Subgroups — C₄×C₄⋊Dic₂₂₃ #28383432

Order 14272.

194 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Dic₂₂₃	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₂₃	C₂	1	2	yes	maximal
3	C₂₂₃⋊#95	C₂	1	2	yes	maximal
4	C₂×C₄²×C₂₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₂₃	C₂²	1	4	yes	—
6	C₄⋊Dic₂₂₃	C₄	1	8	yes	direct product factor
7	C₄×Dic₂₂₃	—	2	4	no	—
8	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
11	C₄²×C₂₂₃	C₄	1	4	yes	complemented
12	C₂×Dic₂₂₃	—	2	4	no	—
13	C₂×Dic₂₂₃	—	2	4	no	—
14	C₂×Dic₂₂₃	C₂×C₄	1	8	yes	—
15	C₂³×C₂₂₃	C₂³	1	1	yes	characteristic
16	C₂×C₄×C₂₂₃	—	2	2	no	—
17	C₂×C₄×C₂₂₃	C₂×C₄	1	4	yes	—
18	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₂₂₃	D₄	1	2	yes	—
22	C₂×C₄×C₂₂₃	Q₈	1	2	yes	—
23	C₂²×C₂₂₃	C₂²×C₄	1	2	yes	—
24	C₂²×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
25	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
26	C₂²×C₂₂₃	C₂×Q₈	1	1	yes	characteristic
27	C₂²×C₂₂₃	#41	1	2	yes	—
28	C₄×C₂₂₃	—	2	2	no	—
29	C₄×C₂₂₃	C₄²	1	4	yes	complemented
30	C₄×C₂₂₃	C₄⋊C₄	1	4	yes	complemented
31	Dic₂₂₃	—	2	8	no	—
32	C₂×C₂₂₃	C₂×C₄²	1	1	yes	characteristic
33	C₂×C₂₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₂₂₃	#117	1	1	yes	characteristic
35	C₂×C₂₂₃	C₄×D₄	1	2	yes	—
36	C₂×C₂₂₃	C₄×Q₈	1	2	yes	—
37	C₂₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
38	C₄×C₄⋊C₄	—	223	1	no	maximal, Sylow
39	C₂×C₄⋊C₄	—	223	2	no	—
40	C₂×C₄²	—	223	2	no	—
41	C₂×C₄²	D₂₂₃	1	1	yes	characteristic
42	#95	—	223	2	no	—
43	C₂²×C₄	—	223	4	no	—
44	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
45	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
46	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
47	C₄⋊C₄	—	223	8	no	—
48	C₄²	—	446	4	no	—
49	C₄²	Dic₂₂₃	1	4	yes	complemented
50	C₂³	C₂²×D₂₂₃	1	1	yes	characteristic
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	446	4	no	—
53	C₂×C₄	—	446	4	no	—
54	C₂×C₄	—	223	8	no	—
55	C₂×C₄	C₄×D₂₂₃	1	4	yes	—
56	C₂×C₄	C₂×Dic₂₂₃	1	2	yes	—
57	C₂×C₄	C₂×Dic₂₂₃	1	2	yes	—
58	C₂×C₄	C₂×Dic₂₂₃	1	2	yes	—
59	C₂×C₄	D₈₉₂	1	2	yes	—
60	C₂×C₄	Dic₄₄₆	1	2	yes	—
61	C₂²	C₂×C₄×D₂₂₃	1	2	yes	—
62	C₂²	C₂²×Dic₂₂₃	1	1	yes	characteristic
63	C₂²	C₂×D₈₉₂	1	1	yes	characteristic
64	C₂²	C₂×Dic₄₄₆	1	1	yes	characteristic
65	C₂²	#16813170	1	2	yes	—
66	C₄	—	2	2	no	—
67	C₄	—	446	8	no	—
68	C₄	C₄×Dic₂₂₃	1	4	yes	complemented
69	C₄	C₄⋊Dic₂₂₃	1	4	yes	direct product factor
70	C₂	C₂×C₄×Dic₂₂₃	1	1	yes	characteristic
71	C₂	C₂×C₄⋊Dic₂₂₃	1	1	yes	characteristic
72	C₂	#20297263	1	1	yes	characteristic
73	C₂	C₄×D₈₉₂	1	2	yes	—
74	C₂	C₄×Dic₄₄₆	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₂₃
commutator	C₂×C₂₂₃	C₂×C₄²
Frattini	C₂³	C₂²×D₂₂₃
Fitting	C₂×C₄²×C₂₂₃	C₂
socle, prime radical, cosocle	C₂³×C₂₂₃	C₂³
hypercenter	C₂×C₄²	D₂₂₃
hypocenter	C₂₂₃	C₄×C₄⋊C₄
LCS	G ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
6. C₄⋊Dic₂₂₃	C₄	direct (N × H)
11. C₄²×C₂₂₃	C₄	semidirect
29. C₄×C₂₂₃	C₄²	semidirect
30. C₄×C₂₂₃	C₄⋊C₄	semidirect
37. C₂₂₃	C₄×C₄⋊C₄	semidirect
49. C₄²	Dic₂₂₃	semidirect
68. C₄	C₄×Dic₂₂₃	semidirect
69. C₄	C₄⋊Dic₂₂₃	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×C₄⋊Dic₂₂₃	2
C₂×C₄×Dic₂₂₃	2
C₂₂₃⋊#95	2
C₂×C₄²×C₂₂₃	2
C₄×C₄⋊C₄	223

Subgroups — C4×C4⋊Dic223 #28383432

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₄⋊Dic₂₂₃ #28383432