Subgroups — C₁₂₇×#2253 #30635205

Order 16256.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#499×C₁₂₇	C₂	1	2	yes	maximal
2	#367×C₁₂₇	C₂	1	2	yes	maximal
3	#366×C₁₂₇	C₂	1	2	yes	maximal
4	C₄×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	#125×C₁₂₇	—	2	2	no	—
6	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
8	#107×C₁₂₇	C₂²	1	4	yes	—
9	#106×C₁₂₇	—	2	2	no	—
10	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
12	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
13	C₄⋊C₄×C₁₂₇	—	2	4	no	—
14	C₄⋊C₄×C₁₂₇	—	4	4	no	—
15	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
16	C₁₆×C₁₂₇	Q₈	1	2	yes	—
17	C₁₆×C₁₂₇	Q₈	1	2	yes	—
18	C₂×C₄×C₁₂₇	—	4	4	no	—
19	C₂×C₄×C₁₂₇	D₈	1	2	yes	—
20	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
21	C₈×C₁₂₇	Q₁₆	1	2	yes	—
22	C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
23	C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
24	C₂²×C₁₂₇	C₂×D₈	1	1	yes	characteristic
25	C₄×C₁₂₇	—	2	2	no	—
26	C₄×C₁₂₇	—	8	4	no	—
27	C₄×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
28	C₄×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
29	C₂×C₁₂₇	#500	1	1	yes	characteristic
30	C₂×C₁₂₇	#508	1	2	yes	—
31	#2253	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₁₂₇	#2253	1	1	yes	characteristic, direct product factor, Sylow
33	#499	C₂×C₁₂₇	1	2	yes	—
34	#367	C₂×C₁₂₇	1	2	yes	—
35	#366	C₂×C₁₂₇	1	2	yes	—
36	C₄×C₁₆	C₂×C₁₂₇	1	1	yes	characteristic
37	#125	—	2	2	no	—
38	C₂×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
39	C₂×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
40	#107	C₂²×C₁₂₇	1	4	yes	—
41	#106	—	2	2	no	—
42	C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
43	C₂×C₈	D₄×C₁₂₇	1	1	yes	characteristic
44	C₂×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
45	C₄⋊C₄	—	2	4	no	—
46	C₄⋊C₄	—	4	4	no	—
47	C₄²	D₄×C₁₂₇	1	1	yes	characteristic
48	C₁₆	Q₈×C₁₂₇	1	2	yes	—
49	C₁₆	Q₈×C₁₂₇	1	2	yes	—
50	C₂×C₄	—	4	4	no	—
51	C₂×C₄	D₈×C₁₂₇	1	2	yes	—
52	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
53	C₈	Q₁₆×C₁₂₇	1	2	yes	—
54	C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
55	C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
56	C₂²	C₂×D₈×C₁₂₇	1	1	yes	characteristic
57	C₄	—	2	2	no	—
58	C₄	—	8	4	no	—
59	C₄	C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
60	C₄	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
61	C₂	#500×C₁₂₇	1	1	yes	characteristic
62	C₂	#508×C₁₂₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₂₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₁₂₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₄²×C₁₂₇ ◁ C₄×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
31. #2253	C₁₂₇	direct (N × H)
32. C₁₂₇	#2253	direct (N × H)

Maximal subgroups

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

subgroup	index
#499×C₁₂₇	2
#367×C₁₂₇	2
#366×C₁₂₇	2
C₄×C₁₆×C₁₂₇	2
#2253	127

Subgroups — C127×#2253 #30635205

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#2253 #30635205