Subgroups — #12087251

Order 1280.

750 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10613665	C₂	1	1	yes	maximal, characteristic
2	#10613624	C₂	1	2	yes	maximal
3	D₅×#494	C₂	1	1	yes	maximal, characteristic
4	#10613339	C₂	1	2	yes	maximal
5	C₅×#1707	C₂	1	1	yes	maximal, characteristic, complemented
6	#10602153	C₂	1	1	yes	maximal, characteristic, complemented
7	#10602134	C₂	1	2	yes	maximal, complemented
8	#10602091	C₂	1	1	yes	maximal, characteristic, complemented
9	#10602020	C₂	1	2	yes	maximal, complemented
10	#10598858	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×#676	C₂	1	1	yes	maximal, characteristic
12	C₂×D₅×Q₁₆	—	2	2	no	—
13	#66062	C₂²	1	2	yes	—
14	D₅×C₄⋊Q₈	C₂²	1	2	yes	—
15	C₅×#500	C₂²	1	1	yes	characteristic
16	C₅×#498	C₂²	1	2	yes	—
17	C₅×#494	C₂²	1	1	yes	characteristic
18	C₅×#487	C₂²	1	2	yes	—
19	#65470	—	2	2	no	—
20	#65382	C₂²	1	2	yes	—
21	#65381	C₂²	1	2	yes	—
22	#65370	C₂²	1	2	yes	—
23	#65368	C₂²	1	2	yes	—
24	#65367	C₂²	1	2	yes	—
25	#65367	C₂²	1	2	yes	—
26	#65176	—	2	2	no	—
27	#65162	—	2	2	no	—
28	#65150	—	2	4	no	—
29	#65145	—	2	4	no	—
30	#65098	—	2	4	no	—
31	#65083	—	2	4	no	—
32	#64988	C₂²	1	1	yes	characteristic
33	#64988	C₂²	1	1	yes	characteristic
34	C₄×#676	C₂²	1	1	yes	characteristic
35	#64974	C₂²	1	1	yes	characteristic
36	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
37	#64971	C₂²	1	2	yes	—
38	#64971	C₂²	1	2	yes	—
39	C₅×C₈²	C₂²	1	1	yes	characteristic
40	#64675	C₂²	1	1	yes	characteristic
41	#64674	C₂²	1	1	yes	characteristic
42	C₈×#168	C₂²	1	1	yes	characteristic
43	#19632	—	5	1	no	maximal, Sylow
44	C₂×Q₈×D₅	—	2	2	no	—
45	C₂×C₅×Q₁₆	—	2	2	no	—
46	C₅×C₄⋊Q₈	C₂³	1	2	yes	—
47	C₅×C₄⋊Q₈	C₂³	1	2	yes	—
48	#4092	—	2	2	no	—
49	Q₈×Dic₅	—	2	2	no	—
50	#4090	—	2	2	no	—
51	C₂×#689	—	2	4	no	—
52	D₅×Q₁₆	—	4	8	no	—
53	C₂×Dic₂₀	—	2	2	no	—
54	C₂×Dic₂₀	—	2	2	no	—
55	C₂×#676	D₄	1	2	yes	—
56	C₂×C₈×D₅	D₄	1	2	yes	—
57	#4043	—	2	2	no	—
58	#4038	—	2	2	no	—
59	D₅×C₄⋊C₄	—	2	2	no	—
60	D₅×C₄⋊C₄	—	2	2	no	—
61	#4036	—	2	2	no	—
62	#4034	—	2	2	no	—
63	#4034	—	2	2	no	—
64	#4033	—	2	2	no	—
65	C₄²×D₅	C₂³	1	1	yes	characteristic
66	#4015	C₂³	1	2	yes	—
67	#4015	C₂³	1	2	yes	—
68	C₅×#107	—	2	2	no	—
69	C₅×#106	—	2	4	no	—
70	C₅×#103	—	2	4	no	—
71	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
72	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
73	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
74	#3967	—	2	4	no	—
75	#3952	D₄	1	2	yes	—
76	#3950	—	2	4	no	—
77	#3950	—	2	4	no	—
78	#3947	D₄	1	2	yes	—
79	#3946	D₄	1	2	yes	—
80	C₈×Dic₅	D₄	1	2	yes	—
