Subgroups — #13043662

Order 1280.

526 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10611056	C₂	1	1	yes	maximal, characteristic
2	#10604152	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×#674	C₂	1	1	yes	maximal, characteristic, complemented
4	#10604025	C₂	1	1	yes	maximal, characteristic, complemented
5	#10604024	C₂	1	1	yes	maximal, characteristic, complemented
6	#10604023	C₂	1	1	yes	maximal, characteristic, complemented
7	#10604021	C₂	1	1	yes	maximal, characteristic, complemented
8	#10603904	C₂	1	2	yes	maximal, complemented
9	#10603903	C₂	1	2	yes	maximal, complemented
10	#10603870	C₂	1	2	yes	maximal, complemented
11	#10603868	C₂	1	2	yes	maximal, complemented
12	#65813	C₄	1	1	yes	characteristic
13	C₂²×#674	C₂²	1	1	yes	characteristic
14	C₂²×#674	C₂²	1	1	yes	characteristic
15	C₂×#3990	C₂²	1	2	yes	—
16	C₂×D₅⋊C₁₆	C₂²	1	2	yes	—
17	C₅×#432	C₄	1	1	yes	characteristic
18	C₂×C₄×#168	C₂²	1	1	yes	characteristic
19	#65127	C₄	1	1	yes	characteristic
20	#65123	C₂²	1	1	yes	characteristic
21	#65120	C₄	1	1	yes	characteristic
22	#65119	C₂²	1	1	yes	characteristic
23	#65019	C₄	1	2	yes	—
24	#65014	C₂²	1	2	yes	—
25	#64998	C₄	1	2	yes	—
26	#64997	C₂²	1	2	yes	—
27	C₂²⋊#674	C₂²	1	2	yes	—
28	#64888	C₂²	1	2	yes	—
29	#64888	C₂²	1	2	yes	—
30	#64887	C₂²	1	2	yes	—
31	#64887	C₂²	1	2	yes	—
32	#64886	C₂²	1	2	yes	—
33	#64885	C₂²	1	2	yes	—
34	#64859	C₂²	1	2	yes	—
35	C₄⋊#674	C₂²	1	1	yes	characteristic
36	C₄⋊#674	C₂²	1	1	yes	characteristic
37	C₄×#674	C₂²	1	1	yes	characteristic
38	C₄×#674	C₂²	1	1	yes	characteristic
39	C₄×#674	C₂²	1	1	yes	characteristic
40	C₄×#674	C₂²	1	1	yes	characteristic
41	#33480	—	5	1	no	maximal, Sylow
42	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
43	#4076	C₈	1	1	yes	characteristic
44	#4075	C₈	1	2	yes	—
45	C₄×C₂²⋊D₅	C₂×C₄	1	2	yes	—
46	#4070	C₈	1	1	yes	characteristic
47	C₂²×#168	C₂×C₄	1	1	yes	characteristic
48	C₂²×#168	C₂³	1	1	yes	characteristic
49	C₂×#676	C₂×C₄	1	2	yes	—
50	C₂×C₈×D₅	C₂³	1	2	yes	—
51	#4022	C₈	1	2	yes	—
52	#4021	C₈	1	1	yes	characteristic
53	C₄×D₂₀	C₂×C₄	1	1	yes	characteristic
54	#4018	C₂×C₄	1	2	yes	—
55	#4016	C₈	1	1	yes	characteristic
56	C₄×Dic₁₀	C₂×C₄	1	1	yes	characteristic
57	C₂×#674	—	2	2	no	—
58	C₂×#674	—	2	2	no	—
59	C₂×#674	C₂³	1	1	yes	characteristic
60	C₂×#674	C₂³	1	1	yes	characteristic
61	C₂×#674	C₂³	1	2	yes	—
62	C₂×#674	C₂³	1	2	yes	—
63	C₂×#674	C₂³	1	1	yes	characteristic
64	C₂×#674	C₂³	1	1	yes	characteristic
65	#3990	—	2	4	no	—
66	D₅⋊C₁₆	—	2	4	no	—
67	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
68	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
69	C₅×C₂²⋊C₈	C₂×C₄	1	2	yes	—
70	C₅×#97	C₂×C₄	1	2	yes	—
71	#3952	C₂×C₄	1	2	yes	—
72	#3952	C₂×C₄	1	2	yes	—
73	#3947	C₂×C₄	1	2	yes	—
74	#3946	C₂×C₄	1	2	yes	—
