Subgroups — C₄×#16864412 #31771567

Order 17216.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16864412	C₂	1	2	yes	maximal
2	#22575508	C₂	1	2	yes	maximal, complemented
3	C₄×#14524137	C₂	1	2	yes	maximal, complemented
4	C₄²×D₂₆₉	C₂	1	1	yes	maximal, characteristic
5	#16864412	C₄	1	8	yes	direct product factor
6	C₂×#14524137	C₂²	1	4	yes	—
7	C₂×C₄×D₂₆₉	C₄	1	1	yes	characteristic
8	C₂×C₄×D₂₆₉	C₄	1	1	yes	characteristic
9	C₂×C₄×D₂₆₉	C₂²	1	1	yes	characteristic
10	C₄×Dic₂₆₉	C₄	1	1	yes	characteristic
11	C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
12	C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
13	C₄²×C₂₆₉	C₄	1	1	yes	characteristic
14	#14524137	C₂×C₄	1	8	yes	complemented
15	C₂²×D₂₆₉	C₂×C₄	1	1	yes	characteristic
16	C₂×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
17	C₂×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₂₆₉	C₂³	1	1	yes	characteristic
19	C₄×D₂₆₉	—	2	2	no	—
20	C₄×D₂₆₉	—	2	2	no	—
21	C₄×D₂₆₉	C₂×C₄	1	4	yes	—
22	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
26	C₂×D₂₆₉	—	2	1	no	—
27	C₂×D₂₆₉	—	2	1	no	—
28	C₂×D₂₆₉	C₄²	1	2	yes	—
29	C₄×C₂₆₉	—	2	1	no	—
30	C₄×C₂₆₉	C₄²	1	2	yes	—
31	C₄×C₂₆₉	M₁₆	1	2	yes	complemented
32	Dic₂₆₉	—	2	1	no	—
33	Dic₂₆₉	M₁₆	1	2	yes	complemented
34	Dic₂₆₉	C₂²×C₄	1	2	yes	—
35	C₂×C₂₆₉	C₂×C₄²	1	1	yes	characteristic
36	C₂×C₂₆₉	C₂×M₁₆	1	1	yes	characteristic
37	C₂×C₂₆₉	C₂×M₁₆	1	1	yes	characteristic
38	D₂₆₉	—	2	2	no	—
39	C₂₆₉	C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
40	C₄×M₁₆	—	269	1	no	maximal, Sylow
41	C₂×M₁₆	—	269	2	no	—
42	C₂×C₄²	—	269	1	no	—
43	#97	—	269	2	no	—
44	C₄×C₈	—	269	2	no	—
45	C₂²×C₄	—	269	1	no	—
46	C₂²×C₄	—	269	1	no	—
47	C₂²×C₄	—	269	1	no	—
48	M₁₆	—	269	8	no	—
49	C₂×C₈	—	269	4	no	—
50	C₄²	—	269	1	no	—
51	C₄²	—	269	1	no	—
52	C₄²	—	269	1	no	—
53	C₄²	#11761640	1	1	yes	characteristic
54	C₂³	—	269	1	no	—
55	C₂×C₄	—	269	1	no	—
56	C₂×C₄	—	269	1	no	—
57	C₂×C₄	—	269	4	no	—
58	C₂×C₄	—	269	1	no	—
59	C₂×C₄	—	538	2	no	—
60	C₂×C₄	—	538	2	no	—
61	C₂×C₄	C₂×#11761640	1	1	yes	characteristic
62	C₂×C₄	C₂×#11761640	1	1	yes	characteristic
63	C₂×C₄	C₂×#11761640	1	1	yes	characteristic
64	C₈	—	269	8	no	—
65	C₂²	—	269	2	no	—
66	C₂²	—	538	1	no	—
67	C₂²	—	538	1	no	—
68	C₂²	C₂²×#11761640	1	1	yes	characteristic
69	C₄	—	2	1	no	—
70	C₄	—	269	2	no	—
71	C₄	—	269	2	no	—
72	C₄	—	538	1	no	—
73	C₄	C₄×#11761640	1	2	yes	—
74	C₄	#16864412	1	2	yes	direct product factor
75	C₂	—	538	2	no	—
76	C₂	C₂×C₄×#11761640	1	1	yes	characteristic
77	C₂	D₂₆₉⋊M₁₆	1	1	yes	characteristic
78	C₂	C₂×#16864412	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11761640
commutator	C₂×C₂₆₉	C₂×C₄²
Frattini	C₂²	C₂²×#11761640
Fitting	C₄²×C₂₆₉	C₄
socle	C₂²×C₂₆₉	C₂²×C₄
prime radical	C₂²×D₂₆₉	C₂×C₄
cosocle	C₂×Dic₂₆₉	C₂³
hypercenter	C₄²	#11761640
hypocenter	C₂₆₉	C₄×M₁₆
LCS	G ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. #22575508	C₂	semidirect
3. C₄×#14524137	C₂	semidirect
5. #16864412	C₄	direct (N × H)
14. #14524137	C₂×C₄	semidirect
31. C₄×C₂₆₉	M₁₆	semidirect
33. Dic₂₆₉	M₁₆	semidirect
39. C₂₆₉	C₄×M₁₆	semidirect
74. C₄	#16864412	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16864412	2
#22575508	2
C₄×#14524137	2
C₄²×D₂₆₉	2
C₄×M₁₆	269

Subgroups — C4×#16864412 #31771567

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16864412 #31771567