Subgroups — C₂×C₃×C₄×C₈×C₁₂₇ #72201680

Order 24384.

324 conjugacy classes of subgroups.

The subgroup lattice has 108 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄²×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₈×C₁₂₇	C₂	1	2	yes	maximal
3	C₃×C₄×C₈×C₁₂₇	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₃×C₄×C₁₂₇	C₄	1	2	yes	—
5	C₂²×C₃×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
6	C₃×C₄²×C₁₂₇	C₄	1	2	yes	—
7	C₃×C₄²×C₁₂₇	C₂²	1	2	yes	—
8	C₂×C₃×C₈×C₁₂₇	C₄	1	8	yes	direct product factor
9	C₂×C₃×C₈×C₁₂₇	C₂²	1	4	yes	—
10	C₂³×C₃×C₁₂₇	C₂×C₄	1	1	yes	characteristic
11	C₂×C₃×C₄×C₁₂₇	C₈	1	8	yes	direct product factor
12	C₂×C₃×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₃×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
14	C₂×C₃×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
15	C₂×C₃×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
16	C₃×C₈×C₁₂₇	C₂×C₄	1	8	yes	direct product factor
17	C₂×C₄×C₈×C₁₂₇	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂²×C₈×C₁₂₇	C₂×C₃	1	2	yes	—
19	C₂×C₄²×C₁₂₇	C₂×C₃	1	1	yes	characteristic
20	C₄×C₈×C₁₂₇	C₂×C₃	1	4	yes	direct product factor
21	C₂²×C₄×C₁₂₇	C₃×C₄	1	2	yes	—
22	C₂²×C₄×C₁₂₇	C₂²×C₃	1	1	yes	characteristic
23	C₂×C₈×C₁₂₇	C₃×C₄	1	8	yes	direct product factor
24	C₂×C₈×C₁₂₇	C₂²×C₃	1	4	yes	—
25	C₄²×C₁₂₇	C₃×C₄	1	2	yes	—
26	C₄²×C₁₂₇	C₂²×C₃	1	2	yes	—
27	C₂²×C₃×C₁₂₇	C₄²	1	2	yes	—
28	C₂²×C₃×C₁₂₇	C₂×C₈	1	4	yes	—
29	C₂²×C₃×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
30	C₃×C₄×C₁₂₇	C₄²	1	2	yes	—
31	C₃×C₄×C₁₂₇	C₂×C₈	1	8	yes	direct product factor
32	C₃×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
33	C₂³×C₁₂₇	C₂×C₃×C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₂₇	C₃×C₈	1	8	yes	direct product factor
35	C₂×C₄×C₁₂₇	C₂×C₃×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₂₇	C₂×C₃×C₄	1	4	yes	—
37	C₂×C₄×C₁₂₇	C₂×C₃×C₄	1	4	yes	—
38	C₂×C₄×C₁₂₇	C₂³×C₃	1	1	yes	characteristic
39	C₈×C₁₂₇	C₂×C₃×C₄	1	8	yes	direct product factor
40	C₂×C₃×C₁₂₇	C₄×C₈	1	4	yes	direct product factor
41	C₂×C₃×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
42	C₂×C₃×C₁₂₇	C₂²×C₈	1	2	yes	—
43	C₂²×C₁₂₇	C₃×C₄²	1	2	yes	—
