Subgroups — C₅×#20806 #11943526

Order 1280.

994 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#430	C₂	1	4	yes	maximal
2	C₂×C₅×#423	C₂	1	2	yes	maximal
3	C₂×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
4	C₅×#1849	C₂	1	4	yes	maximal, complemented
5	C₅×#1848	C₂	1	4	yes	maximal, complemented
6	C₂²×C₅×M₁₆	C₂²	1	4	yes	—
7	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₅×#430	—	2	16	no	—
10	C₂×C₅×#108	C₂²	1	16	yes	—
11	C₅×#423	—	2	4	no	—
12	C₅×#423	C₂²	1	4	yes	—
13	C₅×#423	C₂²	1	4	yes	—
14	C₂×C₅×C₄⋊C₈	—	2	4	no	—
15	C₄×C₅×M₁₆	C₄	1	4	yes	—
16	C₄×C₅×M₁₆	C₄	1	4	yes	—
17	C₂×C₅×#97	C₂²	1	2	yes	—
18	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
19	#20806	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂³×C₄×C₅	Q₈	1	2	yes	—
21	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
22	C₂×C₅×M₁₆	—	2	4	no	—
23	C₂×C₅×M₁₆	—	2	16	no	—
24	C₂×C₅×M₁₆	C₂×C₄	1	8	yes	—
25	C₂×C₅×M₁₆	D₄	1	8	yes	—
26	C₂×C₅×M₁₆	Q₈	1	8	yes	—
27	C₂×C₅×M₁₆	C₂³	1	8	yes	—
28	C₂²×C₅×C₈	—	2	4	no	—
29	C₂²×C₅×C₈	C₂³	1	4	yes	—
30	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅	D₄	1	4	yes	—
33	C₂×C₄²×C₅	Q₈	1	2	yes	—
34	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
36	C₅×#108	—	2	32	no	—
37	C₅×C₄⋊C₈	—	2	8	no	—
38	C₅×C₄⋊C₈	—	2	8	no	—
39	C₅×#97	C₂×C₄	1	8	yes	—
40	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
41	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
42	C₂×#430	C₂×C₅	1	4	yes	—
43	C₂×#423	C₂×C₅	1	2	yes	—
44	C₂×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic
45	#1849	C₂×C₅	1	4	yes	complemented
46	#1848	C₂×C₅	1	4	yes	complemented
47	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₅	—	4	4	no	—
49	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
50	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
51	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
53	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
55	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
56	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
58	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
59	C₅×M₁₆	—	2	16	no	—
60	C₅×M₁₆	—	4	8	no	—
61	C₅×M₁₆	C₄⋊C₄	1	32	yes	—
62	C₂×C₅×C₈	—	2	8	no	—
63	C₂×C₅×C₈	—	4	8	no	—
64	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
65	C₂×C₅×C₈	C₂×D₄	1	8	yes	—
66	C₂×C₅×C₈	C₂×Q₈	1	8	yes	—
67	C₄²×C₅	C₄⋊C₄	1	4	yes	—
68	C₄²×C₅	C₄⋊C₄	1	4	yes	—
69	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₄²×C₅	C₂²×C₄	1	2	yes	—
72	C₄²×C₅	C₂×D₄	1	2	yes	—
73	C₄²×C₅	C₂×D₄	1	2	yes	—
74	C₂²×M₁₆	C₂²×C₅	1	4	yes	—
75	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
76	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
77	#430	—	2	16	no	—
78	C₂×#108	C₂²×C₅	1	16	yes	—
79	#423	—	2	4	no	—
80	#423	C₂²×C₅	1	4	yes	—
81	#423	C₂²×C₅	1	4	yes	—
82	C₂×C₄⋊C₈	—	2	4	no	—
83	C₄×M₁₆	C₄×C₅	1	4	yes	—
84	C₄×M₁₆	C₄×C₅	1	4	yes	—
85	C₂×#97	C₂²×C₅	1	2	yes	—
86	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
87	C₂³×C₅	—	4	2	no	—
88	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
89	C₂³×C₅	C₄⋊Q₈	1	4	yes	—
90	C₂³×C₅	C₂²×Q₈	1	1	yes	characteristic
91	C₂×C₄×C₅	—	2	8	no	—
92	C₂×C₄×C₅	—	2	4	no	—
93	C₂×C₄×C₅	—	4	4	no	—
94	C₂×C₄×C₅	—	4	2	no	—
95	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
96	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
97	C₂×C₄×C₅	C₂×C₄⋊C₄	1	8	yes	—
98	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₅	#127	1	4	yes	—
101	C₂×C₄×C₅	C₄⋊Q₈	1	4	yes	—
102	C₂×C₄×C₅	C₄⋊Q₈	1	4	yes	—
103	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
105	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅	C₂²×Q₈	1	2	yes	—
107	C₅×C₈	—	4	8	no	—
108	C₅×C₈	C₂×C₄⋊C₄	1	16	yes	—
109	C₂³×C₄	C₅×Q₈	1	2	yes	—
110	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
111	C₂×M₁₆	—	2	4	no	—
112	C₂×M₁₆	—	2	16	no	—
113	C₂×M₁₆	C₂×C₄×C₅	1	8	yes	—
114	C₂×M₁₆	C₅×D₄	1	8	yes	—
115	C₂×M₁₆	C₅×Q₈	1	8	yes	—
116	C₂×M₁₆	C₂³×C₅	1	8	yes	—
117	C₂²×C₈	—	2	4	no	—
118	C₂²×C₈	C₂³×C₅	1	4	yes	—
119	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
