Subgroups — C₅×#8279 #11930999

Order 1280.

394 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1767	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₅×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#335	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅×#334	C₂	1	1	yes	maximal, characteristic
6	C₅×#1277	C₂	1	1	yes	maximal, characteristic
7	C₅×#1277	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₅×#97	—	2	1	no	—
13	C₂×C₄×C₅×C₈	—	2	1	no	—
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₄³×C₅	C₂²	1	1	yes	characteristic
17	C₅×#336	—	2	1	no	—
18	C₅×#336	—	2	1	no	—
19	C₅×#336	C₄	1	4	yes	—
20	C₅×#335	C₄	1	4	yes	complemented
21	C₅×#334	C₄	1	4	yes	complemented
22	#8279	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂²×C₅×C₈	—	2	1	no	—
24	C₂²×C₅×C₈	—	2	1	no	—
25	C₂²×C₅×C₈	—	2	1	no	—
26	C₂²×C₅×C₈	—	2	1	no	—
27	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
28	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₅	Q₈	1	2	yes	—
33	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
34	C₅×C₄⋊C₈	—	2	1	no	—
35	C₅×C₄⋊C₈	—	2	1	no	—
36	C₅×C₄⋊C₈	—	2	1	no	—
37	C₅×C₄⋊C₈	—	2	1	no	—
38	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
39	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
40	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
41	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
42	C₅×#97	—	4	2	no	—
43	C₄×C₅×C₈	—	2	1	no	—
44	C₄×C₅×C₈	—	2	1	no	—
45	C₄×C₅×C₈	—	4	2	no	—
46	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
47	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
48	#1767	C₂×C₅	1	1	yes	characteristic
49	C₄×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic
50	#1744	C₂×C₅	1	1	yes	characteristic
51	C₂×#335	C₂×C₅	1	1	yes	characteristic
52	C₂×#334	C₂×C₅	1	1	yes	characteristic
53	#1277	C₂×C₅	1	1	yes	characteristic
54	#1277	C₂×C₅	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₅×C₈	—	2	4	no	—
61	C₂×C₅×C₈	—	2	2	no	—
62	C₂×C₅×C₈	—	2	2	no	—
63	C₂×C₅×C₈	—	2	2	no	—
64	C₂×C₅×C₈	—	2	2	no	—
65	C₂×C₅×C₈	—	2	2	no	—
66	C₂×C₅×C₈	—	2	4	no	—
67	C₂×C₅×C₈	—	4	1	no	—
68	C₂×C₅×C₈	—	4	1	no	—
69	C₂×C₅×C₈	—	4	1	no	—
70	C₂×C₅×C₈	—	4	1	no	—
71	C₂×C₅×C₈	—	4	1	no	—
72	C₂×C₅×C₈	—	4	1	no	—
73	C₂×C₅×C₈	—	4	1	no	—
74	C₂×C₅×C₈	—	4	1	no	—
75	C₂×C₅×C₈	C₄²	1	4	yes	—
76	C₄²×C₅	—	2	4	no	—
77	C₄²×C₅	—	2	4	no	—
78	C₄²×C₅	—	2	1	no	—
79	C₄²×C₅	—	2	1	no	—
80	C₄²×C₅	—	2	1	no	—
81	C₄²×C₅	—	2	1	no	—
82	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₅	#41	1	1	yes	characteristic
85	C₄²×C₅	#41	1	1	yes	characteristic
86	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
87	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
88	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
89	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
90	C₂×#97	—	2	1	no	—
91	C₂×C₄×C₈	—	2	1	no	—
92	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
93	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
94	C₄³	C₂²×C₅	1	1	yes	characteristic
95	#336	—	2	1	no	—
96	#336	—	2	1	no	—
97	#336	C₄×C₅	1	4	yes	—
98	#335	C₄×C₅	1	4	yes	complemented
99	#334	C₄×C₅	1	4	yes	complemented
100	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅	—	2	2	no	—
102	C₂×C₄×C₅	—	2	2	no	—
103	C₂×C₄×C₅	—	2	2	no	—
104	C₂×C₄×C₅	—	2	2	no	—
105	C₂×C₄×C₅	—	2	2	no	—
106	C₂×C₄×C₅	—	2	2	no	—
107	C₂×C₄×C₅	#108	1	4	yes	—
108	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₅	#117	1	1	yes	characteristic
110	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₅	#131	1	4	yes	—
115	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₂×C₄×C₅	#137	1	1	yes	characteristic
118	C₂×C₄×C₅	#137	1	1	yes	characteristic
119	C₅×C₈	—	2	4	no	—
120	C₅×C₈	—	4	2	no	—
121	C₅×C₈	—	4	2	no	—
122	C₅×C₈	—	4	2	no	—
123	C₅×C₈	—	4	2	no	—
124	C₅×C₈	—	4	2	no	—
125	C₂²×C₈	—	2	1	no	—
126	C₂²×C₈	—	2	1	no	—
127	C₂²×C₈	—	2	1	no	—
128	C₂²×C₈	—	2	1	no	—
129	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
130	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
131	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
132	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
