Subgroups — C₂×C₄×C₃₂×C₁₂₇ #44577219

Order 32512.

270 conjugacy classes of subgroups.

The subgroup lattice has 114 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₂×C₁₂₇	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₂×C₁₂₇	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₆×C₁₂₇	C₄	1	1	yes	characteristic
5	C₂²×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₈×C₁₂₇	C₄	1	1	yes	characteristic
7	C₂×C₃₂×C₁₂₇	C₄	1	8	yes	direct product factor
8	C₂×C₃₂×C₁₂₇	C₂²	1	4	yes	—
9	C₄×C₁₆×C₁₂₇	C₄	1	2	yes	—
10	C₄×C₁₆×C₁₂₇	C₂²	1	2	yes	—
11	C₂²×C₈×C₁₂₇	C₈	1	1	yes	characteristic
12	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄²×C₁₂₇	C₈	1	1	yes	characteristic
14	C₂×C₁₆×C₁₂₇	C₈	1	4	yes	—
15	C₂×C₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
17	C₂×C₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
18	C₂×C₁₆×C₁₂₇	C₂³	1	1	yes	characteristic
19	C₄×C₈×C₁₂₇	C₈	1	2	yes	—
20	C₄×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
21	C₃₂×C₁₂₇	C₂×C₄	1	8	yes	direct product factor
22	C₂²×C₄×C₁₂₇	C₁₆	1	2	yes	—
23	C₂²×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
24	C₂×C₈×C₁₂₇	C₁₆	1	4	yes	—
25	C₂×C₈×C₁₂₇	C₄²	1	2	yes	—
26	C₂×C₈×C₁₂₇	C₂×C₈	1	1	yes	characteristic
27	C₂×C₈×C₁₂₇	C₂×C₈	1	2	yes	—
28	C₂×C₈×C₁₂₇	C₂×C₈	1	2	yes	—
29	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
30	C₄²×C₁₂₇	C₁₆	1	2	yes	—
31	C₄²×C₁₂₇	C₂×C₈	1	2	yes	—
32	C₁₆×C₁₂₇	C₄²	1	2	yes	—
33	C₁₆×C₁₂₇	C₂×C₈	1	4	yes	—
34	C₁₆×C₁₂₇	C₂²×C₄	1	2	yes	—
35	C₂³×C₁₂₇	C₂×C₁₆	1	1	yes	characteristic
36	C₂×C₄×C₁₂₇	C₃₂	1	8	yes	direct product factor
37	C₂×C₄×C₁₂₇	C₄×C₈	1	2	yes	—
38	C₂×C₄×C₁₂₇	C₂×C₁₆	1	1	yes	characteristic
39	C₂×C₄×C₁₂₇	C₂×C₁₆	1	2	yes	—
40	C₂×C₄×C₁₂₇	C₂×C₁₆	1	4	yes	—
41	C₂×C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
42	C₈×C₁₂₇	C₄×C₈	1	2	yes	—
43	C₈×C₁₂₇	C₂×C₁₆	1	4	yes	—
44	C₈×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
45	C₈×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
46	C₂²×C₁₂₇	C₄×C₁₆	1	2	yes	—
47	C₂²×C₁₂₇	C₂×C₃₂	1	4	yes	—
48	C₂²×C₁₂₇	C₂²×C₁₆	1	1	yes	characteristic
49	C₄×C₁₂₇	C₄×C₁₆	1	2	yes	—
50	C₄×C₁₂₇	C₂×C₃₂	1	8	yes	direct product factor
51	C₄×C₁₂₇	C₂×C₄×C₈	1	1	yes	characteristic
52	C₄×C₁₂₇	C₂²×C₁₆	1	1	yes	characteristic
53	C₂×C₄×C₃₂	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂×C₁₂₇	C₄×C₃₂	1	4	yes	direct product factor
55	C₂×C₁₂₇	C₂×C₄×C₁₆	1	1	yes	characteristic
56	C₂×C₁₂₇	C₂²×C₃₂	1	2	yes	—
57	C₂²×C₃₂	C₂×C₁₂₇	1	2	yes	—
58	C₂×C₄×C₁₆	C₂×C₁₂₇	1	1	yes	characteristic
59	C₄×C₃₂	C₂×C₁₂₇	1	4	yes	direct product factor
60	C₁₂₇	C₂×C₄×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
61	C₂²×C₁₆	C₄×C₁₂₇	1	1	yes	characteristic
62	C₂²×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
63	C₂×C₄×C₈	C₄×C₁₂₇	1	1	yes	characteristic
