Subgroups — C₂×#22575418 #31771069

Order 17216.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₅₃₈	C₂	1	1	yes	maximal, characteristic
2	#22575418	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16864381	C₂	1	2	yes	maximal
4	C₂×C₄×Dic₂₆₉	C₂	1	2	yes	maximal
5	C₂×C₄×Dic₂₆₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊C₄×C₂₆₉	C₂	1	1	yes	maximal, characteristic
7	C₂×Dic₅₃₈	C₄	1	8	yes	complemented
8	C₂×Dic₅₃₈	C₂²	1	4	yes	—
9	C₂²×Dic₂₆₉	C₂²	1	1	yes	characteristic
10	C₂²×Dic₂₆₉	C₂²	1	2	yes	—
11	C₂²×Dic₂₆₉	C₂²	1	1	yes	characteristic
12	#16864381	C₂²	1	8	yes	—
13	C₄×Dic₂₆₉	C₂²	1	8	yes	—
14	C₄×Dic₂₆₉	C₂²	1	4	yes	—
15	C₂²×C₄×C₂₆₉	C₂²	1	2	yes	—
16	C₂²×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₂₆₉	C₂²	1	4	yes	—
18	Dic₅₃₈	C₂×C₄	1	16	yes	complemented
19	C₂×Dic₂₆₉	—	2	2	no	—
20	C₂×Dic₂₆₉	C₂×C₄	1	8	yes	—
21	C₂×Dic₂₆₉	Q₈	1	4	yes	complemented
22	C₂×Dic₂₆₉	C₂³	1	2	yes	—
23	C₂×Dic₂₆₉	C₂³	1	4	yes	—
24	C₂×Dic₂₆₉	C₂³	1	2	yes	—
25	C₂³×C₂₆₉	C₂³	1	1	yes	characteristic
26	C₂×C₄×C₂₆₉	—	2	4	no	—
27	C₂×C₄×C₂₆₉	C₂×C₄	1	4	yes	—
28	C₂×C₄×C₂₆₉	C₂³	1	4	yes	—
29	C₂×C₄×C₂₆₉	C₂³	1	2	yes	—
30	C₂²×C₂₆₉	C₂²×C₄	1	2	yes	—
31	C₂²×C₂₆₉	C₂×Q₈	1	2	yes	—
32	C₂²×C₂₆₉	#41	1	2	yes	—
33	C₂²×C₂₆₉	C₂⁴	1	1	yes	characteristic
34	C₄×C₂₆₉	—	2	4	no	—
35	C₄×C₂₆₉	C₂²×C₄	1	4	yes	—
36	Dic₂₆₉	—	2	2	no	—
37	Dic₂₆₉	C₂²×C₄	1	8	yes	—
38	Dic₂₆₉	C₂×Q₈	1	4	yes	complemented
39	C₂×C₂₆₉	C₄×Q₈	1	4	yes	complemented
40	C₂×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
41	C₂×C₂₆₉	C₂²×Q₈	1	1	yes	characteristic
42	C₂×C₂₆₉	C₂×#41	1	1	yes	characteristic
43	C₂₆₉	C₂×C₄×Q₈	1	1	yes	characteristic, complemented, Sylow
44	C₂×C₄×Q₈	—	269	1	no	maximal, Sylow
45	C₂²×Q₈	—	269	1	no	—
46	C₄×Q₈	—	269	8	no	—
47	C₂×C₄⋊C₄	—	269	2	no	—
48	C₂×C₄⋊C₄	D₂₆₉	1	1	yes	characteristic
49	C₂×C₄²	—	269	2	no	—
50	C₂×C₄²	—	269	1	no	—
51	C₂×Q₈	—	269	8	no	—
52	C₂×Q₈	—	269	4	no	—
53	C₂²×C₄	—	269	1	no	—
54	C₂²×C₄	—	269	2	no	—
55	C₂²×C₄	—	269	1	no	—
56	C₂²×C₄	C₂×D₂₆₉	1	2	yes	—
57	C₂²×C₄	C₂×D₂₆₉	1	1	yes	characteristic
58	C₄⋊C₄	—	269	8	no	—
59	C₄⋊C₄	C₂×D₂₆₉	1	4	yes	—
60	C₄²	—	269	8	no	—
61	C₄²	—	269	4	no	—
62	C₂³	C₂²×D₂₆₉	1	1	yes	characteristic
63	Q₈	—	269	16	no	—
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	—	269	8	no	—
66	C₂×C₄	—	269	4	no	—
67	C₂×C₄	—	269	2	no	—
68	C₂×C₄	—	269	4	no	—
69	C₂×C₄	—	269	2	no	—
70	C₂×C₄	—	538	2	no	—
71	C₂×C₄	C₄×D₂₆₉	1	4	yes	—
72	C₂×C₄	C₂²×D₂₆₉	1	4	yes	—
73	C₂×C₄	C₂²×D₂₆₉	1	2	yes	—
74	C₂²	C₂×C₄×D₂₆₉	1	2	yes	—
75	C₂²	Q₈×D₂₆₉	1	2	yes	—
76	C₂²	#16864401	1	2	yes	—
77	C₂²	C₂³×D₂₆₉	1	1	yes	characteristic
78	C₄	—	2	4	no	—
79	C₄	—	269	8	no	—
80	C₄	—	269	4	no	—
81	C₄	—	538	2	no	—
82	C₄	C₂×C₄×D₂₆₉	1	4	yes	—
83	C₂	#22575418	1	4	yes	direct product factor
84	C₂	C₂²×C₄×D₂₆₉	1	1	yes	characteristic
85	C₂	C₂×Q₈×D₂₆₉	1	1	yes	characteristic
86	C₂	C₂×#16864401	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₂₆₉
commutator	C₂×C₂₆₉	C₂³×C₄
Frattini	C₂²	C₂³×D₂₆₉
Fitting	C₂×C₄⋊C₄×C₂₆₉	C₂
socle, prime radical	C₂³×C₂₆₉	C₂³
cosocle	C₂²×C₂₆₉	C₂⁴
hypercenter	C₂×C₄⋊C₄	D₂₆₉
hypocenter	C₂₆₉	C₂×C₄×Q₈
LCS	G ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #22575418	C₂	direct (N × H)
7. C₂×Dic₅₃₈	C₄	semidirect
18. Dic₅₃₈	C₂×C₄	semidirect
21. C₂×Dic₂₆₉	Q₈	semidirect
38. Dic₂₆₉	C₂×Q₈	semidirect
39. C₂×C₂₆₉	C₄×Q₈	semidirect
43. C₂₆₉	C₂×C₄×Q₈	semidirect
83. C₂	#22575418	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Dic₅₃₈	2
#22575418	2
C₂×#16864381	2
C₂×C₄×Dic₂₆₉	2
C₂×C₄⋊C₄×C₂₆₉	2
C₂×C₄×Q₈	269

Subgroups — C2×#22575418 #31771069

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22575418 #31771069