Subgroups — #13098714

Order 1600.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11716469	C₂	1	1	yes	maximal, characteristic
2	#11716414	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#87709	C₂	1	1	yes	maximal, characteristic, complemented
4	#11716406	C₂	1	2	yes	maximal, complemented
5	D₂₅⋊C₁₆	C₂	1	2	yes	maximal, complemented
6	#87744	C₄	1	1	yes	characteristic
7	M₁₆×C₂₅	C₄	1	1	yes	characteristic
8	C₂×#6014	C₂²	1	1	yes	characteristic
9	#87712	C₄	1	2	yes	—
10	C₈×D₂₅	C₂²	1	2	yes	—
11	#87709	C₂²	1	1	yes	characteristic
12	#87709	C₂²	1	2	yes	—
13	#87709	C₂²	1	1	yes	characteristic
14	#65725	—	5	1	no	maximal
15	C₂×C₄×C₂₅	C₂×C₄	1	1	yes	characteristic
16	C₂²⋊D₂₅	C₈	1	2	yes	—
17	D₁₀₀	C₈	1	1	yes	characteristic
18	C₄×D₂₅	C₂×C₄	1	2	yes	—
19	Dic₅₀	C₈	1	1	yes	characteristic
20	C₈×C₂₅	C₂×C₄	1	2	yes	—
21	#6014	C₂×C₄	1	1	yes	characteristic
22	#6014	C₂³	1	1	yes	characteristic
23	#4053	—	5	1	no	—
24	#3998	—	5	1	no	—
25	C₂×#674	—	5	1	no	—
26	#3990	—	5	2	no	—
27	D₅⋊C₁₆	—	5	2	no	—
28	C₂²×C₂₅	C₂×C₈	1	1	yes	characteristic
29	C₂×D₂₅	C₂×C₈	1	2	yes	—
30	C₄×C₂₅	C₂×C₈	1	1	yes	characteristic
31	C₄×C₂₅	C₂²×C₄	1	1	yes	characteristic
32	Dic₂₅	C₂×C₈	1	2	yes	—
33	#709	—	5	1	no	—
34	C₅×M₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic
35	C₂×#168	—	5	1	no	—
36	#676	—	5	2	no	—
37	C₈×D₅	—	5	2	no	—
38	#674	—	5	1	no	—
39	#674	—	5	2	no	—
40	#674	—	5	1	no	—
41	#504	—	25	1	no	Sylow
42	C₂×C₂₅	—	2	1	no	—
43	C₂×C₂₅	C₂²×C₈	1	1	yes	characteristic
44	D₂₅	—	2	2	no	—
45	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
46	C₂²⋊D₅	—	5	2	no	—
47	D₂₀	—	5	1	no	—
48	C₄×D₅	—	5	2	no	—
49	Dic₁₀	—	5	1	no	—
50	C₅×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
51	#168	—	5	1	no	—
52	#168	—	5	1	no	—
53	#131	—	25	1	no	—
54	M₃₂	—	25	2	no	—
55	M₃₂	—	25	1	no	—
56	C₂×C₁₆	—	25	2	no	—
57	C₂×C₁₆	—	25	1	no	—
58	C₂₅	#504	1	1	yes	characteristic, complemented, Sylow
59	C₂²×C₅	D₅⋊C₈	1	1	yes	characteristic
60	C₂×D₅	—	5	2	no	—
61	C₄×C₅	D₅⋊C₈	1	1	yes	characteristic
62	C₄×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
63	Dic₅	—	5	2	no	—
64	#41	—	25	1	no	—
65	M₁₆	—	25	2	no	—
66	M₁₆	#1035	1	1	yes	characteristic
67	C₂×C₈	—	25	1	no	—
68	C₂×C₈	—	25	2	no	—
69	C₁₆	—	25	1	no	—
70	C₁₆	—	25	2	no	—
71	C₁₆	—	25	1	no	—
72	C₂×C₅	—	2	1	no	—
73	C₂×C₅	C₂×D₅⋊C₈	1	1	yes	characteristic
74	D₅	—	10	2	no	—
75	Q₈	—	25	1	no	—
76	D₄	—	25	2	no	—
77	D₄	—	25	1	no	—
78	C₂×C₄	—	25	2	no	—
79	C₂×C₄	C₂×#1035	1	1	yes	characteristic
80	C₈	—	25	1	no	—
81	C₈	—	25	1	no	—
82	C₈	C₂×#1035	1	2	yes	—
83	C₅	#65725	1	1	yes	characteristic
84	C₂²	—	25	2	no	—
85	C₂²	D₂₅⋊C₈	1	1	yes	characteristic
86	C₄	—	25	2	no	—
87	C₄	D₂₅⋊C₈	1	1	yes	characteristic
88	C₄	C₂²×#1035	1	1	yes	characteristic
89	C₂	—	2	1	no	—
90	C₂	—	50	2	no	—
91	C₂	C₂×D₂₅⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×#1035
commutator	C₂×C₂₅	C₂²×C₈
Frattini	C₄×C₅	C₂²×AGL(1,𝔽₅)
Fitting	M₁₆×C₂₅	C₄
socle	C₂×C₅	C₂×D₅⋊C₈
prime radical	#87744	C₄
cosocle	#6014	C₂³
hypercenter	M₁₆	#1035
hypocenter	C₂₅	#504
LCS	G ▷ C₂×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
2. #11716414	C₂	semidirect
3. C₂×#87709	C₂	semidirect
4. #11716406	C₂	semidirect
5. D₂₅⋊C₁₆	C₂	semidirect
58. C₂₅	#504	semidirect

Maximal subgroups

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

subgroup	index
#11716469	2
#11716414	2
C₂×#87709	2
#11716406	2
D₂₅⋊C₁₆	2
#65725	5