Subgroups — C₅×#7720 #11930440

Order 1280.

388 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2859	C₂	1	1	yes	maximal, characteristic
2	C₅×#1839	C₂	1	1	yes	maximal, characteristic
3	C₅×#1831	C₂	1	1	yes	maximal, characteristic
4	C₅×#1271	C₂	1	4	yes	maximal
5	C₂×C₅×#125	—	2	1	no	—
6	C₂×C₅×C₄⋊C₈	—	2	1	no	—
7	C₅×#389	—	2	1	no	—
8	C₅×#389	—	2	1	no	—
9	C₅×#389	C₄	1	2	yes	—
10	C₅×#389	C₂²	1	2	yes	—
11	C₅×#384	C₄	1	1	yes	characteristic
12	C₅×#384	C₄	1	1	yes	characteristic
13	C₅×#382	—	2	1	no	—
14	C₄³×C₅	C₂²	1	1	yes	characteristic
15	C₅×#336	—	2	1	no	—
16	C₅×#336	—	2	1	no	—
17	C₅×#336	—	2	1	no	—
18	C₅×#336	C₂²	1	2	yes	—
19	C₅×#336	C₂²	1	2	yes	—
20	#7720	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₅×C₈	—	2	2	no	—
22	C₂²×C₅×C₈	—	2	2	no	—
23	C₅×#125	—	4	4	no	—
24	C₂×C₅×C₄⋊C₄	—	2	1	no	—
25	C₂×C₅×C₄⋊C₄	—	2	1	no	—
26	C₂×C₅×C₄⋊C₄	—	2	1	no	—
27	C₂×C₅×C₄⋊C₄	—	2	1	no	—
28	C₂×C₅×C₄⋊C₄	—	2	2	no	—
29	C₂×C₅×C₄⋊C₄	—	2	2	no	—
30	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
31	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
39	C₅×C₄⋊C₈	—	4	2	no	—
40	C₅×#95	—	2	2	no	—
41	C₅×#95	—	2	2	no	—
42	#2859	C₂×C₅	1	1	yes	characteristic
43	#1839	C₂×C₅	1	1	yes	characteristic
44	#1831	C₂×C₅	1	1	yes	characteristic
45	#1271	C₂×C₅	1	4	yes	—
46	C₂²×C₄×C₅	—	2	2	no	—
47	C₂²×C₄×C₅	—	2	2	no	—
48	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂×C₅×C₈	—	4	2	no	—
56	C₂×C₅×C₈	—	4	2	no	—
57	C₂×C₅×C₈	—	4	2	no	—
58	C₂×C₅×C₈	—	4	2	no	—
59	C₂×C₅×C₈	—	4	2	no	—
60	C₂×C₅×C₈	—	4	2	no	—
61	C₅×C₄⋊C₄	—	2	4	no	—
62	C₅×C₄⋊C₄	—	4	2	no	—
63	C₅×C₄⋊C₄	—	4	2	no	—
64	C₅×C₄⋊C₄	—	4	2	no	—
65	C₅×C₄⋊C₄	—	4	2	no	—
66	C₅×C₄⋊C₄	—	4	2	no	—
67	C₅×C₄⋊C₄	—	4	4	no	—
68	C₅×C₄⋊C₄	—	4	4	no	—
69	C₄²×C₅	—	2	2	no	—
70	C₄²×C₅	—	2	2	no	—
71	C₄²×C₅	—	2	2	no	—
72	C₄²×C₅	—	4	1	no	—
73	C₄²×C₅	—	4	1	no	—
74	C₄²×C₅	—	4	1	no	—
75	C₄²×C₅	SD₁₆	1	2	yes	—
76	C₄²×C₅	Q₁₆	1	2	yes	—
77	C₂×#125	—	2	1	no	—
78	C₂×C₄⋊C₈	—	2	1	no	—
79	#389	—	2	1	no	—
80	#389	—	2	1	no	—
81	#389	C₄×C₅	1	2	yes	—
82	#389	C₂²×C₅	1	2	yes	—
83	#384	C₄×C₅	1	1	yes	characteristic
84	#384	C₄×C₅	1	1	yes	characteristic
85	#382	—	2	1	no	—
86	C₄³	C₂²×C₅	1	1	yes	characteristic
87	#336	—	2	1	no	—
88	#336	—	2	1	no	—
89	#336	—	2	1	no	—
90	#336	C₂²×C₅	1	2	yes	—
91	#336	C₂²×C₅	1	2	yes	—
92	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₅	—	2	1	no	—
94	C₂×C₄×C₅	—	2	1	no	—
95	C₂×C₄×C₅	—	2	1	no	—
96	C₂×C₄×C₅	—	2	1	no	—
97	C₂×C₄×C₅	—	2	1	no	—
98	C₂×C₄×C₅	—	2	1	no	—
99	C₂×C₄×C₅	—	2	1	no	—
100	C₂×C₄×C₅	—	2	1	no	—
101	C₂×C₄×C₅	—	2	1	no	—
102	C₂×C₄×C₅	—	2	1	no	—
103	C₂×C₄×C₅	—	2	1	no	—
104	C₂×C₄×C₅	—	2	1	no	—
105	C₂×C₄×C₅	—	2	1	no	—
106	C₂×C₄×C₅	—	2	1	no	—
107	C₂×C₄×C₅	—	2	1	no	—
108	C₂×C₄×C₅	—	4	2	no	—
109	C₂×C₄×C₅	—	4	2	no	—
110	C₂×C₄×C₅	—	4	2	no	—
111	C₂×C₄×C₅	—	4	2	no	—
112	C₂×C₄×C₅	—	4	2	no	—
113	C₂×C₄×C₅	—	4	2	no	—
114	C₂×C₄×C₅	#101	1	2	yes	—
115	C₂×C₄×C₅	#103	1	2	yes	—
116	C₂×C₄×C₅	#103	1	2	yes	—
117	C₂×C₄×C₅	#131	1	2	yes	—
118	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
120	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
121	C₅×C₈	—	8	2	no	—
122	C₅×C₈	—	8	2	no	—
123	C₂²×C₈	—	2	2	no	—
124	C₂²×C₈	—	2	2	no	—
125	#125	—	4	4	no	—
126	C₂×C₄⋊C₄	—	2	1	no	—
127	C₂×C₄⋊C₄	—	2	1	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	1	no	—
130	C₂×C₄⋊C₄	—	2	1	no	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
133	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
