Subgroups — C₂×#22575377 #31770596

Order 17216.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14524130	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₂₆₉	C₂	1	1	yes	maximal, characteristic
3	#22575377	C₂	1	4	yes	maximal, direct product factor
4	C₂²×Q₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₆₉	—	2	1	no	—
6	C₂×#14524130	—	2	2	no	—
7	C₂×#14524130	C₂²	1	2	yes	—
8	C₄⋊Dic₂₆₉	—	2	1	no	—
9	C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
10	C₂×Q₈×C₂₆₉	—	2	2	no	—
11	C₂×Q₈×C₂₆₉	—	2	1	no	—
12	C₂×Q₈×C₂₆₉	C₄	1	4	yes	complemented
13	C₂×Q₈×C₂₆₉	C₂²	1	2	yes	—
14	C₂²×C₄×C₂₆₉	—	2	1	no	—
15	C₂²×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₆₉	—	2	2	no	—
17	C₂×Dic₂₆₉	—	4	2	no	—
18	#14524130	—	2	2	no	—
19	C₂³×C₂₆₉	D₄	1	1	yes	characteristic
20	Q₈×C₂₆₉	—	2	4	no	—
21	Q₈×C₂₆₉	—	2	2	no	—
22	Q₈×C₂₆₉	C₂×C₄	1	4	yes	complemented
23	C₂×C₄×C₂₆₉	—	2	4	no	—
24	C₂×C₄×C₂₆₉	—	2	2	no	—
25	C₂×C₄×C₂₆₉	C₂×C₄	1	2	yes	—
26	C₂×C₄×C₂₆₉	D₄	1	1	yes	characteristic
27	C₂×C₄×C₂₆₉	D₄	1	2	yes	—
28	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
29	C₂²×C₂₆₉	C₂²⋊C₄	1	2	yes	—
30	C₂²×C₂₆₉	SD₁₆	1	2	yes	—
31	C₂²×C₂₆₉	Q₁₆	1	2	yes	—
32	C₂²×C₂₆₉	C₂×D₄	1	1	yes	characteristic
33	C₄×C₂₆₉	—	2	4	no	—
34	C₄×C₂₆₉	C₂²⋊C₄	1	2	yes	—
35	C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
36	C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
37	Dic₂₆₉	—	4	2	no	—
38	C₂×C₂₆₉	#103	1	4	yes	complemented
39	C₂×C₂₆₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₂₆₉	C₂×SD₁₆	1	1	yes	characteristic
41	C₂×C₂₆₉	C₂×Q₁₆	1	1	yes	characteristic
42	C₂₆₉	C₂×#103	1	1	yes	characteristic, complemented, Sylow
43	C₂×#103	—	269	1	no	maximal, Sylow
44	C₂²×Q₈	D₂₆₉	1	1	yes	characteristic
45	C₂²×C₈	—	269	1	no	—
46	C₂×C₄⋊C₄	—	269	1	no	—
47	#103	—	269	4	no	—
48	C₂×Q₈	—	2	2	no	—
49	C₂×Q₈	—	2	1	no	—
50	C₂×Q₈	Dic₂₆₉	1	4	yes	complemented
51	C₂×Q₈	C₂×D₂₆₉	1	2	yes	—
52	C₂²×C₄	—	2	1	no	—
53	C₂²×C₄	—	538	1	no	—
54	C₂²×C₄	C₂×D₂₆₉	1	1	yes	characteristic
55	C₂×C₈	—	269	2	no	—
56	C₂×C₈	—	538	2	no	—
57	C₄⋊C₄	—	269	2	no	—
58	C₄⋊C₄	—	538	1	no	—
59	C₂³	C₂²⋊D₂₆₉	1	1	yes	characteristic
60	Q₈	—	2	4	no	—
61	Q₈	—	2	2	no	—
62	Q₈	C₂×Dic₂₆₉	1	4	yes	complemented
63	C₂×C₄	—	2	4	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	538	2	no	—
66	C₂×C₄	—	1076	2	no	—
67	C₂×C₄	C₂×Dic₂₆₉	1	2	yes	—
68	C₂×C₄	C₂²⋊D₂₆₉	1	1	yes	characteristic
69	C₂×C₄	C₂²⋊D₂₆₉	1	2	yes	—
70	C₂×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
71	C₈	—	538	2	no	—
72	C₂²	C₂²⋊Dic₂₆₉	1	2	yes	—
73	C₂²	#16864389	1	2	yes	—
74	C₂²	#16864392	1	2	yes	—
75	C₂²	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
76	C₄	—	2	4	no	—
77	C₄	—	1076	2	no	—
78	C₄	C₂²⋊Dic₂₆₉	1	2	yes	—
79	C₄	C₂²×Dic₂₆₉	1	1	yes	characteristic
80	C₄	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
81	C₂	#22575377	1	4	yes	direct product factor
82	C₂	C₂×C₂²⋊Dic₂₆₉	1	1	yes	characteristic
83	C₂	C₂×#16864389	1	1	yes	characteristic
84	C₂	C₂×#16864392	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊D₂₆₉
commutator	C₄×C₂₆₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₆₉
Fitting	C₂²×Q₈×C₂₆₉	C₂
socle, prime radical	C₂³×C₂₆₉	D₄
cosocle	C₂×C₄×C₂₆₉	C₂³
hypercenter	C₂²×Q₈	D₂₆₉
hypocenter	C₂₆₉	C₂×#103
LCS	G ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
3. #22575377	C₂	direct (N × H)
12. C₂×Q₈×C₂₆₉	C₄	semidirect
22. Q₈×C₂₆₉	C₂×C₄	semidirect
38. C₂×C₂₆₉	#103	semidirect
42. C₂₆₉	C₂×#103	semidirect
50. C₂×Q₈	Dic₂₆₉	semidirect
62. Q₈	C₂×Dic₂₆₉	semidirect
81. C₂	#22575377	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×#14524130	2
C₂×C₄⋊Dic₂₆₉	2
#22575377	2
C₂²×Q₈×C₂₆₉	2
C₂×#103	269

Subgroups — C2×#22575377 #31770596

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22575377 #31770596