Subgroups — #1384×C₂₂₃ #43250399

Order 28544.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	#427×C₂₂₃	C₂	1	2	yes	maximal
3	C₂²×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
4	#117×C₂₂₃	—	2	2	no	—
5	M₃₂×C₂₂₃	—	2	2	no	—
6	C₂×C₁₆×C₂₂₃	C₄	1	2	yes	complemented
7	#107×C₂₂₃	—	2	2	no	—
8	#106×C₂₂₃	C₄	1	4	yes	complemented
9	C₂²×C₄×C₂₂₃	D₄	1	1	yes	characteristic
10	C₂×C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
11	C₂×C₈×C₂₂₃	C₂×C₄	1	2	yes	—
12	C₂×C₈×C₂₂₃	D₄	1	2	yes	—
13	C₂×C₈×C₂₂₃	Q₈	1	1	yes	characteristic
14	C₄⋊C₄×C₂₂₃	—	2	4	no	—
15	C₂²⋊C₄×C₂₂₃	—	4	2	no	—
16	C₄²×C₂₂₃	—	4	2	no	—
17	C₁₆×C₂₂₃	—	2	2	no	—
18	C₂³×C₂₂₃	D₈	1	1	yes	characteristic
19	C₂×C₄×C₂₂₃	—	4	4	no	—
20	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	2	yes	—
21	C₂×C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₂₂₃	SD₁₆	1	2	yes	—
23	C₂×C₄×C₂₂₃	Q₁₆	1	1	yes	characteristic
24	C₈×C₂₂₃	C₄²	1	1	yes	characteristic
25	C₈×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₂₂₃	C₄⋊C₄	1	2	yes	—
27	C₂²×C₂₂₃	—	4	1	no	—
28	C₂²×C₂₂₃	#102	1	2	yes	—
29	C₂²×C₂₂₃	#107	1	1	yes	characteristic
30	C₄×C₂₂₃	—	8	4	no	—
31	C₄×C₂₂₃	#95	1	1	yes	characteristic
32	C₄×C₂₂₃	#103	1	2	yes	—
33	C₄×C₂₂₃	#106	1	1	yes	characteristic
34	C₂×C₂₂₃	—	2	2	no	—
35	C₂×C₂₂₃	—	2	1	no	—
36	C₂×C₂₂₃	#340	1	1	yes	characteristic
37	C₂₂₃	#1384	1	1	yes	characteristic, direct product factor, Sylow
38	#1384	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂×M₃₂	C₂×C₂₂₃	1	1	yes	characteristic
40	#427	C₂×C₂₂₃	1	2	yes	—
41	C₂²×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
42	#117	—	2	2	no	—
43	M₃₂	—	2	2	no	—
44	C₂×C₁₆	C₄×C₂₂₃	1	2	yes	complemented
45	#107	—	2	2	no	—
46	#106	C₄×C₂₂₃	1	4	yes	complemented
47	C₂²×C₄	D₄×C₂₂₃	1	1	yes	characteristic
48	C₂×C₈	C₂×C₄×C₂₂₃	1	1	yes	characteristic
49	C₂×C₈	C₂×C₄×C₂₂₃	1	2	yes	—
50	C₂×C₈	D₄×C₂₂₃	1	2	yes	—
51	C₂×C₈	Q₈×C₂₂₃	1	1	yes	characteristic
52	C₄⋊C₄	—	2	4	no	—
53	C₂²⋊C₄	—	4	2	no	—
54	C₄²	—	4	2	no	—
55	C₁₆	—	2	2	no	—
56	C₂³	D₈×C₂₂₃	1	1	yes	characteristic
57	C₂×C₄	—	4	4	no	—
58	C₂×C₄	C₂²⋊C₄×C₂₂₃	1	2	yes	—
59	C₂×C₄	C₄⋊C₄×C₂₂₃	1	1	yes	characteristic
60	C₂×C₄	SD₁₆×C₂₂₃	1	2	yes	—
61	C₂×C₄	Q₁₆×C₂₂₃	1	1	yes	characteristic
62	C₈	C₄²×C₂₂₃	1	1	yes	characteristic
63	C₈	C₂²⋊C₄×C₂₂₃	1	1	yes	characteristic
64	C₈	C₄⋊C₄×C₂₂₃	1	2	yes	—
65	C₂²	—	4	1	no	—
66	C₂²	#102×C₂₂₃	1	2	yes	—
67	C₂²	#107×C₂₂₃	1	1	yes	characteristic
68	C₄	—	8	4	no	—
69	C₄	#95×C₂₂₃	1	1	yes	characteristic
70	C₄	#103×C₂₂₃	1	2	yes	—
71	C₄	#106×C₂₂₃	1	1	yes	characteristic
72	C₂	—	2	2	no	—
73	C₂	—	2	1	no	—
74	C₂	#340×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₂₃	#106
commutator	C₈	C₄²×C₂₂₃
Frattini, cosocle	C₂²×C₈	C₂²×C₂₂₃
socle	C₂×C₂₂₃	#340
prime radical	C₂³×C₂₂₃	D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₂₃ ◁ C₂²×C₄×C₂₂₃ ◁ C₂²×C₈×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₂×C₁₆×C₂₂₃	C₄	semidirect
8. #106×C₂₂₃	C₄	semidirect
37. C₂₂₃	#1384	direct (N × H)
38. #1384	C₂₂₃	direct (N × H)
44. C₂×C₁₆	C₄×C₂₂₃	semidirect
46. #106	C₄×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₃₂×C₂₂₃	2
#427×C₂₂₃	2
#1384	223

Subgroups — #1384×C223 #43250399

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1384×C₂₂₃ #43250399