Subgroups — #12083320

Order 1280.

658 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3170	C₂	1	1	yes	maximal, characteristic, complemented
2	#10615081	C₂	1	2	yes	maximal
3	C₂×#64977	C₂	1	1	yes	maximal, characteristic
4	#10602623	C₂	1	2	yes	maximal, complemented
5	#10602167	C₂	1	2	yes	maximal, complemented
6	#10602099	C₂	1	2	yes	maximal, complemented
7	#10602028	C₂	1	4	yes	maximal, complemented
8	#10600523	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#533	C₂²	1	2	yes	—
10	C₂²×#676	—	2	1	no	—
11	C₂×C₄×D₂₀	C₂²	1	1	yes	characteristic
12	C₅×#500	C₂²	1	1	yes	characteristic, complemented
13	C₅×#494	C₂²	1	1	yes	characteristic, complemented
14	C₅×#487	C₂²	1	2	yes	complemented
15	C₅×#467	—	2	1	no	—
16	C₅×#467	—	2	1	no	—
17	C₄×C₅×Q₁₆	—	2	1	no	—
18	C₄×C₅×Q₁₆	—	2	1	no	—
19	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₂×#3952	—	2	1	no	—
21	#65376	C₂²	1	4	yes	—
22	#65373	C₂²	1	4	yes	—
23	#65355	—	2	4	no	—
24	#65290	—	2	2	no	—
25	#65284	—	2	2	no	—
26	#65213	C₂²	1	2	yes	—
27	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
28	#64977	—	2	1	no	—
29	#64977	—	2	1	no	—
30	#64977	C₂²	1	2	yes	—
31	#64977	C₂²	1	2	yes	—
32	#64767	C₂²	1	4	yes	complemented
33	#64766	C₂²	1	4	yes	complemented
34	#64677	C₂²	1	2	yes	complemented
35	#64677	C₂²	1	2	yes	complemented
36	#19281	—	5	1	no	maximal, Sylow
37	C₂²×D₂₀	D₄	1	1	yes	characteristic
38	C₂²×C₄×D₅	—	2	1	no	—
39	C₂×C₅×Q₁₆	—	2	1	no	—
40	C₂×C₅×Q₁₆	—	2	1	no	—
41	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
42	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
43	C₅×C₄⋊Q₈	C₂³	1	2	yes	—
44	C₅×C₄⋊Q₈	C₂³	1	2	yes	—
45	C₅×#125	—	2	2	no	—
46	C₅×C₂²⋊Q₈	—	2	2	no	—
47	C₅×C₂²⋊Q₈	—	2	2	no	—
48	C₄×C₅×Q₈	—	2	2	no	—
49	C₄×C₅×Q₈	—	2	2	no	—
50	C₅×#117	—	2	2	no	—
51	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
52	C₂×#685	—	2	1	no	—
53	C₂×C₄⋊Dic₅	D₄	1	1	yes	characteristic
54	C₂×#681	—	2	2	no	—
55	C₂²×#168	—	2	1	no	—
56	C₂×#676	—	2	2	no	—
57	C₂×#676	—	2	2	no	—
58	C₂×#676	—	4	4	no	—
59	C₄×D₂₀	—	2	1	no	—
60	C₄×D₂₀	—	2	1	no	—
61	C₄×D₂₀	D₄	1	2	yes	—
62	C₄×D₂₀	C₂³	1	2	yes	—
63	C₅×#107	—	2	1	no	—
64	C₅×#107	—	2	1	no	—
65	C₅×#103	—	2	2	no	—
66	C₅×#103	—	2	2	no	—
67	C₄×C₅×C₈	D₄	1	1	yes	characteristic
68	C₄×C₅×C₈	D₄	1	1	yes	characteristic
69	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
70	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
71	#3952	—	2	2	no	—
72	#3952	—	2	2	no	—
73	#3941	—	2	4	no	—
74	#3941	—	2	4	no	—
75	#3940	—	2	4	no	—
76	#3939	—	2	4	no	—
77	C₄⋊#168	—	2	2	no	—
78	C₄⋊#168	—	2	1	no	—
79	C₄⋊#168	C₂³	1	4	yes	—
80	C₄⋊#168	C₂³	1	2	yes	—
81	#3170	D₅	1	1	yes	characteristic, complemented
82	#3077	—	5	2	no	—
83	C₂×#436	—	5	1	no	—
84	#1703	—	5	4	no	—
85	#1695	—	5	2	no	—
