Subgroups — C₂₃×#235 #11764264

Order 1104.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#67	C₂	1	2	yes	maximal, complemented
3	C₂₃×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×D₃×C₂₃	C₂	1	2	yes	maximal, complemented
5	C₂₃×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#41×C₂₃	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×D₃×C₂₃	C₂²	1	2	yes	—
9	C₃×C₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₃×C₄×C₂₃	C₂²	1	1	yes	characteristic
11	Dic₃×C₂₃	C₂²	1	2	yes	—
12	Q₈×C₂₃	—	3	1	no	—
13	D₄×C₂₃	—	3	2	no	—
14	D₄×C₂₃	—	3	1	no	—
15	C₂×C₄×C₂₃	—	3	2	no	—
16	C₂×C₄×C₂₃	D₃	1	1	yes	characteristic, complemented
17	C₂×C₃×C₂₃	—	2	1	no	—
18	C₂×C₃×C₂₃	C₂³	1	1	yes	characteristic
19	D₃×C₂₃	—	2	2	no	—
20	C₂²×C₂₃	—	3	2	no	—
21	C₂²×C₂₃	C₂×D₃	1	1	yes	characteristic
22	C₄×C₂₃	—	3	2	no	—
23	C₄×C₂₃	C₂×D₃	1	1	yes	characteristic
24	C₄×C₂₃	C₂×D₃	1	1	yes	characteristic
25	C₃×C₂₃	#41	1	1	yes	characteristic, complemented, Hall
26	#235	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
27	C₂×C₂₃	—	2	1	no	—
28	C₂×C₂₃	—	6	2	no	—
29	C₂×C₂₃	C₂²×D₃	1	1	yes	characteristic
30	C₂×C₃×C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
31	#67	C₂×C₂₃	1	2	yes	complemented
32	D₁₂	C₂×C₂₃	1	1	yes	characteristic, complemented
33	C₄×D₃	C₂×C₂₃	1	2	yes	complemented
34	Dic₆	C₂×C₂₃	1	1	yes	characteristic, complemented
35	C₂₃	#235	1	1	yes	characteristic, direct product factor, Sylow
36	#41	—	3	1	no	Sylow
37	C₂²×C₃	C₂²×C₂₃	1	1	yes	characteristic
38	C₂×D₃	C₂²×C₂₃	1	2	yes	—
39	C₃×C₄	C₂²×C₂₃	1	1	yes	characteristic
40	C₃×C₄	C₂²×C₂₃	1	1	yes	characteristic
41	Dic₃	C₂²×C₂₃	1	2	yes	—
42	Q₈	—	3	1	no	—
43	D₄	—	3	2	no	—
44	D₄	—	3	1	no	—
45	C₂×C₄	—	3	2	no	—
46	C₂×C₄	D₃×C₂₃	1	1	yes	characteristic, complemented
47	C₂×C₃	—	2	1	no	—
48	C₂×C₃	C₂³×C₂₃	1	1	yes	characteristic
49	D₃	—	2	2	no	—
50	C₂²	—	3	2	no	—
51	C₂²	C₂×D₃×C₂₃	1	1	yes	characteristic
52	C₄	—	3	2	no	—
53	C₄	C₂×D₃×C₂₃	1	1	yes	characteristic
54	C₄	C₂×D₃×C₂₃	1	1	yes	characteristic
55	C₃	#41×C₂₃	1	1	yes	characteristic, complemented, Sylow
56	C₂	—	2	1	no	—
57	C₂	—	6	2	no	—
58	C₂	C₂²×D₃×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₃	C₂×D₃
commutator, cosocle	C₂×C₃	C₂³×C₂₃
Frattini	C₂	C₂²×D₃×C₂₃
Fitting	C₂×C₃×C₄×C₂₃	C₂
socle	C₂×C₃×C₂₃	C₂³
hypercenter	C₂×C₄×C₂₃	D₃
hypocenter	C₃	#41×C₂₃
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₂₃ ◁ C₂×C₄×C₂₃

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₄×C₂₃	C₂	semidirect
2. C₂₃×#67	C₂	semidirect
3. C₂₃×D₁₂	C₂	semidirect
4. C₄×D₃×C₂₃	C₂	semidirect
5. C₂₃×Dic₆	C₂	semidirect
16. C₂×C₄×C₂₃	D₃	semidirect
25. C₃×C₂₃	#41	semidirect
26. #235	C₂₃	direct (N × H)
30. C₂×C₃×C₄	C₂×C₂₃	semidirect
31. #67	C₂×C₂₃	semidirect
32. D₁₂	C₂×C₂₃	semidirect
33. C₄×D₃	C₂×C₂₃	semidirect
34. Dic₆	C₂×C₂₃	semidirect
35. C₂₃	#235	direct (N × H)
46. C₂×C₄	D₃×C₂₃	semidirect
55. C₃	#41×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₂₃	2
C₂₃×#67	2
C₂₃×D₁₂	2
C₄×D₃×C₂₃	2
C₂₃×Dic₆	2
#41×C₂₃	3
#235	23

Subgroups — C23×#235 #11764264

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#235 #11764264