Subgroups — C₁₂₇×#7052 #44570725

Order 32512.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×C₁₆×C₁₂₇	C₂	1	2	yes	maximal
2	C₈×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₆×C₁₂₇	C₄	1	4	yes	—
4	C₄×C₁₆×C₁₂₇	C₄	1	2	yes	—
5	C₈²×C₁₂₇	C₂²	1	1	yes	characteristic
6	C₂×C₁₆×C₁₂₇	C₈	1	8	yes	—
7	C₂×C₁₆×C₁₂₇	C₈	1	4	yes	—
8	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
9	C₄×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
10	C₂×C₈×C₁₂₇	C₂×C₈	1	2	yes	—
11	C₂×C₈×C₁₂₇	C₂×C₈	1	4	yes	—
12	C₄²×C₁₂₇	C₄²	1	1	yes	characteristic
13	C₁₆×C₁₂₇	—	2	8	no	—
14	C₁₆×C₁₂₇	C₁₆	1	8	yes	complemented
15	C₂×C₄×C₁₂₇	C₄×C₈	1	1	yes	characteristic
16	C₂×C₄×C₁₂₇	C₄×C₈	1	2	yes	—
17	C₈×C₁₂₇	C₂×C₁₆	1	4	yes	—
18	C₈×C₁₂₇	M₃₂	1	8	yes	—
19	C₂²×C₁₂₇	C₈²	1	1	yes	characteristic
20	C₄×C₁₂₇	C₄×C₁₆	1	2	yes	—
21	C₄×C₁₂₇	#346	1	4	yes	—
22	#7052	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂×C₁₂₇	C₈×C₁₆	1	1	yes	characteristic
24	C₂×C₁₂₇	#1311	1	2	yes	—
25	C₈×C₁₆	C₂×C₁₂₇	1	2	yes	—
26	C₈×C₁₆	C₂×C₁₂₇	1	1	yes	characteristic
27	C₁₂₇	#7052	1	1	yes	characteristic, direct product factor, Sylow
28	C₄×C₁₆	C₄×C₁₂₇	1	4	yes	—
29	C₄×C₁₆	C₄×C₁₂₇	1	2	yes	—
30	C₈²	C₂²×C₁₂₇	1	1	yes	characteristic
31	C₂×C₁₆	C₈×C₁₂₇	1	8	yes	—
32	C₂×C₁₆	C₈×C₁₂₇	1	4	yes	—
33	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
34	C₄×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
35	C₂×C₈	C₂×C₈×C₁₂₇	1	2	yes	—
36	C₂×C₈	C₂×C₈×C₁₂₇	1	4	yes	—
37	C₄²	C₄²×C₁₂₇	1	1	yes	characteristic
38	C₁₆	—	2	8	no	—
39	C₁₆	C₁₆×C₁₂₇	1	8	yes	complemented
40	C₂×C₄	C₄×C₈×C₁₂₇	1	1	yes	characteristic
41	C₂×C₄	C₄×C₈×C₁₂₇	1	2	yes	—
42	C₈	C₂×C₁₆×C₁₂₇	1	4	yes	—
43	C₈	M₃₂×C₁₂₇	1	8	yes	—
44	C₂²	C₈²×C₁₂₇	1	1	yes	characteristic
45	C₄	C₄×C₁₆×C₁₂₇	1	2	yes	—
46	C₄	#346×C₁₂₇	1	4	yes	—
47	C₂	C₈×C₁₆×C₁₂₇	1	1	yes	characteristic
48	C₂	C₁₂₇×#1311	1	2	yes	—

	subgroup	G/H
center	C₈²×C₁₂₇	C₂²
commutator	C₂	C₈×C₁₆×C₁₂₇
Frattini, cosocle	C₈²	C₂²×C₁₂₇
socle, prime radical	C₂²×C₁₂₇	C₈²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈²×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
14. C₁₆×C₁₂₇	C₁₆	semidirect
22. #7052	C₁₂₇	direct (N × H)
27. C₁₂₇	#7052	direct (N × H)
39. C₁₆	C₁₆×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₈×C₁₆×C₁₂₇	2
#7052	127

Subgroups — C127×#7052 #44570725

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#7052 #44570725