Subgroups — C₂³×#16864385 #46825586

Order 34432.

1720 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×D₂₆₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×#16864385	C₂	1	28	yes	maximal, direct product factor
3	C₂³×#14524130	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×C₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×D₂₆₉	C₄	1	1	yes	characteristic
6	C₂³×Dic₂₆₉	C₄	1	1	yes	characteristic
7	C₂²×C₄×D₂₆₉	C₄	1	14	yes	—
8	C₂²×C₄×D₂₆₉	C₂²	1	14	yes	—
9	C₂×#16864385	C₂²	1	112	yes	direct product factor
10	C₂²×#14524130	C₂²	1	14	yes	complemented
11	C₂³×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₂₆₉	C₂²	1	14	yes	complemented
13	C₂³×D₂₆₉	—	2	8	no	—
14	C₂³×D₂₆₉	C₂×C₄	1	14	yes	—
15	C₂²×Dic₂₆₉	C₂×C₄	1	14	yes	—
16	C₂×C₄×D₂₆₉	C₂×C₄	1	84	yes	—
17	C₂×C₄×D₂₆₉	C₂³	1	28	yes	—
18	#16864385	C₂³	1	64	yes	direct product factor
19	C₂×#14524130	C₂³	1	28	yes	complemented
20	C₂⁴×C₂₆₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₂₆₉	C₂×C₄	1	7	yes	—
22	C₂²×C₄×C₂₆₉	C₂³	1	7	yes	—
23	C₂×C₈×C₂₆₉	C₂³	1	28	yes	complemented
24	C₂²×D₂₆₉	—	2	56	no	—
25	C₂²×D₂₆₉	C₂²×C₄	1	28	yes	—
26	C₂×Dic₂₆₉	C₂²×C₄	1	28	yes	—
27	C₄×D₂₆₉	C₂²×C₄	1	56	yes	—
28	C₄×D₂₆₉	C₂⁴	1	8	yes	—
29	#14524130	C₂⁴	1	8	yes	complemented
30	C₂³×C₂₆₉	M₁₆	1	8	yes	complemented
31	C₂³×C₂₆₉	C₂²×C₄	1	7	yes	—
32	C₂×C₄×C₂₆₉	C₂²×C₄	1	21	yes	—
33	C₂×C₄×C₂₆₉	C₂⁴	1	7	yes	—
34	C₈×C₂₆₉	C₂⁴	1	8	yes	complemented
35	C₂²×C₂₆₉	C₂×M₁₆	1	28	yes	complemented
36	C₂²×C₂₆₉	C₂³×C₄	1	7	yes	—
37	C₂×D₂₆₉	—	2	56	no	—
38	C₂×D₂₆₉	C₂³×C₄	1	8	yes	—
39	C₄×C₂₆₉	C₂³×C₄	1	7	yes	—
40	C₄×C₂₆₉	C₂⁵	1	1	yes	characteristic
41	Dic₂₆₉	C₂³×C₄	1	8	yes	—
42	C₂×C₂₆₉	C₂²×M₁₆	1	14	yes	complemented
43	C₂×C₂₆₉	C₂⁴×C₄	1	1	yes	characteristic
44	D₂₆₉	—	2	8	no	—
45	C₂₆₉	C₂³×M₁₆	1	1	yes	characteristic, complemented, Sylow
46	C₂³×M₁₆	—	269	1	no	maximal, Sylow
47	C₂⁴×C₄	—	269	1	no	—
48	C₂²×M₁₆	—	269	28	no	—
49	C₂³×C₈	—	269	1	no	—
50	C₂³×C₈	D₂₆₉	1	1	yes	characteristic, complemented
51	C₂⁵	—	269	1	no	—
52	C₂³×C₄	—	269	1	no	—
53	C₂³×C₄	—	269	14	no	—
54	C₂³×C₄	—	269	14	no	—
55	C₂³×C₄	C₂×D₂₆₉	1	1	yes	characteristic
56	C₂×M₁₆	—	269	112	no	—
57	C₂²×C₈	—	269	14	no	—
