Subgroups — C₂²⋊C₃₂×C₂₂₃ #43250414

Order 28544.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₂×C₂₂₃	C₂	1	2	yes	maximal, complemented
3	C₂²×C₈×C₂₂₃	C₄	1	1	yes	characteristic
4	C₂×C₁₆×C₂₂₃	—	2	2	no	—
5	C₂×C₁₆×C₂₂₃	C₄	1	1	yes	characteristic
6	C₂×C₁₆×C₂₂₃	C₂²	1	1	yes	characteristic
7	C₃₂×C₂₂₃	—	2	2	no	—
8	C₂²×C₄×C₂₂₃	C₈	1	1	yes	characteristic
9	C₂×C₈×C₂₂₃	—	2	1	no	—
10	C₂×C₈×C₂₂₃	—	2	1	no	—
11	C₂×C₈×C₂₂₃	C₈	1	1	yes	characteristic
12	C₂×C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
13	C₁₆×C₂₂₃	—	2	1	no	—
14	C₁₆×C₂₂₃	D₄	1	2	yes	—
15	C₂³×C₂₂₃	C₁₆	1	1	yes	characteristic
16	C₂×C₄×C₂₂₃	—	2	1	no	—
17	C₂×C₄×C₂₂₃	—	2	1	no	—
18	C₂×C₄×C₂₂₃	C₁₆	1	1	yes	characteristic
19	C₂×C₄×C₂₂₃	C₂×C₈	1	1	yes	characteristic
20	C₈×C₂₂₃	—	2	1	no	—
21	C₈×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₈×C₂₂₃	M₁₆	1	1	yes	characteristic
23	C₂²×C₂₂₃	—	2	1	no	—
24	C₂²×C₂₂₃	—	2	1	no	—
25	C₂²×C₂₂₃	C₃₂	1	2	yes	complemented
26	C₂²×C₂₂₃	C₂×C₁₆	1	1	yes	characteristic
27	C₄×C₂₂₃	—	2	1	no	—
28	C₄×C₂₂₃	C₂²⋊C₈	1	1	yes	characteristic
29	C₄×C₂₂₃	M₃₂	1	1	yes	characteristic
30	C₂×C₂₂₃	—	2	2	no	—
31	C₂×C₂₂₃	C₂²⋊C₁₆	1	1	yes	characteristic
32	C₂×C₂₂₃	C₂×C₃₂	1	1	yes	characteristic
33	C₂×C₂₂₃	M₆₄	1	1	yes	characteristic
34	C₂₂₃	C₂²⋊C₃₂	1	1	yes	characteristic, direct product factor, Sylow
35	C₂²⋊C₃₂	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂²×C₁₆	C₂×C₂₂₃	1	1	yes	characteristic
37	C₂×C₃₂	C₂×C₂₂₃	1	2	yes	complemented
38	C₂²×C₈	C₄×C₂₂₃	1	1	yes	characteristic
39	C₂×C₁₆	—	2	2	no	—
40	C₂×C₁₆	C₄×C₂₂₃	1	1	yes	characteristic
41	C₂×C₁₆	C₂²×C₂₂₃	1	1	yes	characteristic
42	C₃₂	—	2	2	no	—
43	C₂²×C₄	C₈×C₂₂₃	1	1	yes	characteristic
44	C₂×C₈	—	2	1	no	—
45	C₂×C₈	—	2	1	no	—
46	C₂×C₈	C₈×C₂₂₃	1	1	yes	characteristic
47	C₂×C₈	C₂×C₄×C₂₂₃	1	1	yes	characteristic
48	C₁₆	—	2	1	no	—
49	C₁₆	D₄×C₂₂₃	1	2	yes	—
50	C₂³	C₁₆×C₂₂₃	1	1	yes	characteristic
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	C₁₆×C₂₂₃	1	1	yes	characteristic
54	C₂×C₄	C₂×C₈×C₂₂₃	1	1	yes	characteristic
55	C₈	—	2	1	no	—
56	C₈	C₂²⋊C₄×C₂₂₃	1	1	yes	characteristic
57	C₈	M₁₆×C₂₂₃	1	1	yes	characteristic
58	C₂²	—	2	1	no	—
59	C₂²	—	2	1	no	—
60	C₂²	C₃₂×C₂₂₃	1	2	yes	complemented
61	C₂²	C₂×C₁₆×C₂₂₃	1	1	yes	characteristic
62	C₄	—	2	1	no	—
63	C₄	C₂²⋊C₈×C₂₂₃	1	1	yes	characteristic
64	C₄	M₃₂×C₂₂₃	1	1	yes	characteristic
65	C₂	—	2	2	no	—
66	C₂	C₂²⋊C₁₆×C₂₂₃	1	1	yes	characteristic
67	C₂	C₂×C₃₂×C₂₂₃	1	1	yes	characteristic
68	C₂	M₆₄×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆×C₂₂₃	C₂²
commutator	C₂	C₂×C₃₂×C₂₂₃
Frattini, cosocle	C₂×C₁₆	C₂²×C₂₂₃
socle	C₂²×C₂₂₃	C₂×C₁₆
prime radical	C₂³×C₂₂₃	C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₂×C₂₂₃	C₂	semidirect
25. C₂²×C₂₂₃	C₃₂	semidirect
34. C₂₂₃	C₂²⋊C₃₂	direct (N × H)
35. C₂²⋊C₃₂	C₂₂₃	direct (N × H)
37. C₂×C₃₂	C₂×C₂₂₃	semidirect
60. C₂²	C₃₂×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₆×C₂₂₃	2
C₂×C₃₂×C₂₂₃	2
C₂²⋊C₃₂	223

Subgroups — C22⋊C32×C223 #43250414

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²⋊C₃₂×C₂₂₃ #43250414