Subgroups — #20297262

Order 7136.

76 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	(C₂×D₂₂₃)⋊C₄	C₂	1	2	yes	maximal
4	C₄×Dic₂₂₃	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₄×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	C₄	1	4	yes	complemented
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	C₂×D₂₂₃	—	2	1	no	—
14	C₂×D₂₂₃	—	2	1	no	—
15	C₂×D₂₂₃	C₂×C₄	1	2	yes	—
16	C₄×C₂₂₃	—	2	2	no	—
17	C₄×C₂₂₃	C₂×C₄	1	2	yes	—
18	Dic₂₂₃	—	2	2	no	—
19	Dic₂₂₃	C₂×C₄	1	2	yes	—
20	C₂×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
21	C₂×C₂₂₃	#41	1	1	yes	characteristic
22	C₂×C₂₂₃	#41	1	1	yes	characteristic
23	D₂₂₃	—	2	2	no	—
24	C₂₂₃	#117	1	1	yes	characteristic, complemented, Sylow
25	#117	—	223	1	no	maximal, Sylow
26	C₂²×C₄	—	223	1	no	—
27	C₄⋊C₄	—	223	1	no	—
28	C₄⋊C₄	D₂₂₃	1	1	yes	characteristic, complemented
29	C₂²⋊C₄	—	223	2	no	—
30	C₄²	—	223	2	no	—
31	C₂³	—	223	1	no	—
32	C₂×C₄	—	223	2	no	—
33	C₂×C₄	—	223	1	no	—
34	C₂×C₄	—	223	4	no	—
35	C₂×C₄	C₂×D₂₂₃	1	2	yes	—
36	C₂×C₄	C₂×D₂₂₃	1	1	yes	characteristic
37	C₂²	—	446	1	no	—
38	C₂²	—	446	1	no	—
39	C₂²	—	223	2	no	—
40	C₂²	C₂²×D₂₂₃	1	1	yes	characteristic
41	C₄	—	2	2	no	—
42	C₄	—	446	2	no	—
43	C₄	—	223	2	no	—
44	C₄	C₄×D₂₂₃	1	2	yes	—
45	C₂	—	446	2	no	—
46	C₂	C₂×C₄×D₂₂₃	1	1	yes	characteristic
47	C₂	#16813169	1	1	yes	characteristic
48	C₂	#16813171	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	#117
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
2. C₄⋊Dic₂₂₃	C₂	semidirect
4. C₄×Dic₂₂₃	C₂	semidirect
5. C₄⋊C₄×C₂₂₃	C₂	semidirect
9. C₄×D₂₂₃	C₄	semidirect
24. C₂₂₃	#117	semidirect
28. C₄⋊C₄	D₂₂₃	semidirect