Subgroups — #11998549

Order 1280.

522 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#64983	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#334	C₂	1	1	yes	maximal, characteristic, complemented
3	#10601252	C₂	1	1	yes	maximal, characteristic, complemented
4	#10601250	C₂	1	1	yes	maximal, characteristic
5	C₂×#64701	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#64705	C₂	1	1	yes	maximal, characteristic
7	C₈×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#4020	C₂²	1	1	yes	characteristic
9	C₂×#4015	C₂²	1	1	yes	characteristic
10	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₂×#3953	—	2	2	no	—
14	C₂×#3953	—	2	1	no	—
15	C₂×#3953	—	2	1	no	—
16	C₂×C₈×Dic₅	—	2	1	no	—
17	C₄×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
19	#64983	—	2	2	no	—
20	#64983	C₄	1	4	yes	complemented
21	C₅×#334	—	2	2	no	—
22	#64771	—	2	2	no	—
23	#64770	—	2	1	no	—
24	#64705	—	2	2	no	—
25	#64701	—	2	2	no	—
26	#9735	—	5	1	no	maximal, Sylow
27	C₂²×D₂₀	—	2	1	no	—
28	C₂²×D₂₀	—	2	1	no	—
29	C₂²×D₂₀	—	2	1	no	—
30	C₂²×Dic₁₀	—	2	1	no	—
31	C₂²×C₅×C₈	—	2	1	no	—
32	C₂²×C₅×C₈	—	2	1	no	—
33	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
36	C₂×C₄⋊Dic₅	—	2	1	no	—
37	C₂×C₄⋊Dic₅	—	2	1	no	—
38	C₂×C₄⋊Dic₅	D₄	1	2	yes	—
39	C₂×C₄×Dic₅	—	2	1	no	—
40	C₂²×#168	—	2	1	no	—
41	#4020	—	2	1	no	—
42	#4020	—	2	1	no	—
43	#4020	—	2	1	no	—
44	#4020	C₂×C₄	1	2	yes	—
45	#4015	—	2	1	no	—
46	#4015	—	2	1	no	—
47	#4015	—	2	1	no	—
48	#4015	C₂×C₄	1	2	yes	—
49	C₅×C₄⋊C₈	—	2	1	no	—
50	C₅×C₄⋊C₈	—	2	1	no	—
51	C₅×C₄⋊C₈	—	2	1	no	—
52	C₅×C₄⋊C₈	—	2	1	no	—
53	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
54	C₄×C₅×C₈	D₄	1	2	yes	—
55	#3963	—	2	1	no	—
56	#3953	—	2	4	no	—
57	#3953	—	2	4	no	—
58	#3953	—	4	4	no	—
59	C₈×Dic₅	—	4	2	no	—
60	C₄⋊#168	—	2	1	no	—
61	C₄⋊#168	D₄	1	2	yes	—
62	C₂×#486	—	5	1	no	—
63	#1987	—	5	1	no	—
64	#1982	—	5	1	no	—
65	C₈×C₄⋊C₄	—	5	1	no	—
66	C₂×#334	D₅	1	1	yes	characteristic, complemented
67	C₂×#333	—	5	1	no	—
68	C₂×#331	—	5	1	no	—
69	C₂³×D₅	—	4	1	no	—
70	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	#41	1	1	yes	characteristic
73	C₂²×Dic₅	—	2	2	no	—
74	C₂²×Dic₅	—	4	1	no	—
75	C₂×D₂₀	—	2	2	no	—
76	C₂×D₂₀	—	2	2	no	—
77	C₂×D₂₀	—	2	2	no	—
78	C₂×D₂₀	—	4	2	no	—
79	C₂×D₂₀	—	4	1	no	—
80	C₂×D₂₀	—	4	1	no	—
81	C₂×D₂₀	—	4	2	no	—
82	C₂×D₂₀	—	4	1	no	—
83	C₂×D₂₀	—	4	1	no	—
84	C₂×D₂₀	—	4	1	no	—
85	C₂×D₂₀	—	4	1	no	—
86	C₂×D₂₀	—	4	1	no	—
87	C₂×D₂₀	—	8	2	no	—
88	C₂×Dic₁₀	—	2	2	no	—
89	C₂×Dic₁₀	—	4	1	no	—
90	C₂×Dic₁₀	—	8	2	no	—
91	C₂×C₅×C₈	—	2	1	no	—
92	C₂×C₅×C₈	—	2	1	no	—
