Subgroups — #31771649

Order 17216.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575541	C₂	1	2	yes	maximal, complemented
2	#22575524	C₂	1	1	yes	maximal, characteristic, complemented
3	#22575518	C₂	1	2	yes	maximal, complemented
4	C₈×#11761640	C₂	1	1	yes	maximal, characteristic, complemented
5	#22575476	C₂	1	1	yes	maximal, characteristic
6	#16864412	—	2	2	no	—
7	#16864412	C₂²	1	2	yes	—
8	D₂₆₉⋊C₈	—	2	2	no	—
9	D₂₆₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₄×#11761640	C₂²	1	1	yes	characteristic
11	#16864403	C₂²	1	2	yes	—
12	#16864389	C₄	1	2	yes	complemented
13	D₂₁₅₂	C₄	1	1	yes	characteristic, complemented
14	C₈×D₂₆₉	C₂²	1	1	yes	characteristic
15	Q₁₆×C₂₆₉	C₄	1	1	yes	characteristic, complemented
16	C₂×#11761640	—	2	1	no	—
17	#14524137	—	2	2	no	—
18	#14524137	D₄	1	1	yes	characteristic
19	#14524137	D₄	1	1	yes	characteristic
20	D₁₀₇₆	—	2	2	no	—
21	D₁₀₇₆	C₂×C₄	1	2	yes	—
22	C₄×D₂₆₉	—	2	2	no	—
23	C₄×D₂₆₉	C₂³	1	1	yes	characteristic
24	#14524130	C₂×C₄	1	1	yes	characteristic
25	Q₈×C₂₆₉	C₂×C₄	1	2	yes	—
26	C₈×C₂₆₉	C₂×C₄	1	1	yes	characteristic
27	C₂×D₂₆₉	—	2	2	no	—
28	C₂×D₂₆₉	#41	1	1	yes	characteristic
29	#11761640	—	2	2	no	—
30	C₄×C₂₆₉	—	2	2	no	—
31	C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
32	Dic₂₆₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₂₆₉	C₄×D₄	1	1	yes	characteristic
34	D₂₆₉	—	2	1	no	—
35	D₂₆₉	—	4	2	no	—
36	C₂₆₉	#443	1	1	yes	characteristic, complemented, Sylow
37	#443	—	269	1	no	maximal, Sylow
38	#135	—	269	1	no	—
39	#131	—	269	2	no	—
40	#108	—	269	1	no	—
41	#104	—	269	2	no	—
42	C₄×C₈	—	269	1	no	—
43	#41	—	269	2	no	—
44	Q₁₆	#11761640	1	1	yes	characteristic, complemented
45	SD₁₆	—	269	2	no	—
46	D₈	—	269	1	no	—
47	M₁₆	—	269	2	no	—
48	M₁₆	—	538	2	no	—
49	C₂×C₈	—	269	1	no	—
50	C₂×C₈	—	269	1	no	—
51	C₂×C₈	—	538	2	no	—
52	C₄²	—	269	1	no	—
53	Q₈	C₂×#11761640	1	2	yes	—
54	D₄	—	269	2	no	—
55	D₄	—	538	2	no	—
56	C₂×C₄	—	269	1	no	—
57	C₂×C₄	—	538	2	no	—
58	C₂×C₄	—	538	1	no	—
59	C₈	—	269	1	no	—
60	C₈	—	269	1	no	—
61	C₈	—	269	1	no	—
62	C₈	—	538	2	no	—
63	C₈	C₂×#11761640	1	1	yes	characteristic
64	C₂²	—	269	1	no	—
65	C₂²	—	538	2	no	—
66	C₄	—	2	2	no	—
67	C₄	—	269	1	no	—
68	C₄	—	538	2	no	—
69	C₄	C₂²×#11761640	1	1	yes	characteristic
70	C₂	—	538	1	no	—
71	C₂	—	1076	2	no	—
72	C₂	D₄×#11761640	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×#11761640
commutator	C₄×C₂₆₉	C₂²×C₄
Frattini	C₄	C₂²×#11761640
Fitting	Q₁₆×C₂₆₉	C₄
socle	C₂×C₂₆₉	C₄×D₄
prime radical	#22575476	C₂
cosocle	C₄×D₂₆₉	C₂³
hypercenter	Q₁₆	#11761640
hypocenter	C₂₆₉	#443
LCS	G ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
1. #22575541	C₂	semidirect
2. #22575524	C₂	semidirect
3. #22575518	C₂	semidirect
4. C₈×#11761640	C₂	semidirect
12. #16864389	C₄	semidirect
13. D₂₁₅₂	C₄	semidirect
15. Q₁₆×C₂₆₉	C₄	semidirect
36. C₂₆₉	#443	semidirect
44. Q₁₆	#11761640	semidirect

Maximal subgroups

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

subgroup	index
#22575541	2
#22575524	2
#22575518	2
C₈×#11761640	2
#22575476	2
#443	269