Subgroups — #28384013

Order 14272.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297327	C₂	1	2	yes	maximal, complemented
2	#20297247	C₂	1	2	yes	maximal, complemented
3	D₃₅₆₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₆×D₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₃₂×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#16813171	—	2	2	no	—
7	#16813157	—	2	2	no	—
8	D₁₇₈₄	C₂²	1	2	yes	—
9	C₈×D₂₂₃	C₂²	1	1	yes	characteristic
10	C₂₂₃⋊C₁₆	C₂²	1	1	yes	characteristic
11	Q₁₆×C₂₂₃	C₂²	1	2	yes	—
12	C₁₆×C₂₂₃	C₂²	1	1	yes	characteristic
13	D₈₉₂	—	2	2	no	—
14	D₈₉₂	—	4	2	no	—
15	C₄×D₂₂₃	—	4	2	no	—
16	C₄×D₂₂₃	D₄	1	1	yes	characteristic
17	C₂₂₃⋊C₈	D₄	1	1	yes	characteristic
18	Q₈×C₂₂₃	—	2	2	no	—
19	C₈×C₂₂₃	C₂³	1	1	yes	characteristic
20	C₂×D₂₂₃	—	4	2	no	—
21	C₂×D₂₂₃	D₈	1	1	yes	characteristic
22	C₄×C₂₂₃	—	4	2	no	—
23	C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
24	Dic₂₂₃	D₈	1	1	yes	characteristic
25	C₂×C₂₂₃	C₂×D₈	1	1	yes	characteristic
26	D₂₂₃	—	2	1	no	—
27	D₂₂₃	—	8	2	no	—
28	C₂₂₃	#508	1	1	yes	characteristic, complemented, Sylow
29	#508	—	223	1	no	maximal, Sylow
30	#135	—	223	2	no	—
31	Q₃₂	D₂₂₃	1	1	yes	characteristic, complemented
32	SD₃₂	—	223	2	no	—
33	D₁₆	—	223	1	no	—
34	C₂×C₁₆	—	223	1	no	—
35	#41	—	446	2	no	—
36	Q₁₆	C₂×D₂₂₃	1	2	yes	—
37	SD₁₆	—	446	2	no	—
38	D₈	—	223	2	no	—
39	C₂×C₈	—	223	1	no	—
40	C₁₆	—	223	1	no	—
41	C₁₆	C₂×D₂₂₃	1	1	yes	characteristic
42	Q₈	—	2	2	no	—
43	D₄	—	892	2	no	—
44	D₄	—	446	2	no	—
45	C₂×C₄	—	892	2	no	—
46	C₂×C₄	—	223	1	no	—
47	C₈	—	223	1	no	—
48	C₈	C₂²×D₂₂₃	1	1	yes	characteristic
49	C₂²	—	892	2	no	—
50	C₂²	—	223	1	no	—
51	C₄	—	4	2	no	—
52	C₄	—	223	1	no	—
53	C₄	D₄×D₂₂₃	1	1	yes	characteristic
54	C₂	—	446	1	no	—
55	C₂	—	1784	2	no	—
56	C₂	D₈×D₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×D₂₂₃
commutator, cosocle	C₈×C₂₂₃	C₂³
Frattini	C₈	C₂²×D₂₂₃
Fitting	Q₃₂×C₂₂₃	C₂
socle	C₂×C₂₂₃	C₂×D₈
hypercenter	Q₃₂	D₂₂₃
hypocenter	C₂₂₃	#508
LCS	G ▷ C₈×C₂₂₃ ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
1. #20297327	C₂	semidirect
2. #20297247	C₂	semidirect
3. D₃₅₆₈	C₂	semidirect
4. C₁₆×D₂₂₃	C₂	semidirect
5. Q₃₂×C₂₂₃	C₂	semidirect
28. C₂₂₃	#508	semidirect
31. Q₃₂	D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#20297327	2
#20297247	2
D₃₅₆₈	2
C₁₆×D₂₂₃	2
Q₃₂×C₂₂₃	2
#508	223