Subgroups — #13099596

Order 1600.

264 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×D₈	C₂	1	1	yes	maximal, characteristic
2	#11716685	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#87756	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅²×D₄	—	2	1	no	—
5	C₅²×D₈	—	2	1	no	—
6	C₅²×D₈	C₄	1	2	yes	complemented
7	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
8	C₄⋊#1039	—	2	1	no	—
9	#87756	—	2	1	no	—
10	#64782	—	5	6	no	maximal
11	C₂³×C₅²	—	4	1	no	—
12	C₅²×D₄	—	2	2	no	—
13	C₅²×D₄	—	4	1	no	—
14	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
15	C₂×#1039	—	4	1	no	—
16	C₅²×C₈	C₂×C₄	1	1	yes	characteristic
17	C₅²×C₈	D₄	1	1	yes	characteristic
18	C₂×C₅×D₈	D₅	1	6	yes	—
19	#3950	—	5	6	no	—
20	C₂×#672	—	5	6	no	—
21	C₂²×C₅²	—	4	2	no	—
22	C₂²×C₅²	—	8	1	no	—
23	C₂²×C₅²	—	8	1	no	—
24	C₂²×C₅²	D₈	1	1	yes	characteristic
25	C₄×C₅²	C₂²⋊C₄	1	1	yes	characteristic
26	C₄×C₅²	SD₁₆	1	1	yes	characteristic
27	#1039	—	8	1	no	—
28	C₂×C₅×D₄	—	2	6	no	—
29	C₅×D₈	—	2	6	no	—
30	C₅×D₈	Dic₅	1	12	yes	complemented
31	C₂×C₅×C₈	C₂×D₅	1	6	yes	—
32	C₄⋊Dic₅	—	10	6	no	—
33	#672	—	10	6	no	—
34	#357	—	25	1	no	Sylow
35	C₂×C₅²	—	8	2	no	—
36	C₂×C₅²	#102	1	1	yes	characteristic
37	C₂×C₅²	D₁₆	1	1	yes	characteristic
38	C₂×C₅²	SD₃₂	1	1	yes	characteristic
39	C₂³×C₅	—	4	6	no	—
40	C₅×D₄	—	2	12	no	—
41	C₅×D₄	—	4	6	no	—
42	C₂×C₄×C₅	C₂²⋊D₅	1	6	yes	—
43	C₂×Dic₅	—	20	6	no	—
44	C₅×C₈	C₂×Dic₅	1	6	yes	—
45	C₅×C₈	C₂²⋊D₅	1	6	yes	—
46	C₂×D₈	C₅⋊D₅	1	1	yes	characteristic
47	C₂×C₁₆	—	25	1	no	—
48	#107	—	25	1	no	—
49	C₅²	#357	1	1	yes	characteristic, complemented, Sylow
50	C₂²×C₅	—	4	12	no	—
51	C₂²×C₅	—	8	6	no	—
52	C₂²×C₅	—	8	6	no	—
53	C₂²×C₅	#686	1	6	yes	—
54	C₄×C₅	#687	1	6	yes	—
55	C₄×C₅	C₂²⋊Dic₅	1	6	yes	—
56	Dic₅	—	40	6	no	—
57	C₂×D₄	—	2	1	no	—
58	D₈	—	2	1	no	—
59	D₈	#1039	1	2	yes	complemented
60	C₂×C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
61	C₄⋊C₄	—	50	1	no	—
62	C₁₆	—	50	1	no	—
63	C₂×C₅	—	8	12	no	—
64	C₂×C₅	#3958	1	6	yes	—
65	C₂×C₅	#3959	1	6	yes	—
66	C₂×C₅	#3964	1	6	yes	—
67	C₂³	—	4	1	no	—
68	D₄	—	2	2	no	—
69	D₄	—	4	1	no	—
70	C₂×C₄	—	100	1	no	—
71	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
72	C₈	C₂×#1039	1	1	yes	characteristic
73	C₈	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
74	C₅	#64782	1	6	yes	complemented
75	C₂²	—	4	2	no	—
76	C₂²	—	8	1	no	—
77	C₂²	—	8	1	no	—
78	C₂²	#87809	1	1	yes	characteristic
79	C₄	—	200	1	no	—
80	C₄	#87810	1	1	yes	characteristic
81	C₄	C₂²⋊#1039	1	1	yes	characteristic
82	C₂	—	8	2	no	—
83	C₂	#11716693	1	1	yes	characteristic
84	C₂	#11716694	1	1	yes	characteristic
85	C₂	#11716699	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#87809
commutator	C₅²×C₈	C₂×C₄
Frattini	C₂×C₈	C₂×C₅⋊D₅
Fitting, prime radical	C₂×C₅²×D₈	C₂
socle	C₂²×C₅²	D₈
cosocle	C₂×C₅²×C₈	C₂²
hypercenter	C₂×D₈	C₅⋊D₅
hypocenter	C₅²	#357
LCS	G ▷ C₅²×C₈ ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
2. #11716685	C₂	semidirect
3. C₂×#87756	C₂	semidirect
6. C₅²×D₈	C₄	semidirect
30. C₅×D₈	Dic₅	semidirect
49. C₅²	#357	semidirect
59. D₈	#1039	semidirect
74. C₅	#64782	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅²×D₈	2
#11716685	2
C₂×#87756	2
#64782	5