Subgroups — #13043668

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	#10611056	C₂	1	1	yes	maximal, characteristic
2	#10604152	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#3998	C₂	1	1	yes	maximal, characteristic, complemented
4	#10604026	C₂	1	2	yes	maximal, complemented
5	#10604022	C₂	1	2	yes	maximal, complemented
6	#10603906	C₂	1	2	yes	maximal, complemented
7	#10603905	C₂	1	2	yes	maximal, complemented
8	#10603870	C₂	1	2	yes	maximal, complemented
9	#10603868	C₂	1	2	yes	maximal, complemented
10	#65813	C₄	1	1	yes	characteristic
11	C₂×#3998	C₂²	1	2	yes	—
12	C₂×#3990	C₂²	1	2	yes	—
13	C₂×D₅⋊C₁₆	C₂²	1	2	yes	—
14	C₅×#432	C₄	1	1	yes	characteristic
15	C₂×C₄×#168	C₂²	1	1	yes	characteristic
16	#65127	C₄	1	1	yes	characteristic
17	#65123	C₂²	1	1	yes	characteristic
18	#65120	C₄	1	1	yes	characteristic
19	#65119	C₂²	1	1	yes	characteristic
20	#65019	C₄	1	2	yes	—
21	#65014	C₂²	1	2	yes	—
22	#64998	C₄	1	2	yes	—
23	#64997	C₂²	1	2	yes	—
24	C₂²⋊#674	C₂²	1	1	yes	characteristic
25	C₂²⋊#674	C₂²	1	1	yes	characteristic
26	#64888	C₂²	1	2	yes	—
27	#64888	C₂²	1	2	yes	—
28	#64887	C₂²	1	2	yes	—
29	#64887	C₂²	1	2	yes	—
30	#64886	C₂²	1	2	yes	—
31	#64885	C₂²	1	2	yes	—
32	#64859	C₂²	1	2	yes	—
33	#64859	C₂²	1	2	yes	—
34	C₄⋊#674	C₂²	1	2	yes	—
35	C₄×#674	C₂²	1	1	yes	characteristic
36	C₄×#674	C₂²	1	1	yes	characteristic
37	#33481	—	5	1	no	maximal, Sylow
38	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
39	#4076	C₈	1	1	yes	characteristic
40	#4075	C₈	1	2	yes	—
41	C₄×C₂²⋊D₅	C₂×C₄	1	2	yes	—
42	#4070	C₈	1	1	yes	characteristic
43	C₂²×#168	C₂×C₄	1	1	yes	characteristic
44	C₂²×#168	C₂³	1	1	yes	characteristic
45	C₂×#676	C₂×C₄	1	2	yes	—
46	C₂×C₈×D₅	C₂³	1	2	yes	—
47	#4022	C₈	1	2	yes	—
48	#4021	C₈	1	1	yes	characteristic
49	C₄×D₂₀	C₂×C₄	1	1	yes	characteristic
50	#4018	C₂×C₄	1	2	yes	—
51	#4016	C₈	1	1	yes	characteristic
52	C₄×Dic₁₀	C₂×C₄	1	1	yes	characteristic
53	#3998	—	2	4	no	—
54	C₂×#674	C₂³	1	2	yes	—
55	C₂×#674	C₂³	1	2	yes	—
56	C₂×#674	C₂³	1	2	yes	—
57	C₂×#674	C₂³	1	2	yes	—
58	#3990	—	2	4	no	—
59	D₅⋊C₁₆	—	2	4	no	—
60	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
61	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
62	C₅×C₂²⋊C₈	C₂×C₄	1	2	yes	—
63	C₅×#97	C₂×C₄	1	2	yes	—
64	#3952	C₂×C₄	1	2	yes	—
65	#3952	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	C₂³	1	2	yes	—
70	C₄×#168	C₂×C₄	1	1	yes	characteristic
71	C₄×#168	C₂×C₄	1	1	yes	characteristic
72	C₄×#168	C₂³	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	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₂²⋊D₅	C₂×C₈	1	2	yes	—
87	C₂×D₂₀	C₂×C₈	1	1	yes	characteristic
88	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
89	C₂×Dic₁₀	C₂×C₈	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
91	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
92	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
93	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
94	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
96	C₂²⋊Dic₅	C₂×C₈	1	2	yes	—
97	#685	C₂×C₈	1	2	yes	—
98	#685	C₂×C₈	1	2	yes	—
99	#685	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	1	yes	characteristic
103	#683	C₂×C₈	1	2	yes	—
104	#683	C₂×C₈	1	2	yes	—
105	#683	C₂×C₈	1	2	yes	—
106	#683	C₂×C₈	1	2	yes	—
107	C₄×Dic₅	C₂²×C₄	1	2	yes	—
108	C₂×#168	—	2	2	no	—
109	C₂×#168	—	2	2	no	—
110	C₂×#168	C₂²×C₄	1	1	yes	characteristic
111	C₂×#168	C₂²×C₄	1	1	yes	characteristic
112	C₂×#168	C₂²×C₄	1	1	yes	characteristic
113	C₂×#168	C₂⁴	1	1	yes	characteristic
114	#676	—	2	4	no	—
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	—	5	1	no	—
128	#435	—	5	2	no	—
129	C₈×D₄	—	5	1	no	—
130	C₈×D₄	—	5	2	no	—
131	#433	—	5	2	no	—
132	#432	AGL(1,𝔽₅)	1	1	yes	characteristic
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	1	yes	characteristic
149	C₂²×D₅	C₂²×C₈	1	2	yes	—
150	C₂×C₄×C₅	—	2	1	no	—
151	C₂×C₄×C₅	—	2	1	no	—
152	C₂×C₄×C₅	—	4	1	no	—
153	C₂×C₄×C₅	—	4	1	no	—
154	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
155	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
