Subgroups — #13097981

Order 1600.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₀₀	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₅₀	C₂	1	2	yes	maximal
3	C₂₅×#97	C₂	1	1	yes	maximal, characteristic
4	#11716366	C₂	1	1	yes	maximal, characteristic
5	#11716365	C₂	1	2	yes	maximal
6	C₂×Dic₅₀	C₂²	1	2	yes	—
7	C₂×C₈×C₂₅	C₂²	1	1	yes	characteristic
8	C₂×C₈×C₂₅	C₂²	1	1	yes	characteristic
9	C₄²×C₂₅	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₂₅	C₂²	1	2	yes	—
11	#87718	—	2	2	no	—
12	C₄×Dic₂₅	—	2	2	no	—
13	Dic₁₀₀	C₄	1	4	yes	complemented
14	#65005	—	5	1	no	maximal
15	C₂×C₄×C₂₅	D₄	1	2	yes	—
16	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
17	C₂×Dic₂₅	—	2	2	no	—
18	C₂×Dic₂₅	—	2	2	no	—
19	Dic₅₀	—	2	2	no	—
20	Dic₅₀	C₂×C₄	1	4	yes	—
21	C₈×C₂₅	—	2	1	no	—
22	C₈×C₂₅	C₂×C₄	1	2	yes	—
23	C₂×Dic₂₀	—	5	1	no	—
24	C₄×Dic₁₀	—	5	2	no	—
25	C₅×#97	D₅	1	1	yes	characteristic
26	#3949	—	5	1	no	—
27	#3948	—	5	2	no	—
28	C₂²×C₂₅	C₂×D₄	1	1	yes	characteristic
29	C₄×C₂₅	—	2	2	no	—
30	C₄×C₂₅	C₂²×C₄	1	1	yes	characteristic
31	C₄×C₂₅	#41	1	1	yes	characteristic
32	Dic₂₅	—	2	4	no	—
33	Dic₂₅	—	4	2	no	—
34	C₂×Dic₁₀	—	5	2	no	—
35	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic
36	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic
37	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
38	C₄⋊Dic₅	—	5	2	no	—
39	#683	—	10	2	no	—
40	C₄×Dic₅	—	10	2	no	—
41	Dic₂₀	—	5	4	no	—
42	#441	—	25	1	no	Sylow
43	C₂×C₂₅	C₄×D₄	1	1	yes	characteristic
44	C₂×C₂₅	#137	1	2	yes	—
45	C₂×C₄×C₅	D₂₀	1	2	yes	—
46	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
47	C₂×Dic₅	—	10	2	no	—
48	C₂×Dic₅	—	10	2	no	—
49	Dic₁₀	—	5	4	no	—
50	Dic₁₀	—	10	2	no	—
51	C₅×C₈	—	2	1	no	—
52	C₅×C₈	C₄×D₅	1	2	yes	—
53	C₂×Q₁₆	—	25	1	no	—
54	C₄×Q₈	—	25	2	no	—
55	#106	—	25	1	no	—
56	#103	—	25	2	no	—
57	#97	D₂₅	1	1	yes	characteristic
58	C₂₅	#441	1	1	yes	characteristic, complemented, Sylow
59	C₂²×C₅	C₂×D₂₀	1	1	yes	characteristic
60	C₄×C₅	—	2	2	no	—
61	C₄×C₅	C₂×C₄×D₅	1	1	yes	characteristic
62	C₄×C₅	#709	1	1	yes	characteristic
63	Dic₅	—	10	4	no	—
64	Dic₅	—	20	2	no	—
65	C₂×Q₈	—	25	2	no	—
66	Q₁₆	—	25	4	no	—
67	C₂×C₈	C₂×D₂₅	1	1	yes	characteristic
68	C₂×C₈	C₂×D₂₅	1	1	yes	characteristic
69	C₄⋊C₄	—	25	2	no	—
70	C₄⋊C₄	—	50	2	no	—
71	C₄²	—	50	2	no	—
72	C₄²	C₂×D₂₅	1	1	yes	characteristic
73	C₂×C₅	C₄×D₂₀	1	1	yes	characteristic
74	C₂×C₅	#4055	1	2	yes	—
75	Q₈	—	25	4	no	—
76	Q₈	—	50	2	no	—
77	C₂×C₄	—	50	2	no	—
78	C₂×C₄	—	50	2	no	—
79	C₂×C₄	D₁₀₀	1	2	yes	—
80	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
81	C₈	—	2	1	no	—
82	C₈	C₄×D₂₅	1	2	yes	—
83	C₅	#65005	1	1	yes	characteristic
84	C₂²	C₂×D₁₀₀	1	1	yes	characteristic
85	C₄	—	2	2	no	—
86	C₄	—	50	4	no	—
87	C₄	—	100	2	no	—
88	C₄	C₂×C₄×D₂₅	1	1	yes	characteristic
89	C₄	#87744	1	1	yes	characteristic
90	C₂	C₄×D₁₀₀	1	1	yes	characteristic
91	C₂	#11716471	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₀₀
commutator	C₄×C₂₅	C₂²×C₄
Frattini	C₂×C₄×C₅	C₂²×D₅
Fitting	C₂₅×#97	C₂
socle, prime radical	C₂²×C₅	C₂×D₂₀
cosocle	C₂×C₄×C₂₅	C₂³
hypercenter	#97	D₂₅
hypocenter	C₂₅	#441
LCS	G ▷ C₄×C₂₅ ▷ C₂×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
13. Dic₁₀₀	C₄	semidirect
58. C₂₅	#441	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₀₀	2
C₄×Dic₅₀	2
C₂₅×#97	2
#11716366	2
#11716365	2
#65005	5