Subgroups — C₅×#12826 #11935546

Order 1280.

412 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2948	C₂	1	1	yes	maximal, characteristic
2	C₅×#2153	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#361	C₂	1	1	yes	maximal, characteristic
4	C₅×#2140	C₂	1	1	yes	maximal, characteristic
5	C₅×#1833	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1389	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1384	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
9	C₅×#518	—	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	4	yes	complemented
13	C₄×C₅×D₈	—	2	2	no	—
14	C₂×C₅×#108	C₂²	1	1	yes	characteristic
15	C₅×#427	C₂²	1	1	yes	characteristic
16	C₅×#427	C₂²	1	1	yes	characteristic
17	C₅×#418	—	2	1	no	—
18	C₅×#405	C₂²	1	1	yes	characteristic
19	C₅×#364	—	2	2	no	—
20	C₅×#361	—	2	2	no	—
21	C₅×#357	—	2	2	no	—
22	C₅×#344	—	2	1	no	—
23	C₅×H(3,ℤ/4)	—	2	1	no	—
24	#12826	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂²×C₅×D₄	—	2	1	no	—
26	C₂×C₅×D₈	—	2	1	no	—
27	C₂×C₅×D₈	—	2	1	no	—
28	C₂×C₅×D₈	—	2	1	no	—
29	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
30	C₂×C₅×D₈	D₄	1	2	yes	—
31	C₂×C₅×D₈	Q₈	1	2	yes	—
32	C₂×C₅×M₁₆	—	2	1	no	—
33	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₄×C₅×D₄	—	2	4	no	—
36	C₄×C₅×D₄	—	4	2	no	—
37	C₅×#117	—	2	1	no	—
38	C₅×#117	—	2	1	no	—
39	C₅×#117	D₄	1	1	yes	characteristic
40	C₂×C₅×C₂²⋊C₄	—	4	2	no	—
41	C₅×M₃₂	—	2	1	no	—
42	C₅×M₃₂	—	2	1	no	—
43	C₅×M₃₂	—	2	1	no	—
44	C₅×M₃₂	—	2	1	no	—
45	C₂×C₅×C₁₆	—	2	1	no	—
46	C₂×C₅×C₁₆	—	2	1	no	—
47	C₅×#108	—	2	1	no	—
48	C₅×#108	—	2	1	no	—
49	C₅×#107	—	2	1	no	—
50	C₅×#107	—	2	1	no	—
51	C₅×#106	—	2	1	no	—
52	C₅×#106	C₂×C₄	1	2	yes	—
53	C₅×#102	—	2	4	no	—
54	C₅×#97	C₂×C₄	1	2	yes	—
55	C₄×C₅×C₈	D₄	1	2	yes	—
56	#2948	C₂×C₅	1	1	yes	characteristic
57	#2153	C₂×C₅	1	1	yes	characteristic, complemented
58	C₂×#361	C₂×C₅	1	1	yes	characteristic
59	#2140	C₂×C₅	1	1	yes	characteristic
60	#1833	C₂×C₅	1	1	yes	characteristic, complemented
61	#1389	C₂×C₅	1	1	yes	characteristic, complemented
62	#1384	C₂×C₅	1	1	yes	characteristic, complemented
63	C₂⁴×C₅	—	4	1	no	—
64	C₂×C₅×D₄	—	2	2	no	—
65	C₂×C₅×D₄	—	2	2	no	—
66	C₂×C₅×D₄	—	2	2	no	—
67	C₂×C₅×D₄	—	4	1	no	—
68	C₂×C₅×D₄	—	4	1	no	—
69	C₂×C₅×D₄	—	4	1	no	—
70	C₂²×C₄×C₅	—	4	4	no	—
71	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₅×D₈	—	2	2	no	—
73	C₅×D₈	—	2	2	no	—
74	C₅×D₈	—	2	2	no	—
75	C₅×D₈	C₄⋊C₄	1	4	yes	—
76	C₅×M₁₆	—	2	1	no	—
77	C₅×M₁₆	—	2	1	no	—
78	C₅×M₁₆	—	4	1	no	—
79	C₅×M₁₆	—	4	1	no	—
80	C₂×C₅×C₈	—	4	1	no	—
81	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
82	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
84	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
85	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₅×C₈	#41	1	1	yes	characteristic
87	C₂×C₅×C₈	#41	1	1	yes	characteristic
88	C₅×C₄⋊C₄	—	2	1	no	—
89	C₅×C₄⋊C₄	—	2	2	no	—
90	C₅×C₄⋊C₄	—	4	1	no	—
91	C₅×C₂²⋊C₄	—	2	1	no	—
92	C₅×C₂²⋊C₄	—	4	4	no	—
93	C₅×C₂²⋊C₄	—	4	1	no	—
94	C₅×C₂²⋊C₄	—	4	1	no	—