93	C₂×C₅×C₈	—	2	1	no	—
94	C₂×C₅×C₈	—	2	1	no	—
95	C₂×C₅×C₈	—	4	1	no	—
96	C₂×C₅×C₈	—	4	1	no	—
97	C₂×C₅×C₈	—	4	2	no	—
98	C₂×C₅×C₈	—	4	2	no	—
99	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
100	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
101	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₅	#41	1	1	yes	characteristic
105	C₄⋊Dic₅	—	2	2	no	—
106	C₄⋊Dic₅	—	2	2	no	—
107	C₄⋊Dic₅	—	2	2	no	—
108	C₄⋊Dic₅	—	4	1	no	—
109	C₄⋊Dic₅	—	4	1	no	—
110	C₄⋊Dic₅	SD₁₆	1	4	yes	—
111	C₄×Dic₅	—	4	1	no	—
112	C₄×Dic₅	—	4	1	no	—
113	C₂×#168	—	2	2	no	—
114	C₂×#168	—	4	1	no	—
115	C₂×#168	—	4	1	no	—
116	C₂×C₄⋊Q₈	—	5	1	no	—
117	C₂×#127	—	5	1	no	—
118	#486	—	5	4	no	—
119	#486	—	10	2	no	—
120	C₂×C₄⋊C₈	—	5	1	no	—
121	C₂×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
122	C₂×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
123	C₂×#102	—	10	1	no	—
124	C₂×#102	—	10	1	no	—
125	C₂×#102	—	10	2	no	—
126	C₂×C₄×C₈	—	10	1	no	—
127	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
128	C₄×C₄⋊C₄	—	5	1	no	—
129	#340	—	10	2	no	—
130	#336	—	10	1	no	—
131	#334	—	2	2	no	—
132	#333	—	10	2	no	—
133	#331	—	10	2	no	—
134	C₂³×C₅	#121	1	1	yes	characteristic
135	C₂²×D₅	—	4	2	no	—
136	C₂²×D₅	—	8	1	no	—
137	C₂²×D₅	—	8	1	no	—
138	C₂²×D₅	—	8	2	no	—
139	C₂²×D₅	—	8	2	no	—
140	C₂×C₄×C₅	—	2	2	no	—
141	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₅	#121	1	1	yes	characteristic
145	C₂×C₄×C₅	#124	1	1	yes	characteristic
146	C₂×C₄×C₅	#127	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
148	C₂×C₄×C₅	#135	1	1	yes	characteristic
149	C₂×C₄×C₅	#135	1	1	yes	characteristic
150	C₂×C₄×C₅	#135	1	2	yes	—
151	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂×C₄×C₅	#137	1	1	yes	characteristic
153	C₂×Dic₅	—	2	4	no	—
154	C₂×Dic₅	—	4	2	no	—
155	C₂×Dic₅	—	4	2	no	—
156	C₂×Dic₅	—	4	2	no	—
157	C₂×Dic₅	—	8	2	no	—
158	D₂₀	—	4	2	no	—
159	D₂₀	—	4	2	no	—
160	D₂₀	—	4	4	no	—
161	D₂₀	—	8	2	no	—
162	D₂₀	—	8	2	no	—
163	D₂₀	—	8	1	no	—
164	D₂₀	—	8	1	no	—
165	D₂₀	—	8	1	no	—
166	D₂₀	—	8	1	no	—
167	Dic₁₀	—	8	2	no	—
168	Dic₁₀	—	8	2	no	—
169	C₅×C₈	—	2	2	no	—
170	C₅×C₈	—	2	2	no	—
171	C₅×C₈	—	8	1	no	—
172	C₅×C₈	—	8	1	no	—
173	#168	—	4	2	no	—
174	C₂²×Q₈	—	10	1	no	—
175	C₂²×D₄	—	10	1	no	—
176	C₂²×D₄	—	10	1	no	—
177	C₂²×D₄	—	10	1	no	—
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	—	10	1	no	—
181	C₂²×C₈	D₂₀	1	1	yes	characteristic
182	C₂²×C₈	D₂₀	1	1	yes	characteristic
183	C₄⋊Q₈	—	5	2	no	—
184	C₄⋊Q₈	—	10	1	no	—
185	C₄⋊Q₈	—	10	1	no	—
186	C₄⋊Q₈	—	10	1	no	—
187	#127	—	5	2	no	—
188	#127	—	10	1	no	—
