Subgroups — #43252926

Order 28544.

68 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	C₁₆×Dic₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₂×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₂₃⋊C₁₆	C₄	1	1	yes	characteristic
5	C₈×Dic₂₂₃	C₄	1	1	yes	characteristic
6	C₂₂₃⋊C₃₂	C₄	1	2	yes	complemented
7	C₂×C₁₆×C₂₂₃	C₂²	1	1	yes	characteristic
8	C₃₂×C₂₂₃	C₄	1	2	yes	complemented
9	C₂×C₂₂₃⋊C₈	C₈	1	1	yes	characteristic
10	C₄×Dic₂₂₃	C₈	1	1	yes	characteristic
11	C₂₂₃⋊C₁₆	C₈	1	2	yes	—
12	C₂×C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
13	C₁₆×C₂₂₃	C₂×C₄	1	1	yes	characteristic
14	C₁₆×C₂₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂×Dic₂₂₃	C₁₆	1	2	yes	—
16	C₂₂₃⋊C₈	C₁₆	1	2	yes	—
17	C₂×C₄×C₂₂₃	C₂×C₈	1	1	yes	characteristic
18	C₈×C₂₂₃	C₄²	1	1	yes	characteristic
19	C₈×C₂₂₃	C₂×C₈	1	1	yes	characteristic
20	C₂²×C₂₂₃	C₂×C₁₆	1	1	yes	characteristic
21	C₄×C₂₂₃	C₄×C₈	1	1	yes	characteristic
22	C₄×C₂₂₃	C₂×C₁₆	1	1	yes	characteristic
23	Dic₂₂₃	—	2	2	no	—
24	C₂×C₂₂₃	C₄×C₁₆	1	1	yes	characteristic
25	C₂×C₂₂₃	M₆₄	1	2	yes	—
26	C₂₂₃	#1397	1	1	yes	characteristic, complemented, Sylow
27	#1397	—	223	1	no	maximal, Sylow
28	C₂×C₃₂	—	223	1	no	—
29	C₂×C₃₂	D₂₂₃	1	1	yes	characteristic
30	C₄×C₁₆	—	223	1	no	—
31	C₂×C₁₆	—	223	1	no	—
32	C₂×C₁₆	C₂×D₂₂₃	1	1	yes	characteristic
33	C₄×C₈	—	223	1	no	—
34	C₃₂	—	223	2	no	—
35	C₃₂	Dic₂₂₃	1	2	yes	complemented
36	C₂×C₈	—	223	1	no	—
37	C₂×C₈	C₄×D₂₂₃	1	1	yes	characteristic
38	C₄²	—	223	1	no	—
39	C₁₆	—	223	2	no	—
40	C₁₆	C₄×D₂₂₃	1	1	yes	characteristic
41	C₁₆	C₂×Dic₂₂₃	1	1	yes	characteristic
42	C₂×C₄	—	223	2	no	—
43	C₂×C₄	C₈×D₂₂₃	1	1	yes	characteristic
44	C₈	—	223	2	no	—
45	C₈	C₄×Dic₂₂₃	1	1	yes	characteristic
46	C₈	C₈×D₂₂₃	1	1	yes	characteristic
47	C₂²	C₁₆×D₂₂₃	1	1	yes	characteristic
48	C₄	—	446	2	no	—
49	C₄	C₈×Dic₂₂₃	1	1	yes	characteristic
50	C₄	C₁₆×D₂₂₃	1	1	yes	characteristic
51	C₂	C₁₆×Dic₂₂₃	1	1	yes	characteristic
52	C₂	#28383413	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂×D₂₂₃
commutator	C₂×C₂₂₃	C₄×C₁₆
Fitting	C₂×C₃₂×C₂₂₃	C₂
socle, prime radical	C₂²×C₂₂₃	C₂×C₁₆
cosocle	C₂×C₁₆×C₂₂₃	C₂²
hypercenter	C₂×C₃₂	D₂₂₃
hypocenter	C₂₂₃	#1397
LCS	G ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
6. C₂₂₃⋊C₃₂	C₄	semidirect
8. C₃₂×C₂₂₃	C₄	semidirect
26. C₂₂₃	#1397	semidirect
35. C₃₂	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
C₁₆×Dic₂₂₃	2
C₂×C₃₂×C₂₂₃	2
#1397	223