Subgroups — C₂²×C₅×#519 #11985366

Order 1280.

11230 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₅×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#519	C₂	1	48	yes	maximal, direct product factor
3	C₂²×C₄×C₅×Q₈	C₂	1	8	yes	maximal, complemented
4	C₂²×C₅×#117	C₂	1	6	yes	maximal
5	C₂³×C₅×Q₈	C₄	1	32	yes	complemented
6	C₂³×C₅×Q₈	C₂²	1	4	yes	—
7	C₂³×C₅×Q₈	C₂²	1	24	yes	—
8	C₂⁴×C₄×C₅	C₂²	1	3	yes	—
9	C₅×#519	C₂²	1	256	yes	direct product factor
10	C₂×C₄×C₅×Q₈	C₂²	1	192	yes	complemented
11	C₂×C₅×#117	C₂²	1	144	yes	—
12	C₂²×C₅×C₄⋊C₄	C₂²	1	12	yes	—
13	C₂²×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
14	C₂²×C₄²×C₅	C₂²	1	12	yes	—
15	C₂²×#519	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂⁵×C₅	C₂³	1	1	yes	characteristic
17	C₂²×C₅×Q₈	C₂×C₄	1	64	yes	—
18	C₂²×C₅×Q₈	C₂×C₄	1	384	yes	complemented
19	C₂²×C₅×Q₈	C₂³	1	48	yes	—
20	C₂²×C₅×Q₈	C₂³	1	64	yes	—
21	C₂³×C₄×C₅	C₂×C₄	1	48	yes	—
22	C₂³×C₄×C₅	C₂³	1	8	yes	—
23	C₂³×C₄×C₅	C₂³	1	36	yes	—
24	C₂³×C₄×C₅	C₂³	1	6	yes	—
25	C₄×C₅×Q₈	C₂³	1	512	yes	complemented
26	C₅×#117	C₂³	1	384	yes	—
27	C₂×C₅×C₄⋊C₄	C₂³	1	144	yes	—
28	C₂×C₅×C₂²⋊C₄	C₂³	1	48	yes	—
29	C₂×C₄²×C₅	C₂³	1	144	yes	—
30	C₂⁴×Q₈	C₂×C₅	1	1	yes	characteristic
31	C₂×#519	C₂×C₅	1	48	yes	direct product factor
32	C₂²×C₄×Q₈	C₂×C₅	1	8	yes	complemented
33	C₂²×#117	C₂×C₅	1	6	yes	—
34	C₂⁴×C₅	—	2	8	no	—
35	C₂⁴×C₅	C₂²×C₄	1	8	yes	—
36	C₂⁴×C₅	C₂⁴	1	6	yes	—
37	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
38	C₂×C₅×Q₈	C₂²×C₄	1	384	yes	—
39	C₂×C₅×Q₈	C₂²×C₄	1	512	yes	complemented
40	C₂×C₅×Q₈	C₂⁴	1	64	yes	—
41	C₂²×C₄×C₅	—	2	32	no	—
42	C₂²×C₄×C₅	C₂²×C₄	1	48	yes	—
43	C₂²×C₄×C₅	C₂²×C₄	1	288	yes	—
44	C₂²×C₄×C₅	C₂⁴	1	48	yes	—
45	C₂²×C₄×C₅	C₂⁴	1	36	yes	—
46	C₂²×C₄×C₅	C₂⁴	1	48	yes	—
47	C₅×C₄⋊C₄	C₂⁴	1	192	yes	—
48	C₅×C₂²⋊C₄	C₂⁴	1	64	yes	—
49	C₄²×C₅	C₂⁴	1	192	yes	—
50	C₂³×Q₈	C₄×C₅	1	32	yes	complemented
51	C₂³×Q₈	C₂²×C₅	1	4	yes	—
52	C₂³×Q₈	C₂²×C₅	1	24	yes	—