95	C₄²×C₅	—	4	1	no	—
96	C₄²×C₅	—	4	1	no	—
97	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
98	C₅×C₁₆	—	4	1	no	—
99	C₅×C₁₆	—	4	1	no	—
100	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
101	#518	—	2	1	no	—
102	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
103	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
104	#442	C₄×C₅	1	4	yes	complemented
105	C₄×D₈	—	2	2	no	—
106	C₂×#108	C₂²×C₅	1	1	yes	characteristic
107	#427	C₂²×C₅	1	1	yes	characteristic
108	#427	C₂²×C₅	1	1	yes	characteristic
109	#418	—	2	1	no	—
110	#405	C₂²×C₅	1	1	yes	characteristic
111	#364	—	2	2	no	—
112	#361	—	2	2	no	—
113	#357	—	2	2	no	—
114	#344	—	2	1	no	—
115	H(3,ℤ/4)	—	2	1	no	—
116	C₂³×C₅	—	4	2	no	—
117	C₂³×C₅	—	4	2	no	—
118	C₂³×C₅	—	4	2	no	—
119	C₂³×C₅	—	16	2	no	—
120	C₂³×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₅×D₄	—	2	4	no	—
122	C₅×D₄	—	4	2	no	—
123	C₅×D₄	—	4	2	no	—
124	C₅×D₄	—	4	2	no	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	—	4	2	no	—
127	C₂×C₄×C₅	—	8	1	no	—
128	C₂×C₄×C₅	—	16	4	no	—
129	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₅	#121	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₅	#135	1	1	yes	characteristic
133	C₂×C₄×C₅	#135	1	1	yes	characteristic
134	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₅×C₈	—	2	2	no	—
136	C₅×C₈	—	8	1	no	—
137	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₅×C₈	#117	1	1	yes	characteristic
139	C₅×C₈	#121	1	1	yes	characteristic
140	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
141	C₂²×D₄	—	2	1	no	—
142	C₂×D₈	—	2	1	no	—
143	C₂×D₈	—	2	1	no	—
144	C₂×D₈	—	2	1	no	—
145	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
146	C₂×D₈	C₅×D₄	1	2	yes	—
147	C₂×D₈	C₅×Q₈	1	2	yes	—
148	C₂×M₁₆	—	2	1	no	—
149	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
150	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
151	C₄×D₄	—	2	4	no	—
152	C₄×D₄	—	4	2	no	—
153	#117	—	2	1	no	—
154	#117	—	2	1	no	—
155	#117	C₅×D₄	1	1	yes	characteristic
156	C₂×C₂²⋊C₄	—	4	2	no	—
157	M₃₂	—	2	1	no	—
158	M₃₂	—	2	1	no	—
159	M₃₂	—	2	1	no	—
160	M₃₂	—	2	1	no	—
161	C₂×C₁₆	—	2	1	no	—
162	C₂×C₁₆	—	2	1	no	—
163	#108	—	2	1	no	—
164	#108	—	2	1	no	—
165	#107	—	2	1	no	—
166	#107	—	2	1	no	—
167	#106	—	2	1	no	—
168	#106	C₂×C₄×C₅	1	2	yes	—
169	#102	—	2	4	no	—
170	#97	C₂×C₄×C₅	1	2	yes	—
171	C₄×C₈	C₅×D₄	1	2	yes	—
172	C₂²×C₅	—	4	1	no	—
173	C₂²×C₅	—	4	4	no	—
174	C₂²×C₅	—	8	4	no	—
175	C₂²×C₅	—	16	2	no	—
176	C₂²×C₅	—	16	2	no	—
177	C₂²×C₅	#418	1	1	yes	characteristic
178	C₂²×C₅	#463	1	1	yes	characteristic
179	C₂²×C₅	#478	1	1	yes	characteristic
180	C₄×C₅	—	4	2	no	—
181	C₄×C₅	—	8	2	no	—
182	C₄×C₅	—	16	1	no	—
183	C₄×C₅	#380	1	1	yes	characteristic
184	C₄×C₅	#416	1	1	yes	characteristic
185	C₄×C₅	#463	1	1	yes	characteristic
186	C₄×C₅	#478	1	1	yes	characteristic
187	C₂⁴	—	4	1	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	4	1	no	—
192	C₂×D₄	—	4	1	no	—
193	C₂×D₄	—	4	1	no	—
194	C₂²×C₄	—	4	4	no	—
195	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
