Subgroups — #22558409

Order 8576.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863222	C₂	1	2	yes	maximal
2	C₂×C₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
3	#14523977	—	2	2	no	—
4	C₂×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
5	C₃₂×C₆₇	C₄	1	2	yes	complemented
6	C₄⋊Dic₆₇	—	4	2	no	—
7	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
8	C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
9	C₁₆×C₆₇	Q₈	1	1	yes	characteristic
10	C₂×C₄×C₆₇	D₈	1	1	yes	characteristic
11	C₂×Dic₆₇	—	8	2	no	—
12	C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
13	C₈×C₆₇	Q₁₆	1	1	yes	characteristic
14	C₂²×C₆₇	D₁₆	1	1	yes	characteristic
15	C₄×C₆₇	#107	1	1	yes	characteristic
16	C₄×C₆₇	Q₃₂	1	1	yes	characteristic
17	Dic₆₇	—	16	2	no	—
18	C₂×C₆₇	#366	1	1	yes	characteristic
19	C₂×C₆₇	D₃₂	1	1	yes	characteristic
20	C₂×C₆₇	Q₆₄	1	1	yes	characteristic
21	#1423	—	67	1	no	maximal, Sylow
22	C₆₇	#1423	1	1	yes	characteristic, complemented, Sylow
23	C₂×C₃₂	D₆₇	1	1	yes	characteristic
24	#366	—	67	2	no	—
25	C₂×C₁₆	C₂×D₆₇	1	1	yes	characteristic
26	#107	—	134	2	no	—
27	C₃₂	Dic₆₇	1	2	yes	complemented
28	C₂×C₈	D₂₆₈	1	1	yes	characteristic
29	C₄⋊C₄	—	268	2	no	—
30	C₁₆	Dic₁₃₄	1	1	yes	characteristic
31	C₁₆	C₂×Dic₆₇	1	1	yes	characteristic
32	C₂×C₄	—	536	2	no	—
33	C₂×C₄	D₅₃₆	1	1	yes	characteristic
34	C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
35	C₈	Dic₂₆₈	1	1	yes	characteristic
36	C₂²	D₁₀₇₂	1	1	yes	characteristic
37	C₄	—	1072	2	no	—
38	C₄	#14523977	1	1	yes	characteristic
39	C₄	Dic₅₃₆	1	1	yes	characteristic
40	C₂	#16863222	1	1	yes	characteristic
41	C₂	D₂₁₄₄	1	1	yes	characteristic
42	C₂	Dic₁₀₇₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₀₇₂
commutator	C₁₆×C₆₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₆₇
Fitting	C₂×C₃₂×C₆₇	C₂
socle	C₂²×C₆₇	D₁₆
hypercenter	C₂×C₃₂	D₆₇
hypocenter	C₆₇	#1423
LCS	G ▷ C₁₆×C₆₇ ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
5. C₃₂×C₆₇	C₄	semidirect
22. C₆₇	#1423	semidirect
27. C₃₂	Dic₆₇	semidirect