Subgroups — C₂×#22575523 #31771617

Order 17216.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16864412	C₂	1	2	yes	maximal
2	#22575523	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₈×D₂₆₉	C₂	1	1	yes	maximal, characteristic
4	#16864412	—	2	2	no	—
5	#16864412	C₂²	1	4	yes	—
6	C₂×#14524137	—	2	2	no	—
7	C₂×C₄×D₂₆₉	C₂²	1	1	yes	characteristic
8	C₂×#14524130	C₄	1	1	yes	characteristic
9	C₈×D₂₆₉	C₄	1	2	yes	—
10	C₈×D₂₆₉	C₂²	1	2	yes	—
11	C₂×C₈×C₂₆₉	C₄	1	1	yes	characteristic
12	#14524137	—	2	4	no	—
13	C₂²×D₂₆₉	Q₈	1	1	yes	characteristic
14	C₂×Dic₂₆₉	D₄	1	1	yes	characteristic
15	C₄×D₂₆₉	C₂×C₄	1	1	yes	characteristic
16	C₄×D₂₆₉	D₄	1	1	yes	characteristic
17	C₄×D₂₆₉	Q₈	1	1	yes	characteristic
18	C₄×D₂₆₉	C₂³	1	1	yes	characteristic
19	#14524130	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
21	C₈×C₂₆₉	C₂×C₄	1	2	yes	—
22	C₂²×C₂₆₉	C₄⋊C₄	1	1	yes	characteristic
23	C₂×D₂₆₉	—	2	2	no	—
24	C₂×D₂₆₉	C₄⋊C₄	1	1	yes	characteristic
25	C₂×D₂₆₉	C₂×Q₈	1	1	yes	characteristic
26	C₄×C₂₆₉	C₄⋊C₄	1	1	yes	characteristic
27	C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
28	Dic₂₆₉	C₄⋊C₄	1	1	yes	characteristic
29	Dic₂₆₉	C₂×D₄	1	1	yes	characteristic
30	C₂×C₂₆₉	#108	1	2	yes	complemented
31	C₂×C₂₆₉	C₂×C₄⋊C₄	1	1	yes	characteristic
32	D₂₆₉	—	2	1	no	—
33	D₂₆₉	—	2	1	no	—
34	C₂₆₉	C₂×#108	1	1	yes	characteristic, complemented, Sylow
35	C₂×#108	—	269	1	no	maximal, Sylow
36	C₂×M₁₆	—	269	2	no	—
37	C₂²×C₈	—	269	1	no	—
38	#108	—	269	4	no	—
39	C₂²×C₄	—	269	1	no	—
40	M₁₆	—	269	4	no	—
41	M₁₆	—	538	2	no	—
42	C₂×C₈	—	269	1	no	—
43	C₂×C₈	—	269	2	no	—
44	C₂×C₈	—	269	2	no	—
45	C₂×C₈	—	538	2	no	—
46	C₂×C₈	#11761640	1	1	yes	characteristic
47	C₂³	—	269	1	no	—
48	C₂×C₄	—	269	1	no	—
49	C₂×C₄	—	269	1	no	—
50	C₂×C₄	—	269	1	no	—
51	C₂×C₄	—	269	1	no	—
52	C₂×C₄	—	269	1	no	—
53	C₂×C₄	C₂×#11761640	1	1	yes	characteristic
54	C₈	—	269	2	no	—
55	C₈	—	538	4	no	—
56	C₈	C₂×#11761640	1	2	yes	—
57	C₂²	—	269	1	no	—
58	C₂²	—	269	1	no	—
59	C₂²	—	538	2	no	—
60	C₂²	C₄⋊#11761640	1	1	yes	characteristic
61	C₄	—	269	1	no	—
62	C₄	—	269	1	no	—
63	C₄	C₄⋊#11761640	1	1	yes	characteristic
64	C₄	C₂²×#11761640	1	1	yes	characteristic
65	C₂	—	538	1	no	—
66	C₂	—	538	1	no	—
67	C₂	#22575523	1	2	yes	direct product factor
68	C₂	C₂×C₄⋊#11761640	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₄⋊#11761640
commutator	C₄×C₂₆₉	C₂²×C₄
Frattini	C₄	C₂²×#11761640
Fitting	C₂×C₈×C₂₆₉	C₄
socle	C₂²×C₂₆₉	C₄⋊C₄
prime radical	C₂²×D₂₆₉	Q₈
cosocle	C₄×D₂₆₉	C₂³
hypercenter	C₂×C₈	#11761640
hypocenter	C₂₆₉	C₂×#108
LCS	G ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
2. #22575523	C₂	direct (N × H)
30. C₂×C₂₆₉	#108	semidirect
34. C₂₆₉	C₂×#108	semidirect
67. C₂	#22575523	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16864412	2
#22575523	2
C₂×C₈×D₂₆₉	2
C₂×#108	269

Subgroups — C2×#22575523 #31771617

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22575523 #31771617