Subgroups — #2262×C₂₂₃ #43251277

Order 28544.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#508×C₂₂₃	C₂	1	2	yes	maximal, complemented
2	Q₆₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	SD₆₄×C₂₂₃	C₂	1	2	yes	maximal, complemented
4	D₃₂×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₂×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#135×C₂₂₃	—	2	2	no	—
7	Q₃₂×C₂₂₃	C₂²	1	2	yes	—
8	SD₃₂×C₂₂₃	—	2	2	no	—
9	D₁₆×C₂₂₃	C₂²	1	2	yes	—
10	C₂×C₁₆×C₂₂₃	C₂²	1	1	yes	characteristic
11	C₃₂×C₂₂₃	C₂²	1	1	yes	characteristic
12	C₃₂×C₂₂₃	C₂²	1	1	yes	characteristic
13	#41×C₂₂₃	—	4	2	no	—
14	Q₁₆×C₂₂₃	—	2	2	no	—
15	SD₁₆×C₂₂₃	—	4	2	no	—
16	D₈×C₂₂₃	—	2	2	no	—
17	C₂×C₈×C₂₂₃	D₄	1	1	yes	characteristic
18	C₁₆×C₂₂₃	D₄	1	1	yes	characteristic
19	C₁₆×C₂₂₃	C₂³	1	1	yes	characteristic
20	Q₈×C₂₂₃	—	4	2	no	—
21	D₄×C₂₂₃	—	4	2	no	—
22	D₄×C₂₂₃	—	8	2	no	—
23	C₂×C₄×C₂₂₃	—	8	2	no	—
24	C₂×C₄×C₂₂₃	D₈	1	1	yes	characteristic
25	C₈×C₂₂₃	D₈	1	1	yes	characteristic
26	C₈×C₂₂₃	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₂₂₃	—	8	2	no	—
28	C₂²×C₂₂₃	D₁₆	1	1	yes	characteristic
29	C₄×C₂₂₃	—	8	2	no	—
30	C₄×C₂₂₃	D₁₆	1	1	yes	characteristic
31	C₄×C₂₂₃	C₂×D₈	1	1	yes	characteristic
32	C₂×C₂₂₃	—	2	1	no	—
33	C₂×C₂₂₃	—	16	2	no	—
34	C₂×C₂₂₃	C₂×D₁₆	1	1	yes	characteristic
35	C₂₂₃	#2262	1	1	yes	characteristic, direct product factor, Sylow
36	#2262	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	#508	C₂×C₂₂₃	1	2	yes	complemented
38	Q₆₄	C₂×C₂₂₃	1	1	yes	characteristic, complemented
39	SD₆₄	C₂×C₂₂₃	1	2	yes	complemented
40	D₃₂	C₂×C₂₂₃	1	1	yes	characteristic, complemented
41	C₂×C₃₂	C₂×C₂₂₃	1	1	yes	characteristic, complemented
42	#135	—	2	2	no	—
43	Q₃₂	C₂²×C₂₂₃	1	2	yes	—
44	SD₃₂	—	2	2	no	—
45	D₁₆	C₂²×C₂₂₃	1	2	yes	—
46	C₂×C₁₆	C₂²×C₂₂₃	1	1	yes	characteristic
47	C₃₂	C₂²×C₂₂₃	1	1	yes	characteristic
48	C₃₂	C₂²×C₂₂₃	1	1	yes	characteristic
49	#41	—	4	2	no	—
50	Q₁₆	—	2	2	no	—
51	SD₁₆	—	4	2	no	—
52	D₈	—	2	2	no	—
53	C₂×C₈	D₄×C₂₂₃	1	1	yes	characteristic
54	C₁₆	D₄×C₂₂₃	1	1	yes	characteristic
55	C₁₆	C₂³×C₂₂₃	1	1	yes	characteristic
56	Q₈	—	4	2	no	—
57	D₄	—	4	2	no	—
58	D₄	—	8	2	no	—
59	C₂×C₄	—	8	2	no	—
60	C₂×C₄	D₈×C₂₂₃	1	1	yes	characteristic
61	C₈	D₈×C₂₂₃	1	1	yes	characteristic
62	C₈	C₂×D₄×C₂₂₃	1	1	yes	characteristic
63	C₂²	—	8	2	no	—
64	C₂²	D₁₆×C₂₂₃	1	1	yes	characteristic
65	C₄	—	8	2	no	—
66	C₄	D₁₆×C₂₂₃	1	1	yes	characteristic
67	C₄	C₂×D₈×C₂₂₃	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	16	2	no	—
70	C₂	C₂×D₁₆×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₂₃	D₁₆
commutator, Frattini, cosocle	C₁₆	C₂³×C₂₂₃
socle	C₂×C₂₂₃	C₂×D₁₆
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₂₃ ◁ C₂×C₄×C₂₂₃ ◁ C₂×C₈×C₂₂₃ ◁ C₂×C₁₆×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. #508×C₂₂₃	C₂	semidirect
2. Q₆₄×C₂₂₃	C₂	semidirect
3. SD₆₄×C₂₂₃	C₂	semidirect
4. D₃₂×C₂₂₃	C₂	semidirect
5. C₂×C₃₂×C₂₂₃	C₂	semidirect
35. C₂₂₃	#2262	direct (N × H)
36. #2262	C₂₂₃	direct (N × H)
37. #508	C₂×C₂₂₃	semidirect
38. Q₆₄	C₂×C₂₂₃	semidirect
39. SD₆₄	C₂×C₂₂₃	semidirect
40. D₃₂	C₂×C₂₂₃	semidirect
41. C₂×C₃₂	C₂×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#508×C₂₂₃	2
Q₆₄×C₂₂₃	2
SD₆₄×C₂₂₃	2
D₃₂×C₂₂₃	2
C₂×C₃₂×C₂₂₃	2
#2262	223

Subgroups — #2262×C223 #43251277

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2262×C₂₂₃ #43251277