Subgroups — #31771620

Order 17216.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16864412	C₂	1	1	yes	maximal, characteristic
2	D₂₆₉⋊M₁₆	C₂	1	1	yes	maximal, characteristic
3	#22575524	C₂	1	2	yes	maximal, complemented
4	#22575523	C₂	1	2	yes	maximal, complemented
5	M₁₆×D₂₆₉	C₂	1	1	yes	maximal, characteristic
6	#16864412	—	2	1	no	—
7	#16864412	C₂²	1	1	yes	characteristic
8	#16864412	C₂²	1	2	yes	—
9	#16864412	C₂²	1	1	yes	characteristic
10	#16864411	—	2	1	no	—
11	D₂₆₉⋊C₈	—	2	1	no	—
12	C₂×#14524137	—	2	1	no	—
13	C₂×C₄×D₂₆₉	C₂²	1	1	yes	characteristic
14	#16864386	C₄	1	1	yes	characteristic
15	#16864385	C₄	1	2	yes	—
16	C₈×D₂₆₉	C₂²	1	2	yes	—
17	M₁₆×C₂₆₉	C₄	1	1	yes	characteristic
18	#14524137	—	2	2	no	—
19	#14524137	—	2	1	no	—
20	#14524137	—	2	1	no	—
21	C₂²×D₂₆₉	Q₈	1	1	yes	characteristic
22	C₂×Dic₂₆₉	D₄	1	1	yes	characteristic
23	C₄×D₂₆₉	C₂×C₄	1	1	yes	characteristic
24	C₄×D₂₆₉	D₄	1	1	yes	characteristic
25	C₄×D₂₆₉	Q₈	1	1	yes	characteristic
26	C₄×D₂₆₉	C₂³	1	1	yes	characteristic
27	#14524130	C₂×C₄	1	2	yes	—
28	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
29	C₈×C₂₆₉	C₂×C₄	1	2	yes	—
30	C₂²×C₂₆₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂×D₂₆₉	—	4	1	no	—
32	C₂×D₂₆₉	C₄⋊C₄	1	1	yes	characteristic
33	C₂×D₂₆₉	C₂×Q₈	1	1	yes	characteristic
34	C₄×C₂₆₉	C₄⋊C₄	1	1	yes	characteristic
35	C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
36	Dic₂₆₉	C₄⋊C₄	1	1	yes	characteristic
37	Dic₂₆₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₂₆₉	—	2	1	no	—
39	C₂×C₂₆₉	C₂×C₄⋊C₄	1	1	yes	characteristic
40	D₂₆₉	—	2	1	no	—
41	D₂₆₉	—	2	1	no	—
42	C₂₆₉	#430	1	1	yes	characteristic, complemented, Sylow
43	#430	—	269	1	no	maximal, Sylow
44	C₂×M₁₆	—	269	1	no	—
45	C₂×M₁₆	—	269	1	no	—
46	C₂×M₁₆	—	269	1	no	—
47	#108	—	269	2	no	—
48	#108	—	269	2	no	—
49	C₂²×C₄	—	269	1	no	—
50	M₁₆	—	269	1	no	—
51	M₁₆	—	269	2	no	—
52	M₁₆	—	269	1	no	—
53	M₁₆	—	269	2	no	—
54	M₁₆	—	269	1	no	—
55	M₁₆	—	538	1	no	—
56	M₁₆	—	538	1	no	—
57	M₁₆	#11761640	1	1	yes	characteristic
58	C₂×C₈	—	269	2	no	—
59	C₂×C₈	—	538	1	no	—
60	C₂×C₈	—	538	1	no	—
61	C₂³	—	269	1	no	—
62	C₂×C₄	—	269	1	no	—
63	C₂×C₄	—	269	1	no	—
64	C₂×C₄	—	269	1	no	—
65	C₂×C₄	—	269	1	no	—
66	C₂×C₄	—	269	1	no	—
67	C₂×C₄	C₂×#11761640	1	1	yes	characteristic
68	C₈	—	269	2	no	—
69	C₈	—	538	2	no	—
70	C₈	—	538	1	no	—
71	C₈	—	538	1	no	—
72	C₈	C₂×#11761640	1	2	yes	—
73	C₂²	—	269	1	no	—
74	C₂²	—	269	1	no	—
75	C₂²	—	1076	1	no	—
76	C₂²	C₄⋊#11761640	1	1	yes	characteristic
77	C₄	—	269	1	no	—
78	C₄	—	269	1	no	—
79	C₄	C₄⋊#11761640	1	1	yes	characteristic
80	C₄	C₂²×#11761640	1	1	yes	characteristic
81	C₂	—	2	1	no	—
82	C₂	—	538	1	no	—
83	C₂	—	538	1	no	—
84	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	Q₈
cosocle	C₄×D₂₆₉	C₂³
hypercenter	M₁₆	#11761640
hypocenter	C₂₆₉	#430
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. #22575524	C₂	semidirect
4. #22575523	C₂	semidirect
42. C₂₆₉	#430	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16864412	2
D₂₆₉⋊M₁₆	2
#22575524	2
#22575523	2
M₁₆×D₂₆₉	2
#430	269