Subgroups — #13043667

Order 1280.

508 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	#10611249	C₂	1	1	yes	maximal, characteristic
2	#10604747	C₂	1	2	yes	maximal, complemented
3	#10604741	C₂	1	2	yes	maximal, complemented
4	#10604413	C₂	1	2	yes	maximal, complemented
5	#10604026	C₂	1	2	yes	maximal, complemented
6	#10603906	C₂	1	2	yes	maximal, complemented
7	#10603904	C₂	1	2	yes	maximal, complemented
8	#10603831	C₂	1	1	yes	maximal, characteristic, complemented
9	#10603825	C₂	1	1	yes	maximal, characteristic, complemented
10	#65891	C₄	1	1	yes	characteristic
11	C₂×#3998	C₂²	1	2	yes	—
12	C₂²×#674	C₂²	1	2	yes	—
13	C₂×#3990	C₂²	1	2	yes	—
14	C₅×#436	C₄	1	1	yes	characteristic
15	#65416	C₄	1	2	yes	—
16	#65411	C₂²	1	2	yes	—
17	D₄×#168	C₂²	1	1	yes	characteristic
18	#65124	C₄	1	1	yes	characteristic
19	#65119	C₂²	1	1	yes	characteristic
20	#65014	C₂²	1	2	yes	—
21	#65009	C₄	1	2	yes	—
22	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
23	#64968	C₄	1	1	yes	characteristic
24	C₂²⋊#674	C₂²	1	2	yes	—
25	C₂²⋊#674	C₂²	1	2	yes	—
26	#64888	C₂²	1	2	yes	—
27	#64888	C₂²	1	2	yes	—
28	#64887	C₂²	1	1	yes	characteristic
29	#64887	C₂²	1	1	yes	characteristic
30	#64886	C₂²	1	2	yes	—
31	#64886	C₂²	1	2	yes	—
32	#64885	C₂²	1	1	yes	characteristic
33	#64885	C₂²	1	1	yes	characteristic
34	#64885	C₂²	1	2	yes	—
35	#64859	C₂²	1	2	yes	—
36	C₄⋊#674	C₂²	1	2	yes	—
37	#33481	—	5	1	no	maximal, Sylow
38	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
39	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
40	#4084	C₈	1	1	yes	characteristic
41	#4082	C₈	1	1	yes	characteristic
42	C₄×C₂²⋊D₅	C₂×C₄	1	2	yes	—
43	#4072	C₂×C₄	1	2	yes	—
44	C₂²×#168	C₂³	1	2	yes	—
45	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×D₅	C₂³	1	1	yes	characteristic
47	#4043	C₈	1	1	yes	characteristic
48	#4036	C₈	1	1	yes	characteristic
49	#4029	C₈	1	2	yes	—
50	#4025	C₈	1	2	yes	—
51	C₄²×D₅	C₂×C₄	1	1	yes	characteristic
52	C₄×Dic₁₀	C₂×C₄	1	1	yes	characteristic
53	#3998	—	2	4	no	—
54	C₂×#674	—	2	4	no	—
55	C₂×#674	C₂³	1	2	yes	—
56	C₂×#674	C₂³	1	2	yes	—
57	C₂×#674	C₂³	1	2	yes	—
58	C₂×#674	C₂³	1	2	yes	—
59	#3990	—	2	4	no	—
60	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
61	C₅×C₂²⋊C₈	C₂×C₄	1	2	yes	—
62	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
63	C₂²⋊#168	C₂×C₄	1	2	yes	—
64	#3952	C₂×C₄	1	2	yes	—
65	#3947	C₂×C₄	1	2	yes	—
66	#3947	C₂×C₄	1	2	yes	—
67	#3946	C₂×C₄	1	2	yes	—
68	#3946	C₂×C₄	1	2	yes	—
69	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
70	C₈×Dic₅	C₂³	1	2	yes	—