75	#3946	C₂×C₄	1	2	yes	—
76	C₈×Dic₅	C₂³	1	2	yes	—
77	C₄×#168	C₂×C₄	1	1	yes	characteristic
78	C₄×#168	C₂×C₄	1	1	yes	characteristic
79	C₄×#168	C₂³	1	1	yes	characteristic
80	C₄×#168	C₂³	1	1	yes	characteristic
81	#2966	—	5	1	no	—
82	#2183	—	5	1	no	—
83	Q₈×C₁₆	—	5	1	no	—
84	#2169	—	5	1	no	—
85	#2168	—	5	2	no	—
86	D₄×C₁₆	—	5	2	no	—
87	D₄×C₁₆	—	5	1	no	—
88	#2164	—	5	2	no	—
89	#2163	—	5	1	no	—
90	#2108	—	5	2	no	—
91	C₂×C₄×C₁₆	—	5	1	no	—
92	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
93	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
94	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₂²⋊D₅	C₂×C₈	1	2	yes	—
96	C₂×D₂₀	C₂×C₈	1	1	yes	characteristic
97	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
98	C₂×Dic₁₀	C₂×C₈	1	1	yes	characteristic
99	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
100	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
101	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
102	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
103	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
105	C₂²⋊Dic₅	C₂×C₈	1	2	yes	—
106	#685	C₂×C₈	1	2	yes	—
107	#685	C₂×C₈	1	2	yes	—
108	#685	C₂×C₈	1	2	yes	—
109	#685	C₂×C₈	1	2	yes	—
110	C₄⋊Dic₅	C₂×C₈	1	1	yes	characteristic
111	C₄⋊Dic₅	C₂×C₈	1	1	yes	characteristic
112	#683	C₂×C₈	1	2	yes	—
113	#683	C₂×C₈	1	2	yes	—
114	#683	C₂×C₈	1	2	yes	—
115	#683	C₂×C₈	1	2	yes	—
116	C₄×Dic₅	C₂²×C₄	1	2	yes	—
117	C₂×#168	—	2	2	no	—
118	C₂×#168	—	2	1	no	—
119	C₂×#168	—	2	1	no	—
120	C₂×#168	C₂²×C₄	1	1	yes	characteristic
121	C₂×#168	C₂²×C₄	1	1	yes	characteristic
122	C₂×#168	C₂²×C₄	1	1	yes	characteristic
123	C₂×#168	C₂⁴	1	1	yes	characteristic
124	#676	—	2	4	no	—
125	C₈×D₅	—	2	2	no	—
126	C₈×D₅	—	2	2	no	—
127	#674	—	2	2	no	—
128	#674	—	2	2	no	—
129	#674	—	2	1	no	—
130	#674	—	2	1	no	—
131	#674	#41	1	2	yes	—
132	#674	#41	1	2	yes	—
133	#529	—	5	1	no	—
134	C₂×M₃₂	—	5	2	no	—
135	C₂²×C₁₆	—	5	1	no	—
136	C₂²×C₁₆	—	5	1	no	—
137	C₂²×C₁₆	—	5	2	no	—
138	#446	—	5	1	no	—
139	C₈×Q₈	—	5	1	no	—
140	#436	—	5	1	no	—
141	#435	—	5	2	no	—
142	C₈×D₄	—	5	1	no	—
143	C₈×D₄	—	5	2	no	—
144	#433	—	5	2	no	—
145	#432	AGL(1,𝔽₅)	1	1	yes	characteristic
146	#405	—	5	2	no	—
147	C₂×C₄×C₈	—	5	1	no	—
148	C₄⋊C₁₆	—	5	1	no	—
149	C₄⋊C₁₆	—	5	1	no	—
150	C₄⋊C₁₆	—	5	2	no	—
151	C₄⋊C₁₆	—	5	2	no	—
152	C₂²⋊C₁₆	—	5	2	no	—
153	C₂²⋊C₁₆	—	5	2	no	—
154	C₂²⋊C₁₆	—	5	2	no	—
155	#346	—	5	2	no	—
156	#346	—	5	2	no	—
157	C₄×C₁₆	—	5	1	no	—
158	C₄×C₁₆	—	5	1	no	—
159	C₄×C₁₆	—	5	1	no	—
160	C₄×C₁₆	—	5	1	no	—
161	C₄×C₁₆	—	5	2	no	—
162	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
163	C₂²×D₅	C₂²×C₈	1	2	yes	—
164	C₂×C₄×C₅	—	2	2	no	—
165	C₂×C₄×C₅	—	2	1	no	—
166	C₂×C₄×C₅	—	2	1	no	—
167	C₂×C₄×C₅	—	2	2	no	—
168	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
