Subgroups — C₁₂₇×#2866 #30635818

Order 16256.

418 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#388×C₁₂₇	C₂	1	7	yes	maximal
2	#387×C₁₂₇	C₂	1	7	yes	maximal, complemented
3	C₄³×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	7	yes	—
5	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	7	yes	—
6	C₂×C₄²×C₁₂₇	C₂²	1	7	yes	—
7	#95×C₁₂₇	C₂²	1	14	yes	—
8	C₂⁴×C₁₂₇	C₂³	1	1	yes	characteristic
9	C₂²×C₄×C₁₂₇	C₂³	1	7	yes	—
10	C₂²×C₄×C₁₂₇	C₂³	1	7	yes	—
11	C₄⋊C₄×C₁₂₇	—	2	14	no	—
12	C₂²⋊C₄×C₁₂₇	—	2	14	no	—
13	C₄²×C₁₂₇	—	2	14	no	—
14	C₂³×C₁₂₇	—	2	7	no	—
15	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
16	C₂×C₄×C₁₂₇	—	2	14	no	—
17	C₂×C₄×C₁₂₇	—	2	7	no	—
18	C₂×C₄×C₁₂₇	—	2	14	no	—
19	C₂×C₄×C₁₂₇	#41	1	14	yes	—
20	C₂²×C₁₂₇	—	4	7	no	—
21	C₂²×C₁₂₇	C₂×#41	1	7	yes	—
22	C₄×C₁₂₇	—	2	14	no	—
23	C₄×C₁₂₇	—	4	7	no	—
24	C₂×C₁₂₇	—	8	1	no	—
25	C₂×C₁₂₇	#529	1	7	yes	—
26	#2866	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₁₂₇	#2866	1	1	yes	characteristic, direct product factor, Sylow
28	#388	C₂×C₁₂₇	1	7	yes	—
29	#387	C₂×C₁₂₇	1	7	yes	complemented
30	C₄³	C₂×C₁₂₇	1	1	yes	characteristic, complemented
31	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	7	yes	—
32	C₂×C₂²⋊C₄	C₂²×C₁₂₇	1	7	yes	—
33	C₂×C₄²	C₂²×C₁₂₇	1	7	yes	—
34	#95	C₂²×C₁₂₇	1	14	yes	—
35	C₂⁴	C₂³×C₁₂₇	1	1	yes	characteristic
36	C₂²×C₄	C₂³×C₁₂₇	1	7	yes	—
37	C₂²×C₄	C₂³×C₁₂₇	1	7	yes	—
38	C₄⋊C₄	—	2	14	no	—
39	C₂²⋊C₄	—	2	14	no	—
40	C₄²	—	2	14	no	—
41	C₂³	—	2	7	no	—
42	C₂³	C₂⁴×C₁₂₇	1	1	yes	characteristic
43	C₂×C₄	—	2	14	no	—
44	C₂×C₄	—	2	7	no	—
45	C₂×C₄	—	2	14	no	—
46	C₂×C₄	#41×C₁₂₇	1	14	yes	—
47	C₂²	—	4	7	no	—
48	C₂²	C₂×#41×C₁₂₇	1	7	yes	—
49	C₄	—	2	14	no	—
50	C₄	—	4	7	no	—
51	C₂	—	8	1	no	—
52	C₂	#529×C₁₂₇	1	7	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₂₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₂₇
prime radical	C₂⁴×C₁₂₇	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #387×C₁₂₇	C₂	semidirect
3. C₄³×C₁₂₇	C₂	semidirect
26. #2866	C₁₂₇	direct (N × H)
27. C₁₂₇	#2866	direct (N × H)
29. #387	C₂×C₁₂₇	semidirect
30. C₄³	C₂×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#388×C₁₂₇	2
#387×C₁₂₇	2
C₄³×C₁₂₇	2
#2866	127

Subgroups — C127×#2866 #30635818

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#2866 #30635818