81	#3942	—	2	4	no	—
82	#3940	—	2	4	no	—
83	#3939	—	2	4	no	—
84	#3939	—	2	4	no	—
85	C₄×#168	C₂³	1	1	yes	characteristic
86	C₄×#168	C₂³	1	1	yes	characteristic
87	C₄×#168	C₂³	1	1	yes	characteristic
88	#3230	—	5	2	no	—
89	#3170	—	5	1	no	—
90	#3165	—	5	2	no	—
91	C₂×#494	—	5	1	no	—
92	#1723	—	5	2	no	—
93	#1723	—	5	2	no	—
94	#1713	—	5	1	no	—
95	#1713	—	5	1	no	—
96	#1707	—	5	1	no	—
97	#1707	D₅	1	1	yes	characteristic, complemented
98	C₈×M₁₆	—	5	1	no	—
99	C₂×C₅×Q₈	—	2	2	no	—
100	C₂×C₅×Q₈	—	2	2	no	—
101	Q₈×D₅	—	4	8	no	—
102	C₂×C₄×D₅	—	4	2	no	—
103	C₂×C₄×D₅	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×D₅	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×D₅	C₂×D₄	1	1	yes	characteristic
106	C₂×Dic₁₀	—	2	2	no	—
107	C₂×Dic₁₀	—	2	2	no	—
108	C₂×Dic₁₀	—	4	2	no	—
109	C₅×Q₁₆	—	4	4	no	—
110	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
111	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
112	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
113	C₅×C₄⋊C₄	—	2	2	no	—
114	C₅×C₄⋊C₄	—	2	2	no	—
115	C₅×C₄⋊C₄	—	2	2	no	—
116	C₅×C₄⋊C₄	—	2	2	no	—
117	C₄²×C₅	C₂⁴	1	1	yes	characteristic
118	#689	—	4	8	no	—
119	#685	—	4	2	no	—
120	C₄⋊Dic₅	—	2	2	no	—
121	C₄⋊Dic₅	—	2	2	no	—
122	C₄⋊Dic₅	—	2	2	no	—
123	C₄⋊Dic₅	—	2	2	no	—
124	C₄⋊Dic₅	—	4	2	no	—
125	#683	—	4	2	no	—
126	#683	—	4	2	no	—
127	#683	—	4	2	no	—
128	C₄×Dic₅	—	4	2	no	—
129	C₄×Dic₅	C₂×D₄	1	1	yes	characteristic
130	C₄×Dic₅	C₂×D₄	1	1	yes	characteristic
131	C₄×Dic₅	C₂×D₄	1	1	yes	characteristic
132	C₂×#168	C₂×D₄	1	2	yes	—
133	C₂×#168	C₂×D₄	1	2	yes	—
134	C₂×#168	C₂×D₄	1	2	yes	—
135	Dic₂₀	—	4	4	no	—
136	Dic₂₀	—	4	4	no	—
137	#676	Q₁₆	1	8	yes	complemented
138	C₈×D₅	—	2	4	no	—
139	C₂²×Q₁₆	—	10	2	no	—
140	#533	—	5	2	no	—
141	C₂×C₄⋊Q₈	—	5	2	no	—
142	#501	—	10	4	no	—
143	#500	—	5	2	no	—
144	#500	—	5	2	no	—
145	#500	—	5	2	no	—
146	#500	—	5	2	no	—
147	#500	C₂×D₅	1	1	yes	characteristic
148	#498	—	5	2	no	—
149	#498	C₂×D₅	1	2	yes	—
150	#494	—	5	2	no	—
151	#494	—	5	1	no	—
152	#494	—	5	1	no	—
153	#494	—	10	2	no	—
154	#494	C₂×D₅	1	1	yes	characteristic
155	#487	—	5	2	no	—
156	#487	—	5	2	no	—
157	#487	C₂×D₅	1	2	yes	—
158	#484	—	10	4	no	—
159	#477	—	10	4	no	—
160	#467	—	10	2	no	—
161	C₄×Q₁₆	—	10	2	no	—
162	#431	—	5	1	no	—
163	#428	—	10	4	no	—
164	C₄×M₁₆	—	5	1	no	—
165	C₂×C₄×C₈	—	5	1	no	—
166	#322	—	5	1	no	—
167	#322	—	5	1	no	—
168	C₈²	—	5	1	no	—
169	C₈²	C₂×D₅	1	1	yes	characteristic
170	C₂²×D₅	#127	1	1	yes	characteristic
171	C₅×Q₈	—	4	4	no	—
172	C₅×Q₈	—	4	4	no	—
173	C₂×C₄×C₅	—	4	2	no	—
174	C₂×C₄×C₅	—	4	2	no	—
