Subgroups — C₂×#131×C₂₂₃ #28383279

Order 14272.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	#131×C₂₂₃	C₂	1	8	yes	maximal, direct product factor
3	C₂×M₁₆×C₂₂₃	C₂	1	3	yes	maximal, complemented
4	C₂²×C₈×C₂₂₃	C₂	1	3	yes	maximal, complemented
5	#41×C₂₂₃	C₄	1	4	yes	—
6	#41×C₂₂₃	C₂²	1	4	yes	—
7	C₂×Q₈×C₂₂₃	C₄	1	1	yes	characteristic
8	C₂×D₄×C₂₂₃	C₄	1	3	yes	—
9	C₂²×C₄×C₂₂₃	C₂²	1	3	yes	—
10	M₁₆×C₂₂₃	C₂²	1	12	yes	complemented
11	C₂×C₈×C₂₂₃	C₂²	1	3	yes	—
12	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₈×C₂₂₃	C₂²	1	12	yes	complemented
14	C₂³×C₂₂₃	C₂×C₄	1	3	yes	—
15	Q₈×C₂₂₃	C₂×C₄	1	4	yes	—
16	D₄×C₂₂₃	C₂×C₄	1	12	yes	—
17	C₂×C₄×C₂₂₃	C₂×C₄	1	3	yes	—
18	C₂×C₄×C₂₂₃	C₂×C₄	1	6	yes	—
19	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
20	C₂×C₄×C₂₂₃	C₂³	1	6	yes	—
21	C₈×C₂₂₃	C₂³	1	6	yes	—
22	C₈×C₂₂₃	C₂³	1	2	yes	—
23	C₂²×C₂₂₃	—	2	6	no	—
24	C₂²×C₂₂₃	C₂²×C₄	1	6	yes	—
25	C₂²×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
26	C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
27	C₄×C₂₂₃	C₂²×C₄	1	6	yes	—
28	C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
29	C₂×C₂₂₃	—	2	6	no	—
30	C₂×C₂₂₃	#131	1	2	yes	direct product factor
31	C₂×C₂₂₃	C₂³×C₄	1	1	yes	characteristic
32	C₂₂₃	C₂×#131	1	1	yes	characteristic, direct product factor, Sylow
33	C₂×#131	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂×#41	C₂×C₂₂₃	1	1	yes	characteristic
35	#131	C₂×C₂₂₃	1	8	yes	direct product factor
36	C₂×M₁₆	C₂×C₂₂₃	1	3	yes	complemented
37	C₂²×C₈	C₂×C₂₂₃	1	3	yes	complemented
38	#41	C₄×C₂₂₃	1	4	yes	—
39	#41	C₂²×C₂₂₃	1	4	yes	—
40	C₂×Q₈	C₄×C₂₂₃	1	1	yes	characteristic
41	C₂×D₄	C₄×C₂₂₃	1	3	yes	—
42	C₂²×C₄	C₂²×C₂₂₃	1	3	yes	—
43	M₁₆	C₂²×C₂₂₃	1	12	yes	complemented
44	C₂×C₈	C₂²×C₂₂₃	1	3	yes	—
45	C₂×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
46	C₂×C₈	C₂²×C₂₂₃	1	12	yes	complemented
47	C₂³	C₂×C₄×C₂₂₃	1	3	yes	—
48	Q₈	C₂×C₄×C₂₂₃	1	4	yes	—
49	D₄	C₂×C₄×C₂₂₃	1	12	yes	—
50	C₂×C₄	C₂×C₄×C₂₂₃	1	3	yes	—
51	C₂×C₄	C₂×C₄×C₂₂₃	1	6	yes	—
52	C₂×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
53	C₂×C₄	C₂³×C₂₂₃	1	6	yes	—
54	C₈	C₂³×C₂₂₃	1	2	yes	—
55	C₈	C₂³×C₂₂₃	1	6	yes	—
56	C₂²	—	2	6	no	—
57	C₂²	C₂²×C₄×C₂₂₃	1	6	yes	—
58	C₂²	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
59	C₄	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
60	C₄	C₂²×C₄×C₂₂₃	1	6	yes	—
61	C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
62	C₂	—	2	6	no	—
63	C₂	#131×C₂₂₃	1	2	yes	direct product factor
64	C₂	C₂³×C₄×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₂₂₃	C₂²
commutator	C₂	C₂³×C₄×C₂₂₃
Frattini, cosocle	C₄	C₂⁴×C₂₂₃
socle	C₂²×C₂₂₃	C₂²×C₄
prime radical	C₂×#41×C₂₂₃	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. #131×C₂₂₃	C₂	direct (N × H)
3. C₂×M₁₆×C₂₂₃	C₂	semidirect
4. C₂²×C₈×C₂₂₃	C₂	semidirect
10. M₁₆×C₂₂₃	C₂²	semidirect
13. C₂×C₈×C₂₂₃	C₂²	semidirect
30. C₂×C₂₂₃	#131	direct (N × H)
32. C₂₂₃	C₂×#131	direct (N × H)
33. C₂×#131	C₂₂₃	direct (N × H)
35. #131	C₂×C₂₂₃	direct (N × H)
36. C₂×M₁₆	C₂×C₂₂₃	semidirect
37. C₂²×C₈	C₂×C₂₂₃	semidirect
43. M₁₆	C₂²×C₂₂₃	semidirect
46. C₂×C₈	C₂²×C₂₂₃	semidirect
63. C₂	#131×C₂₂₃	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×#41×C₂₂₃	2
#131×C₂₂₃	2
C₂×M₁₆×C₂₂₃	2
C₂²×C₈×C₂₂₃	2
C₂×#131	223

Subgroups — C2×#131×C223 #28383279

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#131×C₂₂₃ #28383279