174	#131	—	20	8	no	—
175	C₂×M₁₆	—	10	4	no	—
176	C₂×M₁₆	D₂₀	1	1	yes	characteristic
177	C₂×M₁₆	D₂₀	1	1	yes	characteristic
178	C₂²×C₈	—	10	2	no	—
179	C₂²×C₈	—	10	4	no	—
180	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
181	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
182	C₄⋊Q₈	—	5	2	no	—
183	#127	—	5	2	no	—
184	#124	—	10	2	no	—
185	#121	—	5	4	no	—
186	#121	—	10	2	no	—
187	C₄×D₄	—	5	4	no	—
188	C₄×D₄	—	10	2	no	—
189	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
190	#107	—	5	2	no	—
191	#107	—	5	2	no	—
192	C₄⋊C₈	—	2	1	no	—
193	C₄⋊C₈	—	5	2	no	—
194	C₄⋊C₈	—	5	2	no	—
195	C₄⋊C₈	—	10	2	no	—
196	#102	—	5	4	no	—
197	#102	—	5	4	no	—
198	C₂²⋊C₈	—	2	1	no	—
199	C₂²⋊C₈	—	10	4	no	—
200	C₂²⋊C₈	—	10	4	no	—
201	#97	D₂₀	1	2	yes	—
202	C₄×C₈	C₄×D₅	1	1	yes	characteristic
203	C₄×C₈	C₄×D₅	1	1	yes	characteristic
204	C₄×C₈	C₂²×D₅	1	2	yes	—
205	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
206	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₅	1	2	yes	—
208	C₂²×C₅	—	2	1	no	—
209	C₂²×C₅	—	2	1	no	—
210	C₂²×C₅	C₄×D₈	1	2	yes	—
211	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
212	C₂×D₅	—	4	4	no	—
213	C₂×D₅	—	4	4	no	—
214	C₂×D₅	—	4	4	no	—
215	C₄×C₅	—	2	1	no	—
216	C₄×C₅	—	2	1	no	—
217	C₄×C₅	—	2	1	no	—
218	C₄×C₅	C₄×D₈	1	2	yes	—
219	C₄×C₅	C₂×#131	1	1	yes	characteristic
220	C₄×C₅	#568	1	1	yes	characteristic
221	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
222	C₄×C₅	C₂×#135	1	1	yes	characteristic
223	Dic₅	—	4	4	no	—
224	#41	—	20	4	no	—
225	#41	—	20	4	no	—
226	C₂×Q₈	—	10	2	no	—
227	C₂×D₄	—	5	4	no	—
228	C₂×D₄	—	10	4	no	—
229	C₂×D₄	—	10	2	no	—
230	C₂×D₄	—	10	2	no	—
231	C₂²×C₄	—	10	4	no	—
232	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
234	C₂²×C₄	#709	1	1	yes	characteristic
235	D₈	—	10	4	no	—
236	D₈	—	20	2	no	—
237	M₁₆	—	2	2	no	—
238	M₁₆	—	20	8	no	—
239	M₁₆	D₄₀	1	4	yes	complemented
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	10	2	no	—
244	C₂×C₈	—	10	2	no	—
245	C₂×C₈	—	20	4	no	—
246	C₂×C₈	—	20	8	no	—
247	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
248	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
249	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
250	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
251	C₂×C₈	C₂×D₂₀	1	2	yes	—
252	C₂×C₈	C₂×D₂₀	1	2	yes	—
253	C₄⋊C₄	—	5	4	no	—
254	C₄⋊C₄	—	10	2	no	—
255	C₂²⋊C₄	—	10	4	no	—
256	C₂²⋊C₄	—	10	4	no	—
257	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
258	C₄²	C₂×D₂₀	1	1	yes	characteristic
259	C₄²	#709	1	1	yes	characteristic
260	C₄²	C₂³×D₅	1	1	yes	characteristic
261	C₂×C₅	—	2	2	no	—
262	C₂×C₅	#2935	1	1	yes	characteristic
263	C₂×C₅	C₂×C₄×D₈	1	1	yes	characteristic
264	C₂×C₅	#2957	1	1	yes	characteristic
265	D₅	—	8	4	no	—
266	C₂³	—	10	4	no	—
267	C₂³	C₄×D₂₀	1	1	yes	characteristic
268	Q₈	—	20	4	no	—
269	D₄	—	10	4	no	—
270	D₄	—	10	4	no	—
271	D₄	—	20	8	no	—
272	D₄	—	20	4	no	—
273	D₄	—	40	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	10	4	no	—
278	C₂×C₄	—	20	4	no	—
279	C₂×C₄	—	20	4	no	—
280	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
281	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
282	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
283	C₂×C₄	C₂×D₄₀	1	2	yes	—
284	C₂×C₄	#4050	1	2	yes	—
285	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
286	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
287	C₂×C₄	C₂×#709	1	1	yes	characteristic
288	C₈	—	2	2	no	—
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	4	2	no	—
293	C₈	—	20	2	no	—
294	C₈	—	20	2	no	—
295	C₈	C₂×D₄₀	1	4	yes	—
296	C₈	#4053	1	2	yes	complemented
297	C₅	#18066	1	1	yes	characteristic, complemented, Sylow
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	20	4	no	—
301	C₂²	—	20	4	no	—
302	C₂²	—	20	4	no	—
303	C₂²	C₄×D₄₀	1	2	yes	—
304	C₂²	C₂×C₄×D₂₀	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	20	4	no	—
309	C₄	C₄×D₄₀	1	2	yes	—
310	C₄	C₂²×D₄₀	1	1	yes	characteristic
311	C₄	C₂×#4050	1	1	yes	characteristic
312	C₄	C₂×#4053	1	1	yes	characteristic
313	C₄	#66079	1	1	yes	characteristic
314	C₂	—	2	2	no	—
315	C₂	—	40	4	no	—
316	C₂	C₂×C₄×D₄₀	1	1	yes	characteristic
317	C₂	#10610253	1	1	yes	characteristic
318	C₂	#10610311	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₀
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₄²	C₂³×D₅
Fitting	C₅×C₈×M₁₆	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	#10610201	C₂
cosocle	C₄²×C₅	C₂⁴
hypercenter	C₈×M₁₆	D₅
hypocenter	C₅	#18066
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
1. #10610253	C₂	semidirect
3. C₅×C₈×M₁₆	C₂	semidirect
4. #10601551	C₂	semidirect
5. C₂×#64678	C₂	semidirect
6. #10598871	C₂	semidirect
7. #10598845	C₂	semidirect
8. C₈×D₄₀	C₂	semidirect
78. C₈×M₁₆	D₅	semidirect
91. C₅×M₁₆	D₈	semidirect
167. C₅×C₈	#131	semidirect
239. M₁₆	D₄₀	semidirect
296. C₈	#4053	semidirect
297. C₅	#18066	semidirect

Maximal subgroups

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

subgroup	index
#10610253	2
#10610201	2
C₅×C₈×M₁₆	2
#10601551	2
C₂×#64678	2
#10598871	2
#10598845	2
C₈×D₄₀	2
#18066	5