Subgroups — #30636641

Order 16256.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549933	C₂	1	2	yes	maximal
2	#423×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₁₂₇⋊C₈)	—	2	2	no	—
4	C₂²⋊(C₁₂₇⋊C₈)	C₄	1	4	yes	—
5	#16861120	—	2	2	no	—
6	C₂×M₁₆×C₁₂₇	C₄	1	2	yes	—
7	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₄⋊C₈×C₁₂₇	—	2	2	no	—
9	C₂×C₁₂₇⋊C₈	—	2	4	no	—
10	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
11	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
12	M₁₆×C₁₂₇	—	4	2	no	—
13	C₂×C₈×C₁₂₇	—	2	2	no	—
14	C₄²×C₁₂₇	D₄	1	2	yes	—
15	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
16	C₄²×C₁₂₇	Q₈	1	1	yes	characteristic
17	C₁₂₇⋊C₈	—	4	4	no	—
18	C₂³×C₁₂₇	C₄²	1	1	yes	characteristic
19	C₂×C₄×C₁₂₇	—	2	1	no	—
20	C₂×C₄×C₁₂₇	—	2	1	no	—
21	C₂×C₄×C₁₂₇	—	4	2	no	—
22	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
23	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
25	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
26	C₈×C₁₂₇	—	4	2	no	—
27	C₂²×C₁₂₇	—	2	1	no	—
28	C₂²×C₁₂₇	—	2	2	no	—
29	C₂²×C₁₂₇	#95	1	1	yes	characteristic
30	C₄×C₁₂₇	—	2	1	no	—
31	C₄×C₁₂₇	—	2	2	no	—
32	C₄×C₁₂₇	—	2	2	no	—
33	C₄×C₁₂₇	#100	1	1	yes	characteristic
34	C₄×C₁₂₇	#101	1	1	yes	characteristic
35	C₂×C₁₂₇	—	4	1	no	—
36	C₂×C₁₂₇	#343	1	1	yes	characteristic
37	C₂×C₁₂₇	#344	1	2	yes	—
38	#1307	—	127	1	no	maximal, Sylow
39	C₁₂₇	#1307	1	1	yes	characteristic, complemented, Sylow
40	#432	—	127	2	no	—
41	#423	D₁₂₇	1	1	yes	characteristic
42	C₂×M₁₆	Dic₁₂₇	1	2	yes	—
43	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
44	C₄⋊C₈	—	2	2	no	—
45	C₄⋊C₈	—	254	2	no	—
46	C₂²⋊C₈	—	127	4	no	—
47	#97	—	254	2	no	—
48	C₂²×C₄	C₄×D₁₂₇	1	2	yes	—
49	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
50	M₁₆	—	4	2	no	—
51	C₂×C₈	—	2	2	no	—
52	C₂×C₈	—	254	4	no	—
53	C₄²	C₂²⋊D₁₂₇	1	2	yes	—
54	C₄²	D₅₀₈	1	1	yes	characteristic
55	C₄²	Dic₂₅₄	1	1	yes	characteristic
56	C₂³	C₄×Dic₁₂₇	1	1	yes	characteristic
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	4	2	no	—
60	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
61	C₂×C₄	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
62	C₂×C₄	Dic₁₂₇⋊C₄	1	2	yes	—
63	C₂×C₄	C₄⋊Dic₁₂₇	1	1	yes	characteristic
64	C₈	—	4	2	no	—
65	C₈	—	508	4	no	—
66	C₂²	—	2	1	no	—
67	C₂²	—	2	2	no	—
68	C₂²	C₁₂₇⋊#95	1	1	yes	characteristic
69	C₄	—	2	1	no	—
70	C₄	—	2	2	no	—
71	C₄	—	2	2	no	—
72	C₄	#16861126	1	1	yes	characteristic
73	C₄	#16861128	1	1	yes	characteristic
74	C₂	—	4	1	no	—
75	C₂	#22549645	1	1	yes	characteristic
76	C₂	#22549648	1	2	yes	—

	subgroup	G/H
center	C₂²	C₁₂₇⋊#95
commutator, prime radical	C₂³×C₁₂₇	C₄²
Frattini	C₂×C₄²	C₂×D₁₂₇
Fitting	#423×C₁₂₇	C₂
socle	C₂²×C₁₂₇	#95
cosocle	C₂×C₄²×C₁₂₇	C₂²
hypercenter	#423	D₁₂₇
hypocenter	C₁₂₇	#1307
LCS	G ▷ C₂³×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
39. C₁₂₇	#1307	semidirect

Maximal subgroups

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

subgroup	index
#22549933	2
#423×C₁₂₇	2
#1307	127