Subgroups — C₂×#22575520 #31771607

Order 17216.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14524137	C₂	1	2	yes	maximal
2	#22575520	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×Dic₂₆₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#14524137	—	2	4	no	—
5	C₂×#14524137	C₂²	1	4	yes	—
6	C₂²×Dic₂₆₉	C₄	1	1	yes	characteristic
7	C₂²×Dic₂₆₉	C₂²	1	1	yes	characteristic
8	C₄×Dic₂₆₉	C₄	1	2	yes	—
9	C₄×Dic₂₆₉	C₂²	1	2	yes	—
10	C₂²×C₄×C₂₆₉	C₄	1	1	yes	characteristic
11	#14524137	—	2	4	no	—
12	C₂×Dic₂₆₉	C₈	1	4	yes	complemented
13	C₂×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
14	C₂×Dic₂₆₉	C₂×C₄	1	2	yes	—
15	C₂×Dic₂₆₉	D₄	1	2	yes	—
16	C₂×Dic₂₆₉	Q₈	1	2	yes	—
17	C₂×Dic₂₆₉	C₂³	1	1	yes	characteristic
18	C₂³×C₂₆₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₂₆₉	—	2	2	no	—
20	C₂×C₄×C₂₆₉	C₂×C₄	1	2	yes	—
21	C₂²×C₂₆₉	C₄⋊C₄	1	2	yes	—
22	C₂²×C₂₆₉	C₂×C₈	1	2	yes	—
23	C₂²×C₂₆₉	M₁₆	1	2	yes	—
24	C₂²×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
25	C₄×C₂₆₉	—	2	2	no	—
26	Dic₂₆₉	C₄⋊C₄	1	2	yes	—
27	Dic₂₆₉	C₂×C₈	1	4	yes	complemented
28	Dic₂₆₉	C₂×D₄	1	1	yes	characteristic
29	Dic₂₆₉	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₂₆₉	C₄⋊C₈	1	4	yes	complemented
31	C₂×C₂₆₉	C₂×C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₂₆₉	C₂²×C₈	1	1	yes	characteristic
33	C₂×C₂₆₉	C₂×M₁₆	1	1	yes	characteristic
34	C₂₆₉	C₂×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
35	C₂×C₄⋊C₈	—	269	1	no	maximal, Sylow
36	C₂²×C₈	—	269	2	no	—
37	C₂×C₄²	—	269	1	no	—
38	C₄⋊C₈	—	269	4	no	—
39	C₂²×C₄	—	269	1	no	—
40	C₂²×C₄	—	269	1	no	—
41	C₂²×C₄	#11761640	1	1	yes	characteristic
42	C₂×C₈	—	269	4	no	—
43	C₂×C₈	—	538	4	no	—
44	C₄²	—	269	2	no	—
45	C₄²	—	269	2	no	—
46	C₂³	C₂×#11761640	1	1	yes	characteristic
47	C₂×C₄	—	2	2	no	—
48	C₂×C₄	—	269	2	no	—
49	C₂×C₄	—	269	2	no	—
50	C₂×C₄	—	269	4	no	—
51	C₂×C₄	—	269	1	no	—
52	C₂×C₄	—	269	1	no	—
53	C₂×C₄	—	269	2	no	—
54	C₂×C₄	C₂×#11761640	1	2	yes	—
55	C₈	—	538	4	no	—
56	C₂²	C₄⋊#11761640	1	2	yes	—
57	C₂²	D₂₆₉⋊C₈	1	2	yes	—
58	C₂²	#16864411	1	2	yes	—
59	C₂²	C₂²×#11761640	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₄	—	269	1	no	—
62	C₄	—	269	2	no	—
63	C₄	—	269	1	no	—
64	C₄	—	269	4	no	—
65	C₂	#22575520	1	4	yes	direct product factor
66	C₂	C₂×C₄⋊#11761640	1	1	yes	characteristic
67	C₂	C₂×D₂₆₉⋊C₈	1	1	yes	characteristic
68	C₂	C₂×#16864411	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11761640
commutator	C₂×C₂₆₉	C₂²×C₈
Frattini	C₂²	C₂²×#11761640
Fitting	C₂²×C₄×C₂₆₉	C₄
socle, prime radical	C₂³×C₂₆₉	C₂×C₄
cosocle	C₂×Dic₂₆₉	C₂³
hypercenter	C₂²×C₄	#11761640
hypocenter	C₂₆₉	C₂×C₄⋊C₈
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. #22575520	C₂	direct (N × H)
12. C₂×Dic₂₆₉	C₈	semidirect
27. Dic₂₆₉	C₂×C₈	semidirect
30. C₂×C₂₆₉	C₄⋊C₈	semidirect
34. C₂₆₉	C₂×C₄⋊C₈	semidirect
65. C₂	#22575520	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×#14524137	2
#22575520	2
C₂×C₄×Dic₂₆₉	2
C₂×C₄⋊C₈	269

Subgroups — C2×#22575520 #31771607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22575520 #31771607