Subgroups — #72242134

Order 3200.

320 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13100264	C₂	1	1	yes	maximal, characteristic
2	#13100258	C₂	1	1	yes	maximal, characteristic
3	#13099536	C₂	1	1	yes	maximal, characteristic
4	C₄×#6033	C₄	1	2	yes	—
5	C₂×#87764	C₄	1	2	yes	—
6	C₂×C₅²×C₁₆	C₄	1	1	yes	characteristic
7	C₈×#1039	C₂²	1	1	yes	characteristic
8	C₂×#87756	C₄	1	1	yes	characteristic
9	#10598639	—	5	6	no	maximal
10	C₂×#6033	—	2	2	no	—
11	C₂×C₅²×C₈	C₂×C₄	1	1	yes	characteristic
12	C₄×#1039	C₂×C₄	1	1	yes	characteristic
13	C₂×#6029	C₂×C₄	1	1	yes	characteristic
14	#87764	—	2	2	no	—
15	C₅²×C₁₆	C₈	1	2	yes	complemented
16	#87756	C₈	1	2	yes	complemented
17	#64886	—	5	6	no	—
18	#64879	—	5	6	no	—
19	#64722	—	5	6	no	—
20	C₂×C₄×C₅²	C₄²	1	1	yes	characteristic
21	C₂×#1039	M₁₆	1	2	yes	—
22	#6033	—	4	2	no	—
23	C₅²×C₈	C₂×C₈	1	1	yes	characteristic
24	C₅²×C₈	C₂×C₈	1	1	yes	characteristic
25	#6029	M₁₆	1	2	yes	—
26	C₄×#170	—	5	12	no	—
27	C₂×#674	—	5	12	no	—
28	C₂×C₅×C₁₆	AGL(1,𝔽₅)	1	6	yes	—
29	C₈×Dic₅	—	5	6	no	—
30	C₂×#672	—	5	6	no	—
31	#1313	—	25	1	no	Sylow
32	C₂²×C₅²	#97	1	1	yes	characteristic
33	C₄×C₅²	C₄×C₈	1	1	yes	characteristic
34	C₄×C₅²	#97	1	1	yes	characteristic
35	#1039	—	2	2	no	—
36	C₂×#170	—	10	12	no	—
37	C₂×C₅×C₈	C₂×AGL(1,𝔽₅)	1	6	yes	—
38	C₄×Dic₅	—	5	6	no	—
39	C₂×#168	—	5	6	no	—
40	#674	—	10	12	no	—
41	C₅×C₁₆	#170	1	12	yes	complemented
42	#672	—	5	12	no	—
43	#346	—	25	1	no	—
44	#346	—	25	1	no	—
45	#322	—	25	1	no	—
46	C₂×C₅²	#322	1	1	yes	characteristic
47	C₂×C₅²	#347	1	2	yes	—
48	C₂×C₄×C₅	C₄×AGL(1,𝔽₅)	1	6	yes	—
49	C₂×Dic₅	—	5	12	no	—
50	#170	—	20	12	no	—
51	C₅×C₈	D₅⋊C₈	1	6	yes	—
52	C₅×C₈	C₂×#170	1	6	yes	—
53	#168	—	5	12	no	—
54	C₂×C₁₆	—	25	1	no	—
55	C₂×C₁₆	—	25	2	no	—
56	C₂×C₁₆	#1043	1	1	yes	characteristic
57	C₄×C₈	—	25	1	no	—
58	C₄×C₈	—	25	2	no	—
59	C₅²	#1313	1	1	yes	characteristic, complemented, Sylow
60	C₂²×C₅	#3992	1	6	yes	—
61	C₄×C₅	#3992	1	6	yes	—
62	C₄×C₅	C₄×#170	1	6	yes	—
63	Dic₅	—	10	12	no	—
64	C₂×C₈	—	25	1	no	—
65	C₂×C₈	—	50	2	no	—
66	C₂×C₈	C₂×#1043	1	1	yes	characteristic
67	C₄²	—	25	1	no	—
68	C₁₆	—	25	2	no	—
69	C₁₆	—	50	2	no	—
70	C₁₆	#6033	1	2	yes	complemented
71	C₂×C₅	#64845	1	12	yes	—
72	C₂×C₅	#64879	1	6	yes	—
73	C₂×C₄	—	25	2	no	—
74	C₂×C₄	C₄×#1043	1	1	yes	characteristic
75	C₈	—	25	2	no	—
76	C₈	—	100	2	no	—
77	C₈	#87855	1	1	yes	characteristic
78	C₈	C₂×#6033	1	1	yes	characteristic
79	C₅	#10598639	1	6	yes	complemented
80	C₂²	#11716860	1	1	yes	characteristic
81	C₄	—	50	2	no	—
82	C₄	#11716860	1	1	yes	characteristic
83	C₄	C₄×#6033	1	1	yes	characteristic
84	C₂	#13100223	1	2	yes	—
85	C₂	#13100258	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄×#1043
commutator	C₄×C₅²	C₄×C₈
Frattini	C₂×C₈	C₂×#1043
Fitting	C₂×C₅²×C₁₆	C₄
socle, prime radical	C₂²×C₅²	#97
cosocle	C₈×#1039	C₂²
hypercenter	C₂×C₁₆	#1043
hypocenter	C₅²	#1313
LCS	G ▷ C₄×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
15. C₅²×C₁₆	C₈	semidirect
16. #87756	C₈	semidirect
41. C₅×C₁₆	#170	semidirect
59. C₅²	#1313	semidirect
70. C₁₆	#6033	semidirect
79. C₅	#10598639	semidirect

Maximal subgroups

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

subgroup	index
#13100264	2
#13100258	2
#13099536	2
#10598639	5