189	#127	—	10	1	no	—
190	#127	—	10	1	no	—
191	C₂×C₄⋊C₄	—	5	2	no	—
192	C₂×C₄⋊C₄	—	10	1	no	—
193	C₂×C₄⋊C₄	—	10	1	no	—
194	C₂×C₄²	—	10	1	no	—
195	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
196	C₄⋊C₈	—	2	1	no	—
197	C₄⋊C₈	—	2	1	no	—
198	C₄⋊C₈	—	2	1	no	—
199	C₄⋊C₈	—	2	1	no	—
200	C₄⋊C₈	—	5	2	no	—
201	C₄⋊C₈	—	10	1	no	—
202	#102	—	10	4	no	—
203	#102	—	10	4	no	—
204	#102	—	20	4	no	—
205	C₄×C₈	—	20	2	no	—
206	C₄×C₈	C₄×D₅	1	2	yes	—
207	C₄×C₈	C₂²⋊D₅	1	2	yes	—
208	#95	—	10	1	no	—
209	C₂²×C₅	#390	1	1	yes	characteristic
210	C₂²×C₅	#463	1	1	yes	characteristic
211	C₂²×C₅	#464	1	1	yes	characteristic
212	C₂²×C₅	#465	1	2	yes	—
213	C₂²×C₅	#472	1	2	yes	—
214	C₂×D₅	—	8	2	no	—
215	C₂×D₅	—	8	2	no	—
216	C₂×D₅	—	8	2	no	—
217	C₂×D₅	—	16	2	no	—
218	C₂×D₅	—	16	2	no	—
219	C₄×C₅	—	2	1	no	—
220	C₄×C₅	—	2	1	no	—
221	C₄×C₅	#416	1	1	yes	characteristic
222	C₄×C₅	#417	1	1	yes	characteristic
223	C₄×C₅	C₄×SD₁₆	1	2	yes	—
224	C₄×C₅	#463	1	1	yes	characteristic
225	C₄×C₅	#464	1	1	yes	characteristic
226	C₄×C₅	#492	1	1	yes	characteristic
227	C₄×C₅	#495	1	1	yes	characteristic
228	Dic₅	—	4	4	no	—
229	Dic₅	—	8	2	no	—
230	C₂⁴	—	20	1	no	—
231	C₂×Q₈	—	10	2	no	—
232	C₂×Q₈	—	20	1	no	—
233	C₂×Q₈	—	40	2	no	—
234	C₂×D₄	—	10	2	no	—
235	C₂×D₄	—	10	2	no	—
236	C₂×D₄	—	10	2	no	—
237	C₂×D₄	—	20	1	no	—
238	C₂×D₄	—	20	1	no	—
239	C₂×D₄	—	20	1	no	—
240	C₂×D₄	—	20	1	no	—
241	C₂×D₄	—	20	1	no	—
242	C₂×D₄	—	20	2	no	—
243	C₂×D₄	—	20	2	no	—
244	C₂×D₄	—	20	1	no	—
245	C₂×D₄	—	20	1	no	—
246	C₂×D₄	—	40	2	no	—
247	C₂²×C₄	—	10	2	no	—
248	C₂²×C₄	—	20	1	no	—
249	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
250	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
251	C₂²×C₄	#713	1	1	yes	characteristic
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	4	2	no	—
257	C₂×C₈	—	4	2	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	—	10	2	no	—
261	C₂×C₈	—	20	1	no	—
262	C₂×C₈	—	20	1	no	—
263	C₂×C₈	#685	1	2	yes	—
264	C₂×C₈	#685	1	2	yes	—
265	C₄⋊C₄	—	5	4	no	—
266	C₄⋊C₄	—	10	2	no	—
267	C₄⋊C₄	—	10	2	no	—
268	C₄⋊C₄	—	10	2	no	—
269	C₄⋊C₄	—	20	1	no	—
270	C₄⋊C₄	—	20	1	no	—
271	C₄²	—	20	1	no	—
272	C₄²	—	20	1	no	—
273	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
274	C₄²	D₄×D₅	1	1	yes	characteristic
275	C₄²	#713	1	1	yes	characteristic
276	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
277	C₂×C₅	#1677	1	2	yes	—
278	C₂×C₅	#1678	1	2	yes	—
279	C₂×C₅	#1968	1	1	yes	characteristic
280	C₂×C₅	#1978	1	1	yes	characteristic
281	C₂×C₅	#1983	1	1	yes	characteristic
282	D₅	—	16	2	no	—
283	C₂³	—	20	2	no	—
