Subgroups — C₂×C₂₃×Q₁₂₈ #19161366

Order 5888.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×Q₆₄	C₂	1	2	yes	maximal
2	C₂₃×Q₁₂₈	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₂₃×C₆₄	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×Q₃₂	—	2	2	no	—
5	C₂₃×Q₆₄	—	2	2	no	—
6	C₂₃×Q₆₄	C₂²	1	4	yes	—
7	C₂×C₂₃×C₃₂	C₂²	1	1	yes	characteristic
8	C₂₃×C₆₄	C₂²	1	2	yes	—
9	C₂×Q₁₆×C₂₃	—	4	2	no	—
10	C₂₃×Q₃₂	—	2	4	no	—
11	C₂₃×Q₃₂	—	4	2	no	—
12	C₂×C₁₆×C₂₃	D₄	1	1	yes	characteristic
13	C₂₃×C₃₂	D₄	1	1	yes	characteristic
14	C₂₃×C₃₂	C₂³	1	1	yes	characteristic
15	C₂×Q₈×C₂₃	—	8	2	no	—
16	Q₁₆×C₂₃	—	4	4	no	—
17	Q₁₆×C₂₃	—	8	2	no	—
18	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
19	C₁₆×C₂₃	D₈	1	1	yes	characteristic
20	C₁₆×C₂₃	C₂×D₄	1	1	yes	characteristic
21	C₂×Q₁₂₈	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	Q₈×C₂₃	—	8	4	no	—
23	Q₈×C₂₃	—	16	2	no	—
24	C₂×C₄×C₂₃	—	16	2	no	—
25	C₂×C₄×C₂₃	D₁₆	1	1	yes	characteristic
26	C₈×C₂₃	D₁₆	1	1	yes	characteristic
27	C₈×C₂₃	C₂×D₈	1	1	yes	characteristic
28	C₂×Q₆₄	C₂×C₂₃	1	2	yes	—
29	Q₁₂₈	C₂×C₂₃	1	4	yes	direct product factor
30	C₂×C₆₄	C₂×C₂₃	1	1	yes	characteristic
31	C₂²×C₂₃	D₃₂	1	1	yes	characteristic
32	C₄×C₂₃	—	16	4	no	—
33	C₄×C₂₃	D₃₂	1	1	yes	characteristic
34	C₄×C₂₃	C₂×D₁₆	1	1	yes	characteristic
35	C₂×Q₃₂	—	2	2	no	—
36	Q₆₄	—	2	2	no	—
37	Q₆₄	C₂²×C₂₃	1	4	yes	—
38	C₂×C₃₂	C₂²×C₂₃	1	1	yes	characteristic
39	C₆₄	C₂²×C₂₃	1	2	yes	—
40	C₂×C₂₃	Q₁₂₈	1	2	yes	direct product factor
41	C₂×C₂₃	C₂×D₃₂	1	1	yes	characteristic
42	C₂×Q₁₆	—	4	2	no	—
43	Q₃₂	—	2	4	no	—
44	Q₃₂	—	4	2	no	—
45	C₂×C₁₆	D₄×C₂₃	1	1	yes	characteristic
46	C₃₂	D₄×C₂₃	1	1	yes	characteristic
47	C₃₂	C₂³×C₂₃	1	1	yes	characteristic
48	C₂₃	C₂×Q₁₂₈	1	1	yes	characteristic, direct product factor, Sylow
49	C₂×Q₈	—	8	2	no	—
50	Q₁₆	—	4	4	no	—
51	Q₁₆	—	8	2	no	—
52	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
53	C₁₆	D₈×C₂₃	1	1	yes	characteristic
54	C₁₆	C₂×D₄×C₂₃	1	1	yes	characteristic
55	Q₈	—	8	4	no	—
56	Q₈	—	16	2	no	—
57	C₂×C₄	—	16	2	no	—
58	C₂×C₄	C₂₃×D₁₆	1	1	yes	characteristic
59	C₈	C₂₃×D₁₆	1	1	yes	characteristic
60	C₈	C₂×D₈×C₂₃	1	1	yes	characteristic
61	C₂²	C₂₃×D₃₂	1	1	yes	characteristic
62	C₄	—	16	4	no	—
63	C₄	C₂₃×D₃₂	1	1	yes	characteristic
64	C₄	C₂×C₂₃×D₁₆	1	1	yes	characteristic
65	C₂	C₂₃×Q₁₂₈	1	2	yes	direct product factor
66	C₂	C₂×C₂₃×D₃₂	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₂₃	D₃₂
commutator, Frattini, cosocle	C₃₂	C₂³×C₂₃
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄×C₂₃ ◁ C₂×C₈×C₂₃ ◁ C₂×C₁₆×C₂₃ ◁ C₂×C₂₃×C₃₂ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×Q₁₂₈	C₂	direct (N × H)
21. C₂×Q₁₂₈	C₂₃	direct (N × H)
29. Q₁₂₈	C₂×C₂₃	direct (N × H)
40. C₂×C₂₃	Q₁₂₈	direct (N × H)
48. C₂₃	C₂×Q₁₂₈	direct (N × H)
65. C₂	C₂₃×Q₁₂₈	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×C₂₃×Q₆₄	2
C₂₃×Q₁₂₈	2
C₂×C₂₃×C₆₄	2
C₂×Q₁₂₈	23

Subgroups — C2×C23×Q128 #19161366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×Q₁₂₈ #19161366