Subgroups — #15675706

Order 2944.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092849	C₂	1	2	yes	maximal, complemented
2	Dic₃₆₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092256	C₂	1	2	yes	maximal, complemented
4	D₇₃₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
6	#10625075	—	2	2	no	—
7	C₂×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
8	Dic₁₈₄	C₂²	1	2	yes	—
9	#10624982	—	2	2	no	—
10	D₃₆₈	C₂²	1	2	yes	—
11	C₂₃×C₃₂	C₂²	1	1	yes	characteristic
12	C₂₃×C₃₂	C₂²	1	1	yes	characteristic
13	#67288	—	4	2	no	—
14	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
15	Dic₉₂	—	2	2	no	—
16	D₁₈₄	—	2	2	no	—
17	#67263	—	4	2	no	—
18	C₁₆×C₂₃	D₄	1	1	yes	characteristic
19	C₁₆×C₂₃	C₂³	1	1	yes	characteristic
20	C₂×C₄×C₂₃	D₈	1	1	yes	characteristic
21	C₂²⋊D₂₃	—	8	2	no	—
22	D₉₂	—	4	2	no	—
23	C₄×D₂₃	—	8	2	no	—
24	Dic₄₆	—	4	2	no	—
25	C₈×C₂₃	D₈	1	1	yes	characteristic
26	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
27	#2262	—	23	1	no	maximal, Sylow
28	C₂²×C₂₃	D₁₆	1	1	yes	characteristic
29	C₂×D₂₃	—	8	2	no	—
30	C₄×C₂₃	D₁₆	1	1	yes	characteristic
31	C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
32	Dic₂₃	—	8	2	no	—
33	#508	—	23	2	no	—
34	Q₆₄	—	23	1	no	—
35	SD₆₄	—	23	2	no	—
36	D₃₂	—	23	1	no	—
37	C₂×C₃₂	D₂₃	1	1	yes	characteristic, complemented
38	C₂×C₂₃	—	2	1	no	—
39	C₂×C₂₃	C₂×D₁₆	1	1	yes	characteristic
40	D₂₃	—	16	2	no	—
41	#135	—	46	2	no	—
42	Q₃₂	—	23	2	no	—
43	SD₃₂	—	46	2	no	—
44	D₁₆	—	23	2	no	—
45	C₂×C₁₆	C₂×D₂₃	1	1	yes	characteristic
46	C₃₂	C₂×D₂₃	1	1	yes	characteristic
47	C₃₂	C₂×D₂₃	1	1	yes	characteristic
48	C₂₃	#2262	1	1	yes	characteristic, complemented, Sylow
49	#41	—	92	2	no	—
50	Q₁₆	—	46	2	no	—
51	SD₁₆	—	92	2	no	—
52	D₈	—	46	2	no	—
53	C₂×C₈	D₉₂	1	1	yes	characteristic
54	C₁₆	D₉₂	1	1	yes	characteristic
55	C₁₆	C₂²×D₂₃	1	1	yes	characteristic
56	Q₈	—	92	2	no	—
57	D₄	—	92	2	no	—
58	D₄	—	184	2	no	—
59	C₂×C₄	—	184	2	no	—
60	C₂×C₄	D₁₈₄	1	1	yes	characteristic
61	C₈	D₁₈₄	1	1	yes	characteristic
62	C₈	C₂×D₉₂	1	1	yes	characteristic
63	C₂²	—	184	2	no	—
64	C₂²	D₃₆₈	1	1	yes	characteristic
65	C₄	—	184	2	no	—
66	C₄	D₃₆₈	1	1	yes	characteristic
67	C₄	C₂×D₁₈₄	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	368	2	no	—
70	C₂	C₂×D₃₆₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₃₆₈
commutator, cosocle	C₁₆×C₂₃	C₂³
Frattini	C₁₆	C₂²×D₂₃
Fitting	C₂×C₂₃×C₃₂	C₂
socle	C₂×C₂₃	C₂×D₁₆
hypercenter	C₂×C₃₂	D₂₃
hypocenter	C₂₃	#2262
LCS	G ▷ C₁₆×C₂₃ ▷ C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
1. #13092849	C₂	semidirect
2. Dic₃₆₈	C₂	semidirect
3. #13092256	C₂	semidirect
4. D₇₃₆	C₂	semidirect
5. C₂×C₂₃×C₃₂	C₂	semidirect
37. C₂×C₃₂	D₂₃	semidirect
48. C₂₃	#2262	semidirect

Maximal subgroups

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

subgroup	index
#13092849	2
Dic₃₆₈	2
#13092256	2
D₇₃₆	2
C₂×C₂₃×C₃₂	2
#2262	23