44	C₂²×C₁₂₇	C₂×C₃×C₈	1	4	yes	—
45	C₂²×C₁₂₇	C₂²×C₃×C₄	1	1	yes	characteristic
46	C₄×C₁₂₇	C₃×C₄²	1	2	yes	—
47	C₄×C₁₂₇	C₂×C₃×C₈	1	8	yes	direct product factor
48	C₄×C₁₂₇	C₂²×C₃×C₄	1	2	yes	—
49	C₃×C₁₂₇	C₂×C₄×C₈	1	1	yes	characteristic, direct product factor, Hall
50	C₂×C₁₂₇	C₃×C₄×C₈	1	4	yes	direct product factor
51	C₂×C₁₂₇	C₂×C₃×C₄²	1	1	yes	characteristic
52	C₂×C₁₂₇	C₂²×C₃×C₈	1	2	yes	—
53	C₂×C₃×C₄×C₈	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Hall
54	C₁₂₇	C₂×C₃×C₄×C₈	1	1	yes	characteristic, direct product factor, Sylow
55	C₂²×C₃×C₈	C₂×C₁₂₇	1	2	yes	—
56	C₂×C₃×C₄²	C₂×C₁₂₇	1	1	yes	characteristic
57	C₃×C₄×C₈	C₂×C₁₂₇	1	4	yes	direct product factor
58	C₂×C₄×C₈	C₃×C₁₂₇	1	1	yes	characteristic, direct product factor, Sylow
59	C₂²×C₃×C₄	C₄×C₁₂₇	1	2	yes	—
60	C₂²×C₃×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
61	C₂×C₃×C₈	C₄×C₁₂₇	1	8	yes	direct product factor
62	C₂×C₃×C₈	C₂²×C₁₂₇	1	4	yes	—
63	C₃×C₄²	C₄×C₁₂₇	1	2	yes	—
64	C₃×C₄²	C₂²×C₁₂₇	1	2	yes	—
65	C₂²×C₈	C₂×C₃×C₁₂₇	1	2	yes	—
66	C₂×C₄²	C₂×C₃×C₁₂₇	1	1	yes	characteristic
67	C₄×C₈	C₂×C₃×C₁₂₇	1	4	yes	direct product factor
68	C₂³×C₃	C₂×C₄×C₁₂₇	1	1	yes	characteristic
69	C₂×C₃×C₄	C₈×C₁₂₇	1	8	yes	direct product factor
70	C₂×C₃×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
71	C₂×C₃×C₄	C₂×C₄×C₁₂₇	1	4	yes	—
72	C₂×C₃×C₄	C₂×C₄×C₁₂₇	1	4	yes	—
73	C₂×C₃×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
74	C₃×C₈	C₂×C₄×C₁₂₇	1	8	yes	direct product factor
75	C₂²×C₄	C₃×C₄×C₁₂₇	1	2	yes	—
76	C₂²×C₄	C₂²×C₃×C₁₂₇	1	1	yes	characteristic
77	C₂×C₈	C₃×C₄×C₁₂₇	1	8	yes	direct product factor
78	C₂×C₈	C₂²×C₃×C₁₂₇	1	4	yes	—
79	C₄²	C₃×C₄×C₁₂₇	1	2	yes	—
80	C₄²	C₂²×C₃×C₁₂₇	1	2	yes	—
81	C₂²×C₃	C₄²×C₁₂₇	1	2	yes	—
82	C₂²×C₃	C₂×C₈×C₁₂₇	1	4	yes	—
83	C₂²×C₃	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
84	C₃×C₄	C₄²×C₁₂₇	1	2	yes	—
85	C₃×C₄	C₂×C₈×C₁₂₇	1	8	yes	direct product factor
86	C₃×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
87	C₂³	C₂×C₃×C₄×C₁₂₇	1	1	yes	characteristic
88	C₂×C₄	C₃×C₈×C₁₂₇	1	8	yes	direct product factor
89	C₂×C₄	C₂×C₃×C₄×C₁₂₇	1	1	yes	characteristic
90	C₂×C₄	C₂×C₃×C₄×C₁₂₇	1	4	yes	—