156	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
158	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
160	C₂²⋊D₅	—	4	2	no	—
161	C₂×Dic₅	C₂²×C₈	1	2	yes	—
162	C₂×Dic₅	C₂²×C₈	1	2	yes	—
163	C₂×Dic₅	C₂²×C₈	1	2	yes	—
164	D₂₀	—	4	1	no	—
165	C₄×D₅	—	2	2	no	—
166	C₄×D₅	—	2	2	no	—
167	Dic₁₀	—	4	1	no	—
168	C₅×C₈	—	2	2	no	—
169	C₅×C₈	—	2	2	no	—
170	#168	—	2	1	no	—
171	#168	—	2	1	no	—
172	#168	#131	1	2	yes	—
173	#168	C₂×#41	1	2	yes	—
174	C₂×M₁₆	—	5	2	no	—
175	C₂²×C₈	—	5	1	no	—
176	C₂²×C₈	—	5	1	no	—
177	C₂²×C₈	—	5	2	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	—	5	1	no	—
187	#117	—	5	2	no	—
188	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
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	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
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	C₂×AGL(1,𝔽₅)	1	2	yes	—
205	C₄×C₈	—	5	2	no	—
206	C₄×C₈	—	5	1	no	—
207	C₄×C₈	—	5	1	no	—
208	C₄×C₈	—	5	1	no	—
209	C₄×C₈	—	5	1	no	—
210	C₂²×C₅	—	2	1	no	—
211	C₂²×C₅	—	2	1	no	—
212	C₂²×C₅	—	2	1	no	—
213	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
214	C₂×D₅	—	2	2	no	—
215	C₂×D₅	—	2	2	no	—
216	C₂×D₅	—	2	2	no	—
217	C₄×C₅	—	2	1	no	—
218	C₄×C₅	—	2	1	no	—
219	C₄×C₅	—	2	1	no	—
220	C₄×C₅	—	2	1	no	—
221	C₄×C₅	—	2	1	no	—
222	C₄×C₅	C₄×#41	1	1	yes	characteristic
223	C₄×C₅	C₂×#131	1	1	yes	characteristic
224	Dic₅	—	2	2	no	—
225	Dic₅	—	2	2	no	—
226	Dic₅	—	2	2	no	—
227	C₂×Q₈	—	5	1	no	—
228	C₂×D₄	—	5	2	no	—
229	C₂×D₄	—	5	1	no	—
230	C₂²×C₄	—	5	2	no	—
231	C₂²×C₄	D₅⋊C₈	1	1	yes	characteristic
232	C₂²×C₄	D₅⋊C₈	1	1	yes	characteristic
233	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
234	M₁₆	—	10	4	no	—
235	C₂×C₈	—	5	1	no	—
236	C₂×C₈	—	5	1	no	—
237	C₂×C₈	—	5	1	no	—
238	C₂×C₈	—	5	1	no	—
239	C₂×C₈	—	10	2	no	—
240	C₂×C₈	—	10	2	no	—
241	C₂×C₈	—	10	2	no	—
242	C₂×C₈	—	10	2	no	—
243	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	2	yes	—
244	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	2	yes	—
245	C₄⋊C₄	—	5	2	no	—
246	C₄⋊C₄	—	5	2	no	—
247	C₄⋊C₄	—	5	1	no	—
248	C₄⋊C₄	—	5	2	no	—
249	C₄⋊C₄	—	5	1	no	—
250	C₄⋊C₄	—	5	2	no	—
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	—	5	2	no	—
256	C₄²	—	5	2	no	—
257	C₄²	D₅⋊C₈	1	1	yes	characteristic
258	C₄²	D₅⋊C₈	1	1	yes	characteristic
259	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
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	1	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	2	no	—
270	C₂³	—	5	2	no	—
271	C₂³	C₂×D₅⋊C₈	1	1	yes	characteristic
272	Q₈	—	20	1	no	—
273	D₄	—	20	1	no	—
274	D₄	—	20	2	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	5	2	no	—
280	C₂×C₄	—	5	2	no	—
281	C₂×C₄	—	5	2	no	—
282	C₂×C₄	—	10	2	no	—
283	C₂×C₄	—	10	2	no	—
284	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
285	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
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	1	yes	characteristic
289	C₂×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
290	C₈	—	2	2	no	—
291	C₈	—	2	2	no	—
292	C₈	—	5	2	no	—
293	C₈	—	5	2	no	—
294	C₈	—	10	1	no	—
295	C₈	—	10	1	no	—
296	C₅	#33481	1	1	yes	characteristic, complemented, Sylow
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	10	2	no	—
301	C₂²	—	10	2	no	—
302	C₂²	—	10	2	no	—
303	C₂²	C₂²×D₅⋊C₈	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	10	2	no	—
310	C₄	—	10	2	no	—
311	C₄	—	10	2	no	—
312	C₄	D₅⋊#131	1	1	yes	characteristic
313	C₄	#41×AGL(1,𝔽₅)	1	1	yes	characteristic
314	C₂	—	4	1	no	—
315	C₂	—	20	2	no	—
316	C₂	#10617114	1	1	yes	characteristic
317	C₂	#10617404	1	1	yes	characteristic
318	C₂	#10617404	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	#33481
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. C₄×#3998	C₂	semidirect
4. #10604026	C₂	semidirect
5. #10604022	C₂	semidirect
6. #10603906	C₂	semidirect
7. #10603905	C₂	semidirect
8. #10603870	C₂	semidirect
9. #10603868	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
#10611056	2
#10604152	2
C₄×#3998	2
#10604026	2
#10604022	2
#10603906	2
#10603905	2
#10603870	2
#10603868	2
#33481	5