Subgroups — #43252851

Order 28544.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₂₂₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	#28383333	C₂	1	1	yes	maximal, characteristic
3	#500×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₂₃⋊C₁₆	—	2	1	no	—
5	C₄⋊(C₂₂₃⋊C₈)	—	2	1	no	—
6	C₄⋊Q₈×C₂₂₃	—	2	1	no	—
7	#107×C₂₂₃	—	2	1	no	—
8	#107×C₂₂₃	C₄	1	2	yes	—
9	C₄×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₂₂₃⋊C₈	—	4	1	no	—
11	C₂₂₃⋊C₁₆	—	4	1	no	—
12	C₂×Q₈×C₂₂₃	—	4	1	no	—
13	C₂×C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₂₂₃	D₄	1	1	yes	characteristic
15	C₄⋊C₄×C₂₂₃	—	2	2	no	—
16	C₄⋊C₄×C₂₂₃	—	4	1	no	—
17	C₄²×C₂₂₃	D₄	1	1	yes	characteristic
18	C₂₂₃⋊C₈	—	8	1	no	—
19	Q₈×C₂₂₃	—	8	2	no	—
20	C₂×C₄×C₂₂₃	—	4	2	no	—
21	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₂₂₃	D₈	1	1	yes	characteristic
23	C₂×C₄×C₂₂₃	SD₁₆	1	1	yes	characteristic
24	C₈×C₂₂₃	—	2	1	no	—
25	C₈×C₂₂₃	SD₁₆	1	1	yes	characteristic
26	C₈×C₂₂₃	Q₁₆	1	1	yes	characteristic
27	C₂²×C₂₂₃	#102	1	1	yes	characteristic
28	C₄×C₂₂₃	—	2	1	no	—
29	C₄×C₂₂₃	—	8	2	no	—
30	C₄×C₂₂₃	#101	1	1	yes	characteristic
31	C₄×C₂₂₃	#103	1	1	yes	characteristic
32	C₄×C₂₂₃	SD₃₂	1	1	yes	characteristic
33	C₄×C₂₂₃	Q₃₂	1	1	yes	characteristic
34	C₂×C₂₂₃	#332	1	1	yes	characteristic
35	C₂×C₂₂₃	#358	1	1	yes	characteristic
36	C₂×C₂₂₃	#361	1	1	yes	characteristic
37	C₂₂₃	#1365	1	1	yes	characteristic, complemented, Sylow
38	#1365	—	223	1	no	maximal, Sylow
39	#500	D₂₂₃	1	1	yes	characteristic
40	C₄⋊C₁₆	—	223	1	no	—
41	#335	—	223	1	no	—
42	C₄⋊Q₈	—	2	1	no	—
43	C₂×C₁₆	—	446	1	no	—
44	#107	—	2	1	no	—
45	#107	Dic₂₂₃	1	2	yes	—
46	C₄⋊C₈	—	446	1	no	—
47	C₄×C₈	C₂×D₂₂₃	1	1	yes	characteristic
48	C₂×Q₈	—	4	1	no	—
49	C₂×C₈	—	892	1	no	—
50	C₂×C₈	C₂×Dic₂₂₃	1	1	yes	characteristic
51	C₂×C₈	C₂²⋊D₂₂₃	1	1	yes	characteristic
52	C₄⋊C₄	—	2	2	no	—
53	C₄⋊C₄	—	4	1	no	—
54	C₄²	C₂²⋊D₂₂₃	1	1	yes	characteristic
55	C₁₆	—	892	1	no	—
56	Q₈	—	8	2	no	—
57	C₂×C₄	—	4	2	no	—
58	C₂×C₄	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
59	C₂×C₄	#16813155	1	1	yes	characteristic
60	C₂×C₄	#16813158	1	1	yes	characteristic
61	C₈	—	2	1	no	—
62	C₈	—	1784	1	no	—
63	C₈	#16813157	1	1	yes	characteristic
64	C₈	#16813160	1	1	yes	characteristic
65	C₂²	#20297225	1	1	yes	characteristic
66	C₄	—	2	1	no	—
67	C₄	—	8	2	no	—
68	C₄	#20297223	1	1	yes	characteristic
69	C₄	#20297228	1	1	yes	characteristic
70	C₄	#20297247	1	1	yes	characteristic
71	C₄	#20297250	1	1	yes	characteristic
72	C₂	#28383326	1	1	yes	characteristic
73	C₂	#28383384	1	1	yes	characteristic
74	C₂	#28383393	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#20297225
commutator	C₂×C₈×C₂₂₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₂₂₃
Fitting	#500×C₂₂₃	C₂
socle, prime radical	C₂²×C₂₂₃	#102
cosocle	C₄×C₈×C₂₂₃	C₂²
hypercenter	#500	D₂₂₃
hypocenter	C₂₂₃	#1365
LCS	G ▷ C₂×C₈×C₂₂₃ ▷ C₂×C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

Complements

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

N	H ≅ G/N	type
37. C₂₂₃	#1365	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₂₂₃⋊C₁₆)	2
#28383333	2
#500×C₂₂₃	2
#1365	223