53	C₂⁴×C₄	C₂²×C₅	1	3	yes	—
54	#519	C₂²×C₅	1	256	yes	direct product factor
55	C₂×C₄×Q₈	C₂²×C₅	1	192	yes	complemented
56	C₂×#117	C₂²×C₅	1	144	yes	—
57	C₂²×C₄⋊C₄	C₂²×C₅	1	12	yes	—
58	C₂²×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
59	C₂²×C₄²	C₂²×C₅	1	12	yes	—
60	C₂³×C₅	—	2	32	no	—
61	C₂³×C₅	—	2	24	no	—
62	C₂³×C₅	C₂³×C₄	1	24	yes	—
63	C₂³×C₅	C₂³×C₄	1	4	yes	—
64	C₂³×C₅	2₋¹⁺⁴	1	8	yes	—
65	C₂³×C₅	C₂⁵	1	4	yes	—
66	C₂³×C₅	C₂⁵	1	3	yes	—
67	C₅×Q₈	C₂³×C₄	1	256	yes	—
68	C₂×C₄×C₅	—	2	96	no	—
69	C₂×C₄×C₅	C₂³×C₄	1	144	yes	—
70	C₂×C₄×C₅	C₂³×C₄	1	192	yes	—
71	C₂×C₄×C₅	C₂⁵	1	32	yes	—
72	C₂×C₄×C₅	C₂⁵	1	24	yes	—
73	C₂⁵	C₂³×C₅	1	1	yes	characteristic
74	C₂²×Q₈	C₂×C₄×C₅	1	64	yes	—
75	C₂²×Q₈	C₂×C₄×C₅	1	384	yes	complemented
76	C₂²×Q₈	C₂³×C₅	1	48	yes	—
77	C₂²×Q₈	C₂³×C₅	1	64	yes	—
78	C₂³×C₄	C₂×C₄×C₅	1	48	yes	—
79	C₂³×C₄	C₂³×C₅	1	8	yes	—
80	C₂³×C₄	C₂³×C₅	1	6	yes	—
81	C₂³×C₄	C₂³×C₅	1	36	yes	—
82	C₄×Q₈	C₂³×C₅	1	512	yes	complemented
83	#117	C₂³×C₅	1	384	yes	—
84	C₂×C₄⋊C₄	C₂³×C₅	1	144	yes	—
85	C₂×C₂²⋊C₄	C₂³×C₅	1	48	yes	—
86	C₂×C₄²	C₂³×C₅	1	144	yes	—
87	C₂²×C₅	—	2	8	no	—
88	C₂²×C₅	—	2	48	no	—
89	C₂²×C₅	#519	1	16	yes	direct product factor
90	C₂²×C₅	C₂⁴×C₄	1	8	yes	—
91	C₂²×C₅	C₂⁴×C₄	1	6	yes	—
92	C₂²×C₅	C₂×2₋¹⁺⁴	1	12	yes	—
93	C₂²×C₅	C₂⁶	1	1	yes	characteristic
94	C₄×C₅	—	2	32	no	—
95	C₄×C₅	C₂⁴×C₄	1	48	yes	—
96	C₂⁴	—	2	8	no	—
97	C₂⁴	C₂²×C₄×C₅	1	8	yes	—
98	C₂⁴	C₂⁴×C₅	1	6	yes	—
99	C₂⁴	C₂⁴×C₅	1	1	yes	characteristic
100	C₂×Q₈	C₂²×C₄×C₅	1	384	yes	—
101	C₂×Q₈	C₂²×C₄×C₅	1	512	yes	complemented
102	C₂×Q₈	C₂⁴×C₅	1	64	yes	—
103	C₂²×C₄	—	2	32	no	—
104	C₂²×C₄	C₂²×C₄×C₅	1	48	yes	—
105	C₂²×C₄	C₂²×C₄×C₅	1	288	yes	—
106	C₂²×C₄	C₂⁴×C₅	1	48	yes	—
107	C₂²×C₄	C₂⁴×C₅	1	36	yes	—
108	C₂²×C₄	C₂⁴×C₅	1	48	yes	—
109	C₄⋊C₄	C₂⁴×C₅	1	192	yes	—
110	C₂²⋊C₄	C₂⁴×C₅	1	64	yes	—
111	C₄²	C₂⁴×C₅	1	192	yes	—
112	C₂×C₅	—	2	8	no	—
113	C₂×C₅	C₂×#519	1	12	yes	direct product factor
114	C₂×C₅	C₂⁵×C₄	1	1	yes	characteristic
