Subgroups — #43252976

Order 28544.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28384012	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₂₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#368×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#20297326	—	2	1	no	—
5	Dic₁₇₈₄	C₄	1	1	yes	characteristic
6	#20297249	—	2	1	no	—
7	D₃₅₆₈	C₄	1	1	yes	characteristic
8	C₂₂₃⋊C₃₂	—	2	1	no	—
9	C₂×C₁₆×C₂₂₃	C₂²	1	1	yes	characteristic
10	#108×C₂₂₃	—	2	1	no	—
11	#16813170	—	4	1	no	—
12	Dic₈₉₂	—	2	1	no	—
13	#16813159	—	4	1	no	—
14	D₁₇₈₄	—	2	1	no	—
15	M₁₆×C₂₂₃	—	4	1	no	—
16	C₂×C₈×C₂₂₃	D₄	1	1	yes	characteristic
17	C₁₆×C₂₂₃	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₂₂₃	D₄	1	1	yes	characteristic
19	Dic₄₄₆	—	4	1	no	—
20	D₈₉₂	—	4	1	no	—
21	C₂²⋊D₂₂₃	—	8	1	no	—
22	C₄×D₂₂₃	—	8	1	no	—
23	C₂×C₄×C₂₂₃	D₈	1	1	yes	characteristic
24	C₈×C₂₂₃	—	8	1	no	—
25	C₈×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₂₂₃	SD₁₆	1	1	yes	characteristic
27	C₂²×C₂₂₃	SD₃₂	1	1	yes	characteristic
28	C₂×D₂₂₃	—	8	1	no	—
29	C₄×C₂₂₃	#102	1	1	yes	characteristic
30	C₄×C₂₂₃	D₁₆	1	1	yes	characteristic
31	Dic₂₂₃	—	8	1	no	—
32	C₂×C₂₂₃	—	2	1	no	—
33	C₂×C₂₂₃	#357	1	1	yes	characteristic
34	D₂₂₃	—	16	1	no	—
35	C₂₂₃	#1417	1	1	yes	characteristic, complemented, Sylow
36	#1417	—	223	1	no	maximal, Sylow
37	#508	—	223	1	no	—
38	C₂×C₃₂	—	223	1	no	—
39	#368	D₂₂₃	1	1	yes	characteristic, complemented
40	#135	—	446	1	no	—
41	Q₃₂	—	223	1	no	—
42	SD₃₂	—	446	1	no	—
43	D₁₆	—	223	1	no	—
44	C₂×C₁₆	C₂×D₂₂₃	1	1	yes	characteristic
45	#108	—	2	1	no	—
46	C₃₂	—	446	1	no	—
47	#41	—	892	1	no	—
48	Q₁₆	—	446	1	no	—
49	SD₁₆	—	892	1	no	—
50	D₈	—	446	1	no	—
51	M₁₆	—	4	1	no	—
52	C₂×C₈	C₂²⋊D₂₂₃	1	1	yes	characteristic
53	C₁₆	C₄×D₂₂₃	1	1	yes	characteristic
54	C₁₆	D₈₉₂	1	1	yes	characteristic
55	Q₈	—	892	1	no	—
56	D₄	—	892	1	no	—
57	D₄	—	1784	1	no	—
58	C₂×C₄	—	1784	1	no	—
59	C₂×C₄	#16813155	1	1	yes	characteristic
60	C₈	—	8	1	no	—
61	C₈	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
62	C₈	#16813157	1	1	yes	characteristic
63	C₂²	—	1784	1	no	—
64	C₂²	#20297247	1	1	yes	characteristic
65	C₄	—	1784	1	no	—
66	C₄	#20297224	1	1	yes	characteristic
67	C₄	#20297245	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	3568	1	no	—
70	C₂	#28383380	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#20297245
commutator	C₁₆×C₂₂₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₂₃
Fitting	#368×C₂₂₃	C₂
socle	C₂×C₂₂₃	#357
prime radical	#28384012	C₂
cosocle	C₂×C₁₆×C₂₂₃	C₂²
hypercenter	#368	D₂₂₃
hypocenter	C₂₂₃	#1417
LCS	G ▷ C₁₆×C₂₂₃ ▷ C₈×C₂₂₃ ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #368

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₂₃⋊C₃₂	C₂	semidirect
3. #368×C₂₂₃	C₂	semidirect
35. C₂₂₃	#1417	semidirect
39. #368	D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#28384012	2
C₂×C₂₂₃⋊C₃₂	2
#368×C₂₂₃	2
#1417	223