Subgroups — #13102968

Order 1600.

456 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅²×C₈	C₂	1	1	yes	maximal, characteristic
2	#11717246	C₂	1	2	yes	maximal
3	#11716685	C₂	1	2	yes	maximal, complemented
4	#11716684	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
6	C₂×C₅²×C₈	C₄	1	4	yes	complemented
7	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
9	C₂²⋊#1039	—	2	2	no	—
10	C₄⋊#1039	C₂²	1	4	yes	—
11	C₄×#1039	—	2	2	no	—
12	#65395	—	5	6	no	maximal
13	C₂³×C₅²	D₄	1	1	yes	characteristic
14	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
15	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
16	C₂×C₄×C₅²	Q₈	1	2	yes	—
17	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
18	C₂×#1039	—	2	4	no	—
19	C₅²×C₈	C₂×C₄	1	2	yes	—
20	C₅²×C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₅×C₈	D₅	1	6	yes	—
22	#4072	—	5	12	no	—
23	#3950	—	5	12	no	—
24	#3949	—	5	12	no	—
25	C₂²×C₅²	—	2	2	no	—
26	C₂²×C₅²	C₄⋊C₄	1	2	yes	—
27	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
28	C₄×C₅²	C₄⋊C₄	1	2	yes	—
29	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₅²	C₂×Q₈	1	1	yes	characteristic
31	#1039	—	4	4	no	—
32	C₂²×C₄×C₅	C₂×D₅	1	6	yes	—
33	C₂×C₅×C₈	Dic₅	1	24	yes	complemented
34	C₂×C₅×C₈	C₂×D₅	1	6	yes	—
35	C₂×C₅×C₈	C₂×D₅	1	6	yes	—
36	C₂²⋊Dic₅	—	10	12	no	—
37	C₄⋊Dic₅	—	5	24	no	—
38	C₄×Dic₅	—	10	12	no	—
39	#427	—	25	1	no	Sylow
40	C₂×C₅²	—	2	2	no	—
41	C₂×C₅²	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₅²	#135	1	2	yes	—
43	C₂³×C₅	D₂₀	1	6	yes	—
44	C₂×C₄×C₅	Dic₁₀	1	12	yes	—
45	C₂×C₄×C₅	D₂₀	1	6	yes	—
46	C₂×C₄×C₅	C₂×Dic₅	1	12	yes	—
47	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
48	C₂×Dic₅	—	10	24	no	—
49	C₅×C₈	C₂×Dic₅	1	12	yes	—
50	C₅×C₈	C₂×Dic₅	1	12	yes	—
51	C₂²×C₈	C₅⋊D₅	1	1	yes	characteristic
52	#117	—	25	2	no	—
53	#107	—	25	2	no	—
54	#106	—	25	2	no	—
55	C₅²	#427	1	1	yes	characteristic, complemented, Sylow
56	C₂²×C₅	—	2	12	no	—
57	C₂²×C₅	C₄⋊Dic₅	1	12	yes	—
58	C₂²×C₅	C₂×D₂₀	1	6	yes	—
59	C₄×C₅	C₄⋊Dic₅	1	12	yes	—
60	C₄×C₅	C₂×Dic₁₀	1	6	yes	—
61	C₄×C₅	C₂²×Dic₅	1	6	yes	—
62	Dic₅	—	20	24	no	—
63	C₂²×C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
64	C₂×C₈	#1039	1	4	yes	complemented
65	C₂×C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
66	C₂×C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
67	C₄⋊C₄	—	25	4	no	—
68	C₂²⋊C₄	—	50	2	no	—
69	C₄²	—	50	2	no	—
70	C₂×C₅	—	2	12	no	—
71	C₂×C₅	#4050	1	12	yes	—
72	C₂×C₅	C₂×C₄⋊Dic₅	1	6	yes	—
73	C₂³	C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
74	C₂×C₄	—	50	4	no	—
75	C₂×C₄	#6045	1	2	yes	—
76	C₂×C₄	C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
77	C₂×C₄	C₂×#1039	1	2	yes	—
78	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
79	C₈	C₂×#1039	1	2	yes	—
80	C₈	C₂×#1039	1	2	yes	—
81	C₅	#65395	1	6	yes	complemented
82	C₂²	—	2	2	no	—
83	C₂²	C₄⋊#1039	1	2	yes	—
84	C₂²	C₂×C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
85	C₄	—	100	4	no	—
86	C₄	C₄⋊#1039	1	2	yes	—
87	C₄	C₂×#6045	1	1	yes	characteristic
88	C₄	C₂²×#1039	1	1	yes	characteristic
89	C₂	—	2	2	no	—
90	C₂	#11717224	1	2	yes	—
91	C₂	C₂×C₄⋊#1039	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄⋊(C₅⋊D₅)
commutator	C₄×C₅²	C₂²×C₄
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₂²×C₅²×C₈	C₂
socle	C₂²×C₅²	C₂×D₄
prime radical	C₂³×C₅²	D₄
cosocle	C₂×C₄×C₅²	C₂³
hypercenter	C₂²×C₈	C₅⋊D₅
hypocenter	C₅²	#427
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
3. #11716685	C₂	semidirect
4. #11716684	C₂	semidirect
6. C₂×C₅²×C₈	C₄	semidirect
33. C₂×C₅×C₈	Dic₅	semidirect
55. C₅²	#427	semidirect
64. C₂×C₈	#1039	semidirect
81. C₅	#65395	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅²×C₈	2
#11717246	2
#11716685	2
#11716684	2
#65395	5