Subgroups — #13043651

Order 1280.

602 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10611007	C₂	1	1	yes	maximal, characteristic
2	#10604151	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×#674	C₂	1	1	yes	maximal, characteristic, complemented
4	#10604024	C₂	1	2	yes	maximal, complemented
5	#10604021	C₂	1	2	yes	maximal, complemented
6	#10603903	C₂	1	4	yes	maximal, complemented
7	#10603828	C₂	1	2	yes	maximal, complemented
8	C₄×D₅⋊C₁₆	C₂	1	2	yes	maximal, complemented
9	C₄×#709	C₄	1	1	yes	characteristic
10	C₂²×#674	C₂²	1	2	yes	—
11	C₂×D₅⋊C₁₆	C₂²	1	4	yes	—
12	C₅×#431	C₄	1	1	yes	characteristic
13	C₂×C₄×#168	C₂²	1	1	yes	characteristic
14	#65123	C₄	1	1	yes	characteristic
15	#65123	C₂²	1	1	yes	characteristic
16	#65119	C₄	1	1	yes	characteristic
17	#65119	C₂²	1	1	yes	characteristic
18	#65014	C₄	1	2	yes	—
19	#65014	C₂²	1	2	yes	—
20	#64974	C₄	1	2	yes	—
21	C₄×C₈×D₅	C₂²	1	2	yes	—
22	C₂²⋊#674	C₂²	1	2	yes	—
23	#64888	C₂²	1	4	yes	—
24	#64887	C₂²	1	4	yes	—
25	#64885	C₂²	1	4	yes	—
26	C₄⋊#674	C₂²	1	2	yes	—
27	C₄×#674	C₂²	1	1	yes	characteristic
28	C₄×#674	C₂²	1	2	yes	—
29	C₄×#674	C₂²	1	1	yes	characteristic
30	C₄×#674	C₂²	1	2	yes	—
31	C₁₆×#41	—	5	1	no	maximal, Sylow
32	C₂×#709	C₈	1	1	yes	characteristic
33	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	C₄×C₂²⋊D₅	C₈	1	2	yes	—
35	C₄×C₂²⋊D₅	C₂×C₄	1	2	yes	—
36	#4072	C₈	1	1	yes	characteristic
37	C₂²×#168	C₂×C₄	1	1	yes	characteristic
38	C₂²×#168	C₂³	1	1	yes	characteristic
39	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₅	C₂³	1	2	yes	—
41	C₄×D₂₀	C₈	1	1	yes	characteristic
42	C₄×D₂₀	C₂×C₄	1	1	yes	characteristic
43	#4018	C₈	1	2	yes	—
44	C₄²×D₅	C₂×C₄	1	2	yes	—
45	C₄×Dic₁₀	C₈	1	1	yes	characteristic
46	C₄×Dic₁₀	C₂×C₄	1	1	yes	characteristic
47	C₂×#674	—	2	4	no	—
48	C₂×#674	C₂³	1	2	yes	—
49	C₂×#674	C₂³	1	4	yes	—
50	C₂×#674	C₂³	1	2	yes	—
51	D₅⋊C₁₆	—	2	8	no	—
52	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
54	C₅×C₂²⋊C₈	C₂×C₄	1	2	yes	—
55	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
56	#3952	C₂×C₄	1	2	yes	—
57	#3952	C₂×C₄	1	2	yes	—
58	#3946	C₂×C₄	1	2	yes	—
59	#3946	C₂×C₄	1	2	yes	—
60	C₈×Dic₅	C₂×C₄	1	2	yes	—
61	C₈×Dic₅	C₂³	1	2	yes	—
62	C₄×#168	C₂×C₄	1	1	yes	characteristic
63	C₄×#168	C₂×C₄	1	1	yes	characteristic
64	C₄×#168	C₂³	1	1	yes	characteristic
65	C₄×#168	C₂³	1	1	yes	characteristic
66	C₈×#41	—	5	1	no	—
67	Q₈×C₁₆	—	5	2	no	—
68	D₄×C₁₆	—	5	4	no	—
69	D₄×C₁₆	—	5	2	no	—
70	#2162	—	5	2	no	—
71	#2162	—	5	1	no	—
72	C₂×C₄×C₁₆	—	5	2	no	—
73	C₂×C₄×C₁₆	—	5	1	no	—
