Subgroups — C₂²×#87904 #13107883

Order 1600.

3560 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅²×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#87904	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄⋊(C₅⋊D₅)	C₂	1	3	yes	maximal, complemented
4	C₂²×C₄×C₅⋊D₅	C₂	1	3	yes	maximal, complemented
5	C₂³×C₅⋊D₅	C₂²	1	3	yes	—
6	C₂×C₅²×Q₈	C₂²	1	12	yes	complemented
7	C₂²×C₄×C₅²	C₂²	1	3	yes	—
8	C₂²×#1039	C₂²	1	1	yes	characteristic
9	#87904	C₂²	1	64	yes	direct product factor
10	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	36	yes	complemented
11	C₂×C₄×C₅⋊D₅	C₂²	1	36	yes	complemented
12	C₂²×#713	—	5	6	no	maximal
13	C₂³×C₅²	C₂³	1	1	yes	characteristic
14	C₂²×C₅⋊D₅	—	2	12	no	—
15	C₂²×C₅⋊D₅	C₂³	1	18	yes	—
16	C₅²×Q₈	C₂³	1	16	yes	complemented
17	C₂×C₄×C₅²	C₂³	1	18	yes	—
18	C₂×#1039	C₂³	1	6	yes	—
19	C₄⋊(C₅⋊D₅)	C₂³	1	48	yes	complemented
20	C₄×C₅⋊D₅	C₂³	1	48	yes	complemented
21	C₂²×C₅×Q₈	D₅	1	6	yes	complemented
22	C₂×#713	—	5	144	no	—
23	C₂²×D₂₀	—	5	18	no	—
24	C₂²×C₄×D₅	—	5	18	no	—
25	C₂²×C₅²	#41	1	4	yes	complemented
26	C₂²×C₅²	C₂⁴	1	3	yes	—
27	C₂×C₅⋊D₅	—	2	36	no	—
28	C₂×C₅⋊D₅	C₂⁴	1	12	yes	—
29	C₄×C₅²	C₂⁴	1	12	yes	—
30	#1039	C₂⁴	1	4	yes	—
31	C₂³×D₅	—	5	18	no	—
32	C₂×C₅×Q₈	C₂×D₅	1	72	yes	complemented
33	C₂²×C₄×C₅	C₂×D₅	1	18	yes	—
34	C₂²×Dic₅	—	5	6	no	—
35	#713	—	5	384	no	—
36	C₂×D₂₀	—	5	216	no	—
37	C₂×C₄×D₅	—	5	216	no	—
38	C₂²×#41	—	25	1	no	Sylow
39	C₂×C₅²	C₂×#41	1	6	yes	complemented
40	C₂×C₅²	C₂⁵	1	1	yes	characteristic
41	C₅⋊D₅	—	2	12	no	—
42	C₂³×C₅	C₂²×D₅	1	6	yes	—
43	C₂²×D₅	—	5	108	no	—
44	C₂²×D₅	—	10	72	no	—
45	C₅×Q₈	C₂²×D₅	1	96	yes	complemented
46	C₂×C₄×C₅	C₂²×D₅	1	108	yes	—
47	C₂×Dic₅	—	5	36	no	—
48	D₂₀	—	5	288	no	—
49	C₄×D₅	—	5	288	no	—
50	C₂×#41	—	25	24	no	—
51	C₂²×Q₈	C₅⋊D₅	1	1	yes	characteristic, complemented
52	C₂²×D₄	—	25	3	no	—
53	C₂³×C₄	—	25	3	no	—
54	C₅²	C₂²×#41	1	1	yes	characteristic, complemented, Sylow
55	C₂²×C₅	#713	1	24	yes	complemented
56	C₂²×C₅	C₂³×D₅	1	18	yes	—
57	C₂×D₅	—	5	72	no	—
58	C₂×D₅	—	10	216	no	—
