Subgroups — #30636652

Order 16256.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549652	C₂	1	1	yes	maximal, characteristic
2	#22549597	C₂	1	1	yes	maximal, characteristic
3	#346×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₂₇⋊C₁₆	C₄	1	2	yes	—
5	C₄×C₁₂₇⋊C₈	C₄	1	2	yes	—
6	C₂×C₁₆×C₁₂₇	C₄	1	2	yes	—
7	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₂×C₁₂₇⋊C₈	—	2	2	no	—
9	C₁₂₇⋊C₁₆	C₈	1	4	yes	complemented
10	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
11	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
12	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
13	C₁₆×C₁₂₇	—	2	2	no	—
14	C₁₂₇⋊C₈	—	4	2	no	—
15	C₂×C₄×C₁₂₇	C₄²	1	1	yes	characteristic
16	C₂×C₄×C₁₂₇	M₁₆	1	2	yes	—
17	C₈×C₁₂₇	C₂×C₈	1	2	yes	—
18	C₈×C₁₂₇	M₁₆	1	2	yes	—
19	C₂²×C₁₂₇	#97	1	1	yes	characteristic
20	C₄×C₁₂₇	—	2	2	no	—
21	C₄×C₁₂₇	C₄×C₈	1	1	yes	characteristic
22	C₄×C₁₂₇	#97	1	1	yes	characteristic
23	C₂×C₁₂₇	#322	1	1	yes	characteristic
24	C₂×C₁₂₇	#347	1	2	yes	—
25	#1313	—	127	1	no	maximal, Sylow
26	C₁₂₇	#1313	1	1	yes	characteristic, complemented, Sylow
27	#346	—	127	1	no	—
28	#346	D₁₂₇	1	1	yes	characteristic
29	#322	—	127	1	no	—
30	C₂×C₁₆	—	127	2	no	—
31	C₂×C₁₆	Dic₁₂₇	1	2	yes	—
32	C₄×C₈	—	127	2	no	—
33	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
34	C₂×C₈	—	254	2	no	—
35	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
36	C₂×C₈	C₂×Dic₁₂₇	1	1	yes	characteristic
37	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
38	C₁₆	—	2	2	no	—
39	C₁₆	—	127	4	no	—
40	C₂×C₄	C₄×Dic₁₂₇	1	1	yes	characteristic
41	C₂×C₄	#14523231	1	2	yes	—
42	C₈	—	508	2	no	—
43	C₈	C₈×D₁₂₇	1	2	yes	—
44	C₈	#14523232	1	2	yes	—
45	C₂²	#16861119	1	1	yes	characteristic
46	C₄	—	2	2	no	—
47	C₄	C₈×Dic₁₂₇	1	1	yes	characteristic
48	C₄	#16861120	1	1	yes	characteristic
49	C₂	#22549598	1	1	yes	characteristic
50	C₂	#22549653	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄×Dic₁₂₇
commutator	C₄×C₁₂₇	C₄×C₈
Frattini	C₄×C₈	C₂×D₁₂₇
Fitting	#346×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	#97
cosocle	C₄×C₈×C₁₂₇	C₂²
hypercenter	#346	D₁₂₇
hypocenter	C₁₂₇	#1313
LCS	G ▷ C₄×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ #346

Complements

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

N	H ≅ G/N	type
9. C₁₂₇⋊C₁₆	C₈	semidirect
26. C₁₂₇	#1313	semidirect

Maximal subgroups

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

subgroup	index
#22549652	2
#22549597	2
#346×C₁₂₇	2
#1313	127