Subgroups — #15671586

Order 2944.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃₆₈	C₂	1	1	yes	maximal, characteristic
2	#13092251	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#368	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₈₄	—	2	1	no	—
5	C₂×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
6	C₂₃×#108	—	2	1	no	—
7	D₃₆₈	—	2	1	no	—
8	D₃₆₈	C₄	1	2	yes	—
9	C₂₃⋊C₃₂	—	2	1	no	—
10	C₂×D₉₂	—	4	1	no	—
11	M₁₆×C₂₃	—	4	1	no	—
12	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
13	D₁₈₄	—	2	2	no	—
14	D₁₈₄	—	4	1	no	—
15	C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₂₃	D₄	1	1	yes	characteristic
17	C₂²×D₂₃	—	8	1	no	—
18	C₂×C₄×C₂₃	D₈	1	1	yes	characteristic
19	D₉₂	—	4	2	no	—
20	D₉₂	—	8	1	no	—
21	C₈×C₂₃	—	8	1	no	—
22	C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₂₃	SD₁₆	1	1	yes	characteristic
24	#1419	—	23	1	no	maximal, Sylow
25	C₂²×C₂₃	SD₃₂	1	1	yes	characteristic
26	C₂×D₂₃	—	8	2	no	—
27	C₂×D₂₃	—	16	2	no	—
28	C₄×C₂₃	#102	1	1	yes	characteristic
29	C₄×C₂₃	D₁₆	1	1	yes	characteristic
30	C₂×D₁₆	—	23	1	no	—
31	M₆₄	—	23	1	no	—
32	#368	D₂₃	1	1	yes	characteristic, complemented
33	C₂×C₂₃	—	2	1	no	—
34	C₂×C₂₃	#357	1	1	yes	characteristic
35	D₂₃	—	16	2	no	—
36	C₂×D₈	—	46	1	no	—
37	D₁₆	—	23	2	no	—
38	D₁₆	—	46	1	no	—
39	C₂×C₁₆	C₂×D₂₃	1	1	yes	characteristic
40	#108	—	2	1	no	—
41	C₃₂	—	46	1	no	—
42	C₂₃	#1419	1	1	yes	characteristic, complemented, Sylow
43	C₂×D₄	—	92	1	no	—
44	D₈	—	46	2	no	—
45	D₈	—	92	1	no	—
46	M₁₆	—	4	1	no	—
47	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
48	C₁₆	C₄×D₂₃	1	1	yes	characteristic
49	C₁₆	D₉₂	1	1	yes	characteristic
50	C₂³	—	184	1	no	—
51	D₄	—	92	2	no	—
52	D₄	—	184	1	no	—
53	C₂×C₄	#67272	1	1	yes	characteristic
54	C₈	—	8	1	no	—
55	C₈	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
56	C₈	#67274	1	1	yes	characteristic
57	C₂²	—	368	2	no	—
58	C₂²	—	184	2	no	—
59	C₂²	#10625010	1	1	yes	characteristic
60	C₄	#10624991	1	1	yes	characteristic
61	C₄	#10625008	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	368	2	no	—
64	C₂	#13092217	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#13092217
commutator	C₁₆×C₂₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₃
Fitting	C₂₃×#368	C₂
socle	C₂×C₂₃	#357
prime radical	C₂×D₃₆₈	C₂
cosocle	C₂×C₁₆×C₂₃	C₂²
hypercenter	#368	D₂₃
hypocenter	C₂₃	#1419
LCS	G ▷ C₁₆×C₂₃ ▷ C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #368

Complements

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

N	H ≅ G/N	type
2. #13092251	C₂	semidirect
3. C₂₃×#368	C₂	semidirect
32. #368	D₂₃	semidirect
42. C₂₃	#1419	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃₆₈	2
#13092251	2
C₂₃×#368	2
#1419	23