Subgroups — #30636822

Order 16256.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861143	C₂	1	2	yes	maximal
2	C₂²×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊Dic₁₂₇	—	2	2	no	—
4	#16861143	—	2	2	no	—
5	#16861143	C₄	1	4	yes	complemented
6	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₂×C₁₆×C₁₂₇	—	2	2	no	—
8	C₂×C₁₆×C₁₂₇	C₄	1	2	yes	complemented
9	C₂²×Dic₁₂₇	—	4	2	no	—
10	C₄⋊Dic₁₂₇	—	2	4	no	—
11	C₄⋊Dic₁₂₇	—	4	2	no	—
12	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
13	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
14	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
16	C₂×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
17	C₁₆×C₁₂₇	—	2	2	no	—
18	C₂×Dic₁₂₇	—	4	4	no	—
19	C₂×Dic₁₂₇	—	8	2	no	—
20	C₂×Dic₁₂₇	—	8	2	no	—
21	C₂³×C₁₂₇	D₈	1	1	yes	characteristic
22	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
23	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
24	C₂×C₄×C₁₂₇	SD₁₆	1	2	yes	—
25	C₂×C₄×C₁₂₇	Q₁₆	1	1	yes	characteristic
26	C₈×C₁₂₇	C₄²	1	1	yes	characteristic
27	C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
28	C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
29	C₂²×C₁₂₇	#102	1	2	yes	—
30	C₂²×C₁₂₇	#107	1	1	yes	characteristic
31	C₂²×C₁₂₇	D₁₆	1	1	yes	characteristic
32	C₂²×C₁₂₇	SD₃₂	1	2	yes	—
33	C₂²×C₁₂₇	Q₃₂	1	1	yes	characteristic
34	C₄×C₁₂₇	#95	1	1	yes	characteristic
35	C₄×C₁₂₇	#103	1	2	yes	—
36	C₄×C₁₂₇	#106	1	1	yes	characteristic
37	Dic₁₂₇	—	8	4	no	—
38	C₂×C₁₂₇	#340	1	1	yes	characteristic
39	C₂×C₁₂₇	#357	1	2	yes	—
40	C₂×C₁₂₇	#358	1	2	yes	—
41	C₂×C₁₂₇	#366	1	1	yes	characteristic
42	C₂×C₁₂₇	#367	1	1	yes	characteristic
43	#1380	—	127	1	no	maximal, Sylow
44	C₁₂₇	#1380	1	1	yes	characteristic, complemented, Sylow
45	C₂²×C₁₆	D₁₂₇	1	1	yes	characteristic
46	C₂×#107	—	127	2	no	—
47	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
48	C₂×C₄⋊C₄	—	254	2	no	—
49	C₂×C₁₆	—	2	2	no	—
50	C₂×C₁₆	Dic₁₂₇	1	2	yes	complemented
51	#107	—	127	4	no	—
52	#107	—	254	2	no	—
53	C₂²×C₄	—	508	2	no	—
54	C₂²×C₄	D₅₀₈	1	1	yes	characteristic
55	C₂×C₈	C₄×D₁₂₇	1	2	yes	—
56	C₂×C₈	C₂×Dic₁₂₇	1	1	yes	characteristic
57	C₂×C₈	C₂²⋊D₁₂₇	1	2	yes	—
58	C₂×C₈	Dic₂₅₄	1	1	yes	characteristic
59	C₄⋊C₄	—	508	2	no	—
60	C₄⋊C₄	—	254	4	no	—
61	C₁₆	—	2	2	no	—
62	C₂³	D₁₀₁₆	1	1	yes	characteristic
63	C₂×C₄	—	1016	2	no	—
64	C₂×C₄	—	1016	2	no	—
65	C₂×C₄	—	508	4	no	—
66	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
67	C₂×C₄	C₄⋊Dic₁₂₇	1	1	yes	characteristic
68	C₂×C₄	#14523237	1	2	yes	—
69	C₂×C₄	Dic₅₀₈	1	1	yes	characteristic
70	C₈	C₄×Dic₁₂₇	1	1	yes	characteristic
71	C₈	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
72	C₈	Dic₁₂₇⋊C₄	1	2	yes	—
73	C₂²	#16861131	1	2	yes	—
74	C₂²	#16861143	1	1	yes	characteristic
75	C₂²	D₂₀₃₂	1	1	yes	characteristic
76	C₂²	#16861154	1	2	yes	—
77	C₂²	Dic₁₀₁₆	1	1	yes	characteristic
78	C₄	—	1016	4	no	—
79	C₄	C₁₂₇⋊#95	1	1	yes	characteristic
80	C₄	#16861134	1	2	yes	—
81	C₄	#16861141	1	1	yes	characteristic
82	C₂	#22549639	1	1	yes	characteristic
83	C₂	#22549678	1	2	yes	—
84	C₂	#22549681	1	2	yes	—
85	C₂	#22549702	1	1	yes	characteristic
86	C₂	#22549704	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₁₀₁₆
commutator	C₈×C₁₂₇	C₄²
Frattini	C₂²×C₈	C₂×D₁₂₇
Fitting	C₂²×C₁₆×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	D₈
cosocle	C₂²×C₈×C₁₂₇	C₂²
hypercenter	C₂²×C₁₆	D₁₂₇
hypocenter	C₁₂₇	#1380
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
5. #16861143	C₄	semidirect
8. C₂×C₁₆×C₁₂₇	C₄	semidirect
44. C₁₂₇	#1380	semidirect
50. C₂×C₁₆	Dic₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861143	2
C₂²×C₁₆×C₁₂₇	2
#1380	127