Subgroups — #43252863

Order 28544.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₂₂₃⋊C₁₆)	C₂	1	2	yes	maximal
2	C₈²×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₂₃⋊C₁₆	—	2	2	no	—
4	C₄×C₈×C₂₂₃	C₄	1	1	yes	characteristic
5	C₄×C₈×C₂₂₃	C₄	1	1	yes	characteristic
6	C₄×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
7	C₂₂₃⋊C₁₆	—	4	2	no	—
8	C₂×C₈×C₂₂₃	—	2	1	no	—
9	C₂×C₈×C₂₂₃	C₈	1	1	yes	characteristic
10	C₂×C₈×C₂₂₃	C₈	1	1	yes	characteristic
11	C₂×C₈×C₂₂₃	D₄	1	1	yes	characteristic
12	C₂×C₈×C₂₂₃	Q₈	1	1	yes	characteristic
13	C₄²×C₂₂₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₄×C₂₂₃	C₂×C₈	1	1	yes	characteristic
16	C₂×C₄×C₂₂₃	M₁₆	1	1	yes	characteristic
17	C₈×C₂₂₃	—	2	2	no	—
18	C₈×C₂₂₃	—	2	1	no	—
19	C₈×C₂₂₃	—	2	1	no	—
20	C₈×C₂₂₃	C₁₆	1	2	yes	complemented
21	C₈×C₂₂₃	D₈	1	1	yes	characteristic
22	C₈×C₂₂₃	Q₁₆	1	1	yes	characteristic
23	C₂²×C₂₂₃	C₄⋊C₈	1	1	yes	characteristic
24	C₄×C₂₂₃	—	2	1	no	—
25	C₄×C₂₂₃	#107	1	1	yes	characteristic
26	C₄×C₂₂₃	#108	1	1	yes	characteristic
27	C₄×C₂₂₃	C₂×C₁₆	1	1	yes	characteristic
28	C₄×C₂₂₃	M₃₂	1	1	yes	characteristic
29	C₂×C₂₂₃	#335	1	1	yes	characteristic
30	C₂×C₂₂₃	C₄⋊C₁₆	1	1	yes	characteristic
31	C₂×C₂₂₃	#364	1	1	yes	characteristic
32	C₂₂₃	#1370	1	1	yes	characteristic, complemented, Sylow
33	#1370	—	223	1	no	maximal, Sylow
34	C₄⋊C₁₆	—	223	2	no	—
35	C₈²	D₂₂₃	1	1	yes	characteristic
36	C₂×C₁₆	—	446	2	no	—
37	C₄×C₈	Dic₂₂₃	1	1	yes	characteristic
38	C₄×C₈	Dic₂₂₃	1	1	yes	characteristic
39	C₄×C₈	C₂×D₂₂₃	1	1	yes	characteristic
40	C₂×C₈	—	2	1	no	—
41	C₂×C₈	C₂₂₃⋊C₈	1	1	yes	characteristic
42	C₂×C₈	C₂₂₃⋊C₈	1	1	yes	characteristic
43	C₂×C₈	D₈₉₂	1	1	yes	characteristic
44	C₂×C₈	Dic₄₄₆	1	1	yes	characteristic
45	C₄²	C₂×Dic₂₂₃	1	1	yes	characteristic
46	C₁₆	—	892	2	no	—
47	C₂×C₄	C₄⋊Dic₂₂₃	1	1	yes	characteristic
48	C₂×C₄	C₂×C₂₂₃⋊C₈	1	1	yes	characteristic
49	C₂×C₄	#16813154	1	1	yes	characteristic
50	C₈	—	2	2	no	—
51	C₈	—	2	1	no	—
52	C₈	—	2	1	no	—
53	C₈	C₂₂₃⋊C₁₆	1	2	yes	complemented
54	C₈	D₁₇₈₄	1	1	yes	characteristic
55	C₈	Dic₈₉₂	1	1	yes	characteristic
56	C₂²	C₄⋊(C₂₂₃⋊C₈)	1	1	yes	characteristic
57	C₄	—	2	1	no	—
58	C₄	#20297238	1	1	yes	characteristic
59	C₄	#20297240	1	1	yes	characteristic
60	C₄	C₂×C₂₂₃⋊C₁₆	1	1	yes	characteristic
61	C₄	#20297244	1	1	yes	characteristic
62	C₂	#28383333	1	1	yes	characteristic
63	C₂	C₄⋊(C₂₂₃⋊C₁₆)	1	1	yes	characteristic
64	C₂	#28383401	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₈₉₂
commutator	C₄×C₂₂₃	C₂×C₁₆
Frattini	C₄×C₈	C₂×D₂₂₃
Fitting	C₈²×C₂₂₃	C₂
socle, prime radical	C₂²×C₂₂₃	C₄⋊C₈
cosocle	C₄×C₈×C₂₂₃	C₂²
hypercenter	C₈²	D₂₂₃
hypocenter	C₂₂₃	#1370
LCS	G ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈²

Complements

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

N	H ≅ G/N	type
20. C₈×C₂₂₃	C₁₆	semidirect
32. C₂₂₃	#1370	semidirect
53. C₈	C₂₂₃⋊C₁₆	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
C₈²×C₂₂₃	2
#1370	223