Subgroups — #43252862

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	1	yes	maximal, characteristic
2	C₈×C₂₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₂₃⋊C₁₆	—	2	1	no	—
5	C₄×C₂₂₃⋊C₈	C₄	1	1	yes	characteristic
6	C₄×C₂₂₃⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₂₂₃	—	2	1	no	—
8	C₄×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₂₂₃⋊C₈	—	2	1	no	—
10	C₂×C₂₂₃⋊C₈	C₈	1	1	yes	characteristic
11	C₂×C₂₂₃⋊C₈	C₈	1	1	yes	characteristic
12	C₂₂₃⋊C₁₆	—	4	1	no	—
13	C₂×C₈×C₂₂₃	D₄	1	1	yes	characteristic
14	C₂×C₈×C₂₂₃	Q₈	1	1	yes	characteristic
15	C₄²×C₂₂₃	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₂₂₃	—	4	1	no	—
17	C₂₂₃⋊C₈	—	2	2	no	—
18	C₂₂₃⋊C₈	—	2	1	no	—
19	C₂₂₃⋊C₈	C₁₆	1	2	yes	complemented
20	C₂×C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
21	C₂×C₄×C₂₂₃	C₂×C₈	1	1	yes	characteristic
22	C₂×C₄×C₂₂₃	M₁₆	1	1	yes	characteristic
23	C₈×C₂₂₃	—	2	1	no	—
24	C₈×C₂₂₃	D₈	1	1	yes	characteristic
25	C₈×C₂₂₃	Q₁₆	1	1	yes	characteristic
26	C₂²×C₂₂₃	C₄⋊C₈	1	1	yes	characteristic
27	C₄×C₂₂₃	—	2	1	no	—
28	C₄×C₂₂₃	#107	1	1	yes	characteristic
29	C₄×C₂₂₃	#108	1	1	yes	characteristic
30	C₄×C₂₂₃	C₂×C₁₆	1	1	yes	characteristic
31	C₄×C₂₂₃	M₃₂	1	1	yes	characteristic
32	C₂×C₂₂₃	#335	1	1	yes	characteristic
33	C₂×C₂₂₃	C₄⋊C₁₆	1	1	yes	characteristic
34	C₂×C₂₂₃	#364	1	1	yes	characteristic
35	C₂₂₃	#1370	1	1	yes	characteristic, complemented, Sylow
36	#1370	—	223	1	no	maximal, Sylow
37	C₄⋊C₁₆	—	223	1	no	—
38	C₄⋊C₁₆	D₂₂₃	1	1	yes	characteristic
39	C₈²	—	223	1	no	—
40	C₂×C₁₆	—	2	1	no	—
41	C₂×C₁₆	—	446	1	no	—
42	C₄×C₈	—	223	1	no	—
43	C₄×C₈	—	223	1	no	—
44	C₄×C₈	C₂×D₂₂₃	1	1	yes	characteristic
45	C₂×C₈	—	446	1	no	—
46	C₂×C₈	—	223	1	no	—
47	C₂×C₈	—	223	1	no	—
48	C₂×C₈	C₂²⋊D₂₂₃	1	1	yes	characteristic
49	C₂×C₈	Dic₄₄₆	1	1	yes	characteristic
50	C₄²	C₄×D₂₂₃	1	1	yes	characteristic
51	C₁₆	—	4	1	no	—
52	C₁₆	—	892	1	no	—
53	C₂×C₄	Dic₂₂₃⋊C₄	1	1	yes	characteristic
54	C₂×C₄	C₈×D₂₂₃	1	1	yes	characteristic
55	C₂×C₄	#16813153	1	1	yes	characteristic
56	C₈	—	2	1	no	—
57	C₈	—	446	2	no	—
58	C₈	—	446	1	no	—
59	C₈	—	223	2	no	—
60	C₈	#16813155	1	1	yes	characteristic
61	C₈	#16813160	1	1	yes	characteristic
62	C₂²	Dic₂₂₃⋊C₈	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	#20297237	1	1	yes	characteristic
65	C₄	#20297239	1	1	yes	characteristic
66	C₄	C₁₆×D₂₂₃	1	1	yes	characteristic
67	C₄	#20297243	1	1	yes	characteristic
68	C₂	#28383332	1	1	yes	characteristic
69	C₂	Dic₂₂₃⋊C₁₆	1	1	yes	characteristic
70	C₂	#28383400	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
19. C₂₂₃⋊C₈	C₁₆	semidirect
35. C₂₂₃	#1370	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
C₈×C₂₂₃⋊C₈	2
C₄⋊C₁₆×C₂₂₃	2
#1370	223