Subgroups — C₂²×C₄×#11761640 #31771665

Order 17216.

498 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11761640	C₂	1	2	yes	maximal
2	C₂×C₄×#11761640	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₄×D₂₆₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11761640	C₄	1	16	yes	direct product factor
5	C₂²×#11761640	C₂²	1	12	yes	—
6	C₄×#11761640	C₂²	1	16	yes	direct product factor
7	C₂³×D₂₆₉	C₂²	1	1	yes	characteristic
8	C₂²×Dic₂₆₉	C₄	1	1	yes	characteristic, complemented
9	C₂×C₄×D₂₆₉	C₄	1	6	yes	complemented
10	C₂×C₄×D₂₆₉	C₂²	1	6	yes	—
11	C₂²×C₄×C₂₆₉	C₄	1	1	yes	characteristic, complemented
12	C₂×#11761640	C₂×C₄	1	48	yes	direct product factor
13	C₂×#11761640	C₂³	1	8	yes	—
14	C₂²×D₂₆₉	C₂×C₄	1	3	yes	—
15	C₂²×D₂₆₉	C₂×C₄	1	8	yes	—
16	C₂²×D₂₆₉	C₂³	1	3	yes	—
17	C₂×Dic₂₆₉	C₂×C₄	1	6	yes	complemented
18	C₄×D₂₆₉	C₂×C₄	1	12	yes	complemented
19	C₄×D₂₆₉	C₂³	1	4	yes	—
20	C₂³×C₂₆₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₂₆₉	C₂×C₄	1	6	yes	complemented
22	C₂²×C₂₆₉	C₄²	1	4	yes	complemented
23	C₂²×C₂₆₉	C₂²×C₄	1	3	yes	—
24	C₂×D₂₆₉	C₄²	1	12	yes	complemented
25	C₂×D₂₆₉	C₂²×C₄	1	3	yes	—
26	C₂×D₂₆₉	C₂²×C₄	1	12	yes	—
27	C₂×D₂₆₉	C₂⁴	1	1	yes	characteristic
28	#11761640	C₂²×C₄	1	16	yes	direct product factor
29	C₄×C₂₆₉	C₂²×C₄	1	4	yes	complemented
30	Dic₂₆₉	C₂²×C₄	1	4	yes	complemented
31	C₂×C₂₆₉	C₂×C₄²	1	6	yes	complemented
32	C₂×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
33	D₂₆₉	C₂×C₄²	1	6	yes	complemented
34	D₂₆₉	C₂³×C₄	1	2	yes	—
35	C₂₆₉	C₂²×C₄²	1	1	yes	characteristic, complemented, Sylow
36	C₂²×C₄²	—	269	1	no	maximal, Sylow
37	C₂³×C₄	—	269	2	no	—
38	C₂³×C₄	—	269	1	no	—
39	C₂×C₄²	—	269	12	no	—
40	C₂⁴	—	269	1	no	—
41	C₂²×C₄	—	269	12	no	—
42	C₂²×C₄	—	269	16	no	—
43	C₂²×C₄	—	269	1	no	—
44	C₂²×C₄	—	269	6	no	—
45	C₂²×C₄	—	269	6	no	—
46	C₂²×C₄	#11761640	1	1	yes	characteristic, direct product factor
47	C₄²	—	269	16	no	—
48	C₂³	—	269	8	no	—
49	C₂³	—	269	3	no	—
50	C₂³	—	269	3	no	—
51	C₂³	C₂×#11761640	1	1	yes	characteristic
52	C₂×C₄	—	269	8	no	—
53	C₂×C₄	—	269	48	no	—
54	C₂×C₄	—	269	6	no	—
55	C₂×C₄	—	269	4	no	—
56	C₂×C₄	—	269	12	no	—
57	C₂×C₄	C₂×#11761640	1	6	yes	direct product factor
58	C₂²	—	269	12	no	—
59	C₂²	—	269	12	no	—
60	C₂²	—	269	1	no	—
61	C₂²	—	269	3	no	—
62	C₂²	C₄×#11761640	1	4	yes	direct product factor
63	C₂²	C₂²×#11761640	1	3	yes	—
64	C₄	—	269	16	no	—
65	C₄	—	269	4	no	—
66	C₄	C₂²×#11761640	1	4	yes	direct product factor
67	C₂	—	269	2	no	—
68	C₂	—	269	6	no	—
69	C₂	C₂×C₄×#11761640	1	6	yes	direct product factor
70	C₂	C₂³×#11761640	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₄	#11761640
commutator, hypocenter	C₂₆₉	C₂²×C₄²
Frattini	C₂	C₂³×#11761640
Fitting	C₂²×C₄×C₂₆₉	C₄
socle	C₂³×C₂₆₉	C₂×C₄
prime radical	C₂³×D₂₆₉	C₂²
cosocle	C₂×D₂₆₉	C₂⁴
LCS	G ▷ C₂₆₉
UCS	C₁ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×#11761640	C₂	direct (N × H)
4. C₂²×#11761640	C₄	direct (N × H)
6. C₄×#11761640	C₂²	direct (N × H)
8. C₂²×Dic₂₆₉	C₄	semidirect
9. C₂×C₄×D₂₆₉	C₄	semidirect
11. C₂²×C₄×C₂₆₉	C₄	semidirect
12. C₂×#11761640	C₂×C₄	direct (N × H)
17. C₂×Dic₂₆₉	C₂×C₄	semidirect
18. C₄×D₂₆₉	C₂×C₄	semidirect
21. C₂×C₄×C₂₆₉	C₂×C₄	semidirect
22. C₂²×C₂₆₉	C₄²	semidirect
24. C₂×D₂₆₉	C₄²	semidirect
28. #11761640	C₂²×C₄	direct (N × H)
29. C₄×C₂₆₉	C₂²×C₄	semidirect
30. Dic₂₆₉	C₂²×C₄	semidirect
31. C₂×C₂₆₉	C₂×C₄²	semidirect
33. D₂₆₉	C₂×C₄²	semidirect
35. C₂₆₉	C₂²×C₄²	semidirect
46. C₂²×C₄	#11761640	direct (N × H)
57. C₂×C₄	C₂×#11761640	direct (N × H)
62. C₂²	C₄×#11761640	direct (N × H)
66. C₄	C₂²×#11761640	direct (N × H)
69. C₂	C₂×C₄×#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
C₂³×#11761640	2
C₂×C₄×#11761640	2
C₂²×C₄×D₂₆₉	2
C₂²×C₄²	269

Subgroups — C22×C4×#11761640 #31771665

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×#11761640 #31771665