Subgroups — #43252974

Order 28544.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₂₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic
2	#28383406	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₃₂×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	#20297238	—	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	complemented
9	C₂×C₁₆×C₂₂₃	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₂₂₃	—	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	C₂×Dic₂₂₃	—	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	D₁₆	1	1	yes	characteristic
25	C₄×C₂₂₃	—	8	2	no	—
26	C₄×C₂₂₃	#102	1	1	yes	characteristic
27	C₄×C₂₂₃	SD₃₂	1	1	yes	characteristic
28	Dic₂₂₃	—	16	1	no	—
29	C₂×C₂₂₃	#357	1	1	yes	characteristic
30	C₂×C₂₂₃	SD₆₄	1	1	yes	characteristic
31	C₂×C₂₂₃	Q₆₄	1	1	yes	characteristic
32	C₂₂₃	#1416	1	1	yes	characteristic, complemented, Sylow
33	#1416	—	223	1	no	maximal, Sylow
34	C₂×Q₃₂	D₂₂₃	1	1	yes	characteristic
35	C₂×C₃₂	—	223	1	no	—
36	#366	—	223	1	no	—
37	C₂×Q₁₆	—	2	1	no	—
38	Q₃₂	—	2	1	no	—
39	Q₃₂	Dic₂₂₃	1	2	yes	complemented
40	C₂×C₁₆	C₂×D₂₂₃	1	1	yes	characteristic
41	#107	—	446	1	no	—
42	C₃₂	—	446	1	no	—
43	C₂×Q₈	—	4	1	no	—
44	Q₁₆	—	2	2	no	—
45	Q₁₆	—	4	1	no	—
46	C₂×C₈	C₂²⋊D₂₂₃	1	1	yes	characteristic
47	C₄⋊C₄	—	892	1	no	—
48	C₁₆	C₂×Dic₂₂₃	1	1	yes	characteristic
49	C₁₆	C₂²⋊D₂₂₃	1	1	yes	characteristic
50	Q₈	—	4	2	no	—
51	Q₈	—	8	1	no	—
52	C₂×C₄	—	8	1	no	—
53	C₂×C₄	—	1784	1	no	—
54	C₂×C₄	#16813155	1	1	yes	characteristic
55	C₈	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
56	C₈	#16813158	1	1	yes	characteristic
57	C₂²	#20297245	1	1	yes	characteristic
58	C₄	—	8	2	no	—
59	C₄	—	3568	1	no	—
60	C₄	#20297225	1	1	yes	characteristic
61	C₄	#20297248	1	1	yes	characteristic
62	C₂	#28383381	1	1	yes	characteristic
63	C₂	#28383417	1	1	yes	characteristic
64	C₂	#28383420	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#20297245
commutator	C₁₆×C₂₂₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₂₃
Fitting	C₂×Q₃₂×C₂₂₃	C₂
socle, prime radical	C₂²×C₂₂₃	D₁₆
cosocle	C₂×C₁₆×C₂₂₃	C₂²
hypercenter	C₂×Q₃₂	D₂₂₃
hypocenter	C₂₂₃	#1416
LCS	G ▷ C₁₆×C₂₂₃ ▷ C₈×C₂₂₃ ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
8. Q₃₂×C₂₂₃	C₄	semidirect
32. C₂₂₃	#1416	semidirect
39. Q₃₂	Dic₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₂₃⋊C₃₂	2
#28383406	2
C₂×Q₃₂×C₂₂₃	2
#1416	223