Subgroups — C₄⋊Q₈×D₂₆₉ #31771146

Order 17216.

290 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₂₆₉	C₂	1	2	yes	maximal
2	#22575453	C₂	1	1	yes	maximal, characteristic, complemented
3	#22575452	C₂	1	4	yes	maximal, complemented
4	#22575451	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₄×D₂₆₉	C₂	1	4	yes	maximal
6	C₄²×D₂₆₉	C₂	1	1	yes	maximal, characteristic
7	C₄⋊Q₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₅₃₈	C₂²	1	2	yes	—
9	C₂×Dic₅₃₈	C₂²	1	4	yes	—
10	Q₈×D₂₆₉	—	2	8	no	—
11	C₂×C₄×D₂₆₉	C₂²	1	4	yes	—
12	C₂×C₄×D₂₆₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₂₆₉	C₂²	1	2	yes	—
14	C₄⋊Dic₂₆₉	C₂²	1	4	yes	—
15	#16864381	C₂²	1	8	yes	—
16	C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
17	C₄×Dic₂₆₉	C₂²	1	2	yes	—
18	C₂×Q₈×C₂₆₉	C₂²	1	2	yes	—
19	C₄⋊C₄×C₂₆₉	C₂²	1	4	yes	—
20	C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
21	C₂²×D₂₆₉	C₂³	1	1	yes	characteristic
22	Dic₅₃₈	—	2	4	no	—
23	Dic₅₃₈	—	2	8	no	—
24	C₂×Dic₂₆₉	C₂³	1	4	yes	—
25	C₂×Dic₂₆₉	C₂³	1	1	yes	characteristic
26	C₂×Dic₂₆₉	C₂³	1	2	yes	—
27	C₄×D₂₆₉	—	2	8	no	—
28	C₄×D₂₆₉	D₄	1	4	yes	—
29	C₄×D₂₆₉	Q₈	1	8	yes	complemented
30	Q₈×C₂₆₉	—	2	4	no	—
31	C₂×C₄×C₂₆₉	C₂³	1	4	yes	—
32	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
33	C₂×C₄×C₂₆₉	C₂³	1	2	yes	—
34	C₂²×C₂₆₉	C₂⁴	1	1	yes	characteristic
35	C₂×D₂₆₉	C₂×D₄	1	2	yes	—
36	C₂×D₂₆₉	C₂×Q₈	1	4	yes	—
37	C₄×C₂₆₉	—	2	4	no	—
38	C₄×C₂₆₉	C₂×D₄	1	2	yes	—
39	C₄×C₂₆₉	C₂×Q₈	1	4	yes	complemented
40	Dic₂₆₉	—	2	4	no	—
41	Dic₂₆₉	C₂×D₄	1	2	yes	—
42	Dic₂₆₉	C₂×Q₈	1	4	yes	complemented
43	C₂×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
44	C₂×C₂₆₉	C₂²×Q₈	1	2	yes	—
45	D₂₆₉	C₄⋊Q₈	1	4	yes	direct product factor
46	C₂₆₉	C₂×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₄⋊Q₈	—	269	1	no	maximal, Sylow
48	C₂²×Q₈	—	269	2	no	—
49	C₄⋊Q₈	—	269	2	no	—
50	C₄⋊Q₈	—	269	4	no	—
51	C₄⋊Q₈	—	269	1	no	—
52	C₄⋊Q₈	D₂₆₉	1	1	yes	characteristic, direct product factor
53	C₂×C₄⋊C₄	—	269	4	no	—
54	C₂×C₄²	—	269	1	no	—
55	C₂×Q₈	—	269	4	no	—
56	C₂×Q₈	—	269	2	no	—
57	C₂×Q₈	—	538	8	no	—
58	C₂×Q₈	C₂×D₂₆₉	1	2	yes	—
59	C₂²×C₄	—	269	2	no	—
60	C₂²×C₄	—	269	1	no	—
61	C₂²×C₄	—	269	4	no	—
62	C₄⋊C₄	—	269	8	no	—
63	C₄⋊C₄	—	269	4	no	—
64	C₄⋊C₄	C₂×D₂₆₉	1	4	yes	—
65	C₄²	—	269	1	no	—
66	C₄²	—	269	2	no	—
67	C₄²	C₂×D₂₆₉	1	1	yes	characteristic
68	C₂³	—	269	1	no	—
69	Q₈	—	2	4	no	—
70	Q₈	—	538	8	no	—
71	Q₈	—	538	4	no	—
72	C₂×C₄	—	269	2	no	—
73	C₂×C₄	—	269	1	no	—
74	C₂×C₄	—	269	4	no	—
75	C₂×C₄	—	269	8	no	—
76	C₂×C₄	—	269	4	no	—
77	C₂×C₄	—	538	8	no	—
78	C₂×C₄	C₂²×D₂₆₉	1	4	yes	—
79	C₂×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
80	C₂×C₄	C₂²×D₂₆₉	1	2	yes	—
81	C₂²	—	269	2	no	—
82	C₂²	—	269	4	no	—
83	C₂²	C₂³×D₂₆₉	1	1	yes	characteristic
84	C₄	—	2	4	no	—
85	C₄	—	269	2	no	—
86	C₄	—	269	4	no	—
87	C₄	—	538	4	no	—
88	C₄	D₄×D₂₆₉	1	2	yes	—
89	C₄	Q₈×D₂₆₉	1	4	yes	complemented
90	C₂	—	269	4	no	—
91	C₂	C₂×D₄×D₂₆₉	1	1	yes	characteristic
92	C₂	C₂×Q₈×D₂₆₉	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. #22575453	C₂	semidirect
3. #22575452	C₂	semidirect
4. #22575451	C₂	semidirect
7. C₄⋊Q₈×C₂₆₉	C₂	semidirect
29. C₄×D₂₆₉	Q₈	semidirect
39. C₄×C₂₆₉	C₂×Q₈	semidirect
42. Dic₂₆₉	C₂×Q₈	semidirect
45. D₂₆₉	C₄⋊Q₈	direct (N × H)
46. C₂₆₉	C₂×C₄⋊Q₈	semidirect
52. C₄⋊Q₈	D₂₆₉	direct (N × H)
89. C₄	Q₈×D₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×D₂₆₉	2
#22575453	2
#22575452	2
#22575451	2
C₄⋊C₄×D₂₆₉	2
C₄²×D₂₆₉	2
C₄⋊Q₈×C₂₆₉	2
C₂×C₄⋊Q₈	269

Subgroups — C4⋊Q8×D269 #31771146

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊Q₈×D₂₆₉ #31771146