Subgroups — C₂×#22575541 #31771682

Order 17216.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575541	C₂	1	8	yes	maximal, direct product factor
2	C₂×#16864412	C₂	1	3	yes	maximal, complemented
3	C₂×D₂₆₉⋊C₈	C₂	1	3	yes	maximal, complemented
4	C₂×#16864403	C₂	1	1	yes	maximal, characteristic
5	#16864412	C₂²	1	12	yes	complemented
6	D₂₆₉⋊C₈	C₂²	1	12	yes	complemented
7	C₂×#14524137	C₂²	1	1	yes	characteristic
8	C₂×#14524137	C₂²	1	3	yes	—
9	#16864403	C₄	1	4	yes	—
10	#16864403	C₂²	1	4	yes	—
11	C₂×D₁₀₇₆	C₄	1	3	yes	—
12	C₂×C₄×D₂₆₉	C₂²	1	3	yes	—
13	C₂×Q₈×C₂₆₉	C₄	1	1	yes	characteristic
14	#14524137	C₂³	1	2	yes	—
15	#14524137	C₂³	1	6	yes	—
16	C₂²×D₂₆₉	C₂×C₄	1	3	yes	—
17	D₁₀₇₆	C₂×C₄	1	12	yes	—
18	C₂×Dic₂₆₉	C₂³	1	1	yes	characteristic
19	C₄×D₂₆₉	C₂×C₄	1	6	yes	—
20	C₄×D₂₆₉	C₂³	1	6	yes	—
21	Q₈×C₂₆₉	C₂×C₄	1	4	yes	—
22	C₂×C₄×C₂₆₉	C₂×C₄	1	3	yes	—
23	C₂²×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
24	C₂×D₂₆₉	—	2	6	no	—
25	C₂×D₂₆₉	C₂²×C₄	1	6	yes	—
26	C₄×C₂₆₉	C₂²×C₄	1	6	yes	—
27	Dic₂₆₉	C₂²×C₄	1	1	yes	characteristic
28	Dic₂₆₉	C₂⁴	1	1	yes	characteristic
29	C₂×C₂₆₉	#131	1	2	yes	complemented
30	C₂×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
31	D₂₆₉	—	2	6	no	—
32	C₂₆₉	C₂×#131	1	1	yes	characteristic, complemented, Sylow
33	C₂×#131	—	269	1	no	maximal, Sylow
34	C₂×#41	—	269	1	no	—
35	#131	—	269	8	no	—
36	C₂×M₁₆	—	269	3	no	—
37	C₂²×C₈	—	269	3	no	—
38	#41	—	269	4	no	—
39	#41	—	269	4	no	—
40	C₂×Q₈	#11761640	1	1	yes	characteristic
41	C₂×D₄	—	269	3	no	—
42	C₂²×C₄	—	269	3	no	—
43	M₁₆	—	269	12	no	—
44	C₂×C₈	—	269	1	no	—
45	C₂×C₈	—	269	3	no	—
46	C₂×C₈	—	269	12	no	—
47	C₂³	—	269	3	no	—
48	Q₈	C₂×#11761640	1	4	yes	—
49	D₄	—	269	12	no	—
50	C₂×C₄	—	269	1	no	—
51	C₂×C₄	—	269	6	no	—
52	C₂×C₄	—	269	6	no	—
53	C₂×C₄	C₂×#11761640	1	3	yes	—
54	C₈	—	269	2	no	—
55	C₈	—	269	6	no	—
56	C₂²	—	269	6	no	—
57	C₂²	—	538	6	no	—
58	C₂²	C₂²×#11761640	1	1	yes	characteristic
59	C₄	—	269	1	no	—
60	C₄	—	269	1	no	—
61	C₄	C₂²×#11761640	1	6	yes	—
62	C₂	—	538	6	no	—
63	C₂	#22575541	1	2	yes	direct product factor
64	C₂	C₂³×#11761640	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11761640
commutator	C₂×C₂₆₉	C₂³×C₄
Frattini	C₂	C₂³×#11761640
Fitting	C₂×Q₈×C₂₆₉	C₄
socle	C₂²×C₂₆₉	C₂²×C₄
prime radical	C₂×#16864403	C₂
cosocle	Dic₂₆₉	C₂⁴
hypercenter	C₂×Q₈	#11761640
hypocenter	C₂₆₉	C₂×#131
LCS	G ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. #22575541	C₂	direct (N × H)
2. C₂×#16864412	C₂	semidirect
3. C₂×D₂₆₉⋊C₈	C₂	semidirect
5. #16864412	C₂²	semidirect
6. D₂₆₉⋊C₈	C₂²	semidirect
29. C₂×C₂₆₉	#131	semidirect
32. C₂₆₉	C₂×#131	semidirect
63. C₂	#22575541	direct (N × H)

Maximal subgroups

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

subgroup	index
#22575541	2
C₂×#16864412	2
C₂×D₂₆₉⋊C₈	2
C₂×#16864403	2
C₂×#131	269

Subgroups — C2×#22575541 #31771682

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22575541 #31771682