Subgroups — #13102717

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×#106	C₂	1	1	yes	maximal, characteristic
3	#11716684	C₂	1	1	yes	maximal, characteristic
4	C₈×#1039	C₂	1	1	yes	maximal, characteristic
5	#11716675	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	#65144	—	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	2	yes	—
18	#6045	—	4	4	no	—
19	C₅²×C₈	Q₈	1	2	yes	complemented
20	#6029	Q₈	1	2	yes	complemented
21	#4034	—	5	12	no	—
22	C₅×#106	D₅	1	6	yes	—
23	#3949	—	5	6	no	—
24	C₈×Dic₅	—	5	6	no	—
25	#3940	—	5	12	no	—
26	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
27	C₄×C₅²	—	4	2	no	—
28	C₄×C₅²	C₂×Q₈	1	1	yes	characteristic
29	C₄×C₅²	C₂×Q₈	1	1	yes	characteristic
30	#1039	—	4	2	no	—
31	#1039	SD₁₆	1	4	yes	—
32	C₂×Dic₁₀	—	10	12	no	—
33	C₂×C₅×C₈	C₂×D₅	1	6	yes	—
34	C₅×C₄⋊C₄	C₂×D₅	1	12	yes	—
35	C₄⋊Dic₅	—	5	12	no	—
36	#683	—	10	12	no	—
37	C₄×Dic₅	—	5	6	no	—
38	C₂×#168	—	5	6	no	—
39	#498	—	25	1	no	Sylow
40	C₂×C₅²	C₄⋊Q₈	1	1	yes	characteristic
41	C₂×C₅²	C₂×SD₁₆	1	2	yes	—
42	C₂×C₄×C₅	—	2	12	no	—
43	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
44	C₂×Dic₅	—	5	12	no	—
45	C₂×Dic₅	—	10	12	no	—
46	Dic₁₀	—	20	24	no	—
47	C₅×C₈	Dic₁₀	1	12	yes	complemented
48	#168	—	5	12	no	—
49	C₄⋊Q₈	—	25	2	no	—
50	#106	—	25	1	no	—
51	#106	—	25	2	no	—
52	#106	C₅⋊D₅	1	1	yes	characteristic
53	C₄×C₈	—	25	1	no	—
54	C₅²	#498	1	1	yes	characteristic, complemented, Sylow
55	C₂²×C₅	D₄×D₅	1	6	yes	—
56	C₄×C₅	—	4	12	no	—
57	C₄×C₅	C₂×Dic₁₀	1	6	yes	—
58	C₄×C₅	Q₈×D₅	1	6	yes	—
59	Dic₅	—	5	24	no	—
60	Dic₅	—	20	12	no	—
61	C₂×Q₈	—	50	2	no	—
62	C₂×C₈	—	25	1	no	—
63	C₂×C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
64	C₄⋊C₄	—	25	2	no	—
65	C₄⋊C₄	—	50	2	no	—
66	C₄⋊C₄	C₂×C₅⋊D₅	1	2	yes	—
67	C₄²	—	25	1	no	—
68	C₂×C₅	#4034	1	6	yes	—
69	C₂×C₅	D₅×SD₁₆	1	12	yes	—
70	Q₈	—	100	4	no	—
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	25	2	no	—
73	C₂×C₄	—	50	2	no	—
74	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
75	C₈	—	25	2	no	—
76	C₈	#6045	1	2	yes	complemented
77	C₅	#65144	1	6	yes	complemented
78	C₂²	D₄×C₅⋊D₅	1	1	yes	characteristic
79	C₄	—	4	2	no	—
80	C₄	—	25	4	no	—
81	C₄	—	100	2	no	—
82	C₄	C₂×#6045	1	1	yes	characteristic
83	C₄	Q₈×C₅⋊D₅	1	1	yes	characteristic
84	C₂	#11717208	1	1	yes	characteristic
85	C₂	SD₁₆×C₅⋊D₅	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×C₅⋊D₅
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₅²×#106	C₂
socle, prime radical	C₂²×C₅²	C₂×D₄
hypercenter	#106	C₅⋊D₅
hypocenter	C₅²	#498
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
19. C₅²×C₈	Q₈	semidirect
20. #6029	Q₈	semidirect
47. C₅×C₈	Dic₁₀	semidirect
54. C₅²	#498	semidirect
76. C₈	#6045	semidirect
77. C₅	#65144	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×#106	2
#11716684	2
C₈×#1039	2
#11716675	2
#65144	5