Subgroups — C₅×#13440 #11936160

Order 1280.

398 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×SD₃₂	C₂	1	1	yes	maximal, characteristic
2	C₅×#2151	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#362	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#361	C₂	1	1	yes	maximal, characteristic
5	C₅×#1979	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1978	C₂	1	1	yes	maximal, characteristic
7	C₅×#1381	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₅×SD₃₂	—	2	2	no	—
11	C₂×C₅×SD₃₂	—	2	2	no	—
12	C₂×C₅×SD₃₂	—	2	2	no	—
13	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
14	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₅×#108	C₂²	1	1	yes	characteristic
16	C₂×C₅×#108	C₂²	1	1	yes	characteristic
17	C₅×#419	—	2	1	no	—
18	C₅×#418	—	2	1	no	—
19	C₂×C₅×#101	—	2	1	no	—
20	C₂×C₅×#100	—	2	1	no	—
21	C₅×#405	C₂²	1	1	yes	characteristic
22	C₅×C₄⋊C₁₆	—	2	2	no	—
23	C₅×#362	—	2	2	no	—
24	C₅×#361	—	2	2	no	—
25	#13440	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂²×C₅×Q₈	—	2	1	no	—
27	C₂²×C₅×D₄	—	2	1	no	—
28	C₂×C₅×Q₁₆	—	2	1	no	—
29	C₂×C₅×Q₁₆	—	2	1	no	—
30	C₂×C₅×Q₁₆	—	2	1	no	—
31	C₂×C₅×Q₁₆	—	2	1	no	—
32	C₂×C₅×Q₁₆	—	2	1	no	—
33	C₂×C₅×Q₁₆	D₄	1	2	yes	—
34	C₂×C₅×D₈	—	2	1	no	—
35	C₂×C₅×D₈	—	2	1	no	—
36	C₂×C₅×D₈	—	2	1	no	—
37	C₂×C₅×D₈	—	2	1	no	—
38	C₂×C₅×D₈	—	2	1	no	—
39	C₂×C₅×D₈	D₄	1	2	yes	—
40	C₂×C₅×M₁₆	—	2	1	no	—
41	C₂×C₅×M₁₆	—	2	1	no	—
42	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
43	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
44	C₅×#117	D₄	1	1	yes	characteristic
45	C₅×SD₃₂	—	2	8	no	—
46	C₅×M₃₂	—	2	1	no	—
47	C₅×M₃₂	—	2	1	no	—
48	C₂×C₅×C₁₆	—	2	1	no	—
49	C₂×C₅×C₁₆	—	2	1	no	—
50	C₂×C₅×C₁₆	—	2	1	no	—
51	C₂×C₅×C₁₆	D₄	1	2	yes	—
52	C₅×#108	—	2	1	no	—
53	C₅×#108	—	2	1	no	—
54	C₅×#108	—	2	1	no	—
55	C₅×#108	—	2	1	no	—
56	C₅×#103	—	4	2	no	—
57	C₅×#102	—	4	2	no	—
58	C₅×#101	—	4	2	no	—
59	C₅×#100	—	4	2	no	—
60	C₅×#97	—	2	1	no	—
61	C₄×C₅×C₈	—	2	1	no	—
62	C₂²×SD₃₂	C₂×C₅	1	1	yes	characteristic
63	#2151	C₂×C₅	1	1	yes	characteristic, complemented
64	C₂×#362	C₂×C₅	1	1	yes	characteristic, complemented
65	C₂×#361	C₂×C₅	1	1	yes	characteristic
66	#1979	C₂×C₅	1	1	yes	characteristic, complemented
67	#1978	C₂×C₅	1	1	yes	characteristic
68	#1381	C₂×C₅	1	1	yes	characteristic, complemented
69	C₂⁴×C₅	—	4	1	no	—
70	C₂×C₅×Q₈	—	2	2	no	—
