Subgroups — C₂₃×#12899 #19160525

Order 5888.

370 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×C₄⋊C₁₆	C₂	1	6	yes	maximal
2	C₂₃×#1726	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₆×C₂₃	C₂²	1	4	yes	—
4	C₂×C₂₃×#97	C₄	1	3	yes	—
5	C₂×C₄×C₈×C₂₃	C₂²	1	3	yes	—
6	C₂₃×#376	C₄	1	1	yes	characteristic
7	C₂₃×C₄⋊C₁₆	—	2	12	no	—
8	C₂²×C₈×C₂₃	C₂×C₄	1	3	yes	—
9	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
10	C₂×C₄²×C₂₃	C₂×C₄	1	3	yes	—
11	C₂×C₁₆×C₂₃	—	2	12	no	—
12	C₂₃×#97	—	2	6	no	—
13	C₄×C₈×C₂₃	D₄	1	6	yes	—
14	C₄×C₈×C₂₃	Q₈	1	6	yes	—
15	C₂₃×#95	C₈	1	4	yes	—
16	C₂²×C₄×C₂₃	C₂×C₈	1	6	yes	—
17	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
18	C₂×C₈×C₂₃	—	2	6	no	—
19	C₂×C₈×C₂₃	C₄⋊C₄	1	6	yes	—
20	C₂×C₈×C₂₃	C₂×D₄	1	3	yes	—
21	C₂×C₈×C₂₃	C₂×Q₈	1	3	yes	—
22	C₄²×C₂₃	C₄⋊C₄	1	6	yes	—
23	C₄²×C₂₃	M₁₆	1	6	yes	—
24	C₁₆×C₂₃	—	4	4	no	—
25	#12899	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂³×C₂₃	C₂²×C₈	1	1	yes	characteristic
27	C₂×C₄×C₂₃	—	2	12	no	—
28	C₂×C₄×C₂₃	C₄⋊C₈	1	12	yes	—
29	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	3	yes	—
30	C₂×C₄×C₂₃	C₂×M₁₆	1	3	yes	—
31	C₈×C₂₃	—	2	6	no	—
32	C₈×C₂₃	#127	1	1	yes	characteristic
33	C₈×C₂₃	C₄⋊Q₈	1	3	yes	—
34	C₂×C₄⋊C₁₆	C₂×C₂₃	1	6	yes	—
35	#1726	C₂×C₂₃	1	1	yes	characteristic
36	C₂²×C₂₃	C₂×C₄⋊C₈	1	3	yes	—
37	C₂²×C₂₃	#504	1	4	yes	—
38	C₄×C₂₃	—	2	12	no	—
39	C₄×C₂₃	#384	1	1	yes	characteristic
40	C₄×C₂₃	#423	1	3	yes	—
41	C₂²×C₁₆	C₂²×C₂₃	1	4	yes	—
42	C₂×#97	C₄×C₂₃	1	3	yes	—
43	C₂×C₄×C₈	C₂²×C₂₃	1	3	yes	—
44	#376	C₄×C₂₃	1	1	yes	characteristic
45	C₄⋊C₁₆	—	2	12	no	—
46	C₂×C₂₃	#1842	1	1	yes	characteristic
47	C₂×C₂₃	#2151	1	6	yes	—
48	C₂²×C₈	C₂×C₄×C₂₃	1	3	yes	—
49	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
50	C₂×C₄²	C₂×C₄×C₂₃	1	3	yes	—
51	C₂×C₁₆	—	2	12	no	—
52	#97	—	2	6	no	—
53	C₄×C₈	D₄×C₂₃	1	6	yes	—
54	C₄×C₈	Q₈×C₂₃	1	6	yes	—
55	#95	C₈×C₂₃	1	4	yes	—
56	C₂₃	#12899	1	1	yes	characteristic, direct product factor, Sylow
57	C₂²×C₄	C₂×C₈×C₂₃	1	6	yes	—
58	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
59	C₂×C₈	—	2	6	no	—
60	C₂×C₈	C₄⋊C₄×C₂₃	1	6	yes	—
61	C₂×C₈	C₂×D₄×C₂₃	1	3	yes	—
62	C₂×C₈	C₂×Q₈×C₂₃	1	3	yes	—
63	C₄²	C₄⋊C₄×C₂₃	1	6	yes	—
64	C₄²	M₁₆×C₂₃	1	6	yes	—
65	C₁₆	—	4	4	no	—
66	C₂³	C₂²×C₈×C₂₃	1	1	yes	characteristic
67	C₂×C₄	—	2	12	no	—
68	C₂×C₄	C₂₃×C₄⋊C₈	1	12	yes	—
69	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	3	yes	—
70	C₂×C₄	C₂×M₁₆×C₂₃	1	3	yes	—
71	C₈	—	2	6	no	—
72	C₈	C₂₃×#127	1	1	yes	characteristic
73	C₈	C₂₃×C₄⋊Q₈	1	3	yes	—
74	C₂²	C₂×C₂₃×C₄⋊C₈	1	3	yes	—
75	C₂²	C₂₃×#504	1	4	yes	—
76	C₄	—	2	12	no	—
77	C₄	C₂₃×#384	1	1	yes	characteristic
78	C₄	C₂₃×#423	1	3	yes	—
79	C₂	C₂₃×#1842	1	1	yes	characteristic
80	C₂	C₂₃×#2151	1	6	yes	—

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

Complements

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

N	H ≅ G/N	type
25. #12899	C₂₃	direct (N × H)
56. C₂₃	#12899	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×C₄⋊C₁₆	2
C₂₃×#1726	2
#12899	23

Subgroups — C23×#12899 #19160525

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12899 #19160525