Subgroups — C₅×#12828 #11935548

Order 1280.

390 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2952	C₂	1	1	yes	maximal, characteristic
2	C₅×#2153	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2145	C₂	1	1	yes	maximal, characteristic
4	C₅×#2143	C₂	1	1	yes	maximal, characteristic
5	C₅×#1833	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1386	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1383	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#135	C₂²	1	1	yes	characteristic
9	C₅×#520	—	2	1	no	—
10	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
12	C₅×#442	C₄	1	2	yes	complemented
13	C₅×#441	C₄	1	2	yes	complemented
14	C₄×C₅×SD₁₆	—	2	2	no	—
15	C₂×C₅×#108	C₂²	1	1	yes	characteristic
16	C₅×#427	C₂²	1	1	yes	characteristic
17	C₂×C₅×#106	C₂²	1	1	yes	characteristic
18	C₅×#417	—	2	1	no	—
19	C₅×#405	C₂²	1	1	yes	characteristic
20	C₅×#364	—	2	2	no	—
21	C₅×#360	—	2	2	no	—
22	C₅×#359	—	2	2	no	—
23	C₅×#344	—	2	1	no	—
24	C₅×H(3,ℤ/4)	—	2	1	no	—
25	#12828	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂×C₅×#41	—	2	1	no	—
27	C₅×#135	—	2	1	no	—
28	C₅×#135	—	2	1	no	—
29	C₅×#135	D₄	1	2	yes	—
30	C₅×#135	Q₈	1	2	yes	—
31	C₂×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂×C₅×SD₁₆	—	2	1	no	—
33	C₂×C₅×D₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₅×M₁₆	—	2	1	no	—
35	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
36	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
37	C₄×C₅×Q₈	—	2	2	no	—
38	C₄×C₅×D₄	—	2	2	no	—
39	C₄×C₅×D₄	—	4	2	no	—
40	C₅×#117	—	2	1	no	—
41	C₅×#117	—	4	1	no	—
42	C₅×#117	D₄	1	1	yes	characteristic
43	C₂×C₅×C₄⋊C₄	—	2	1	no	—
44	C₂×C₅×C₄⋊C₄	—	4	1	no	—
45	C₅×M₃₂	—	2	1	no	—
46	C₅×M₃₂	—	2	1	no	—
47	C₅×M₃₂	—	2	1	no	—
48	C₅×M₃₂	—	2	1	no	—
49	C₂×C₅×C₁₆	—	2	1	no	—
50	C₂×C₅×C₁₆	—	2	1	no	—
51	C₅×#108	—	2	1	no	—
52	C₅×#108	—	2	1	no	—
53	C₅×#107	—	2	1	no	—
54	C₅×#106	—	2	1	no	—
55	C₅×#106	—	2	1	no	—
56	C₅×#106	C₂×C₄	1	2	yes	—
57	C₅×#103	—	2	2	no	—
58	C₅×#102	—	2	2	no	—
59	C₅×#97	C₂×C₄	1	2	yes	—
60	C₄×C₅×C₈	D₄	1	2	yes	—
61	#2952	C₂×C₅	1	1	yes	characteristic
62	#2153	C₂×C₅	1	1	yes	characteristic, complemented
63	#2145	C₂×C₅	1	1	yes	characteristic
64	#2143	C₂×C₅	1	1	yes	characteristic
65	#1833	C₂×C₅	1	1	yes	characteristic, complemented
66	#1386	C₂×C₅	1	1	yes	characteristic, complemented
67	#1383	C₂×C₅	1	1	yes	characteristic, complemented
68	C₅×#41	—	2	2	no	—
69	C₅×#41	—	2	2	no	—
70	C₅×#41	—	4	1	no	—
71	C₅×#41	—	4	1	no	—
72	C₂×C₅×Q₈	—	2	1	no	—
73	C₂×C₅×D₄	—	2	1	no	—
74	C₂×C₅×D₄	—	4	1	no	—
75	C₂²×C₄×C₅	—	4	2	no	—
76	C₂²×C₄×C₅	—	4	1	no	—
77	C₂²×C₄×C₅	—	4	1	no	—
78	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₅×Q₁₆	—	2	1	no	—
80	C₅×Q₁₆	C₄⋊C₄	1	2	yes	—
81	C₅×SD₁₆	—	2	2	no	—
82	C₅×SD₁₆	—	2	2	no	—
83	C₅×D₈	—	2	1	no	—
84	C₅×D₈	C₄⋊C₄	1	2	yes	—
85	C₅×M₁₆	—	2	1	no	—
86	C₅×M₁₆	—	2	1	no	—
87	C₅×M₁₆	—	4	1	no	—
88	C₅×M₁₆	—	4	1	no	—
89	C₂×C₅×C₈	—	4	1	no	—
90	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
91	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
95	C₂×C₅×C₈	#41	1	1	yes	characteristic
96	C₂×C₅×C₈	#41	1	1	yes	characteristic
97	C₅×C₄⋊C₄	—	2	1	no	—
98	C₅×C₄⋊C₄	—	2	2	no	—
99	C₅×C₄⋊C₄	—	4	2	no	—
100	C₅×C₄⋊C₄	—	4	1	no	—
101	C₅×C₄⋊C₄	—	4	1	no	—
102	C₅×C₂²⋊C₄	—	2	1	no	—
103	C₅×C₂²⋊C₄	—	4	2	no	—
