Subgroups — C₂×#22549714 #30641031

Order 16256.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₀₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#22549714	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₁₂₇⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₅₀₈	—	2	1	no	—
6	Dic₁₀₁₆	—	2	1	no	—
7	Dic₁₀₁₆	C₂²	1	2	yes	complemented
8	C₁₂₇⋊C₃₂	C₂²	1	2	yes	complemented
9	C₂×D₈×C₁₂₇	—	2	1	no	—
10	D₁₆×C₁₂₇	—	2	1	no	—
11	D₁₆×C₁₂₇	C₂²	1	2	yes	—
12	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×Dic₂₅₄	—	4	1	no	—
14	Dic₅₀₈	—	2	2	no	—
15	Dic₅₀₈	—	4	1	no	—
16	C₂×D₄×C₁₂₇	—	4	1	no	—
17	D₈×C₁₂₇	—	2	2	no	—
18	D₈×C₁₂₇	—	4	1	no	—
19	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
20	C₁₆×C₁₂₇	D₄	1	1	yes	characteristic
21	C₁₆×C₁₂₇	C₂³	1	1	yes	characteristic
22	Dic₂₅₄	—	4	2	no	—
23	Dic₂₅₄	—	8	1	no	—
24	C₂×Dic₁₂₇	—	8	1	no	—
25	C₂³×C₁₂₇	—	8	1	no	—
26	D₄×C₁₂₇	—	4	2	no	—
27	D₄×C₁₂₇	—	8	1	no	—
28	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
29	C₈×C₁₂₇	D₈	1	1	yes	characteristic
30	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₁₂₇	—	8	2	no	—
32	C₂²×C₁₂₇	—	16	2	no	—
33	C₂²×C₁₂₇	D₁₆	1	1	yes	characteristic
34	C₄×C₁₂₇	D₁₆	1	1	yes	characteristic
35	C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
36	Dic₁₂₇	—	8	2	no	—
37	C₂×C₁₂₇	—	16	2	no	—
38	C₂×C₁₂₇	SD₆₄	1	2	yes	complemented
39	C₂×C₁₂₇	C₂×D₁₆	1	1	yes	characteristic
40	C₂×SD₆₄	—	127	1	no	maximal, Sylow
41	C₁₂₇	C₂×SD₆₄	1	1	yes	characteristic, complemented, Sylow
42	C₂×Q₃₂	—	127	1	no	—
43	C₂×D₁₆	D₁₂₇	1	1	yes	characteristic
44	SD₆₄	—	127	4	no	—
45	C₂×C₃₂	—	127	1	no	—
46	C₂×Q₁₆	—	254	1	no	—
47	C₂×D₈	—	2	1	no	—
48	Q₃₂	—	127	2	no	—
49	Q₃₂	—	254	1	no	—
50	D₁₆	—	2	1	no	—
51	D₁₆	C₂×D₁₂₇	1	2	yes	—
52	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
53	C₃₂	—	127	2	no	—
54	C₂×Q₈	—	508	1	no	—
55	C₂×D₄	—	4	1	no	—
56	Q₁₆	—	508	1	no	—
57	Q₁₆	—	254	2	no	—
58	D₈	—	2	2	no	—
59	D₈	—	4	1	no	—
60	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
61	C₁₆	C₂²⋊D₁₂₇	1	1	yes	characteristic
62	C₁₆	C₂²×D₁₂₇	1	1	yes	characteristic
63	C₂³	—	8	1	no	—
64	Q₈	—	1016	1	no	—
65	Q₈	—	508	2	no	—
66	D₄	—	4	2	no	—
67	D₄	—	8	1	no	—
68	C₂×C₄	—	1016	1	no	—
69	C₂×C₄	#14523233	1	1	yes	characteristic
70	C₈	#14523233	1	1	yes	characteristic
71	C₈	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
72	C₂²	—	8	2	no	—
73	C₂²	—	16	2	no	—
74	C₂²	#16861150	1	1	yes	characteristic
75	C₄	—	1016	2	no	—
76	C₄	#16861150	1	1	yes	characteristic
77	C₄	C₂×#14523233	1	1	yes	characteristic
78	C₂	—	16	2	no	—
79	C₂	#22549714	1	2	yes	direct product factor
80	C₂	C₂×#16861150	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16861150
commutator, cosocle	C₁₆×C₁₂₇	C₂³
Frattini	C₁₆	C₂²×D₁₂₇
Fitting, prime radical	C₂×D₁₆×C₁₂₇	C₂
socle	C₂²×C₁₂₇	D₁₆
hypercenter	C₂×D₁₆	D₁₂₇
hypocenter	C₁₂₇	C₂×SD₆₄
LCS	G ▷ C₁₆×C₁₂₇ ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×D₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₁₀₁₆	C₂	semidirect
2. #22549714	C₂	direct (N × H)
3. C₂×C₁₂₇⋊C₃₂	C₂	semidirect
7. Dic₁₀₁₆	C₂²	semidirect
8. C₁₂₇⋊C₃₂	C₂²	semidirect
38. C₂×C₁₂₇	SD₆₄	semidirect
41. C₁₂₇	C₂×SD₆₄	semidirect
79. C₂	#22549714	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×Dic₁₀₁₆	2
#22549714	2
C₂×C₁₂₇⋊C₃₂	2
C₂×D₁₆×C₁₂₇	2
C₂×SD₆₄	127

Subgroups — C2×#22549714 #30641031

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22549714 #30641031