Subgroups — C₂×#22549710 #30641022

Order 16256.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549710	C₂	1	4	yes	maximal, direct product factor
2	C₂×C₁₂₇⋊C₃₂	C₂	1	2	yes	maximal, complemented
3	C₂²×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇⋊C₃₂	C₂²	1	4	yes	complemented
5	C₂²×C₈×C₁₂₇	C₄	1	1	yes	characteristic
6	C₂×C₁₆×C₁₂₇	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₁₂₇	C₄	1	2	yes	—
8	C₂×C₁₆×C₁₂₇	C₂²	1	2	yes	—
9	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₁₂₇	C₈	1	1	yes	characteristic
11	C₂×C₈×C₁₂₇	C₈	1	2	yes	—
12	C₂×C₈×C₁₂₇	C₈	1	1	yes	characteristic
13	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
14	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
15	C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
16	C₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
17	C₁₆×C₁₂₇	C₂³	1	1	yes	characteristic
18	C₂³×C₁₂₇	C₁₆	1	1	yes	characteristic
19	C₂×C₄×C₁₂₇	C₁₆	1	2	yes	—
20	C₂×C₄×C₁₂₇	C₁₆	1	1	yes	characteristic
21	C₂×C₄×C₁₂₇	C₂×C₈	1	2	yes	—
22	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
23	C₈×C₁₂₇	C₂×C₈	1	2	yes	—
24	C₈×C₁₂₇	C₂×C₈	1	1	yes	characteristic
25	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₁₂₇	—	2	2	no	—
27	C₂²×C₁₂₇	C₂×C₁₆	1	2	yes	—
28	C₂²×C₁₂₇	C₂×C₁₆	1	1	yes	characteristic
29	C₄×C₁₂₇	C₂×C₁₆	1	1	yes	characteristic
30	C₄×C₁₂₇	C₂×C₁₆	1	2	yes	—
31	C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
32	C₂×C₁₂₇	—	2	2	no	—
33	C₂×C₁₂₇	M₆₄	1	2	yes	complemented
34	C₂×C₁₂₇	C₂²×C₁₆	1	1	yes	characteristic
35	C₂×M₆₄	—	127	1	no	maximal, Sylow
36	C₁₂₇	C₂×M₆₄	1	1	yes	characteristic, complemented, Sylow
37	C₂²×C₁₆	D₁₂₇	1	1	yes	characteristic
38	M₆₄	—	127	4	no	—
39	C₂×C₃₂	—	127	2	no	—
40	C₂²×C₈	Dic₁₂₇	1	1	yes	characteristic
41	C₂×C₁₆	Dic₁₂₇	1	1	yes	characteristic
42	C₂×C₁₆	Dic₁₂₇	1	2	yes	—
43	C₂×C₁₆	C₂×D₁₂₇	1	2	yes	—
44	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
45	C₃₂	—	127	4	no	—
46	C₂²×C₄	C₁₂₇⋊C₈	1	1	yes	characteristic
47	C₂×C₈	C₁₂₇⋊C₈	1	1	yes	characteristic
48	C₂×C₈	C₁₂₇⋊C₈	1	2	yes	—
49	C₂×C₈	C₂×Dic₁₂₇	1	2	yes	—
50	C₂×C₈	C₂×Dic₁₂₇	1	1	yes	characteristic
51	C₁₆	C₂×Dic₁₂₇	1	2	yes	—
52	C₁₆	C₂×Dic₁₂₇	1	1	yes	characteristic
53	C₁₆	C₂²×D₁₂₇	1	1	yes	characteristic
54	C₂³	C₁₂₇⋊C₁₆	1	1	yes	characteristic
55	C₂×C₄	C₁₂₇⋊C₁₆	1	2	yes	—
56	C₂×C₄	C₁₂₇⋊C₁₆	1	1	yes	characteristic
57	C₂×C₄	C₂×C₁₂₇⋊C₈	1	2	yes	—
58	C₂×C₄	C₂×C₁₂₇⋊C₈	1	1	yes	characteristic
59	C₈	C₂×C₁₂₇⋊C₈	1	2	yes	—
60	C₈	C₂×C₁₂₇⋊C₈	1	1	yes	characteristic
61	C₈	C₂²×Dic₁₂₇	1	1	yes	characteristic
62	C₂²	—	2	2	no	—
63	C₂²	C₂×C₁₂₇⋊C₁₆	1	2	yes	—
64	C₂²	C₂×C₁₂₇⋊C₁₆	1	1	yes	characteristic
65	C₄	C₂×C₁₂₇⋊C₁₆	1	1	yes	characteristic
66	C₄	C₂×C₁₂₇⋊C₁₆	1	2	yes	—
67	C₄	C₂²×C₁₂₇⋊C₈	1	1	yes	characteristic
68	C₂	—	2	2	no	—
69	C₂	#22549710	1	2	yes	direct product factor
70	C₂	C₂²×C₁₂₇⋊C₁₆	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. #22549710	C₂	direct (N × H)
2. C₂×C₁₂₇⋊C₃₂	C₂	semidirect
4. C₁₂₇⋊C₃₂	C₂²	semidirect
33. C₂×C₁₂₇	M₆₄	semidirect
36. C₁₂₇	C₂×M₆₄	semidirect
69. C₂	#22549710	direct (N × H)

Maximal subgroups

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

subgroup	index
#22549710	2
C₂×C₁₂₇⋊C₃₂	2
C₂²×C₁₆×C₁₂₇	2
C₂×M₆₄	127

Subgroups — C2×#22549710 #30641022

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22549710 #30641022