Subgroups — #13110233

Order 1656.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃⋊D₆₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×D₆₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×Dic₃×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	D₂₇₆	—	3	1	no	maximal
5	#10587771	—	3	1	no	maximal
6	C₂×C₃²×C₂₃	C₂²	1	1	yes	characteristic
7	C₃⋊D₆₉	—	2	1	no	—
8	C₃×D₆₉	—	2	1	no	—
9	C₂×D₆₉	—	3	1	no	—
10	C₂×D₆₉	—	3	1	no	—
11	C₂×D₆₉	—	6	1	no	—
12	C₂×D₆₉	D₃	1	1	yes	characteristic, complemented
13	C₂×C₃×D₂₃	—	3	1	no	—
14	C₃×C₄×C₂₃	—	3	1	no	—
15	Dic₃×C₂₃	D₃	1	1	yes	characteristic, complemented
16	C₃²×C₂₃	D₄	1	1	yes	characteristic, complemented, Hall
17	D₉₂	—	9	1	no	Hall
18	C₂×C₃×C₂₃	—	2	1	no	—
19	C₂×C₃×C₂₃	C₂×D₃	1	1	yes	characteristic
20	C₂×C₃×C₂₃	C₂×D₃	1	1	yes	characteristic
21	D₆₉	—	2	1	no	—
22	D₆₉	—	6	2	no	—
23	D₆₉	—	6	1	no	—
24	D₆₉	—	6	1	no	—
25	C₃×D₂₃	—	6	1	no	—
26	C₂×D₂₃	—	3	1	no	—
27	C₂×D₂₃	—	9	1	no	—
28	C₄×C₂₃	—	3	1	no	—
29	#626	—	23	1	no	maximal, Hall
30	C₃×C₂₃	—	2	1	no	—
31	C₃×C₂₃	D₁₂	1	1	yes	characteristic, complemented
32	C₃×C₂₃	#67	1	1	yes	characteristic, complemented
33	C₂×C₂₃	D₃²	1	1	yes	characteristic
34	D₂₃	—	6	1	no	—
35	D₂₃	—	18	1	no	—
36	C₂×C₃⋊D₃	—	23	1	no	—
37	C₂×C₃×D₃	—	23	1	no	—
38	C₃×Dic₃	D₂₃	1	1	yes	characteristic, complemented
39	#67	—	69	1	no	—
40	D₁₂	—	69	1	no	—
41	C₂₃	#626	1	1	yes	characteristic, complemented, Sylow
42	C₂×C₃²	C₂×D₂₃	1	1	yes	characteristic
43	C₃⋊D₃	—	46	1	no	—
44	C₃×D₃	—	46	1	no	—
45	C₂²×C₃	—	69	1	no	—
46	C₂×D₃	—	23	1	no	—
47	C₂×D₃	—	69	1	no	—
48	C₂×D₃	—	69	1	no	—
49	C₂×D₃	—	138	1	no	—
50	C₃×C₄	—	3	1	no	—
51	Dic₃	D₆₉	1	1	yes	characteristic, complemented
52	C₃²	D₉₂	1	1	yes	characteristic, complemented, Sylow
53	D₄	—	207	1	no	Sylow
54	C₂×C₃	—	2	1	no	—
55	C₂×C₃	—	138	1	no	—
56	C₂×C₃	D₃×D₂₃	1	1	yes	characteristic
57	C₂×C₃	C₂×D₆₉	1	1	yes	characteristic
58	D₃	—	46	1	no	—
59	D₃	—	138	2	no	—
60	D₃	—	138	1	no	—
61	D₃	—	138	1	no	—
62	C₂²	—	69	1	no	—
63	C₂²	—	207	1	no	—
64	C₄	—	3	1	no	—
65	C₃	—	2	1	no	—
66	C₃	#10587771	1	1	yes	characteristic, complemented
67	C₃	D₂₇₆	1	1	yes	characteristic, complemented
68	C₂	—	138	1	no	—
69	C₂	—	414	1	no	—
70	C₂	D₃×D₆₉	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×D₆₉
commutator, Fitting, socle, cosocle	C₂×C₃²×C₂₃	C₂²
hypocenter	C₃²×C₂₃	D₄
LCS	G ▷ C₂×C₃²×C₂₃ ▷ C₃²×C₂₃
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂×C₃⋊D₆₉	C₂	semidirect
2. C₂×C₃×D₆₉	C₂	semidirect
3. C₃×Dic₃×C₂₃	C₂	semidirect
12. C₂×D₆₉	D₃	semidirect
15. Dic₃×C₂₃	D₃	semidirect
16. C₃²×C₂₃	D₄	semidirect
31. C₃×C₂₃	D₁₂	semidirect
32. C₃×C₂₃	#67	semidirect
38. C₃×Dic₃	D₂₃	semidirect
41. C₂₃	#626	semidirect
51. Dic₃	D₆₉	semidirect
52. C₃²	D₉₂	semidirect
66. C₃	#10587771	semidirect
67. C₃	D₂₇₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃⋊D₆₉	2
C₂×C₃×D₆₉	2
C₃×Dic₃×C₂₃	2
D₂₇₆	3
#10587771	3
#626	23