Subgroups — #45688314

Order 33344.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅₂₁⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	#30659787	C₂	1	1	yes	maximal, characteristic
3	C₂×#22552347	C₂	1	1	yes	maximal, characteristic
4	#22552374	—	2	1	no	—
5	#22552373	C₄	1	2	yes	complemented
6	#22552372	—	2	1	no	—
7	C₄⋊#14523438	—	2	1	no	—
8	C₂²⋊#14523438	C₄	1	2	yes	—
9	C₄×#14523438	—	2	1	no	—
10	C₂×C₄×D₅₂₁	C₂²	1	1	yes	characteristic
11	#22552347	—	2	1	no	—
12	#22552347	—	2	1	no	—
13	C₂×#16861960	C₄	1	1	yes	characteristic, complemented
14	C₂×C₈×C₅₂₁	C₄	1	1	yes	characteristic, complemented
15	C₂×#14523438	—	2	2	no	—
16	#16861967	—	2	2	no	—
17	C₂²×D₅₂₁	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₅₂₁	C₂×C₄	1	1	yes	characteristic
19	C₄×D₅₂₁	D₄	1	1	yes	characteristic
20	C₄×D₅₂₁	D₄	1	1	yes	characteristic
21	C₄×D₅₂₁	D₄	1	1	yes	characteristic
22	C₄×D₅₂₁	Q₈	1	1	yes	characteristic
23	#16861960	—	2	1	no	—
24	C₂×C₄×C₅₂₁	C₂×C₄	1	1	yes	characteristic
25	C₈×C₅₂₁	—	2	1	no	—
26	C₂²×C₅₂₁	C₄²	1	1	yes	characteristic
27	C₂×D₅₂₁	—	4	1	no	—
28	C₂×D₅₂₁	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×D₅₂₁	C₄⋊C₄	1	1	yes	characteristic
30	#14523438	—	4	2	no	—
31	C₄×C₅₂₁	C₂²⋊C₄	1	1	yes	characteristic
32	C₄×C₅₂₁	C₄⋊C₄	1	1	yes	characteristic
33	Dic₅₂₁	C₂²⋊C₄	1	1	yes	characteristic
34	Dic₅₂₁	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₅₂₁	—	2	1	no	—
36	C₂×C₅₂₁	#95	1	1	yes	characteristic
37	D₅₂₁	—	2	1	no	—
38	D₅₂₁	—	2	1	no	—
39	C₅₂₁	#344	1	1	yes	characteristic, complemented, Sylow
40	#344	—	521	1	no	maximal, Sylow
41	C₂×M₁₆	—	521	1	no	—
42	C₂×M₁₆	—	521	1	no	—
43	#117	—	521	1	no	—
44	C₂²×C₄	—	521	1	no	—
45	M₁₆	—	521	2	no	—
46	M₁₆	—	1042	1	no	—
47	M₁₆	—	1042	1	no	—
48	M₁₆	—	1042	1	no	—
49	C₂×C₈	—	521	1	no	—
50	C₂×C₈	—	1042	1	no	—
51	C₂×C₈	#14523438	1	1	yes	characteristic, complemented
52	C₄⋊C₄	—	1042	1	no	—
53	C₂²⋊C₄	—	521	2	no	—
54	C₄²	—	1042	1	no	—
55	C₂³	—	521	1	no	—
56	C₂×C₄	—	521	1	no	—
57	C₂×C₄	—	521	1	no	—
58	C₂×C₄	—	521	1	no	—
59	C₂×C₄	—	521	1	no	—
60	C₂×C₄	—	521	1	no	—
61	C₂×C₄	—	1042	2	no	—
62	C₂×C₄	C₂×#14523438	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	1042	2	no	—
65	C₈	—	1042	1	no	—
66	C₂²	—	521	1	no	—
67	C₂²	—	521	1	no	—
68	C₂²	—	2084	1	no	—
69	C₂²	C₄×#14523438	1	1	yes	characteristic
70	C₄	—	521	1	no	—
71	C₄	—	521	1	no	—
72	C₄	—	2084	2	no	—
73	C₄	C₂²⋊#14523438	1	1	yes	characteristic
74	C₄	C₄⋊#14523438	1	1	yes	characteristic
75	C₂	—	2	1	no	—
76	C₂	—	1042	1	no	—
77	C₂	—	1042	1	no	—
78	C₂	(C₂×#14523438)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊#14523438
commutator	C₂²×C₅₂₁	C₄²
Frattini	C₂×C₄	C₂×#14523438
Fitting	C₂×C₈×C₅₂₁	C₄
socle	C₂×C₅₂₁	#95
prime radical	C₂²×D₅₂₁	C₂×C₄
cosocle	C₂×C₄×D₅₂₁	C₂²
hypercenter	C₂×C₈	#14523438
hypocenter	C₅₂₁	#344
LCS	G ▷ C₂²×C₅₂₁ ▷ C₂×C₅₂₁ ▷ C₅₂₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
5. #22552373	C₄	semidirect
13. C₂×#16861960	C₄	semidirect
14. C₂×C₈×C₅₂₁	C₄	semidirect
39. C₅₂₁	#344	semidirect
51. C₂×C₈	#14523438	semidirect

Maximal subgroups

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

subgroup	index
D₅₂₁⋊M₁₆	2
#30659787	2
C₂×#22552347	2
#344	521