Subgroups — C₂×#22575510 #31771574

Order 17216.

202 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14524137	C₂	1	2	yes	maximal, complemented
2	#22575510	C₂	1	4	yes	maximal, direct product factor
3	C₂²×C₄×D₂₆₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#14524137	—	2	4	no	—
5	C₂×#14524137	C₂²	1	4	yes	complemented
6	C₂³×D₂₆₉	C₄	1	1	yes	characteristic
7	C₂²×Dic₂₆₉	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₂₆₉	—	2	4	no	—
9	C₂×C₄×D₂₆₉	C₄	1	2	yes	—
10	C₂×C₄×D₂₆₉	C₂²	1	2	yes	—
11	C₂²×C₄×C₂₆₉	C₄	1	1	yes	characteristic
12	#14524137	—	2	4	no	—
13	C₂²×D₂₆₉	—	2	2	no	—
14	C₂²×D₂₆₉	—	2	2	no	—
15	C₂²×D₂₆₉	C₈	1	4	yes	complemented
16	C₂²×D₂₆₉	C₂×C₄	1	2	yes	—
17	C₂×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₂₆₉	D₄	1	4	yes	—
19	C₂×Dic₂₆₉	C₂³	1	1	yes	characteristic
20	C₄×D₂₆₉	—	2	4	no	—
21	C₄×D₂₆₉	—	2	4	no	—
22	C₂³×C₂₆₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₂₆₉	—	2	2	no	—
24	C₂×C₄×C₂₆₉	C₂×C₄	1	2	yes	—
25	C₂²×C₂₆₉	C₂²⋊C₄	1	2	yes	—
26	C₂²×C₂₆₉	C₂×C₈	1	2	yes	—
27	C₂²×C₂₆₉	M₁₆	1	2	yes	—
28	C₂²×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
29	C₂×D₂₆₉	—	2	8	no	—
30	C₂×D₂₆₉	—	2	2	no	—
31	C₂×D₂₆₉	—	2	2	no	—
32	C₂×D₂₆₉	C₂×C₈	1	4	yes	complemented
33	C₄×C₂₆₉	—	2	2	no	—
34	Dic₂₆₉	C₂²⋊C₄	1	2	yes	—
35	Dic₂₆₉	C₂×D₄	1	2	yes	—
36	C₂×C₂₆₉	C₂²⋊C₈	1	4	yes	complemented
37	C₂×C₂₆₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₂₆₉	C₂²×C₈	1	1	yes	characteristic
39	C₂×C₂₆₉	C₂×M₁₆	1	1	yes	characteristic
40	D₂₆₉	—	2	4	no	—
41	C₂₆₉	C₂×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
42	C₂×C₂²⋊C₈	—	269	1	no	maximal, Sylow
43	C₂³×C₄	—	269	1	no	—
44	C₂²×C₈	—	269	2	no	—
45	C₂²⋊C₈	—	269	4	no	—
46	C₂⁴	—	269	1	no	—
47	C₂²×C₄	—	269	1	no	—
48	C₂²×C₄	—	269	2	no	—
49	C₂²×C₄	—	269	2	no	—
50	C₂²×C₄	—	538	4	no	—
51	C₂²×C₄	#11761640	1	1	yes	characteristic
52	C₂×C₈	—	269	4	no	—
53	C₂×C₈	—	538	4	no	—
54	C₂³	—	269	4	no	—
55	C₂³	—	269	2	no	—
56	C₂³	—	538	2	no	—
57	C₂³	—	538	2	no	—
58	C₂³	C₂×#11761640	1	1	yes	characteristic
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	269	4	no	—
61	C₂×C₄	—	269	1	no	—
62	C₂×C₄	—	269	1	no	—
63	C₂×C₄	—	538	4	no	—
64	C₂×C₄	—	538	4	no	—
65	C₂×C₄	C₂×#11761640	1	2	yes	—
66	C₈	—	538	4	no	—
67	C₂²	—	269	4	no	—
68	C₂²	—	538	8	no	—
69	C₂²	—	538	2	no	—
70	C₂²	—	538	2	no	—
71	C₂²	C₂²⋊#11761640	1	2	yes	—
72	C₂²	D₂₆₉⋊C₈	1	2	yes	—
73	C₂²	#16864412	1	2	yes	—
74	C₂²	C₂²×#11761640	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	269	2	no	—
77	C₄	—	269	2	no	—
78	C₂	—	538	4	no	—
79	C₂	#22575510	1	4	yes	direct product factor
80	C₂	C₂×C₂²⋊#11761640	1	1	yes	characteristic
81	C₂	C₂×D₂₆₉⋊C₈	1	1	yes	characteristic
82	C₂	C₂×#16864412	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11761640
commutator	C₂×C₂₆₉	C₂²×C₈
Frattini	C₂²	C₂²×#11761640
Fitting	C₂²×C₄×C₂₆₉	C₄
socle	C₂³×C₂₆₉	C₂×C₄
prime radical	C₂³×D₂₆₉	C₄
cosocle	C₂×Dic₂₆₉	C₂³
hypercenter	C₂²×C₄	#11761640
hypocenter	C₂₆₉	C₂×C₂²⋊C₈
LCS	G ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#14524137	C₂	semidirect
2. #22575510	C₂	direct (N × H)
5. C₂×#14524137	C₂²	semidirect
15. C₂²×D₂₆₉	C₈	semidirect
32. C₂×D₂₆₉	C₂×C₈	semidirect
36. C₂×C₂₆₉	C₂²⋊C₈	semidirect
41. C₂₆₉	C₂×C₂²⋊C₈	semidirect
79. C₂	#22575510	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14524137	2
#22575510	2
C₂²×C₄×D₂₆₉	2
C₂×C₂²⋊C₈	269

Subgroups — C2×#22575510 #31771574

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22575510 #31771574