169	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
170	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
172	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
174	C₂²⋊D₅	—	2	4	no	—
175	C₂×Dic₅	C₂²×C₈	1	2	yes	—
176	C₂×Dic₅	C₂²×C₈	1	2	yes	—
177	C₂×Dic₅	C₂²×C₈	1	2	yes	—
178	D₂₀	—	2	2	no	—
179	C₄×D₅	—	2	2	no	—
180	C₄×D₅	—	2	2	no	—
181	Dic₁₀	—	2	2	no	—
182	C₅×C₈	—	2	2	no	—
183	C₅×C₈	—	2	2	no	—
184	#168	—	2	1	no	—
185	#168	—	2	1	no	—
186	#168	#131	1	2	yes	—
187	#168	C₂×#41	1	1	yes	characteristic
188	#168	C₂×#41	1	1	yes	characteristic
189	C₂×M₁₆	—	5	2	no	—
190	C₂²×C₈	—	5	1	no	—
191	C₂²×C₈	—	5	1	no	—
192	C₂²×C₈	—	5	2	no	—
193	#126	—	5	2	no	—
194	#125	—	5	1	no	—
195	#124	—	5	1	no	—
196	#123	—	5	2	no	—
197	C₂²⋊Q₈	—	5	1	no	—
198	#121	—	5	1	no	—
199	C₄×Q₈	—	5	1	no	—
200	C₄×D₄	—	5	2	no	—
201	C₄×D₄	—	5	1	no	—
202	#117	—	5	2	no	—
203	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
204	M₃₂	—	10	4	no	—
205	C₂×C₁₆	—	5	1	no	—
206	C₂×C₁₆	—	5	1	no	—
207	C₂×C₁₆	—	5	1	no	—
208	C₂×C₁₆	—	5	1	no	—
209	C₂×C₁₆	—	5	2	no	—
210	C₂×C₁₆	—	5	2	no	—
211	C₂×C₁₆	—	10	4	no	—
212	C₂×C₁₆	—	10	2	no	—
213	C₂×C₁₆	—	10	2	no	—
214	C₄⋊C₈	—	5	2	no	—
215	C₄⋊C₈	—	5	2	no	—
216	C₄⋊C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
217	C₄⋊C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
218	C₂²⋊C₈	—	5	2	no	—
219	C₂²⋊C₈	—	5	2	no	—
220	C₂²⋊C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
221	#97	—	5	2	no	—
222	#97	C₂×AGL(1,𝔽₅)	1	2	yes	—
223	C₄×C₈	—	5	2	no	—
224	C₄×C₈	—	5	1	no	—
225	C₄×C₈	—	5	1	no	—
226	C₄×C₈	—	5	1	no	—
227	C₄×C₈	—	5	1	no	—
228	C₂²×C₅	—	2	1	no	—
229	C₂²×C₅	—	2	1	no	—
230	C₂²×C₅	#504	1	2	yes	—
231	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
232	C₂×D₅	—	2	2	no	—
233	C₂×D₅	—	2	2	no	—
234	C₂×D₅	—	2	2	no	—
235	C₄×C₅	—	2	1	no	—
236	C₄×C₅	—	2	1	no	—
237	C₄×C₅	—	2	1	no	—
238	C₄×C₅	—	2	1	no	—
239	C₄×C₅	#504	1	2	yes	—
240	C₄×C₅	C₄×#41	1	1	yes	characteristic
241	C₄×C₅	C₂×#131	1	1	yes	characteristic
242	Dic₅	—	2	2	no	—
243	Dic₅	—	2	2	no	—
244	Dic₅	—	2	2	no	—
245	C₂×Q₈	—	5	1	no	—
246	C₂×D₄	—	5	2	no	—
247	C₂×D₄	—	5	1	no	—
248	C₂²×C₄	—	5	2	no	—
249	C₂²×C₄	D₅⋊C₈	1	1	yes	characteristic
250	C₂²×C₄	D₅⋊C₈	1	1	yes	characteristic
251	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
252	M₁₆	—	10	4	no	—
253	C₂×C₈	—	5	1	no	—
254	C₂×C₈	—	5	1	no	—
255	C₂×C₈	—	5	1	no	—
256	C₂×C₈	—	5	1	no	—
257	C₂×C₈	—	10	2	no	—
258	C₂×C₈	—	10	2	no	—
259	C₂×C₈	—	10	1	no	—
260	C₂×C₈	—	10	1	no	—
261	C₂×C₈	—	10	2	no	—
262	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	2	yes	—
263	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	2	yes	—
264	C₄⋊C₄	—	5	2	no	—
265	C₄⋊C₄	—	5	2	no	—
266	C₄⋊C₄	—	5	1	no	—
267	C₄⋊C₄	—	5	2	no	—
268	C₄⋊C₄	—	5	1	no	—
269	C₄⋊C₄	—	5	2	no	—
270	C₂²⋊C₄	—	5	2	no	—
