Subgroups — #13102736

Order 1600.

392 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11717208	C₂	1	2	yes	maximal
2	C₅²×#107	C₂	1	1	yes	maximal, characteristic
3	#11716685	C₂	1	1	yes	maximal, characteristic
4	C₈×#1039	C₂	1	1	yes	maximal, characteristic
5	#11716674	C₂	1	2	yes	maximal
6	C₂×#6045	—	2	2	no	—
7	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
8	C₅²×C₄⋊C₄	C₂²	1	2	yes	—
9	C₄⋊#1039	C₂²	1	2	yes	—
10	#87806	—	2	2	no	—
11	C₄×#1039	C₂²	1	1	yes	characteristic
12	C₂×#6029	C₂²	1	1	yes	characteristic
13	#65163	—	5	6	no	maximal
14	C₂×C₄×C₅²	—	2	2	no	—
15	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
16	C₂×#1039	—	2	2	no	—
17	C₂×#1039	D₄	1	1	yes	characteristic
18	C₂×#1039	D₄	1	1	yes	characteristic
19	#6045	—	4	4	no	—
20	C₅²×C₈	Q₈	1	2	yes	complemented
21	#6029	Q₈	1	2	yes	complemented
22	#4034	—	5	12	no	—
23	C₅×#107	D₅	1	6	yes	—
24	#3950	—	5	6	no	—
25	C₈×Dic₅	—	5	6	no	—
26	#3939	—	5	12	no	—
27	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
28	C₄×C₅²	—	4	2	no	—
29	C₄×C₅²	C₂×Q₈	1	1	yes	characteristic
30	C₄×C₅²	C₂×Q₈	1	1	yes	characteristic
31	#1039	—	4	2	no	—
32	#1039	D₈	1	2	yes	—
33	#1039	Q₁₆	1	2	yes	—
34	C₂×Dic₁₀	—	10	12	no	—
35	C₂×C₅×C₈	C₂×D₅	1	6	yes	—
36	C₅×C₄⋊C₄	C₂×D₅	1	12	yes	—
37	C₄⋊Dic₅	—	5	12	no	—
38	#683	—	10	12	no	—
39	C₄×Dic₅	—	5	6	no	—
40	C₂×#168	—	5	6	no	—
41	#500	—	25	1	no	Sylow
42	C₂×C₅²	C₄⋊Q₈	1	1	yes	characteristic
43	C₂×C₅²	C₂×D₈	1	1	yes	characteristic
44	C₂×C₅²	C₂×Q₁₆	1	1	yes	characteristic
45	C₂×C₄×C₅	—	2	12	no	—
46	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
47	C₂×Dic₅	—	5	6	no	—
48	C₂×Dic₅	—	5	6	no	—
49	C₂×Dic₅	—	10	12	no	—
50	Dic₁₀	—	20	24	no	—
51	C₅×C₈	Dic₁₀	1	12	yes	complemented
52	#168	—	5	12	no	—
53	C₄⋊Q₈	—	25	2	no	—
54	#107	—	25	1	no	—
55	#107	—	25	2	no	—
56	#107	C₅⋊D₅	1	1	yes	characteristic
57	C₄×C₈	—	25	1	no	—
58	C₅²	#500	1	1	yes	characteristic, complemented, Sylow
59	C₂²×C₅	D₄×D₅	1	6	yes	—
60	C₄×C₅	—	4	12	no	—
61	C₄×C₅	C₂×Dic₁₀	1	6	yes	—
62	C₄×C₅	Q₈×D₅	1	6	yes	—
63	Dic₅	—	5	12	no	—
64	Dic₅	—	5	12	no	—
65	Dic₅	—	20	12	no	—
66	C₂×Q₈	—	50	2	no	—
67	C₂×C₈	—	25	1	no	—
68	C₂×C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
69	C₄⋊C₄	—	25	2	no	—
70	C₄⋊C₄	—	50	2	no	—
71	C₄⋊C₄	C₂×C₅⋊D₅	1	2	yes	—
72	C₄²	—	25	1	no	—
73	C₂×C₅	#4034	1	6	yes	—
74	C₂×C₅	D₅×D₈	1	6	yes	—
75	C₂×C₅	D₅×Q₁₆	1	6	yes	—
76	Q₈	—	100	4	no	—
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	25	1	no	—
79	C₂×C₄	—	25	1	no	—
80	C₂×C₄	—	50	2	no	—
81	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
82	C₈	—	25	2	no	—
83	C₈	#6045	1	2	yes	complemented
84	C₅	#65163	1	6	yes	complemented
85	C₂²	D₄×C₅⋊D₅	1	1	yes	characteristic
86	C₄	—	4	2	no	—
87	C₄	—	25	2	no	—
88	C₄	—	25	2	no	—
89	C₄	—	100	2	no	—
90	C₄	C₂×#6045	1	1	yes	characteristic
91	C₄	Q₈×C₅⋊D₅	1	1	yes	characteristic
92	C₂	#11717208	1	1	yes	characteristic
93	C₂	D₈×C₅⋊D₅	1	1	yes	characteristic
94	C₂	Q₁₆×C₅⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×C₅⋊D₅
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₅²×#107	C₂
socle, prime radical	C₂²×C₅²	C₂×D₄
hypercenter	#107	C₅⋊D₅
hypocenter	C₅²	#500
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
20. C₅²×C₈	Q₈	semidirect
21. #6029	Q₈	semidirect
51. C₅×C₈	Dic₁₀	semidirect
58. C₅²	#500	semidirect
83. C₈	#6045	semidirect
84. C₅	#65163	semidirect

Maximal subgroups

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

subgroup	index
#11717208	2
C₅²×#107	2
#11716685	2
C₈×#1039	2
#11716674	2
#65163	5