Subgroups — C₅×#12666 #11935386

Order 1280.

318 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2150	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#347	C₂	1	2	yes	maximal
3	C₄×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
4	C₅×#1770	C₂	1	1	yes	maximal, characteristic
5	C₅×#1375	C₂	1	2	yes	maximal
6	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
7	C₂×C₅×M₃₂	C₂²	1	2	yes	—
8	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
9	C₂×C₅×C₄⋊C₈	C₄	1	1	yes	characteristic
10	C₂×C₅×#97	C₂²	1	2	yes	—
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₄×C₅×C₄⋊C₄	C₄	1	1	yes	characteristic
13	C₅×C₄⋊C₁₆	C₄	1	2	yes	complemented
14	C₅×C₄⋊C₁₆	C₄	1	2	yes	complemented
15	C₅×#347	—	2	4	no	—
16	C₅×#346	C₄	1	2	yes	complemented
17	C₄×C₅×C₁₆	C₄	1	2	yes	complemented
18	C₅×#336	C₄	1	2	yes	—
19	#12666	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
23	C₂×C₅×C₄⋊C₄	—	2	1	no	—
24	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
26	C₅×M₃₂	—	2	2	no	—
27	C₅×M₃₂	—	2	1	no	—
28	C₅×M₃₂	—	2	1	no	—
29	C₅×M₃₂	—	2	2	no	—
30	C₅×M₃₂	D₄	1	2	yes	—
31	C₅×M₃₂	Q₈	1	2	yes	—
32	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
33	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
34	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
35	C₅×C₄⋊C₈	—	2	1	no	—
36	C₅×C₄⋊C₈	—	2	1	no	—
37	C₅×#97	—	2	2	no	—
38	C₅×#97	—	2	2	no	—
39	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
40	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
41	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
42	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
43	C₅×#95	—	2	1	no	—
44	#2150	C₂×C₅	1	1	yes	characteristic
45	C₂×#347	C₂×C₅	1	2	yes	—
46	C₄×M₃₂	C₂×C₅	1	1	yes	characteristic
47	#1770	C₂×C₅	1	1	yes	characteristic
48	#1375	C₂×C₅	1	2	yes	—
49	C₂²×C₄×C₅	—	2	2	no	—
50	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₅×C₈	—	2	1	no	—
54	C₂×C₅×C₈	—	2	1	no	—
55	C₂×C₅×C₈	—	2	2	no	—
56	C₂×C₅×C₈	—	2	2	no	—
57	C₂×C₅×C₈	—	2	2	no	—
58	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
59	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
60	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
63	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₅×C₈	#41	1	1	yes	characteristic
65	C₂×C₅×C₈	#41	1	1	yes	characteristic
66	C₅×C₄⋊C₄	—	4	2	no	—
67	C₄²×C₅	—	2	2	no	—
68	C₄²×C₅	—	2	2	no	—
69	C₄²×C₅	C₄²	1	1	yes	characteristic
70	C₄²×C₅	C₄²	1	1	yes	characteristic
71	C₄²×C₅	M₁₆	1	1	yes	characteristic
72	C₄²×C₅	M₁₆	1	1	yes	characteristic
73	C₅×C₁₆	—	2	2	no	—
74	C₅×C₁₆	—	2	4	no	—
75	C₅×C₁₆	C₄⋊C₄	1	4	yes	complemented
76	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
77	C₂×M₃₂	C₂²×C₅	1	2	yes	—
78	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
79	C₂×C₄⋊C₈	C₄×C₅	1	1	yes	characteristic
80	C₂×#97	C₂²×C₅	1	2	yes	—
81	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
82	C₄×C₄⋊C₄	C₄×C₅	1	1	yes	characteristic
83	C₄⋊C₁₆	C₄×C₅	1	2	yes	complemented
84	C₄⋊C₁₆	C₄×C₅	1	2	yes	complemented
85	#347	—	2	4	no	—
86	#346	C₄×C₅	1	2	yes	complemented
87	C₄×C₁₆	C₄×C₅	1	2	yes	complemented
88	#336	C₄×C₅	1	2	yes	—
89	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₅	—	2	2	no	—
91	C₂×C₄×C₅	—	4	1	no	—
92	C₂×C₄×C₅	—	4	2	no	—
93	C₂×C₄×C₅	—	4	2	no	—
94	C₂×C₄×C₅	—	4	2	no	—
95	C₂×C₄×C₅	#97	1	1	yes	characteristic
96	C₂×C₄×C₅	#97	1	1	yes	characteristic
97	C₂×C₄×C₅	#97	1	1	yes	characteristic
98	C₂×C₄×C₅	#97	1	1	yes	characteristic
99	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
100	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
102	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
103	C₂×C₄×C₅	#131	1	1	yes	characteristic
104	C₂×C₄×C₅	#131	1	1	yes	characteristic
105	C₅×C₈	—	2	1	no	—
106	C₅×C₈	—	2	1	no	—
107	C₅×C₈	—	4	2	no	—
108	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₅×C₈	#117	1	1	yes	characteristic
110	C₅×C₈	C₄×D₄	1	1	yes	characteristic
111	C₅×C₈	C₄×Q₈	1	1	yes	characteristic
112	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
113	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
114	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
115	C₂×C₄⋊C₄	—	2	1	no	—
116	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
117	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
118	M₃₂	—	2	2	no	—
119	M₃₂	—	2	1	no	—
120	M₃₂	—	2	1	no	—
121	M₃₂	—	2	2	no	—