271	C₂²⋊C₄	—	5	2	no	—
272	C₂²⋊C₄	—	5	2	no	—
273	C₂²⋊C₄	—	5	2	no	—
274	C₂²⋊C₄	—	5	2	no	—
275	C₄²	—	5	2	no	—
276	C₄²	D₅⋊C₈	1	1	yes	characteristic
277	C₄²	D₅⋊C₈	1	1	yes	characteristic
278	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
279	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
280	C₁₆	—	5	2	no	—
281	C₁₆	—	5	2	no	—
282	C₁₆	—	10	1	no	—
283	C₁₆	—	10	1	no	—
284	C₁₆	—	10	2	no	—
285	C₁₆	—	10	2	no	—
286	C₂×C₅	—	2	2	no	—
287	C₂×C₅	C₈×#41	1	1	yes	characteristic
288	C₂×C₅	C₂×#504	1	1	yes	characteristic
289	C₂×C₅	C₂×#504	1	1	yes	characteristic
290	D₅	—	4	2	no	—
291	C₂³	—	5	2	no	—
292	C₂³	C₂×D₅⋊C₈	1	1	yes	characteristic
293	Q₈	—	10	2	no	—
294	D₄	—	10	4	no	—
295	D₄	—	10	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	5	2	no	—
301	C₂×C₄	—	5	2	no	—
302	C₂×C₄	—	5	2	no	—
303	C₂×C₄	—	10	2	no	—
304	C₂×C₄	—	10	2	no	—
305	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
306	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
307	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
308	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
309	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
310	C₂×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
311	C₈	—	2	2	no	—
312	C₈	—	2	2	no	—
313	C₈	—	5	2	no	—
314	C₈	—	5	1	no	—
315	C₈	—	5	1	no	—
316	C₈	—	10	1	no	—
317	C₈	—	10	1	no	—
318	C₅	#33480	1	1	yes	characteristic, complemented, Sylow
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	10	2	no	—
322	C₂²	—	10	2	no	—
323	C₂²	—	10	2	no	—
324	C₂²	#65725	1	2	yes	—
325	C₂²	C₂²×D₅⋊C₈	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	10	2	no	—
331	C₄	—	10	2	no	—
332	C₄	—	10	2	no	—
333	C₄	#65725	1	2	yes	—
334	C₄	D₅⋊#131	1	1	yes	characteristic
335	C₄	#41×AGL(1,𝔽₅)	1	1	yes	characteristic
336	C₂	—	2	2	no	—
337	C₂	—	20	2	no	—
338	C₂	#10617114	1	1	yes	characteristic
339	C₂	C₂×#65725	1	1	yes	characteristic
340	C₂	C₂×#65725	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂³×C₈
Fitting	C₅×#432	C₄
prime radical	#4076	C₈
cosocle	C₂×#168	C₂⁴
hypercenter	#432	AGL(1,𝔽₅)
hypocenter	C₅	#33480
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #432

Complements

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

N	H ≅ G/N	type
2. #10604152	C₂	semidirect
3. C₂×C₄×#674	C₂	semidirect
4. #10604025	C₂	semidirect
5. #10604024	C₂	semidirect
6. #10604023	C₂	semidirect
7. #10604021	C₂	semidirect
8. #10603904	C₂	semidirect
9. #10603903	C₂	semidirect
10. #10603870	C₂	semidirect
11. #10603868	C₂	semidirect
318. C₅	#33480	semidirect

Maximal subgroups

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

subgroup	index
#10611056	2
#10604152	2
C₂×C₄×#674	2
#10604025	2
#10604024	2
#10604023	2
#10604021	2
#10603904	2
#10603903	2
#10603870	2
#10603868	2
#33480	5