Subgroups — #20297280

Order 7136.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₂₂₃	C₂	1	1	yes	maximal, characteristic
2	C₄⋊Dic₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₂₂₃⋊C₄	C₂	1	2	yes	maximal, complemented
4	(C₂×D₂₂₃)⋊C₄	C₂	1	2	yes	maximal
5	C₂×Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₂₃	C₂²	1	1	yes	characteristic
7	C₂×Dic₂₂₃	C₂²	1	2	yes	—
8	C₂×Dic₂₂₃	C₂²	1	1	yes	characteristic
9	C₄×D₂₂₃	—	2	2	no	—
10	Q₈×C₂₂₃	—	2	2	no	—
11	C₂×C₄×C₂₂₃	C₂²	1	2	yes	—
12	C₂×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
13	C₂²×C₂₂₃	C₂³	1	1	yes	characteristic
14	C₂×D₂₂₃	—	2	1	no	—
15	C₂×D₂₂₃	—	2	1	no	—
16	C₂×D₂₂₃	Q₈	1	2	yes	complemented
17	C₄×C₂₂₃	—	2	2	no	—
18	C₄×C₂₂₃	D₄	1	2	yes	—
19	Dic₂₂₃	—	2	1	no	—
20	Dic₂₂₃	—	2	2	no	—
21	C₂×C₂₂₃	C₂×D₄	1	1	yes	characteristic
22	C₂×C₂₂₃	C₂×Q₈	1	1	yes	characteristic
23	C₂×C₂₂₃	#41	1	1	yes	characteristic
24	D₂₂₃	—	2	2	no	—
25	C₂₂₃	C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
26	C₂²⋊Q₈	—	223	1	no	maximal, Sylow
27	C₂×Q₈	D₂₂₃	1	1	yes	characteristic, complemented
28	C₂²×C₄	—	223	1	no	—
29	C₄⋊C₄	—	223	2	no	—
30	C₄⋊C₄	—	223	1	no	—
31	C₂²⋊C₄	—	223	2	no	—
32	C₂³	—	223	1	no	—
33	Q₈	—	2	2	no	—
34	C₂×C₄	—	446	2	no	—
35	C₂×C₄	—	223	2	no	—
36	C₂×C₄	—	223	1	no	—
37	C₂×C₄	C₂×D₂₂₃	1	2	yes	—
38	C₂×C₄	C₂×D₂₂₃	1	1	yes	characteristic
39	C₂²	—	446	1	no	—
40	C₂²	—	446	1	no	—
41	C₂²	—	223	2	no	—
42	C₂²	C₂²×D₂₂₃	1	1	yes	characteristic
43	C₄	—	2	2	no	—
44	C₄	—	446	1	no	—
45	C₄	—	446	2	no	—
46	C₄	C₂²⋊D₂₂₃	1	2	yes	—
47	C₂	—	446	2	no	—
48	C₂	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
49	C₂	Q₈×D₂₂₃	1	1	yes	characteristic
50	C₂	#16813171	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₄⋊Dic₂₂₃	C₂	semidirect
3. Dic₂₂₃⋊C₄	C₂	semidirect
5. C₂×Q₈×C₂₂₃	C₂	semidirect
16. C₂×D₂₂₃	Q₈	semidirect
25. C₂₂₃	C₂²⋊Q₈	semidirect
27. C₂×Q₈	D₂₂₃	semidirect