Subgroups — #45688308

Order 33344.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22552374	C₂	1	2	yes	maximal
2	C₂×C₈×D₅₂₁	C₂	1	1	yes	maximal, characteristic
3	#22552374	—	2	2	no	—
4	#22552374	C₄	1	4	yes	—
5	C₂×#16861967	—	2	2	no	—
6	C₂×C₄×D₅₂₁	C₂²	1	1	yes	characteristic
7	C₂×#16861960	C₄	1	1	yes	characteristic
8	C₈×D₅₂₁	—	2	2	no	—
9	C₂×C₈×C₅₂₁	C₄	1	1	yes	characteristic
10	#16861967	—	2	4	no	—
11	C₂²×D₅₂₁	Q₈	1	1	yes	characteristic
12	C₂×Dic₅₂₁	D₄	1	1	yes	characteristic
13	C₄×D₅₂₁	C₂×C₄	1	2	yes	—
14	C₄×D₅₂₁	D₄	1	2	yes	—
15	#16861960	—	2	1	no	—
16	C₂×C₄×C₅₂₁	C₂×C₄	1	1	yes	characteristic
17	C₈×C₅₂₁	—	2	1	no	—
18	C₂²×C₅₂₁	C₄⋊C₄	1	1	yes	characteristic
19	C₂×D₅₂₁	—	2	1	no	—
20	C₂×D₅₂₁	—	2	1	no	—
21	C₂×D₅₂₁	C₄⋊C₄	1	2	yes	—
22	C₄×C₅₂₁	C₄²	1	1	yes	characteristic
23	C₄×C₅₂₁	C₂²⋊C₄	1	1	yes	characteristic
24	Dic₅₂₁	C₂²⋊C₄	1	2	yes	—
25	C₂×C₅₂₁	#95	1	1	yes	characteristic
26	C₂×C₅₂₁	#108	1	1	yes	characteristic
27	C₂×C₅₂₁	#108	1	1	yes	characteristic
28	D₅₂₁	—	2	2	no	—
29	C₅₂₁	#341	1	1	yes	characteristic, complemented, Sylow
30	#341	—	521	1	no	maximal, Sylow
31	C₂×M₁₆	—	521	2	no	—
32	C₂²×C₈	—	521	1	no	—
33	C₂²×C₄	—	521	1	no	—
34	M₁₆	—	521	4	no	—
35	M₁₆	—	1042	2	no	—
36	C₂×C₈	—	521	1	no	—
37	C₂×C₈	—	1042	2	no	—
38	C₂×C₈	—	1042	2	no	—
39	C₂×C₈	#14523438	1	1	yes	characteristic
40	C₂³	—	521	1	no	—
41	C₂×C₄	—	521	1	no	—
42	C₂×C₄	—	521	2	no	—
43	C₂×C₄	—	521	2	no	—
44	C₂×C₄	C₂×#14523438	1	1	yes	characteristic
45	C₈	—	2	1	no	—
46	C₈	—	1042	4	no	—
47	C₈	—	1042	1	no	—
48	C₂²	—	521	2	no	—
49	C₂²	—	1042	1	no	—
50	C₂²	—	1042	1	no	—
51	C₂²	C₄⋊#14523438	1	1	yes	characteristic
52	C₄	—	521	2	no	—
53	C₄	C₄×#14523438	1	1	yes	characteristic
54	C₄	C₂²⋊#14523438	1	1	yes	characteristic
55	C₂	—	1042	2	no	—
56	C₂	(C₂×#14523438)⋊C₄	1	1	yes	characteristic
57	C₂	#30659778	1	1	yes	characteristic
58	C₂	#30659779	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₄⋊#14523438
commutator	C₄×C₅₂₁	C₄²
Frattini	C₂×C₄	C₂×#14523438
Fitting	C₂×C₈×C₅₂₁	C₄
socle	C₂²×C₅₂₁	C₄⋊C₄
prime radical	C₂²×D₅₂₁	Q₈
cosocle	C₂×C₄×D₅₂₁	C₂²
hypercenter	C₂×C₈	#14523438
hypocenter	C₅₂₁	#341
LCS	G ▷ C₄×C₅₂₁ ▷ C₂×C₅₂₁ ▷ C₅₂₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
29. C₅₂₁	#341	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22552374	2
C₂×C₈×D₅₂₁	2
#341	521