Subgroups — #43252986

Order 28544.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28383414	C₂	1	1	yes	maximal, characteristic
2	#28383410	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₃₂×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	#20297240	—	2	1	no	—
5	C₂₂₃⋊C₃₂	—	2	1	no	—
6	C₂×Q₁₆×C₂₂₃	—	2	1	no	—
7	Q₃₂×C₂₂₃	—	2	1	no	—
8	Q₃₂×C₂₂₃	C₄	1	2	yes	—
9	C₂×C₁₆×C₂₂₃	C₂²	1	1	yes	characteristic
10	#16813154	—	4	1	no	—
11	C₂×Q₈×C₂₂₃	—	4	1	no	—
12	Q₁₆×C₂₂₃	—	2	2	no	—
13	Q₁₆×C₂₂₃	—	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	C₂₂₃⋊C₈	—	8	1	no	—
18	Q₈×C₂₂₃	—	4	2	no	—
19	Q₈×C₂₂₃	—	8	1	no	—
20	C₂×C₄×C₂₂₃	—	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	C₄×C₂₂₃	—	8	2	no	—
26	C₄×C₂₂₃	#102	1	1	yes	characteristic
27	C₄×C₂₂₃	D₁₆	1	1	yes	characteristic
28	C₂×C₂₂₃	—	2	1	no	—
29	C₂×C₂₂₃	#357	1	1	yes	characteristic
30	C₂₂₃	#1420	1	1	yes	characteristic, complemented, Sylow
31	#1420	—	223	1	no	maximal, Sylow
32	C₂×Q₃₂	D₂₂₃	1	1	yes	characteristic
33	M₆₄	—	223	1	no	—
34	#368	—	223	1	no	—
35	C₂×Q₁₆	—	2	1	no	—
36	Q₃₂	—	2	1	no	—
37	Q₃₂	Dic₂₂₃	1	2	yes	—
38	C₂×C₁₆	C₂×D₂₂₃	1	1	yes	characteristic
39	#108	—	446	1	no	—
40	C₃₂	—	446	1	no	—
41	C₂×Q₈	—	4	1	no	—
42	Q₁₆	—	2	2	no	—
43	Q₁₆	—	4	1	no	—
44	M₁₆	—	892	1	no	—
45	C₂×C₈	C₂²⋊D₂₂₃	1	1	yes	characteristic
46	C₁₆	C₂×Dic₂₂₃	1	1	yes	characteristic
47	C₁₆	C₂²⋊D₂₂₃	1	1	yes	characteristic
48	Q₈	—	4	2	no	—
49	Q₈	—	8	1	no	—
50	C₂×C₄	—	8	1	no	—
51	C₂×C₄	#16813155	1	1	yes	characteristic
52	C₈	—	1784	1	no	—
53	C₈	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
54	C₈	#16813158	1	1	yes	characteristic
55	C₂²	#20297248	1	1	yes	characteristic
56	C₄	—	8	2	no	—
57	C₄	#20297225	1	1	yes	characteristic
58	C₄	#20297245	1	1	yes	characteristic
59	C₂	—	2	1	no	—
60	C₂	#28383381	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#28383381
commutator	C₁₆×C₂₂₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₂₃
Fitting	C₂×Q₃₂×C₂₂₃	C₂
socle	C₂×C₂₂₃	#357
prime radical	C₂²×C₂₂₃	SD₃₂
cosocle	C₂×C₁₆×C₂₂₃	C₂²
hypercenter	C₂×Q₃₂	D₂₂₃
hypocenter	C₂₂₃	#1420
LCS	G ▷ C₁₆×C₂₂₃ ▷ C₈×C₂₂₃ ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
30. C₂₂₃	#1420	semidirect

Maximal subgroups

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

subgroup	index
#28383414	2
#28383410	2
C₂×Q₃₂×C₂₂₃	2
#1420	223