Subgroups — D₆₉² #34039971

Order 19044.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⋊(C₃⋊(C₂₃⋊D₂₃))	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₂₃×D₆₉	C₂	1	2	yes	maximal, complemented
3	D₂₃×D₆₉	—	3	2	no	maximal
4	C₃²×C₂₃²	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₃⋊(C₂₃⋊D₂₃)	—	3	2	no	—
6	C₃⋊(C₂₃⋊D₂₃)	—	6	1	no	—
7	C₂₃×D₆₉	D₃	1	2	yes	complemented
8	C₃×C₂₃×D₂₃	—	3	2	no	—
9	D₂₃²	—	9	1	no	Hall
10	C₃×C₂₃²	—	2	1	no	—
11	C₃×C₂₃²	C₂×D₃	1	2	yes	complemented
12	C₂₃⋊D₂₃	—	9	1	no	—
13	C₂₃×D₂₃	—	3	2	no	—
14	D₃×D₆₉	—	23	2	no	maximal
15	C₂₃²	D₃²	1	1	yes	characteristic, complemented, Sylow
16	C₃⋊D₆₉	—	23	2	no	—
17	C₃⋊D₆₉	—	46	11	no	—
18	C₃×D₆₉	D₂₃	1	2	yes	complemented
19	C₃×D₃×C₂₃	—	23	2	no	—
20	C₂×D₆₉	—	69	2	no	—
21	D₃×D₂₃	—	69	2	no	—
22	C₃²×C₂₃	—	2	11	no	—
23	C₃²×C₂₃	C₂×D₂₃	1	2	yes	complemented
24	C₂×C₃×C₂₃	—	69	2	no	—
25	D₆₉	—	69	2	no	—
26	D₆₉	—	69	2	no	—
27	D₆₉	—	138	22	no	—
28	D₆₉	—	138	22	no	—
29	D₆₉	—	138	2	no	—
30	D₆₉	D₆₉	1	2	yes	direct product factor
31	C₃×D₂₃	—	3	2	no	—
32	D₃×C₂₃	—	23	2	no	—
33	C₂×D₂₃	—	207	2	no	—
34	C₃×C₂₃	—	2	22	no	—
35	C₃×C₂₃	—	2	2	no	—
36	C₃×C₂₃	—	2	22	no	—
37	C₃×C₂₃	D₃×D₂₃	1	2	yes	complemented
38	C₃×C₂₃	C₂×D₆₉	1	2	yes	complemented
39	C₂×C₂₃	—	69	2	no	—
40	D₂₃	—	3	2	no	—
41	D₂₃	—	414	11	no	—
42	D₂₃	—	207	2	no	—
43	D₃²	—	529	1	no	Hall
44	C₂₃	—	2	11	no	—
45	C₂₃	D₃×D₆₉	1	2	yes	complemented
46	C₃⋊D₃	—	529	1	no	—
47	C₃×D₃	—	23	2	no	—
48	C₂×D₃	—	1587	2	no	—
49	C₃²	D₂₃²	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₃	—	69	2	no	—
51	D₃	—	23	2	no	—
52	D₃	—	1587	2	no	—
53	D₃	—	3174	1	no	—
54	C₂²	—	4761	1	no	Sylow
55	C₃	—	2	1	no	—
56	C₃	D₂₃×D₆₉	1	2	yes	complemented
57	C₂	—	69	2	no	—
58	C₂	—	4761	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃²×C₂₃²	C₂²
LCS	G ▷ C₃²×C₂₃²
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₃⋊(C₃⋊(C₂₃⋊D₂₃))	C₂	semidirect
2. C₃×C₂₃×D₆₉	C₂	semidirect
4. C₃²×C₂₃²	C₂²	semidirect
7. C₂₃×D₆₉	D₃	semidirect
11. C₃×C₂₃²	C₂×D₃	semidirect
15. C₂₃²	D₃²	semidirect
18. C₃×D₆₉	D₂₃	semidirect
23. C₃²×C₂₃	C₂×D₂₃	semidirect
30. D₆₉	D₆₉	direct (N × H)
37. C₃×C₂₃	D₃×D₂₃	semidirect
38. C₃×C₂₃	C₂×D₆₉	semidirect
45. C₂₃	D₃×D₆₉	semidirect
49. C₃²	D₂₃²	semidirect
56. C₃	D₂₃×D₆₉	semidirect

Maximal subgroups

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

subgroup	index
C₃⋊(C₃⋊(C₂₃⋊D₂₃))	2
C₃×C₂₃×D₆₉	2
D₂₃×D₆₉	3
D₃×D₆₉	23

Subgroups — D692 #34039971

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₆₉² #34039971