Subgroups — #30636935

Order 16256.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549710	C₂	1	1	yes	maximal, characteristic
2	#22549706	C₂	1	1	yes	maximal, characteristic
3	#367×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	#16861145	—	2	1	no	—
5	C₁₂₇⋊C₃₂	C₄	1	2	yes	complemented
6	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
7	#107×C₁₂₇	—	2	1	no	—
8	#14523232	—	4	1	no	—
9	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
10	C₄⋊C₄×C₁₂₇	—	4	1	no	—
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	—	8	1	no	—
15	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
16	C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
17	C₈×C₁₂₇	Q₁₆	1	1	yes	characteristic
18	C₂²×C₁₂₇	SD₃₂	1	1	yes	characteristic
19	C₄×C₁₂₇	—	16	1	no	—
20	C₄×C₁₂₇	#107	1	1	yes	characteristic
21	C₄×C₁₂₇	SD₃₂	1	1	yes	characteristic
22	C₂×C₁₂₇	—	2	1	no	—
23	C₂×C₁₂₇	#367	1	1	yes	characteristic
24	#1426	—	127	1	no	maximal, Sylow
25	C₁₂₇	#1426	1	1	yes	characteristic, complemented, Sylow
26	M₆₄	—	127	1	no	—
27	#368	—	127	1	no	—
28	#367	D₁₂₇	1	1	yes	characteristic
29	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
30	#108	—	254	1	no	—
31	#107	—	2	1	no	—
32	C₃₂	—	127	2	no	—
33	M₁₆	—	508	1	no	—
34	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
35	C₄⋊C₄	—	4	1	no	—
36	C₁₆	C₄×D₁₂₇	1	1	yes	characteristic
37	C₁₆	Dic₂₅₄	1	1	yes	characteristic
38	C₂×C₄	—	8	1	no	—
39	C₂×C₄	#14523233	1	1	yes	characteristic
40	C₈	—	1016	1	no	—
41	C₈	Dic₁₂₇⋊C₄	1	1	yes	characteristic
42	C₈	#14523238	1	1	yes	characteristic
43	C₂²	#16861153	1	1	yes	characteristic
44	C₄	—	16	1	no	—
45	C₄	#16861142	1	1	yes	characteristic
46	C₄	#16861152	1	1	yes	characteristic
47	C₂	—	2	1	no	—
48	C₂	#22549703	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22549703
commutator	C₁₆×C₁₂₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₂₇
Fitting	#367×C₁₂₇	C₂
socle	C₂×C₁₂₇	#367
prime radical	C₂²×C₁₂₇	SD₃₂
cosocle	C₂×C₁₆×C₁₂₇	C₂²
hypercenter	#367	D₁₂₇
hypocenter	C₁₂₇	#1426
LCS	G ▷ C₁₆×C₁₂₇ ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #367

Complements

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

N	H ≅ G/N	type
5. C₁₂₇⋊C₃₂	C₄	semidirect
25. C₁₂₇	#1426	semidirect

Maximal subgroups

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

subgroup	index
#22549710	2
#22549706	2
#367×C₁₂₇	2
#1426	127