Subgroups — #30636682

Order 16256.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549904	C₂	1	1	yes	maximal, characteristic
2	#22549658	C₂	1	1	yes	maximal, characteristic
3	#349×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₂₇⋊C₈	C₄	1	1	yes	characteristic
5	#16861168	C₄	1	1	yes	characteristic
6	#16861149	—	2	1	no	—
7	Dic₁₂₇⋊C₈	—	2	1	no	—
8	Dic₁₂₇⋊C₈	—	2	1	no	—
9	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
10	M₃₂×C₁₂₇	—	2	1	no	—
11	C₂×C₁₂₇⋊C₈	—	2	1	no	—
12	C₂×C₁₂₇⋊C₈	—	4	1	no	—
13	C₄⋊Dic₁₂₇	—	2	1	no	—
14	C₂²⋊Dic₁₂₇	C₈	1	2	yes	—
15	C₄×Dic₁₂₇	—	2	1	no	—
16	C₁₂₇⋊C₁₆	—	4	1	no	—
17	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
18	M₁₆×C₁₂₇	—	4	1	no	—
19	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
20	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
21	C₁₆×C₁₂₇	—	4	1	no	—
22	C₂×Dic₁₂₇	—	2	2	no	—
23	C₁₂₇⋊C₈	—	4	1	no	—
24	C₂³×C₁₂₇	C₂×C₈	1	1	yes	characteristic
25	C₂×C₄×C₁₂₇	—	2	1	no	—
26	C₂×C₄×C₁₂₇	—	2	1	no	—
27	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₁₂₇	M₁₆	1	1	yes	characteristic
29	C₈×C₁₂₇	—	2	1	no	—
30	C₈×C₁₂₇	—	2	1	no	—
31	C₂²×C₁₂₇	—	2	1	no	—
32	C₂²×C₁₂₇	—	4	1	no	—
33	C₂²×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
34	C₄×C₁₂₇	—	2	1	no	—
35	C₄×C₁₂₇	#99	1	1	yes	characteristic
36	C₄×C₁₂₇	#101	1	1	yes	characteristic
37	Dic₁₂₇	—	4	2	no	—
38	C₂×C₁₂₇	—	2	1	no	—
39	C₂×C₁₂₇	—	4	1	no	—
40	C₂×C₁₂₇	#324	1	1	yes	characteristic
41	#1328	—	127	1	no	maximal, Sylow
42	C₁₂₇	#1328	1	1	yes	characteristic, complemented, Sylow
43	#424	—	127	1	no	—
44	#349	—	127	1	no	—
45	#349	D₁₂₇	1	1	yes	characteristic
46	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
47	C₂²×C₈	—	127	1	no	—
48	#117	—	127	1	no	—
49	M₃₂	—	2	1	no	—
50	M₃₂	—	254	1	no	—
51	C₄⋊C₈	—	254	1	no	—
52	C₄⋊C₈	—	254	1	no	—
53	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
54	M₁₆	—	4	1	no	—
55	C₂×C₈	—	508	1	no	—
56	C₂×C₈	—	254	1	no	—
57	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
58	C₂×C₈	D₅₀₈	1	1	yes	characteristic
59	C₄⋊C₄	—	254	1	no	—
60	C₂²⋊C₄	—	127	2	no	—
61	C₄²	—	254	1	no	—
62	C₁₆	—	4	1	no	—
63	C₁₆	—	508	1	no	—
64	C₂³	C₈×D₁₂₇	1	1	yes	characteristic
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	254	2	no	—
68	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
69	C₂×C₄	#14523231	1	1	yes	characteristic
70	C₈	—	2	1	no	—
71	C₈	—	2	1	no	—
72	C₈	—	508	1	no	—
73	C₂²	—	2	1	no	—
74	C₂²	—	4	1	no	—
75	C₂²	(C₂×D₁₂₇)⋊C₈	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	508	2	no	—
78	C₄	#16861123	1	1	yes	characteristic
79	C₄	#16861127	1	1	yes	characteristic
80	C₂	—	2	1	no	—
81	C₂	—	4	1	no	—
82	C₂	#22549601	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16861123
commutator, prime radical	C₂³×C₁₂₇	C₂×C₈
Frattini	C₂×M₁₆	C₂×D₁₂₇
Fitting	#349×C₁₂₇	C₂
socle	C₂×C₁₂₇	#324
cosocle	C₂×M₁₆×C₁₂₇	C₂²
hypercenter	#349	D₁₂₇
hypocenter	C₁₂₇	#1328
LCS	G ▷ C₂³×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #349

Complements

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

N	H ≅ G/N	type
42. C₁₂₇	#1328	semidirect

Maximal subgroups

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

subgroup	index
#22549904	2
#22549658	2
#349×C₁₂₇	2
#1328	127