59	C₄×C₅	C₂³×D₅	1	72	yes	—
60	Dic₅	—	5	24	no	—
61	C₂⁴	—	25	3	no	—
62	#41	—	25	64	no	—
63	C₂×Q₈	C₂×C₅⋊D₅	1	12	yes	complemented
64	C₂×D₄	—	25	36	no	—
65	C₂²×C₄	—	25	1	no	—
66	C₂²×C₄	—	25	36	no	—
67	C₂²×C₄	C₂×C₅⋊D₅	1	3	yes	—
68	C₂×C₅	C₂×#713	1	36	yes	complemented
69	C₂×C₅	C₂⁴×D₅	1	6	yes	—
70	D₅	—	10	72	no	—
71	C₂³	—	25	18	no	—
72	C₂³	—	50	12	no	—
73	C₂³	C₂²×C₅⋊D₅	1	1	yes	characteristic
74	Q₈	C₂²×C₅⋊D₅	1	16	yes	complemented
75	D₄	—	25	48	no	—
76	C₂×C₄	—	25	6	no	—
77	C₂×C₄	—	25	48	no	—
78	C₂×C₄	C₂²×C₅⋊D₅	1	18	yes	—
79	C₅	C₂²×#713	1	6	yes	complemented
80	C₂²	—	25	12	no	—
81	C₂²	—	50	36	no	—
82	C₂²	#87904	1	4	yes	direct product factor
83	C₂²	C₂³×C₅⋊D₅	1	3	yes	—
84	C₄	—	25	4	no	—
85	C₄	C₂³×C₅⋊D₅	1	12	yes	—
86	C₂	—	50	12	no	—
87	C₂	C₂×#87904	1	6	yes	direct product factor
88	C₂	C₂⁴×C₅⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₅⋊D₅
commutator, cosocle	C₂×C₅²	C₂⁵
Frattini	C₂	C₂⁴×C₅⋊D₅
Fitting	C₂²×C₅²×Q₈	C₂
socle	C₂³×C₅²	C₂³
hypercenter	C₂²×Q₈	C₅⋊D₅
hypocenter	C₅²	C₂²×#41
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅²×Q₈	C₂	semidirect
2. C₂×#87904	C₂	direct (N × H)
3. C₂²×C₄⋊(C₅⋊D₅)	C₂	semidirect
4. C₂²×C₄×C₅⋊D₅	C₂	semidirect
6. C₂×C₅²×Q₈	C₂²	semidirect
9. #87904	C₂²	direct (N × H)
10. C₂×C₄⋊(C₅⋊D₅)	C₂²	semidirect
11. C₂×C₄×C₅⋊D₅	C₂²	semidirect
16. C₅²×Q₈	C₂³	semidirect
19. C₄⋊(C₅⋊D₅)	C₂³	semidirect
20. C₄×C₅⋊D₅	C₂³	semidirect
21. C₂²×C₅×Q₈	D₅	semidirect
25. C₂²×C₅²	#41	semidirect
32. C₂×C₅×Q₈	C₂×D₅	semidirect
39. C₂×C₅²	C₂×#41	semidirect
45. C₅×Q₈	C₂²×D₅	semidirect
51. C₂²×Q₈	C₅⋊D₅	semidirect
54. C₅²	C₂²×#41	semidirect
55. C₂²×C₅	#713	semidirect
63. C₂×Q₈	C₂×C₅⋊D₅	semidirect
68. C₂×C₅	C₂×#713	semidirect
74. Q₈	C₂²×C₅⋊D₅	semidirect
79. C₅	C₂²×#713	semidirect
82. C₂²	#87904	direct (N × H)
87. C₂	C₂×#87904	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×C₅²×Q₈	2
C₂×#87904	2
C₂²×C₄⋊(C₅⋊D₅)	2
C₂²×C₄×C₅⋊D₅	2
C₂²×#713	5

Subgroups — C22×#87904 #13107883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#87904 #13107883