Subgroups — #13092771

Order 1472.

96 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	#10625041	C₂	1	1	yes	maximal, characteristic, complemented
3	#10625017	C₂	1	2	yes	maximal, complemented
4	#10625014	C₂	1	2	yes	maximal
5	C₄×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×D₄×C₂₃	C₂²	1	1	yes	characteristic
8	C₂²⋊C₄×C₂₃	—	2	2	no	—
9	C₄²×C₂₃	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₂₃	C₂²	1	2	yes	—
11	Dic₂₃⋊C₄	—	2	2	no	—
12	C₂×C₂₃⋊C₈	C₂²	1	2	yes	—
13	C₂³×C₂₃	—	2	1	no	—
14	Q₈×C₂₃	—	2	2	no	—
15	D₄×C₂₃	—	2	2	no	—
16	C₂×C₄×C₂₃	—	2	1	no	—
17	C₂×C₄×C₂₃	D₄	1	2	yes	—
18	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
19	C₂×Dic₂₃	—	2	2	no	—
20	C₂₃⋊C₈	—	2	2	no	—
21	C₂²×C₂₃	—	4	1	no	—
22	C₂²×C₂₃	—	4	2	no	—
23	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
24	C₄×C₂₃	—	2	2	no	—
25	C₄×C₂₃	—	4	1	no	—
26	C₄×C₂₃	#41	1	2	yes	—
27	Dic₂₃	—	4	2	no	—
28	#488	—	23	1	no	maximal, Sylow
29	C₂×C₂₃	—	8	1	no	—
30	C₂×C₂₃	#124	1	1	yes	characteristic
31	C₂×C₂₃	#135	1	2	yes	—
32	#125	—	23	1	no	—
33	#124	D₂₃	1	1	yes	characteristic
34	#103	—	23	2	no	—
35	#102	—	23	2	no	—
36	C₄×C₈	—	23	1	no	—
37	C₂₃	#488	1	1	yes	characteristic, complemented, Sylow
38	C₂×Q₈	C₂×D₂₃	1	1	yes	characteristic
39	C₂×D₄	C₂×D₂₃	1	1	yes	characteristic
40	C₂×C₈	—	23	2	no	—
41	C₄⋊C₄	—	23	2	no	—
42	C₄⋊C₄	—	46	2	no	—
43	C₂²⋊C₄	—	2	2	no	—
44	C₄²	C₂×D₂₃	1	1	yes	characteristic
45	C₂³	—	2	1	no	—
46	Q₈	—	2	2	no	—
47	D₄	—	2	2	no	—
48	C₂×C₄	—	2	1	no	—
49	C₂×C₄	—	46	2	no	—
50	C₂×C₄	C₂²⋊D₂₃	1	2	yes	—
51	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
52	C₈	—	46	2	no	—
53	C₂²	—	4	1	no	—
54	C₂²	—	4	2	no	—
55	C₂²	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	—	4	1	no	—
58	C₄	—	92	2	no	—
59	C₄	#67290	1	2	yes	—
60	C₂	—	8	1	no	—
61	C₂	#10625107	1	1	yes	characteristic
62	C₂	#10625121	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₃
Fitting	C₂₃×#124	C₂
socle	C₂²×C₂₃	C₂×D₄
prime radical	C₂×D₄×C₂₃	C₂²
hypercenter	#124	D₂₃
hypocenter	C₂₃	#488
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #124

Complements

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

N	H ≅ G/N	type
2. #10625041	C₂	semidirect
3. #10625017	C₂	semidirect
5. C₄×C₂₃⋊C₈	C₂	semidirect
37. C₂₃	#488	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
#10625041	2
#10625017	2
#10625014	2
C₄×C₂₃⋊C₈	2
#488	23