Subgroups — #1370×C₂₂₃ #43250385

Order 28544.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊C₁₆×C₂₂₃	C₂	1	2	yes	maximal
2	C₈²×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×C₂₂₃	—	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	C₂×C₈×C₂₂₃	—	2	1	no	—
8	C₂×C₈×C₂₂₃	C₈	1	1	yes	characteristic
9	C₂×C₈×C₂₂₃	C₈	1	1	yes	characteristic
10	C₂×C₈×C₂₂₃	D₄	1	1	yes	characteristic
11	C₂×C₈×C₂₂₃	Q₈	1	1	yes	characteristic
12	C₄²×C₂₂₃	C₂×C₄	1	1	yes	characteristic
13	C₁₆×C₂₂₃	—	4	2	no	—
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, direct product factor, Sylow
33	#1370	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₄⋊C₁₆	C₂×C₂₂₃	1	2	yes	—
35	C₈²	C₂×C₂₂₃	1	1	yes	characteristic
36	C₂×C₁₆	—	2	2	no	—
37	C₄×C₈	C₄×C₂₂₃	1	1	yes	characteristic
38	C₄×C₈	C₄×C₂₂₃	1	1	yes	characteristic
39	C₄×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
40	C₂×C₈	—	2	1	no	—
41	C₂×C₈	C₈×C₂₂₃	1	1	yes	characteristic
42	C₂×C₈	C₈×C₂₂₃	1	1	yes	characteristic
43	C₂×C₈	D₄×C₂₂₃	1	1	yes	characteristic
44	C₂×C₈	Q₈×C₂₂₃	1	1	yes	characteristic
45	C₄²	C₂×C₄×C₂₂₃	1	1	yes	characteristic
46	C₁₆	—	4	2	no	—
47	C₂×C₄	C₄⋊C₄×C₂₂₃	1	1	yes	characteristic
48	C₂×C₄	C₂×C₈×C₂₂₃	1	1	yes	characteristic
49	C₂×C₄	M₁₆×C₂₂₃	1	1	yes	characteristic
50	C₈	—	2	2	no	—
51	C₈	—	2	1	no	—
52	C₈	—	2	1	no	—
53	C₈	C₁₆×C₂₂₃	1	2	yes	complemented
54	C₈	D₈×C₂₂₃	1	1	yes	characteristic
55	C₈	Q₁₆×C₂₂₃	1	1	yes	characteristic
56	C₂²	C₄⋊C₈×C₂₂₃	1	1	yes	characteristic
57	C₄	—	2	1	no	—
58	C₄	#107×C₂₂₃	1	1	yes	characteristic
59	C₄	#108×C₂₂₃	1	1	yes	characteristic
60	C₄	C₂×C₁₆×C₂₂₃	1	1	yes	characteristic
61	C₄	M₃₂×C₂₂₃	1	1	yes	characteristic
62	C₂	#335×C₂₂₃	1	1	yes	characteristic
63	C₂	C₄⋊C₁₆×C₂₂₃	1	1	yes	characteristic
64	C₂	#364×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₂₂₃	D₄
commutator	C₄	C₂×C₁₆×C₂₂₃
Frattini, cosocle	C₄×C₈	C₂²×C₂₂₃
socle, prime radical	C₂²×C₂₂₃	C₄⋊C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₂₃ ◁ C₄×C₈×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
20. C₈×C₂₂₃	C₁₆	semidirect
32. C₂₂₃	#1370	direct (N × H)
33. #1370	C₂₂₃	direct (N × H)
53. C₈	C₁₆×C₂₂₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄⋊C₁₆×C₂₂₃	2
C₈²×C₂₂₃	2
#1370	223

Subgroups — #1370×C223 #43250385

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1370×C₂₂₃ #43250385