Subgroups — C₂₃×#13726 #19161352

Order 5888.

230 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×Q₆₄	C₂	1	4	yes	maximal
2	C₂₃×#2247	C₂	1	2	yes	maximal
3	C₄×C₂₃×C₃₂	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×Q₃₂	—	2	2	no	—
5	C₂×C₂₃×Q₃₂	C₂²	1	4	yes	—
6	C₂₃×#494	—	2	2	no	—
7	C₂₃×Q₆₄	—	2	8	no	—
8	C₂×C₂₃×C₃₂	C₂²	1	2	yes	—
9	C₄×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×Q₁₆×C₂₃	—	2	4	no	—
11	C₂×Q₁₆×C₂₃	—	4	2	no	—
12	C₂₃×C₄⋊Q₈	—	4	2	no	—
13	C₂₃×Q₃₂	—	2	8	no	—
14	C₂₃×Q₃₂	—	4	4	no	—
15	C₂×C₁₆×C₂₃	D₄	1	1	yes	characteristic
16	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
17	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
18	C₂₃×C₃₂	D₄	1	4	yes	—
19	C₂×Q₈×C₂₃	—	4	4	no	—
20	Q₁₆×C₂₃	—	4	4	no	—
21	Q₁₆×C₂₃	—	4	4	no	—
22	Q₁₆×C₂₃	—	8	4	no	—
23	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
24	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
25	C₄⋊C₄×C₂₃	—	8	4	no	—
26	C₄²×C₂₃	D₈	1	1	yes	characteristic
27	C₁₆×C₂₃	D₈	1	2	yes	—
28	C₁₆×C₂₃	C₂×D₄	1	2	yes	—
29	#13726	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	Q₈×C₂₃	—	8	4	no	—
31	Q₈×C₂₃	—	8	4	no	—
32	C₂×C₄×C₂₃	—	8	4	no	—
33	C₂×C₄×C₂₃	D₁₆	1	2	yes	—
34	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
35	C₈×C₂₃	D₁₆	1	2	yes	—
36	C₈×C₂₃	#127	1	1	yes	characteristic
37	C₈×C₂₃	C₂×D₈	1	1	yes	characteristic
38	C₂×Q₆₄	C₂×C₂₃	1	4	yes	—
39	#2247	C₂×C₂₃	1	2	yes	—
40	C₄×C₃₂	C₂×C₂₃	1	1	yes	characteristic
41	C₂²×C₂₃	C₂×D₁₆	1	1	yes	characteristic
42	C₄×C₂₃	—	16	4	no	—
43	C₄×C₂₃	Q₆₄	1	4	yes	complemented
44	C₄×C₂₃	#493	1	1	yes	characteristic
45	C₄×C₂₃	C₂×D₁₆	1	1	yes	characteristic
46	C₂×Q₃₂	—	2	2	no	—
47	C₂×Q₃₂	C₂²×C₂₃	1	4	yes	—
48	#494	—	2	2	no	—
49	Q₆₄	—	2	8	no	—
50	C₂×C₃₂	C₂²×C₂₃	1	2	yes	—
51	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
52	C₂×C₂₃	#2246	1	1	yes	characteristic
53	C₂×C₂₃	C₂×Q₆₄	1	2	yes	—
54	C₂×Q₁₆	—	2	4	no	—
55	C₂×Q₁₆	—	4	2	no	—
56	C₄⋊Q₈	—	4	2	no	—
57	Q₃₂	—	2	8	no	—
58	Q₃₂	—	4	4	no	—
59	C₂×C₁₆	D₄×C₂₃	1	1	yes	characteristic
60	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
61	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
62	C₃₂	D₄×C₂₃	1	4	yes	—
63	C₂₃	#13726	1	1	yes	characteristic, direct product factor, Sylow
64	C₂×Q₈	—	4	4	no	—
65	Q₁₆	—	4	4	no	—
66	Q₁₆	—	4	4	no	—
67	Q₁₆	—	8	4	no	—
68	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
69	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
70	C₄⋊C₄	—	8	4	no	—
71	C₄²	D₈×C₂₃	1	1	yes	characteristic
72	C₁₆	D₈×C₂₃	1	2	yes	—
73	C₁₆	C₂×D₄×C₂₃	1	2	yes	—
74	Q₈	—	8	4	no	—
75	Q₈	—	8	4	no	—
76	C₂×C₄	—	8	4	no	—
77	C₂×C₄	C₂₃×D₁₆	1	2	yes	—
78	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
79	C₈	C₂₃×D₁₆	1	2	yes	—
80	C₈	C₂₃×#127	1	1	yes	characteristic
81	C₈	C₂×D₈×C₂₃	1	1	yes	characteristic
82	C₂²	C₂×C₂₃×D₁₆	1	1	yes	characteristic
83	C₄	—	16	4	no	—
84	C₄	C₂₃×Q₆₄	1	4	yes	complemented
85	C₄	C₂₃×#493	1	1	yes	characteristic
86	C₄	C₂×C₂₃×D₁₆	1	1	yes	characteristic
87	C₂	C₂₃×#2246	1	1	yes	characteristic
88	C₂	C₂×C₂₃×Q₆₄	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₂₃	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₂₃
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄²×C₂₃ ◁ C₄×C₈×C₂₃ ◁ C₄×C₁₆×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
29. #13726	C₂₃	direct (N × H)
43. C₄×C₂₃	Q₆₄	semidirect
63. C₂₃	#13726	direct (N × H)
84. C₄	C₂₃×Q₆₄	semidirect

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×#2247	2
C₄×C₂₃×C₃₂	2
#13726	23

Subgroups — C23×#13726 #19161352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13726 #19161352