Subgroups — C₂×C₄⋊C₈×C₂₂₃ #28383134

Order 14272.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×C₂₂₃	C₂	1	2	yes	maximal
2	C₂×C₄²×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×C₂₂₃	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₄×C₂₂₃	C₄	1	1	yes	characteristic
5	C₂²×C₄×C₂₂₃	C₄	1	1	yes	characteristic
6	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₈×C₂₂₃	—	2	4	no	—
8	C₂×C₈×C₂₂₃	C₂²	1	4	yes	—
9	C₄²×C₂₂₃	C₄	1	2	yes	—
10	C₄²×C₂₂₃	C₂²	1	2	yes	—
11	C₂³×C₂₂₃	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄×C₂₂₃	—	2	2	no	—
13	C₂×C₄×C₂₂₃	C₈	1	4	yes	complemented
14	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
15	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₂₂₃	D₄	1	2	yes	—
18	C₂×C₄×C₂₂₃	Q₈	1	2	yes	—
19	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
20	C₈×C₂₂₃	—	2	4	no	—
21	C₂²×C₂₂₃	C₄⋊C₄	1	2	yes	—
22	C₂²×C₂₂₃	C₂×C₈	1	2	yes	—
23	C₂²×C₂₂₃	M₁₆	1	2	yes	—
24	C₂²×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
25	C₄×C₂₂₃	—	2	2	no	—
26	C₄×C₂₂₃	C₄⋊C₄	1	2	yes	—
27	C₄×C₂₂₃	C₂×C₈	1	4	yes	complemented
28	C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
29	C₄×C₂₂₃	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₂₂₃	C₄⋊C₈	1	4	yes	direct product factor
31	C₂×C₂₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₂₂₃	C₂²×C₈	1	1	yes	characteristic
33	C₂×C₂₂₃	C₂×M₁₆	1	1	yes	characteristic
34	C₂₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic, direct product factor, Sylow
35	C₂×C₄⋊C₈	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂²×C₈	C₂×C₂₂₃	1	2	yes	—
37	C₂×C₄²	C₂×C₂₂₃	1	1	yes	characteristic
38	C₄⋊C₈	C₂×C₂₂₃	1	4	yes	direct product factor
39	C₂²×C₄	C₄×C₂₂₃	1	1	yes	characteristic
40	C₂²×C₄	C₄×C₂₂₃	1	1	yes	characteristic
41	C₂²×C₄	C₂²×C₂₂₃	1	1	yes	characteristic
42	C₂×C₈	—	2	4	no	—
43	C₂×C₈	C₂²×C₂₂₃	1	4	yes	—
44	C₄²	C₄×C₂₂₃	1	2	yes	—
45	C₄²	C₂²×C₂₂₃	1	2	yes	—
46	C₂³	C₂×C₄×C₂₂₃	1	1	yes	characteristic
47	C₂×C₄	—	2	2	no	—
48	C₂×C₄	C₈×C₂₂₃	1	4	yes	complemented
49	C₂×C₄	C₂×C₄×C₂₂₃	1	2	yes	—
50	C₂×C₄	C₂×C₄×C₂₂₃	1	1	yes	characteristic
51	C₂×C₄	C₂×C₄×C₂₂₃	1	2	yes	—
52	C₂×C₄	D₄×C₂₂₃	1	2	yes	—
53	C₂×C₄	Q₈×C₂₂₃	1	2	yes	—
54	C₂×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
55	C₈	—	2	4	no	—
56	C₂²	C₄⋊C₄×C₂₂₃	1	2	yes	—
57	C₂²	C₂×C₈×C₂₂₃	1	2	yes	—
58	C₂²	M₁₆×C₂₂₃	1	2	yes	—
59	C₂²	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₄	C₄⋊C₄×C₂₂₃	1	2	yes	—
62	C₄	C₂×C₈×C₂₂₃	1	4	yes	complemented
63	C₄	C₂×D₄×C₂₂₃	1	1	yes	characteristic
64	C₄	C₂×Q₈×C₂₂₃	1	1	yes	characteristic
65	C₂	C₄⋊C₈×C₂₂₃	1	4	yes	direct product factor
66	C₂	C₂×C₄⋊C₄×C₂₂₃	1	1	yes	characteristic
67	C₂	C₂²×C₈×C₂₂₃	1	1	yes	characteristic
68	C₂	C₂×M₁₆×C₂₂₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₄⋊C₈×C₂₂₃	C₂	direct (N × H)
13. C₂×C₄×C₂₂₃	C₈	semidirect
27. C₄×C₂₂₃	C₂×C₈	semidirect
30. C₂×C₂₂₃	C₄⋊C₈	direct (N × H)
34. C₂₂₃	C₂×C₄⋊C₈	direct (N × H)
35. C₂×C₄⋊C₈	C₂₂₃	direct (N × H)
38. C₄⋊C₈	C₂×C₂₂₃	direct (N × H)
48. C₂×C₄	C₈×C₂₂₃	semidirect
62. C₄	C₂×C₈×C₂₂₃	semidirect
65. C₂	C₄⋊C₈×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
C₂²×C₈×C₂₂₃	2
C₂×C₄²×C₂₂₃	2
C₄⋊C₈×C₂₂₃	2
C₂×C₄⋊C₈	223

Subgroups — C2×C4⋊C8×C223 #28383134

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊C₈×C₂₂₃ #28383134