Subgroups — #44450004

Order 29824.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₈₆₄	C₂	1	1	yes	maximal, characteristic
2	#28416265	C₂	1	1	yes	maximal, characteristic
3	M₆₄×C₂₃₃	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₉₃₂	—	2	1	no	—
5	Dic₁₈₆₄	—	2	1	no	—
6	Dic₁₈₆₄	C₄	1	2	yes	—
7	#21418471	—	2	1	no	—
8	C₂×C₁₆×C₂₃₃	C₂²	1	1	yes	characteristic
9	C₃₂×C₂₃₃	—	2	1	no	—
10	C₂×Dic₄₆₆	—	4	1	no	—
11	Dic₉₃₂	—	2	2	no	—
12	Dic₉₃₂	—	4	1	no	—
13	#16836578	—	4	1	no	—
14	C₂×C₈×C₂₃₃	D₄	1	1	yes	characteristic
15	C₁₆×C₂₃₃	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₂₃₃	D₄	1	1	yes	characteristic
17	Dic₄₆₆	—	4	2	no	—
18	Dic₄₆₆	—	8	1	no	—
19	C₂×Dic₂₃₃	—	8	1	no	—
20	#14272338	—	8	1	no	—
21	C₂×C₄×C₂₃₃	D₈	1	1	yes	characteristic
22	C₈×C₂₃₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₂₃₃	SD₁₆	1	1	yes	characteristic
24	C₂²×C₂₃₃	SD₃₂	1	1	yes	characteristic
25	C₄×C₂₃₃	#102	1	1	yes	characteristic
26	C₄×C₂₃₃	D₁₆	1	1	yes	characteristic
27	Dic₂₃₃	—	8	2	no	—
28	C₂×C₂₃₃	—	2	1	no	—
29	C₂×C₂₃₃	#357	1	1	yes	characteristic
30	C₂₃₃	#1420	1	1	yes	characteristic, complemented, Sylow
31	#1420	—	233	1	no	maximal, Sylow
32	C₂×Q₃₂	—	233	1	no	—
33	M₆₄	D₂₃₃	1	1	yes	characteristic
34	#368	—	233	1	no	—
35	C₂×Q₁₆	—	466	1	no	—
36	Q₃₂	—	466	1	no	—
37	Q₃₂	—	233	2	no	—
38	C₂×C₁₆	C₂×D₂₃₃	1	1	yes	characteristic
39	#108	—	466	1	no	—
40	C₃₂	—	2	1	no	—
41	C₂×Q₈	—	932	1	no	—
42	Q₁₆	—	932	1	no	—
43	Q₁₆	—	466	2	no	—
44	M₁₆	—	932	1	no	—
45	C₂×C₈	D₉₃₂	1	1	yes	characteristic
46	C₁₆	C₄×D₂₃₃	1	1	yes	characteristic
47	C₁₆	C₂²⋊D₂₃₃	1	1	yes	characteristic
48	Q₈	—	1864	1	no	—
49	Q₈	—	932	2	no	—
50	C₂×C₄	—	1864	1	no	—
51	C₂×C₄	D₁₈₆₄	1	1	yes	characteristic
52	C₈	—	1864	1	no	—
53	C₈	#16836571	1	1	yes	characteristic
54	C₈	#16836583	1	1	yes	characteristic
55	C₂²	#21418480	1	1	yes	characteristic
56	C₄	—	1864	2	no	—
57	C₄	#21418457	1	1	yes	characteristic
58	C₄	D₃₇₂₈	1	1	yes	characteristic
59	C₂	—	2	1	no	—
60	C₂	#28416237	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#28416237
commutator	C₁₆×C₂₃₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₃₃
Fitting	M₆₄×C₂₃₃	C₂
socle	C₂×C₂₃₃	#357
prime radical	C₂²×C₂₃₃	SD₃₂
cosocle	C₂×C₁₆×C₂₃₃	C₂²
hypercenter	M₆₄	D₂₃₃
hypocenter	C₂₃₃	#1420
LCS	G ▷ C₁₆×C₂₃₃ ▷ C₈×C₂₃₃ ▷ C₄×C₂₃₃ ▷ C₂×C₂₃₃ ▷ C₂₃₃
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
30. C₂₃₃	#1420	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₈₆₄	2
#28416265	2
M₆₄×C₂₃₃	2
#1420	233