Subgroups — #43252711

Order 28544.

92 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	C₄	1	4	yes	—
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	—	2	4	no	—
8	C₂×C₈×C₂₂₃	C₈	1	2	yes	—
9	C₂×C₈×C₂₂₃	C₈	1	2	yes	—
10	C₂×C₈×C₂₂₃	C₂×C₄	1	2	yes	—
11	C₄²×C₂₂₃	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄×C₂₂₃	C₄²	1	1	yes	characteristic
13	C₂×C₄×C₂₂₃	C₂×C₈	1	1	yes	characteristic
14	C₂×C₄×C₂₂₃	C₂×C₈	1	1	yes	characteristic
15	C₈×C₂₂₃	—	2	2	no	—
16	C₈×C₂₂₃	C₁₆	1	4	yes	complemented
17	C₈×C₂₂₃	M₁₆	1	4	yes	—
18	C₂²×C₂₂₃	C₄×C₈	1	1	yes	characteristic
19	C₄×C₂₂₃	#97	1	2	yes	—
20	C₄×C₂₂₃	C₂×C₁₆	1	2	yes	—
21	C₄×C₂₂₃	M₃₂	1	2	yes	—
22	C₂×C₂₂₃	#322	1	1	yes	characteristic
23	C₂×C₂₂₃	C₄×C₁₆	1	1	yes	characteristic
24	C₂×C₂₂₃	#346	1	1	yes	characteristic
25	C₂₂₃	#1312	1	1	yes	characteristic, complemented, Sylow
26	#1312	—	223	1	no	maximal, Sylow
27	C₄×C₁₆	—	223	2	no	—
28	C₈²	D₂₂₃	1	1	yes	characteristic
29	C₂×C₁₆	—	223	4	no	—
30	C₄×C₈	Dic₂₂₃	1	1	yes	characteristic
31	C₄×C₈	Dic₂₂₃	1	1	yes	characteristic
32	C₄×C₈	C₂×D₂₂₃	1	1	yes	characteristic
33	C₂×C₈	C₂₂₃⋊C₈	1	2	yes	—
34	C₂×C₈	C₂₂₃⋊C₈	1	2	yes	—
35	C₂×C₈	C₄×D₂₂₃	1	2	yes	—
36	C₄²	C₂×Dic₂₂₃	1	1	yes	characteristic
37	C₁₆	—	446	4	no	—
38	C₂×C₄	C₄×Dic₂₂₃	1	1	yes	characteristic
39	C₂×C₄	C₂×C₂₂₃⋊C₈	1	1	yes	characteristic
40	C₂×C₄	C₂×C₂₂₃⋊C₈	1	1	yes	characteristic
41	C₈	—	2	2	no	—
42	C₈	C₂₂₃⋊C₁₆	1	4	yes	complemented
43	C₈	#16813153	1	4	yes	—
44	C₂²	C₄×C₂₂₃⋊C₈	1	1	yes	characteristic
45	C₄	#20297214	1	2	yes	—
46	C₄	C₂×C₂₂₃⋊C₁₆	1	2	yes	—
47	C₄	#20297244	1	2	yes	—
48	C₂	#28383301	1	1	yes	characteristic
49	C₂	C₄×C₂₂₃⋊C₁₆	1	1	yes	characteristic
50	C₂	#28383356	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄×C₈	C₂×D₂₂₃
commutator	C₂×C₂₂₃	C₄×C₁₆
Fitting	C₈²×C₂₂₃	C₂
socle, prime radical	C₂²×C₂₂₃	C₄×C₈
cosocle	C₄×C₈×C₂₂₃	C₂²
hypercenter	C₈²	D₂₂₃
hypocenter	C₂₂₃	#1312
LCS	G ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₄×C₈ ◁ C₈²

Complements

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

N	H ≅ G/N	type
16. C₈×C₂₂₃	C₁₆	semidirect
25. C₂₂₃	#1312	semidirect
42. 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
#1312	223