86	#1689	—	5	2	no	—
87	#1566	—	5	1	no	—
88	#1550	—	5	2	no	—
89	C₂³×D₅	—	2	1	no	—
90	C₂×C₅×Q₈	—	2	2	no	—
91	C₂×C₅×Q₈	—	2	2	no	—
92	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₅	#41	1	1	yes	characteristic
95	C₂²×Dic₅	—	2	1	no	—
96	C₂×D₂₀	—	2	4	no	—
97	C₂×D₂₀	—	2	1	no	—
98	C₂×D₂₀	C₂×D₄	1	2	yes	—
99	C₂×C₄×D₅	—	2	2	no	—
100	C₂×C₄×D₅	—	2	2	no	—
101	C₂×C₄×D₅	—	4	2	no	—
102	C₂×C₄×D₅	—	4	2	no	—
103	C₅×Q₁₆	—	4	2	no	—
104	C₅×Q₁₆	—	4	2	no	—
105	C₂×C₅×C₈	—	2	2	no	—
106	C₂×C₅×C₈	—	4	1	no	—
107	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
110	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
111	C₅×C₄⋊C₄	—	2	2	no	—
112	C₅×C₄⋊C₄	—	2	2	no	—
113	C₅×C₄⋊C₄	—	2	2	no	—
114	C₅×C₄⋊C₄	—	2	2	no	—
115	C₅×C₄⋊C₄	—	4	2	no	—
116	C₅×C₄⋊C₄	—	4	2	no	—
117	C₅×C₂²⋊C₄	—	4	2	no	—
118	C₄²×C₅	—	4	2	no	—
119	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₅	#41	1	1	yes	characteristic
122	C₄²×C₅	C₂⁴	1	1	yes	characteristic
123	#685	—	2	2	no	—
124	#685	—	2	2	no	—
125	C₄⋊Dic₅	—	2	1	no	—
126	C₄⋊Dic₅	C₂×D₄	1	2	yes	—
127	#681	—	4	4	no	—
128	C₂×#168	—	2	4	no	—
129	C₂×#168	—	2	2	no	—
130	C₂×#168	—	4	2	no	—
131	#676	—	4	4	no	—
132	#676	—	4	4	no	—
133	C₂²×M₁₆	—	10	1	no	—
134	#533	C₂×D₅	1	2	yes	—
135	C₂×C₄×D₄	—	5	1	no	—
136	#500	C₂×D₅	1	1	yes	characteristic, complemented
137	#494	C₂×D₅	1	1	yes	characteristic, complemented
138	#487	C₂×D₅	1	2	yes	complemented
139	#478	—	10	4	no	—
140	#476	—	5	4	no	—
141	#474	—	5	4	no	—
142	#467	—	2	1	no	—
143	#467	—	2	1	no	—
144	C₄×Q₁₆	—	2	1	no	—
145	C₄×Q₁₆	—	2	1	no	—
146	#436	—	5	2	no	—
147	#436	—	5	2	no	—
148	#436	—	10	1	no	—
149	#436	—	10	1	no	—
150	#428	—	10	2	no	—
151	#423	—	5	2	no	—
152	C₂×C₄⋊C₈	—	5	1	no	—
153	#418	—	10	2	no	—
154	C₂×C₂²⋊C₈	—	10	1	no	—
155	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
156	#334	—	5	2	no	—
157	#334	—	5	2	no	—
158	#333	—	5	4	no	—
159	#332	—	5	4	no	—
160	C₂³×C₅	#121	1	1	yes	characteristic
161	C₂²×D₅	—	2	2	no	—
162	C₂²×D₅	—	4	2	no	—
163	C₂²×D₅	—	4	2	no	—
164	C₂²×D₅	—	4	2	no	—
165	C₅×Q₈	—	4	2	no	—
166	C₅×Q₈	—	4	2	no	—
167	C₅×Q₈	—	8	2	no	—
168	C₂×C₄×C₅	—	2	1	no	—
169	C₂×C₄×C₅	—	2	1	no	—
170	C₂×C₄×C₅	—	4	1	no	—
171	C₂×C₄×C₅	—	4	2	no	—
172	C₂×C₄×C₅	—	4	2	no	—
173	C₂×C₄×C₅	#121	1	1	yes	characteristic
174	C₂×C₄×C₅	#121	1	1	yes	characteristic
175	C₂×C₄×C₅	#121	1	1	yes	characteristic
176	C₂×C₄×C₅	#137	1	4	yes	—
177	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
179	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
180	C₂×Dic₅	—	2	2	no	—
181	C₂×Dic₅	—	4	2	no	—
182	D₂₀	—	2	4	no	—
183	D₂₀	—	4	2	no	—
184	C₄×D₅	—	4	4	no	—
185	C₄×D₅	—	4	2	no	—
186	C₄×D₅	—	4	2	no	—
187	C₅×C₈	—	2	1	no	—
188	C₅×C₈	—	2	1	no	—
189	C₅×C₈	—	2	1	no	—
190	C₅×C₈	AGL(1,ℤ/8)	1	2	yes	—
191	#168	—	4	2	no	—
