Subgroups — #13092212

Order 1472.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#124	C₂	1	1	yes	maximal, characteristic
2	#10625016	C₂	1	2	yes	maximal
3	C₂×Q₈×C₂₃	C₄	1	1	yes	characteristic, complemented
4	C₂×D₄×C₂₃	C₂²	1	1	yes	characteristic
5	C₂²⋊C₄×C₂₃	—	2	2	no	—
6	C₄²×C₂₃	C₄	1	1	yes	characteristic, complemented
7	C₂²⋊Dic₂₃	—	2	2	no	—
8	C₂³×C₂₃	D₄	1	2	yes	—
9	Q₈×C₂₃	—	4	1	no	—
10	D₄×C₂₃	—	4	1	no	—
11	C₂×C₄×C₂₃	—	2	1	no	—
12	C₂×C₄×C₂₃	—	2	1	no	—
13	C₂×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
14	C₂×Dic₂₃	—	4	2	no	—
15	C₂²×C₂₃	—	2	2	no	—
16	C₂²×C₂₃	—	4	2	no	—
17	C₂²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
18	C₄×C₂₃	—	2	1	no	—
19	C₄×C₂₃	—	4	1	no	—
20	C₄×C₂₃	—	4	1	no	—
21	Dic₂₃	—	8	2	no	—
22	#354	—	23	1	no	maximal, Sylow
23	C₂×C₂₃	—	2	1	no	—
24	C₂×C₂₃	—	4	2	no	—
25	C₂×C₂₃	#99	1	1	yes	characteristic
26	#124	D₂₃	1	1	yes	characteristic
27	#99	—	23	2	no	—
28	C₂₃	#354	1	1	yes	characteristic, complemented, Sylow
29	C₂×Q₈	Dic₂₃	1	1	yes	characteristic, complemented
30	C₂×D₄	C₂×D₂₃	1	1	yes	characteristic
31	C₂²⋊C₄	—	2	2	no	—
32	C₂²⋊C₄	—	46	2	no	—
33	C₄²	Dic₂₃	1	1	yes	characteristic, complemented
34	C₂³	C₂²⋊D₂₃	1	2	yes	—
35	Q₈	—	4	1	no	—
36	D₄	—	4	1	no	—
37	C₂×C₄	—	2	1	no	—
38	C₂×C₄	—	2	1	no	—
39	C₂×C₄	—	92	2	no	—
40	C₂×C₄	C₂×Dic₂₃	1	1	yes	characteristic
41	C₂²	—	2	2	no	—
42	C₂²	—	4	2	no	—
43	C₂²	C₂²⋊Dic₂₃	1	1	yes	characteristic
44	C₄	—	2	1	no	—
45	C₄	—	4	1	no	—
46	C₄	—	4	1	no	—
47	C₄	—	184	2	no	—
48	C₂	—	2	1	no	—
49	C₂	—	4	2	no	—
50	C₂	#10625016	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#10625016
commutator	C₂×C₄×C₂₃	C₂×C₄
Frattini	C₂×D₄	C₂×D₂₃
Fitting	C₂₃×#124	C₂
socle	C₂×C₂₃	#99
prime radical, cosocle	C₂×D₄×C₂₃	C₂²
hypercenter	#124	D₂₃
hypocenter	C₂₃	#354
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #124

Complements

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

N	H ≅ G/N	type
3. C₂×Q₈×C₂₃	C₄	semidirect
6. C₄²×C₂₃	C₄	semidirect
28. C₂₃	#354	semidirect
29. C₂×Q₈	Dic₂₃	semidirect
33. C₄²	Dic₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#124	2
#10625016	2
#354	23