122	M₃₂	C₅×D₄	1	2	yes	—
123	M₃₂	C₅×Q₈	1	2	yes	—
124	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
125	C₂×C₁₆	C₂×C₄×C₅	1	4	yes	—
126	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
127	C₄⋊C₈	—	2	1	no	—
128	C₄⋊C₈	—	2	1	no	—
129	#97	—	2	2	no	—
130	#97	—	2	2	no	—
131	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
132	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
133	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
134	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
135	#95	—	2	1	no	—
136	C₂²×C₅	—	2	1	no	—
137	C₂²×C₅	—	2	1	no	—
138	C₂²×C₅	C₂×#97	1	1	yes	characteristic
139	C₂²×C₅	#436	1	1	yes	characteristic
140	C₂²×C₅	#446	1	1	yes	characteristic
141	C₄×C₅	—	2	1	no	—
142	C₄×C₅	—	2	1	no	—
143	C₄×C₅	—	2	1	no	—
144	C₄×C₅	—	8	2	no	—
145	C₄×C₅	#347	1	2	yes	complemented
146	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
147	C₄×C₅	#405	1	1	yes	characteristic
148	C₄×C₅	#436	1	1	yes	characteristic
149	C₄×C₅	#446	1	1	yes	characteristic
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	C₅×M₁₆	1	1	yes	characteristic
152	C₂²×C₄	C₅×M₁₆	1	1	yes	characteristic
153	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	C₄²×C₅	1	1	yes	characteristic
160	C₂×C₈	C₄²×C₅	1	1	yes	characteristic
161	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
162	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
163	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
164	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
165	C₂×C₈	C₅×#41	1	1	yes	characteristic
166	C₂×C₈	C₅×#41	1	1	yes	characteristic
167	C₄⋊C₄	—	4	2	no	—
168	C₄²	—	2	2	no	—
169	C₄²	—	2	2	no	—
170	C₄²	C₄²×C₅	1	1	yes	characteristic
171	C₄²	C₄²×C₅	1	1	yes	characteristic
172	C₄²	C₅×M₁₆	1	1	yes	characteristic
173	C₄²	C₅×M₁₆	1	1	yes	characteristic
174	C₁₆	—	2	2	no	—
175	C₁₆	—	2	4	no	—
176	C₁₆	C₅×C₄⋊C₄	1	4	yes	complemented
177	C₂×C₅	—	2	1	no	—
178	C₂×C₅	—	2	1	no	—
179	C₂×C₅	#1770	1	1	yes	characteristic
180	C₂×C₅	C₂×#347	1	1	yes	characteristic
181	C₂×C₅	#2110	1	1	yes	characteristic
182	C₂³	C₂×C₅×M₁₆	1	1	yes	characteristic
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	4	1	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	C₅×#97	1	1	yes	characteristic
189	C₂×C₄	C₅×#97	1	1	yes	characteristic
190	C₂×C₄	C₅×#97	1	1	yes	characteristic
191	C₂×C₄	C₅×#97	1	1	yes	characteristic
192	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
193	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
194	C₂×C₄	C₄×C₅×Q₈	1	1	yes	characteristic
195	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
196	C₂×C₄	C₅×#131	1	1	yes	characteristic
197	C₂×C₄	C₅×#131	1	1	yes	characteristic
198	C₈	—	2	1	no	—
199	C₈	—	2	1	no	—
200	C₈	—	4	2	no	—
201	C₈	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
202	C₈	C₅×#117	1	1	yes	characteristic
203	C₈	C₄×C₅×D₄	1	1	yes	characteristic
204	C₈	C₄×C₅×Q₈	1	1	yes	characteristic
205	C₅	#12666	1	1	yes	characteristic, direct product factor, Sylow
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	C₂×C₅×#97	1	1	yes	characteristic
209	C₂²	C₅×#436	1	1	yes	characteristic
210	C₂²	C₅×#446	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	8	2	no	—
215	C₄	C₅×#347	1	2	yes	complemented
216	C₄	C₄×C₅×C₄⋊C₄	1	1	yes	characteristic
217	C₄	C₅×#405	1	1	yes	characteristic
218	C₄	C₅×#436	1	1	yes	characteristic
219	C₄	C₅×#446	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	C₅×#1770	1	1	yes	characteristic
223	C₂	C₂×C₅×#347	1	1	yes	characteristic
224	C₂	C₅×#2110	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂×C₄²
commutator	C₂×C₄	C₂×C₄²×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂²×C₅	C₂×#97
prime radical	C₂³×C₅	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₅×C₄⋊C₁₆	C₄	semidirect
14. C₅×C₄⋊C₁₆	C₄	semidirect
16. C₅×#346	C₄	semidirect
17. C₄×C₅×C₁₆	C₄	semidirect
19. #12666	C₅	direct (N × H)
75. C₅×C₁₆	C₄⋊C₄	semidirect
83. C₄⋊C₁₆	C₄×C₅	semidirect
84. C₄⋊C₁₆	C₄×C₅	semidirect
86. #346	C₄×C₅	semidirect
87. C₄×C₁₆	C₄×C₅	semidirect
145. C₄×C₅	#347	semidirect
176. C₁₆	C₅×C₄⋊C₄	semidirect
205. C₅	#12666	direct (N × H)
215. C₄	C₅×#347	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2150	2
C₂×C₅×#347	2
C₄×C₅×M₃₂	2
C₅×#1770	2
C₅×#1375	2
#12666	5

Subgroups — C5×#12666 #11935386

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12666 #11935386