Subgroups — C₂×#22552364 #30659751

Order 16672.

164 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂×C₄	C₂×D₅₂₁
commutator, cosocle	C₂×C₅₂₁	C₂⁴
Frattini	C₂	C₂³×D₅₂₁
Fitting	C₂²×C₄×C₅₂₁	C₂
socle	C₂²×C₅₂₁	C₂³
hypercenter	C₂²×C₄	D₅₂₁
hypocenter	C₅₂₁	C₂×#41
LCS	G ▷ C₂×C₅₂₁ ▷ C₅₂₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. #22552364	C₂	direct (N × H)
2. C₂×Dic₁₀₄₂	C₂	semidirect
3. C₂×D₂₀₈₄	C₂	semidirect
4. C₂×C₂²⋊D₅₂₁	C₂	semidirect
5. C₂×C₄×D₅₂₁	C₂	semidirect
6. C₂²×C₄×C₅₂₁	C₂	semidirect
8. Dic₁₀₄₂	C₂²	semidirect
9. D₂₀₈₄	C₂²	semidirect
10. C₂²⋊D₅₂₁	C₂²	semidirect
12. C₄×D₅₂₁	C₂²	semidirect
16. C₂×C₄×C₅₂₁	C₂²	semidirect
26. C₂×C₅₂₁	#41	semidirect
29. C₅₂₁	C₂×#41	semidirect
36. C₂²×C₄	D₅₂₁	semidirect
46. C₂×C₄	C₂×D₅₂₁	semidirect
57. C₂	#22552364	direct (N × H)

Maximal subgroups

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

subgroup	index
#22552364	2
C₂×Dic₁₀₄₂	2
C₂×D₂₀₈₄	2
C₂×C₂²⋊D₅₂₁	2
C₂×C₄×D₅₂₁	2
C₂²×C₄×C₅₂₁	2
C₂×#41	521

Subgroups — C2×#22552364 #30659751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22552364 #30659751