Subgroups — #13092539

Order 1472.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#135	C₂	1	1	yes	maximal, characteristic
2	#10625119	C₂	1	2	yes	maximal, complemented
3	#10625019	C₂	1	2	yes	maximal, complemented
4	#10625001	C₂	1	1	yes	maximal, characteristic, complemented
5	#10624997	C₂	1	1	yes	maximal, characteristic, complemented
6	#41×C₂₃	C₂²	1	2	yes	—
7	Q₁₆×C₂₃	C₄	1	1	yes	characteristic, complemented
8	SD₁₆×C₂₃	C₄	1	2	yes	complemented
9	D₈×C₂₃	C₄	1	1	yes	characteristic, complemented
10	C₂×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₄×Dic₂₃	C₂²	1	1	yes	characteristic
12	#67267	—	2	2	no	—
13	#67267	C₂²	1	2	yes	—
14	C₂×C₂₃⋊C₈	—	2	2	no	—
15	C₂×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
16	Q₈×C₂₃	C₂×C₄	1	2	yes	—
17	D₄×C₂₃	—	2	2	no	—
18	D₄×C₂₃	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₂₃	—	2	2	no	—
20	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
21	C₂×Dic₂₃	—	2	1	no	—
22	C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂₃⋊C₈	—	2	2	no	—
25	C₂₃⋊C₈	D₄	1	2	yes	—
26	C₂²×C₂₃	—	2	2	no	—
27	C₂²×C₂₃	#41	1	1	yes	characteristic
28	C₄×C₂₃	—	2	2	no	—
29	C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
31	Dic₂₃	—	4	1	no	—
32	#444	—	23	1	no	maximal, Sylow
33	C₂×C₂₃	—	2	1	no	—
34	C₂×C₂₃	—	4	2	no	—
35	C₂×C₂₃	C₄×D₄	1	1	yes	characteristic
36	#135	D₂₃	1	1	yes	characteristic
37	#131	—	23	2	no	—
38	#108	—	23	1	no	—
39	#104	—	23	2	no	—
40	#97	—	23	1	no	—
41	C₂₃	#444	1	1	yes	characteristic, complemented, Sylow
42	#41	C₂×D₂₃	1	2	yes	—
43	Q₁₆	Dic₂₃	1	1	yes	characteristic, complemented
44	SD₁₆	Dic₂₃	1	2	yes	complemented
45	D₈	Dic₂₃	1	1	yes	characteristic, complemented
46	M₁₆	—	23	2	no	—
47	M₁₆	—	46	2	no	—
48	C₂×C₈	—	23	1	no	—
49	C₂×C₈	—	46	2	no	—
50	C₂×C₈	C₂×D₂₃	1	1	yes	characteristic
51	C₄²	—	23	1	no	—
52	Q₈	C₂×Dic₂₃	1	2	yes	—
53	D₄	—	2	2	no	—
54	D₄	C₂×Dic₂₃	1	2	yes	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	46	1	no	—
57	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
58	C₈	—	23	2	no	—
59	C₈	—	46	2	no	—
60	C₈	C₂×Dic₂₃	1	1	yes	characteristic
61	C₈	C₂×Dic₂₃	1	1	yes	characteristic
62	C₂²	—	2	2	no	—
63	C₂²	#67290	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	92	1	no	—
66	C₄	D₄×D₂₃	1	1	yes	characteristic
67	C₄	C₂²×Dic₂₃	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	4	2	no	—
70	C₂	D₄×Dic₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₂₃
commutator	C₄×C₂₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₃
Fitting, prime radical	C₂₃×#135	C₂
socle	C₂×C₂₃	C₄×D₄
cosocle	C₂×C₄×C₂₃	C₂³
hypercenter	#135	D₂₃
hypocenter	C₂₃	#444
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
2. #10625119	C₂	semidirect
3. #10625019	C₂	semidirect
4. #10625001	C₂	semidirect
5. #10624997	C₂	semidirect
7. Q₁₆×C₂₃	C₄	semidirect
8. SD₁₆×C₂₃	C₄	semidirect
9. D₈×C₂₃	C₄	semidirect
41. C₂₃	#444	semidirect
43. Q₁₆	Dic₂₃	semidirect
44. SD₁₆	Dic₂₃	semidirect
45. D₈	Dic₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#135	2
#10625119	2
#10625019	2
#10625001	2
#10624997	2
#444	23