104	C₅×C₂²⋊C₄	—	4	1	no	—
105	C₄²×C₅	—	4	2	no	—
106	C₄²×C₅	—	4	1	no	—
107	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
108	C₅×C₁₆	—	4	1	no	—
109	C₅×C₁₆	—	4	1	no	—
110	C₂×#135	C₂²×C₅	1	1	yes	characteristic
111	#520	—	2	1	no	—
112	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
113	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
114	#442	C₄×C₅	1	2	yes	complemented
115	#441	C₄×C₅	1	2	yes	complemented
116	C₄×SD₁₆	—	2	2	no	—
117	C₂×#108	C₂²×C₅	1	1	yes	characteristic
118	#427	C₂²×C₅	1	1	yes	characteristic
119	C₂×#106	C₂²×C₅	1	1	yes	characteristic
120	#417	—	2	1	no	—
121	#405	C₂²×C₅	1	1	yes	characteristic
122	#364	—	2	2	no	—
123	#360	—	2	2	no	—
124	#359	—	2	2	no	—
125	#344	—	2	1	no	—
126	H(3,ℤ/4)	—	2	1	no	—
127	C₂³×C₅	—	4	1	no	—
128	C₂³×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₅×Q₈	—	2	2	no	—
130	C₅×Q₈	—	4	1	no	—
131	C₅×D₄	—	2	2	no	—
132	C₅×D₄	—	4	2	no	—
133	C₅×D₄	—	4	2	no	—
134	C₅×D₄	—	4	1	no	—
135	C₂×C₄×C₅	—	2	2	no	—
136	C₂×C₄×C₅	—	4	1	no	—
137	C₂×C₄×C₅	—	4	2	no	—
138	C₂×C₄×C₅	—	4	2	no	—
139	C₂×C₄×C₅	—	4	2	no	—
140	C₂×C₄×C₅	—	8	1	no	—
141	C₂×C₄×C₅	—	16	2	no	—
142	C₂×C₄×C₅	—	16	1	no	—
143	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₅	#121	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
146	C₂×C₄×C₅	#135	1	1	yes	characteristic
147	C₂×C₄×C₅	#135	1	1	yes	characteristic
148	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₅×C₈	—	2	2	no	—
150	C₅×C₈	—	8	1	no	—
151	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₅×C₈	#117	1	1	yes	characteristic
153	C₅×C₈	#121	1	1	yes	characteristic
154	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
155	C₂×#41	—	2	1	no	—
156	#135	—	2	1	no	—
157	#135	—	2	1	no	—
158	#135	C₅×D₄	1	2	yes	—
159	#135	C₅×Q₈	1	2	yes	—
160	C₂×Q₁₆	C₂×C₄×C₅	1	1	yes	characteristic
161	C₂×SD₁₆	—	2	1	no	—
162	C₂×D₈	C₂×C₄×C₅	1	1	yes	characteristic
163	C₂×M₁₆	—	2	1	no	—
164	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
165	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
166	C₄×Q₈	—	2	2	no	—
167	C₄×D₄	—	2	2	no	—
168	C₄×D₄	—	4	2	no	—
169	#117	—	2	1	no	—
170	#117	—	4	1	no	—
171	#117	C₅×D₄	1	1	yes	characteristic
172	C₂×C₄⋊C₄	—	2	1	no	—
173	C₂×C₄⋊C₄	—	4	1	no	—
174	M₃₂	—	2	1	no	—
175	M₃₂	—	2	1	no	—
176	M₃₂	—	2	1	no	—
177	M₃₂	—	2	1	no	—
178	C₂×C₁₆	—	2	1	no	—
179	C₂×C₁₆	—	2	1	no	—
180	#108	—	2	1	no	—
181	#108	—	2	1	no	—
182	#107	—	2	1	no	—
183	#106	—	2	1	no	—
184	#106	—	2	1	no	—
185	#106	C₂×C₄×C₅	1	2	yes	—
186	#103	—	2	2	no	—
187	#102	—	2	2	no	—
188	#97	C₂×C₄×C₅	1	2	yes	—
189	C₄×C₈	C₅×D₄	1	2	yes	—
190	C₂²×C₅	—	4	1	no	—
191	C₂²×C₅	—	4	2	no	—
192	C₂²×C₅	—	8	1	no	—
193	C₂²×C₅	—	8	1	no	—
194	C₂²×C₅	#418	1	1	yes	characteristic
195	C₂²×C₅	#463	1	1	yes	characteristic
196	C₂²×C₅	#478	1	1	yes	characteristic
197	C₄×C₅	—	4	2	no	—
198	C₄×C₅	—	4	2	no	—
199	C₄×C₅	—	8	2	no	—
200	C₄×C₅	—	16	1	no	—
201	C₄×C₅	#380	1	1	yes	characteristic
202	C₄×C₅	#416	1	1	yes	characteristic
203	C₄×C₅	#463	1	1	yes	characteristic
204	C₄×C₅	#478	1	1	yes	characteristic
205	#41	—	2	2	no	—
206	#41	—	2	2	no	—
207	#41	—	4	1	no	—
208	#41	—	4	1	no	—
209	C₂×Q₈	—	2	1	no	—
210	C₂×D₄	—	2	1	no	—
211	C₂×D₄	—	4	1	no	—
212	C₂²×C₄	—	4	2	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
216	Q₁₆	—	2	1	no	—
