Subgroups — #13092439

Order 1472.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#67262	C₂	1	2	yes	maximal
2	C₄²×D₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	Dic₂₃⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₂₃	C₄	1	1	yes	characteristic
7	C₂×C₄×D₂₃	C₄	1	1	yes	characteristic
8	C₂×C₄×D₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₈×C₂₃	C₂²	1	2	yes	—
10	C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	C₄×Dic₂₃	C₄	1	1	yes	characteristic
12	C₄×Dic₂₃	C₄	1	1	yes	characteristic
13	C₄×Dic₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₂₃⋊C₈	C₂²	1	2	yes	—
15	#67262	—	2	4	no	—
16	C₂²×D₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
20	C₂×Dic₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂×Dic₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂×Dic₂₃	C₂×C₄	1	1	yes	characteristic
23	C₄×D₂₃	—	2	2	no	—
24	C₄×D₂₃	—	2	2	no	—
25	C₄×D₂₃	D₄	1	2	yes	—
26	C₄×D₂₃	Q₈	1	2	yes	—
27	C₈×C₂₃	—	2	2	no	—
28	C₂₃⋊C₈	—	2	2	no	—
29	C₂²×C₂₃	C₂²×C₄	1	1	yes	characteristic
30	C₂×D₂₃	—	2	1	no	—
31	C₂×D₂₃	—	2	1	no	—
32	C₂×D₂₃	C₄⋊C₄	1	2	yes	—
33	C₄×C₂₃	—	2	1	no	—
34	C₄×C₂₃	M₁₆	1	2	yes	complemented
35	C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
36	C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
37	Dic₂₃	—	2	1	no	—
38	Dic₂₃	C₄⋊C₄	1	2	yes	—
39	Dic₂₃	M₁₆	1	2	yes	complemented
40	#423	—	23	1	no	maximal, Sylow
41	C₂×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₂₃	C₂×M₁₆	1	1	yes	characteristic
43	C₂×C₂₃	C₂×M₁₆	1	1	yes	characteristic
44	D₂₃	—	2	2	no	—
45	C₂×M₁₆	—	23	2	no	—
46	C₂×C₄²	—	23	1	no	—
47	C₄⋊C₈	—	23	2	no	—
48	C₄⋊C₈	—	23	1	no	—
49	C₄⋊C₈	D₂₃	1	1	yes	characteristic, complemented
50	C₂₃	#423	1	1	yes	characteristic, complemented, Sylow
51	C₂²×C₄	—	23	1	no	—
52	C₂²×C₄	—	23	1	no	—
53	C₂²×C₄	—	23	1	no	—
54	M₁₆	—	46	4	no	—
55	C₂×C₈	—	23	2	no	—
56	C₂×C₈	C₂×D₂₃	1	2	yes	—
57	C₄²	—	23	1	no	—
58	C₄²	—	23	1	no	—
59	C₄²	—	23	1	no	—
60	C₄²	C₂×D₂₃	1	1	yes	characteristic
61	C₂³	—	23	1	no	—
62	C₂×C₄	—	23	1	no	—
63	C₂×C₄	—	23	1	no	—
64	C₂×C₄	—	23	2	no	—
65	C₂×C₄	—	23	2	no	—
66	C₂×C₄	—	23	1	no	—
67	C₂×C₄	—	46	2	no	—
68	C₂×C₄	—	46	2	no	—
69	C₂×C₄	C₄×D₂₃	1	1	yes	characteristic
70	C₂×C₄	C₄×D₂₃	1	1	yes	characteristic
71	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
72	C₈	—	2	2	no	—
73	C₈	—	46	2	no	—
74	C₂²	—	23	2	no	—
75	C₂²	—	46	1	no	—
76	C₂²	—	46	1	no	—
77	C₂²	C₂×C₄×D₂₃	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	—	23	2	no	—
80	C₄	—	23	2	no	—
81	C₄	—	46	1	no	—
82	C₄	#67262	1	2	yes	complemented
83	C₄	D₄×D₂₃	1	1	yes	characteristic
84	C₄	Q₈×D₂₃	1	1	yes	characteristic
85	C₂	—	46	2	no	—
86	C₂	C₄⋊C₄×D₂₃	1	1	yes	characteristic
87	C₂	C₂×#67262	1	1	yes	characteristic
88	C₂	M₁₆×D₂₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₃
commutator, socle	C₂²×C₂₃	C₂²×C₄
Fitting	C₂₃×C₄⋊C₈	C₂
prime radical	C₂²×D₂₃	C₂×C₄
cosocle	C₂×C₄×C₂₃	C₂³
hypercenter	C₄⋊C₈	D₂₃
hypocenter	C₂₃	#423
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. C₂₃×C₄⋊C₈	C₂	semidirect
4. Dic₂₃⋊C₈	C₂	semidirect
5. C₄⋊(C₂₃⋊C₈)	C₂	semidirect
34. C₄×C₂₃	M₁₆	semidirect
39. Dic₂₃	M₁₆	semidirect
49. C₄⋊C₈	D₂₃	semidirect
50. C₂₃	#423	semidirect
82. C₄	#67262	semidirect

Maximal subgroups

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

subgroup	index
C₂×#67262	2
C₄²×D₂₃	2
C₂₃×C₄⋊C₈	2
Dic₂₃⋊C₈	2
C₄⋊(C₂₃⋊C₈)	2
#423	23