Subgroups — #13104242

Order 1600.

188 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11717037	C₂	1	2	yes	maximal
2	C₄×#6034	C₂	1	1	yes	maximal, characteristic
3	#11716801	C₂	1	4	yes	maximal
4	C₂×#6039	—	2	2	no	—
5	C₂×#6034	C₂²	1	2	yes	—
6	C₄×#1039	C₂²	1	1	yes	characteristic
7	C₅×C₄⋊Dic₅	—	2	2	no	—
8	#87781	C₂²	1	4	yes	—
9	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
10	C₂×#1039	D₄	1	1	yes	characteristic
11	C₂×#1039	C₂³	1	1	yes	characteristic
12	C₂×C₅×Dic₅	—	2	4	no	—
13	#6039	—	4	4	no	—
14	#6034	Q₈	1	4	yes	complemented
15	#4034	—	10	2	no	—
16	C₄×#170	—	10	1	no	—
17	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
18	C₄×C₅²	Q₁₆	1	2	yes	—
19	#1039	D₈	1	2	yes	—
20	#1039	C₂×Q₈	1	2	yes	—
21	C₅×Dic₅	—	4	4	no	—
22	C₂×Dic₁₀	—	10	4	no	—
23	C₂×#170	—	10	2	no	—
24	C₅×C₄⋊C₄	—	10	2	no	—
25	C₄⋊Dic₅	—	2	2	no	—
26	#683	—	10	4	no	—
27	C₄×Dic₅	—	10	1	no	—
28	C₄×Dic₅	—	10	2	no	—
29	#500	—	25	1	no	maximal, Sylow
30	C₂×C₅²	C₄⋊Q₈	1	1	yes	characteristic
31	C₂×C₅²	C₂×D₈	1	1	yes	characteristic
32	C₂×C₅²	C₂×Q₁₆	1	1	yes	characteristic
33	C₂×C₄×C₅	—	2	1	no	—
34	C₂×C₄×C₅	—	2	2	no	—
35	C₂×C₄×C₅	—	10	4	no	—
36	C₂×Dic₅	—	2	4	no	—
37	C₂×Dic₅	—	10	1	no	—
38	C₂×Dic₅	—	10	2	no	—
39	C₂×Dic₅	—	10	2	no	—
40	C₂×Dic₅	—	10	1	no	—
41	Dic₁₀	—	20	8	no	—
42	#170	—	10	4	no	—
43	C₄⋊Q₈	—	25	2	no	—
44	#107	—	25	4	no	—
45	C₄×C₈	—	25	1	no	—
46	C₅²	#500	1	1	yes	characteristic, complemented, Sylow
47	C₂²×C₅	—	2	2	no	—
48	C₂²×C₅	—	2	1	no	—
49	C₄×C₅	—	2	2	no	—
50	C₄×C₅	—	2	4	no	—
51	C₄×C₅	—	20	4	no	—
52	Dic₅	—	4	4	no	—
53	Dic₅	—	10	4	no	—
54	Dic₅	—	10	2	no	—
55	Dic₅	—	10	2	no	—
56	Dic₅	—	10	4	no	—
57	C₂×Q₈	—	50	2	no	—
58	C₂×C₈	—	25	2	no	—
59	C₄⋊C₄	—	10	2	no	—
60	C₄⋊C₄	—	25	4	no	—
61	C₄²	—	25	1	no	—
62	C₂×C₅	—	2	1	no	—
63	C₂×C₅	—	2	2	no	—
64	C₂×C₅	—	2	2	no	—
65	C₂×C₅	—	2	1	no	—
66	C₂×C₅	—	2	2	no	—
67	C₂×C₅	—	2	1	no	—
68	Q₈	—	100	4	no	—
69	C₂×C₄	—	10	4	no	—
70	C₂×C₄	—	25	1	no	—
71	C₂×C₄	—	25	1	no	—
72	C₂×C₄	D₅²⋊C₂	1	1	yes	characteristic
73	C₈	—	25	4	no	—
74	C₅	—	2	2	no	—
75	C₅	—	2	1	no	—
76	C₂²	C₂×D₅²⋊C₂	1	1	yes	characteristic
77	C₄	—	20	4	no	—
78	C₄	—	25	2	no	—
79	C₄	—	25	2	no	—
80	C₄	#87839	1	2	yes	—
81	C₂	#11717380	1	1	yes	characteristic
82	C₂	#11717383	1	1	yes	characteristic
83	C₂	C₂×#87839	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×D₅²⋊C₂
commutator, cosocle	C₂×#1039	C₂³
Fitting	C₂×C₄×C₅²	D₄
socle, prime radical	C₂²×C₅²	C₂×D₄
hypercenter	C₂×C₄	D₅²⋊C₂
hypocenter	C₅²	#500
LCS	G ▷ C₂×#1039 ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
14. #6034	Q₈	semidirect
46. C₅²	#500	semidirect

Maximal subgroups

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

subgroup	index
#11717037	2
C₄×#6034	2
#11716801	2
#500	25