284	C₂³	—	40	1	no	—
285	C₂³	—	40	1	no	—
286	C₂³	—	40	2	no	—
287	C₂³	—	40	2	no	—
288	C₂³	#4041	1	1	yes	characteristic
289	Q₈	—	40	2	no	—
290	Q₈	—	40	2	no	—
291	D₄	—	20	4	no	—
292	D₄	—	20	2	no	—
293	D₄	—	20	2	no	—
294	D₄	—	40	2	no	—
295	D₄	—	40	2	no	—
296	D₄	—	40	1	no	—
297	D₄	—	40	1	no	—
298	D₄	—	40	1	no	—
299	D₄	—	40	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	10	4	no	—
302	C₂×C₄	—	20	2	no	—
303	C₂×C₄	—	20	2	no	—
304	C₂×C₄	—	20	2	no	—
305	C₂×C₄	—	40	2	no	—
306	C₂×C₄	#4039	1	1	yes	characteristic
307	C₂×C₄	#4039	1	1	yes	characteristic
308	C₂×C₄	#4041	1	1	yes	characteristic
309	C₂×C₄	#4050	1	1	yes	characteristic
310	C₂×C₄	#4050	1	1	yes	characteristic
311	C₂×C₄	#4054	1	1	yes	characteristic
312	C₂×C₄	#4055	1	1	yes	characteristic
313	C₂×C₄	D₅×SD₁₆	1	2	yes	—
314	C₂×C₄	#4062	1	2	yes	—
315	C₂×C₄	C₂×#685	1	1	yes	characteristic
316	C₂×C₄	#4086	1	1	yes	characteristic
317	C₂×C₄	#4093	1	1	yes	characteristic
318	C₈	—	2	2	no	—
319	C₈	—	2	2	no	—
320	C₈	—	8	1	no	—
321	C₈	—	8	1	no	—
322	C₈	—	20	2	no	—
323	C₅	#9735	1	1	yes	characteristic, complemented, Sylow
324	C₂²	—	40	2	no	—
325	C₂²	—	40	2	no	—
326	C₂²	—	40	2	no	—
327	C₂²	—	80	2	no	—
328	C₂²	—	80	2	no	—
329	C₂²	#65133	1	1	yes	characteristic
330	C₂²	#65134	1	1	yes	characteristic
331	C₂²	#65153	1	2	yes	—
332	C₂²	#65186	1	2	yes	—
333	C₂²	#65277	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	20	4	no	—
337	C₄	—	40	2	no	—
338	C₄	#65133	1	1	yes	characteristic
339	C₄	#65134	1	1	yes	characteristic
340	C₄	#65141	1	2	yes	—
341	C₄	#65402	1	1	yes	characteristic
342	C₄	#65422	1	1	yes	characteristic
343	C₄	#65457	1	1	yes	characteristic
344	C₄	#65464	1	1	yes	characteristic
345	C₂	—	80	2	no	—
346	C₂	#10599331	1	2	yes	—
347	C₂	#10599341	1	2	yes	—
348	C₂	#10601251	1	1	yes	characteristic
349	C₂	#10601367	1	1	yes	characteristic
350	C₂	#10601502	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#4041
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₂×C₅×#334	C₂
socle	C₂³×C₅	#121
prime radical	C₂×#4020	C₂²
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂×#334	D₅
hypocenter	C₅	#9735
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#334	C₂	semidirect
3. #10601252	C₂	semidirect
5. C₂×#64701	C₂	semidirect
7. C₈×C₄⋊Dic₅	C₂	semidirect
20. #64983	C₄	semidirect
66. C₂×#334	D₅	semidirect
323. C₅	#9735	semidirect

Maximal subgroups

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

subgroup	index
C₂×#64983	2
C₂×C₅×#334	2
#10601252	2
#10601250	2
C₂×#64701	2
C₂×#64705	2
C₈×C₄⋊Dic₅	2
#9735	5