Subgroups — C₂×#22549709 #30641021

Order 16256.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₆×D₁₂₇	C₂	1	1	yes	maximal, characteristic
2	#22549709	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₁₂₇⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₈×D₁₂₇	C₄	1	1	yes	characteristic
6	C₂×C₁₂₇⋊C₁₆	C₄	1	1	yes	characteristic
7	C₁₆×D₁₂₇	C₄	1	2	yes	—
8	C₁₆×D₁₂₇	C₂²	1	2	yes	—
9	C₁₂₇⋊C₃₂	C₂²	1	2	yes	complemented
10	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₃₂×C₁₂₇	C₂²	1	2	yes	complemented
12	C₂×C₄×D₁₂₇	C₈	1	1	yes	characteristic
13	C₂×C₁₂₇⋊C₈	C₈	1	1	yes	characteristic
14	C₈×D₁₂₇	C₈	1	2	yes	—
15	C₈×D₁₂₇	C₂×C₄	1	2	yes	—
16	C₁₂₇⋊C₁₆	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
19	C₁₆×C₁₂₇	C₂³	1	1	yes	characteristic
20	C₂²×D₁₂₇	C₁₆	1	1	yes	characteristic
21	C₂×Dic₁₂₇	C₁₆	1	1	yes	characteristic
22	C₄×D₁₂₇	C₁₆	1	2	yes	—
23	C₄×D₁₂₇	C₂×C₈	1	2	yes	—
24	C₁₂₇⋊C₈	C₂×C₈	1	2	yes	—
25	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
26	C₈×C₁₂₇	C₂×C₈	1	1	yes	characteristic
27	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₁₂₇	C₂×C₁₆	1	1	yes	characteristic
29	C₂×D₁₂₇	—	2	2	no	—
30	C₂×D₁₂₇	C₂×C₁₆	1	2	yes	—
31	C₄×C₁₂₇	C₂×C₁₆	1	1	yes	characteristic
32	C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
33	Dic₁₂₇	C₂×C₁₆	1	2	yes	—
34	C₂×C₁₂₇	M₆₄	1	2	yes	complemented
35	C₂×C₁₂₇	C₂²×C₁₆	1	1	yes	characteristic
36	D₁₂₇	—	2	2	no	—
37	C₂×M₆₄	—	127	1	no	maximal, Sylow
38	C₁₂₇	C₂×M₆₄	1	1	yes	characteristic, complemented, Sylow
39	C₂²×C₁₆	—	127	1	no	—
40	M₆₄	—	127	4	no	—
41	C₂×C₃₂	—	127	1	no	—
42	C₂×C₃₂	D₁₂₇	1	1	yes	characteristic, complemented
43	C₂²×C₈	—	127	1	no	—
44	C₂×C₁₆	—	127	1	no	—
45	C₂×C₁₆	—	127	2	no	—
46	C₂×C₁₆	—	127	2	no	—
47	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
48	C₃₂	—	127	2	no	—
49	C₃₂	C₂×D₁₂₇	1	2	yes	complemented
50	C₂²×C₄	—	127	1	no	—
51	C₂×C₈	—	127	1	no	—
52	C₂×C₈	—	127	2	no	—
53	C₂×C₈	—	127	2	no	—
54	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
55	C₁₆	—	127	2	no	—
56	C₁₆	C₄×D₁₂₇	1	1	yes	characteristic
57	C₁₆	C₂²×D₁₂₇	1	1	yes	characteristic
58	C₂³	—	127	1	no	—
59	C₂×C₄	—	127	1	no	—
60	C₂×C₄	—	127	2	no	—
61	C₂×C₄	—	127	2	no	—
62	C₂×C₄	C₈×D₁₂₇	1	1	yes	characteristic
63	C₈	—	127	2	no	—
64	C₈	C₈×D₁₂₇	1	1	yes	characteristic
65	C₈	C₂×C₄×D₁₂₇	1	1	yes	characteristic
66	C₂²	—	127	2	no	—
67	C₂²	—	254	2	no	—
68	C₂²	C₁₆×D₁₂₇	1	1	yes	characteristic
69	C₄	—	127	2	no	—
70	C₄	C₁₆×D₁₂₇	1	1	yes	characteristic
71	C₄	C₂×C₈×D₁₂₇	1	1	yes	characteristic
72	C₂	—	254	2	no	—
73	C₂	#22549709	1	2	yes	direct product factor
74	C₂	C₂×C₁₆×D₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂×D₁₂₇
commutator	C₂×C₁₂₇	C₂²×C₁₆
Frattini	C₁₆	C₂²×D₁₂₇
Fitting	C₂×C₃₂×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×C₁₆
prime radical	C₂²×D₁₂₇	C₁₆
cosocle	C₁₆×C₁₂₇	C₂³
hypercenter	C₂×C₃₂	D₁₂₇
hypocenter	C₁₂₇	C₂×M₆₄
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
2. #22549709	C₂	direct (N × H)
3. C₂×C₁₂₇⋊C₃₂	C₂	semidirect
4. C₂×C₃₂×C₁₂₇	C₂	semidirect
9. C₁₂₇⋊C₃₂	C₂²	semidirect
11. C₃₂×C₁₂₇	C₂²	semidirect
34. C₂×C₁₂₇	M₆₄	semidirect
38. C₁₂₇	C₂×M₆₄	semidirect
42. C₂×C₃₂	D₁₂₇	semidirect
49. C₃₂	C₂×D₁₂₇	semidirect
73. C₂	#22549709	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₆×D₁₂₇	2
#22549709	2
C₂×C₁₂₇⋊C₃₂	2
C₂×C₃₂×C₁₂₇	2
C₂×M₆₄	127

Subgroups — C2×#22549709 #30641021

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22549709 #30641021