Subgroups — C₂²×C₅×#490 #11983327

Order 1280.

2830 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#127	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#490	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₅×#102	C₂	1	4	yes	maximal, complemented
5	C₂²×C₅×#97	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₅×D₄	—	2	2	no	—
7	C₂³×C₅×D₄	C₂²	1	2	yes	—
8	C₂³×C₅×C₈	C₂²	1	2	yes	—
9	C₂×C₅×#127	—	2	6	no	—
10	C₂×C₅×#127	C₂²	1	12	yes	—
11	C₂×C₅×#125	—	2	6	no	—
12	C₂×C₅×#125	C₂²	1	12	yes	complemented
13	C₂²×C₅×C₄⋊C₄	—	2	2	no	—
14	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
16	C₅×#490	C₂²	1	64	yes	direct product factor
17	C₂×C₅×#102	C₂²	1	48	yes	complemented
18	C₂×C₅×#97	C₂²	1	12	yes	complemented
19	C₂²×#490	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂⁵×C₅	—	2	2	no	—
21	C₂²×C₅×D₄	—	2	6	no	—
22	C₂²×C₅×D₄	—	2	24	no	—
23	C₂²×C₅×D₄	—	2	16	no	—
24	C₂²×C₅×D₄	—	4	16	no	—
25	C₂²×C₅×D₄	C₂³	1	12	yes	—
26	C₂³×C₄×C₅	—	2	2	no	—
27	C₂³×C₄×C₅	D₄	1	2	yes	—
28	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂²×C₅×C₈	—	2	8	no	—
30	C₂²×C₅×C₈	C₂³	1	12	yes	—
31	C₅×#127	—	2	24	no	—
32	C₅×#127	C₂³	1	16	yes	—
33	C₅×#125	—	2	24	no	—
34	C₅×#125	C₂³	1	16	yes	complemented
35	C₂×C₅×C₄⋊C₄	—	2	24	no	—
36	C₂×C₅×C₄⋊C₄	—	2	6	no	—
37	C₂×C₅×C₄⋊C₄	C₂³	1	12	yes	—
38	C₂×C₄²×C₅	D₄	1	6	yes	—
39	C₂×C₄²×C₅	C₂³	1	6	yes	—
40	C₅×#102	C₂³	1	64	yes	complemented
41	C₅×#97	C₂³	1	16	yes	complemented
42	C₂²×#127	C₂×C₅	1	1	yes	characteristic
43	C₂²×#125	C₂×C₅	1	1	yes	characteristic, complemented
44	C₂×#490	C₂×C₅	1	24	yes	direct product factor
45	C₂²×#102	C₂×C₅	1	4	yes	complemented
46	C₂²×#97	C₂×C₅	1	1	yes	characteristic, complemented
47	C₂⁴×C₅	—	2	12	no	—
48	C₂⁴×C₅	—	4	8	no	—
49	C₂⁴×C₅	—	4	16	no	—
50	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
51	C₂×C₅×D₄	—	2	12	no	—
52	C₂×C₅×D₄	—	2	32	no	—
53	C₂×C₅×D₄	—	2	48	no	—
54	C₂×C₅×D₄	—	4	96	no	—
55	C₂×C₅×D₄	—	4	24	no	—
56	C₂×C₅×D₄	C₂⁴	1	8	yes	—
57	C₂²×C₄×C₅	—	2	8	no	—
58	C₂²×C₄×C₅	—	2	12	no	—
59	C₂²×C₄×C₅	—	4	8	no	—
60	C₂²×C₄×C₅	C₂×D₄	1	12	yes	—
61	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
62	C₂²×C₄×C₅	#41	1	8	yes	—
63	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
64	C₂×C₅×C₈	—	2	24	no	—
65	C₂×C₅×C₈	C₂⁴	1	8	yes	—
66	C₅×C₄⋊C₄	—	2	32	no	—
67	C₅×C₄⋊C₄	—	2	12	no	—
68	C₅×C₄⋊C₄	C₂⁴	1	8	yes	—
69	C₄²×C₅	C₂×D₄	1	12	yes	—
70	C₄²×C₅	C₂⁴	1	4	yes	—
71	C₂³×D₄	—	2	2	no	—
72	C₂³×D₄	C₂²×C₅	1	2	yes	—
73	C₂³×C₈	C₂²×C₅	1	2	yes	—
