Subgroups — #13102947

Order 1600.

488 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
2	C₂×#87812	C₂	1	4	yes	maximal
3	#11717189	C₂	1	1	yes	maximal, characteristic
4	C₄×#6029	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅²×Q₈	C₂²	1	2	yes	—
6	C₂×#6045	C₂²	1	2	yes	—
7	C₅²×C₄⋊C₄	—	2	2	no	—
8	C₄²×C₅²	C₂²	1	1	yes	characteristic
9	#87812	—	2	8	no	—
10	C₄⋊#1039	—	2	2	no	—
11	C₂×#6029	C₂²	1	2	yes	—
12	#65381	—	5	6	no	maximal
13	C₅²×Q₈	—	2	4	no	—
14	C₂×C₄×C₅²	—	2	2	no	—
15	C₂×C₄×C₅²	D₄	1	2	yes	—
16	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
17	C₂×#1039	—	2	2	no	—
18	#6045	—	2	4	no	—
19	#6029	D₄	1	4	yes	—
20	C₅×C₄⋊Q₈	D₅	1	6	yes	—
21	C₂×#689	—	5	24	no	—
22	#4015	—	5	6	no	—
23	C₄×#168	—	5	6	no	—
24	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
25	C₄×C₅²	—	4	2	no	—
26	C₄×C₅²	Q₁₆	1	4	yes	complemented
27	C₄×C₅²	C₂×D₄	1	2	yes	—
28	#1039	—	4	2	no	—
29	C₂×C₅×Q₈	C₂×D₅	1	12	yes	—
30	C₂×Dic₁₀	—	5	12	no	—
31	C₅×C₄⋊C₄	—	2	12	no	—
32	C₄²×C₅	C₂×D₅	1	6	yes	—
33	#689	—	10	48	no	—
34	C₄⋊Dic₅	—	10	12	no	—
35	C₂×#168	—	5	12	no	—
36	#494	—	25	1	no	Sylow
37	C₂×C₅²	#127	1	1	yes	characteristic
38	C₂×C₅²	C₂×Q₁₆	1	2	yes	—
39	C₅×Q₈	—	2	24	no	—
40	C₂×C₄×C₅	—	2	12	no	—
41	C₂×C₄×C₅	C₂²⋊D₅	1	12	yes	—
42	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
43	C₂×Dic₅	—	10	12	no	—
44	Dic₁₀	—	10	24	no	—
45	#168	—	5	24	no	—
46	C₂×Q₁₆	—	25	4	no	—
47	C₄⋊Q₈	—	25	1	no	—
48	C₄⋊Q₈	C₅⋊D₅	1	1	yes	characteristic
49	C₄×C₈	—	25	1	no	—
50	C₅²	#494	1	1	yes	characteristic, complemented, Sylow
51	C₂²×C₅	C₂×C₂²⋊D₅	1	6	yes	—
52	C₄×C₅	—	4	12	no	—
53	C₄×C₅	#689	1	24	yes	complemented
54	C₄×C₅	D₄×D₅	1	12	yes	—
55	Dic₅	—	20	12	no	—
56	C₂×Q₈	—	25	2	no	—
57	C₂×Q₈	C₂×C₅⋊D₅	1	2	yes	—
58	Q₁₆	—	50	8	no	—
59	C₂×C₈	—	25	2	no	—
60	C₄⋊C₄	—	2	2	no	—
61	C₄⋊C₄	—	50	2	no	—
62	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
63	C₂×C₅	#4086	1	6	yes	—
64	C₂×C₅	C₂×#689	1	12	yes	—
65	Q₈	—	2	4	no	—
66	Q₈	—	50	4	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	50	2	no	—
69	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	2	yes	—
70	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
71	C₈	—	25	4	no	—
72	C₅	#65381	1	6	yes	complemented
73	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
74	C₄	—	4	2	no	—
75	C₄	—	100	2	no	—
76	C₄	#87812	1	4	yes	complemented
77	C₄	D₄×C₅⋊D₅	1	2	yes	—
78	C₂	#11717260	1	1	yes	characteristic
79	C₂	C₂×#87812	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, prime radical	C₂²×C₅²	C₂×D₄
hypercenter	C₄⋊Q₈	C₅⋊D₅
hypocenter	C₅²	#494
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
26. C₄×C₅²	Q₁₆	semidirect
50. C₅²	#494	semidirect
53. C₄×C₅	#689	semidirect
72. C₅	#65381	semidirect
76. C₄	#87812	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×#87812	2
#11717189	2
C₄×#6029	2
#65381	5