Subgroups — #13103044

Order 1600.

432 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	Q₈×#1039	C₂	1	2	yes	maximal
3	#11716702	C₂	1	2	yes	maximal
4	#11716684	C₂	1	1	yes	maximal, characteristic
5	#11716682	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅²×Q₈	C₂²	1	2	yes	—
7	C₅²×Q₁₆	C₄	1	4	yes	complemented
8	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
9	C₄⋊#1039	—	2	2	no	—
10	C₄⋊#1039	C₂²	1	2	yes	—
11	C₄×#1039	—	2	2	no	—
12	C₄×#1039	C₂²	1	1	yes	characteristic
13	C₂×#6029	C₂²	1	1	yes	characteristic
14	#65473	—	5	6	no	maximal
15	C₅²×Q₈	—	2	2	no	—
16	C₅²×Q₈	C₂×C₄	1	4	yes	—
17	C₂×C₄×C₅²	—	2	2	no	—
18	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
19	C₂×#1039	—	2	2	no	—
20	C₂×#1039	D₄	1	2	yes	—
21	C₅²×C₈	C₂×C₄	1	2	yes	—
22	#6029	—	2	1	no	—
23	C₂×C₅×Q₁₆	D₅	1	6	yes	—
24	Q₈×Dic₅	—	5	12	no	—
25	#3967	—	5	12	no	—
26	#3949	—	5	6	no	—
27	#3947	—	5	6	no	—
28	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
29	C₄×C₅²	—	2	4	no	—
30	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
31	C₄×C₅²	#41	1	1	yes	characteristic
32	#1039	—	2	2	no	—
33	#1039	—	4	2	no	—
34	C₂×C₅×Q₈	C₂×D₅	1	12	yes	—
35	C₅×Q₁₆	Dic₅	1	24	yes	complemented
36	C₂×C₅×C₈	C₂×D₅	1	6	yes	—
37	C₄⋊Dic₅	—	5	12	no	—
38	C₄⋊Dic₅	—	10	12	no	—
39	C₄×Dic₅	—	5	6	no	—
40	C₄×Dic₅	—	10	12	no	—
41	C₂×#168	—	5	6	no	—
42	#441	—	25	1	no	Sylow
43	C₂×C₅²	C₄×D₄	1	1	yes	characteristic
44	C₂×C₅²	#137	1	2	yes	—
45	C₅×Q₈	—	2	12	no	—
46	C₅×Q₈	C₂×Dic₅	1	24	yes	—
47	C₂×C₄×C₅	—	2	12	no	—
48	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
49	C₂×Dic₅	—	5	12	no	—
50	C₂×Dic₅	—	10	12	no	—
51	C₅×C₈	C₂×Dic₅	1	12	yes	—
52	#168	—	10	6	no	—
53	C₂×Q₁₆	C₅⋊D₅	1	1	yes	characteristic
54	C₄×Q₈	—	25	2	no	—
55	#106	—	25	1	no	—
56	#103	—	25	2	no	—
57	#97	—	25	1	no	—
58	C₅²	#441	1	1	yes	characteristic, complemented, Sylow
59	C₂²×C₅	D₄×D₅	1	6	yes	—
60	C₄×C₅	—	2	24	no	—
61	C₄×C₅	#711	1	6	yes	—
62	C₄×C₅	C₂²×Dic₅	1	6	yes	—
63	Dic₅	—	10	12	no	—
64	Dic₅	—	20	12	no	—
65	C₂×Q₈	C₂×C₅⋊D₅	1	2	yes	—
66	Q₁₆	#1039	1	4	yes	complemented
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	C₄²	—	25	1	no	—
72	C₄²	—	50	2	no	—
73	C₂×C₅	#4064	1	12	yes	—
74	C₂×C₅	D₄×Dic₅	1	6	yes	—
75	Q₈	—	2	2	no	—
76	Q₈	C₂×#1039	1	4	yes	—
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	25	2	no	—
79	C₂×C₄	—	50	2	no	—
80	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
81	C₈	—	50	1	no	—
82	C₈	C₂×#1039	1	2	yes	—
83	C₅	#65473	1	6	yes	complemented
84	C₂²	D₄×C₅⋊D₅	1	1	yes	characteristic
85	C₄	—	2	4	no	—
86	C₄	—	50	2	no	—
87	C₄	—	100	2	no	—
88	C₄	#87902	1	1	yes	characteristic
89	C₄	C₂²×#1039	1	1	yes	characteristic
90	C₂	#11717238	1	2	yes	—
91	C₂	D₄×#1039	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×C₅⋊D₅
commutator	C₄×C₅²	C₂²×C₄
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₂×C₅²×Q₁₆	C₂
socle, prime radical	C₂²×C₅²	C₂×D₄
cosocle	C₂×C₄×C₅²	C₂³
hypercenter	C₂×Q₁₆	C₅⋊D₅
hypocenter	C₅²	#441
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
7. C₅²×Q₁₆	C₄	semidirect
35. C₅×Q₁₆	Dic₅	semidirect
58. C₅²	#441	semidirect
66. Q₁₆	#1039	semidirect
83. C₅	#65473	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅²×Q₁₆	2
Q₈×#1039	2
#11716702	2
#11716684	2
#11716682	2
#65473	5