133	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
134	C₂×C₄²	C₅×Q₈	1	2	yes	—
135	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
136	C₄⋊C₈	—	2	1	no	—
137	C₄⋊C₈	—	2	1	no	—
138	C₄⋊C₈	—	2	1	no	—
139	C₄⋊C₈	—	2	1	no	—
140	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
141	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
142	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
143	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
144	#97	—	4	2	no	—
145	C₄×C₈	—	2	1	no	—
146	C₄×C₈	—	2	1	no	—
147	C₄×C₈	—	4	2	no	—
148	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
149	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
150	C₂²×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
151	C₂²×C₅	#424	1	2	yes	—
152	C₂²×C₅	#428	1	1	yes	characteristic
153	C₂²×C₅	#428	1	1	yes	characteristic
154	C₂²×C₅	C₂×#108	1	2	yes	—
155	C₄×C₅	—	2	4	no	—
156	C₄×C₅	—	2	1	no	—
157	C₄×C₅	—	2	1	no	—
158	C₄×C₅	—	4	2	no	—
159	C₄×C₅	#405	1	2	yes	—
160	C₄×C₅	#433	1	2	yes	—
161	C₄×C₅	#440	1	1	yes	characteristic
162	C₄×C₅	#440	1	1	yes	characteristic
163	C₄×C₅	#441	1	1	yes	characteristic
164	C₄×C₅	#442	1	1	yes	characteristic
165	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
166	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
167	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
168	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
169	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
170	C₂×C₈	—	2	4	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	4	1	no	—
178	C₂×C₈	—	4	1	no	—
179	C₂×C₈	—	4	1	no	—
180	C₂×C₈	—	4	1	no	—
181	C₂×C₈	—	4	1	no	—
182	C₂×C₈	—	4	1	no	—
183	C₂×C₈	—	4	1	no	—
184	C₂×C₈	—	4	1	no	—
185	C₂×C₈	C₄²×C₅	1	4	yes	—
186	C₄²	—	2	4	no	—
187	C₄²	—	2	1	no	—
188	C₄²	—	2	1	no	—
189	C₄²	—	2	1	no	—
190	C₄²	—	2	1	no	—
191	C₄²	—	2	4	no	—
192	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
193	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
194	C₄²	C₅×#41	1	1	yes	characteristic
195	C₄²	C₅×#41	1	1	yes	characteristic
196	C₂×C₅	#1573	1	2	yes	—
197	C₂×C₅	#1574	1	2	yes	—
198	C₂×C₅	#1768	1	1	yes	characteristic
199	C₂×C₅	#1776	1	1	yes	characteristic
200	C₂×C₅	C₄×#108	1	1	yes	characteristic
201	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	C₅×#108	1	4	yes	—
209	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
210	C₂×C₄	C₅×#117	1	1	yes	characteristic
211	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
212	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
213	C₂×C₄	C₄×C₅×Q₈	1	1	yes	characteristic
214	C₂×C₄	C₄×C₅×Q₈	1	1	yes	characteristic
215	C₂×C₄	C₅×#131	1	4	yes	—
216	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
217	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
218	C₂×C₄	C₅×#137	1	1	yes	characteristic
219	C₂×C₄	C₅×#137	1	1	yes	characteristic
220	C₈	—	2	4	no	—
221	C₈	—	4	2	no	—
222	C₈	—	4	2	no	—
223	C₈	—	4	2	no	—
224	C₈	—	4	2	no	—
225	C₈	—	4	2	no	—
226	C₅	#8279	1	1	yes	characteristic, direct product factor, Sylow
227	C₂²	C₄×C₅×C₄⋊C₄	1	1	yes	characteristic
228	C₂²	C₅×#424	1	2	yes	—
229	C₂²	C₅×#428	1	1	yes	characteristic
230	C₂²	C₅×#428	1	1	yes	characteristic
231	C₂²	C₂×C₅×#108	1	2	yes	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	4	no	—
235	C₄	—	4	2	no	—
236	C₄	C₅×#405	1	2	yes	—
237	C₄	C₅×#433	1	2	yes	—
238	C₄	C₅×#440	1	1	yes	characteristic
239	C₄	C₅×#440	1	1	yes	characteristic
240	C₄	C₅×#441	1	1	yes	characteristic
241	C₄	C₅×#442	1	1	yes	characteristic
242	C₂	C₅×#1573	1	2	yes	—
243	C₂	C₅×#1574	1	2	yes	—
244	C₂	C₅×#1768	1	1	yes	characteristic
245	C₂	C₅×#1776	1	1	yes	characteristic
246	C₂	C₄×C₅×#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle, prime radical	C₂³×C₅	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
20. C₅×#335	C₄	semidirect
21. C₅×#334	C₄	semidirect
22. #8279	C₅	direct (N × H)
98. #335	C₄×C₅	semidirect
99. #334	C₄×C₅	semidirect
226. C₅	#8279	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×#1767	2
C₄×C₅×C₄⋊C₈	2
C₅×#1744	2
C₂×C₅×#335	2
C₂×C₅×#334	2
C₅×#1277	2
#8279	5

Subgroups — C5×#8279 #11930999

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8279 #11930999