Subgroups — #30636559

Order 16256.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549929	C₂	1	2	yes	maximal
2	#376×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₁₂₇⋊C₈)	—	2	2	no	—
4	C₂²⋊(C₁₂₇⋊C₈)	—	2	2	no	—
5	C₄×C₁₂₇⋊C₈	C₄	1	4	yes	complemented
6	C₂×C₄²×C₁₂₇	C₄	1	2	yes	—
7	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
8	#95×C₁₂₇	—	2	2	no	—
9	C₂×C₁₂₇⋊C₈	—	2	4	no	—
10	C₂²×C₄×C₁₂₇	—	2	2	no	—
11	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
13	C₄²×C₁₂₇	—	2	2	no	—
14	C₄²×C₁₂₇	C₈	1	4	yes	complemented
15	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
16	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
17	C₄²×C₁₂₇	Q₈	1	1	yes	characteristic
18	C₁₂₇⋊C₈	—	4	4	no	—
19	C₂³×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
20	C₂×C₄×C₁₂₇	—	2	4	no	—
21	C₂×C₄×C₁₂₇	—	4	2	no	—
22	C₂×C₄×C₁₂₇	—	4	4	no	—
23	C₂×C₄×C₁₂₇	C₄²	1	1	yes	characteristic
24	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
25	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
26	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₂₇	C₂×C₈	1	2	yes	—
28	C₂×C₄×C₁₂₇	M₁₆	1	2	yes	—
29	C₂²×C₁₂₇	—	2	2	no	—
30	C₂²×C₁₂₇	#95	1	1	yes	characteristic
31	C₂²×C₁₂₇	C₂²⋊C₈	1	2	yes	—
32	C₄×C₁₂₇	—	2	2	no	—
33	C₄×C₁₂₇	—	2	2	no	—
34	C₄×C₁₂₇	—	4	4	no	—
35	C₄×C₁₂₇	C₄×C₈	1	1	yes	characteristic
36	C₄×C₁₂₇	#97	1	1	yes	characteristic
37	C₄×C₁₂₇	C₄⋊C₈	1	2	yes	—
38	C₂×C₁₂₇	—	2	2	no	—
39	C₂×C₁₂₇	#336	1	1	yes	characteristic
40	C₂×C₁₂₇	H(3,ℤ/4)	1	2	yes	—
41	#1274	—	127	1	no	maximal, Sylow
42	C₁₂₇	#1274	1	1	yes	characteristic, complemented, Sylow
43	#431	—	127	2	no	—
44	#376	D₁₂₇	1	1	yes	characteristic
45	C₂×C₄²	Dic₁₂₇	1	2	yes	—
46	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
47	C₄⋊C₈	—	254	2	no	—
48	C₂²⋊C₈	—	254	2	no	—
49	C₄×C₈	—	127	4	no	—
50	#95	—	2	2	no	—
51	C₂²×C₄	—	2	2	no	—
52	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
53	C₂²×C₄	C₂²⋊D₁₂₇	1	2	yes	—
54	C₂×C₈	—	254	4	no	—
55	C₄²	—	2	2	no	—
56	C₄²	C₁₂₇⋊C₈	1	4	yes	complemented
57	C₄²	C₄×D₁₂₇	1	2	yes	—
58	C₄²	D₅₀₈	1	1	yes	characteristic
59	C₄²	Dic₂₅₄	1	1	yes	characteristic
60	C₂³	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
61	C₂×C₄	—	2	4	no	—
62	C₂×C₄	—	4	2	no	—
63	C₂×C₄	—	4	4	no	—
64	C₂×C₄	C₄×Dic₁₂₇	1	1	yes	characteristic
65	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
66	C₂×C₄	Dic₁₂₇⋊C₄	1	2	yes	—
67	C₂×C₄	C₄⋊Dic₁₂₇	1	1	yes	characteristic
68	C₂×C₄	C₂×C₁₂₇⋊C₈	1	2	yes	—
69	C₂×C₄	#14523232	1	2	yes	—
70	C₈	—	508	4	no	—
71	C₂²	—	2	2	no	—
72	C₂²	C₁₂₇⋊#95	1	1	yes	characteristic
73	C₂²	C₂²⋊(C₁₂₇⋊C₈)	1	2	yes	—
74	C₄	—	2	2	no	—
75	C₄	—	2	2	no	—
76	C₄	—	4	4	no	—
77	C₄	C₄×C₁₂₇⋊C₈	1	1	yes	characteristic
78	C₄	#16861120	1	1	yes	characteristic
79	C₄	C₄⋊(C₁₂₇⋊C₈)	1	2	yes	—
80	C₂	—	2	2	no	—
81	C₂	#22549631	1	1	yes	characteristic
82	C₂	#22549632	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄×Dic₁₂₇
commutator	C₄×C₁₂₇	C₄×C₈
Frattini	C₂×C₄²	C₂×D₁₂₇
Fitting	#376×C₁₂₇	C₂
socle	C₂²×C₁₂₇	#95
prime radical	C₂³×C₁₂₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₂₇	C₂²
hypercenter	#376	D₁₂₇
hypocenter	C₁₂₇	#1274
LCS	G ▷ C₄×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ #376

Complements

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

N	H ≅ G/N	type
5. C₄×C₁₂₇⋊C₈	C₄	semidirect
14. C₄²×C₁₂₇	C₈	semidirect
42. C₁₂₇	#1274	semidirect
56. C₄²	C₁₂₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22549929	2
#376×C₁₂₇	2
#1274	127