Subgroups — C₂²×#87756 #13102959

Order 1600.

392 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅²×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#87756	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₄×C₅²	C₄	1	1	yes	characteristic
4	C₂×C₅²×C₈	C₄	1	3	yes	—
5	C₂×C₅²×C₈	C₂²	1	3	yes	—
6	#87756	C₂²	1	4	yes	direct product factor
7	C₂²×#672	—	5	6	no	maximal
8	C₂³×C₅²	C₈	1	1	yes	characteristic
9	C₂×C₄×C₅²	C₈	1	3	yes	—
10	C₂×C₄×C₅²	C₂×C₄	1	3	yes	—
11	C₅²×C₈	C₂×C₄	1	3	yes	—
12	C₅²×C₈	C₂³	1	1	yes	characteristic
13	C₂²×C₅×C₈	D₅	1	6	yes	—
14	C₂×#672	—	5	36	no	—
15	C₂²×C₅²	C₁₆	1	4	yes	complemented
16	C₂²×C₅²	C₂×C₈	1	3	yes	—
17	C₄×C₅²	C₂×C₈	1	3	yes	—
18	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₅	Dic₅	1	6	yes	—
20	C₂×C₅×C₈	Dic₅	1	18	yes	—
21	C₂×C₅×C₈	C₂×D₅	1	18	yes	—
22	#672	—	5	24	no	—
23	C₂²×C₁₆	—	25	1	no	Sylow
24	C₂×C₅²	C₂×C₁₆	1	6	yes	complemented
25	C₂×C₅²	C₂²×C₈	1	1	yes	characteristic
26	C₂³×C₅	#168	1	6	yes	—
27	C₂×C₄×C₅	#168	1	18	yes	—
28	C₂×C₄×C₅	C₂×Dic₅	1	18	yes	—
29	C₅×C₈	C₂×Dic₅	1	18	yes	—
30	C₅×C₈	C₂²×D₅	1	6	yes	—
31	C₂²×C₈	C₅⋊D₅	1	1	yes	characteristic
32	C₂×C₁₆	—	25	6	no	—
33	C₅²	C₂²×C₁₆	1	1	yes	characteristic, complemented, Sylow
34	C₂²×C₅	#672	1	24	yes	complemented
35	C₂²×C₅	C₂×#168	1	18	yes	—
36	C₄×C₅	C₂×#168	1	18	yes	—
37	C₄×C₅	C₂²×Dic₅	1	6	yes	—
38	C₂²×C₄	#1039	1	1	yes	characteristic
39	C₂×C₈	#1039	1	3	yes	—
40	C₂×C₈	C₂×C₅⋊D₅	1	3	yes	—
41	C₁₆	—	25	4	no	—
42	C₂×C₅	C₂×#672	1	36	yes	complemented
43	C₂×C₅	C₂²×#168	1	6	yes	—
44	C₂³	#6029	1	1	yes	characteristic
45	C₂×C₄	#6029	1	3	yes	—
46	C₂×C₄	C₂×#1039	1	3	yes	—
47	C₈	C₂×#1039	1	3	yes	—
48	C₈	C₂²×C₅⋊D₅	1	1	yes	characteristic
49	C₅	C₂²×#672	1	6	yes	complemented
50	C₂²	#87756	1	4	yes	direct product factor
51	C₂²	C₂×#6029	1	3	yes	—
52	C₄	C₂×#6029	1	3	yes	—
53	C₄	C₂²×#1039	1	1	yes	characteristic
54	C₂	C₂×#87756	1	6	yes	direct product factor
55	C₂	C₂²×#6029	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₈	C₅⋊D₅
commutator, hypocenter	C₅²	C₂²×C₁₆
Frattini	C₈	C₂²×C₅⋊D₅
Fitting	C₂²×C₅²×C₈	C₂
socle, prime radical	C₂³×C₅²	C₈
cosocle	C₅²×C₈	C₂³
LCS	G ▷ C₅²
UCS	C₁ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#87756	C₂	direct (N × H)
6. #87756	C₂²	direct (N × H)
15. C₂²×C₅²	C₁₆	semidirect
24. C₂×C₅²	C₂×C₁₆	semidirect
33. C₅²	C₂²×C₁₆	semidirect
34. C₂²×C₅	#672	semidirect
42. C₂×C₅	C₂×#672	semidirect
49. C₅	C₂²×#672	semidirect
50. C₂²	#87756	direct (N × H)
54. C₂	C₂×#87756	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅²×C₈	2
C₂×#87756	2
C₂²×#672	5

Subgroups — C22×#87756 #13102959

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#87756 #13102959