Subgroups — C₂²×C₄⋊#11761640 #31771667

Order 17216.

418 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	—	2	8	no	—
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	—	2	24	no	—
13	C₂×#11761640	C₂³	1	8	yes	—
14	C₂²×D₂₆₉	C₂×C₄	1	3	yes	—
15	C₂²×D₂₆₉	D₄	1	4	yes	—
16	C₂²×D₂₆₉	Q₈	1	4	yes	—
17	C₂²×D₂₆₉	C₂³	1	3	yes	—
18	C₂×Dic₂₆₉	C₂×C₄	1	6	yes	complemented
19	C₄×D₂₆₉	C₂×C₄	1	12	yes	complemented
20	C₄×D₂₆₉	C₂³	1	4	yes	—
21	C₂³×C₂₆₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₂₆₉	C₂×C₄	1	6	yes	complemented
23	C₂²×C₂₆₉	C₄⋊C₄	1	4	yes	complemented
24	C₂²×C₂₆₉	C₂²×C₄	1	3	yes	—
25	C₂×D₂₆₉	C₄⋊C₄	1	12	yes	complemented
26	C₂×D₂₆₉	C₂²×C₄	1	3	yes	—
27	C₂×D₂₆₉	C₂×D₄	1	6	yes	—
28	C₂×D₂₆₉	C₂×Q₈	1	6	yes	—
29	C₂×D₂₆₉	C₂⁴	1	1	yes	characteristic
30	#11761640	—	2	8	no	—
31	C₄×C₂₆₉	C₂²×C₄	1	4	yes	complemented
32	Dic₂₆₉	C₂²×C₄	1	4	yes	complemented
33	C₂×C₂₆₉	C₂×C₄⋊C₄	1	6	yes	complemented
34	C₂×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
35	D₂₆₉	C₂×C₄⋊C₄	1	6	yes	complemented
36	D₂₆₉	C₂²×D₄	1	1	yes	characteristic
37	D₂₆₉	C₂²×Q₈	1	1	yes	characteristic
38	C₂₆₉	C₂²×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
39	C₂²×C₄⋊C₄	—	269	1	no	maximal, Sylow
40	C₂³×C₄	—	269	1	no	—
41	C₂³×C₄	—	269	2	no	—
42	C₂×C₄⋊C₄	—	269	12	no	—
43	C₂⁴	—	269	1	no	—
44	C₂²×C₄	—	269	1	no	—
45	C₂²×C₄	—	269	6	no	—
46	C₂²×C₄	—	269	6	no	—
47	C₂²×C₄	—	269	12	no	—
48	C₂²×C₄	—	538	8	no	—
49	C₂²×C₄	#11761640	1	1	yes	characteristic, complemented
50	C₄⋊C₄	—	269	16	no	—
51	C₂³	—	269	4	no	—
52	C₂³	—	269	4	no	—
53	C₂³	—	269	3	no	—
54	C₂³	—	269	3	no	—
55	C₂³	C₂×#11761640	1	1	yes	characteristic
56	C₂×C₄	—	269	6	no	—
57	C₂×C₄	—	269	4	no	—
58	C₂×C₄	—	269	12	no	—
59	C₂×C₄	—	269	8	no	—
60	C₂×C₄	—	538	24	no	—
61	C₂×C₄	C₂×#11761640	1	6	yes	complemented
62	C₂²	—	269	6	no	—
63	C₂²	—	269	12	no	—
64	C₂²	—	269	6	no	—
65	C₂²	—	269	1	no	—
66	C₂²	—	269	3	no	—
67	C₂²	C₄⋊#11761640	1	4	yes	direct product factor
68	C₂²	C₂²×#11761640	1	3	yes	—
69	C₄	—	269	4	no	—
70	C₄	—	538	8	no	—
71	C₄	C₂²×#11761640	1	4	yes	complemented
72	C₂	—	269	1	no	—
73	C₂	—	269	6	no	—
74	C₂	—	269	1	no	—
75	C₂	C₂×C₄⋊#11761640	1	6	yes	direct product factor
76	C₂	C₂³×#11761640	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11761640
commutator	C₂×C₂₆₉	C₂³×C₄
Frattini	C₂	C₂³×#11761640
Fitting	C₂²×C₄×C₂₆₉	C₄
socle	C₂³×C₂₆₉	C₂×C₄
prime radical	C₂³×D₂₆₉	C₂²
cosocle	C₂×D₂₆₉	C₂⁴
hypercenter	C₂²×C₄	#11761640
hypocenter	C₂₆₉	C₂²×C₄⋊C₄
LCS	G ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ 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)
6. C₄⋊#11761640	C₂²	direct (N × H)
8. C₂²×Dic₂₆₉	C₄	semidirect
9. C₂×C₄×D₂₆₉	C₄	semidirect
11. C₂²×C₄×C₂₆₉	C₄	semidirect
18. C₂×Dic₂₆₉	C₂×C₄	semidirect
19. C₄×D₂₆₉	C₂×C₄	semidirect
22. C₂×C₄×C₂₆₉	C₂×C₄	semidirect
23. C₂²×C₂₆₉	C₄⋊C₄	semidirect
25. C₂×D₂₆₉	C₄⋊C₄	semidirect
31. C₄×C₂₆₉	C₂²×C₄	semidirect
32. Dic₂₆₉	C₂²×C₄	semidirect
33. C₂×C₂₆₉	C₂×C₄⋊C₄	semidirect
35. D₂₆₉	C₂×C₄⋊C₄	semidirect
38. C₂₆₉	C₂²×C₄⋊C₄	semidirect
49. C₂²×C₄	#11761640	semidirect
61. C₂×C₄	C₂×#11761640	semidirect
67. C₂²	C₄⋊#11761640	direct (N × H)
71. C₄	C₂²×#11761640	semidirect
75. 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×C₄⋊C₄	269

Subgroups — C22×C4⋊#11761640 #31771667

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄⋊#11761640 #31771667