Subgroups — C₂²×C₄⋊C₄×C₂₂₃ #28383225

Order 14272.

418 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

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

Complements

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

N	H ≅ G/N	type
3. C₂×C₄⋊C₄×C₂₂₃	C₂	direct (N × H)
6. C₂²×C₄×C₂₂₃	C₄	semidirect
9. C₄⋊C₄×C₂₂₃	C₂²	direct (N × H)
15. C₂×C₄×C₂₂₃	C₂×C₄	semidirect
18. C₂²×C₂₂₃	C₄⋊C₄	direct (N × H)
24. C₄×C₂₂₃	C₂²×C₄	semidirect
25. C₂×C₂₂₃	C₂×C₄⋊C₄	direct (N × H)
29. C₂₂₃	C₂²×C₄⋊C₄	direct (N × H)
30. C₂²×C₄⋊C₄	C₂₂₃	direct (N × H)
33. C₂×C₄⋊C₄	C₂×C₂₂₃	direct (N × H)
36. C₂²×C₄	C₄×C₂₂₃	semidirect
39. C₄⋊C₄	C₂²×C₂₂₃	direct (N × H)
45. C₂×C₄	C₂×C₄×C₂₂₃	semidirect
48. C₂²	C₄⋊C₄×C₂₂₃	direct (N × H)
54. C₄	C₂²×C₄×C₂₂₃	semidirect
55. C₂	C₂×C₄⋊C₄×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
C₂³×C₄×C₂₂₃	2
C₂×C₄⋊C₄×C₂₂₃	2
C₂²×C₄⋊C₄	223

Subgroups — C22×C4⋊C4×C223 #28383225

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄⋊C₄×C₂₂₃ #28383225