Subgroups — #13102949

Order 1600.

568 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#87811	C₂	1	4	yes	maximal, complemented
3	#11717194	C₂	1	1	yes	maximal, characteristic
4	C₄×#6029	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅²×Q₈	C₂²	1	2	yes	—
6	C₂×C₄⋊(C₅⋊D₅)	—	2	2	no	—
7	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	2	yes	—
8	C₅²×C₄⋊C₄	—	2	2	no	—
9	C₄²×C₅²	C₂²	1	1	yes	characteristic
10	#87811	—	2	8	no	—
11	C₂×#6029	C₂²	1	2	yes	—
12	#65374	—	5	6	no	maximal
13	C₂²×C₅⋊D₅	—	2	2	no	—
14	C₅²×Q₈	—	2	4	no	—
15	C₂×C₄×C₅²	—	2	2	no	—
16	C₂×C₄×C₅²	D₄	1	2	yes	—
17	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
18	C₄⋊(C₅⋊D₅)	—	2	4	no	—
19	C₄⋊(C₅⋊D₅)	—	4	4	no	—
20	#6029	D₄	1	4	yes	complemented
21	C₅×C₄⋊Q₈	D₅	1	6	yes	complemented
22	C₂×#688	—	5	24	no	—
23	#4020	—	5	6	no	—
24	C₄×#168	—	5	6	no	—
25	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
26	C₂×C₅⋊D₅	—	4	2	no	—
27	C₂×C₅⋊D₅	—	4	4	no	—
28	C₄×C₅²	—	4	2	no	—
29	C₄×C₅²	SD₁₆	1	4	yes	complemented
30	C₄×C₅²	C₂×D₄	1	2	yes	—
31	C₂×C₅×Q₈	C₂×D₅	1	12	yes	—
32	C₂×D₂₀	—	5	12	no	—
33	C₂×D₂₀	—	10	12	no	—
34	C₅×C₄⋊C₄	—	2	12	no	—
35	C₄²×C₅	C₂×D₅	1	6	yes	—
36	#688	—	10	48	no	—
37	C₂×#168	—	5	12	no	—
38	#492	—	25	1	no	Sylow
39	C₂×C₅²	#127	1	1	yes	characteristic
40	C₂×C₅²	C₂×SD₁₆	1	2	yes	—
41	C₅⋊D₅	—	8	2	no	—
42	C₂²×D₅	—	10	12	no	—
43	C₅×Q₈	—	2	24	no	—
44	C₂×C₄×C₅	—	2	12	no	—
45	C₂×C₄×C₅	C₂²⋊D₅	1	12	yes	—
46	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
47	D₂₀	—	10	24	no	—
48	D₂₀	—	20	24	no	—
49	#168	—	5	24	no	—
50	C₂×SD₁₆	—	25	4	no	—
51	C₄⋊Q₈	C₅⋊D₅	1	1	yes	characteristic, complemented
52	#127	—	25	1	no	—
53	C₄×C₈	—	25	1	no	—
54	C₅²	#492	1	1	yes	characteristic, complemented, Sylow
55	C₂²×C₅	C₂×C₂²⋊D₅	1	6	yes	—
56	C₂×D₅	—	20	12	no	—
57	C₂×D₅	—	20	24	no	—
58	C₄×C₅	—	4	12	no	—
59	C₄×C₅	#688	1	24	yes	complemented
60	C₄×C₅	D₄×D₅	1	12	yes	—
61	C₂×Q₈	C₂×C₅⋊D₅	1	2	yes	—
62	C₂×D₄	—	25	2	no	—
63	C₂×D₄	—	50	2	no	—
64	SD₁₆	—	50	8	no	—
65	C₂×C₈	—	25	2	no	—
66	C₄⋊C₄	—	2	2	no	—
67	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
68	C₂×C₅	#4086	1	6	yes	—
69	C₂×C₅	C₂×#688	1	12	yes	—
70	D₅	—	40	12	no	—
71	C₂³	—	50	2	no	—
72	Q₈	—	2	4	no	—
73	D₄	—	50	4	no	—
74	D₄	—	100	4	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	2	yes	—
77	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
78	C₈	—	25	4	no	—
79	C₅	#65374	1	6	yes	complemented
80	C₂²	—	100	2	no	—
81	C₂²	—	100	4	no	—
82	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
83	C₄	—	4	2	no	—
84	C₄	#87811	1	4	yes	complemented
85	C₄	D₄×C₅⋊D₅	1	2	yes	—
86	C₂	—	200	2	no	—
87	C₂	#11717260	1	1	yes	characteristic
88	C₂	C₂×#87811	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊(C₅⋊D₅)
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₅²×C₄⋊Q₈	C₂
socle	C₂²×C₅²	C₂×D₄
prime radical	#11717194	C₂
hypercenter	C₄⋊Q₈	C₅⋊D₅
hypocenter	C₅²	#492
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₅²×C₄⋊Q₈	C₂	semidirect
2. C₂×#87811	C₂	semidirect
4. C₄×#6029	C₂	semidirect
20. #6029	D₄	semidirect
21. C₅×C₄⋊Q₈	D₅	semidirect
29. C₄×C₅²	SD₁₆	semidirect
51. C₄⋊Q₈	C₅⋊D₅	semidirect
54. C₅²	#492	semidirect
59. C₄×C₅	#688	semidirect
79. C₅	#65374	semidirect
84. C₄	#87811	semidirect

Maximal subgroups

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

subgroup	index
C₅²×C₄⋊Q₈	2
C₂×#87811	2
#11717194	2
C₄×#6029	2
#65374	5