192	#168	—	4	4	no	—
193	C₂²×D₄	—	5	1	no	—
194	C₂³×C₄	—	10	1	no	—
195	C₂×Q₁₆	—	2	1	no	—
196	C₂×Q₁₆	—	2	1	no	—
197	C₂×M₁₆	—	10	2	no	—
198	C₂×M₁₆	—	10	2	no	—
199	C₂×M₁₆	—	10	2	no	—
200	C₂×M₁₆	—	20	4	no	—
201	C₂²×C₈	—	10	1	no	—
202	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
203	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
204	C₄⋊Q₈	C₂²×D₅	1	2	yes	—
205	C₄⋊Q₈	C₂²×D₅	1	2	yes	—
206	#125	—	2	2	no	—
207	C₂²⋊Q₈	—	2	2	no	—
208	C₂²⋊Q₈	—	2	2	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₄×Q₈	—	2	2	no	—
211	C₄×D₄	—	5	2	no	—
212	C₄×D₄	—	5	2	no	—
213	C₄×D₄	—	10	1	no	—
214	C₄×D₄	—	10	1	no	—
215	#117	—	2	2	no	—
216	C₂×C₄⋊C₄	—	5	1	no	—
217	C₂×C₂²⋊C₄	—	10	1	no	—
218	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
219	#107	—	2	1	no	—
220	#107	—	2	1	no	—
221	#107	—	10	4	no	—
222	#106	—	10	4	no	—
223	C₄⋊C₈	—	5	2	no	—
224	C₄⋊C₈	—	5	4	no	—
225	C₄⋊C₈	—	10	2	no	—
226	C₄⋊C₈	—	10	1	no	—
227	#103	—	2	2	no	—
228	#103	—	2	2	no	—
229	#102	—	10	4	no	—
230	#102	—	10	4	no	—
231	C₂²⋊C₈	—	10	2	no	—
232	C₂²⋊C₈	—	10	2	no	—
233	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
234	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
235	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
237	C₂²×C₅	—	2	1	no	—
238	C₂²×C₅	—	2	1	no	—
239	C₂²×C₅	#470	1	2	yes	—
240	C₂²×C₅	C₂×#121	1	1	yes	characteristic
241	C₂×D₅	—	4	2	no	—
242	C₂×D₅	—	4	2	no	—
243	C₂×D₅	—	4	2	no	—
244	C₂×D₅	—	8	4	no	—
245	C₄×C₅	—	2	1	no	—
246	C₄×C₅	—	2	1	no	—
247	C₄×C₅	—	2	1	no	—
248	C₄×C₅	—	8	2	no	—
249	C₄×C₅	—	8	2	no	—
250	C₄×C₅	#470	1	2	yes	—
251	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
252	C₄×C₅	C₂×#137	1	2	yes	—
253	C₄×C₅	#575	1	1	yes	characteristic
254	Dic₅	—	4	2	no	—
255	C₂⁴	—	10	1	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×D₄	—	5	2	no	—
259	C₂×D₄	—	10	4	no	—
260	C₂×D₄	—	10	1	no	—
261	C₂²×C₄	—	10	1	no	—
262	C₂²×C₄	—	10	2	no	—
263	C₂²×C₄	—	10	2	no	—
264	C₂²×C₄	—	20	2	no	—
265	C₂²×C₄	—	20	2	no	—
266	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
267	C₂²×C₄	#711	1	1	yes	characteristic
268	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
269	Q₁₆	—	4	2	no	—
270	Q₁₆	—	4	2	no	—
271	M₁₆	—	20	4	no	—
272	M₁₆	—	20	4	no	—
273	M₁₆	—	20	4	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	4	1	no	—
276	C₂×C₈	—	10	4	no	—
277	C₂×C₈	—	10	2	no	—
278	C₂×C₈	—	20	2	no	—
279	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
280	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
281	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
282	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	4	2	no	—
288	C₄⋊C₄	—	4	2	no	—
289	C₄⋊C₄	—	5	2	no	—
290	C₄⋊C₄	—	10	1	no	—
291	C₂²⋊C₄	—	4	2	no	—
292	C₂²⋊C₄	—	10	2	no	—
293	C₂²⋊C₄	—	10	2	no	—