74	C₂×#127	—	2	6	no	—
75	C₂×#127	C₂²×C₅	1	12	yes	—
76	C₂×#125	—	2	6	no	—
77	C₂×#125	C₂²×C₅	1	12	yes	complemented
78	C₂²×C₄⋊C₄	—	2	2	no	—
79	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	—
80	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
81	#490	C₂²×C₅	1	64	yes	direct product factor
82	C₂×#102	C₂²×C₅	1	48	yes	complemented
83	C₂×#97	C₂²×C₅	1	12	yes	complemented
84	C₂³×C₅	—	2	8	no	—
85	C₂³×C₅	—	4	24	no	—
86	C₂³×C₅	—	4	48	no	—
87	C₂³×C₅	—	8	32	no	—
88	C₂³×C₅	#124	1	4	yes	—
89	C₂³×C₅	AGL(1,ℤ/8)	1	8	yes	—
90	C₂³×C₅	C₂²×D₄	1	3	yes	—
91	C₅×D₄	—	2	16	no	—
92	C₅×D₄	—	4	64	no	—
93	C₅×D₄	—	4	24	no	—
94	C₂×C₄×C₅	—	2	24	no	—
95	C₂×C₄×C₅	—	2	8	no	—
96	C₂×C₄×C₅	—	4	24	no	—
97	C₂×C₄×C₅	#124	1	12	yes	—
98	C₂×C₄×C₅	C₂²×D₄	1	8	yes	—
99	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
100	C₂×C₄×C₅	C₂×#41	1	12	yes	—
101	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
102	C₅×C₈	—	2	8	no	—
103	C₂⁵	—	2	2	no	—
104	C₂²×D₄	—	2	6	no	—
105	C₂²×D₄	—	2	24	no	—
106	C₂²×D₄	—	2	16	no	—
107	C₂²×D₄	—	4	16	no	—
108	C₂²×D₄	C₂³×C₅	1	12	yes	—
109	C₂³×C₄	—	2	2	no	—
110	C₂³×C₄	C₅×D₄	1	2	yes	—
111	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
112	C₂²×C₈	—	2	8	no	—
113	C₂²×C₈	C₂³×C₅	1	12	yes	—
114	#127	—	2	24	no	—
115	#127	C₂³×C₅	1	16	yes	—
116	#125	—	2	24	no	—
117	#125	C₂³×C₅	1	16	yes	complemented
118	C₂×C₄⋊C₄	—	2	24	no	—
119	C₂×C₄⋊C₄	—	2	6	no	—
120	C₂×C₄⋊C₄	C₂³×C₅	1	12	yes	—
121	C₂×C₄²	C₅×D₄	1	6	yes	—
122	C₂×C₄²	C₂³×C₅	1	6	yes	—
123	#102	C₂³×C₅	1	64	yes	complemented
124	#97	C₂³×C₅	1	16	yes	complemented
125	C₂²×C₅	—	4	8	no	—
126	C₂²×C₅	—	4	16	no	—
127	C₂²×C₅	—	8	48	no	—
128	C₂²×C₅	#490	1	16	yes	direct product factor
129	C₂²×C₅	C₂×#124	1	6	yes	—
130	C₂²×C₅	C₂×AGL(1,ℤ/8)	1	12	yes	—
131	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
132	C₄×C₅	—	2	8	no	—
133	C₄×C₅	—	4	8	no	—
134	C₄×C₅	C₂×#124	1	6	yes	—
135	C₄×C₅	C₂²×#41	1	2	yes	—
136	C₂⁴	—	2	12	no	—
137	C₂⁴	—	4	8	no	—
138	C₂⁴	—	4	16	no	—
139	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
140	C₂×D₄	—	2	12	no	—
141	C₂×D₄	—	2	32	no	—
142	C₂×D₄	—	2	48	no	—
143	C₂×D₄	—	4	96	no	—
144	C₂×D₄	—	4	24	no	—
145	C₂×D₄	C₂⁴×C₅	1	8	yes	—
146	C₂²×C₄	—	2	8	no	—
147	C₂²×C₄	—	2	12	no	—
148	C₂²×C₄	—	4	8	no	—
149	C₂²×C₄	C₂×C₅×D₄	1	12	yes	—
150	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
151	C₂²×C₄	C₅×#41	1	8	yes	—
152	C₂²×C₄	C₂⁴×C₅	1	3	yes	—
153	C₂×C₈	—	2	24	no	—
154	C₂×C₈	C₂⁴×C₅	1	8	yes	—
155	C₄⋊C₄	—	2	32	no	—
