Subgroups — #20297269

Order 7136.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₄₄₆	C₂	1	1	yes	maximal, characteristic
2	Dic₂₂₃⋊C₄	C₂	1	2	yes	maximal
3	C₄×Dic₂₂₃	C₂	1	2	yes	maximal
4	C₄×Dic₂₂₃	C₂	1	1	yes	maximal, characteristic
5	C₄⋊C₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic
6	Dic₄₄₆	C₄	1	4	yes	complemented
7	C₂×Dic₂₂₃	C₂²	1	1	yes	characteristic
8	C₂×Dic₂₂₃	C₂²	1	2	yes	—
9	C₂×Dic₂₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₂₂₃	C₂²	1	2	yes	—
11	C₂×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
12	C₂²×C₂₂₃	C₂³	1	1	yes	characteristic
13	C₄×C₂₂₃	—	2	2	no	—
14	C₄×C₂₂₃	C₂×C₄	1	2	yes	—
15	Dic₂₂₃	—	2	1	no	—
16	Dic₂₂₃	C₂×C₄	1	4	yes	—
17	Dic₂₂₃	Q₈	1	2	yes	complemented
18	C₂×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
19	C₂×C₂₂₃	C₂×Q₈	1	1	yes	characteristic
20	C₂×C₂₂₃	#41	1	1	yes	characteristic
21	C₂₂₃	C₄×Q₈	1	1	yes	characteristic, complemented, Sylow
22	C₄×Q₈	—	223	1	no	maximal, Sylow
23	C₂×Q₈	—	223	1	no	—
24	C₄⋊C₄	—	223	2	no	—
25	C₄⋊C₄	D₂₂₃	1	1	yes	characteristic
26	C₄²	—	223	1	no	—
27	C₄²	—	223	2	no	—
28	Q₈	—	223	4	no	—
29	C₂×C₄	—	223	1	no	—
30	C₂×C₄	—	223	1	no	—
31	C₂×C₄	—	223	2	no	—
32	C₂×C₄	C₂×D₂₂₃	1	2	yes	—
33	C₂×C₄	C₂×D₂₂₃	1	1	yes	characteristic
34	C₂²	C₂²×D₂₂₃	1	1	yes	characteristic
35	C₄	—	2	2	no	—
36	C₄	—	446	1	no	—
37	C₄	—	223	4	no	—
38	C₄	—	223	2	no	—
39	C₄	C₄×D₂₂₃	1	2	yes	—
40	C₂	C₂×C₄×D₂₂₃	1	1	yes	characteristic
41	C₂	Q₈×D₂₂₃	1	1	yes	characteristic
42	C₂	#16813169	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₂₂₃
commutator	C₂×C₂₂₃	C₂²×C₄
Fitting	C₄⋊C₄×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂³
hypercenter	C₄⋊C₄	D₂₂₃
hypocenter	C₂₂₃	C₄×Q₈
LCS	G ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
6. Dic₄₄₆	C₄	semidirect
17. Dic₂₂₃	Q₈	semidirect
21. C₂₂₃	C₄×Q₈	semidirect