Subgroups — C₂₃×#12894 #19160520

Order 5888.

342 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1374	C₂	1	6	yes	maximal
3	C₂²×C₁₆×C₂₃	C₂²	1	4	yes	—
4	C₂×C₄×C₈×C₂₃	C₄	1	3	yes	—
5	C₂×C₄×C₈×C₂₃	C₂²	1	3	yes	—
6	C₄³×C₂₃	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₂₃	C₂×C₄	1	3	yes	—
8	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
9	C₂×C₄²×C₂₃	C₈	1	3	yes	—
10	C₂×C₄²×C₂₃	C₈	1	1	yes	characteristic
11	C₂×C₄²×C₂₃	C₂×C₄	1	3	yes	—
12	C₂×C₁₆×C₂₃	—	2	12	no	—
13	C₄×C₈×C₂₃	—	2	6	no	—
14	C₄×C₈×C₂₃	—	2	6	no	—
15	C₂²×C₄×C₂₃	C₂×C₈	1	3	yes	—
16	C₂²×C₄×C₂₃	C₂×C₈	1	3	yes	—
17	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
18	C₂×C₈×C₂₃	—	2	3	no	—
19	C₂×C₈×C₂₃	—	2	3	no	—
20	C₂×C₈×C₂₃	M₁₆	1	6	yes	—
21	C₂×C₈×C₂₃	#41	1	6	yes	—
22	C₄²×C₂₃	—	2	6	no	—
23	C₄²×C₂₃	—	2	3	no	—
24	C₄²×C₂₃	—	2	3	no	—
25	C₄²×C₂₃	C₁₆	1	4	yes	complemented
26	C₁₆×C₂₃	—	4	4	no	—
27	#12894	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂³×C₂₃	C₂²×C₈	1	1	yes	characteristic
29	C₂×C₄×C₂₃	—	2	3	no	—
30	C₂×C₄×C₂₃	—	2	3	no	—
31	C₂×C₄×C₂₃	—	2	3	no	—
32	C₂×C₄×C₂₃	—	2	3	no	—
33	C₂×C₄×C₂₃	C₂×C₁₆	1	6	yes	—
34	C₂×C₄×C₂₃	M₃₂	1	6	yes	—
35	C₂×C₄×C₂₃	#117	1	3	yes	—
36	C₂×C₄×C₂₃	C₂×M₁₆	1	3	yes	—
37	C₈×C₂₃	—	2	6	no	—
38	C₈×C₂₃	#126	1	4	yes	—
39	C₄²×C₈	C₂×C₂₃	1	1	yes	characteristic
40	#1374	C₂×C₂₃	1	6	yes	—
41	C₂²×C₂₃	#431	1	3	yes	—
42	C₂²×C₂₃	C₂²×C₁₆	1	1	yes	characteristic
43	C₂²×C₂₃	C₂×M₃₂	1	3	yes	—
44	C₄×C₂₃	—	2	6	no	—
45	C₄×C₂₃	—	2	6	no	—
46	C₄×C₂₃	#383	1	1	yes	characteristic
47	C₄×C₂₃	#432	1	3	yes	—
48	C₂²×C₁₆	C₂²×C₂₃	1	4	yes	—
49	C₂×C₄×C₈	C₄×C₂₃	1	3	yes	—
50	C₂×C₄×C₈	C₂²×C₂₃	1	3	yes	—
51	C₄³	C₄×C₂₃	1	1	yes	characteristic
52	C₂×C₂₃	#1839	1	1	yes	characteristic
53	C₂×C₂₃	#2162	1	3	yes	—
54	C₂×C₂₃	#2163	1	3	yes	—
55	C₂²×C₈	C₂×C₄×C₂₃	1	3	yes	—
56	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
57	C₂×C₄²	C₈×C₂₃	1	3	yes	—
58	C₂×C₄²	C₈×C₂₃	1	1	yes	characteristic
59	C₂×C₄²	C₂×C₄×C₂₃	1	3	yes	—
60	C₂×C₁₆	—	2	12	no	—
61	C₄×C₈	—	2	6	no	—
62	C₄×C₈	—	2	6	no	—
63	C₂₃	#12894	1	1	yes	characteristic, direct product factor, Sylow
64	C₂²×C₄	C₂×C₈×C₂₃	1	3	yes	—
65	C₂²×C₄	C₂×C₈×C₂₃	1	3	yes	—
66	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
67	C₂×C₈	—	2	3	no	—
68	C₂×C₈	—	2	3	no	—
69	C₂×C₈	M₁₆×C₂₃	1	6	yes	—
70	C₂×C₈	#41×C₂₃	1	6	yes	—
71	C₄²	—	2	6	no	—
72	C₄²	—	2	3	no	—
73	C₄²	—	2	3	no	—
74	C₄²	C₁₆×C₂₃	1	4	yes	complemented
75	C₁₆	—	4	4	no	—
76	C₂³	C₂²×C₈×C₂₃	1	1	yes	characteristic
77	C₂×C₄	—	2	3	no	—
78	C₂×C₄	—	2	3	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	C₂×C₁₆×C₂₃	1	6	yes	—
82	C₂×C₄	C₂₃×M₃₂	1	6	yes	—
83	C₂×C₄	C₂₃×#117	1	3	yes	—
84	C₂×C₄	C₂×M₁₆×C₂₃	1	3	yes	—
85	C₈	—	2	6	no	—
86	C₈	C₂₃×#126	1	4	yes	—
87	C₂²	C₂₃×#431	1	3	yes	—
88	C₂²	C₂²×C₁₆×C₂₃	1	1	yes	characteristic
89	C₂²	C₂×C₂₃×M₃₂	1	3	yes	—
90	C₄	—	2	6	no	—
91	C₄	—	2	6	no	—
92	C₄	C₂₃×#383	1	1	yes	characteristic
93	C₄	C₂₃×#432	1	3	yes	—
94	C₂	C₂₃×#1839	1	1	yes	characteristic
95	C₂	C₂₃×#2162	1	3	yes	—
96	C₂	C₂₃×#2163	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
25. C₄²×C₂₃	C₁₆	semidirect
27. #12894	C₂₃	direct (N × H)
63. C₂₃	#12894	direct (N × H)
74. C₄²	C₁₆×C₂₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄²×C₈×C₂₃	2
C₂₃×#1374	2
#12894	23

Subgroups — C23×#12894 #19160520

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12894 #19160520