Subgroups — C₂×#20297298 #28384105

Order 14272.

246 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297298	C₂	1	8	yes	maximal, direct product factor
2	C₂×Dic₂₂₃⋊C₄	C₂	1	3	yes	maximal, complemented
3	C₂×(C₂×D₂₂₃)⋊C₄	C₂	1	3	yes	maximal
4	C₂×C₄²×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₂₂₃	C₂²	1	1	yes	characteristic
6	C₂²×Dic₂₂₃	C₂²	1	3	yes	—
7	Dic₂₂₃⋊C₄	C₂²	1	12	yes	complemented
8	(C₂×D₂₂₃)⋊C₄	C₂²	1	12	yes	—
9	C₂²×C₄×C₂₂₃	C₂²	1	3	yes	—
10	C₄²×C₂₂₃	C₂²	1	4	yes	complemented
11	C₂²×D₂₂₃	—	2	6	no	—
12	C₂²×D₂₂₃	C₂³	1	2	yes	—
13	C₂×Dic₂₂₃	—	2	6	no	—
14	C₂×Dic₂₂₃	C₂³	1	6	yes	—
15	C₂³×C₂₂₃	C₂³	1	1	yes	characteristic
16	C₂×C₄×C₂₂₃	—	2	6	no	—
17	C₂×C₄×C₂₂₃	C₂³	1	6	yes	—
18	C₂²×C₂₂₃	#41	1	6	yes	—
19	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
20	C₂×D₂₂₃	—	2	6	no	—
21	C₂×D₂₂₃	—	4	4	no	—
22	C₄×C₂₂₃	—	2	6	no	—
23	Dic₂₂₃	—	2	6	no	—
24	C₂×C₂₂₃	#126	1	4	yes	complemented
25	C₂×C₂₂₃	C₂×#41	1	3	yes	—
26	D₂₂₃	—	4	2	no	—
27	C₂₂₃	C₂×#126	1	1	yes	characteristic, complemented, Sylow
28	C₂×#126	—	223	1	no	maximal, Sylow
29	#126	—	223	8	no	—
30	C₂×C₄⋊C₄	—	223	3	no	—
31	C₂×C₂²⋊C₄	—	223	3	no	—
32	C₂×C₄²	D₂₂₃	1	1	yes	characteristic, complemented
33	C₂⁴	—	223	1	no	—
34	C₂²×C₄	—	223	3	no	—
35	C₂²×C₄	C₂×D₂₂₃	1	3	yes	—
36	C₄⋊C₄	—	223	12	no	—
37	C₂²⋊C₄	—	223	12	no	—
38	C₄²	C₂×D₂₂₃	1	4	yes	complemented
39	C₂³	—	446	6	no	—
40	C₂³	—	223	2	no	—
41	C₂³	C₂²×D₂₂₃	1	1	yes	characteristic
42	C₂×C₄	—	2	6	no	—
43	C₂×C₄	—	446	6	no	—
44	C₂×C₄	—	223	6	no	—
45	C₂×C₄	C₂²×D₂₂₃	1	6	yes	—
46	C₂²	—	892	4	no	—
47	C₂²	—	446	6	no	—
48	C₂²	#16813170	1	6	yes	—
49	C₂²	C₂³×D₂₂₃	1	1	yes	characteristic
50	C₄	—	2	6	no	—
51	C₄	—	446	6	no	—
52	C₂	—	892	2	no	—
53	C₂	#20297298	1	4	yes	direct product factor
54	C₂	C₂×#16813170	1	3	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₂₂₃
commutator, cosocle	C₂²×C₂₂₃	C₂⁴
Frattini	C₂²	C₂³×D₂₂₃
Fitting	C₂×C₄²×C₂₂₃	C₂
socle	C₂³×C₂₂₃	C₂³
prime radical	C₂³×D₂₂₃	C₂²
hypercenter	C₂×C₄²	D₂₂₃
hypocenter	C₂₂₃	C₂×#126
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
1. #20297298	C₂	direct (N × H)
2. C₂×Dic₂₂₃⋊C₄	C₂	semidirect
4. C₂×C₄²×C₂₂₃	C₂	semidirect
7. Dic₂₂₃⋊C₄	C₂²	semidirect
10. C₄²×C₂₂₃	C₂²	semidirect
24. C₂×C₂₂₃	#126	semidirect
27. C₂₂₃	C₂×#126	semidirect
32. C₂×C₄²	D₂₂₃	semidirect
38. C₄²	C₂×D₂₂₃	semidirect
53. C₂	#20297298	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20297298	2
C₂×Dic₂₂₃⋊C₄	2
C₂×(C₂×D₂₂₃)⋊C₄	2
C₂×C₄²×C₂₂₃	2
C₂×#126	223

Subgroups — C2×#20297298 #28384105

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20297298 #28384105