Subgroups — #11716513

Order 800.

93 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×C₂₅	C₂	1	1	yes	maximal, characteristic, complemented
2	#87744	C₂	1	1	yes	maximal, characteristic, complemented
3	#87724	C₂	1	1	yes	maximal, characteristic, complemented
4	#87723	C₂	1	1	yes	maximal, characteristic, complemented
5	#87722	C₂	1	1	yes	maximal, characteristic, complemented
6	#87721	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#6014	C₂	1	1	yes	maximal, characteristic, complemented
8	Q₈×C₂₅	C₂²	1	1	yes	characteristic
9	D₄×C₂₅	—	2	1	no	—
10	D₄×C₂₅	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₂₅	—	2	1	no	—
12	C₂×C₄×C₂₅	C₂²	1	1	yes	characteristic
13	C₂²⋊D₂₅	—	2	1	no	—
14	D₁₀₀	C₂²	1	1	yes	characteristic
15	C₄×D₂₅	—	2	1	no	—
16	Dic₅₀	C₂²	1	1	yes	characteristic
17	#6014	C₂²	1	1	yes	characteristic
18	#6014	C₂²	1	1	yes	characteristic
19	#4096	—	5	1	no	maximal
20	C₂²×C₂₅	—	2	1	no	—
21	C₂²×C₂₅	D₄	1	1	yes	characteristic
22	C₂×D₂₅	—	2	1	no	—
23	C₄×C₂₅	—	2	1	no	—
24	C₄×C₂₅	D₄	1	1	yes	characteristic
25	C₄×C₂₅	C₂³	1	1	yes	characteristic
26	Dic₂₅	—	2	1	no	—
27	C₅×#41	D₅	1	1	yes	characteristic
28	#709	—	5	1	no	—
29	#689	—	5	1	no	—
30	#688	—	5	1	no	—
31	#687	—	5	1	no	—
32	#686	—	5	1	no	—
33	C₂×#168	—	5	1	no	—
34	C₂×C₂₅	—	2	1	no	—
35	C₂×C₂₅	—	4	1	no	—
36	C₂×C₂₅	C₂×D₄	1	1	yes	characteristic
37	D₂₅	—	4	1	no	—
38	C₅×Q₈	C₂×D₅	1	1	yes	characteristic
39	C₅×D₄	—	2	1	no	—
40	C₅×D₄	C₂×D₅	1	1	yes	characteristic
41	C₂×C₄×C₅	—	2	1	no	—
42	C₂×C₄×C₅	C₂×D₅	1	1	yes	characteristic
43	C₂²⋊D₅	—	10	1	no	—
44	D₂₀	—	5	1	no	—
45	C₄×D₅	—	10	1	no	—
46	Dic₁₀	—	5	1	no	—
47	#168	—	5	1	no	—
48	#168	—	5	1	no	—
49	#135	—	25	1	no	Sylow
50	C₂₅	#135	1	1	yes	characteristic, complemented, Sylow
51	C₂²×C₅	—	2	1	no	—
52	C₂²×C₅	C₂²⋊D₅	1	1	yes	characteristic
53	C₂×D₅	—	10	1	no	—
54	C₄×C₅	—	2	1	no	—
55	C₄×C₅	C₂²⋊D₅	1	1	yes	characteristic
56	C₄×C₅	C₂²×D₅	1	1	yes	characteristic
57	Dic₅	—	10	1	no	—
58	#41	—	25	1	no	—
59	#41	D₂₅	1	1	yes	characteristic, complemented
60	Q₁₆	—	25	1	no	—
61	SD₁₆	—	25	1	no	—
62	SD₁₆	—	25	1	no	—
63	D₈	—	25	1	no	—
64	C₂×C₈	—	25	1	no	—
65	C₂×C₅	—	2	1	no	—
66	C₂×C₅	—	4	1	no	—
67	C₂×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
68	D₅	—	20	1	no	—
69	Q₈	—	25	1	no	—
70	Q₈	C₂×D₂₅	1	1	yes	characteristic
71	D₄	—	2	1	no	—
72	D₄	—	25	1	no	—
73	D₄	—	50	1	no	—
74	D₄	C₂×D₂₅	1	1	yes	characteristic
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	50	1	no	—
77	C₂×C₄	C₂×D₂₅	1	1	yes	characteristic
78	C₈	—	25	1	no	—
79	C₈	—	25	1	no	—
80	C₅	#4096	1	1	yes	characteristic
81	C₂²	—	2	1	no	—
82	C₂²	—	50	1	no	—
83	C₂²	C₂²⋊D₂₅	1	1	yes	characteristic
84	C₄	—	2	1	no	—
85	C₄	—	50	1	no	—
86	C₄	C₂²⋊D₂₅	1	1	yes	characteristic
87	C₄	C₂²×D₂₅	1	1	yes	characteristic
88	C₂	—	2	1	no	—
89	C₂	—	4	1	no	—
90	C₂	—	100	1	no	—
91	C₂	C₂×C₂²⋊D₂₅	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊D₂₅
commutator, cosocle	C₄×C₂₅	C₂³
Frattini	C₄×C₅	C₂²×D₅
Fitting	#41×C₂₅	C₂
socle	C₂×C₅	C₂×C₂²⋊D₅
hypercenter	#41	D₂₅
hypocenter	C₂₅	#135
LCS	G ▷ C₄×C₂₅ ▷ C₂×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
1. #41×C₂₅	C₂	semidirect
2. #87744	C₂	semidirect
3. #87724	C₂	semidirect
4. #87723	C₂	semidirect
5. #87722	C₂	semidirect
6. #87721	C₂	semidirect
7. C₂×#6014	C₂	semidirect
50. C₂₅	#135	semidirect
59. #41	D₂₅	semidirect

Maximal subgroups

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

subgroup	index
#41×C₂₅	2
#87744	2
#87724	2
#87723	2
#87722	2
#87721	2
C₂×#6014	2
#4096	5