Subgroups — #13099674

Order 1600.

399 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₂₅	C₂	1	1	yes	maximal, characteristic
2	Q₈×#1035	C₂	1	8	yes	maximal, complemented
3	#11716549	C₂	1	6	yes	maximal
4	C₂×Q₈×C₂₅	C₄	1	1	yes	characteristic, complemented
5	Q₈×D₂₅	C₄	1	4	yes	complemented
6	Q₈×D₂₅	C₂²	1	4	yes	—
7	C₂×C₄×D₂₅	C₂²	1	3	yes	—
8	C₂×Dic₅₀	C₄	1	3	yes	complemented
9	C₂²⋊#1035	C₂²	1	4	yes	—
10	C₄⋊#1035	C₂²	1	12	yes	—
11	C₄×#1035	C₂²	1	12	yes	—
12	#66262	—	5	1	no	maximal
13	C₂²×D₂₅	C₂³	1	1	yes	characteristic
14	C₂×#1035	C₂³	1	8	yes	—
15	Q₈×C₂₅	C₂×C₄	1	4	yes	—
16	C₂×C₄×C₂₅	C₂×C₄	1	3	yes	—
17	C₂×Dic₂₅	C₂×C₄	1	3	yes	—
18	C₄×D₂₅	C₂×C₄	1	6	yes	—
19	C₄×D₂₅	C₂³	1	6	yes	—
20	Dic₅₀	C₂×C₄	1	12	yes	—
21	C₂×Q₈×D₅	—	5	1	no	—
22	Q₈×AGL(1,𝔽₅)	—	5	8	no	—
23	#4130	—	5	6	no	—
24	C₂²×C₂₅	C₂²×C₄	1	1	yes	characteristic
25	C₂×D₂₅	—	2	2	no	—
26	C₂×D₂₅	C₂²×C₄	1	1	yes	characteristic
27	C₂×D₂₅	C₂⁴	1	1	yes	characteristic
28	#1035	—	2	8	no	—
29	C₄×C₂₅	C₂²×C₄	1	6	yes	—
30	Dic₂₅	C₂²×C₄	1	6	yes	—
31	C₂×C₅×Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic
32	Q₈×D₅	—	5	4	no	—
33	Q₈×D₅	—	5	4	no	—
34	C₂×C₄×D₅	—	5	3	no	—
35	C₂×Dic₁₀	—	5	3	no	—
36	C₂²⋊AGL(1,𝔽₅)	—	5	4	no	—
37	C₄⋊AGL(1,𝔽₅)	—	5	12	no	—
38	C₄×AGL(1,𝔽₅)	—	5	12	no	—
39	#519	—	25	1	no	Sylow
40	C₂×C₂₅	—	2	1	no	—
41	C₂×C₂₅	C₂³×C₄	1	1	yes	characteristic
42	D₂₅	—	2	1	no	—
43	D₂₅	2₋¹⁺⁴	1	2	yes	—
44	C₂²×D₅	—	5	1	no	—
45	C₂×AGL(1,𝔽₅)	—	5	8	no	—
46	C₅×Q₈	C₂×AGL(1,𝔽₅)	1	4	yes	—
47	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	3	yes	—
48	C₂×Dic₅	—	5	3	no	—
49	C₄×D₅	—	5	6	no	—
50	C₄×D₅	—	5	6	no	—
51	Dic₁₀	—	5	12	no	—
52	C₂²×Q₈	—	25	1	no	—
53	C₄×Q₈	—	25	8	no	—
54	#117	—	25	6	no	—
55	C₂₅	#519	1	1	yes	characteristic, complemented, Sylow
56	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
57	C₂×D₅	—	5	1	no	—
58	C₂×D₅	—	5	1	no	—
59	C₂×D₅	—	10	2	no	—
60	AGL(1,𝔽₅)	—	10	8	no	—
61	C₄×C₅	C₂²×AGL(1,𝔽₅)	1	6	yes	—
62	Dic₅	—	5	6	no	—
63	C₂×Q₈	—	25	4	no	—
64	C₂×Q₈	—	25	4	no	—
65	C₂×Q₈	—	25	3	no	—
66	C₂×Q₈	#1035	1	1	yes	characteristic, complemented
67	C₂²×C₄	—	25	3	no	—
68	C₄⋊C₄	—	25	12	no	—
69	C₂²⋊C₄	—	25	4	no	—
70	C₄²	—	25	12	no	—
71	C₂×C₅	—	2	1	no	—
72	C₂×C₅	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
73	D₅	—	5	2	no	—
74	D₅	—	10	1	no	—
75	C₂³	—	25	1	no	—
76	Q₈	—	25	12	no	—
77	Q₈	C₂×#1035	1	4	yes	—
78	C₂×C₄	—	25	3	no	—
79	C₂×C₄	—	25	6	no	—
80	C₂×C₄	—	25	6	no	—
81	C₂×C₄	—	25	8	no	—
82	C₂×C₄	C₂×#1035	1	3	yes	—
83	C₅	#66262	1	1	yes	characteristic
84	C₂²	—	25	1	no	—
85	C₂²	—	25	1	no	—
86	C₂²	—	50	2	no	—
87	C₂²	C₂²×#1035	1	1	yes	characteristic
88	C₄	—	25	6	no	—
89	C₄	—	50	8	no	—
90	C₄	C₂²×#1035	1	6	yes	—
91	C₂	—	2	1	no	—
92	C₂	—	25	2	no	—
93	C₂	—	50	1	no	—
94	C₂	C₂³×#1035	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂³×#1035
commutator	C₂×C₂₅	C₂³×C₄
Frattini, socle	C₂×C₅	C₂³×AGL(1,𝔽₅)
Fitting	C₂×Q₈×C₂₅	C₄
prime radical	C₂²×D₂₅	C₂³
cosocle	C₂×D₂₅	C₂⁴
hypercenter	C₂×Q₈	#1035
hypocenter	C₂₅	#519
LCS	G ▷ C₂×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
2. Q₈×#1035	C₂	semidirect
4. C₂×Q₈×C₂₅	C₄	semidirect
5. Q₈×D₂₅	C₄	semidirect
8. C₂×Dic₅₀	C₄	semidirect
55. C₂₅	#519	semidirect
66. C₂×Q₈	#1035	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×D₂₅	2
Q₈×#1035	2
#11716549	2
#66262	5