74	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₂²⋊D₅	C₂×C₈	1	2	yes	—
78	#709	C₁₆	1	8	yes	complemented
79	C₂×D₂₀	C₂×C₈	1	1	yes	characteristic
80	C₂×C₄×D₅	C₂×C₈	1	2	yes	—
81	C₂×C₄×D₅	C₂×C₈	1	2	yes	—
82	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
83	C₂×Dic₁₀	C₂×C₈	1	1	yes	characteristic
84	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
85	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
86	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
87	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
88	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
90	C₂²⋊Dic₅	C₂×C₈	1	2	yes	—
91	#685	C₂×C₈	1	2	yes	—
92	#685	C₂×C₈	1	2	yes	—
93	C₄⋊Dic₅	C₂×C₈	1	1	yes	characteristic
94	C₄⋊Dic₅	C₂×C₈	1	1	yes	characteristic
95	#683	C₂×C₈	1	2	yes	—
96	#683	C₂×C₈	1	2	yes	—
97	C₄×Dic₅	C₂×C₈	1	2	yes	—
98	C₄×Dic₅	C₂×C₈	1	2	yes	—
99	C₄×Dic₅	C₂²×C₄	1	2	yes	—
100	C₂×#168	—	2	2	no	—
101	C₂×#168	—	2	2	no	—
102	C₂×#168	C₂²×C₄	1	1	yes	characteristic
103	C₂×#168	C₂²×C₄	1	1	yes	characteristic
104	C₂×#168	C₂²×C₄	1	1	yes	characteristic
105	C₂×#168	C₂⁴	1	1	yes	characteristic
106	C₈×D₅	—	2	4	no	—
107	C₈×D₅	—	2	4	no	—
108	#674	—	2	4	no	—
109	#674	—	2	2	no	—
110	#674	#41	1	4	yes	complemented
111	C₄×#41	—	5	1	no	—
112	C₂²×C₁₆	—	5	4	no	—
113	C₂²×C₁₆	—	5	2	no	—
114	C₈×Q₈	—	5	1	no	—
115	C₈×Q₈	—	5	1	no	—
116	C₈×D₄	—	5	1	no	—
117	C₈×D₄	—	5	1	no	—
118	C₈×D₄	—	5	2	no	—
119	C₈×D₄	—	5	2	no	—
120	#431	—	5	2	no	—
121	#431	AGL(1,𝔽₅)	1	1	yes	characteristic
122	C₂×C₄×C₈	—	5	1	no	—
123	C₂×C₄×C₈	—	5	2	no	—
124	C₄⋊C₁₆	—	5	2	no	—
125	C₄⋊C₁₆	—	5	4	no	—
126	C₂²⋊C₁₆	—	5	4	no	—
127	C₂²⋊C₁₆	—	5	2	no	—
128	C₄×C₁₆	—	5	1	no	—
129	C₄×C₁₆	—	5	2	no	—
130	C₄×C₁₆	—	5	1	no	—
131	C₄×C₁₆	—	5	2	no	—
132	C₄×C₁₆	—	5	4	no	—
133	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
134	C₂²×D₅	C₂²×C₈	1	2	yes	—
135	C₂×C₄×C₅	—	2	1	no	—
136	C₂×C₄×C₅	—	2	1	no	—
137	C₂×C₄×C₅	—	2	2	no	—
138	C₂×C₄×C₅	C₂×C₁₆	1	4	yes	complemented
139	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
145	C₂²⋊D₅	C₂×C₁₆	1	8	yes	complemented
146	C₂×Dic₅	C₂²×C₈	1	2	yes	—
147	C₂×Dic₅	C₂²×C₈	1	2	yes	—
148	C₂×Dic₅	C₂²×C₈	1	2	yes	—
149	D₂₀	C₂×C₁₆	1	4	yes	complemented
150	C₄×D₅	—	2	2	no	—
151	C₄×D₅	—	2	2	no	—
152	C₄×D₅	—	2	4	no	—
153	C₄×D₅	C₂×C₁₆	1	8	yes	complemented
154	Dic₁₀	C₂×C₁₆	1	4	yes	complemented
155	C₅×C₈	—	2	2	no	—
156	C₅×C₈	—	2	2	no	—
157	#168	—	2	1	no	—
158	#168	—	2	1	no	—
159	#168	#131	1	2	yes	—
160	#168	C₂×#41	1	2	yes	—
161	C₂×#41	—	5	1	no	—
162	C₂²×C₈	—	5	1	no	—
163	C₂²×C₈	—	5	1	no	—
164	C₂²×C₈	—	5	2	no	—
165	C₂²×C₈	—	5	2	no	—
