Subgroups — C₂×#22575451 #31771147

Order 17216.

290 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₅₃₈	C₂	1	1	yes	maximal, characteristic
2	#22575451	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16864381	C₂	1	4	yes	maximal
4	C₂×C₄×Dic₂₆₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₃₈	—	2	4	no	—
7	C₂×Dic₅₃₈	C₂²	1	4	yes	—
8	C₂²×Dic₂₆₉	C₂²	1	2	yes	—
9	C₂²×Dic₂₆₉	C₂²	1	2	yes	—
10	#16864381	C₂²	1	16	yes	—
11	C₄×Dic₂₆₉	C₂²	1	4	yes	—
12	C₂×Q₈×C₂₆₉	—	2	4	no	—
13	C₂×Q₈×C₂₆₉	C₂²	1	4	yes	—
14	C₂²×C₄×C₂₆₉	C₂²	1	2	yes	—
15	C₂²×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
16	Dic₅₃₈	—	2	8	no	—
17	C₂×Dic₂₆₉	—	2	4	no	—
18	C₂×Dic₂₆₉	Q₈	1	8	yes	complemented
19	C₂×Dic₂₆₉	C₂³	1	4	yes	—
20	C₂×Dic₂₆₉	C₂³	1	4	yes	—
21	C₂³×C₂₆₉	C₂³	1	1	yes	characteristic
22	Q₈×C₂₆₉	—	2	8	no	—
23	C₂×C₄×C₂₆₉	—	2	4	no	—
24	C₂×C₄×C₂₆₉	D₄	1	4	yes	—
25	C₂×C₄×C₂₆₉	C₂³	1	4	yes	—
26	C₂×C₄×C₂₆₉	C₂³	1	2	yes	—
27	C₂²×C₂₆₉	C₂×D₄	1	2	yes	—
28	C₂²×C₂₆₉	C₂×Q₈	1	4	yes	—
29	C₂²×C₂₆₉	C₂⁴	1	1	yes	characteristic
30	C₄×C₂₆₉	—	2	4	no	—
31	C₄×C₂₆₉	C₂×D₄	1	4	yes	—
32	Dic₂₆₉	—	2	4	no	—
33	Dic₂₆₉	C₂×Q₈	1	8	yes	complemented
34	C₂×C₂₆₉	C₄⋊Q₈	1	4	yes	complemented
35	C₂×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
36	C₂×C₂₆₉	C₂²×Q₈	1	2	yes	—
37	C₂₆₉	C₂×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
38	C₂×C₄⋊Q₈	—	269	1	no	maximal, Sylow
39	C₂²×Q₈	—	269	1	no	—
40	C₂²×Q₈	D₂₆₉	1	1	yes	characteristic
41	C₄⋊Q₈	—	269	8	no	—
42	C₂×C₄⋊C₄	—	269	4	no	—
43	C₂×C₄²	—	269	1	no	—
44	C₂×Q₈	—	2	4	no	—
45	C₂×Q₈	—	269	4	no	—
46	C₂×Q₈	—	538	4	no	—
47	C₂×Q₈	C₂×D₂₆₉	1	4	yes	—
48	C₂²×C₄	—	269	2	no	—
49	C₂²×C₄	—	269	2	no	—
50	C₂²×C₄	C₂×D₂₆₉	1	2	yes	—
51	C₂²×C₄	C₂×D₂₆₉	1	1	yes	characteristic
52	C₄⋊C₄	—	269	16	no	—
53	C₄²	—	269	4	no	—
54	C₂³	C₂²×D₂₆₉	1	1	yes	characteristic
55	Q₈	—	2	8	no	—
56	Q₈	—	538	8	no	—
57	C₂×C₄	—	2	4	no	—
58	C₂×C₄	—	269	8	no	—
59	C₂×C₄	—	269	4	no	—
60	C₂×C₄	—	269	4	no	—
61	C₂×C₄	—	538	4	no	—
62	C₂×C₄	C₂²⋊D₂₆₉	1	4	yes	—
63	C₂×C₄	C₂²×D₂₆₉	1	4	yes	—
64	C₂×C₄	C₂²×D₂₆₉	1	2	yes	—
65	C₂²	C₂×C₂²⋊D₂₆₉	1	2	yes	—
66	C₂²	Q₈×D₂₆₉	1	4	yes	—
67	C₂²	C₂³×D₂₆₉	1	1	yes	characteristic
68	C₄	—	2	4	no	—
69	C₄	—	269	8	no	—
70	C₄	—	538	4	no	—
71	C₄	C₂×C₂²⋊D₂₆₉	1	4	yes	—
72	C₂	#22575451	1	4	yes	direct product factor
73	C₂	C₂²×C₂²⋊D₂₆₉	1	1	yes	characteristic
74	C₂	C₂×Q₈×D₂₆₉	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₂₆₉
commutator, cosocle	C₂²×C₂₆₉	C₂⁴
Frattini	C₂²	C₂³×D₂₆₉
Fitting	C₂²×Q₈×C₂₆₉	C₂
socle, prime radical	C₂³×C₂₆₉	C₂³
hypercenter	C₂²×Q₈	D₂₆₉
hypocenter	C₂₆₉	C₂×C₄⋊Q₈
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
2. #22575451	C₂	direct (N × H)
18. C₂×Dic₂₆₉	Q₈	semidirect
33. Dic₂₆₉	C₂×Q₈	semidirect
34. C₂×C₂₆₉	C₄⋊Q₈	semidirect
37. C₂₆₉	C₂×C₄⋊Q₈	semidirect
72. C₂	#22575451	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×Dic₅₃₈	2
#22575451	2
C₂×#16864381	2
C₂×C₄×Dic₂₆₉	2
C₂²×Q₈×C₂₆₉	2
C₂×C₄⋊Q₈	269

Subgroups — C2×#22575451 #31771147

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22575451 #31771147