175	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
178	C₂×Dic₅	—	4	2	no	—
179	C₂×Dic₅	—	4	2	no	—
180	C₂×Dic₅	#127	1	1	yes	characteristic
181	C₂×Dic₅	#127	1	1	yes	characteristic
182	C₂×Dic₅	#127	1	1	yes	characteristic
183	C₄×D₅	—	2	2	no	—
184	C₄×D₅	—	2	2	no	—
185	C₄×D₅	—	8	4	no	—
186	C₄×D₅	C₂×Q₁₆	1	4	yes	—
187	Dic₁₀	—	4	4	no	—
188	Dic₁₀	—	4	4	no	—
189	Dic₁₀	—	8	4	no	—
190	C₅×C₈	—	2	2	no	—
191	C₅×C₈	C₂×Q₁₆	1	4	yes	complemented
192	C₅×C₈	AGL(1,ℤ/8)	1	4	yes	—
193	#168	—	2	2	no	—
194	#168	—	2	2	no	—
195	#168	C₂×Q₁₆	1	4	yes	complemented
196	C₂²×Q₈	—	10	2	no	—
197	C₂×Q₁₆	—	2	2	no	—
198	C₂×Q₁₆	—	10	4	no	—
199	C₂×Q₁₆	—	10	2	no	—
200	C₂×Q₁₆	—	10	2	no	—
201	C₂×Q₁₆	—	20	8	no	—
202	C₂×M₁₆	—	5	2	no	—
203	C₂²×C₈	—	5	2	no	—
204	C₄⋊Q₈	—	5	2	no	—
205	C₄⋊Q₈	—	5	2	no	—
206	C₄⋊Q₈	—	10	2	no	—
207	C₄⋊Q₈	—	10	2	no	—
208	C₄⋊Q₈	—	10	2	no	—
209	C₄⋊Q₈	C₂²×D₅	1	2	yes	—
210	C₄⋊Q₈	C₂²×D₅	1	2	yes	—
211	#125	—	10	2	no	—
212	C₂²⋊Q₈	—	10	2	no	—
213	C₂²⋊Q₈	—	10	2	no	—
214	C₄×Q₈	—	10	2	no	—
215	C₄×Q₈	—	10	2	no	—
216	#117	—	10	2	no	—
217	C₂×C₄⋊C₄	—	10	2	no	—
218	C₂×C₄⋊C₄	—	10	2	no	—
219	C₂×C₄²	—	5	1	no	—
220	#107	—	2	2	no	—
221	#107	—	10	4	no	—
222	#107	—	10	4	no	—
223	#107	—	10	4	no	—
224	#107	—	10	4	no	—
225	#106	—	2	4	no	—
226	#106	—	10	4	no	—
227	C₄⋊C₈	—	5	2	no	—
228	#103	—	2	4	no	—
229	#103	—	10	4	no	—
230	#103	—	10	4	no	—
231	C₂²⋊C₈	—	5	2	no	—
232	#97	—	5	2	no	—
233	C₄×C₈	—	5	2	no	—
234	C₄×C₈	—	5	1	no	—
235	C₄×C₈	—	5	1	no	—
236	C₄×C₈	—	5	1	no	—
237	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
238	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
239	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
240	C₂²×C₅	C₂×#127	1	1	yes	characteristic
241	C₂×D₅	—	2	1	no	—
242	C₂×D₅	—	2	1	no	—
243	C₂×D₅	#494	1	2	yes	—
244	C₄×C₅	—	8	2	no	—
245	C₄×C₅	—	8	2	no	—
246	C₄×C₅	C₂²×Q₁₆	1	2	yes	—
247	C₄×C₅	C₂×AGL(1,ℤ/8)	1	2	yes	—
248	C₄×C₅	#578	1	2	yes	—
249	Dic₅	—	2	1	no	—
250	Dic₅	—	2	1	no	—
251	Dic₅	—	8	2	no	—
252	Dic₅	—	8	2	no	—
253	Dic₅	#494	1	2	yes	—
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	—	2	2	no	—
256	C₂×Q₈	—	10	2	no	—
257	C₂×Q₈	—	10	2	no	—
258	C₂×Q₈	—	20	2	no	—
259	C₂×Q₈	—	20	8	no	—
260	C₂²×C₄	—	5	1	no	—
261	C₂²×C₄	—	5	1	no	—
262	C₂²×C₄	—	5	1	no	—
263	C₂²×C₄	—	20	2	no	—
264	Q₁₆	—	4	4	no	—
265	Q₁₆	—	20	8	no	—
266	Q₁₆	—	20	4	no	—
267	Q₁₆	—	20	4	no	—
268	M₁₆	—	5	8	no	—
269	C₂×C₈	—	5	2	no	—
270	C₂×C₈	—	5	2	no	—
271	C₂×C₈	—	5	2	no	—
272	C₂×C₈	—	10	4	no	—
273	C₂×C₈	D₄×D₅	1	2	yes	—
274	C₂×C₈	D₄×D₅	1	2	yes	—
275	C₂×C₈	D₄×D₅	1	2	yes	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	10	2	no	—
