Subgroups — #30636796

Order 16256.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549627	C₂	1	2	yes	maximal
2	#346×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₁₂₇⋊C₈)	—	2	2	no	—
4	C₂×C₁₆×C₁₂₇	C₄	1	1	yes	characteristic
5	C₂×C₁₆×C₁₂₇	C₄	1	1	yes	characteristic
6	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₂×C₁₂₇⋊C₈	—	4	2	no	—
8	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
9	C₂×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
10	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
11	C₁₆×C₁₂₇	—	2	1	no	—
12	C₁₆×C₁₂₇	C₈	1	2	yes	complemented
13	C₁₂₇⋊C₈	—	8	2	no	—
14	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₄×C₁₂₇	SD₁₆	1	2	yes	—
16	C₈×C₁₂₇	—	2	1	no	—
17	C₈×C₁₂₇	C₂×C₈	1	1	yes	characteristic
18	C₈×C₁₂₇	M₁₆	1	1	yes	characteristic
19	C₂²×C₁₂₇	#106	1	1	yes	characteristic
20	C₄×C₁₂₇	—	2	2	no	—
21	C₄×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
22	C₄×C₁₂₇	#108	1	1	yes	characteristic
23	C₂×C₁₂₇	#334	1	1	yes	characteristic
24	C₂×C₁₂₇	#365	1	2	yes	—
25	#1368	—	127	1	no	maximal, Sylow
26	C₁₂₇	#1368	1	1	yes	characteristic, complemented, Sylow
27	#346	D₁₂₇	1	1	yes	characteristic
28	#334	—	127	2	no	—
29	C₂×C₁₆	Dic₁₂₇	1	1	yes	characteristic
30	C₂×C₁₆	Dic₁₂₇	1	1	yes	characteristic
31	C₄⋊C₈	—	254	2	no	—
32	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
33	C₂×C₈	—	508	2	no	—
34	C₂×C₈	C₂×Dic₁₂₇	1	1	yes	characteristic
35	C₂×C₈	Dic₂₅₄	1	1	yes	characteristic
36	C₄²	D₅₀₈	1	1	yes	characteristic
37	C₁₆	—	2	1	no	—
38	C₁₆	C₁₂₇⋊C₈	1	2	yes	complemented
39	C₂×C₄	C₄⋊Dic₁₂₇	1	1	yes	characteristic
40	C₂×C₄	#14523237	1	2	yes	—
41	C₈	—	2	1	no	—
42	C₈	—	1016	2	no	—
43	C₈	C₂×C₁₂₇⋊C₈	1	1	yes	characteristic
44	C₈	#14523232	1	1	yes	characteristic
45	C₂²	#16861141	1	1	yes	characteristic
46	C₄	—	2	2	no	—
47	C₄	C₄⋊(C₁₂₇⋊C₈)	1	1	yes	characteristic
48	C₄	#16861145	1	1	yes	characteristic
49	C₂	#22549627	1	1	yes	characteristic
50	C₂	#22549699	1	2	yes	—

	subgroup	G/H
center	C₂²	#16861141
commutator	C₈×C₁₂₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₂₇
Fitting	#346×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	#106
cosocle	C₄×C₈×C₁₂₇	C₂²
hypercenter	#346	D₁₂₇
hypocenter	C₁₂₇	#1368
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
12. C₁₆×C₁₂₇	C₈	semidirect
26. C₁₂₇	#1368	semidirect
38. C₁₆	C₁₂₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22549627	2
#346×C₁₂₇	2
#1368	127