294	C₄²	—	4	2	no	—
295	C₄²	D₄×D₅	1	1	yes	characteristic
296	C₄²	#711	1	1	yes	characteristic
297	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
298	C₄²	C₂³×D₅	1	1	yes	characteristic
299	C₂×C₅	—	2	2	no	—
300	C₂×C₅	#3036	1	1	yes	characteristic
301	C₂×C₅	C₂×#470	1	1	yes	characteristic
302	C₂×C₅	#3067	1	1	yes	characteristic
303	D₅	—	8	2	no	—
304	C₂³	—	10	2	no	—
305	C₂³	—	20	2	no	—
306	C₂³	—	20	2	no	—
307	C₂³	—	20	2	no	—
308	C₂³	#4084	1	1	yes	characteristic
309	Q₈	—	4	2	no	—
310	Q₈	—	4	2	no	—
311	Q₈	—	8	2	no	—
312	D₄	—	10	4	no	—
313	D₄	—	20	2	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	10	2	no	—
320	C₂×C₄	—	20	4	no	—
321	C₂×C₄	—	20	2	no	—
322	C₂×C₄	—	20	2	no	—
323	C₂×C₄	—	20	2	no	—
324	C₂×C₄	#4064	1	4	yes	—
325	C₂×C₄	#4084	1	1	yes	characteristic
326	C₂×C₄	#4084	1	1	yes	characteristic
327	C₂×C₄	#4084	1	1	yes	characteristic
328	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
329	C₂×C₄	C₂×#711	1	1	yes	characteristic
330	C₂×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
331	C₈	—	2	1	no	—
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	20	4	no	—
335	C₈	—	20	2	no	—
336	C₈	#4095	1	2	yes	—
337	C₅	#19281	1	1	yes	characteristic, complemented, Sylow
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	20	2	no	—
341	C₂²	—	20	2	no	—
342	C₂²	—	20	2	no	—
343	C₂²	—	40	4	no	—
344	C₂²	#65478	1	2	yes	—
345	C₂²	C₂×#4084	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	8	2	no	—
350	C₄	—	8	2	no	—
351	C₄	—	20	2	no	—
352	C₄	#65478	1	2	yes	—
353	C₄	C₂×#4064	1	2	yes	—
354	C₄	C₂×#4095	1	1	yes	characteristic
355	C₄	#66156	1	1	yes	characteristic
356	C₂	—	2	2	no	—
357	C₂	—	40	2	no	—
358	C₂	#10615028	1	1	yes	characteristic
359	C₂	C₂×#65478	1	1	yes	characteristic
360	C₂	#10615659	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#4084
commutator, cosocle	C₄²×C₅	C₂⁴
Frattini	C₄²	C₂³×D₅
Fitting	C₅×#3170	C₂
socle	C₂²×C₅	C₂×#121
prime radical	C₂²×D₂₀	D₄
hypercenter	#3170	D₅
hypocenter	C₅	#19281
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
1. C₅×#3170	C₂	semidirect
4. #10602623	C₂	semidirect
5. #10602167	C₂	semidirect
6. #10602099	C₂	semidirect
7. #10602028	C₂	semidirect
8. #10600523	C₂	semidirect
12. C₅×#500	C₂²	semidirect
13. C₅×#494	C₂²	semidirect
14. C₅×#487	C₂²	semidirect
32. #64767	C₂²	semidirect
33. #64766	C₂²	semidirect
34. #64677	C₂²	semidirect
35. #64677	C₂²	semidirect
81. #3170	D₅	semidirect
136. #500	C₂×D₅	semidirect
137. #494	C₂×D₅	semidirect
138. #487	C₂×D₅	semidirect
337. C₅	#19281	semidirect

Maximal subgroups

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

subgroup	index
C₅×#3170	2
#10615081	2
C₂×#64977	2
#10602623	2
#10602167	2
#10602099	2
#10602028	2
#10600523	2
#19281	5