Subgroups — #13092733

Order 1472.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#121	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#10625014	C₂	1	2	yes	maximal, complemented
4	C₂²⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	#10624990	C₂	1	2	yes	maximal, complemented
6	C₂×D₄×C₂₃	—	2	1	no	—
7	C₂×D₄×C₂₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×Dic₂₃	—	2	1	no	—
10	C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²⋊C₄×C₂₃	—	2	1	no	—
12	C₄⋊Dic₂₃	—	2	1	no	—
13	C₄⋊Dic₂₃	C₂²	1	2	yes	—
14	C₂×C₂₃⋊C₈	C₂²	1	2	yes	—
15	C₂³×C₂₃	—	2	1	no	—
16	C₂³×C₂₃	D₄	1	1	yes	characteristic
17	D₄×C₂₃	—	2	2	no	—
18	D₄×C₂₃	—	4	2	no	—
19	C₂×C₄×C₂₃	—	2	2	no	—
20	C₂×C₄×C₂₃	—	2	1	no	—
21	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
22	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
23	C₂×Dic₂₃	—	2	2	no	—
24	C₂×Dic₂₃	—	4	2	no	—
25	C₂₃⋊C₈	—	2	2	no	—
26	C₂²×C₂₃	—	2	1	no	—
27	C₂²×C₂₃	—	2	1	no	—
28	C₂²×C₂₃	—	4	1	no	—
29	C₂²×C₂₃	—	4	1	no	—
30	C₂²×C₂₃	—	4	1	no	—
31	C₂²×C₂₃	SD₁₆	1	2	yes	—
32	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
33	C₄×C₂₃	—	2	1	no	—
34	C₄×C₂₃	—	4	1	no	—
35	C₄×C₂₃	#41	1	2	yes	—
36	Dic₂₃	—	4	2	no	—
37	#481	—	23	1	no	maximal, Sylow
38	C₂×C₂₃	—	2	2	no	—
39	C₂×C₂₃	—	8	1	no	—
40	C₂×C₂₃	#123	1	1	yes	characteristic
41	C₂×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
42	C₂×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
43	#121	D₂₃	1	1	yes	characteristic
44	C₂×C₄⋊C₄	—	23	1	no	—
45	#106	—	23	2	no	—
46	#102	—	23	2	no	—
47	C₂²⋊C₈	—	23	1	no	—
48	C₂₃	#481	1	1	yes	characteristic, complemented, Sylow
49	C₂×D₄	—	2	1	no	—
50	C₂×D₄	C₂×D₂₃	1	1	yes	characteristic
51	C₂²×C₄	—	46	1	no	—
52	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
53	C₂×C₈	—	23	2	no	—
54	C₄⋊C₄	—	23	2	no	—
55	C₄⋊C₄	—	46	1	no	—
56	C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
57	C₂²⋊C₄	—	2	1	no	—
58	C₂³	—	2	1	no	—
59	C₂³	C₂²⋊D₂₃	1	1	yes	characteristic
60	D₄	—	2	2	no	—
61	D₄	—	4	2	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	46	2	no	—
65	C₂×C₄	—	92	2	no	—
66	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
67	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
68	C₈	—	46	2	no	—
69	C₂²	—	2	1	no	—
70	C₂²	—	2	1	no	—
71	C₂²	—	4	1	no	—
72	C₂²	—	4	1	no	—
73	C₂²	—	4	1	no	—
74	C₂²	#67273	1	2	yes	—
75	C₂²	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	4	1	no	—
78	C₄	—	92	2	no	—
79	C₄	#67290	1	2	yes	—
80	C₂	—	2	2	no	—
81	C₂	—	8	1	no	—
82	C₂	C₂×#67273	1	1	yes	characteristic
83	C₂	#10625106	1	1	yes	characteristic
84	C₂	#10625120	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₃
Fitting, prime radical	C₂₃×#121	C₂
socle	C₂²×C₂₃	C₂×D₄
hypercenter	#121	D₂₃
hypocenter	C₂₃	#481
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #121

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊Dic₂₃	C₂	semidirect
3. #10625014	C₂	semidirect
4. C₂²⋊(C₂₃⋊C₈)	C₂	semidirect
5. #10624990	C₂	semidirect
48. C₂₃	#481	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#121	2
C₂×C₄⋊Dic₂₃	2
#10625014	2
C₂²⋊(C₂₃⋊C₈)	2
#10624990	2
#481	23