71	C₈×Dic₅	C₂³	1	1	yes	characteristic
72	C₄⋊#168	C₂×C₄	1	1	yes	characteristic
73	C₄×#168	C₂³	1	1	yes	characteristic
74	#2966	—	5	1	no	—
75	#2183	—	5	2	no	—
76	#2169	—	5	2	no	—
77	#2168	—	5	2	no	—
78	#2168	—	5	2	no	—
79	#2164	—	5	2	no	—
80	#2163	—	5	1	no	—
81	#2108	—	5	2	no	—
82	C₄×M₃₂	—	5	1	no	—
83	C₂×C₅×D₄	C₂×C₈	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
85	C₂×C₂²⋊D₅	C₂×C₈	1	2	yes	—
86	C₂×C₄×D₅	C₂×C₈	1	1	yes	characteristic
87	C₂×C₄×D₅	C₂×C₈	1	1	yes	characteristic
88	C₂×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
89	C₂×Dic₁₀	C₂×C₈	1	1	yes	characteristic
90	C₅×M₁₆	—	2	4	no	—
91	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
92	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
94	C₅×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
95	C₅×C₂²⋊C₄	C₂×C₈	1	2	yes	—
96	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
97	C₂²⋊Dic₅	C₂×C₈	1	2	yes	—
98	C₂²⋊Dic₅	C₂×C₈	1	2	yes	—
99	C₂²⋊Dic₅	C₂×C₈	1	2	yes	—
100	#685	C₂×C₈	1	2	yes	—
101	C₄⋊Dic₅	C₂×C₈	1	1	yes	characteristic
102	C₄⋊Dic₅	C₂×C₈	1	2	yes	—
103	C₄⋊Dic₅	C₂×C₈	1	2	yes	—
104	#683	C₂×C₈	1	2	yes	—
105	#683	C₂×C₈	1	2	yes	—
106	C₄×Dic₅	C₂×C₈	1	1	yes	characteristic
107	C₄×Dic₅	C₂×C₈	1	1	yes	characteristic
108	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
109	C₄×Dic₅	C₂²×C₄	1	2	yes	—
110	C₂×#168	—	2	2	no	—
111	C₂×#168	—	2	2	no	—
112	C₂×#168	C₂²×C₄	1	1	yes	characteristic
113	C₂×#168	C₂²×C₄	1	2	yes	—
114	C₂×#168	C₂⁴	1	1	yes	characteristic
115	C₈×D₅	—	2	2	no	—
116	C₈×D₅	—	2	2	no	—
117	#674	—	2	2	no	—
118	#674	—	2	2	no	—
119	#674	—	2	2	no	—
120	#674	#41	1	4	yes	—
121	#529	—	5	1	no	—
122	C₂×M₃₂	—	5	2	no	—
123	C₂×M₃₂	—	5	2	no	—
124	C₂²×C₁₆	—	5	2	no	—
125	#446	—	5	1	no	—
126	C₈×Q₈	—	5	1	no	—
127	#436	AGL(1,𝔽₅)	1	1	yes	characteristic
128	#435	—	5	2	no	—
129	C₈×D₄	—	5	1	no	—
130	C₈×D₄	—	5	2	no	—
131	#433	—	5	2	no	—
132	#432	—	5	1	no	—
133	#405	—	5	2	no	—
134	C₂×C₄×C₈	—	5	1	no	—
135	C₄⋊C₁₆	—	5	2	no	—
136	C₄⋊C₁₆	—	5	2	no	—
137	C₄⋊C₁₆	—	5	2	no	—
138	C₂²⋊C₁₆	—	5	1	no	—
139	C₂²⋊C₁₆	—	5	1	no	—
140	C₂²⋊C₁₆	—	5	2	no	—
141	C₂²⋊C₁₆	—	5	2	no	—
142	#346	—	5	2	no	—
143	#346	—	5	2	no	—
144	#346	—	5	2	no	—
145	C₄×C₁₆	—	5	1	no	—
146	C₄×C₁₆	—	5	1	no	—
147	C₄×C₁₆	—	5	2	no	—
148	C₂³×C₅	C₂²×C₈	1	2	yes	—
149	C₂²×D₅	C₂²×C₈	1	1	yes	characteristic
150	C₅×D₄	—	4	1	no	—
151	C₂×C₄×C₅	—	2	2	no	—
152	C₂×C₄×C₅	—	2	2	no	—
153	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
155	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
156	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
