Subgroups — C₂₃×#2179 #15669798

Order 2944.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#508	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#443	C₂	1	2	yes	maximal, complemented
3	C₂₃×#368	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#359	C₂	1	2	yes	maximal, complemented
5	C₂₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#135	C₂²	1	2	yes	—
7	C₂₃×#131	—	2	2	no	—
8	C₂₃×Q₃₂	C₄	1	1	yes	characteristic, complemented
9	C₂₃×SD₃₂	C₄	1	2	yes	complemented
10	C₂₃×D₁₆	C₄	1	1	yes	characteristic, complemented
11	C₂×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#108	C₂²	1	2	yes	—
14	C₂₃×#104	—	2	2	no	—
15	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	#41×C₂₃	—	2	2	no	—
17	Q₁₆×C₂₃	C₂×C₄	1	2	yes	—
18	SD₁₆×C₂₃	—	2	2	no	—
19	D₈×C₂₃	C₂×C₄	1	2	yes	—
20	M₁₆×C₂₃	—	2	2	no	—
21	M₁₆×C₂₃	—	4	2	no	—
22	C₂×C₈×C₂₃	—	4	2	no	—
23	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
24	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
25	C₄²×C₂₃	D₄	1	1	yes	characteristic
26	C₁₆×C₂₃	—	2	1	no	—
27	C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
29	Q₈×C₂₃	—	2	2	no	—
30	D₄×C₂₃	—	2	2	no	—
31	D₄×C₂₃	—	4	2	no	—
32	C₂×C₄×C₂₃	—	2	1	no	—
33	C₂×C₄×C₂₃	—	4	2	no	—
34	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
35	C₈×C₂₃	—	4	2	no	—
36	C₈×C₂₃	D₈	1	2	yes	—
37	C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
38	C₈×C₂₃	#41	1	1	yes	characteristic
39	#2179	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂²×C₂₃	—	4	2	no	—
41	C₂²×C₂₃	#135	1	1	yes	characteristic
42	C₄×C₂₃	—	4	1	no	—
43	C₄×C₂₃	—	4	2	no	—
44	C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
45	C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
46	#508	C₂×C₂₃	1	1	yes	characteristic
47	#443	C₂×C₂₃	1	2	yes	complemented
48	#368	C₂×C₂₃	1	1	yes	characteristic, complemented
49	#359	C₂×C₂₃	1	2	yes	complemented
50	#346	C₂×C₂₃	1	1	yes	characteristic, complemented
51	C₂×C₂₃	—	2	1	no	—
52	C₂×C₂₃	—	8	2	no	—
53	C₂×C₂₃	C₄×D₈	1	1	yes	characteristic
54	#135	C₂²×C₂₃	1	2	yes	—
55	#131	—	2	2	no	—
56	Q₃₂	C₄×C₂₃	1	1	yes	characteristic, complemented
57	SD₃₂	C₄×C₂₃	1	2	yes	complemented
58	D₁₆	C₄×C₂₃	1	1	yes	characteristic, complemented
59	C₂×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
60	C₂×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
61	#108	C₂²×C₂₃	1	2	yes	—
62	#104	—	2	2	no	—
63	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
64	C₂₃	#2179	1	1	yes	characteristic, direct product factor, Sylow
65	#41	—	2	2	no	—
66	Q₁₆	C₂×C₄×C₂₃	1	2	yes	—
67	SD₁₆	—	2	2	no	—
68	D₈	C₂×C₄×C₂₃	1	2	yes	—
69	M₁₆	—	2	2	no	—
70	M₁₆	—	4	2	no	—
71	C₂×C₈	—	4	2	no	—
72	C₂×C₈	D₄×C₂₃	1	1	yes	characteristic
73	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
74	C₄²	D₄×C₂₃	1	1	yes	characteristic
75	C₁₆	—	2	1	no	—
76	C₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
77	C₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
78	Q₈	—	2	2	no	—
79	D₄	—	2	2	no	—
80	D₄	—	4	2	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
84	C₈	—	4	2	no	—
85	C₈	D₈×C₂₃	1	2	yes	—
86	C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
87	C₈	#41×C₂₃	1	1	yes	characteristic
88	C₂²	—	4	2	no	—
89	C₂²	C₂₃×#135	1	1	yes	characteristic
90	C₄	—	4	1	no	—
91	C₄	—	4	2	no	—
92	C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
93	C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	8	2	no	—
96	C₂	C₄×D₈×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₃	C₂×D₈
commutator	C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₈	C₂³×C₂₃
socle	C₂×C₂₃	C₄×D₈
prime radical	C₂₃×#508	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃ ◁ C₂×C₈×C₂₃ ◁ C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#443	C₂	semidirect
3. C₂₃×#368	C₂	semidirect
4. C₂₃×#359	C₂	semidirect
5. C₂₃×#346	C₂	semidirect
8. C₂₃×Q₃₂	C₄	semidirect
9. C₂₃×SD₃₂	C₄	semidirect
10. C₂₃×D₁₆	C₄	semidirect
39. #2179	C₂₃	direct (N × H)
47. #443	C₂×C₂₃	semidirect
48. #368	C₂×C₂₃	semidirect
49. #359	C₂×C₂₃	semidirect
50. #346	C₂×C₂₃	semidirect
56. Q₃₂	C₄×C₂₃	semidirect
57. SD₃₂	C₄×C₂₃	semidirect
58. D₁₆	C₄×C₂₃	semidirect
64. C₂₃	#2179	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#508	2
C₂₃×#443	2
C₂₃×#368	2
C₂₃×#359	2
C₂₃×#346	2
#2179	23

Subgroups — C23×#2179 #15669798

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2179 #15669798