Subgroups — #43252988

Order 28544.

62 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	—	2	1	no	—
7	C₂×C₁₆×C₂₂₃	C₂²	1	1	yes	characteristic
8	C₃₂×C₂₂₃	—	2	1	no	—
9	C₂×C₂₂₃⋊C₈	C₈	1	1	yes	characteristic
10	C₄×Dic₂₂₃	C₈	1	1	yes	characteristic
11	C₂₂₃⋊C₁₆	—	2	1	no	—
12	C₂×C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
13	C₁₆×C₂₂₃	D₄	1	1	yes	characteristic
14	C₁₆×C₂₂₃	Q₈	1	1	yes	characteristic
15	C₂×Dic₂₂₃	C₁₆	1	1	yes	characteristic
16	C₂×Dic₂₂₃	C₁₆	1	1	yes	characteristic
17	C₂₂₃⋊C₈	—	2	1	no	—
18	C₂×C₄×C₂₂₃	C₂×C₈	1	1	yes	characteristic
19	C₈×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
20	C₈×C₂₂₃	M₁₆	1	1	yes	characteristic
21	C₂²×C₂₂₃	C₂×C₁₆	1	1	yes	characteristic
22	C₄×C₂₂₃	C₄⋊C₈	1	1	yes	characteristic
23	C₄×C₂₂₃	M₃₂	1	1	yes	characteristic
24	Dic₂₂₃	—	2	1	no	—
25	Dic₂₂₃	C₃₂	1	2	yes	complemented
26	C₂×C₂₂₃	C₄⋊C₁₆	1	1	yes	characteristic
27	C₂×C₂₂₃	C₂×C₃₂	1	1	yes	characteristic
28	C₂×C₂₂₃	M₆₄	1	1	yes	characteristic
29	C₂₂₃	C₄⋊C₃₂	1	1	yes	characteristic, complemented, Sylow
30	C₄⋊C₃₂	—	223	1	no	maximal, Sylow
31	C₂×C₃₂	—	223	1	no	—
32	C₂×C₃₂	D₂₂₃	1	1	yes	characteristic
33	C₄×C₁₆	—	223	1	no	—
34	C₂×C₁₆	—	223	1	no	—
35	C₂×C₁₆	C₂×D₂₂₃	1	1	yes	characteristic
36	C₄×C₈	—	223	1	no	—
37	C₃₂	—	2	1	no	—
38	C₃₂	—	446	1	no	—
39	C₂×C₈	—	223	1	no	—
40	C₂×C₈	C₄×D₂₂₃	1	1	yes	characteristic
41	C₄²	—	223	1	no	—
42	C₁₆	—	446	1	no	—
43	C₁₆	C₂²⋊D₂₂₃	1	1	yes	characteristic
44	C₁₆	Dic₄₄₆	1	1	yes	characteristic
45	C₂×C₄	—	223	1	no	—
46	C₂×C₄	—	223	1	no	—
47	C₂×C₄	C₈×D₂₂₃	1	1	yes	characteristic
48	C₈	—	446	1	no	—
49	C₈	Dic₂₂₃⋊C₄	1	1	yes	characteristic
50	C₈	#16813153	1	1	yes	characteristic
51	C₂²	C₁₆×D₂₂₃	1	1	yes	characteristic
52	C₄	—	446	1	no	—
53	C₄	—	223	2	no	—
54	C₄	Dic₂₂₃⋊C₈	1	1	yes	characteristic
55	C₄	#20297243	1	1	yes	characteristic
56	C₂	Dic₂₂₃⋊C₁₆	1	1	yes	characteristic
57	C₂	C₃₂×D₂₂₃	1	1	yes	characteristic
58	C₂	#28383413	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂×D₂₂₃
commutator	C₂×C₂₂₃	C₂×C₃₂
Fitting	C₂×C₃₂×C₂₂₃	C₂
socle, prime radical	C₂²×C₂₂₃	C₂×C₁₆
cosocle	C₂×C₁₆×C₂₂₃	C₂²
hypercenter	C₂×C₃₂	D₂₂₃
hypocenter	C₂₂₃	C₄⋊C₃₂
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
25. Dic₂₂₃	C₃₂	semidirect
29. C₂₂₃	C₄⋊C₃₂	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
C₄⋊C₃₂	223