Subgroups — #45687563

Order 33344.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#16861960	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#16861960	C₂	1	2	yes	maximal
3	#30659619	C₂	1	2	yes	maximal
4	C₄×#16861960	C₂	1	1	yes	maximal, characteristic
5	C₄×Q₈×C₅₂₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#16861960	C₂²	1	1	yes	characteristic
7	C₂×#16861960	C₂²	1	2	yes	—
8	C₂×#16861960	C₂²	1	1	yes	characteristic
9	C₂×Q₈×C₅₂₁	C₄	1	1	yes	characteristic
10	C₄⋊C₄×C₅₂₁	C₄	1	2	yes	—
11	C₄⋊C₄×C₅₂₁	C₄	1	1	yes	characteristic
12	C₄²×C₅₂₁	C₂²	1	1	yes	characteristic
13	C₄²×C₅₂₁	C₂²	1	2	yes	—
14	#16861960	—	2	1	no	—
15	#16861960	—	2	2	no	—
16	#16861960	Q₈	1	2	yes	complemented
17	Q₈×C₅₂₁	—	2	2	no	—
18	C₂×C₄×C₅₂₁	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₅₂₁	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₅₂₁	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₅₂₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₅₂₁	C₂³	1	1	yes	characteristic
23	C₂²×C₅₂₁	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₅₂₁	—	2	2	no	—
25	C₄×C₅₂₁	—	2	1	no	—
26	C₄×C₅₂₁	—	2	2	no	—
27	C₄×C₅₂₁	M₁₆	1	2	yes	—
28	C₄×C₅₂₁	C₂×Q₈	1	1	yes	characteristic
29	C₄×C₅₂₁	#41	1	1	yes	characteristic
30	C₂×C₅₂₁	C₄×Q₈	1	1	yes	characteristic
31	C₂×C₅₂₁	C₂×M₁₆	1	1	yes	characteristic
32	C₂×C₅₂₁	#131	1	1	yes	characteristic
33	C₅₂₁	#446	1	1	yes	characteristic, complemented, Sylow
34	#446	—	521	1	no	maximal, Sylow
35	C₄×Q₈	D₅₂₁	1	1	yes	characteristic
36	C₄⋊C₈	—	521	1	no	—
37	C₄⋊C₈	—	521	2	no	—
38	#97	—	521	2	no	—
39	C₄×C₈	—	521	1	no	—
40	C₂×Q₈	Dic₅₂₁	1	1	yes	characteristic
41	C₂×C₈	—	521	1	no	—
42	C₂×C₈	—	521	1	no	—
43	C₂×C₈	—	521	2	no	—
44	C₄⋊C₄	Dic₅₂₁	1	2	yes	—
45	C₄⋊C₄	Dic₅₂₁	1	1	yes	characteristic
46	C₄²	C₂×D₅₂₁	1	1	yes	characteristic
47	C₄²	C₂×D₅₂₁	1	2	yes	—
48	Q₈	—	2	2	no	—
49	C₂×C₄	C₂×Dic₅₂₁	1	1	yes	characteristic
50	C₂×C₄	C₂×Dic₅₂₁	1	2	yes	—
51	C₂×C₄	C₂×Dic₅₂₁	1	2	yes	—
52	C₂×C₄	C₂×Dic₅₂₁	1	1	yes	characteristic
53	C₂×C₄	C₂²×D₅₂₁	1	1	yes	characteristic
54	C₈	—	521	2	no	—
55	C₈	—	1042	1	no	—
56	C₈	—	1042	2	no	—
57	C₂²	C₂²×Dic₅₂₁	1	1	yes	characteristic
58	C₄	—	2	2	no	—
59	C₄	—	2	1	no	—
60	C₄	—	2	2	no	—
61	C₄	#22552348	1	2	yes	—
62	C₄	Q₈×D₅₂₁	1	1	yes	characteristic
63	C₄	#22552365	1	1	yes	characteristic
64	C₂	Q₈×Dic₅₂₁	1	1	yes	characteristic
65	C₂	C₂×#22552348	1	1	yes	characteristic
66	C₂	#30659715	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₅₂₁
commutator, socle, prime radical	C₂²×C₅₂₁	C₂²×C₄
Fitting	C₄×Q₈×C₅₂₁	C₂
cosocle	C₂×C₄×C₅₂₁	C₂³
hypercenter	C₄×Q₈	D₅₂₁
hypocenter	C₅₂₁	#446
LCS	G ▷ C₂²×C₅₂₁ ▷ C₅₂₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
16. #16861960	Q₈	semidirect
33. C₅₂₁	#446	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#16861960	2
#30659619	2
C₄×#16861960	2
C₄×Q₈×C₅₂₁	2
#446	521