Subgroups — C₂×#30659756 #45688266

Order 33344.

794 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30659756	C₂	1	16	yes	maximal, direct product factor
2	C₂×#22552365	C₂	1	4	yes	maximal, complemented
3	C₂×#22552364	C₂	1	6	yes	maximal, complemented
4	C₂×Q₈×D₅₂₁	C₂	1	4	yes	maximal, complemented
5	C₂²×Q₈×C₅₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#22552365	C₂²	1	32	yes	complemented
7	#22552364	C₂²	1	48	yes	complemented
8	C₂×Dic₁₀₄₂	C₂²	1	6	yes	—
9	Q₈×D₅₂₁	C₂²	1	32	yes	complemented
10	C₂×D₂₀₈₄	C₂²	1	6	yes	—
11	C₂×C₂²⋊D₅₂₁	C₂²	1	4	yes	—
12	C₂×C₄×D₅₂₁	C₂²	1	12	yes	—
13	C₂×Q₈×C₅₂₁	C₂²	1	4	yes	—
14	C₂×Q₈×C₅₂₁	C₂²	1	8	yes	complemented
15	C₂²×C₄×C₅₂₁	C₂²	1	3	yes	—
16	C₂²×D₅₂₁	C₂³	1	4	yes	—
17	Dic₁₀₄₂	C₂³	1	24	yes	—
18	D₂₀₈₄	C₂³	1	24	yes	—
19	C₂²⋊D₅₂₁	C₂³	1	16	yes	—
20	C₂×Dic₅₂₁	C₂³	1	4	yes	—
21	C₄×D₅₂₁	C₂³	1	48	yes	—
22	C₂³×C₅₂₁	C₂³	1	1	yes	characteristic
23	Q₈×C₅₂₁	C₂³	1	16	yes	—
24	C₂×C₄×C₅₂₁	C₂³	1	6	yes	—
25	C₂×C₄×C₅₂₁	C₂³	1	12	yes	—
26	C₂²×C₅₂₁	—	2	2	no	—
27	C₂²×C₅₂₁	C₂⁴	1	2	yes	—
28	C₂²×C₅₂₁	C₂⁴	1	1	yes	characteristic
29	C₂×D₅₂₁	—	2	8	no	—
30	C₂×D₅₂₁	C₂⁴	1	8	yes	—
31	C₄×C₅₂₁	C₂⁴	1	12	yes	—
32	Dic₅₂₁	C₂⁴	1	8	yes	—
33	C₂×C₅₂₁	—	2	2	no	—
34	C₂×C₅₂₁	2₋¹⁺⁴	1	2	yes	complemented
35	C₂×C₅₂₁	C₂⁵	1	1	yes	characteristic
36	D₅₂₁	—	2	8	no	—
37	C₅₂₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
38	C₂×2₋¹⁺⁴	—	521	1	no	maximal, Sylow
39	2₋¹⁺⁴	—	521	16	no	—
40	C₂×#41	—	521	4	no	—
41	C₂×#41	—	521	6	no	—
42	C₂²×Q₈	—	521	4	no	—
43	C₂²×Q₈	D₅₂₁	1	1	yes	characteristic, complemented
44	#41	—	521	32	no	—
45	#41	—	521	48	no	—
46	C₂×Q₈	—	521	32	no	—
47	C₂×Q₈	—	521	6	no	—
48	C₂×Q₈	C₂×D₅₂₁	1	4	yes	—
49	C₂×Q₈	C₂×D₅₂₁	1	8	yes	complemented
50	C₂×D₄	—	521	4	no	—
51	C₂×D₄	—	521	6	no	—
52	C₂²×C₄	—	521	12	no	—
53	C₂²×C₄	C₂×D₅₂₁	1	3	yes	—
54	C₂³	—	521	4	no	—
55	C₂³	C₂²×D₅₂₁	1	1	yes	characteristic
56	Q₈	—	521	24	no	—
57	Q₈	C₂²×D₅₂₁	1	16	yes	—
58	D₄	—	521	16	no	—
59	D₄	—	521	24	no	—
60	C₂×C₄	—	521	4	no	—
61	C₂×C₄	—	521	48	no	—
62	C₂×C₄	C₂²×D₅₂₁	1	12	yes	—
63	C₂×C₄	C₂²×D₅₂₁	1	6	yes	—
64	C₂²	—	2	2	no	—
65	C₂²	—	521	8	no	—
66	C₂²	—	1042	8	no	—
67	C₂²	C₂³×D₅₂₁	1	2	yes	—
68	C₂²	C₂³×D₅₂₁	1	1	yes	characteristic
69	C₄	—	521	8	no	—
70	C₄	C₂³×D₅₂₁	1	12	yes	—
71	C₂	—	2	2	no	—
72	C₂	—	1042	8	no	—
73	C₂	#30659756	1	2	yes	direct product factor
74	C₂	C₂⁴×D₅₂₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×D₅₂₁
commutator, cosocle	C₂×C₅₂₁	C₂⁵
Frattini	C₂	C₂⁴×D₅₂₁
Fitting	C₂²×Q₈×C₅₂₁	C₂
socle	C₂²×C₅₂₁	C₂⁴
hypercenter	C₂²×Q₈	D₅₂₁
hypocenter	C₅₂₁	C₂×2₋¹⁺⁴
LCS	G ▷ C₂×C₅₂₁ ▷ C₅₂₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
1. #30659756	C₂	direct (N × H)
2. C₂×#22552365	C₂	semidirect
3. C₂×#22552364	C₂	semidirect
4. C₂×Q₈×D₅₂₁	C₂	semidirect
5. C₂²×Q₈×C₅₂₁	C₂	semidirect
6. #22552365	C₂²	semidirect
7. #22552364	C₂²	semidirect
9. Q₈×D₅₂₁	C₂²	semidirect
14. C₂×Q₈×C₅₂₁	C₂²	semidirect
34. C₂×C₅₂₁	2₋¹⁺⁴	semidirect
37. C₅₂₁	C₂×2₋¹⁺⁴	semidirect
43. C₂²×Q₈	D₅₂₁	semidirect
49. C₂×Q₈	C₂×D₅₂₁	semidirect
73. C₂	#30659756	direct (N × H)

Maximal subgroups

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

subgroup	index
#30659756	2
C₂×#22552365	2
C₂×#22552364	2
C₂×Q₈×D₅₂₁	2
C₂²×Q₈×C₅₂₁	2
C₂×2₋¹⁺⁴	521

Subgroups — C2×#30659756 #45688266

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#30659756 #45688266