217	Q₁₆	C₅×C₄⋊C₄	1	2	yes	—
218	SD₁₆	—	2	2	no	—
219	SD₁₆	—	2	2	no	—
220	D₈	—	2	1	no	—
221	D₈	C₅×C₄⋊C₄	1	2	yes	—
222	M₁₆	—	2	1	no	—
223	M₁₆	—	2	1	no	—
224	M₁₆	—	4	1	no	—
225	M₁₆	—	4	1	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
228	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
229	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
230	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
231	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
232	C₂×C₈	C₅×#41	1	1	yes	characteristic
233	C₂×C₈	C₅×#41	1	1	yes	characteristic
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	1	no	—
238	C₄⋊C₄	—	4	1	no	—
239	C₂²⋊C₄	—	2	1	no	—
240	C₂²⋊C₄	—	4	2	no	—
241	C₂²⋊C₄	—	4	1	no	—
242	C₄²	—	4	2	no	—
243	C₄²	—	4	1	no	—
244	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
245	C₁₆	—	4	1	no	—
246	C₁₆	—	4	1	no	—
247	C₂×C₅	—	2	1	no	—
248	C₂×C₅	—	2	1	no	—
249	C₂×C₅	—	2	1	no	—
250	C₂×C₅	—	8	2	no	—
251	C₂×C₅	#1804	1	1	yes	characteristic
252	C₂³	—	4	1	no	—
253	C₂³	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
254	Q₈	—	2	2	no	—
255	Q₈	—	4	1	no	—
256	D₄	—	2	2	no	—
257	D₄	—	4	2	no	—
258	D₄	—	4	2	no	—
259	D₄	—	4	1	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	8	1	no	—
266	C₂×C₄	—	16	2	no	—
267	C₂×C₄	—	16	1	no	—
268	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
269	C₂×C₄	C₅×#121	1	1	yes	characteristic
270	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
271	C₂×C₄	C₅×#135	1	1	yes	characteristic
272	C₂×C₄	C₅×#135	1	1	yes	characteristic
273	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
274	C₈	—	2	2	no	—
275	C₈	—	8	1	no	—
276	C₈	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
277	C₈	C₅×#117	1	1	yes	characteristic
278	C₈	C₅×#121	1	1	yes	characteristic
279	C₈	C₅×C₂²⋊Q₈	1	1	yes	characteristic
280	C₅	#12828	1	1	yes	characteristic, direct product factor, Sylow
281	C₂²	—	4	1	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	8	1	no	—
284	C₂²	—	8	1	no	—
285	C₂²	C₅×#418	1	1	yes	characteristic
286	C₂²	C₅×#463	1	1	yes	characteristic
287	C₂²	C₅×#478	1	1	yes	characteristic
288	C₄	—	4	2	no	—
289	C₄	—	4	2	no	—
290	C₄	—	8	2	no	—
291	C₄	—	16	1	no	—
292	C₄	C₅×#380	1	1	yes	characteristic
293	C₄	C₅×#416	1	1	yes	characteristic
294	C₄	C₅×#463	1	1	yes	characteristic
295	C₄	C₅×#478	1	1	yes	characteristic
296	C₂	—	2	1	no	—
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	8	2	no	—
300	C₂	C₅×#1804	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅	#1804
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂×C₅×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×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×#2153	C₂	semidirect
5. C₅×#1833	C₂	semidirect
6. C₅×#1386	C₂	semidirect
7. C₅×#1383	C₂	semidirect
12. C₅×#442	C₄	semidirect
13. C₅×#441	C₄	semidirect
25. #12828	C₅	direct (N × H)
62. #2153	C₂×C₅	semidirect
65. #1833	C₂×C₅	semidirect
66. #1386	C₂×C₅	semidirect
67. #1383	C₂×C₅	semidirect
114. #442	C₄×C₅	semidirect
115. #441	C₄×C₅	semidirect
280. C₅	#12828	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#2952	2
C₅×#2153	2
C₅×#2145	2
C₅×#2143	2
C₅×#1833	2
C₅×#1386	2
C₅×#1383	2
#12828	5

Subgroups — C5×#12828 #11935548

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12828 #11935548