Subgroups — C₁₂₇×#7455 #44571128

Order 32512.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×C₄⋊C₃₂	C₂	1	2	yes	maximal
2	C₈×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₂×C₁₂₇	—	2	2	no	—
4	C₄×C₁₆×C₁₂₇	C₄	1	1	yes	characteristic
5	C₄×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
6	C₈²×C₁₂₇	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₁₂₇	—	2	1	no	—
8	C₂×C₁₆×C₁₂₇	D₄	1	1	yes	characteristic
9	C₂×C₁₆×C₁₂₇	Q₈	1	1	yes	characteristic
10	C₄×C₈×C₁₂₇	C₈	1	1	yes	characteristic
11	C₄×C₈×C₁₂₇	C₈	1	1	yes	characteristic
12	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
13	C₃₂×C₁₂₇	—	4	2	no	—
14	C₂×C₈×C₁₂₇	—	2	1	no	—
15	C₂×C₈×C₁₂₇	C₁₆	1	1	yes	characteristic
16	C₂×C₈×C₁₂₇	C₁₆	1	1	yes	characteristic
17	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₈×C₁₂₇	M₁₆	1	1	yes	characteristic
19	C₄²×C₁₂₇	C₂×C₈	1	1	yes	characteristic
20	C₁₆×C₁₂₇	—	2	2	no	—
21	C₁₆×C₁₂₇	—	2	1	no	—
22	C₁₆×C₁₂₇	SD₁₆	1	2	yes	—
23	C₂×C₄×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
24	C₂×C₄×C₁₂₇	C₂×C₁₆	1	1	yes	characteristic
25	C₂×C₄×C₁₂₇	M₃₂	1	1	yes	characteristic
26	C₈×C₁₂₇	—	2	2	no	—
27	C₈×C₁₂₇	—	2	1	no	—
28	C₈×C₁₂₇	—	2	1	no	—
29	C₈×C₁₂₇	C₃₂	1	2	yes	complemented
30	C₈×C₁₂₇	#106	1	1	yes	characteristic
31	C₈×C₁₂₇	#108	1	1	yes	characteristic
32	C₂²×C₁₂₇	C₄⋊C₁₆	1	1	yes	characteristic
33	C₄×C₁₂₇	—	2	1	no	—
34	C₄×C₁₂₇	#334	1	1	yes	characteristic
35	C₄×C₁₂₇	#364	1	1	yes	characteristic
36	C₄×C₁₂₇	C₂×C₃₂	1	1	yes	characteristic
37	C₄×C₁₂₇	M₆₄	1	1	yes	characteristic
38	#7455	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂×C₁₂₇	#1367	1	1	yes	characteristic
40	C₂×C₁₂₇	C₄⋊C₃₂	1	1	yes	characteristic
41	C₂×C₁₂₇	#1422	1	1	yes	characteristic
42	C₄⋊C₃₂	C₂×C₁₂₇	1	2	yes	—
43	C₈×C₁₆	C₂×C₁₂₇	1	1	yes	characteristic
44	C₁₂₇	#7455	1	1	yes	characteristic, direct product factor, Sylow
45	C₂×C₃₂	—	2	2	no	—
46	C₄×C₁₆	C₄×C₁₂₇	1	1	yes	characteristic
47	C₄×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
48	C₈²	C₄×C₁₂₇	1	1	yes	characteristic
49	C₂×C₁₆	—	2	1	no	—
50	C₂×C₁₆	D₄×C₁₂₇	1	1	yes	characteristic
51	C₂×C₁₆	Q₈×C₁₂₇	1	1	yes	characteristic
52	C₄×C₈	C₈×C₁₂₇	1	1	yes	characteristic
53	C₄×C₈	C₈×C₁₂₇	1	1	yes	characteristic
54	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
55	C₃₂	—	4	2	no	—
56	C₂×C₈	—	2	1	no	—
57	C₂×C₈	C₁₆×C₁₂₇	1	1	yes	characteristic
58	C₂×C₈	C₁₆×C₁₂₇	1	1	yes	characteristic
59	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
60	C₂×C₈	M₁₆×C₁₂₇	1	1	yes	characteristic
61	C₄²	C₂×C₈×C₁₂₇	1	1	yes	characteristic
62	C₁₆	—	2	2	no	—
63	C₁₆	—	2	1	no	—
64	C₁₆	SD₁₆×C₁₂₇	1	2	yes	—
65	C₂×C₄	C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
66	C₂×C₄	C₂×C₁₆×C₁₂₇	1	1	yes	characteristic
67	C₂×C₄	M₃₂×C₁₂₇	1	1	yes	characteristic
68	C₈	—	2	2	no	—
69	C₈	—	2	1	no	—
70	C₈	—	2	1	no	—
71	C₈	C₃₂×C₁₂₇	1	2	yes	complemented
72	C₈	#106×C₁₂₇	1	1	yes	characteristic
73	C₈	#108×C₁₂₇	1	1	yes	characteristic
74	C₂²	C₄⋊C₁₆×C₁₂₇	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	#334×C₁₂₇	1	1	yes	characteristic
77	C₄	#364×C₁₂₇	1	1	yes	characteristic
78	C₄	C₂×C₃₂×C₁₂₇	1	1	yes	characteristic
79	C₄	M₆₄×C₁₂₇	1	1	yes	characteristic
80	C₂	C₁₂₇×#1367	1	1	yes	characteristic
81	C₂	C₁₂₇×C₄⋊C₃₂	1	1	yes	characteristic
82	C₂	C₁₂₇×#1422	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆×C₁₂₇	D₄
commutator	C₄	C₂×C₃₂×C₁₂₇
Frattini, cosocle	C₄×C₁₆	C₂²×C₁₂₇
socle, prime radical	C₂²×C₁₂₇	C₄⋊C₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₁₂₇ ◁ C₄×C₁₆×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
29. C₈×C₁₂₇	C₃₂	semidirect
38. #7455	C₁₂₇	direct (N × H)
44. C₁₂₇	#7455	direct (N × H)
71. C₈	C₃₂×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₂₇×C₄⋊C₃₂	2
C₈×C₁₆×C₁₂₇	2
#7455	127

Subgroups — C127×#7455 #44571128

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#7455 #44571128