Subgroups — #13092835

Order 1472.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10625072	C₂	1	1	yes	maximal, characteristic
2	C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#10624980	C₂	1	2	yes	maximal, complemented
5	C₁₆×D₂₃	C₂	1	2	yes	maximal, complemented
6	#67288	C₄	1	1	yes	characteristic
7	C₂×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	#67267	C₄	1	1	yes	characteristic
9	#67262	C₄	1	2	yes	—
10	C₈×D₂₃	C₂²	1	2	yes	—
11	C₁₆×C₂₃	C₂²	1	2	yes	—
12	C₂₃⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂₃⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂²⋊D₂₃	C₈	1	2	yes	—
16	D₉₂	C₈	1	1	yes	characteristic
17	C₄×D₂₃	C₂×C₄	1	2	yes	—
18	Dic₄₆	C₈	1	1	yes	characteristic
19	C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₈×C₂₃	C₂³	1	1	yes	characteristic
21	C₂₃⋊C₈	C₂×C₄	1	2	yes	—
22	C₂²×C₂₃	C₂×C₈	1	1	yes	characteristic
23	C₂×D₂₃	C₂×C₈	1	2	yes	—
24	C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
25	C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
26	Dic₂₃	C₂×C₈	1	2	yes	—
27	#504	—	23	1	no	maximal, Sylow
28	C₂×C₂₃	—	2	1	no	—
29	C₂×C₂₃	C₂²×C₈	1	1	yes	characteristic
30	D₂₃	—	2	2	no	—
31	#131	—	23	1	no	—
32	M₃₂	—	23	2	no	—
33	M₃₂	D₂₃	1	1	yes	characteristic, complemented
34	C₂×C₁₆	—	23	1	no	—
35	C₂×C₁₆	—	23	2	no	—
36	C₂₃	#504	1	1	yes	characteristic, complemented, Sylow
37	#41	—	23	1	no	—
38	M₁₆	—	23	1	no	—
39	M₁₆	—	23	2	no	—
40	C₂×C₈	—	23	2	no	—
41	C₂×C₈	C₂×D₂₃	1	1	yes	characteristic
42	C₁₆	—	23	1	no	—
43	C₁₆	—	23	1	no	—
44	C₁₆	C₂×D₂₃	1	2	yes	—
45	Q₈	—	23	1	no	—
46	D₄	—	23	2	no	—
47	D₄	—	23	1	no	—
48	C₂×C₄	—	23	2	no	—
49	C₂×C₄	C₄×D₂₃	1	1	yes	characteristic
50	C₈	—	23	2	no	—
51	C₈	C₄×D₂₃	1	1	yes	characteristic
52	C₈	C₂²×D₂₃	1	1	yes	characteristic
53	C₂²	—	23	2	no	—
54	C₂²	C₈×D₂₃	1	1	yes	characteristic
55	C₄	—	23	2	no	—
56	C₄	C₈×D₂₃	1	1	yes	characteristic
57	C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
58	C₂	—	2	1	no	—
59	C₂	—	46	2	no	—
60	C₂	C₂×C₈×D₂₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂²×D₂₃
commutator, socle	C₂×C₂₃	C₂²×C₈
Fitting	C₂₃×M₃₂	C₂
prime radical	#67288	C₄
cosocle	C₈×C₂₃	C₂³
hypercenter	M₃₂	D₂₃
hypocenter	C₂₃	#504
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂₃×M₃₂	C₂	semidirect
3. C₂×C₂₃⋊C₁₆	C₂	semidirect
4. #10624980	C₂	semidirect
5. C₁₆×D₂₃	C₂	semidirect
33. M₃₂	D₂₃	semidirect
36. C₂₃	#504	semidirect

Maximal subgroups

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

subgroup	index
#10625072	2
C₂₃×M₃₂	2
C₂×C₂₃⋊C₁₆	2
#10624980	2
C₁₆×D₂₃	2
#504	23