120	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
121	C₂×C₄²	C₅×D₄	1	4	yes	—
122	C₂×C₄²	C₅×Q₈	1	2	yes	—
123	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
124	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
125	#108	—	2	32	no	—
126	C₄⋊C₈	—	2	8	no	—
127	C₄⋊C₈	—	2	8	no	—
128	#97	C₂×C₄×C₅	1	8	yes	—
129	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
130	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
131	C₂²×C₅	—	2	4	no	—
132	C₂²×C₅	—	2	2	no	—
133	C₂²×C₅	—	4	4	no	—
134	C₂²×C₅	#384	1	4	yes	—
135	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
136	C₂²×C₅	C₂×C₄⋊Q₈	1	2	yes	—
137	C₄×C₅	—	2	4	no	—
138	C₄×C₅	—	2	2	no	—
139	C₄×C₅	#384	1	4	yes	—
140	C₄×C₅	#430	1	4	yes	complemented
141	C₄×C₅	C₂²×C₄⋊C₄	1	2	yes	—
142	C₄×C₅	C₂×#127	1	1	yes	characteristic
143	C₄×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
144	C₂⁴	C₂×C₅×Q₈	1	1	yes	characteristic
145	C₂²×C₄	—	4	4	no	—
146	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
147	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
148	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
149	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
150	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
151	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
152	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
153	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
154	C₂²×C₄	C₂×C₅×Q₈	1	4	yes	—
155	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
156	M₁₆	—	2	16	no	—
157	M₁₆	—	4	8	no	—
158	M₁₆	C₅×C₄⋊C₄	1	32	yes	—
159	C₂×C₈	—	2	8	no	—
160	C₂×C₈	—	4	8	no	—
161	C₂×C₈	C₂²×C₄×C₅	1	8	yes	—
162	C₂×C₈	C₂×C₅×D₄	1	8	yes	—
163	C₂×C₈	C₂×C₅×Q₈	1	8	yes	—
164	C₄²	C₅×C₄⋊C₄	1	4	yes	—
165	C₄²	C₅×C₄⋊C₄	1	4	yes	—
166	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
167	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
168	C₄²	C₂²×C₄×C₅	1	2	yes	—
169	C₄²	C₂×C₅×D₄	1	2	yes	—
170	C₄²	C₂×C₅×D₄	1	2	yes	—
171	C₂×C₅	—	2	2	no	—
172	C₂×C₅	—	2	4	no	—
173	C₂×C₅	C₂×#384	1	1	yes	characteristic
174	C₂×C₅	C₂×#430	1	2	yes	—
175	C₂³	—	4	2	no	—
176	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
177	C₂³	C₅×C₄⋊Q₈	1	4	yes	—
178	C₂³	C₂²×C₅×Q₈	1	1	yes	characteristic
179	C₂×C₄	—	2	8	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
184	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
185	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
186	C₂×C₄	C₂×C₅×C₄⋊C₄	1	8	yes	—
187	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
188	C₂×C₄	C₅×#127	1	4	yes	—
189	C₂×C₄	C₅×C₄⋊Q₈	1	4	yes	—
190	C₂×C₄	C₅×C₄⋊Q₈	1	4	yes	—
191	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
192	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
193	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
194	C₂×C₄	C₂²×C₅×Q₈	1	2	yes	—
195	C₈	—	4	8	no	—
196	C₈	C₂×C₅×C₄⋊C₄	1	16	yes	—
197	C₅	#20806	1	1	yes	characteristic, direct product factor, Sylow
198	C₂²	—	2	4	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	4	4	no	—
201	C₂²	C₅×#384	1	4	yes	—
202	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
203	C₂²	C₂×C₅×C₄⋊Q₈	1	2	yes	—
204	C₄	—	2	4	no	—
205	C₄	—	2	2	no	—
206	C₄	C₅×#384	1	4	yes	—
207	C₄	C₅×#430	1	4	yes	complemented
208	C₄	C₂²×C₅×C₄⋊C₄	1	2	yes	—
209	C₄	C₂×C₅×#127	1	1	yes	characteristic
210	C₄	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
211	C₂	—	2	4	no	—
212	C₂	—	2	2	no	—
213	C₂	C₂×C₅×#384	1	1	yes	characteristic
214	C₂	C₂×C₅×#430	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂²×C₅	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#1849	C₂	semidirect
5. C₅×#1848	C₂	semidirect
19. #20806	C₅	direct (N × H)
45. #1849	C₂×C₅	semidirect
46. #1848	C₂×C₅	semidirect
140. C₄×C₅	#430	semidirect
197. C₅	#20806	direct (N × H)
207. C₄	C₅×#430	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#430	2
C₂×C₅×#423	2
C₂×C₄×C₅×M₁₆	2
C₅×#1849	2
C₅×#1848	2
#20806	5

Subgroups — C5×#20806 #11943526

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#20806 #11943526