Subgroups — #30636648

Order 16256.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₁₂₇⋊C₁₆	C₂	1	2	yes	maximal
2	C₈²×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₂₇⋊C₁₆	C₄	1	4	yes	—
4	C₄×C₈×C₁₂₇	C₄	1	1	yes	characteristic
5	C₄×C₈×C₁₂₇	C₄	1	1	yes	characteristic
6	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₁₂₇⋊C₁₆	—	2	4	no	—
8	C₂×C₈×C₁₂₇	C₈	1	2	yes	—
9	C₂×C₈×C₁₂₇	C₈	1	2	yes	—
10	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
11	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄×C₁₂₇	C₄²	1	1	yes	characteristic
13	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
14	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
15	C₈×C₁₂₇	—	2	2	no	—
16	C₈×C₁₂₇	C₁₆	1	4	yes	complemented
17	C₈×C₁₂₇	M₁₆	1	4	yes	—
18	C₂²×C₁₂₇	C₄×C₈	1	1	yes	characteristic
19	C₄×C₁₂₇	#97	1	2	yes	—
20	C₄×C₁₂₇	C₂×C₁₆	1	2	yes	—
21	C₄×C₁₂₇	M₃₂	1	2	yes	—
22	C₂×C₁₂₇	#322	1	1	yes	characteristic
23	C₂×C₁₂₇	C₄×C₁₆	1	1	yes	characteristic
24	C₂×C₁₂₇	#346	1	1	yes	characteristic
25	#1312	—	127	1	no	maximal, Sylow
26	C₁₂₇	#1312	1	1	yes	characteristic, complemented, Sylow
27	C₄×C₁₆	—	127	2	no	—
28	C₈²	D₁₂₇	1	1	yes	characteristic
29	C₂×C₁₆	—	127	4	no	—
30	C₄×C₈	Dic₁₂₇	1	1	yes	characteristic
31	C₄×C₈	Dic₁₂₇	1	1	yes	characteristic
32	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
33	C₂×C₈	C₁₂₇⋊C₈	1	2	yes	—
34	C₂×C₈	C₁₂₇⋊C₈	1	2	yes	—
35	C₂×C₈	C₄×D₁₂₇	1	2	yes	—
36	C₄²	C₂×Dic₁₂₇	1	1	yes	characteristic
37	C₁₆	—	254	4	no	—
38	C₂×C₄	C₄×Dic₁₂₇	1	1	yes	characteristic
39	C₂×C₄	C₂×C₁₂₇⋊C₈	1	1	yes	characteristic
40	C₂×C₄	C₂×C₁₂₇⋊C₈	1	1	yes	characteristic
41	C₈	—	2	2	no	—
42	C₈	C₁₂₇⋊C₁₆	1	4	yes	complemented
43	C₈	#14523231	1	4	yes	—
44	C₂²	C₄×C₁₂₇⋊C₈	1	1	yes	characteristic
45	C₄	#16861119	1	2	yes	—
46	C₄	C₂×C₁₂₇⋊C₁₆	1	2	yes	—
47	C₄	#16861149	1	2	yes	—
48	C₂	#22549597	1	1	yes	characteristic
49	C₂	C₄×C₁₂₇⋊C₁₆	1	1	yes	characteristic
50	C₂	#22549652	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄×C₈	C₂×D₁₂₇
commutator	C₂×C₁₂₇	C₄×C₁₆
Fitting	C₈²×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	C₄×C₈
cosocle	C₄×C₈×C₁₂₇	C₂²
hypercenter	C₈²	D₁₂₇
hypocenter	C₁₂₇	#1312
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄×C₈ ◁ C₈²

Complements

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

N	H ≅ G/N	type
16. C₈×C₁₂₇	C₁₆	semidirect
26. C₁₂₇	#1312	semidirect
42. C₈	C₁₂₇⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₁₂₇⋊C₁₆	2
C₈²×C₁₂₇	2
#1312	127