Subgroups — C₂×C₄×#14524137 #31771560

Order 17216.

162 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	C₄×#14524137	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×Dic₂₆₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#14524137	C₄	1	8	yes	direct product factor
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	C₂×C₄	1	8	yes	direct product factor
12	C₂×Dic₂₆₉	C₈	1	4	yes	complemented
13	C₂×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
14	C₂×Dic₂₆₉	C₂×C₄	1	4	yes	—
15	C₂×Dic₂₆₉	C₂×C₄	1	2	yes	—
16	C₂×Dic₂₆₉	C₂³	1	1	yes	characteristic
17	C₂³×C₂₆₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₂₆₉	C₈	1	4	yes	complemented
19	C₂×C₄×C₂₆₉	C₂×C₄	1	2	yes	—
20	C₂²×C₂₆₉	C₄²	1	2	yes	—
21	C₂²×C₂₆₉	C₂×C₈	1	2	yes	—
22	C₂²×C₂₆₉	C₂×C₈	1	2	yes	—
23	C₂²×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₂₆₉	C₂×C₈	1	4	yes	complemented
25	Dic₂₆₉	C₄²	1	2	yes	—
26	Dic₂₆₉	C₂×C₈	1	4	yes	complemented
27	Dic₂₆₉	C₂²×C₄	1	2	yes	—
28	C₂×C₂₆₉	C₄×C₈	1	4	yes	complemented
29	C₂×C₂₆₉	C₂×C₄²	1	1	yes	characteristic
30	C₂×C₂₆₉	C₂²×C₈	1	1	yes	characteristic
31	C₂×C₂₆₉	C₂²×C₈	1	1	yes	characteristic
32	C₂₆₉	C₂×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
33	C₂×C₄×C₈	—	269	1	no	maximal, Sylow
34	C₂²×C₈	—	269	2	no	—
35	C₂×C₄²	—	269	1	no	—
36	C₄×C₈	—	269	4	no	—
37	C₂²×C₄	—	269	1	no	—
38	C₂²×C₄	—	269	1	no	—
39	C₂²×C₄	#11761640	1	1	yes	characteristic
40	C₂×C₈	—	269	4	no	—
41	C₂×C₈	—	269	8	no	—
42	C₄²	—	269	2	no	—
43	C₄²	—	269	2	no	—
44	C₂³	C₂×#11761640	1	1	yes	characteristic
45	C₂×C₄	—	269	4	no	—
46	C₂×C₄	—	269	4	no	—
47	C₂×C₄	—	269	1	no	—
48	C₂×C₄	—	269	1	no	—
49	C₂×C₄	—	269	2	no	—
50	C₂×C₄	#14524137	1	4	yes	direct product factor
51	C₂×C₄	C₂×#11761640	1	2	yes	—
52	C₈	—	269	8	no	—
53	C₂²	C₄×#11761640	1	2	yes	—
54	C₂²	C₂×#14524137	1	2	yes	—
55	C₂²	D₂₆₉⋊C₈	1	2	yes	—
56	C₂²	C₂²×#11761640	1	1	yes	characteristic
57	C₄	—	269	2	no	—
58	C₄	—	269	2	no	—
59	C₄	—	269	4	no	—
60	C₄	C₂×#14524137	1	4	yes	direct product factor
61	C₂	C₄×#14524137	1	4	yes	direct product factor
62	C₂	C₂×C₄×#11761640	1	1	yes	characteristic
63	C₂	C₂²×#14524137	1	1	yes	characteristic
64	C₂	C₂×D₂₆₉⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₄	#11761640
commutator, hypocenter	C₂₆₉	C₂×C₄×C₈
Frattini	C₂²	C₂²×#11761640
Fitting	C₂²×C₄×C₂₆₉	C₄
socle, prime radical	C₂³×C₂₆₉	C₂×C₄
cosocle	C₂×Dic₂₆₉	C₂³
LCS	G ▷ C₂₆₉
UCS	C₁ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₄×#14524137	C₂	direct (N × H)
4. C₂×#14524137	C₄	direct (N × H)
11. #14524137	C₂×C₄	direct (N × H)
12. C₂×Dic₂₆₉	C₈	semidirect
18. C₂×C₄×C₂₆₉	C₈	semidirect
24. C₄×C₂₆₉	C₂×C₈	semidirect
26. Dic₂₆₉	C₂×C₈	semidirect
28. C₂×C₂₆₉	C₄×C₈	semidirect
32. C₂₆₉	C₂×C₄×C₈	semidirect
50. C₂×C₄	#14524137	direct (N × H)
60. C₄	C₂×#14524137	direct (N × H)
61. C₂	C₄×#14524137	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
C₄×#14524137	2
C₂×C₄×Dic₂₆₉	2
C₂×C₄×C₈	269

Subgroups — C2×C4×#14524137 #31771560

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#14524137 #31771560