Subgroups — #13099581

Order 1600.

256 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	C₄⋊#6029	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#6029	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅²×Q₈	—	2	1	no	—
5	C₅²×C₄⋊C₄	—	2	1	no	—
6	C₅²×C₄⋊C₄	C₄	1	2	yes	—
7	C₄²×C₅²	C₂²	1	1	yes	characteristic
8	C₂×#6029	—	2	1	no	—
9	C₂×#6029	—	2	1	no	—
10	#64767	—	5	6	no	maximal
11	C₅²×Q₈	—	4	1	no	—
12	C₅²×Q₈	—	4	1	no	—
13	C₂×C₄×C₅²	—	2	2	no	—
14	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
16	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
17	#6029	—	4	1	no	—
18	#6029	—	4	1	no	—
19	C₅×C₄⋊Q₈	D₅	1	6	yes	—
20	C₄⋊#168	—	5	6	no	—
21	C₄⋊#168	—	5	6	no	—
22	C₂²×C₅²	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₅²	—	2	1	no	—
24	C₄×C₅²	—	4	2	no	—
25	C₄×C₅²	D₈	1	1	yes	characteristic
26	C₄×C₅²	SD₁₆	1	1	yes	characteristic
27	C₄×C₅²	SD₁₆	1	1	yes	characteristic
28	C₄×C₅²	Q₁₆	1	1	yes	characteristic
29	C₂×C₅×Q₈	—	2	6	no	—
30	C₅×C₄⋊C₄	—	2	6	no	—
31	C₅×C₄⋊C₄	Dic₅	1	12	yes	—
32	C₄²×C₅	C₂×D₅	1	6	yes	—
33	C₂×#168	—	10	6	no	—
34	C₂×#168	—	10	6	no	—
35	#332	—	25	1	no	Sylow
36	C₂×C₅²	#101	1	1	yes	characteristic
37	C₂×C₅²	#102	1	1	yes	characteristic
38	C₂×C₅²	#103	1	1	yes	characteristic
39	C₅×Q₈	—	4	6	no	—
40	C₅×Q₈	—	4	6	no	—
41	C₂×C₄×C₅	—	2	12	no	—
42	C₂×C₄×C₅	C₂×Dic₅	1	6	yes	—
43	C₂×C₄×C₅	C₂²⋊D₅	1	6	yes	—
44	C₂×C₄×C₅	C₂²⋊D₅	1	6	yes	—
45	#168	—	20	6	no	—
46	#168	—	20	6	no	—
47	C₄⋊Q₈	C₅⋊D₅	1	1	yes	characteristic
48	C₄⋊C₈	—	25	1	no	—
49	C₄⋊C₈	—	25	1	no	—
50	C₅²	#332	1	1	yes	characteristic, complemented, Sylow
51	C₂²×C₅	C₂²⋊Dic₅	1	6	yes	—
52	C₄×C₅	—	2	6	no	—
53	C₄×C₅	—	4	12	no	—
54	C₄×C₅	#686	1	6	yes	—
55	C₄×C₅	#687	1	6	yes	—
56	C₄×C₅	#688	1	6	yes	—
57	C₄×C₅	#689	1	6	yes	—
58	C₂×Q₈	—	2	1	no	—
59	C₂×C₈	—	50	1	no	—
60	C₂×C₈	—	50	1	no	—
61	C₄⋊C₄	—	2	1	no	—
62	C₄⋊C₄	#1039	1	2	yes	—
63	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
64	C₂×C₅	#3964	1	6	yes	—
65	C₂×C₅	#3967	1	6	yes	—
66	C₂×C₅	#3968	1	6	yes	—
67	Q₈	—	4	1	no	—
68	Q₈	—	4	1	no	—
69	C₂×C₄	—	2	2	no	—
70	C₂×C₄	C₂×#1039	1	1	yes	characteristic
71	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
72	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
73	C₈	—	100	1	no	—
74	C₈	—	100	1	no	—
75	C₅	#64767	1	6	yes	complemented
76	C₂²	C₂²⋊#1039	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	4	2	no	—
79	C₄	#87809	1	1	yes	characteristic
80	C₄	#87810	1	1	yes	characteristic
81	C₄	#87811	1	1	yes	characteristic
82	C₄	#87812	1	1	yes	characteristic
83	C₂	#11716699	1	1	yes	characteristic
84	C₂	#11716702	1	1	yes	characteristic
85	C₂	#11716703	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#1039
commutator	C₂×C₄×C₅²	C₂×C₄
Frattini	C₄²	C₂×C₅⋊D₅
Fitting	C₅²×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₅²	C₂²⋊C₄
cosocle	C₄²×C₅²	C₂²
hypercenter	C₄⋊Q₈	C₅⋊D₅
hypocenter	C₅²	#332
LCS	G ▷ C₂×C₄×C₅² ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
50. C₅²	#332	semidirect
75. C₅	#64767	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
C₄⋊#6029	2
#64767	5