Subgroups — #43252984

Order 28544.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃₅₆₈	C₂	1	1	yes	maximal, characteristic
2	#28383410	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₇₈₄	—	2	1	no	—
5	D₃₅₆₈	—	2	1	no	—
6	D₃₅₆₈	C₄	1	2	yes	—
7	#20297240	—	2	1	no	—
8	C₂×C₁₆×C₂₂₃	C₂²	1	1	yes	characteristic
9	C₃₂×C₂₂₃	—	2	1	no	—
10	C₂×D₈₉₂	—	4	1	no	—
11	D₁₇₈₄	—	2	2	no	—
12	D₁₇₈₄	—	4	1	no	—
13	#16813154	—	4	1	no	—
14	C₂×C₈×C₂₂₃	D₄	1	1	yes	characteristic
15	C₁₆×C₂₂₃	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₂₂₃	D₄	1	1	yes	characteristic
17	C₂²×D₂₂₃	—	8	1	no	—
18	D₈₉₂	—	4	2	no	—
19	D₈₉₂	—	8	1	no	—
20	C₂₂₃⋊C₈	—	8	1	no	—
21	C₂×C₄×C₂₂₃	D₈	1	1	yes	characteristic
22	C₈×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₂₂₃	SD₁₆	1	1	yes	characteristic
24	C₂²×C₂₂₃	SD₃₂	1	1	yes	characteristic
25	C₂×D₂₂₃	—	8	2	no	—
26	C₂×D₂₂₃	—	16	2	no	—
27	C₄×C₂₂₃	#102	1	1	yes	characteristic
28	C₄×C₂₂₃	D₁₆	1	1	yes	characteristic
29	C₂×C₂₂₃	—	2	1	no	—
30	C₂×C₂₂₃	#357	1	1	yes	characteristic
31	D₂₂₃	—	16	2	no	—
32	C₂₂₃	#1419	1	1	yes	characteristic, complemented, Sylow
33	#1419	—	223	1	no	maximal, Sylow
34	C₂×D₁₆	—	223	1	no	—
35	M₆₄	D₂₂₃	1	1	yes	characteristic, complemented
36	#368	—	223	1	no	—
37	C₂×D₈	—	446	1	no	—
38	D₁₆	—	446	1	no	—
39	D₁₆	—	223	2	no	—
40	C₂×C₁₆	C₂×D₂₂₃	1	1	yes	characteristic
41	#108	—	446	1	no	—
42	C₃₂	—	2	1	no	—
43	C₂×D₄	—	892	1	no	—
44	D₈	—	892	1	no	—
45	D₈	—	446	2	no	—
46	M₁₆	—	892	1	no	—
47	C₂×C₈	D₈₉₂	1	1	yes	characteristic
48	C₁₆	C₄×D₂₂₃	1	1	yes	characteristic
49	C₁₆	C₂²⋊D₂₂₃	1	1	yes	characteristic
50	C₂³	—	1784	1	no	—
51	D₄	—	892	2	no	—
52	D₄	—	1784	1	no	—
53	C₂×C₄	D₁₇₈₄	1	1	yes	characteristic
54	C₈	—	1784	1	no	—
55	C₈	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
56	C₈	#16813159	1	1	yes	characteristic
57	C₂²	—	3568	2	no	—
58	C₂²	—	1784	2	no	—
59	C₂²	#20297249	1	1	yes	characteristic
60	C₄	#20297226	1	1	yes	characteristic
61	C₄	D₃₅₆₈	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	3568	2	no	—
64	C₂	#28383382	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#28383382
commutator	C₁₆×C₂₂₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₂₃
Fitting	M₆₄×C₂₂₃	C₂
socle	C₂×C₂₂₃	#357
prime radical	C₂×D₃₅₆₈	C₂
cosocle	C₂×C₁₆×C₂₂₃	C₂²
hypercenter	M₆₄	D₂₂₃
hypocenter	C₂₂₃	#1419
LCS	G ▷ C₁₆×C₂₂₃ ▷ C₈×C₂₂₃ ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. #28383410	C₂	semidirect
3. M₆₄×C₂₂₃	C₂	semidirect
32. C₂₂₃	#1419	semidirect
35. M₆₄	D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃₅₆₈	2
#28383410	2
M₆₄×C₂₂₃	2
#1419	223