156	C₄⋊C₄	—	2	12	no	—
157	C₄⋊C₄	C₂⁴×C₅	1	8	yes	—
158	C₄²	C₂×C₅×D₄	1	12	yes	—
159	C₄²	C₂⁴×C₅	1	4	yes	—
160	C₂×C₅	—	8	8	no	—
161	C₂×C₅	C₂×#490	1	12	yes	direct product factor
162	C₂×C₅	C₂²×#124	1	1	yes	characteristic
163	C₂×C₅	C₂²×AGL(1,ℤ/8)	1	2	yes	—
164	C₂³	—	2	8	no	—
165	C₂³	—	4	24	no	—
166	C₂³	—	4	48	no	—
167	C₂³	—	8	32	no	—
168	C₂³	C₅×#124	1	4	yes	—
169	C₂³	C₅×AGL(1,ℤ/8)	1	8	yes	—
170	C₂³	C₂²×C₅×D₄	1	3	yes	—
171	D₄	—	2	16	no	—
172	D₄	—	4	64	no	—
173	D₄	—	4	24	no	—
174	C₂×C₄	—	2	24	no	—
175	C₂×C₄	—	2	8	no	—
176	C₂×C₄	—	4	24	no	—
177	C₂×C₄	C₅×#124	1	12	yes	—
178	C₂×C₄	C₂²×C₅×D₄	1	8	yes	—
179	C₂×C₄	C₂²×C₅×D₄	1	3	yes	—
180	C₂×C₄	C₂×C₅×#41	1	12	yes	—
181	C₂×C₄	C₂⁵×C₅	1	1	yes	characteristic
182	C₈	—	2	8	no	—
183	C₅	C₂²×#490	1	1	yes	characteristic, direct product factor, Sylow
184	C₂²	—	4	8	no	—
185	C₂²	—	4	16	no	—
186	C₂²	—	8	48	no	—
187	C₂²	C₅×#490	1	16	yes	direct product factor
188	C₂²	C₂×C₅×#124	1	6	yes	—
189	C₂²	C₂×C₅×AGL(1,ℤ/8)	1	12	yes	—
190	C₂²	C₂³×C₅×D₄	1	1	yes	characteristic
191	C₄	—	2	8	no	—
192	C₄	—	4	8	no	—
193	C₄	C₂×C₅×#124	1	6	yes	—
194	C₄	C₂²×C₅×#41	1	2	yes	—
195	C₂	—	8	8	no	—
196	C₂	C₂×C₅×#490	1	12	yes	direct product factor
197	C₂	C₂²×C₅×#124	1	1	yes	characteristic
198	C₂	C₂²×C₅×AGL(1,ℤ/8)	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅
prime radical	C₂²×C₅×#127	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₅×#125	C₂	semidirect
3. C₂×C₅×#490	C₂	direct (N × H)
4. C₂²×C₅×#102	C₂	semidirect
5. C₂²×C₅×#97	C₂	semidirect
12. C₂×C₅×#125	C₂²	semidirect
16. C₅×#490	C₂²	direct (N × H)
17. C₂×C₅×#102	C₂²	semidirect
18. C₂×C₅×#97	C₂²	semidirect
19. C₂²×#490	C₅	direct (N × H)
34. C₅×#125	C₂³	semidirect
40. C₅×#102	C₂³	semidirect
41. C₅×#97	C₂³	semidirect
43. C₂²×#125	C₂×C₅	semidirect
44. C₂×#490	C₂×C₅	direct (N × H)
45. C₂²×#102	C₂×C₅	semidirect
46. C₂²×#97	C₂×C₅	semidirect
77. C₂×#125	C₂²×C₅	semidirect
81. #490	C₂²×C₅	direct (N × H)
82. C₂×#102	C₂²×C₅	semidirect
83. C₂×#97	C₂²×C₅	semidirect
117. #125	C₂³×C₅	semidirect
123. #102	C₂³×C₅	semidirect
124. #97	C₂³×C₅	semidirect
128. C₂²×C₅	#490	direct (N × H)
161. C₂×C₅	C₂×#490	direct (N × H)
183. C₅	C₂²×#490	direct (N × H)
187. C₂²	C₅×#490	direct (N × H)
196. C₂	C₂×C₅×#490	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×#127	2
C₂²×C₅×#125	2
C₂×C₅×#490	2
C₂²×C₅×#102	2
C₂²×C₅×#97	2
C₂²×#490	5

Subgroups — C22×C5×#490 #11983327

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#490 #11983327