Subgroups — #13092180

Order 1472.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₂₃	C₂	1	2	yes	maximal
3	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
4	C₂²×Dic₂₃	—	2	2	no	—
5	C₂×C₈×C₂₃	—	2	2	no	—
6	C₂×C₈×C₂₃	C₄	1	2	yes	complemented
7	C₄⋊Dic₂₃	—	2	2	no	—
8	C₄⋊Dic₂₃	C₄	1	4	yes	complemented
9	C₂³×C₂₃	D₄	1	1	yes	characteristic
10	C₂×C₄×C₂₃	C₂×C₄	1	2	yes	—
11	C₂×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄×C₂₃	D₄	1	2	yes	—
13	C₂×C₄×C₂₃	Q₈	1	1	yes	characteristic
14	C₂×Dic₂₃	—	2	4	no	—
15	C₂×Dic₂₃	—	4	2	no	—
16	C₂×Dic₂₃	—	4	2	no	—
17	C₈×C₂₃	—	2	2	no	—
18	C₂²×C₂₃	C₂²⋊C₄	1	2	yes	—
19	C₂²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
20	C₂²×C₂₃	D₈	1	1	yes	characteristic
21	C₂²×C₂₃	SD₁₆	1	2	yes	—
22	C₂²×C₂₃	Q₁₆	1	1	yes	characteristic
23	C₄×C₂₃	C₄²	1	1	yes	characteristic
24	C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₄×C₂₃	C₄⋊C₄	1	2	yes	—
26	Dic₂₃	—	4	4	no	—
27	#340	—	23	1	no	maximal, Sylow
28	C₂×C₂₃	#95	1	1	yes	characteristic
29	C₂×C₂₃	#102	1	2	yes	—
30	C₂×C₂₃	#103	1	2	yes	—
31	C₂×C₂₃	#106	1	1	yes	characteristic
32	C₂×C₂₃	#107	1	1	yes	characteristic
33	C₂²×C₈	D₂₃	1	1	yes	characteristic
34	C₂×C₄⋊C₄	—	23	2	no	—
35	C₂₃	#340	1	1	yes	characteristic, complemented, Sylow
36	C₂²×C₄	—	46	2	no	—
37	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
38	C₂×C₈	—	2	2	no	—
39	C₂×C₈	Dic₂₃	1	2	yes	complemented
40	C₄⋊C₄	—	23	4	no	—
41	C₄⋊C₄	—	46	2	no	—
42	C₂³	D₉₂	1	1	yes	characteristic
43	C₂×C₄	—	46	4	no	—
44	C₂×C₄	—	92	2	no	—
45	C₂×C₄	—	92	2	no	—
46	C₂×C₄	Dic₄₆	1	1	yes	characteristic
47	C₂×C₄	C₄×D₂₃	1	2	yes	—
48	C₂×C₄	C₂×Dic₂₃	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊D₂₃	1	2	yes	—
50	C₈	—	2	2	no	—
51	C₂²	#67263	1	2	yes	—
52	C₂²	D₁₈₄	1	1	yes	characteristic
53	C₂²	Dic₉₂	1	1	yes	characteristic
54	C₂²	C₄⋊Dic₂₃	1	1	yes	characteristic
55	C₂²	(C₂×D₂₃)⋊C₄	1	2	yes	—
56	C₄	—	92	4	no	—
57	C₄	C₄×Dic₂₃	1	1	yes	characteristic
58	C₄	Dic₂₃⋊C₄	1	2	yes	—
59	C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
60	C₂	#10624998	1	2	yes	—
61	C₂	#10624999	1	1	yes	characteristic
62	C₂	#10625000	1	1	yes	characteristic
63	C₂	#10625003	1	2	yes	—
64	C₂	C₂₃⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₉₂
commutator	C₄×C₂₃	C₄²
Frattini	C₂²×C₄	C₂×D₂₃
Fitting	C₂²×C₈×C₂₃	C₂
socle, prime radical	C₂³×C₂₃	D₄
cosocle	C₂²×C₄×C₂₃	C₂²
hypercenter	C₂²×C₈	D₂₃
hypocenter	C₂₃	#340
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
6. C₂×C₈×C₂₃	C₄	semidirect
8. C₄⋊Dic₂₃	C₄	semidirect
35. C₂₃	#340	semidirect
39. C₂×C₈	Dic₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₈×C₂₃	2
C₂×C₄⋊Dic₂₃	2
#340	23