Subgroups — C₂×C₈×#11761640 #31771559

Order 17216.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₆₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11761640	C₂	1	1	yes	maximal, characteristic
3	C₈×#11761640	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₈×D₂₆₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11761640	C₄	1	2	yes	—
6	D₂₆₉⋊C₈	C₄	1	2	yes	complemented
7	D₂₆₉⋊C₈	C₂²	1	2	yes	—
8	C₂×#14524137	C₄	1	2	yes	complemented
9	C₄×#11761640	C₄	1	2	yes	—
10	C₄×#11761640	C₂²	1	2	yes	—
11	C₂×C₄×D₂₆₉	C₂²	1	1	yes	characteristic
12	C₂×#14524130	C₄	1	1	yes	characteristic, complemented
13	C₈×D₂₆₉	C₄	1	2	yes	complemented
14	C₈×D₂₆₉	C₂²	1	2	yes	—
15	C₂×C₈×C₂₆₉	C₄	1	1	yes	characteristic, complemented
16	C₂×#11761640	C₈	1	8	yes	direct product factor
17	C₂×#11761640	C₂×C₄	1	4	yes	—
18	#14524137	C₂×C₄	1	4	yes	complemented
19	C₂²×D₂₆₉	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
21	C₄×D₂₆₉	C₂×C₄	1	1	yes	characteristic
22	C₄×D₂₆₉	C₂×C₄	1	1	yes	characteristic
23	C₄×D₂₆₉	C₂×C₄	1	1	yes	characteristic
24	C₄×D₂₆₉	C₂³	1	1	yes	characteristic
25	#14524130	C₂×C₄	1	2	yes	complemented
26	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
27	C₈×C₂₆₉	C₂×C₄	1	2	yes	complemented
28	C₂²×C₂₆₉	C₄²	1	1	yes	characteristic
29	C₂×D₂₆₉	C₄²	1	1	yes	characteristic
30	C₂×D₂₆₉	C₂×C₈	1	4	yes	—
31	C₂×D₂₆₉	C₂²×C₄	1	1	yes	characteristic
32	#11761640	C₂×C₈	1	8	yes	direct product factor
33	C₄×C₂₆₉	C₄²	1	1	yes	characteristic
34	C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
35	Dic₂₆₉	C₄²	1	1	yes	characteristic
36	Dic₂₆₉	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₂₆₉	C₄×C₈	1	2	yes	complemented
38	C₂×C₂₆₉	C₂×C₄²	1	1	yes	characteristic
39	D₂₆₉	C₄×C₈	1	2	yes	complemented
40	D₂₆₉	C₂²×C₈	1	2	yes	—
41	C₂₆₉	C₂×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
42	C₂×C₄×C₈	—	269	1	no	maximal, Sylow
43	C₂²×C₈	—	269	1	no	—
44	C₂²×C₈	—	269	1	no	—
45	C₂×C₄²	—	269	1	no	—
46	C₄×C₈	—	269	4	no	—
47	C₂²×C₄	—	269	2	no	—
48	C₂²×C₄	—	269	1	no	—
49	C₂×C₈	—	269	2	no	—
50	C₂×C₈	—	269	2	no	—
51	C₂×C₈	—	269	2	no	—
52	C₂×C₈	—	269	1	no	—
53	C₂×C₈	—	269	2	no	—
54	C₂×C₈	—	269	2	no	—
55	C₂×C₈	#11761640	1	1	yes	characteristic, direct product factor
56	C₄²	—	269	2	no	—
57	C₄²	—	269	2	no	—
58	C₂³	—	269	1	no	—
59	C₂×C₄	—	269	4	no	—
60	C₂×C₄	—	269	8	no	—
61	C₂×C₄	—	269	1	no	—
62	C₂×C₄	—	269	1	no	—
63	C₂×C₄	—	269	1	no	—
64	C₂×C₄	—	269	1	no	—
65	C₂×C₄	—	269	1	no	—
66	C₂×C₄	C₂×#11761640	1	1	yes	characteristic
67	C₈	—	269	4	no	—
68	C₈	—	269	2	no	—
69	C₈	C₂×#11761640	1	2	yes	direct product factor
70	C₂²	—	269	4	no	—
71	C₂²	—	269	1	no	—
72	C₂²	—	269	1	no	—
73	C₂²	C₄×#11761640	1	1	yes	characteristic
74	C₄	—	269	8	no	—
75	C₄	—	269	1	no	—
76	C₄	—	269	1	no	—
77	C₄	C₄×#11761640	1	1	yes	characteristic
78	C₄	C₂²×#11761640	1	1	yes	characteristic
79	C₂	—	269	2	no	—
80	C₂	—	269	2	no	—
81	C₂	C₈×#11761640	1	2	yes	direct product factor
82	C₂	C₂×C₄×#11761640	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₈	#11761640
commutator, hypocenter	C₂₆₉	C₂×C₄×C₈
Frattini	C₄	C₂²×#11761640
Fitting	C₂×C₈×C₂₆₉	C₄
socle	C₂²×C₂₆₉	C₄²
prime radical	C₂²×D₂₆₉	C₂×C₄
cosocle	C₄×D₂₆₉	C₂³
LCS	G ▷ C₂₆₉
UCS	C₁ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
3. C₈×#11761640	C₂	direct (N × H)
6. D₂₆₉⋊C₈	C₄	semidirect
8. C₂×#14524137	C₄	semidirect
12. C₂×#14524130	C₄	semidirect
13. C₈×D₂₆₉	C₄	semidirect
15. C₂×C₈×C₂₆₉	C₄	semidirect
16. C₂×#11761640	C₈	direct (N × H)
18. #14524137	C₂×C₄	semidirect
25. #14524130	C₂×C₄	semidirect
27. C₈×C₂₆₉	C₂×C₄	semidirect
32. #11761640	C₂×C₈	direct (N × H)
37. C₂×C₂₆₉	C₄×C₈	semidirect
39. D₂₆₉	C₄×C₈	semidirect
41. C₂₆₉	C₂×C₄×C₈	semidirect
55. C₂×C₈	#11761640	direct (N × H)
69. C₈	C₂×#11761640	direct (N × H)
81. C₂	C₈×#11761640	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×D₂₆₉⋊C₈	2
C₂×C₄×#11761640	2
C₈×#11761640	2
C₂×C₈×D₂₆₉	2
C₂×C₄×C₈	269

Subgroups — C2×C8×#11761640 #31771559

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×#11761640 #31771559