71	C₂×C₅×Q₈	—	4	1	no	—
72	C₂×C₅×Q₈	—	4	1	no	—
73	C₂×C₅×Q₈	—	4	1	no	—
74	C₂×C₅×Q₈	—	4	1	no	—
75	C₂×C₅×Q₈	—	4	1	no	—
76	C₂×C₅×D₄	—	2	2	no	—
77	C₂×C₅×D₄	—	4	1	no	—
78	C₂×C₅×D₄	—	4	1	no	—
79	C₂×C₅×D₄	—	4	1	no	—
80	C₂×C₅×D₄	—	4	1	no	—
81	C₂×C₅×D₄	—	4	1	no	—
82	C₂²×C₄×C₅	—	4	1	no	—
83	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₅×Q₁₆	—	2	2	no	—
85	C₅×Q₁₆	—	2	2	no	—
86	C₅×Q₁₆	—	4	1	no	—
87	C₅×Q₁₆	—	4	1	no	—
88	C₅×D₈	—	2	2	no	—
89	C₅×D₈	—	2	2	no	—
90	C₅×D₈	—	4	1	no	—
91	C₅×D₈	—	4	1	no	—
92	C₅×M₁₆	—	2	1	no	—
93	C₅×M₁₆	—	2	1	no	—
94	C₅×M₁₆	—	4	1	no	—
95	C₅×M₁₆	—	4	1	no	—
96	C₅×M₁₆	—	4	1	no	—
97	C₅×M₁₆	—	4	1	no	—
98	C₂×C₅×C₈	—	2	1	no	—
99	C₂×C₅×C₈	—	4	1	no	—
100	C₂×C₅×C₈	—	4	1	no	—
101	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
102	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
104	C₂×C₅×C₈	#41	1	1	yes	characteristic
105	C₂×C₅×C₈	#41	1	1	yes	characteristic
106	C₂×C₅×C₈	#41	1	1	yes	characteristic
107	C₅×C₄⋊C₄	D₈	1	2	yes	—
108	C₅×C₂²⋊C₄	—	2	1	no	—
109	C₄²×C₅	—	2	1	no	—
110	C₅×C₁₆	—	2	2	no	—
111	C₅×C₁₆	—	4	1	no	—
112	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
113	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
114	C₂×SD₃₂	—	2	2	no	—
115	C₂×SD₃₂	—	2	2	no	—
116	C₂×SD₃₂	—	2	2	no	—
117	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
118	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
119	C₂×#108	C₂²×C₅	1	1	yes	characteristic
120	C₂×#108	C₂²×C₅	1	1	yes	characteristic
121	#419	—	2	1	no	—
122	#418	—	2	1	no	—
123	C₂×#101	—	2	1	no	—
124	C₂×#100	—	2	1	no	—
125	#405	C₂²×C₅	1	1	yes	characteristic
126	C₄⋊C₁₆	—	2	2	no	—
127	#362	—	2	2	no	—
128	#361	—	2	2	no	—
129	C₂³×C₅	—	4	2	no	—
130	C₂³×C₅	—	8	1	no	—
131	C₂³×C₅	—	8	1	no	—
132	C₂³×C₅	—	16	2	no	—
133	C₂³×C₅	#135	1	1	yes	characteristic
134	C₅×Q₈	—	4	2	no	—
135	C₅×Q₈	—	4	2	no	—
136	C₅×Q₈	—	8	1	no	—
137	C₅×Q₈	—	8	1	no	—
138	C₅×D₄	—	4	2	no	—
139	C₅×D₄	—	4	2	no	—
140	C₅×D₄	—	8	1	no	—
141	C₅×D₄	—	8	1	no	—
142	C₂×C₄×C₅	—	2	2	no	—
143	C₂×C₄×C₅	—	4	2	no	—
144	C₂×C₄×C₅	—	8	1	no	—
145	C₂×C₄×C₅	—	8	1	no	—
146	C₂×C₄×C₅	#121	1	1	yes	characteristic
147	C₂×C₄×C₅	#121	1	1	yes	characteristic
148	C₂×C₄×C₅	#121	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₅×C₈	—	4	1	no	—
153	C₅×C₈	—	8	1	no	—
154	C₅×C₈	—	8	1	no	—
155	C₅×C₈	#120	1	1	yes	characteristic
156	C₅×C₈	#123	1	1	yes	characteristic