58	C₂²×C₈	C₂×D₂₆₉	1	14	yes	complemented
59	C₂⁴	—	269	14	no	—
60	C₂⁴	—	538	8	no	—
61	C₂⁴	C₄×D₂₆₉	1	1	yes	characteristic
62	C₂²×C₄	—	269	14	no	—
63	C₂²×C₄	—	269	84	no	—
64	C₂²×C₄	—	269	28	no	—
65	C₂²×C₄	C₄×D₂₆₉	1	7	yes	—
66	C₂²×C₄	C₂²×D₂₆₉	1	7	yes	—
67	M₁₆	—	269	64	no	—
68	C₂×C₈	—	269	28	no	—
69	C₂×C₈	C₂²×D₂₆₉	1	28	yes	complemented
70	C₂³	—	269	28	no	—
71	C₂³	—	538	56	no	—
72	C₂³	#16864385	1	8	yes	direct product factor
73	C₂³	C₂×C₄×D₂₆₉	1	7	yes	—
74	C₂×C₄	—	269	28	no	—
75	C₂×C₄	—	269	56	no	—
76	C₂×C₄	—	269	8	no	—
77	C₂×C₄	C₂×C₄×D₂₆₉	1	21	yes	—
78	C₂×C₄	C₂³×D₂₆₉	1	7	yes	—
79	C₈	—	269	8	no	—
80	C₈	C₂³×D₂₆₉	1	8	yes	complemented
81	C₂²	—	269	8	no	—
82	C₂²	—	538	56	no	—
83	C₂²	C₂×#16864385	1	28	yes	direct product factor
84	C₂²	C₂²×C₄×D₂₆₉	1	7	yes	—
85	C₄	—	269	8	no	—
86	C₄	C₂²×C₄×D₂₆₉	1	7	yes	—
87	C₄	C₂⁴×D₂₆₉	1	1	yes	characteristic
88	C₂	—	538	8	no	—
89	C₂	C₂²×#16864385	1	14	yes	direct product factor
90	C₂	C₂³×C₄×D₂₆₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂×D₂₆₉
commutator	C₂×C₂₆₉	C₂⁴×C₄
Frattini	C₄	C₂⁴×D₂₆₉
Fitting	C₂³×C₈×C₂₆₉	C₂
socle	C₂⁴×C₂₆₉	C₂×C₄
prime radical	C₂⁴×D₂₆₉	C₄
cosocle	C₄×C₂₆₉	C₂⁵
hypercenter	C₂³×C₈	D₂₆₉
hypocenter	C₂₆₉	C₂³×M₁₆
LCS	G ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×#16864385	C₂	direct (N × H)
3. C₂³×#14524130	C₂	semidirect
4. C₂³×C₈×C₂₆₉	C₂	semidirect
9. C₂×#16864385	C₂²	direct (N × H)
10. C₂²×#14524130	C₂²	semidirect
12. C₂²×C₈×C₂₆₉	C₂²	semidirect
18. #16864385	C₂³	direct (N × H)
19. C₂×#14524130	C₂³	semidirect
23. C₂×C₈×C₂₆₉	C₂³	semidirect
29. #14524130	C₂⁴	semidirect
30. C₂³×C₂₆₉	M₁₆	semidirect
34. C₈×C₂₆₉	C₂⁴	semidirect
35. C₂²×C₂₆₉	C₂×M₁₆	semidirect
42. C₂×C₂₆₉	C₂²×M₁₆	semidirect
45. C₂₆₉	C₂³×M₁₆	semidirect
50. C₂³×C₈	D₂₆₉	semidirect
58. C₂²×C₈	C₂×D₂₆₉	semidirect
69. C₂×C₈	C₂²×D₂₆₉	semidirect
72. C₂³	#16864385	direct (N × H)
80. C₈	C₂³×D₂₆₉	semidirect
83. C₂²	C₂×#16864385	direct (N × H)
89. C₂	C₂²×#16864385	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄×D₂₆₉	2
C₂²×#16864385	2
C₂³×#14524130	2
C₂³×C₈×C₂₆₉	2
C₂³×M₁₆	269

Subgroups — C23×#16864385 #46825586

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#16864385 #46825586