Subgroups — #13098381

Order 1600.

171 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×C₂₅	C₂	1	1	yes	maximal, characteristic
2	#11716488	C₂	1	2	yes	maximal
3	#11716367	C₂	1	2	yes	maximal, complemented
4	#11716366	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×C₂₅	C₂²	1	1	yes	characteristic
6	C₂×C₈×C₂₅	C₄	1	4	yes	complemented
7	C₂×C₈×C₂₅	C₂²	1	1	yes	characteristic
8	C₂×C₈×C₂₅	C₂²	1	1	yes	characteristic
9	C₂²⋊Dic₂₅	—	2	2	no	—
10	C₄⋊Dic₂₅	C₂²	1	4	yes	—
11	C₄×Dic₂₅	—	2	2	no	—
12	#65395	—	5	1	no	maximal
13	C₂³×C₂₅	D₄	1	1	yes	characteristic
14	C₂×C₄×C₂₅	C₂×C₄	1	2	yes	—
15	C₂×C₄×C₂₅	D₄	1	1	yes	characteristic
16	C₂×C₄×C₂₅	Q₈	1	2	yes	—
17	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
18	C₂×Dic₂₅	—	2	4	no	—
19	C₈×C₂₅	C₂×C₄	1	2	yes	—
20	C₈×C₂₅	C₂×C₄	1	2	yes	—
21	C₂²×C₅×C₈	D₅	1	1	yes	characteristic
22	#4072	—	5	2	no	—
23	#3950	—	5	2	no	—
24	#3949	—	5	2	no	—
25	C₂²×C₂₅	—	2	2	no	—
26	C₂²×C₂₅	C₄⋊C₄	1	2	yes	—
27	C₂²×C₂₅	C₂×D₄	1	1	yes	characteristic
28	C₄×C₂₅	C₄⋊C₄	1	2	yes	—
29	C₄×C₂₅	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₂₅	C₂×Q₈	1	1	yes	characteristic
31	Dic₂₅	—	4	4	no	—
32	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
33	C₂×C₅×C₈	Dic₅	1	4	yes	—
34	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic
35	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic
36	C₂²⋊Dic₅	—	10	2	no	—
37	C₄⋊Dic₅	—	5	4	no	—
38	C₄×Dic₅	—	10	2	no	—
39	#427	—	25	1	no	Sylow
40	C₂×C₂₅	—	2	2	no	—
41	C₂×C₂₅	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₂₅	#135	1	2	yes	—
43	C₂³×C₅	D₂₀	1	1	yes	characteristic
44	C₂×C₄×C₅	Dic₁₀	1	2	yes	—
45	C₂×C₄×C₅	D₂₀	1	1	yes	characteristic
46	C₂×C₄×C₅	C₂×Dic₅	1	2	yes	—
47	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
48	C₂×Dic₅	—	10	4	no	—
49	C₅×C₈	C₂×Dic₅	1	2	yes	—
50	C₅×C₈	C₂×Dic₅	1	2	yes	—
51	C₂²×C₈	D₂₅	1	1	yes	characteristic
52	#117	—	25	2	no	—
53	#107	—	25	2	no	—
54	#106	—	25	2	no	—
55	C₂₅	#427	1	1	yes	characteristic, complemented, Sylow
56	C₂²×C₅	—	2	2	no	—
57	C₂²×C₅	C₄⋊Dic₅	1	2	yes	—
58	C₂²×C₅	C₂×D₂₀	1	1	yes	characteristic
59	C₄×C₅	C₄⋊Dic₅	1	2	yes	—
60	C₄×C₅	C₂×Dic₁₀	1	1	yes	characteristic
61	C₄×C₅	C₂²×Dic₅	1	1	yes	characteristic
62	Dic₅	—	20	4	no	—
63	C₂²×C₄	C₂×D₂₅	1	1	yes	characteristic
64	C₂×C₈	Dic₂₅	1	4	yes	complemented
65	C₂×C₈	C₂×D₂₅	1	1	yes	characteristic
66	C₂×C₈	C₂×D₂₅	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	2	no	—
71	C₂×C₅	#4050	1	2	yes	—
72	C₂×C₅	C₂×C₄⋊Dic₅	1	1	yes	characteristic
73	C₂³	D₁₀₀	1	1	yes	characteristic
74	C₂×C₄	—	50	4	no	—
75	C₂×C₄	Dic₅₀	1	2	yes	—
76	C₂×C₄	D₁₀₀	1	1	yes	characteristic
77	C₂×C₄	C₂×Dic₂₅	1	2	yes	—
78	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
79	C₈	C₂×Dic₂₅	1	2	yes	—
80	C₈	C₂×Dic₂₅	1	2	yes	—
81	C₅	#65395	1	1	yes	characteristic
82	C₂²	—	2	2	no	—
83	C₂²	C₄⋊Dic₂₅	1	2	yes	—
84	C₂²	C₂×D₁₀₀	1	1	yes	characteristic
85	C₄	—	100	4	no	—
86	C₄	C₄⋊Dic₂₅	1	2	yes	—
87	C₄	C₂×Dic₅₀	1	1	yes	characteristic
88	C₄	C₂²×Dic₂₅	1	1	yes	characteristic
89	C₂	—	2	2	no	—
90	C₂	#11716467	1	2	yes	—
91	C₂	C₂×C₄⋊Dic₂₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₁₀₀
commutator	C₄×C₂₅	C₂²×C₄
Frattini	C₂×C₄×C₅	C₂²×D₅
Fitting	C₂²×C₈×C₂₅	C₂
socle	C₂²×C₅	C₂×D₂₀
prime radical	C₂³×C₅	D₂₀
cosocle	C₂×C₄×C₂₅	C₂³
hypercenter	C₂²×C₈	D₂₅
hypocenter	C₂₅	#427
LCS	G ▷ C₄×C₂₅ ▷ C₂×C₂₅ ▷ C₂₅
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. #11716367	C₂	semidirect
4. #11716366	C₂	semidirect
6. C₂×C₈×C₂₅	C₄	semidirect
55. C₂₅	#427	semidirect
64. C₂×C₈	Dic₂₅	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₈×C₂₅	2
#11716488	2
#11716367	2
#11716366	2
#65395	5