Subgroups — #30636649

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	1	yes	maximal, characteristic
2	C₈×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₂₇⋊C₁₆	C₄	1	2	yes	—
5	C₄×C₁₂₇⋊C₈	C₄	1	1	yes	characteristic
6	C₄×C₁₂₇⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₁₂₇	C₄	1	2	yes	—
8	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₂×C₁₂₇⋊C₈	C₈	1	2	yes	—
10	C₂×C₁₂₇⋊C₈	C₈	1	2	yes	—
11	C₁₂₇⋊C₁₆	—	2	2	no	—
12	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
14	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
15	C₁₆×C₁₂₇	—	2	2	no	—
16	C₁₂₇⋊C₈	—	2	2	no	—
17	C₁₂₇⋊C₈	C₁₆	1	4	yes	complemented
18	C₂×C₄×C₁₂₇	C₄²	1	1	yes	characteristic
19	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
20	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
21	C₈×C₁₂₇	M₁₆	1	2	yes	—
22	C₈×C₁₂₇	M₁₆	1	2	yes	—
23	C₂²×C₁₂₇	C₄×C₈	1	1	yes	characteristic
24	C₄×C₁₂₇	#97	1	1	yes	characteristic
25	C₄×C₁₂₇	#97	1	1	yes	characteristic
26	C₄×C₁₂₇	C₂×C₁₆	1	2	yes	—
27	C₄×C₁₂₇	M₃₂	1	2	yes	—
28	C₂×C₁₂₇	#322	1	1	yes	characteristic
29	C₂×C₁₂₇	C₄×C₁₆	1	1	yes	characteristic
30	C₂×C₁₂₇	#346	1	1	yes	characteristic
31	#1312	—	127	1	no	maximal, Sylow
32	C₁₂₇	#1312	1	1	yes	characteristic, complemented, Sylow
33	C₄×C₁₆	—	127	1	no	—
34	C₄×C₁₆	D₁₂₇	1	1	yes	characteristic
35	C₈²	—	127	1	no	—
36	C₂×C₁₆	—	127	2	no	—
37	C₂×C₁₆	Dic₁₂₇	1	2	yes	—
38	C₄×C₈	—	127	1	no	—
39	C₄×C₈	—	127	1	no	—
40	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
41	C₂×C₈	—	127	2	no	—
42	C₂×C₈	—	127	2	no	—
43	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
44	C₂×C₈	C₂×Dic₁₂₇	1	1	yes	characteristic
45	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
46	C₁₆	—	2	2	no	—
47	C₁₆	—	254	2	no	—
48	C₂×C₄	C₄×Dic₁₂₇	1	1	yes	characteristic
49	C₂×C₄	C₈×D₁₂₇	1	1	yes	characteristic
50	C₂×C₄	C₈×D₁₂₇	1	1	yes	characteristic
51	C₈	—	127	4	no	—
52	C₈	—	254	2	no	—
53	C₈	#14523231	1	2	yes	—
54	C₈	#14523232	1	2	yes	—
55	C₂²	C₈×Dic₁₂₇	1	1	yes	characteristic
56	C₄	#16861119	1	1	yes	characteristic
57	C₄	#16861120	1	1	yes	characteristic
58	C₄	C₁₆×D₁₂₇	1	2	yes	—
59	C₄	#16861148	1	2	yes	—
60	C₂	#22549598	1	1	yes	characteristic
61	C₂	C₁₆×Dic₁₂₇	1	1	yes	characteristic
62	C₂	#22549651	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
17. C₁₂₇⋊C₈	C₁₆	semidirect
32. C₁₂₇	#1312	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
C₈×C₁₂₇⋊C₈	2
C₄×C₁₆×C₁₂₇	2
#1312	127