Subgroups — C₂×#27212305 #41003254

Order 26944.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄₂₁	C₂	1	1	yes	maximal, characteristic
2	#27212305	C₂	1	4	yes	maximal, direct product factor
3	C₂×#20269746	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₆×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄×D₄₂₁	C₄	1	1	yes	characteristic
6	C₂×#16809890	C₄	1	1	yes	characteristic
7	C₈×D₄₂₁	C₄	1	2	yes	—
8	C₈×D₄₂₁	C₂²	1	2	yes	—
9	#20269746	C₂²	1	2	yes	complemented
10	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
11	C₁₆×C₄₂₁	C₂²	1	2	yes	complemented
12	C₂²×D₄₂₁	C₈	1	1	yes	characteristic
13	C₂×Dic₄₂₁	C₈	1	1	yes	characteristic
14	C₄×D₄₂₁	C₈	1	2	yes	—
15	C₄×D₄₂₁	C₂×C₄	1	2	yes	—
16	#16809890	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₄₂₁	C₂×C₄	1	1	yes	characteristic
18	C₈×C₄₂₁	C₂×C₄	1	1	yes	characteristic
19	C₈×C₄₂₁	C₂³	1	1	yes	characteristic
20	C₂²×C₄₂₁	C₂×C₈	1	1	yes	characteristic
21	C₂×D₄₂₁	—	2	2	no	—
22	C₂×D₄₂₁	C₂×C₈	1	2	yes	—
23	C₄×C₄₂₁	C₂×C₈	1	1	yes	characteristic
24	C₄×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
25	Dic₄₂₁	C₂×C₈	1	2	yes	—
26	C₂×C₄₂₁	M₃₂	1	2	yes	complemented
27	C₂×C₄₂₁	C₂²×C₈	1	1	yes	characteristic
28	D₄₂₁	—	2	2	no	—
29	C₄₂₁	C₂×M₃₂	1	1	yes	characteristic, complemented, Sylow
30	C₂×M₃₂	—	421	1	no	maximal, Sylow
31	C₂²×C₈	—	421	1	no	—
32	M₃₂	—	421	4	no	—
33	C₂×C₁₆	—	421	1	no	—
34	C₂×C₁₆	D₄₂₁	1	1	yes	characteristic, complemented
35	C₂²×C₄	—	421	1	no	—
36	C₂×C₈	—	421	1	no	—
37	C₂×C₈	—	421	2	no	—
38	C₂×C₈	—	421	2	no	—
39	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
40	C₁₆	—	421	2	no	—
41	C₁₆	C₂×D₄₂₁	1	2	yes	complemented
42	C₂³	—	421	1	no	—
43	C₂×C₄	—	421	1	no	—
44	C₂×C₄	—	421	2	no	—
45	C₂×C₄	—	421	2	no	—
46	C₂×C₄	C₄×D₄₂₁	1	1	yes	characteristic
47	C₈	—	421	2	no	—
48	C₈	C₄×D₄₂₁	1	1	yes	characteristic
49	C₈	C₂²×D₄₂₁	1	1	yes	characteristic
50	C₂²	—	842	2	no	—
51	C₂²	—	421	2	no	—
52	C₂²	C₈×D₄₂₁	1	1	yes	characteristic
53	C₄	—	421	2	no	—
54	C₄	C₈×D₄₂₁	1	1	yes	characteristic
55	C₄	C₂×C₄×D₄₂₁	1	1	yes	characteristic
56	C₂	—	842	2	no	—
57	C₂	#27212305	1	2	yes	direct product factor
58	C₂	C₂×C₈×D₄₂₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₄₂₁
commutator	C₂×C₄₂₁	C₂²×C₈
Frattini	C₈	C₂²×D₄₂₁
Fitting	C₂×C₁₆×C₄₂₁	C₂
socle	C₂²×C₄₂₁	C₂×C₈
prime radical	C₂²×D₄₂₁	C₈
cosocle	C₈×C₄₂₁	C₂³
hypercenter	C₂×C₁₆	D₄₂₁
hypocenter	C₄₂₁	C₂×M₃₂
LCS	G ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
2. #27212305	C₂	direct (N × H)
3. C₂×#20269746	C₂	semidirect
4. C₂×C₁₆×C₄₂₁	C₂	semidirect
9. #20269746	C₂²	semidirect
11. C₁₆×C₄₂₁	C₂²	semidirect
26. C₂×C₄₂₁	M₃₂	semidirect
29. C₄₂₁	C₂×M₃₂	semidirect
34. C₂×C₁₆	D₄₂₁	semidirect
41. C₁₆	C₂×D₄₂₁	semidirect
57. C₂	#27212305	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×C₈×D₄₂₁	2
#27212305	2
C₂×#20269746	2
C₂×C₁₆×C₄₂₁	2
C₂×M₃₂	421

Subgroups — C2×#27212305 #41003254

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27212305 #41003254