Subgroups — C₂×C₂₃×M₁₂₈ #19161362

Order 5888.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×C₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂₃×M₁₂₈	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₂₃×C₆₄	C₂	1	2	yes	maximal, complemented
4	C₂²×C₁₆×C₂₃	C₄	1	1	yes	characteristic
5	C₂×C₂₃×C₃₂	C₄	1	2	yes	—
6	C₂×C₂₃×C₃₂	C₄	1	1	yes	characteristic
7	C₂×C₂₃×C₃₂	C₂²	1	2	yes	—
8	C₂×C₂₃×C₃₂	C₂²	1	1	yes	characteristic
9	C₂₃×C₆₄	C₂²	1	4	yes	complemented
10	C₂²×C₈×C₂₃	C₈	1	1	yes	characteristic
11	C₂×C₁₆×C₂₃	C₈	1	1	yes	characteristic
12	C₂×C₁₆×C₂₃	C₈	1	2	yes	—
13	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
14	C₂×C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂₃×C₃₂	C₂×C₄	1	2	yes	—
16	C₂₃×C₃₂	C₂×C₄	1	1	yes	characteristic
17	C₂₃×C₃₂	C₂³	1	1	yes	characteristic
18	C₂²×C₄×C₂₃	C₁₆	1	1	yes	characteristic
19	C₂×C₈×C₂₃	C₁₆	1	2	yes	—
20	C₂×C₈×C₂₃	C₁₆	1	1	yes	characteristic
21	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
22	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
23	C₁₆×C₂₃	C₂×C₈	1	2	yes	—
24	C₁₆×C₂₃	C₂×C₈	1	1	yes	characteristic
25	C₁₆×C₂₃	C₂²×C₄	1	1	yes	characteristic
26	C₂×M₁₂₈	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂³×C₂₃	C₃₂	1	1	yes	characteristic
28	C₂×C₄×C₂₃	C₃₂	1	2	yes	—
29	C₂×C₄×C₂₃	C₃₂	1	1	yes	characteristic
30	C₂×C₄×C₂₃	C₂×C₁₆	1	2	yes	—
31	C₂×C₄×C₂₃	C₂×C₁₆	1	1	yes	characteristic
32	C₈×C₂₃	C₂×C₁₆	1	2	yes	—
33	C₈×C₂₃	C₂×C₁₆	1	1	yes	characteristic
34	C₈×C₂₃	C₂²×C₈	1	1	yes	characteristic
35	C₂²×C₃₂	C₂×C₂₃	1	1	yes	characteristic
36	M₁₂₈	C₂×C₂₃	1	4	yes	direct product factor
37	C₂×C₆₄	C₂×C₂₃	1	2	yes	complemented
38	C₂²×C₂₃	—	2	2	no	—
39	C₂²×C₂₃	C₂×C₃₂	1	2	yes	—
40	C₂²×C₂₃	C₂×C₃₂	1	1	yes	characteristic
41	C₄×C₂₃	C₂×C₃₂	1	1	yes	characteristic
42	C₄×C₂₃	C₂×C₃₂	1	2	yes	—
43	C₄×C₂₃	C₂²×C₁₆	1	1	yes	characteristic
44	C₂²×C₁₆	C₄×C₂₃	1	1	yes	characteristic
45	C₂×C₃₂	C₄×C₂₃	1	1	yes	characteristic
46	C₂×C₃₂	C₄×C₂₃	1	2	yes	—
47	C₂×C₃₂	C₂²×C₂₃	1	2	yes	—
48	C₂×C₃₂	C₂²×C₂₃	1	1	yes	characteristic
49	C₆₄	C₂²×C₂₃	1	4	yes	complemented
50	C₂×C₂₃	—	2	2	no	—
51	C₂×C₂₃	M₁₂₈	1	2	yes	direct product factor
52	C₂×C₂₃	C₂²×C₃₂	1	1	yes	characteristic
53	C₂²×C₈	C₈×C₂₃	1	1	yes	characteristic
54	C₂×C₁₆	C₈×C₂₃	1	1	yes	characteristic
55	C₂×C₁₆	C₈×C₂₃	1	2	yes	—
56	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
57	C₂×C₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
58	C₃₂	C₂×C₄×C₂₃	1	2	yes	—
59	C₃₂	C₂×C₄×C₂₃	1	1	yes	characteristic
60	C₃₂	C₂³×C₂₃	1	1	yes	characteristic
61	C₂₃	C₂×M₁₂₈	1	1	yes	characteristic, direct product factor, Sylow
62	C₂²×C₄	C₁₆×C₂₃	1	1	yes	characteristic
63	C₂×C₈	C₁₆×C₂₃	1	2	yes	—
64	C₂×C₈	C₁₆×C₂₃	1	1	yes	characteristic
65	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
66	C₂×C₈	C₂×C₈×C₂₃	1	1	yes	characteristic
67	C₁₆	C₂×C₈×C₂₃	1	2	yes	—
68	C₁₆	C₂×C₈×C₂₃	1	1	yes	characteristic
69	C₁₆	C₂²×C₄×C₂₃	1	1	yes	characteristic
70	C₂³	C₂₃×C₃₂	1	1	yes	characteristic
71	C₂×C₄	C₂₃×C₃₂	1	2	yes	—
72	C₂×C₄	C₂₃×C₃₂	1	1	yes	characteristic
73	C₂×C₄	C₂×C₁₆×C₂₃	1	2	yes	—
74	C₂×C₄	C₂×C₁₆×C₂₃	1	1	yes	characteristic
75	C₈	C₂×C₁₆×C₂₃	1	1	yes	characteristic
76	C₈	C₂×C₁₆×C₂₃	1	2	yes	—
77	C₈	C₂²×C₈×C₂₃	1	1	yes	characteristic
78	C₂²	—	2	2	no	—
79	C₂²	C₂×C₂₃×C₃₂	1	2	yes	—
80	C₂²	C₂×C₂₃×C₃₂	1	1	yes	characteristic
81	C₄	C₂×C₂₃×C₃₂	1	2	yes	—
82	C₄	C₂×C₂₃×C₃₂	1	1	yes	characteristic
83	C₄	C₂²×C₁₆×C₂₃	1	1	yes	characteristic
84	C₂	—	2	2	no	—
85	C₂	C₂₃×M₁₂₈	1	2	yes	direct product factor
86	C₂	C₂²×C₂₃×C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₂₃×C₃₂	C₂²
commutator	C₂	C₂²×C₂₃×C₃₂
Frattini, cosocle	C₃₂	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×C₃₂
prime radical	C₂³×C₂₃	C₃₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₃×C₃₂ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×M₁₂₈	C₂	direct (N × H)
3. C₂×C₂₃×C₆₄	C₂	semidirect
9. C₂₃×C₆₄	C₂²	semidirect
26. C₂×M₁₂₈	C₂₃	direct (N × H)
36. M₁₂₈	C₂×C₂₃	direct (N × H)
37. C₂×C₆₄	C₂×C₂₃	semidirect
49. C₆₄	C₂²×C₂₃	semidirect
51. C₂×C₂₃	M₁₂₈	direct (N × H)
61. C₂₃	C₂×M₁₂₈	direct (N × H)
85. C₂	C₂₃×M₁₂₈	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×C₂₃×C₃₂	2
C₂₃×M₁₂₈	2
C₂×C₂₃×C₆₄	2
C₂×M₁₂₈	23

Subgroups — C2×C23×M128 #19161362

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×M₁₂₈ #19161362