Subgroups — #13093146

Order 1472.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₂₃	C₂	1	2	yes	maximal
3	#10625061	C₂	1	6	yes	maximal
4	#10625040	C₂	1	3	yes	maximal
5	C₄×Dic₄₆	C₂	1	3	yes	maximal
6	C₂×Q₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×Dic₄₆	C₂²	1	3	yes	—
8	C₄⋊C₄×C₂₃	C₂²	1	3	yes	—
9	C₄²×C₂₃	C₂²	1	3	yes	—
10	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₂₃	C₂²	1	6	yes	—
12	C₄⋊Dic₂₃	C₂²	1	6	yes	—
13	Dic₂₃⋊C₄	C₂²	1	6	yes	—
14	C₄×Dic₂₃	C₂²	1	6	yes	—
15	Q₈×C₂₃	Q₈	1	4	yes	complemented
16	C₂×C₄×C₂₃	C₂³	1	3	yes	—
17	C₂×C₄×C₂₃	C₂³	1	3	yes	—
18	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
19	C₂×Dic₂₃	C₂³	1	2	yes	—
20	C₂×Dic₂₃	C₂³	1	6	yes	—
21	Dic₄₆	—	2	6	no	—
22	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
23	C₄×C₂₃	—	2	3	no	—
24	C₄×C₂₃	C₂×Q₈	1	6	yes	—
25	C₄×C₂₃	#41	1	2	yes	—
26	Dic₂₃	—	2	6	no	—
27	Dic₂₃	—	2	2	no	—
28	#557	—	23	1	no	maximal, Sylow
29	C₂×C₂₃	C₂²×Q₈	1	1	yes	characteristic
30	C₂×C₂₃	C₂×#41	1	1	yes	characteristic
31	C₂×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
32	C₄⋊Q₈	—	23	3	no	—
33	#125	—	23	6	no	—
34	C₄×Q₈	—	23	2	no	—
35	C₄×Q₈	—	23	3	no	—
36	C₄×Q₈	D₂₃	1	1	yes	characteristic
37	C₂₃	#557	1	1	yes	characteristic, complemented, Sylow
38	C₂×Q₈	—	23	3	no	—
39	C₂×Q₈	C₂×D₂₃	1	1	yes	characteristic
40	C₄⋊C₄	—	23	1	no	—
41	C₄⋊C₄	—	23	6	no	—
42	C₄⋊C₄	—	23	6	no	—
43	C₄⋊C₄	—	23	6	no	—
44	C₄⋊C₄	C₂×D₂₃	1	3	yes	—
45	C₄²	—	23	6	no	—
46	C₄²	C₂×D₂₃	1	3	yes	—
47	Q₈	—	46	6	no	—
48	Q₈	Dic₄₆	1	4	yes	complemented
49	C₂×C₄	—	23	2	no	—
50	C₂×C₄	—	23	6	no	—
51	C₂×C₄	C₂²×D₂₃	1	3	yes	—
52	C₂×C₄	C₂²×D₂₃	1	3	yes	—
53	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
54	C₂²	C₂³×D₂₃	1	1	yes	characteristic
55	C₄	—	2	3	no	—
56	C₄	—	46	6	no	—
57	C₄	—	46	2	no	—
58	C₄	C₂×Dic₄₆	1	6	yes	—
59	C₄	#67292	1	2	yes	—
60	C₂	C₂²×Dic₄₆	1	1	yes	characteristic
61	C₂	C₂×#67292	1	1	yes	characteristic
62	C₂	#10625161	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator, socle, prime radical, cosocle	C₂²×C₂₃	C₂⁴
Fitting	C₄×Q₈×C₂₃	C₂
hypercenter	C₄×Q₈	D₂₃
hypocenter	C₂₃	#557
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
15. Q₈×C₂₃	Q₈	semidirect
37. C₂₃	#557	semidirect
48. Q₈	Dic₄₆	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₈×C₂₃	2
Q₈×Dic₂₃	2
#10625061	2
#10625040	2
C₄×Dic₄₆	2
#557	23