91	C₂×C₄	C₂×C₃×C₄×C₁₂₇	1	4	yes	—
92	C₂×C₄	C₂³×C₃×C₁₂₇	1	1	yes	characteristic
93	C₈	C₂×C₃×C₄×C₁₂₇	1	8	yes	direct product factor
94	C₂×C₃	C₄×C₈×C₁₂₇	1	4	yes	direct product factor
95	C₂×C₃	C₂×C₄²×C₁₂₇	1	1	yes	characteristic
96	C₂×C₃	C₂²×C₈×C₁₂₇	1	2	yes	—
97	C₂²	C₂×C₃×C₈×C₁₂₇	1	4	yes	—
98	C₂²	C₃×C₄²×C₁₂₇	1	2	yes	—
99	C₂²	C₂²×C₃×C₄×C₁₂₇	1	1	yes	characteristic
100	C₄	C₂×C₃×C₈×C₁₂₇	1	8	yes	direct product factor
101	C₄	C₃×C₄²×C₁₂₇	1	2	yes	—
102	C₄	C₂²×C₃×C₄×C₁₂₇	1	2	yes	—
103	C₃	C₂×C₄×C₈×C₁₂₇	1	1	yes	characteristic, direct product factor, Sylow
104	C₂	C₃×C₄×C₈×C₁₂₇	1	4	yes	direct product factor
105	C₂	C₂²×C₃×C₈×C₁₂₇	1	2	yes	—
106	C₂	C₂×C₃×C₄²×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₄	C₂³×C₃×C₁₂₇
socle, prime radical	C₂³×C₃×C₁₂₇	C₂×C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₄×C₈×C₁₂₇	C₂	direct (N × H)
8. C₂×C₃×C₈×C₁₂₇	C₄	direct (N × H)
11. C₂×C₃×C₄×C₁₂₇	C₈	direct (N × H)
16. C₃×C₈×C₁₂₇	C₂×C₄	direct (N × H)
17. C₂×C₄×C₈×C₁₂₇	C₃	direct (N × H)
20. C₄×C₈×C₁₂₇	C₂×C₃	direct (N × H)
23. C₂×C₈×C₁₂₇	C₃×C₄	direct (N × H)
31. C₃×C₄×C₁₂₇	C₂×C₈	direct (N × H)
34. C₂×C₄×C₁₂₇	C₃×C₈	direct (N × H)
39. C₈×C₁₂₇	C₂×C₃×C₄	direct (N × H)
40. C₂×C₃×C₁₂₇	C₄×C₈	direct (N × H)
47. C₄×C₁₂₇	C₂×C₃×C₈	direct (N × H)
49. C₃×C₁₂₇	C₂×C₄×C₈	direct (N × H)
50. C₂×C₁₂₇	C₃×C₄×C₈	direct (N × H)
53. C₂×C₃×C₄×C₈	C₁₂₇	direct (N × H)
54. C₁₂₇	C₂×C₃×C₄×C₈	direct (N × H)
57. C₃×C₄×C₈	C₂×C₁₂₇	direct (N × H)
58. C₂×C₄×C₈	C₃×C₁₂₇	direct (N × H)
61. C₂×C₃×C₈	C₄×C₁₂₇	direct (N × H)
67. C₄×C₈	C₂×C₃×C₁₂₇	direct (N × H)
69. C₂×C₃×C₄	C₈×C₁₂₇	direct (N × H)
74. C₃×C₈	C₂×C₄×C₁₂₇	direct (N × H)
77. C₂×C₈	C₃×C₄×C₁₂₇	direct (N × H)
85. C₃×C₄	C₂×C₈×C₁₂₇	direct (N × H)
88. C₂×C₄	C₃×C₈×C₁₂₇	direct (N × H)
93. C₈	C₂×C₃×C₄×C₁₂₇	direct (N × H)
94. C₂×C₃	C₄×C₈×C₁₂₇	direct (N × H)
100. C₄	C₂×C₃×C₈×C₁₂₇	direct (N × H)
103. C₃	C₂×C₄×C₈×C₁₂₇	direct (N × H)
104. C₂	C₃×C₄×C₈×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
C₂×C₃×C₄²×C₁₂₇	2
C₂²×C₃×C₈×C₁₂₇	2
C₃×C₄×C₈×C₁₂₇	2
C₂×C₄×C₈×C₁₂₇	3
C₂×C₃×C₄×C₈	127

Subgroups — C2×C3×C4×C8×C127 #72201680

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₄×C₈×C₁₂₇ #72201680