Subgroups — C₁₂₇×#7450 #44571123

Order 32512.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#2252	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1371	C₂	1	1	yes	maximal, characteristic
4	#500×C₁₂₇	—	2	1	no	—
5	C₂×C₃₂×C₁₂₇	—	2	1	no	—
6	#366×C₁₂₇	—	2	1	no	—
7	#366×C₁₂₇	C₄	1	2	yes	—
8	C₄×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
9	#335×C₁₂₇	—	2	1	no	—
10	C₄⋊Q₈×C₁₂₇	—	4	1	no	—
11	C₂×C₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
12	C₂×C₁₆×C₁₂₇	D₄	1	1	yes	characteristic
13	#107×C₁₂₇	—	2	2	no	—
14	#107×C₁₂₇	—	4	1	no	—
15	C₄⋊C₈×C₁₂₇	—	4	1	no	—
16	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
17	C₃₂×C₁₂₇	—	4	1	no	—
18	C₂×Q₈×C₁₂₇	—	8	1	no	—
19	C₂×C₈×C₁₂₇	—	8	1	no	—
20	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
21	C₂×C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
22	C₄⋊C₄×C₁₂₇	—	4	2	no	—
23	C₄⋊C₄×C₁₂₇	—	8	1	no	—
24	C₄²×C₁₂₇	D₈	1	1	yes	characteristic
25	C₁₆×C₁₂₇	—	2	1	no	—
26	C₁₆×C₁₂₇	SD₁₆	1	1	yes	characteristic
27	C₁₆×C₁₂₇	Q₁₆	1	1	yes	characteristic
28	Q₈×C₁₂₇	—	16	1	no	—
29	Q₈×C₁₂₇	—	16	1	no	—
30	C₂×C₄×C₁₂₇	—	8	2	no	—
31	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
32	C₂×C₄×C₁₂₇	D₁₆	1	1	yes	characteristic
33	C₂×C₄×C₁₂₇	SD₃₂	1	1	yes	characteristic
34	C₈×C₁₂₇	—	2	1	no	—
35	C₈×C₁₂₇	—	16	1	no	—
36	C₈×C₁₂₇	#101	1	1	yes	characteristic
37	C₈×C₁₂₇	#103	1	1	yes	characteristic
38	C₂²×C₁₂₇	#357	1	1	yes	characteristic
39	C₄×C₁₂₇	—	2	1	no	—
40	C₄×C₁₂₇	—	16	2	no	—
41	C₄×C₁₂₇	#332	1	1	yes	characteristic
42	C₄×C₁₂₇	#362	1	1	yes	characteristic
43	C₄×C₁₂₇	D₃₂	1	1	yes	characteristic
44	C₄×C₁₂₇	SD₆₄	1	1	yes	characteristic
45	#7450	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂×C₁₂₇	#1364	1	1	yes	characteristic
47	C₂×C₁₂₇	#1415	1	1	yes	characteristic
48	C₂×C₁₂₇	#1420	1	1	yes	characteristic
49	#2252	C₂×C₁₂₇	1	1	yes	characteristic
50	C₄⋊C₃₂	C₂×C₁₂₇	1	1	yes	characteristic
51	#1371	C₂×C₁₂₇	1	1	yes	characteristic
52	C₁₂₇	#7450	1	1	yes	characteristic, direct product factor, Sylow
53	#500	—	2	1	no	—
54	C₂×C₃₂	—	2	1	no	—
55	#366	—	2	1	no	—
56	#366	C₄×C₁₂₇	1	2	yes	—
57	C₄×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
58	#335	—	2	1	no	—
59	C₄⋊Q₈	—	4	1	no	—
60	C₂×C₁₆	C₂×C₄×C₁₂₇	1	1	yes	characteristic
61	C₂×C₁₆	D₄×C₁₂₇	1	1	yes	characteristic
62	#107	—	2	2	no	—
63	#107	—	4	1	no	—
64	C₄⋊C₈	—	4	1	no	—
65	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
66	C₃₂	—	4	1	no	—
67	C₂×Q₈	—	8	1	no	—
68	C₂×C₈	—	8	1	no	—
69	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
70	C₂×C₈	SD₁₆×C₁₂₇	1	1	yes	characteristic
71	C₄⋊C₄	—	4	2	no	—
72	C₄⋊C₄	—	8	1	no	—
73	C₄²	D₈×C₁₂₇	1	1	yes	characteristic
74	C₁₆	—	2	1	no	—
75	C₁₆	SD₁₆×C₁₂₇	1	1	yes	characteristic
76	C₁₆	Q₁₆×C₁₂₇	1	1	yes	characteristic
77	Q₈	—	16	1	no	—
78	Q₈	—	16	1	no	—
79	C₂×C₄	—	8	2	no	—
80	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
81	C₂×C₄	D₁₆×C₁₂₇	1	1	yes	characteristic
82	C₂×C₄	SD₃₂×C₁₂₇	1	1	yes	characteristic
83	C₈	—	2	1	no	—
84	C₈	—	16	1	no	—
85	C₈	#101×C₁₂₇	1	1	yes	characteristic
86	C₈	#103×C₁₂₇	1	1	yes	characteristic
87	C₂²	#357×C₁₂₇	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	16	2	no	—
90	C₄	#332×C₁₂₇	1	1	yes	characteristic
91	C₄	#362×C₁₂₇	1	1	yes	characteristic
92	C₄	D₃₂×C₁₂₇	1	1	yes	characteristic
93	C₄	SD₆₄×C₁₂₇	1	1	yes	characteristic
94	C₂	C₁₂₇×#1364	1	1	yes	characteristic
95	C₂	C₁₂₇×#1415	1	1	yes	characteristic
96	C₂	C₁₂₇×#1420	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₂₇	#357
commutator	C₂×C₁₆	C₂×C₄×C₁₂₇
Frattini, cosocle	C₄×C₁₆	C₂²×C₁₂₇
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₄²×C₁₂₇ ◁ C₄×C₈×C₁₂₇ ◁ C₄×C₁₆×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
45. #7450	C₁₂₇	direct (N × H)
52. C₁₂₇	#7450	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₂₇×#2252	2
C₁₂₇×C₄⋊C₃₂	2
C₁₂₇×#1371	2
#7450	127

Subgroups — C127×#7450 #44571123

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#7450 #44571123