166	C₄×Q₈	—	5	1	no	—
167	C₄×Q₈	—	5	1	no	—
168	C₄×D₄	—	5	2	no	—
169	C₄×D₄	—	5	2	no	—
170	C₄×D₄	—	5	1	no	—
171	C₄×D₄	—	5	1	no	—
172	#117	—	5	1	no	—
173	#117	—	5	2	no	—
174	C₂×C₄²	—	5	2	no	—
175	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
176	C₂×C₁₆	—	5	2	no	—
177	C₂×C₁₆	—	5	2	no	—
178	C₂×C₁₆	—	5	4	no	—
179	C₂×C₁₆	—	10	8	no	—
180	C₂×C₁₆	—	10	4	no	—
181	C₄⋊C₈	—	5	2	no	—
182	C₄⋊C₈	—	5	2	no	—
183	C₄⋊C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
184	C₄⋊C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
185	C₂²⋊C₈	—	5	2	no	—
186	C₂²⋊C₈	—	5	2	no	—
187	C₂²⋊C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
188	C₄×C₈	—	5	2	no	—
189	C₄×C₈	—	5	2	no	—
190	C₄×C₈	—	5	1	no	—
191	C₄×C₈	—	5	1	no	—
192	C₄×C₈	—	5	1	no	—
193	C₄×C₈	—	5	1	no	—
194	C₄×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
195	C₂²×C₅	—	2	1	no	—
196	C₂²×C₅	—	2	1	no	—
197	C₂²×C₅	C₂²×C₁₆	1	2	yes	—
198	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
199	C₂×D₅	—	2	2	no	—
200	C₂×D₅	—	2	2	no	—
201	C₂×D₅	C₂²×C₁₆	1	4	yes	—
202	C₄×C₅	—	2	1	no	—
203	C₄×C₅	—	2	1	no	—
204	C₄×C₅	C₂²×C₁₆	1	2	yes	—
205	C₄×C₅	C₂²×C₁₆	1	2	yes	—
206	C₄×C₅	#504	1	2	yes	—
207	C₄×C₅	C₄×#41	1	1	yes	characteristic
208	C₄×C₅	C₂×#131	1	1	yes	characteristic
209	Dic₅	—	2	2	no	—
210	Dic₅	—	2	2	no	—
211	Dic₅	C₂²×C₁₆	1	4	yes	—
212	#41	—	5	8	no	—
213	C₂×Q₈	—	5	1	no	—
214	C₂×D₄	—	5	2	no	—
215	C₂×D₄	—	5	1	no	—
216	C₂²×C₄	—	5	2	no	—
217	C₂²×C₄	—	5	2	no	—
218	C₂²×C₄	—	5	2	no	—
219	C₂²×C₄	D₅⋊C₈	1	1	yes	characteristic
220	C₂²×C₄	D₅⋊C₈	1	1	yes	characteristic
221	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
222	C₂×C₈	—	5	1	no	—
223	C₂×C₈	—	5	1	no	—
224	C₂×C₈	—	5	1	no	—
225	C₂×C₈	—	5	1	no	—
226	C₂×C₈	—	10	4	no	—
227	C₂×C₈	—	10	4	no	—
228	C₂×C₈	—	10	2	no	—
229	C₂×C₈	—	10	2	no	—
230	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	2	yes	—
231	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	2	yes	—
232	C₄⋊C₄	—	5	2	no	—
233	C₄⋊C₄	—	5	2	no	—
234	C₄⋊C₄	—	5	1	no	—
235	C₄⋊C₄	—	5	1	no	—
236	C₂²⋊C₄	—	5	2	no	—
237	C₂²⋊C₄	—	5	2	no	—
238	C₂²⋊C₄	—	5	2	no	—
239	C₄²	—	5	2	no	—
240	C₄²	—	5	2	no	—
241	C₄²	—	5	2	no	—
242	C₄²	D₅⋊C₈	1	1	yes	characteristic
243	C₄²	D₅⋊C₈	1	1	yes	characteristic
244	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
245	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
246	C₁₆	—	5	4	no	—
247	C₁₆	—	10	4	no	—
248	C₁₆	—	10	2	no	—
249	C₂×C₅	—	2	2	no	—
250	C₂×C₅	C₈×#41	1	1	yes	characteristic
251	C₂×C₅	C₂³×C₁₆	1	1	yes	characteristic
252	C₂×C₅	C₂×#504	1	1	yes	characteristic