64	C₂×C₃₂	C₄×C₁₂₇	1	8	yes	direct product factor
65	C₂×C₃₂	C₂²×C₁₂₇	1	4	yes	—
66	C₄×C₁₆	C₄×C₁₂₇	1	2	yes	—
67	C₄×C₁₆	C₂²×C₁₂₇	1	2	yes	—
68	C₂²×C₈	C₈×C₁₂₇	1	1	yes	characteristic
69	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
70	C₂×C₄²	C₈×C₁₂₇	1	1	yes	characteristic
71	C₂×C₁₆	C₈×C₁₂₇	1	4	yes	—
72	C₂×C₁₆	C₂×C₄×C₁₂₇	1	1	yes	characteristic
73	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
74	C₂×C₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
75	C₂×C₁₆	C₂³×C₁₂₇	1	1	yes	characteristic
76	C₄×C₈	C₈×C₁₂₇	1	2	yes	—
77	C₄×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
78	C₃₂	C₂×C₄×C₁₂₇	1	8	yes	direct product factor
79	C₂²×C₄	C₁₆×C₁₂₇	1	2	yes	—
80	C₂²×C₄	C₂×C₈×C₁₂₇	1	1	yes	characteristic
81	C₂×C₈	C₁₆×C₁₂₇	1	4	yes	—
82	C₂×C₈	C₄²×C₁₂₇	1	2	yes	—
83	C₂×C₈	C₂×C₈×C₁₂₇	1	1	yes	characteristic
84	C₂×C₈	C₂×C₈×C₁₂₇	1	2	yes	—
85	C₂×C₈	C₂×C₈×C₁₂₇	1	2	yes	—
86	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
87	C₄²	C₁₆×C₁₂₇	1	2	yes	—
88	C₄²	C₂×C₈×C₁₂₇	1	2	yes	—
89	C₁₆	C₄²×C₁₂₇	1	2	yes	—
90	C₁₆	C₂×C₈×C₁₂₇	1	4	yes	—
91	C₁₆	C₂²×C₄×C₁₂₇	1	2	yes	—
92	C₂³	C₂×C₁₆×C₁₂₇	1	1	yes	characteristic
93	C₂×C₄	C₃₂×C₁₂₇	1	8	yes	direct product factor
94	C₂×C₄	C₄×C₈×C₁₂₇	1	2	yes	—
95	C₂×C₄	C₂×C₁₆×C₁₂₇	1	1	yes	characteristic
96	C₂×C₄	C₂×C₁₆×C₁₂₇	1	2	yes	—
97	C₂×C₄	C₂×C₁₆×C₁₂₇	1	4	yes	—
98	C₂×C₄	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
99	C₈	C₄×C₈×C₁₂₇	1	2	yes	—
100	C₈	C₂×C₁₆×C₁₂₇	1	4	yes	—
101	C₈	C₂×C₄²×C₁₂₇	1	1	yes	characteristic
102	C₈	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
103	C₂²	C₄×C₁₆×C₁₂₇	1	2	yes	—
104	C₂²	C₂×C₃₂×C₁₂₇	1	4	yes	—
105	C₂²	C₂²×C₁₆×C₁₂₇	1	1	yes	characteristic
106	C₄	C₄×C₁₆×C₁₂₇	1	2	yes	—
107	C₄	C₂×C₃₂×C₁₂₇	1	8	yes	direct product factor
108	C₄	C₂×C₄×C₈×C₁₂₇	1	1	yes	characteristic
109	C₄	C₂²×C₁₆×C₁₂₇	1	1	yes	characteristic
110	C₂	C₄×C₃₂×C₁₂₇	1	4	yes	direct product factor
111	C₂	C₂×C₄×C₁₆×C₁₂₇	1	1	yes	characteristic
112	C₂	C₂²×C₃₂×C₁₂₇	1	2	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₁₆	C₂³×C₁₂₇
socle, prime radical	C₂³×C₁₂₇	C₂×C₁₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×C₃₂×C₁₂₇	C₂	direct (N × H)
7. C₂×C₃₂×C₁₂₇	C₄	direct (N × H)
21. C₃₂×C₁₂₇	C₂×C₄	direct (N × H)
36. C₂×C₄×C₁₂₇	C₃₂	direct (N × H)
50. C₄×C₁₂₇	C₂×C₃₂	direct (N × H)
53. C₂×C₄×C₃₂	C₁₂₇	direct (N × H)
54. C₂×C₁₂₇	C₄×C₃₂	direct (N × H)
59. C₄×C₃₂	C₂×C₁₂₇	direct (N × H)
60. C₁₂₇	C₂×C₄×C₃₂	direct (N × H)
64. C₂×C₃₂	C₄×C₁₂₇	direct (N × H)
78. C₃₂	C₂×C₄×C₁₂₇	direct (N × H)
93. C₂×C₄	C₃₂×C₁₂₇	direct (N × H)
107. C₄	C₂×C₃₂×C₁₂₇	direct (N × H)
110. C₂	C₄×C₃₂×C₁₂₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₂×C₁₂₇	2
C₂×C₄×C₁₆×C₁₂₇	2
C₄×C₃₂×C₁₂₇	2
C₂×C₄×C₃₂	127

Subgroups — C2×C4×C32×C127 #44577219

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₃₂×C₁₂₇ #44577219