157	C₂²⋊D₅	—	4	2	no	—
158	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
159	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
160	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
161	C₂×Dic₅	C₂²×C₈	1	2	yes	—
162	C₂×Dic₅	C₂²×C₈	1	2	yes	—
163	C₄×D₅	—	2	1	no	—
164	C₄×D₅	—	2	1	no	—
165	C₄×D₅	—	4	1	no	—
166	C₄×D₅	—	4	1	no	—
167	Dic₁₀	—	4	1	no	—
168	C₅×C₈	—	2	1	no	—
169	C₅×C₈	—	2	2	no	—
170	C₅×C₈	#131	1	2	yes	—
171	#168	—	2	2	no	—
172	#168	—	2	1	no	—
173	#168	C₂×#41	1	2	yes	—
174	C₂×M₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	—
175	C₂²×C₈	—	5	2	no	—
176	C₂²×C₈	—	5	1	no	—
177	C₂²×C₈	—	5	1	no	—
178	#126	—	5	2	no	—
179	#125	—	5	1	no	—
180	#124	—	5	1	no	—
181	#123	—	5	2	no	—
182	C₂²⋊Q₈	—	5	1	no	—
183	#121	—	5	1	no	—
184	C₄×Q₈	—	5	1	no	—
185	C₄×D₄	—	5	2	no	—
186	C₄×D₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
187	#117	—	5	2	no	—
188	C₂×C₄²	—	5	1	no	—
189	M₃₂	—	10	4	no	—
190	M₃₂	—	10	4	no	—
191	C₂×C₁₆	—	5	2	no	—
192	C₂×C₁₆	—	5	2	no	—
193	C₂×C₁₆	—	5	2	no	—
194	C₂×C₁₆	—	5	2	no	—
195	C₂×C₁₆	—	10	4	no	—
196	C₄⋊C₈	—	5	2	no	—
197	C₄⋊C₈	—	5	2	no	—
198	C₄⋊C₈	—	5	1	no	—
199	C₄⋊C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
200	C₂²⋊C₈	—	5	2	no	—
201	C₂²⋊C₈	—	5	2	no	—
202	C₂²⋊C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
203	#97	—	5	2	no	—
204	#97	—	5	2	no	—
205	C₄×C₈	—	5	1	no	—
206	C₄×C₈	—	5	2	no	—
207	C₄×C₈	—	5	1	no	—
208	C₄×C₈	—	5	1	no	—
209	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
210	C₂²×C₅	—	2	2	no	—
211	C₂²×C₅	—	2	2	no	—
212	C₂²×C₅	—	2	2	no	—
213	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
214	C₂×D₅	—	2	1	no	—
215	C₂×D₅	—	2	1	no	—
216	C₂×D₅	—	2	1	no	—
217	C₄×C₅	—	2	1	no	—
218	C₄×C₅	—	2	1	no	—
219	C₄×C₅	—	2	2	no	—
220	C₄×C₅	C₄×#41	1	1	yes	characteristic
221	C₄×C₅	C₂×#131	1	1	yes	characteristic
222	Dic₅	—	2	2	no	—
223	Dic₅	—	2	1	no	—
224	Dic₅	—	2	1	no	—
225	Dic₅	—	2	2	no	—
226	Dic₅	—	2	1	no	—
227	C₂×Q₈	—	5	1	no	—
228	C₂×D₄	—	5	2	no	—
229	C₂×D₄	D₅⋊C₈	1	1	yes	characteristic
230	C₂²×C₄	—	5	1	no	—
231	C₂²×C₄	—	5	1	no	—
232	C₂²×C₄	—	5	1	no	—
233	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
234	M₁₆	—	2	4	no	—
235	C₂×C₈	—	5	1	no	—
236	C₂×C₈	—	5	1	no	—
237	C₂×C₈	—	5	2	no	—
238	C₂×C₈	—	10	2	no	—
239	C₂×C₈	—	10	2	no	—
240	C₂×C₈	—	10	2	no	—
241	C₂×C₈	—	10	2	no	—
242	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
243	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	2	yes	—
244	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
245	C₄⋊C₄	—	5	2	no	—
