Subgroups — #31771616

Order 17216.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575532	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11761640	C₂	1	1	yes	maximal, characteristic
3	#22575522	C₂	1	2	yes	maximal, complemented
4	#22575521	C₂	1	2	yes	maximal, complemented
5	M₁₆×D₂₆₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761640	—	2	1	no	—
7	C₄⋊#11761640	—	2	1	no	—
8	C₄⋊#11761640	C₂²	1	1	yes	characteristic
9	C₄⋊#11761640	C₂²	1	2	yes	—
10	C₄⋊#11761640	C₂²	1	1	yes	characteristic
11	C₂²⋊#11761640	—	2	1	no	—
12	C₄×#11761640	—	2	1	no	—
13	C₂×C₄×D₂₆₉	C₂²	1	1	yes	characteristic
14	#16864386	C₄	1	1	yes	characteristic, complemented
15	#16864385	C₄	1	2	yes	complemented
16	C₈×D₂₆₉	C₂²	1	2	yes	—
17	M₁₆×C₂₆₉	C₄	1	1	yes	characteristic, complemented
18	C₂×#11761640	—	2	1	no	—
19	C₂×#11761640	—	2	1	no	—
20	C₂×#11761640	—	2	2	no	—
21	C₂×#11761640	—	4	2	no	—
22	C₂²×D₂₆₉	D₄	1	1	yes	characteristic
23	C₂×Dic₂₆₉	Q₈	1	1	yes	characteristic
24	C₄×D₂₆₉	C₂×C₄	1	1	yes	characteristic
25	C₄×D₂₆₉	D₄	1	1	yes	characteristic
26	C₄×D₂₆₉	Q₈	1	1	yes	characteristic
27	C₄×D₂₆₉	C₂³	1	1	yes	characteristic
28	#14524130	C₂×C₄	1	2	yes	complemented
29	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
30	C₈×C₂₆₉	C₂×C₄	1	2	yes	complemented
31	C₂²×C₂₆₉	C₄⋊C₄	1	1	yes	characteristic
32	C₂×D₂₆₉	—	2	1	no	—
33	C₂×D₂₆₉	—	2	1	no	—
34	C₂×D₂₆₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂×D₂₆₉	C₂×D₄	1	1	yes	characteristic
36	#11761640	—	4	1	no	—
37	#11761640	—	4	2	no	—
38	#11761640	—	4	1	no	—
39	C₄×C₂₆₉	C₄⋊C₄	1	1	yes	characteristic
40	C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
41	Dic₂₆₉	C₄⋊C₄	1	1	yes	characteristic
42	Dic₂₆₉	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₂₆₉	—	2	1	no	—
44	C₂×C₂₆₉	C₂×C₄⋊C₄	1	1	yes	characteristic
45	D₂₆₉	—	2	1	no	—
46	D₂₆₉	AGL(1,ℤ/8)	1	1	yes	characteristic
47	D₂₆₉	#137	1	1	yes	characteristic
48	C₂₆₉	#428	1	1	yes	characteristic, complemented, Sylow
49	#428	—	269	1	no	maximal, Sylow
50	C₂×M₁₆	—	269	1	no	—
51	#117	—	269	1	no	—
52	C₂×C₄⋊C₄	—	269	1	no	—
53	#107	—	269	2	no	—
54	#106	—	269	2	no	—
55	C₂²×C₄	—	269	1	no	—
56	C₂²×C₄	—	538	1	no	—
57	M₁₆	—	269	1	no	—
58	M₁₆	—	269	2	no	—
59	M₁₆	#11761640	1	1	yes	characteristic, complemented
60	C₂×C₈	—	269	2	no	—
61	C₄⋊C₄	—	269	1	no	—
62	C₄⋊C₄	—	269	2	no	—
63	C₄⋊C₄	—	269	1	no	—
64	C₄⋊C₄	—	538	1	no	—
65	C₂²⋊C₄	—	538	1	no	—
66	C₄²	—	538	1	no	—
67	C₂³	—	269	1	no	—
68	C₂×C₄	—	269	1	no	—
69	C₂×C₄	—	269	1	no	—
70	C₂×C₄	—	269	1	no	—
71	C₂×C₄	—	269	1	no	—
72	C₂×C₄	—	269	1	no	—
73	C₂×C₄	—	538	1	no	—
74	C₂×C₄	—	538	1	no	—
75	C₂×C₄	—	538	2	no	—
76	C₂×C₄	—	1076	2	no	—
77	C₂×C₄	C₂×#11761640	1	1	yes	characteristic
78	C₈	—	269	2	no	—
79	C₈	C₂×#11761640	1	2	yes	complemented
80	C₂²	—	269	1	no	—
81	C₂²	—	269	1	no	—
82	C₂²	—	538	1	no	—
83	C₂²	—	538	1	no	—
84	C₂²	C₄⋊#11761640	1	1	yes	characteristic
85	C₄	—	269	1	no	—
86	C₄	—	269	1	no	—
87	C₄	—	1076	2	no	—
88	C₄	—	1076	1	no	—
89	C₄	—	1076	1	no	—
90	C₄	C₄⋊#11761640	1	1	yes	characteristic
91	C₄	C₂²×#11761640	1	1	yes	characteristic
92	C₂	—	2	1	no	—
93	C₂	—	269	1	no	—
94	C₂	—	269	1	no	—
95	C₂	—	538	1	no	—
96	C₂	C₂×C₄⋊#11761640	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₄⋊#11761640
commutator	C₄×C₂₆₉	C₂²×C₄
Frattini	C₄	C₂²×#11761640
Fitting	M₁₆×C₂₆₉	C₄
socle	C₂×C₂₆₉	C₂×C₄⋊C₄
prime radical	C₂²×D₂₆₉	D₄
cosocle	C₄×D₂₆₉	C₂³
hypercenter	M₁₆	#11761640
hypocenter	C₂₆₉	#428
LCS	G ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
3. #22575522	C₂	semidirect
4. #22575521	C₂	semidirect
14. #16864386	C₄	semidirect
15. #16864385	C₄	semidirect
17. M₁₆×C₂₆₉	C₄	semidirect
28. #14524130	C₂×C₄	semidirect
30. C₈×C₂₆₉	C₂×C₄	semidirect
48. C₂₆₉	#428	semidirect
59. M₁₆	#11761640	semidirect
79. C₈	C₂×#11761640	semidirect

Maximal subgroups

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

subgroup	index
#22575532	2
C₂×C₄⋊#11761640	2
#22575522	2
#22575521	2
M₁₆×D₂₆₉	2
#428	269