281	C₄⋊C₄	—	10	2	no	—
282	C₄⋊C₄	—	10	2	no	—
283	C₄⋊C₄	—	10	2	no	—
284	C₄⋊C₄	—	20	2	no	—
285	C₄⋊C₄	—	20	2	no	—
286	C₄⋊C₄	—	20	2	no	—
287	C₄⋊C₄	—	20	2	no	—
288	C₂²⋊C₄	—	20	2	no	—
289	C₄²	—	5	1	no	—
290	C₄²	—	5	1	no	—
291	C₄²	—	5	1	no	—
292	C₄²	—	20	2	no	—
293	C₄²	C₂³×D₅	1	1	yes	characteristic
294	C₂×C₅	C₂×#494	1	1	yes	characteristic
295	C₂×C₅	#3151	1	1	yes	characteristic
296	C₂×C₅	#3156	1	1	yes	characteristic
297	D₅	—	2	2	no	—
298	C₂³	—	5	1	no	—
299	Q₈	—	4	4	no	—
300	Q₈	—	4	4	no	—
301	Q₈	—	20	4	no	—
302	Q₈	—	20	4	no	—
303	Q₈	—	40	4	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	5	1	no	—
307	C₂×C₄	—	5	4	no	—
308	C₂×C₄	—	5	1	no	—
309	C₂×C₄	—	5	1	no	—
310	C₂×C₄	—	10	2	no	—
311	C₂×C₄	—	10	2	no	—
312	C₂×C₄	—	20	2	no	—
313	C₂×C₄	—	20	2	no	—
314	C₂×C₄	—	40	4	no	—
315	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
316	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
317	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
318	C₈	—	2	2	no	—
319	C₈	—	5	4	no	—
320	C₈	—	10	2	no	—
321	C₈	—	10	2	no	—
322	C₈	#4060	1	4	yes	—
323	C₈	D₅×Q₁₆	1	4	yes	complemented
324	C₅	#19632	1	1	yes	characteristic, complemented, Sylow
325	C₂²	—	5	2	no	—
326	C₂²	—	10	1	no	—
327	C₂²	—	10	1	no	—
328	C₂²	D₅×#127	1	1	yes	characteristic
329	C₄	—	5	2	no	—
330	C₄	—	8	2	no	—
331	C₄	—	8	2	no	—
332	C₄	—	10	1	no	—
333	C₄	—	10	1	no	—
334	C₄	—	40	2	no	—
335	C₄	—	40	2	no	—
336	C₄	C₂×#4060	1	2	yes	—
337	C₄	C₂×D₅×Q₁₆	1	2	yes	—
338	C₄	#66105	1	2	yes	—
339	C₂	—	10	2	no	—
340	C₂	#10613492	1	1	yes	characteristic
341	C₂	D₅×#494	1	1	yes	characteristic
342	C₂	#10613555	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₅×#127
commutator, cosocle	C₄²×C₅	C₂⁴
Frattini	C₄²	C₂³×D₅
Fitting	C₅×#1707	C₂
socle	C₂²×C₅	C₂×#127
prime radical	C₂²×D₅	#127
hypercenter	#1707	D₅
hypocenter	C₅	#19632
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1707

Complements

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

N	H ≅ G/N	type
5. C₅×#1707	C₂	semidirect
6. #10602153	C₂	semidirect
7. #10602134	C₂	semidirect
8. #10602091	C₂	semidirect
9. #10602020	C₂	semidirect
10. #10598858	C₂	semidirect
97. #1707	D₅	semidirect
137. #676	Q₁₆	semidirect
191. C₅×C₈	C₂×Q₁₆	semidirect
195. #168	C₂×Q₁₆	semidirect
323. C₈	D₅×Q₁₆	semidirect
324. C₅	#19632	semidirect

Maximal subgroups

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

subgroup	index
#10613665	2
#10613624	2
D₅×#494	2
#10613339	2
C₅×#1707	2
#10602153	2
#10602134	2
#10602091	2
#10602020	2
#10598858	2
C₈×#676	2
#19632	5