Subgroups — #31771571

Order 17216.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₆₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#22575532	C₂	1	1	yes	maximal, characteristic
3	#22575507	C₂	1	2	yes	maximal, complemented
4	C₈×#11761640	C₂	1	2	yes	maximal, complemented
5	M₁₆×D₂₆₉	C₂	1	1	yes	maximal, characteristic
6	D₂₆₉⋊C₈	C₄	1	2	yes	complemented
7	D₂₆₉⋊C₈	C₂²	1	1	yes	characteristic
8	D₂₆₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×#14524137	C₄	1	2	yes	complemented
10	C₄⋊#11761640	C₄	1	2	yes	—
11	C₂²⋊#11761640	C₄	1	2	yes	—
12	C₄×#11761640	C₂²	1	2	yes	—
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	C₂×C₄	1	4	yes	—
19	#14524137	C₂×C₄	1	2	yes	—
20	#14524137	C₂×C₄	1	2	yes	—
21	C₂²×D₂₆₉	C₂×C₄	1	1	yes	characteristic
22	C₂×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
23	C₄×D₂₆₉	C₂×C₄	1	1	yes	characteristic
24	C₄×D₂₆₉	C₂×C₄	1	1	yes	characteristic
25	C₄×D₂₆₉	C₂×C₄	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	C₄²	1	1	yes	characteristic
31	C₂×D₂₆₉	—	2	2	no	—
32	C₂×D₂₆₉	C₄²	1	1	yes	characteristic
33	C₂×D₂₆₉	C₂²×C₄	1	1	yes	characteristic
34	#11761640	—	2	4	no	—
35	C₄×C₂₆₉	C₄²	1	1	yes	characteristic
36	C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
37	Dic₂₆₉	C₄²	1	1	yes	characteristic
38	Dic₂₆₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₂₆₉	—	2	1	no	—
40	C₂×C₂₆₉	C₂×C₄²	1	1	yes	characteristic
41	D₂₆₉	—	2	1	no	—
42	D₂₆₉	#131	1	2	yes	—
43	C₂₆₉	#405	1	1	yes	characteristic, complemented, Sylow
44	#405	—	269	1	no	maximal, Sylow
45	C₂×M₁₆	—	269	1	no	—
46	C₂²×C₈	—	269	1	no	—
47	#117	—	269	1	no	—
48	#97	—	269	2	no	—
49	C₄×C₈	—	269	2	no	—
50	C₂²×C₄	—	269	1	no	—
51	M₁₆	—	269	1	no	—
52	M₁₆	—	269	2	no	—
53	M₁₆	#11761640	1	1	yes	characteristic, complemented
54	C₂×C₈	—	269	1	no	—
55	C₂×C₈	—	269	1	no	—
56	C₂×C₈	—	269	2	no	—
57	C₂×C₈	—	269	2	no	—
58	C₂×C₈	—	269	2	no	—
59	C₄⋊C₄	—	269	2	no	—
60	C₂²⋊C₄	—	269	2	no	—
61	C₄²	—	269	2	no	—
62	C₂³	—	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	—	269	1	no	—
68	C₂×C₄	—	269	4	no	—
69	C₂×C₄	C₂×#11761640	1	1	yes	characteristic
70	C₈	—	269	2	no	—
71	C₈	—	269	2	no	—
72	C₈	—	269	2	no	—
73	C₈	C₂×#11761640	1	2	yes	—
74	C₂²	—	269	1	no	—
75	C₂²	—	269	1	no	—
76	C₂²	—	538	2	no	—
77	C₂²	C₄×#11761640	1	1	yes	characteristic
78	C₄	—	269	1	no	—
79	C₄	—	269	1	no	—
80	C₄	—	538	4	no	—
81	C₄	C₄×#11761640	1	1	yes	characteristic
82	C₄	C₂²×#11761640	1	1	yes	characteristic
83	C₂	—	2	1	no	—
84	C₂	—	269	2	no	—
85	C₂	—	538	1	no	—
86	C₂	C₂×C₄×#11761640	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×#11761640
commutator, socle	C₂×C₂₆₉	C₂×C₄²
Fitting	M₁₆×C₂₆₉	C₄
prime radical	C₂²×D₂₆₉	C₂×C₄
cosocle	C₄×D₂₆₉	C₂³
hypercenter	M₁₆	#11761640
hypocenter	C₂₆₉	#405
LCS	G ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
3. #22575507	C₂	semidirect
4. C₈×#11761640	C₂	semidirect
6. D₂₆₉⋊C₈	C₄	semidirect
9. C₂×#14524137	C₄	semidirect
14. #16864386	C₄	semidirect
15. #16864385	C₄	semidirect
17. M₁₆×C₂₆₉	C₄	semidirect
43. C₂₆₉	#405	semidirect
53. M₁₆	#11761640	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₆₉⋊C₈	2
#22575532	2
#22575507	2
C₈×#11761640	2
M₁₆×D₂₆₉	2
#405	269