Subgroups — C₂×2₋¹⁺⁴×C₂₂₃ #28383296

Order 14272.

794 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	2₋¹⁺⁴×C₂₂₃	C₂	1	16	yes	maximal, direct product factor
2	C₂×#41×C₂₂₃	C₂	1	10	yes	maximal, complemented
3	C₂²×Q₈×C₂₂₃	C₂	1	5	yes	maximal, complemented
4	#41×C₂₂₃	C₂²	1	80	yes	complemented
5	C₂×Q₈×C₂₂₃	C₂²	1	10	yes	—
6	C₂×Q₈×C₂₂₃	C₂²	1	40	yes	complemented
7	C₂×D₄×C₂₂₃	C₂²	1	10	yes	—
8	C₂²×C₄×C₂₂₃	C₂²	1	15	yes	—
9	C₂³×C₂₂₃	C₂³	1	5	yes	—
10	Q₈×C₂₂₃	C₂³	1	40	yes	—
11	D₄×C₂₂₃	C₂³	1	40	yes	—
12	C₂×C₄×C₂₂₃	C₂³	1	60	yes	—
13	C₂×C₄×C₂₂₃	C₂³	1	10	yes	—
14	C₂²×C₂₂₃	—	2	10	no	—
15	C₂²×C₂₂₃	C₂⁴	1	10	yes	—
16	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
17	C₄×C₂₂₃	C₂⁴	1	20	yes	—
18	C₂×C₂₂₃	—	2	10	no	—
19	C₂×C₂₂₃	2₋¹⁺⁴	1	2	yes	direct product factor
20	C₂×C₂₂₃	C₂⁵	1	1	yes	characteristic
21	C₂₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic, direct product factor, Sylow
22	C₂×2₋¹⁺⁴	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	2₋¹⁺⁴	C₂×C₂₂₃	1	16	yes	direct product factor
24	C₂×#41	C₂×C₂₂₃	1	10	yes	complemented
25	C₂²×Q₈	C₂×C₂₂₃	1	5	yes	complemented
26	#41	C₂²×C₂₂₃	1	80	yes	complemented
27	C₂×Q₈	C₂²×C₂₂₃	1	10	yes	—
28	C₂×Q₈	C₂²×C₂₂₃	1	40	yes	complemented
29	C₂×D₄	C₂²×C₂₂₃	1	10	yes	—
30	C₂²×C₄	C₂²×C₂₂₃	1	15	yes	—
31	C₂³	C₂³×C₂₂₃	1	5	yes	—
32	Q₈	C₂³×C₂₂₃	1	40	yes	—
33	D₄	C₂³×C₂₂₃	1	40	yes	—
34	C₂×C₄	C₂³×C₂₂₃	1	10	yes	—
35	C₂×C₄	C₂³×C₂₂₃	1	60	yes	—
36	C₂²	—	2	10	no	—
37	C₂²	C₂⁴×C₂₂₃	1	10	yes	—
38	C₂²	C₂⁴×C₂₂₃	1	1	yes	characteristic
39	C₄	C₂⁴×C₂₂₃	1	20	yes	—
40	C₂	—	2	10	no	—
41	C₂	2₋¹⁺⁴×C₂₂₃	1	2	yes	direct product factor
42	C₂	C₂⁵×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₂₃	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁵×C₂₂₃
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. 2₋¹⁺⁴×C₂₂₃	C₂	direct (N × H)
2. C₂×#41×C₂₂₃	C₂	semidirect
3. C₂²×Q₈×C₂₂₃	C₂	semidirect
4. #41×C₂₂₃	C₂²	semidirect
6. C₂×Q₈×C₂₂₃	C₂²	semidirect
19. C₂×C₂₂₃	2₋¹⁺⁴	direct (N × H)
21. C₂₂₃	C₂×2₋¹⁺⁴	direct (N × H)
22. C₂×2₋¹⁺⁴	C₂₂₃	direct (N × H)
23. 2₋¹⁺⁴	C₂×C₂₂₃	direct (N × H)
24. C₂×#41	C₂×C₂₂₃	semidirect
25. C₂²×Q₈	C₂×C₂₂₃	semidirect
26. #41	C₂²×C₂₂₃	semidirect
28. C₂×Q₈	C₂²×C₂₂₃	semidirect
41. C₂	2₋¹⁺⁴×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
2₋¹⁺⁴×C₂₂₃	2
C₂×#41×C₂₂₃	2
C₂²×Q₈×C₂₂₃	2
C₂×2₋¹⁺⁴	223

Subgroups — C2×2−1+4×C223 #28383296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×2₋¹⁺⁴×C₂₂₃ #28383296