Subgroups — C₂₃×#2916 #15670535

Order 2944.

440 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#568	C₂	1	2	yes	maximal
2	C₂×C₂₃×#131	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#430	C₂	1	6	yes	maximal, complemented
4	C₂×C₂₃×#108	C₂	1	6	yes	maximal, complemented
5	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
6	C₂₃×#131	—	2	8	no	—
7	C₂₃×#131	C₄	1	8	yes	—
8	C₂×M₁₆×C₂₃	C₂²	1	12	yes	—
9	C₂×M₁₆×C₂₃	C₂²	1	3	yes	—
10	C₂²×C₈×C₂₃	C₂²	1	3	yes	—
11	C₂₃×#108	C₂²	1	4	yes	—
12	C₂₃×#108	C₂²	1	12	yes	—
13	#41×C₂₃	C₂×C₄	1	4	yes	—
14	#41×C₂₃	D₄	1	4	yes	—
15	C₂×Q₈×C₂₃	Q₈	1	1	yes	characteristic
16	C₂×D₄×C₂₃	Q₈	1	3	yes	—
17	C₂²×C₄×C₂₃	C₂³	1	3	yes	—
18	M₁₆×C₂₃	—	2	12	no	—
19	M₁₆×C₂₃	C₂×C₄	1	12	yes	—
20	M₁₆×C₂₃	C₂³	1	8	yes	—
21	C₂×C₈×C₂₃	—	2	12	no	—
22	C₂×C₈×C₂₃	C₂×C₄	1	12	yes	—
23	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
24	C₂×C₈×C₂₃	C₂³	1	3	yes	—
25	C₂³×C₂₃	C₂×Q₈	1	3	yes	—
26	Q₈×C₂₃	C₄⋊C₄	1	4	yes	—
27	D₄×C₂₃	C₄⋊C₄	1	12	yes	—
28	C₂×C₄×C₂₃	C₂²×C₄	1	6	yes	—
29	C₂×C₄×C₂₃	C₂×D₄	1	6	yes	—
30	C₂×C₄×C₂₃	C₂×Q₈	1	3	yes	—
31	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
32	C₈×C₂₃	—	2	8	no	—
33	C₈×C₂₃	C₂²×C₄	1	2	yes	—
34	C₈×C₂₃	C₂²×C₄	1	6	yes	—
35	#2916	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂²×C₂₃	—	4	3	no	—
37	C₂²×C₂₃	C₂×C₄⋊C₄	1	6	yes	—
38	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
39	C₄×C₂₃	C₂×C₄⋊C₄	1	6	yes	—
40	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
41	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
42	#568	C₂×C₂₃	1	2	yes	—
43	C₂×#131	C₂×C₂₃	1	1	yes	characteristic
44	#430	C₂×C₂₃	1	6	yes	complemented
45	C₂×#108	C₂×C₂₃	1	6	yes	complemented
46	C₂×C₂₃	—	2	1	no	—
47	C₂×C₂₃	—	2	6	no	—
48	C₂×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
49	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
50	#131	—	2	8	no	—
51	#131	C₄×C₂₃	1	8	yes	—
52	C₂×M₁₆	C₂²×C₂₃	1	12	yes	—
53	C₂×M₁₆	C₂²×C₂₃	1	3	yes	—
54	C₂²×C₈	C₂²×C₂₃	1	3	yes	—
55	#108	C₂²×C₂₃	1	4	yes	—
56	#108	C₂²×C₂₃	1	12	yes	—
57	C₂₃	#2916	1	1	yes	characteristic, direct product factor, Sylow
58	#41	C₂×C₄×C₂₃	1	4	yes	—
59	#41	D₄×C₂₃	1	4	yes	—
60	C₂×Q₈	Q₈×C₂₃	1	1	yes	characteristic
61	C₂×D₄	Q₈×C₂₃	1	3	yes	—
62	C₂²×C₄	C₂³×C₂₃	1	3	yes	—
63	M₁₆	—	2	12	no	—
64	M₁₆	C₂×C₄×C₂₃	1	12	yes	—
65	M₁₆	C₂³×C₂₃	1	8	yes	—
66	C₂×C₈	—	2	12	no	—
67	C₂×C₈	C₂×C₄×C₂₃	1	12	yes	—
68	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
69	C₂×C₈	C₂³×C₂₃	1	3	yes	—
70	C₂³	C₂×Q₈×C₂₃	1	3	yes	—
71	Q₈	C₄⋊C₄×C₂₃	1	4	yes	—
72	D₄	C₄⋊C₄×C₂₃	1	12	yes	—
73	C₂×C₄	C₂²×C₄×C₂₃	1	6	yes	—
74	C₂×C₄	C₂×D₄×C₂₃	1	6	yes	—
75	C₂×C₄	C₂×Q₈×C₂₃	1	3	yes	—
76	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
77	C₈	—	2	8	no	—
78	C₈	C₂²×C₄×C₂₃	1	2	yes	—
79	C₈	C₂²×C₄×C₂₃	1	6	yes	—
80	C₂²	—	4	3	no	—
81	C₂²	C₂×C₄⋊C₄×C₂₃	1	6	yes	—
82	C₂²	C₂²×Q₈×C₂₃	1	1	yes	characteristic
83	C₄	C₂×C₄⋊C₄×C₂₃	1	6	yes	—
84	C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
85	C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	2	6	no	—
88	C₂	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₃	C₂²×D₄
commutator	C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
socle	C₂×C₂₃	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₂₃	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃ ◁ C₂×#41×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#430	C₂	semidirect
4. C₂×C₂₃×#108	C₂	semidirect
35. #2916	C₂₃	direct (N × H)
44. #430	C₂×C₂₃	semidirect
45. C₂×#108	C₂×C₂₃	semidirect
57. C₂₃	#2916	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×#568	2
C₂×C₂₃×#131	2
C₂₃×#430	2
C₂×C₂₃×#108	2
#2916	23

Subgroups — C23×#2916 #15670535

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2916 #15670535