Subgroups — #22558412

Order 8576.

50 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₇⋊C₃₂	C₂	1	1	yes	maximal, characteristic
2	#16863226	C₂	1	1	yes	maximal, characteristic
3	#368×C₆₇	C₂	1	1	yes	maximal, characteristic
4	#14523979	—	2	1	no	—
5	C₆₇⋊C₃₂	C₄	1	2	yes	—
6	C₂×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
7	#108×C₆₇	—	2	1	no	—
8	#11761612	—	4	1	no	—
9	M₁₆×C₆₇	—	4	1	no	—
10	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
11	C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₆₇	Q₈	1	1	yes	characteristic
13	C₂×C₄×C₆₇	D₈	1	1	yes	characteristic
14	C₈×C₆₇	—	8	1	no	—
15	C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
16	C₈×C₆₇	Q₁₆	1	1	yes	characteristic
17	C₆₇⋊C₈	—	8	1	no	—
18	C₂²×C₆₇	Q₃₂	1	1	yes	characteristic
19	C₄×C₆₇	#107	1	1	yes	characteristic
20	C₄×C₆₇	D₁₆	1	1	yes	characteristic
21	C₂×C₆₇	—	2	1	no	—
22	C₂×C₆₇	#366	1	1	yes	characteristic
23	#1425	—	67	1	no	maximal, Sylow
24	C₆₇	#1425	1	1	yes	characteristic, complemented, Sylow
25	C₂×C₃₂	—	67	1	no	—
26	#368	—	67	1	no	—
27	#368	D₆₇	1	1	yes	characteristic
28	C₂×C₁₆	C₂×D₆₇	1	1	yes	characteristic
29	#108	—	2	1	no	—
30	#108	—	134	1	no	—
31	C₃₂	—	67	2	no	—
32	M₁₆	—	4	1	no	—
33	M₁₆	—	268	1	no	—
34	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
35	C₁₆	Dic₁₃₄	1	1	yes	characteristic
36	C₁₆	C₄×D₆₇	1	1	yes	characteristic
37	C₂×C₄	#11761613	1	1	yes	characteristic
38	C₈	—	8	1	no	—
39	C₈	—	536	1	no	—
40	C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
41	C₈	#11761618	1	1	yes	characteristic
42	C₂²	#14523989	1	1	yes	characteristic
43	C₄	#14523976	1	1	yes	characteristic
44	C₄	#14523984	1	1	yes	characteristic
45	C₂	—	2	1	no	—
46	C₂	#16863221	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#14523984
commutator	C₁₆×C₆₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₆₇
Fitting	#368×C₆₇	C₂
socle	C₂×C₆₇	#366
hypercenter	#368	D₆₇
hypocenter	C₆₇	#1425
LCS	G ▷ C₁₆×C₆₇ ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #368

Complements

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

N	H ≅ G/N	type
24. C₆₇	#1425	semidirect