Subgroups — #31771623

Order 17216.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575520	C₂	1	2	yes	maximal, complemented
2	#22575510	C₂	1	2	yes	maximal
3	C₄×#14524137	C₂	1	2	yes	maximal, complemented
4	C₄²×D₂₆₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14524137	C₂²	1	4	yes	—
6	C₂×C₄×D₂₆₉	C₄	1	1	yes	characteristic
7	C₂×C₄×D₂₆₉	C₄	1	1	yes	characteristic
8	C₂×C₄×D₂₆₉	C₂²	1	1	yes	characteristic
9	C₄×Dic₂₆₉	C₄	1	1	yes	characteristic
10	C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
11	C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
12	C₄²×C₂₆₉	C₄	1	1	yes	characteristic
13	#14524137	—	2	4	no	—
14	C₂²×D₂₆₉	C₂×C₄	1	1	yes	characteristic
15	C₂×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
16	C₂×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
17	C₂×Dic₂₆₉	C₂³	1	1	yes	characteristic
18	C₄×D₂₆₉	—	2	2	no	—
19	C₄×D₂₆₉	—	2	2	no	—
20	C₄×D₂₆₉	C₈	1	4	yes	complemented
21	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
25	C₂×D₂₆₉	—	2	1	no	—
26	C₂×D₂₆₉	—	2	1	no	—
27	C₂×D₂₆₉	C₂×C₈	1	2	yes	—
28	C₄×C₂₆₉	—	2	1	no	—
29	C₄×C₂₆₉	—	2	1	no	—
30	C₄×C₂₆₉	C₂×C₈	1	2	yes	—
31	Dic₂₆₉	C₂×C₈	1	2	yes	—
32	Dic₂₆₉	M₁₆	1	2	yes	—
33	Dic₂₆₉	#41	1	2	yes	—
34	C₂×C₂₆₉	#117	1	1	yes	characteristic
35	C₂×C₂₆₉	C₂²×C₈	1	1	yes	characteristic
36	C₂×C₂₆₉	C₂×M₁₆	1	1	yes	characteristic
37	D₂₆₉	—	2	2	no	—
38	C₂₆₉	#431	1	1	yes	characteristic, complemented, Sylow
39	#431	—	269	1	no	maximal, Sylow
40	C₂×C₄²	—	269	1	no	—
41	C₄⋊C₈	—	269	2	no	—
42	C₂²⋊C₈	—	269	2	no	—
43	C₄×C₈	—	269	2	no	—
44	C₂²×C₄	—	269	1	no	—
45	C₂²×C₄	—	269	1	no	—
46	C₂²×C₄	—	269	1	no	—
47	C₂×C₈	—	269	4	no	—
48	C₄²	—	269	1	no	—
49	C₄²	—	269	1	no	—
50	C₄²	—	269	1	no	—
51	C₄²	#11761640	1	1	yes	characteristic
52	C₂³	—	269	1	no	—
53	C₂×C₄	—	269	1	no	—
54	C₂×C₄	—	269	1	no	—
55	C₂×C₄	—	269	1	no	—
56	C₂×C₄	—	269	4	no	—
57	C₂×C₄	—	538	2	no	—
58	C₂×C₄	—	538	2	no	—
59	C₂×C₄	C₂×#11761640	1	1	yes	characteristic
60	C₂×C₄	C₂×#11761640	1	1	yes	characteristic
61	C₂×C₄	C₂×#11761640	1	1	yes	characteristic
62	C₈	—	538	4	no	—
63	C₂²	—	269	2	no	—
64	C₂²	—	538	1	no	—
65	C₂²	—	538	1	no	—
66	C₂²	C₂²×#11761640	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	2	1	no	—
69	C₄	—	269	2	no	—
70	C₄	—	269	2	no	—
71	C₄	—	269	2	no	—
72	C₄	D₂₆₉⋊C₈	1	2	yes	—
73	C₂	—	538	2	no	—
74	C₂	#22575532	1	1	yes	characteristic
75	C₂	C₂×D₂₆₉⋊C₈	1	1	yes	characteristic
76	C₂	D₂₆₉⋊M₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11761640
commutator	C₂×C₂₆₉	C₂²×C₈
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	#431
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
1. #22575520	C₂	semidirect
3. C₄×#14524137	C₂	semidirect
20. C₄×D₂₆₉	C₈	semidirect
38. C₂₆₉	#431	semidirect

Maximal subgroups

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

subgroup	index
#22575520	2
#22575510	2
C₄×#14524137	2
C₄²×D₂₆₉	2
#431	269