Subgroups — #1372×C₂₂₃ #43250387

Order 28544.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	#335×C₂₂₃	C₂	1	2	yes	maximal
3	C₂×C₁₆×C₂₂₃	C₄	1	1	yes	characteristic
4	C₂×C₁₆×C₂₂₃	C₄	1	1	yes	characteristic
5	C₄⋊C₈×C₂₂₃	—	2	2	no	—
6	C₄×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₈×C₂₂₃	—	4	2	no	—
8	C₂×C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
9	C₂×C₈×C₂₂₃	Q₈	1	1	yes	characteristic
10	C₄²×C₂₂₃	D₄	1	1	yes	characteristic
11	C₁₆×C₂₂₃	—	2	1	no	—
12	C₁₆×C₂₂₃	C₈	1	2	yes	complemented
13	C₂×C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
14	C₂×C₄×C₂₂₃	D₈	1	1	yes	characteristic
15	C₂×C₄×C₂₂₃	Q₁₆	1	1	yes	characteristic
16	C₈×C₂₂₃	—	2	1	no	—
17	C₈×C₂₂₃	—	8	2	no	—
18	C₈×C₂₂₃	C₂×C₈	1	1	yes	characteristic
19	C₈×C₂₂₃	M₁₆	1	1	yes	characteristic
20	C₂²×C₂₂₃	#107	1	1	yes	characteristic
21	C₄×C₂₂₃	—	2	1	no	—
22	C₄×C₂₂₃	C₄⋊C₈	1	1	yes	characteristic
23	C₄×C₂₂₃	#108	1	1	yes	characteristic
24	C₄×C₂₂₃	SD₃₂	1	2	yes	—
25	C₂×C₂₂₃	#335	1	1	yes	characteristic
26	C₂×C₂₂₃	#367	1	1	yes	characteristic
27	C₂×C₂₂₃	#368	1	1	yes	characteristic
28	C₂₂₃	#1372	1	1	yes	characteristic, direct product factor, Sylow
29	#1372	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₄×C₁₆	C₂×C₂₂₃	1	1	yes	characteristic
31	#335	C₂×C₂₂₃	1	2	yes	—
32	C₂×C₁₆	C₄×C₂₂₃	1	1	yes	characteristic
33	C₂×C₁₆	C₄×C₂₂₃	1	1	yes	characteristic
34	C₄⋊C₈	—	2	2	no	—
35	C₄×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
36	C₂×C₈	—	4	2	no	—
37	C₂×C₈	C₂×C₄×C₂₂₃	1	1	yes	characteristic
38	C₂×C₈	Q₈×C₂₂₃	1	1	yes	characteristic
39	C₄²	D₄×C₂₂₃	1	1	yes	characteristic
40	C₁₆	—	2	1	no	—
41	C₁₆	C₈×C₂₂₃	1	2	yes	complemented
42	C₂×C₄	C₄⋊C₄×C₂₂₃	1	1	yes	characteristic
43	C₂×C₄	D₈×C₂₂₃	1	1	yes	characteristic
44	C₂×C₄	Q₁₆×C₂₂₃	1	1	yes	characteristic
45	C₈	—	2	1	no	—
46	C₈	—	8	2	no	—
47	C₈	C₂×C₈×C₂₂₃	1	1	yes	characteristic
48	C₈	M₁₆×C₂₂₃	1	1	yes	characteristic
49	C₂²	#107×C₂₂₃	1	1	yes	characteristic
50	C₄	—	2	1	no	—
51	C₄	C₄⋊C₈×C₂₂₃	1	1	yes	characteristic
52	C₄	#108×C₂₂₃	1	1	yes	characteristic
53	C₄	SD₃₂×C₂₂₃	1	2	yes	—
54	C₂	#335×C₂₂₃	1	1	yes	characteristic
55	C₂	#367×C₂₂₃	1	1	yes	characteristic
56	C₂	#368×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₂₃	D₈
commutator	C₈	C₂×C₈×C₂₂₃
Frattini, cosocle	C₄×C₈	C₂²×C₂₂₃
socle, prime radical	C₂²×C₂₂₃	#107
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₂₃ ◁ C₄²×C₂₂₃ ◁ C₄×C₈×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₁₆×C₂₂₃	C₈	semidirect
28. C₂₂₃	#1372	direct (N × H)
29. #1372	C₂₂₃	direct (N × H)
41. C₁₆	C₈×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
#335×C₂₂₃	2
#1372	223

Subgroups — #1372×C223 #43250387

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1372×C₂₂₃ #43250387