134	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
135	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
136	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
137	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
138	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
139	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
140	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
141	C₄⋊C₈	—	4	2	no	—
142	#95	—	2	2	no	—
143	#95	—	2	2	no	—
144	C₂²×C₅	#408	1	1	yes	characteristic
145	C₂²×C₅	#410	1	1	yes	characteristic
146	C₂²×C₅	C₂×#101	1	1	yes	characteristic
147	C₂²×C₅	C₂×#103	1	1	yes	characteristic
148	C₂²×C₅	#418	1	1	yes	characteristic
149	C₂²×C₅	#421	1	1	yes	characteristic
150	C₂²×C₅	#421	1	1	yes	characteristic
151	C₄×C₅	—	2	2	no	—
152	C₄×C₅	—	2	2	no	—
153	C₄×C₅	—	2	2	no	—
154	C₄×C₅	—	2	2	no	—
155	C₄×C₅	—	2	1	no	—
156	C₄×C₅	—	2	1	no	—
157	C₄×C₅	—	2	2	no	—
158	C₄×C₅	—	4	1	no	—
159	C₄×C₅	—	8	2	no	—
160	C₄×C₅	—	8	2	no	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
164	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
165	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
166	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
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×D₄	1	1	yes	characteristic
170	C₂×C₈	—	4	2	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	—	4	2	no	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	4	4	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₄⋊C₄	—	4	4	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	2	2	no	—
187	C₄²	—	4	1	no	—
188	C₄²	—	4	1	no	—
189	C₄²	—	4	1	no	—
190	C₄²	C₅×SD₁₆	1	2	yes	—
191	C₄²	C₅×Q₁₆	1	2	yes	—
192	C₂×C₅	#1470	1	1	yes	characteristic
193	C₂×C₅	#1481	1	1	yes	characteristic
194	C₂×C₅	#1496	1	1	yes	characteristic
195	C₂×C₅	#1514	1	1	yes	characteristic
196	C₂×C₅	#1518	1	1	yes	characteristic
197	C₂×C₅	#1534	1	1	yes	characteristic
198	C₂×C₅	#1549	1	1	yes	characteristic
199	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	C₅×#101	1	2	yes	—
222	C₂×C₄	C₅×#103	1	2	yes	—
223	C₂×C₄	C₅×#103	1	2	yes	—
224	C₂×C₄	C₅×#131	1	2	yes	—
225	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
226	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
227	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
228	C₈	—	8	2	no	—
229	C₈	—	8	2	no	—
230	C₅	#7720	1	1	yes	characteristic, direct product factor, Sylow
231	C₂²	C₅×#408	1	1	yes	characteristic
232	C₂²	C₅×#410	1	1	yes	characteristic
233	C₂²	C₂×C₅×#101	1	1	yes	characteristic
234	C₂²	C₂×C₅×#103	1	1	yes	characteristic
235	C₂²	C₅×#418	1	1	yes	characteristic
236	C₂²	C₅×#421	1	1	yes	characteristic
237	C₂²	C₅×#421	1	1	yes	characteristic
238	C₄	—	2	2	no	—
239	C₄	—	2	1	no	—
240	C₄	—	2	1	no	—
241	C₄	—	2	2	no	—
242	C₄	—	2	2	no	—
243	C₄	—	2	2	no	—
244	C₄	—	2	2	no	—
245	C₄	—	4	1	no	—
246	C₄	—	8	2	no	—
247	C₄	—	8	2	no	—
248	C₂	C₅×#1470	1	1	yes	characteristic
249	C₂	C₅×#1481	1	1	yes	characteristic
250	C₂	C₅×#1496	1	1	yes	characteristic
251	C₂	C₅×#1514	1	1	yes	characteristic
252	C₂	C₅×#1518	1	1	yes	characteristic
253	C₂	C₅×#1534	1	1	yes	characteristic
254	C₂	C₅×#1549	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
20. #7720	C₅	direct (N × H)
230. C₅	#7720	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×#2859	2
C₅×#1839	2
C₅×#1831	2
C₅×#1271	2
#7720	5

Subgroups — C5×#7720 #11930440

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#7720 #11930440