253	D₅	—	2	4	no	—
254	C₂³	—	5	2	no	—
255	C₂³	C₂×D₅⋊C₈	1	1	yes	characteristic
256	Q₈	—	5	4	no	—
257	D₄	—	5	8	no	—
258	D₄	—	5	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	5	2	no	—
263	C₂×C₄	—	5	8	no	—
264	C₂×C₄	—	5	2	no	—
265	C₂×C₄	—	5	2	no	—
266	C₂×C₄	—	10	4	no	—
267	C₂×C₄	—	10	2	no	—
268	C₂×C₄	—	10	2	no	—
269	C₂×C₄	D₅⋊C₁₆	1	4	yes	complemented
270	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
271	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
272	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
273	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
274	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
275	C₂×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	—	2	2	no	—
278	C₈	—	5	2	no	—
279	C₈	—	5	2	no	—
280	C₈	—	10	1	no	—
281	C₈	—	10	1	no	—
282	C₅	C₁₆×#41	1	1	yes	characteristic, complemented, Sylow
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	5	4	no	—
286	C₂²	—	10	2	no	—
287	C₂²	—	10	2	no	—
288	C₂²	C₂×D₅⋊C₁₆	1	2	yes	—
289	C₂²	C₂²×D₅⋊C₈	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	5	4	no	—
293	C₄	—	10	2	no	—
294	C₄	—	10	2	no	—
295	C₄	C₂×D₅⋊C₁₆	1	2	yes	—
296	C₄	C₂×D₅⋊C₁₆	1	2	yes	—
297	C₄	#65725	1	2	yes	—
298	C₄	D₅⋊#131	1	1	yes	characteristic
299	C₄	#41×AGL(1,𝔽₅)	1	1	yes	characteristic
300	C₂	—	2	2	no	—
301	C₂	—	10	4	no	—
302	C₂	#10617114	1	1	yes	characteristic
303	C₂	C₂²×D₅⋊C₁₆	1	1	yes	characteristic
304	C₂	C₂×#65725	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂²×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂³×C₁₆
Frattini	C₂×C₄	C₂³×AGL(1,𝔽₅)
Fitting	C₅×#431	C₄
socle	C₂²×C₅	C₂³×C₈
prime radical	C₂×#709	C₈
cosocle	C₂×#168	C₂⁴
hypercenter	#431	AGL(1,𝔽₅)
hypocenter	C₅	C₁₆×#41
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄² ◁ #431

Complements

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

N	H ≅ G/N	type
2. #10604151	C₂	semidirect
3. C₂×C₄×#674	C₂	semidirect
4. #10604024	C₂	semidirect
5. #10604021	C₂	semidirect
6. #10603903	C₂	semidirect
7. #10603828	C₂	semidirect
8. C₄×D₅⋊C₁₆	C₂	semidirect
78. #709	C₁₆	semidirect
110. #674	#41	semidirect
138. C₂×C₄×C₅	C₂×C₁₆	semidirect
145. C₂²⋊D₅	C₂×C₁₆	semidirect
149. D₂₀	C₂×C₁₆	semidirect
153. C₄×D₅	C₂×C₁₆	semidirect
154. Dic₁₀	C₂×C₁₆	semidirect
269. C₂×C₄	D₅⋊C₁₆	semidirect
282. C₅	C₁₆×#41	semidirect

Maximal subgroups

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

subgroup	index
#10611007	2
#10604151	2
C₂×C₄×#674	2
#10604024	2
#10604021	2
#10603903	2
#10603828	2
C₄×D₅⋊C₁₆	2
C₁₆×#41	5