Subgroups — #15671574

Order 2944.

82 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	C₂×C₂₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#366	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₈₄	—	2	1	no	—
5	C₂×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
6	C₂₃×#107	—	2	1	no	—
7	D₃₆₈	—	2	1	no	—
8	D₃₆₈	C₄	1	2	yes	complemented
9	C₂₃⋊C₃₂	—	2	1	no	—
10	C₂×D₉₂	—	4	1	no	—
11	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
12	C₄⋊C₄×C₂₃	—	4	1	no	—
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	—	8	1	no	—
19	C₂×C₄×C₂₃	D₈	1	1	yes	characteristic
20	D₉₂	—	4	2	no	—
21	D₉₂	—	8	1	no	—
22	C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₂₃	SD₁₆	1	1	yes	characteristic
24	#1415	—	23	1	no	maximal, Sylow
25	C₂²×C₂₃	D₁₆	1	1	yes	characteristic
26	C₂×D₂₃	—	8	2	no	—
27	C₂×D₂₃	—	16	1	no	—
28	C₂×D₂₃	—	16	1	no	—
29	C₄×C₂₃	—	16	1	no	—
30	C₄×C₂₃	#102	1	1	yes	characteristic
31	C₄×C₂₃	SD₃₂	1	1	yes	characteristic
32	C₂×D₁₆	—	23	1	no	—
33	C₂×C₃₂	—	23	1	no	—
34	#366	D₂₃	1	1	yes	characteristic, complemented
35	C₂×C₂₃	#357	1	1	yes	characteristic
36	C₂×C₂₃	D₃₂	1	1	yes	characteristic
37	C₂×C₂₃	SD₆₄	1	1	yes	characteristic
38	D₂₃	—	16	2	no	—
39	C₂×D₈	—	46	1	no	—
40	D₁₆	—	23	2	no	—
41	D₁₆	—	46	1	no	—
42	C₂×C₁₆	C₂×D₂₃	1	1	yes	characteristic
43	#107	—	2	1	no	—
44	C₃₂	—	46	1	no	—
45	C₂₃	#1415	1	1	yes	characteristic, complemented, Sylow
46	C₂×D₄	—	92	1	no	—
47	D₈	—	46	2	no	—
48	D₈	—	92	1	no	—
49	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
50	C₄⋊C₄	—	4	1	no	—
51	C₁₆	C₄×D₂₃	1	1	yes	characteristic
52	C₁₆	D₉₂	1	1	yes	characteristic
53	C₂³	—	184	1	no	—
54	D₄	—	92	2	no	—
55	D₄	—	184	1	no	—
56	C₂×C₄	—	8	1	no	—
57	C₂×C₄	#67272	1	1	yes	characteristic
58	C₈	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
59	C₈	#67274	1	1	yes	characteristic
60	C₂²	—	368	1	no	—
61	C₂²	—	368	1	no	—
62	C₂²	—	184	2	no	—
63	C₂²	#10625008	1	1	yes	characteristic
64	C₄	—	16	1	no	—
65	C₄	#10624991	1	1	yes	characteristic
66	C₄	#10625010	1	1	yes	characteristic
67	C₂	—	368	2	no	—
68	C₂	#13092217	1	1	yes	characteristic
69	C₂	#13092252	1	1	yes	characteristic
70	C₂	#13092254	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#10625008
commutator	C₁₆×C₂₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₃
Fitting	C₂₃×#366	C₂
socle	C₂²×C₂₃	D₁₆
prime radical	C₂×D₃₆₈	C₂
cosocle	C₂×C₁₆×C₂₃	C₂²
hypercenter	#366	D₂₃
hypocenter	C₂₃	#1415
LCS	G ▷ C₁₆×C₂₃ ▷ C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #366

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃⋊C₃₂	C₂	semidirect
3. C₂₃×#366	C₂	semidirect
8. D₃₆₈	C₄	semidirect
34. #366	D₂₃	semidirect
45. C₂₃	#1415	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
C₂×C₂₃⋊C₃₂	2
C₂₃×#366	2
#1415	23