246	C₄⋊C₄	—	5	2	no	—
247	C₄⋊C₄	—	5	2	no	—
248	C₄⋊C₄	—	5	1	no	—
249	C₄⋊C₄	—	5	2	no	—
250	C₄⋊C₄	D₅⋊C₈	1	1	yes	characteristic
251	C₂²⋊C₄	—	5	2	no	—
252	C₂²⋊C₄	—	5	2	no	—
253	C₂²⋊C₄	—	5	2	no	—
254	C₂²⋊C₄	—	5	2	no	—
255	C₂²⋊C₄	D₅⋊C₈	1	2	yes	—
256	C₄²	—	5	1	no	—
257	C₄²	—	5	1	no	—
258	C₄²	—	5	1	no	—
259	C₄²	—	5	2	no	—
260	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
261	C₁₆	—	5	4	no	—
262	C₁₆	—	10	2	no	—
263	C₁₆	—	10	2	no	—
264	C₁₆	—	10	2	no	—
265	C₂×C₅	—	4	2	no	—
266	C₂×C₅	C₈×#41	1	1	yes	characteristic
267	C₂×C₅	#3407	1	1	yes	characteristic
268	C₂×C₅	#3407	1	1	yes	characteristic
269	D₅	—	4	1	no	—
270	C₂³	—	5	1	no	—
271	C₂³	C₂×D₅⋊C₈	1	2	yes	—
272	Q₈	—	20	1	no	—
273	D₄	—	4	1	no	—
274	D₄	—	20	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	5	1	no	—
278	C₂×C₄	—	5	1	no	—
279	C₂×C₄	—	5	1	no	—
280	C₂×C₄	—	5	2	no	—
281	C₂×C₄	—	5	2	no	—
282	C₂×C₄	—	10	1	no	—
283	C₂×C₄	—	10	1	no	—
284	C₂×C₄	—	20	1	no	—
285	C₂×C₄	—	20	1	no	—
286	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
287	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
288	C₂×C₄	C₂×D₅⋊C₈	1	2	yes	—
289	C₂×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
290	C₈	—	2	2	no	—
291	C₈	—	2	1	no	—
292	C₈	—	5	2	no	—
293	C₈	—	10	1	no	—
294	C₈	—	10	2	no	—
295	C₈	#4131	1	2	yes	—
296	C₅	#33481	1	1	yes	characteristic, complemented, Sylow
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	10	1	no	—
301	C₂²	—	10	1	no	—
302	C₂²	—	10	1	no	—
303	C₂²	C₂²×D₅⋊C₈	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	1	no	—
307	C₄	—	10	1	no	—
308	C₄	—	10	2	no	—
309	C₄	—	10	1	no	—
310	C₄	—	10	1	no	—
311	C₄	—	10	2	no	—
312	C₄	C₂×#4131	1	1	yes	characteristic
313	C₄	#41×AGL(1,𝔽₅)	1	1	yes	characteristic
314	C₂	—	4	2	no	—
315	C₂	—	20	1	no	—
316	C₂	#10617153	1	1	yes	characteristic
317	C₂	#10617404	1	1	yes	characteristic
318	C₂	#10617413	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂³×C₈
Fitting	C₅×#436	C₄
prime radical	#4084	C₈
cosocle	C₂×#168	C₂⁴
hypercenter	#436	AGL(1,𝔽₅)
hypocenter	C₅	#33481
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
2. #10604747	C₂	semidirect
3. #10604741	C₂	semidirect
4. #10604413	C₂	semidirect
5. #10604026	C₂	semidirect
6. #10603906	C₂	semidirect
7. #10603904	C₂	semidirect
8. #10603831	C₂	semidirect
9. #10603825	C₂	semidirect
296. C₅	#33481	semidirect

Maximal subgroups

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

subgroup	index
#10611249	2
#10604747	2
#10604741	2
#10604413	2
#10604026	2
#10603906	2
#10603904	2
#10603831	2
#10603825	2
#33481	5