157	C₅×C₈	#123	1	1	yes	characteristic
158	C₅×C₈	#124	1	1	yes	characteristic
159	C₂²×Q₈	—	2	1	no	—
160	C₂²×D₄	—	2	1	no	—
161	C₂×Q₁₆	—	2	1	no	—
162	C₂×Q₁₆	—	2	1	no	—
163	C₂×Q₁₆	—	2	1	no	—
164	C₂×Q₁₆	—	2	1	no	—
165	C₂×Q₁₆	—	2	1	no	—
166	C₂×Q₁₆	C₅×D₄	1	2	yes	—
167	C₂×D₈	—	2	1	no	—
168	C₂×D₈	—	2	1	no	—
169	C₂×D₈	—	2	1	no	—
170	C₂×D₈	—	2	1	no	—
171	C₂×D₈	—	2	1	no	—
172	C₂×D₈	C₅×D₄	1	2	yes	—
173	C₂×M₁₆	—	2	1	no	—
174	C₂×M₁₆	—	2	1	no	—
175	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
176	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
177	#117	C₅×D₄	1	1	yes	characteristic
178	SD₃₂	—	2	8	no	—
179	M₃₂	—	2	1	no	—
180	M₃₂	—	2	1	no	—
181	C₂×C₁₆	—	2	1	no	—
182	C₂×C₁₆	—	2	1	no	—
183	C₂×C₁₆	—	2	1	no	—
184	C₂×C₁₆	C₅×D₄	1	2	yes	—
185	#108	—	2	1	no	—
186	#108	—	2	1	no	—
187	#108	—	2	1	no	—
188	#108	—	2	1	no	—
189	#103	—	4	2	no	—
190	#102	—	4	2	no	—
191	#101	—	4	2	no	—
192	#100	—	4	2	no	—
193	#97	—	2	1	no	—
194	C₄×C₈	—	2	1	no	—
195	C₂²×C₅	—	2	2	no	—
196	C₂²×C₅	—	8	2	no	—
197	C₂²×C₅	—	16	2	no	—
198	C₂²×C₅	—	16	2	no	—
199	C₂²×C₅	—	16	2	no	—
200	C₂²×C₅	#463	1	1	yes	characteristic
201	C₂²×C₅	#463	1	1	yes	characteristic
202	C₂²×C₅	#466	1	1	yes	characteristic
203	C₄×C₅	—	4	2	no	—
204	C₄×C₅	—	8	2	no	—
205	C₄×C₅	#394	1	1	yes	characteristic
206	C₄×C₅	#459	1	1	yes	characteristic
207	C₄×C₅	#459	1	1	yes	characteristic
208	C₄×C₅	#469	1	1	yes	characteristic
209	C₂⁴	—	4	1	no	—
210	C₂×Q₈	—	2	2	no	—
211	C₂×Q₈	—	4	1	no	—
212	C₂×Q₈	—	4	1	no	—
213	C₂×Q₈	—	4	1	no	—
214	C₂×Q₈	—	4	1	no	—
215	C₂×Q₈	—	4	1	no	—
216	C₂×D₄	—	2	2	no	—
217	C₂×D₄	—	4	1	no	—
218	C₂×D₄	—	4	1	no	—
219	C₂×D₄	—	4	1	no	—
220	C₂×D₄	—	4	1	no	—
221	C₂×D₄	—	4	1	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
224	Q₁₆	—	2	2	no	—
225	Q₁₆	—	2	2	no	—
226	Q₁₆	—	4	1	no	—
227	Q₁₆	—	4	1	no	—
228	D₈	—	2	2	no	—
229	D₈	—	2	2	no	—
230	D₈	—	4	1	no	—
231	D₈	—	4	1	no	—
232	M₁₆	—	2	1	no	—
233	M₁₆	—	2	1	no	—
234	M₁₆	—	4	1	no	—
235	M₁₆	—	4	1	no	—
236	M₁₆	—	4	1	no	—
237	M₁₆	—	4	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	4	1	no	—
240	C₂×C₈	—	4	1	no	—
241	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
242	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
243	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
244	C₂×C₈	C₅×#41	1	1	yes	characteristic
245	C₂×C₈	C₅×#41	1	1	yes	characteristic
246	C₂×C₈	C₅×#41	1	1	yes	characteristic