115	C₂×C₅	C₂²×2₋¹⁺⁴	1	2	yes	—
116	C₂³	—	2	32	no	—
117	C₂³	—	2	24	no	—
118	C₂³	C₂³×C₄×C₅	1	24	yes	—
119	C₂³	C₂³×C₄×C₅	1	4	yes	—
120	C₂³	C₅×2₋¹⁺⁴	1	8	yes	—
121	C₂³	C₂⁵×C₅	1	4	yes	—
122	C₂³	C₂⁵×C₅	1	3	yes	—
123	Q₈	C₂³×C₄×C₅	1	256	yes	—
124	C₂×C₄	—	2	96	no	—
125	C₂×C₄	C₂³×C₄×C₅	1	144	yes	—
126	C₂×C₄	C₂³×C₄×C₅	1	192	yes	—
127	C₂×C₄	C₂⁵×C₅	1	32	yes	—
128	C₂×C₄	C₂⁵×C₅	1	24	yes	—
129	C₅	C₂²×#519	1	1	yes	characteristic, direct product factor, Sylow
130	C₂²	—	2	8	no	—
131	C₂²	—	2	48	no	—
132	C₂²	C₅×#519	1	16	yes	direct product factor
133	C₂²	C₂⁴×C₄×C₅	1	8	yes	—
134	C₂²	C₂⁴×C₄×C₅	1	6	yes	—
135	C₂²	C₂×C₅×2₋¹⁺⁴	1	12	yes	—
136	C₂²	C₂⁶×C₅	1	1	yes	characteristic
137	C₄	—	2	32	no	—
138	C₄	C₂⁴×C₄×C₅	1	48	yes	—
139	C₂	—	2	8	no	—
140	C₂	C₂×C₅×#519	1	12	yes	direct product factor
141	C₂	C₂⁵×C₄×C₅	1	1	yes	characteristic
142	C₂	C₂²×C₅×2₋¹⁺⁴	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₅	C₂⁴
commutator	C₂	C₂⁵×C₄×C₅
Frattini, cosocle	C₂²	C₂⁶×C₅
prime radical	C₂⁵×C₅	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#519	C₂	direct (N × H)
3. C₂²×C₄×C₅×Q₈	C₂	semidirect
5. C₂³×C₅×Q₈	C₄	semidirect
9. C₅×#519	C₂²	direct (N × H)
10. C₂×C₄×C₅×Q₈	C₂²	semidirect
15. C₂²×#519	C₅	direct (N × H)
18. C₂²×C₅×Q₈	C₂×C₄	semidirect
25. C₄×C₅×Q₈	C₂³	semidirect
31. C₂×#519	C₂×C₅	direct (N × H)
32. C₂²×C₄×Q₈	C₂×C₅	semidirect
39. C₂×C₅×Q₈	C₂²×C₄	semidirect
50. C₂³×Q₈	C₄×C₅	semidirect
54. #519	C₂²×C₅	direct (N × H)
55. C₂×C₄×Q₈	C₂²×C₅	semidirect
75. C₂²×Q₈	C₂×C₄×C₅	semidirect
82. C₄×Q₈	C₂³×C₅	semidirect
89. C₂²×C₅	#519	direct (N × H)
101. C₂×Q₈	C₂²×C₄×C₅	semidirect
113. C₂×C₅	C₂×#519	direct (N × H)
129. C₅	C₂²×#519	direct (N × H)
132. C₂²	C₅×#519	direct (N × H)
140. C₂	C₂×C₅×#519	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₅×Q₈	2
C₂×C₅×#519	2
C₂²×C₄×C₅×Q₈	2
C₂²×C₅×#117	2
C₂²×#519	5

Subgroups — C22×C5×#519 #11985366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#519 #11985366