Subgroups — C₂×#20269977 #27213690

Order 13504.

290 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₄₂₂	C₂	1	2	yes	maximal
2	#20269977	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊Dic₂₁₁	C₂	1	4	yes	maximal
4	C₂×C₄²×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₄₂₂	—	2	8	no	—
6	C₂×Dic₄₂₂	C₂²	1	8	yes	—
7	C₂²×Dic₂₁₁	C₂²	1	4	yes	—
8	C₄⋊Dic₂₁₁	C₂²	1	16	yes	—
9	C₂²×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₂₁₁	C₂²	1	2	yes	—
11	C₄²×C₂₁₁	C₂²	1	4	yes	—
12	Dic₄₂₂	—	2	16	no	—
13	C₂×Dic₂₁₁	—	2	8	no	—
14	C₂×Dic₂₁₁	C₂³	1	8	yes	—
15	C₂³×C₂₁₁	C₂³	1	1	yes	characteristic
16	C₂×C₄×C₂₁₁	D₄	1	4	yes	—
17	C₂×C₄×C₂₁₁	Q₈	1	8	yes	complemented
18	C₂×C₄×C₂₁₁	C₂³	1	2	yes	—
19	C₂×C₄×C₂₁₁	C₂³	1	4	yes	—
20	C₂²×C₂₁₁	C₂×D₄	1	2	yes	—
21	C₂²×C₂₁₁	C₂×Q₈	1	4	yes	—
22	C₂²×C₂₁₁	C₂⁴	1	1	yes	characteristic
23	C₄×C₂₁₁	C₂×D₄	1	4	yes	—
24	C₄×C₂₁₁	C₂×Q₈	1	8	yes	complemented
25	Dic₂₁₁	—	2	8	no	—
26	C₂×C₂₁₁	C₄⋊Q₈	1	4	yes	complemented
27	C₂×C₂₁₁	C₂²×D₄	1	1	yes	characteristic
28	C₂×C₂₁₁	C₂²×Q₈	1	2	yes	—
29	C₂₁₁	C₂×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
30	C₂×C₄⋊Q₈	—	211	1	no	maximal, Sylow
31	C₂²×Q₈	—	211	2	no	—
32	C₄⋊Q₈	—	211	8	no	—
33	C₂×C₄⋊C₄	—	211	4	no	—
34	C₂×C₄²	D₂₁₁	1	1	yes	characteristic
35	C₂×Q₈	—	422	8	no	—
36	C₂×Q₈	—	211	8	no	—
37	C₂²×C₄	—	211	4	no	—
38	C₂²×C₄	C₂×D₂₁₁	1	1	yes	characteristic
39	C₂²×C₄	C₂×D₂₁₁	1	2	yes	—
40	C₄⋊C₄	—	211	16	no	—
41	C₄²	C₂×D₂₁₁	1	4	yes	—
42	C₂³	C₂²×D₂₁₁	1	1	yes	characteristic
43	Q₈	—	422	16	no	—
44	C₂×C₄	—	422	8	no	—
45	C₂×C₄	—	211	8	no	—
46	C₂×C₄	D₈₄₄	1	4	yes	—
47	C₂×C₄	Dic₄₂₂	1	8	yes	complemented
48	C₂×C₄	C₂²×D₂₁₁	1	2	yes	—
49	C₂×C₄	C₂²×D₂₁₁	1	4	yes	—
50	C₂²	C₂×D₈₄₄	1	2	yes	—
51	C₂²	C₂×Dic₄₂₂	1	4	yes	—
52	C₂²	C₂³×D₂₁₁	1	1	yes	characteristic
53	C₄	—	422	8	no	—
54	C₄	C₂×D₈₄₄	1	4	yes	—
55	C₄	C₂×Dic₄₂₂	1	8	yes	complemented
56	C₂	#20269977	1	4	yes	direct product factor
57	C₂	C₂²×D₈₄₄	1	1	yes	characteristic
58	C₂	C₂²×Dic₄₂₂	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₂₁₁
commutator, cosocle	C₂²×C₂₁₁	C₂⁴
Frattini	C₂²	C₂³×D₂₁₁
Fitting	C₂×C₄²×C₂₁₁	C₂
socle, prime radical	C₂³×C₂₁₁	C₂³
hypercenter	C₂×C₄²	D₂₁₁
hypocenter	C₂₁₁	C₂×C₄⋊Q₈
LCS	G ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. #20269977	C₂	direct (N × H)
17. C₂×C₄×C₂₁₁	Q₈	semidirect
24. C₄×C₂₁₁	C₂×Q₈	semidirect
26. C₂×C₂₁₁	C₄⋊Q₈	semidirect
29. C₂₁₁	C₂×C₄⋊Q₈	semidirect
47. C₂×C₄	Dic₄₂₂	semidirect
55. C₄	C₂×Dic₄₂₂	semidirect
56. C₂	#20269977	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Dic₄₂₂	2
#20269977	2
C₂×C₄⋊Dic₂₁₁	2
C₂×C₄²×C₂₁₁	2
C₂×C₄⋊Q₈	211

Subgroups — C2×#20269977 #27213690

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20269977 #27213690