Subgroups — #30636933

Order 16256.

50 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₂₇⋊C₃₂	C₂	1	1	yes	maximal, characteristic
2	#22549706	C₂	1	1	yes	maximal, characteristic
3	#368×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	#16861145	—	2	1	no	—
5	C₁₂₇⋊C₃₂	C₄	1	2	yes	—
6	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
7	#108×C₁₂₇	—	2	1	no	—
8	#14523232	—	4	1	no	—
9	M₁₆×C₁₂₇	—	4	1	no	—
10	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
11	C₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₁₂₇	Q₈	1	1	yes	characteristic
13	C₁₂₇⋊C₈	—	8	1	no	—
14	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
15	C₈×C₁₂₇	—	8	1	no	—
16	C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
17	C₈×C₁₂₇	Q₁₆	1	1	yes	characteristic
18	C₂²×C₁₂₇	Q₃₂	1	1	yes	characteristic
19	C₄×C₁₂₇	#107	1	1	yes	characteristic
20	C₄×C₁₂₇	D₁₆	1	1	yes	characteristic
21	C₂×C₁₂₇	—	2	1	no	—
22	C₂×C₁₂₇	#366	1	1	yes	characteristic
23	#1425	—	127	1	no	maximal, Sylow
24	C₁₂₇	#1425	1	1	yes	characteristic, complemented, Sylow
25	C₂×C₃₂	—	127	1	no	—
26	#368	—	127	1	no	—
27	#368	D₁₂₇	1	1	yes	characteristic
28	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
29	#108	—	2	1	no	—
30	#108	—	254	1	no	—
31	C₃₂	—	127	2	no	—
32	M₁₆	—	4	1	no	—
33	M₁₆	—	508	1	no	—
34	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
35	C₁₆	C₄×D₁₂₇	1	1	yes	characteristic
36	C₁₆	Dic₂₅₄	1	1	yes	characteristic
37	C₂×C₄	#14523233	1	1	yes	characteristic
38	C₈	—	8	1	no	—
39	C₈	—	1016	1	no	—
40	C₈	Dic₁₂₇⋊C₄	1	1	yes	characteristic
41	C₈	#14523238	1	1	yes	characteristic
42	C₂²	#16861155	1	1	yes	characteristic
43	C₄	#16861142	1	1	yes	characteristic
44	C₄	#16861150	1	1	yes	characteristic
45	C₂	—	2	1	no	—
46	C₂	#22549701	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16861150
commutator	C₁₆×C₁₂₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₂₇
Fitting	#368×C₁₂₇	C₂
socle	C₂×C₁₂₇	#366
prime radical	C₂²×C₁₂₇	Q₃₂
cosocle	C₂×C₁₆×C₁₂₇	C₂²
hypercenter	#368	D₁₂₇
hypocenter	C₁₂₇	#1425
LCS	G ▷ C₁₆×C₁₂₇ ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
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. C₁₂₇	#1425	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₂₇⋊C₃₂	2
#22549706	2
#368×C₁₂₇	2
#1425	127