Subgroups — #15673549

Order 2944.

300 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₂₃⋊C₈)	C₂	1	6	yes	maximal
2	C₄³×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄²×C₂₃	C₄	1	3	yes	—
4	C₂×C₄²×C₂₃	C₄	1	1	yes	characteristic
5	C₂×C₄²×C₂₃	C₂²	1	3	yes	—
6	C₂²×C₂₃⋊C₈	C₂²	1	4	yes	—
7	C₄⋊(C₂₃⋊C₈)	—	2	12	no	—
8	C₂²×C₄×C₂₃	C₂×C₄	1	3	yes	—
9	C₂²×C₄×C₂₃	C₂×C₄	1	3	yes	—
10	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
11	C₄²×C₂₃	—	2	6	no	—
12	C₄²×C₂₃	C₈	1	4	yes	complemented
13	C₄²×C₂₃	D₄	1	6	yes	—
14	C₄²×C₂₃	Q₈	1	6	yes	—
15	C₂×C₂₃⋊C₈	—	2	12	no	—
16	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
17	C₂×C₄×C₂₃	—	2	6	no	—
18	C₂×C₄×C₂₃	C₄⋊C₄	1	6	yes	—
19	C₂×C₄×C₂₃	C₄⋊C₄	1	6	yes	—
20	C₂×C₄×C₂₃	C₂×C₈	1	6	yes	—
21	C₂×C₄×C₂₃	M₁₆	1	6	yes	—
22	C₂×C₄×C₂₃	C₂×D₄	1	3	yes	—
23	C₂×C₄×C₂₃	C₂×Q₈	1	3	yes	—
24	C₂₃⋊C₈	—	4	4	no	—
25	#1842	—	23	1	no	maximal, Sylow
26	C₂²×C₂₃	C₂×C₄⋊C₄	1	3	yes	—
27	C₂²×C₂₃	C₂²×C₈	1	1	yes	characteristic
28	C₂²×C₂₃	C₂×M₁₆	1	3	yes	—
29	C₄×C₂₃	—	2	6	no	—
30	C₄×C₂₃	C₄⋊C₈	1	12	yes	complemented
31	C₄×C₂₃	#127	1	1	yes	characteristic
32	C₄×C₂₃	C₄⋊Q₈	1	3	yes	—
33	C₂×C₄⋊C₈	—	23	6	no	—
34	C₄³	D₂₃	1	1	yes	characteristic
35	C₂×C₂₃	#384	1	1	yes	characteristic
36	C₂×C₂₃	C₂×C₄⋊C₈	1	3	yes	—
37	C₂×C₂₃	#423	1	3	yes	—
38	C₂²×C₈	—	23	4	no	—
39	C₂×C₄²	Dic₂₃	1	3	yes	—
40	C₂×C₄²	Dic₂₃	1	1	yes	characteristic
41	C₂×C₄²	C₂×D₂₃	1	3	yes	—
42	C₄⋊C₈	—	46	12	no	—
43	C₂₃	#1842	1	1	yes	characteristic, complemented, Sylow
44	C₂²×C₄	C₂×Dic₂₃	1	3	yes	—
45	C₂²×C₄	C₂×Dic₂₃	1	3	yes	—
46	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
47	C₂×C₈	—	46	12	no	—
48	C₄²	—	2	6	no	—
49	C₄²	C₂₃⋊C₈	1	4	yes	complemented
50	C₄²	Dic₄₆	1	6	yes	—
51	C₄²	D₉₂	1	6	yes	—
52	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
53	C₂×C₄	—	2	6	no	—
54	C₂×C₄	C₂×C₂₃⋊C₈	1	6	yes	—
55	C₂×C₄	#67267	1	6	yes	—
56	C₂×C₄	C₄⋊Dic₂₃	1	6	yes	—
57	C₂×C₄	C₄⋊Dic₂₃	1	6	yes	—
58	C₂×C₄	C₂×Dic₄₆	1	3	yes	—
59	C₂×C₄	C₂×D₉₂	1	3	yes	—
60	C₈	—	92	4	no	—
61	C₂²	C₂²×C₂₃⋊C₈	1	1	yes	characteristic
62	C₂²	C₂×#67267	1	3	yes	—
63	C₂²	C₂×C₄⋊Dic₂₃	1	3	yes	—
64	C₄	—	2	6	no	—
65	C₄	C₄⋊(C₂₃⋊C₈)	1	12	yes	complemented
66	C₄	#10625040	1	3	yes	—
67	C₄	#10625045	1	1	yes	characteristic
68	C₂	#13092287	1	1	yes	characteristic
69	C₂	C₂×C₄⋊(C₂₃⋊C₈)	1	3	yes	—
70	C₂	#13092441	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₃
commutator	C₂²×C₂₃	C₂²×C₈
Fitting	C₄³×C₂₃	C₂
socle, prime radical	C₂³×C₂₃	C₂²×C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₄³	D₂₃
hypocenter	C₂₃	#1842
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

Complements

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

N	H ≅ G/N	type
12. C₄²×C₂₃	C₈	semidirect
30. C₄×C₂₃	C₄⋊C₈	semidirect
43. C₂₃	#1842	semidirect
49. C₄²	C₂₃⋊C₈	semidirect
65. C₄	C₄⋊(C₂₃⋊C₈)	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₂₃⋊C₈)	2
C₄³×C₂₃	2
#1842	23