Subgroups — C₂×Q₈×#11761640 #31771670

Order 17216.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#11761640	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊#11761640	C₂	1	3	yes	maximal
3	C₂×C₄×#11761640	C₂	1	3	yes	maximal
4	C₂×Q₈×D₂₆₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11761640	C₂²	1	3	yes	—
6	C₂²×#11761640	C₂²	1	1	yes	characteristic
7	C₄⋊#11761640	C₂²	1	12	yes	—
8	C₄×#11761640	C₂²	1	12	yes	—
9	C₂×Dic₅₃₈	C₄	1	3	yes	complemented
10	Q₈×D₂₆₉	C₄	1	4	yes	complemented
11	Q₈×D₂₆₉	C₂²	1	4	yes	—
12	C₂×C₄×D₂₆₉	C₂²	1	3	yes	—
13	C₂×Q₈×C₂₆₉	C₄	1	1	yes	characteristic, complemented
14	C₂×#11761640	—	2	6	no	—
15	C₂×#11761640	Q₈	1	4	yes	direct product factor
16	C₂×#11761640	C₂³	1	6	yes	—
17	C₂×#11761640	C₂³	1	2	yes	—
18	C₂²×D₂₆₉	C₂³	1	1	yes	characteristic
19	Dic₅₃₈	C₂×C₄	1	12	yes	complemented
20	C₂×Dic₂₆₉	C₂×C₄	1	3	yes	—
21	C₄×D₂₆₉	C₂×C₄	1	6	yes	—
22	C₄×D₂₆₉	C₂³	1	6	yes	—
23	Q₈×C₂₆₉	C₂×C₄	1	4	yes	complemented
24	C₂×C₄×C₂₆₉	C₂×C₄	1	3	yes	—
25	C₂²×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
26	C₂×D₂₆₉	C₂²×C₄	1	1	yes	characteristic
27	C₂×D₂₆₉	C₂×Q₈	1	2	yes	—
28	C₂×D₂₆₉	#41	1	2	yes	—
29	C₂×D₂₆₉	C₂⁴	1	1	yes	characteristic
30	#11761640	—	2	6	no	—
31	#11761640	C₂×Q₈	1	4	yes	direct product factor
32	C₄×C₂₆₉	C₂²×C₄	1	6	yes	—
33	Dic₂₆₉	C₂²×C₄	1	6	yes	—
34	C₂×C₂₆₉	C₄×Q₈	1	2	yes	complemented
35	C₂×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
36	D₂₆₉	C₄×Q₈	1	2	yes	complemented
37	D₂₆₉	C₂²×Q₈	1	1	yes	characteristic
38	D₂₆₉	C₂×#41	1	1	yes	characteristic
39	C₂₆₉	C₂×C₄×Q₈	1	1	yes	characteristic, complemented, Sylow
40	C₂×C₄×Q₈	—	269	1	no	maximal, Sylow
41	C₂²×Q₈	—	269	1	no	—
42	C₄×Q₈	—	269	8	no	—
43	C₂×C₄⋊C₄	—	269	3	no	—
44	C₂×C₄²	—	269	3	no	—
45	C₂×Q₈	—	269	4	no	—
46	C₂×Q₈	—	269	4	no	—
47	C₂×Q₈	—	269	3	no	—
48	C₂×Q₈	#11761640	1	1	yes	characteristic, direct product factor
49	C₂²×C₄	—	269	1	no	—
50	C₂²×C₄	—	269	3	no	—
51	C₂²×C₄	—	269	3	no	—
52	C₄⋊C₄	—	269	12	no	—
53	C₄²	—	269	12	no	—
54	C₂³	—	269	1	no	—
55	Q₈	—	269	12	no	—
56	Q₈	C₂×#11761640	1	4	yes	direct product factor
57	C₂×C₄	—	269	2	no	—
58	C₂×C₄	—	269	4	no	—
59	C₂×C₄	—	269	3	no	—
60	C₂×C₄	—	269	6	no	—
61	C₂×C₄	—	269	6	no	—
62	C₂×C₄	—	269	6	no	—
63	C₂×C₄	—	538	6	no	—
64	C₂×C₄	C₂×#11761640	1	3	yes	—
65	C₂²	—	269	2	no	—
66	C₂²	—	269	2	no	—
67	C₂²	—	269	1	no	—
68	C₂²	—	269	1	no	—
69	C₂²	C₂²×#11761640	1	1	yes	characteristic
70	C₄	—	269	4	no	—
71	C₄	—	269	6	no	—
72	C₄	—	538	6	no	—
73	C₄	C₂²×#11761640	1	6	yes	—
74	C₂	—	269	1	no	—
75	C₂	—	269	1	no	—
76	C₂	—	269	2	no	—
77	C₂	Q₈×#11761640	1	2	yes	direct product factor
78	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	C₂²×D₂₆₉	C₂³
cosocle	C₂×D₂₆₉	C₂⁴
hypercenter	C₂×Q₈	#11761640
hypocenter	C₂₆₉	C₂×C₄×Q₈
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. Q₈×#11761640	C₂	direct (N × H)
9. C₂×Dic₅₃₈	C₄	semidirect
10. Q₈×D₂₆₉	C₄	semidirect
13. C₂×Q₈×C₂₆₉	C₄	semidirect
15. C₂×#11761640	Q₈	direct (N × H)
19. Dic₅₃₈	C₂×C₄	semidirect
23. Q₈×C₂₆₉	C₂×C₄	semidirect
31. #11761640	C₂×Q₈	direct (N × H)
34. C₂×C₂₆₉	C₄×Q₈	semidirect
36. D₂₆₉	C₄×Q₈	semidirect
39. C₂₆₉	C₂×C₄×Q₈	semidirect
48. C₂×Q₈	#11761640	direct (N × H)
56. Q₈	C₂×#11761640	direct (N × H)
77. C₂	Q₈×#11761640	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×#11761640	2
C₂×C₄⋊#11761640	2
C₂×C₄×#11761640	2
C₂×Q₈×D₂₆₉	2
C₂×C₄×Q₈	269

Subgroups — C2×Q8×#11761640 #31771670

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×#11761640 #31771670