Subgroups — M₃₂×D₂₂₃ #28383993

Order 14272.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₂₂₃	C₂	1	1	yes	maximal, characteristic
2	#20297244	C₂	1	1	yes	maximal, characteristic, complemented
3	#20297243	C₂	1	2	yes	maximal, complemented
4	C₁₆×D₂₂₃	C₂	1	2	yes	maximal, complemented
5	M₃₂×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₂₂₃	C₄	1	1	yes	characteristic
7	C₂×C₂₂₃⋊C₈	C₄	1	1	yes	characteristic
8	C₈×D₂₂₃	C₄	1	1	yes	characteristic
9	C₈×D₂₂₃	C₄	1	1	yes	characteristic
10	C₈×D₂₂₃	C₂²	1	1	yes	characteristic
11	C₈×D₂₂₃	C₂²	1	1	yes	characteristic
12	C₂₂₃⋊C₁₆	C₂²	1	2	yes	complemented
13	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
14	C₁₆×C₂₂₃	C₂²	1	2	yes	complemented
15	C₂²×D₂₂₃	C₈	1	1	yes	characteristic
16	C₂×Dic₂₂₃	C₈	1	1	yes	characteristic
17	C₄×D₂₂₃	C₈	1	1	yes	characteristic
18	C₄×D₂₂₃	C₈	1	1	yes	characteristic
19	C₄×D₂₂₃	C₂×C₄	1	1	yes	characteristic
20	C₄×D₂₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂₂₃⋊C₈	C₂×C₄	1	1	yes	characteristic
22	C₂₂₃⋊C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
24	C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
25	C₈×C₂₂₃	C₂³	1	1	yes	characteristic
26	C₂²×C₂₂₃	C₂×C₈	1	1	yes	characteristic
27	C₂×D₂₂₃	—	2	2	no	—
28	C₂×D₂₂₃	C₂×C₈	1	1	yes	characteristic
29	C₂×D₂₂₃	C₂×C₈	1	1	yes	characteristic
30	C₄×C₂₂₃	C₂×C₈	1	1	yes	characteristic
31	C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
32	Dic₂₂₃	C₂×C₈	1	1	yes	characteristic
33	Dic₂₂₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₂₂₃	—	2	1	no	—
35	C₂×C₂₂₃	C₂²×C₈	1	1	yes	characteristic
36	D₂₂₃	—	2	1	no	—
37	D₂₂₃	M₃₂	1	2	yes	direct product factor
38	C₂₂₃	C₂×M₃₂	1	1	yes	characteristic, complemented, Sylow
39	C₂×M₃₂	—	223	1	no	maximal, Sylow
40	C₂²×C₈	—	223	1	no	—
41	M₃₂	—	223	1	no	—
42	M₃₂	—	223	2	no	—
43	M₃₂	D₂₂₃	1	1	yes	characteristic, direct product factor
44	C₂×C₁₆	—	223	2	no	—
45	C₂²×C₄	—	223	1	no	—
46	C₂×C₈	—	223	1	no	—
47	C₂×C₈	—	223	1	no	—
48	C₂×C₈	—	223	1	no	—
49	C₂×C₈	—	223	1	no	—
50	C₂×C₈	—	223	1	no	—
51	C₂×C₈	C₂×D₂₂₃	1	1	yes	characteristic
52	C₁₆	—	223	2	no	—
53	C₁₆	C₂×D₂₂₃	1	2	yes	complemented
54	C₂³	—	223	1	no	—
55	C₂×C₄	—	223	1	no	—
56	C₂×C₄	—	223	1	no	—
57	C₂×C₄	—	223	1	no	—
58	C₂×C₄	—	223	1	no	—
59	C₂×C₄	—	223	1	no	—
60	C₂×C₄	C₄×D₂₂₃	1	1	yes	characteristic
61	C₈	—	223	1	no	—
62	C₈	—	223	1	no	—
63	C₈	C₄×D₂₂₃	1	1	yes	characteristic
64	C₈	C₂²×D₂₂₃	1	1	yes	characteristic
65	C₂²	—	446	2	no	—
66	C₂²	—	223	1	no	—
67	C₂²	—	223	1	no	—
68	C₂²	C₈×D₂₂₃	1	1	yes	characteristic
69	C₄	—	223	1	no	—
70	C₄	—	223	1	no	—
71	C₄	C₈×D₂₂₃	1	1	yes	characteristic
72	C₄	C₂×C₄×D₂₂₃	1	1	yes	characteristic
73	C₂	—	2	1	no	—
74	C₂	—	446	1	no	—
75	C₂	—	223	2	no	—
76	C₂	C₂×C₈×D₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂²×D₂₂₃
commutator, socle	C₂×C₂₂₃	C₂²×C₈
Fitting	M₃₂×C₂₂₃	C₂
prime radical	C₂²×D₂₂₃	C₈
cosocle	C₈×C₂₂₃	C₂³
hypercenter	M₃₂	D₂₂₃
hypocenter	C₂₂₃	C₂×M₃₂
LCS	G ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. #20297244	C₂	semidirect
3. #20297243	C₂	semidirect
4. C₁₆×D₂₂₃	C₂	semidirect
5. M₃₂×C₂₂₃	C₂	semidirect
12. C₂₂₃⋊C₁₆	C₂²	semidirect
14. C₁₆×C₂₂₃	C₂²	semidirect
37. D₂₂₃	M₃₂	direct (N × H)
38. C₂₂₃	C₂×M₃₂	semidirect
43. M₃₂	D₂₂₃	direct (N × H)
53. C₁₆	C₂×D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₂₂₃	2
#20297244	2
#20297243	2
C₁₆×D₂₂₃	2
M₃₂×C₂₂₃	2
C₂×M₃₂	223

Subgroups — M32×D223 #28383993

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₃₂×D₂₂₃ #28383993