196	D₈	—	2	2	no	—
197	D₈	—	2	2	no	—
198	D₈	—	2	2	no	—
199	D₈	C₅×C₄⋊C₄	1	4	yes	—
200	M₁₆	—	2	1	no	—
201	M₁₆	—	2	1	no	—
202	M₁₆	—	4	1	no	—
203	M₁₆	—	4	1	no	—
204	C₂×C₈	—	4	1	no	—
205	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
206	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
207	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
208	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
209	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
210	C₂×C₈	C₅×#41	1	1	yes	characteristic
211	C₂×C₈	C₅×#41	1	1	yes	characteristic
212	C₄⋊C₄	—	2	1	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	4	1	no	—
215	C₂²⋊C₄	—	2	1	no	—
216	C₂²⋊C₄	—	4	4	no	—
217	C₂²⋊C₄	—	4	1	no	—
218	C₂²⋊C₄	—	4	1	no	—
219	C₄²	—	4	1	no	—
220	C₄²	—	4	1	no	—
221	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
222	C₁₆	—	4	1	no	—
223	C₁₆	—	4	1	no	—
224	C₂×C₅	—	2	1	no	—
225	C₂×C₅	—	2	1	no	—
226	C₂×C₅	—	2	1	no	—
227	C₂×C₅	—	8	4	no	—
228	C₂×C₅	#1804	1	1	yes	characteristic
229	C₂³	—	4	2	no	—
230	C₂³	—	4	2	no	—
231	C₂³	—	4	2	no	—
232	C₂³	—	16	2	no	—
233	C₂³	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
234	D₄	—	2	4	no	—
235	D₄	—	4	2	no	—
236	D₄	—	4	2	no	—
237	D₄	—	4	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	8	1	no	—
241	C₂×C₄	—	16	4	no	—
242	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
243	C₂×C₄	C₅×#121	1	1	yes	characteristic
244	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
245	C₂×C₄	C₅×#135	1	1	yes	characteristic
246	C₂×C₄	C₅×#135	1	1	yes	characteristic
247	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	—	8	1	no	—
250	C₈	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
251	C₈	C₅×#117	1	1	yes	characteristic
252	C₈	C₅×#121	1	1	yes	characteristic
253	C₈	C₅×C₂²⋊Q₈	1	1	yes	characteristic
254	C₅	#12826	1	1	yes	characteristic, direct product factor, Sylow
255	C₂²	—	4	1	no	—
256	C₂²	—	4	4	no	—
257	C₂²	—	8	4	no	—
258	C₂²	—	16	2	no	—
259	C₂²	—	16	2	no	—
260	C₂²	C₅×#418	1	1	yes	characteristic
261	C₂²	C₅×#463	1	1	yes	characteristic
262	C₂²	C₅×#478	1	1	yes	characteristic
263	C₄	—	4	2	no	—
264	C₄	—	8	2	no	—
265	C₄	—	16	1	no	—
266	C₄	C₅×#380	1	1	yes	characteristic
267	C₄	C₅×#416	1	1	yes	characteristic
268	C₄	C₅×#463	1	1	yes	characteristic
269	C₄	C₅×#478	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	—	2	1	no	—
273	C₂	—	8	4	no	—
274	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	C₂²×C₅×D₈	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×#1389	C₂	semidirect
7. C₅×#1384	C₂	semidirect
12. C₅×#442	C₄	semidirect
24. #12826	C₅	direct (N × H)
57. #2153	C₂×C₅	semidirect
60. #1833	C₂×C₅	semidirect
61. #1389	C₂×C₅	semidirect
62. #1384	C₂×C₅	semidirect
104. #442	C₄×C₅	semidirect
254. C₅	#12826	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×#2948	2
C₅×#2153	2
C₂×C₅×#361	2
C₅×#2140	2
C₅×#1833	2
C₅×#1389	2
C₅×#1384	2
#12826	5

Subgroups — C5×#12826 #11935546

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12826 #11935546