Subgroups — #30636871

Order 16256.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550292	C₂	1	1	yes	maximal, characteristic
2	#22549710	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861215	C₄	1	1	yes	characteristic
5	#16861149	C₄	1	1	yes	characteristic
6	#16861148	—	2	1	no	—
7	C₁₆×D₁₂₇	—	2	1	no	—
8	C₁₂₇⋊C₃₂	—	2	1	no	—
9	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₃₂×C₁₂₇	—	2	1	no	—
11	#14523248	C₈	1	1	yes	characteristic
12	#14523232	C₈	1	1	yes	characteristic
13	#14523231	—	2	1	no	—
14	C₈×D₁₂₇	—	2	1	no	—
15	C₁₂₇⋊C₁₆	—	2	1	no	—
16	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
17	C₁₆×C₁₂₇	D₄	1	1	yes	characteristic
18	C₁₆×C₁₂₇	D₄	1	1	yes	characteristic
19	Dic₂₅₄	C₁₆	1	1	yes	characteristic
20	D₅₀₈	C₁₆	1	1	yes	characteristic
21	C₂²⋊D₁₂₇	—	2	1	no	—
22	C₄×D₁₂₇	—	2	1	no	—
23	C₁₂₇⋊C₈	—	2	1	no	—
24	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
25	C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₂₇	M₁₆	1	1	yes	characteristic
27	C₂²×C₁₂₇	M₃₂	1	1	yes	characteristic
28	C₂×D₁₂₇	—	2	1	no	—
29	C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
30	C₄×C₁₂₇	C₂×C₁₆	1	1	yes	characteristic
31	Dic₁₂₇	—	2	1	no	—
32	C₂×C₁₂₇	—	2	1	no	—
33	C₂×C₁₂₇	C₂²⋊C₁₆	1	1	yes	characteristic
34	D₁₂₇	—	4	1	no	—
35	#1401	—	127	1	no	maximal, Sylow
36	C₁₂₇	#1401	1	1	yes	characteristic, complemented, Sylow
37	#504	—	127	1	no	—
38	M₆₄	—	127	1	no	—
39	C₂×C₃₂	D₁₂₇	1	1	yes	characteristic, complemented
40	#131	—	127	1	no	—
41	M₃₂	—	127	1	no	—
42	M₃₂	—	254	1	no	—
43	C₂×C₁₆	—	254	1	no	—
44	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
45	C₃₂	—	2	1	no	—
46	C₃₂	—	254	1	no	—
47	#41	—	127	1	no	—
48	M₁₆	—	127	1	no	—
49	M₁₆	—	254	1	no	—
50	C₂×C₈	—	254	1	no	—
51	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
52	C₁₆	—	254	1	no	—
53	C₁₆	C₂²⋊D₁₂₇	1	1	yes	characteristic
54	C₁₆	D₅₀₈	1	1	yes	characteristic
55	Q₈	—	127	1	no	—
56	D₄	—	127	1	no	—
57	D₄	—	254	1	no	—
58	C₂×C₄	—	254	1	no	—
59	C₂×C₄	C₈×D₁₂₇	1	1	yes	characteristic
60	C₈	—	254	1	no	—
61	C₈	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
62	C₈	#14523231	1	1	yes	characteristic
63	C₂²	—	254	1	no	—
64	C₂²	#16861148	1	1	yes	characteristic
65	C₄	—	254	1	no	—
66	C₄	(C₂×D₁₂₇)⋊C₈	1	1	yes	characteristic
67	C₄	C₁₆×D₁₂₇	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	508	1	no	—
70	C₂	(C₂×D₁₂₇)⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	D₅₀₈
commutator	C₄×C₁₂₇	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₁₂₇
Fitting	C₂×C₃₂×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂²⋊C₁₆
prime radical	#14523248	C₈
cosocle	C₂×C₁₆×C₁₂₇	C₂²
hypercenter	C₂×C₃₂	D₁₂₇
hypocenter	C₁₂₇	#1401
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
2. #22549710	C₂	semidirect
3. C₂×C₃₂×C₁₂₇	C₂	semidirect
36. C₁₂₇	#1401	semidirect
39. C₂×C₃₂	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550292	2
#22549710	2
C₂×C₃₂×C₁₂₇	2
#1401	127