247	C₄⋊C₄	C₅×D₈	1	2	yes	—
248	C₂²⋊C₄	—	2	1	no	—
249	C₄²	—	2	1	no	—
250	C₁₆	—	2	2	no	—
251	C₁₆	—	4	1	no	—
252	C₂×C₅	—	2	1	no	—
253	C₂×C₅	—	2	1	no	—
254	C₂×C₅	—	16	2	no	—
255	C₂×C₅	#2054	1	1	yes	characteristic
256	C₂×C₅	#2223	1	2	yes	—
257	C₂³	—	4	2	no	—
258	C₂³	—	8	1	no	—
259	C₂³	—	8	1	no	—
260	C₂³	—	16	2	no	—
261	C₂³	C₅×#135	1	1	yes	characteristic
262	Q₈	—	4	2	no	—
263	Q₈	—	4	2	no	—
264	Q₈	—	8	1	no	—
265	Q₈	—	8	1	no	—
266	D₄	—	4	2	no	—
267	D₄	—	4	2	no	—
268	D₄	—	8	1	no	—
269	D₄	—	8	1	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	8	1	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	C₅×#121	1	1	yes	characteristic
275	C₂×C₄	C₅×#121	1	1	yes	characteristic
276	C₂×C₄	C₅×#121	1	1	yes	characteristic
277	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
278	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
279	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
280	C₈	—	4	1	no	—
281	C₈	—	8	1	no	—
282	C₈	—	8	1	no	—
283	C₈	C₅×#120	1	1	yes	characteristic
284	C₈	C₅×#123	1	1	yes	characteristic
285	C₈	C₅×#123	1	1	yes	characteristic
286	C₈	C₅×#124	1	1	yes	characteristic
287	C₅	#13440	1	1	yes	characteristic, direct product factor, Sylow
288	C₂²	—	2	2	no	—
289	C₂²	—	8	2	no	—
290	C₂²	—	16	2	no	—
291	C₂²	—	16	2	no	—
292	C₂²	—	16	2	no	—
293	C₂²	C₅×#463	1	1	yes	characteristic
294	C₂²	C₅×#463	1	1	yes	characteristic
295	C₂²	C₅×#466	1	1	yes	characteristic
296	C₄	—	4	2	no	—
297	C₄	—	8	2	no	—
298	C₄	C₅×#394	1	1	yes	characteristic
299	C₄	C₅×#459	1	1	yes	characteristic
300	C₄	C₅×#459	1	1	yes	characteristic
301	C₄	C₅×#469	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	16	2	no	—
305	C₂	C₅×#2054	1	1	yes	characteristic
306	C₂	C₅×#2223	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂²×C₅×D₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2151	C₂	semidirect
3. C₂×C₅×#362	C₂	semidirect
5. C₅×#1979	C₂	semidirect
7. C₅×#1381	C₂	semidirect
25. #13440	C₅	direct (N × H)
63. #2151	C₂×C₅	semidirect
64. C₂×#362	C₂×C₅	semidirect
66. #1979	C₂×C₅	semidirect
68. #1381	C₂×C₅	semidirect
287. C₅	#13440	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×SD₃₂	2
C₅×#2151	2
C₂×C₅×#362	2
C₂×C₅×#361	2
C₅×#1979	2
C₅×#1978	2
C₅×#1381	2
#13440	5

Subgroups — C5×#13440 #11936160

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13440 #11936160