Subgroups — C₂×#22549716 #30641034

Order 16256.

104 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
2	#22549716	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₁₂₇⋊C₃₂	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₅₀₈	—	2	1	no	—
6	Dic₁₀₁₆	—	2	1	no	—
7	Dic₁₀₁₆	C₂²	1	2	yes	—
8	C₁₂₇⋊C₃₂	C₂²	1	2	yes	—
9	C₂×Q₁₆×C₁₂₇	—	2	1	no	—
10	Q₃₂×C₁₂₇	—	2	1	no	—
11	Q₃₂×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×Q₈×C₁₂₇	—	4	1	no	—
17	Q₁₆×C₁₂₇	—	2	2	no	—
18	Q₁₆×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	Q₈×C₁₂₇	—	4	2	no	—
26	Q₈×C₁₂₇	—	8	1	no	—
27	C₂×C₄×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	D₁₆	1	1	yes	characteristic
32	C₄×C₁₂₇	—	8	2	no	—
33	C₄×C₁₂₇	D₁₆	1	1	yes	characteristic
34	C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
35	Dic₁₂₇	—	8	2	no	—
36	C₂×C₁₂₇	Q₆₄	1	2	yes	complemented
37	C₂×C₁₂₇	C₂×D₁₆	1	1	yes	characteristic
38	C₂×Q₆₄	—	127	1	no	maximal, Sylow
39	C₁₂₇	C₂×Q₆₄	1	1	yes	characteristic, complemented, Sylow
40	C₂×Q₃₂	—	127	1	no	—
41	C₂×Q₃₂	D₁₂₇	1	1	yes	characteristic
42	Q₆₄	—	127	4	no	—
43	C₂×C₃₂	—	127	1	no	—
44	C₂×Q₁₆	—	2	1	no	—
45	C₂×Q₁₆	—	254	1	no	—
46	Q₃₂	—	2	1	no	—
47	Q₃₂	—	127	2	no	—
48	Q₃₂	—	254	1	no	—
49	Q₃₂	C₂×D₁₂₇	1	2	yes	—
50	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
51	C₃₂	—	127	2	no	—
52	C₂×Q₈	—	4	1	no	—
53	C₂×Q₈	—	508	1	no	—
54	Q₁₆	—	2	2	no	—
55	Q₁₆	—	4	1	no	—
56	Q₁₆	—	508	1	no	—
57	Q₁₆	—	254	2	no	—
58	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
59	C₁₆	C₂²⋊D₁₂₇	1	1	yes	characteristic
60	C₁₆	C₂²×D₁₂₇	1	1	yes	characteristic
61	Q₈	—	4	2	no	—
62	Q₈	—	8	1	no	—
63	Q₈	—	1016	1	no	—
64	Q₈	—	508	2	no	—
65	C₂×C₄	—	8	1	no	—
66	C₂×C₄	—	1016	1	no	—
67	C₂×C₄	#14523233	1	1	yes	characteristic
68	C₈	#14523233	1	1	yes	characteristic
69	C₈	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
70	C₂²	#16861150	1	1	yes	characteristic
71	C₄	—	8	2	no	—
72	C₄	—	1016	2	no	—
73	C₄	#16861150	1	1	yes	characteristic
74	C₄	C₂×#14523233	1	1	yes	characteristic
75	C₂	#22549716	1	2	yes	direct product factor
76	C₂	C₂×#16861150	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16861150
commutator, cosocle	C₁₆×C₁₂₇	C₂³
Frattini	C₁₆	C₂²×D₁₂₇
Fitting	C₂×Q₃₂×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	D₁₆
hypercenter	C₂×Q₃₂	D₁₂₇
hypocenter	C₁₂₇	C₂×Q₆₄
LCS	G ▷ C₁₆×C₁₂₇ ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
2. #22549716	C₂	direct (N × H)
36. C₂×C₁₂₇	Q₆₄	semidirect
39. C₁₂₇	C₂×Q₆₄	semidirect
75. C₂	#22549716	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
#22549716	2
C₂×C₁₂₇⋊C₃₂	2
C₂×Q₃₂×C₁₂₇	2
C₂×Q₆₄	127

Subgroups — C2×#22549716 #30641034

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22549716 #30641034