Subgroups — C₂×#27212309 #41003263

Order 26944.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃₃₆₈	C₂	1	1	yes	maximal, characteristic
2	#27212309	C₂	1	4	yes	maximal, direct product factor
3	C₂×#20269746	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×Q₁₆×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₆₈₄	—	2	1	no	—
6	D₃₃₆₈	—	2	1	no	—
7	D₃₃₆₈	C₂²	1	2	yes	—
8	#20269746	C₂²	1	2	yes	complemented
9	C₂×Q₈×C₄₂₁	—	2	1	no	—
10	Q₁₆×C₄₂₁	—	2	1	no	—
11	Q₁₆×C₄₂₁	C₂²	1	2	yes	complemented
12	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
13	C₂²×D₄₂₁	—	4	1	no	—
14	D₁₆₈₄	—	2	2	no	—
15	D₁₆₈₄	—	4	1	no	—
16	Q₈×C₄₂₁	—	2	2	no	—
17	Q₈×C₄₂₁	—	4	1	no	—
18	C₂×C₄×C₄₂₁	—	4	1	no	—
19	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
20	C₈×C₄₂₁	D₄	1	1	yes	characteristic
21	C₈×C₄₂₁	C₂³	1	1	yes	characteristic
22	C₂²×C₄₂₁	D₈	1	1	yes	characteristic
23	C₂×D₄₂₁	—	4	2	no	—
24	C₂×D₄₂₁	—	8	2	no	—
25	C₄×C₄₂₁	—	4	2	no	—
26	C₄×C₄₂₁	D₈	1	1	yes	characteristic
27	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄₂₁	SD₃₂	1	2	yes	complemented
29	C₂×C₄₂₁	C₂×D₈	1	1	yes	characteristic
30	D₄₂₁	—	8	2	no	—
31	C₄₂₁	C₂×SD₃₂	1	1	yes	characteristic, complemented, Sylow
32	C₂×SD₃₂	—	421	1	no	maximal, Sylow
33	C₂×Q₁₆	D₄₂₁	1	1	yes	characteristic, complemented
34	C₂×D₈	—	421	1	no	—
35	SD₃₂	—	421	4	no	—
36	C₂×C₁₆	—	421	1	no	—
37	C₂×Q₈	—	2	1	no	—
38	C₂×D₄	—	842	1	no	—
39	Q₁₆	—	2	1	no	—
40	Q₁₆	C₂×D₄₂₁	1	2	yes	complemented
41	D₈	—	842	1	no	—
42	D₈	—	421	2	no	—
43	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
44	C₁₆	—	421	2	no	—
45	C₂³	—	1684	1	no	—
46	Q₈	—	2	2	no	—
47	Q₈	—	4	1	no	—
48	D₄	—	842	2	no	—
49	D₄	—	1684	1	no	—
50	C₂×C₄	—	4	1	no	—
51	C₂×C₄	C₂²⋊D₄₂₁	1	1	yes	characteristic
52	C₈	C₂²⋊D₄₂₁	1	1	yes	characteristic
53	C₈	C₂²×D₄₂₁	1	1	yes	characteristic
54	C₂²	—	3368	2	no	—
55	C₂²	—	1684	2	no	—
56	C₂²	#20269756	1	1	yes	characteristic
57	C₄	—	4	2	no	—
58	C₄	#20269756	1	1	yes	characteristic
59	C₄	C₂×C₂²⋊D₄₂₁	1	1	yes	characteristic
60	C₂	—	3368	2	no	—
61	C₂	#27212309	1	2	yes	direct product factor
62	C₂	C₂×#20269756	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#20269756
commutator, cosocle	C₈×C₄₂₁	C₂³
Frattini	C₈	C₂²×D₄₂₁
Fitting	C₂×Q₁₆×C₄₂₁	C₂
socle	C₂²×C₄₂₁	D₈
prime radical	C₂×D₃₃₆₈	C₂
hypercenter	C₂×Q₁₆	D₄₂₁
hypocenter	C₄₂₁	C₂×SD₃₂
LCS	G ▷ C₈×C₄₂₁ ▷ C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #27212309	C₂	direct (N × H)
3. C₂×#20269746	C₂	semidirect
4. C₂×Q₁₆×C₄₂₁	C₂	semidirect
8. #20269746	C₂²	semidirect
11. Q₁₆×C₄₂₁	C₂²	semidirect
28. C₂×C₄₂₁	SD₃₂	semidirect
31. C₄₂₁	C₂×SD₃₂	semidirect
33. C₂×Q₁₆	D₄₂₁	semidirect
40. Q₁₆	C₂×D₄₂₁	semidirect
61. C₂	#27212309	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×D₃₃₆₈	2
#27212309	2
C₂×#20269746	2
C₂×Q₁₆×C₄₂₁	2
C₂×SD₃₂	421

Subgroups — C2×#27212309 #41003263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27212309 #41003263