Subgroups — #31770813

Order 17216.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16864392	C₂	1	1	yes	maximal, characteristic
2	#22575451	C₂	1	1	yes	maximal, characteristic
3	Q₈×Dic₂₆₉	C₂	1	1	yes	maximal, characteristic
4	#22575382	C₂	1	1	yes	maximal, characteristic
5	#22575378	C₂	1	1	yes	maximal, characteristic
6	#22575377	C₂	1	1	yes	maximal, characteristic
7	C₂×Q₁₆×C₂₆₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₅₃₈	C₂²	1	1	yes	characteristic
9	#16864392	—	2	2	no	—
10	C₂×#14524130	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₂₆₉	—	2	1	no	—
12	C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
13	#16864381	—	2	1	no	—
14	#16864381	—	2	1	no	—
15	C₄×Dic₂₆₉	—	2	1	no	—
16	C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₂₆₉	C₂²	1	1	yes	characteristic
18	C₂×Q₈×C₂₆₉	C₂²	1	1	yes	characteristic
19	Q₁₆×C₂₆₉	—	2	2	no	—
20	C₂×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
21	Dic₅₃₈	—	2	1	no	—
22	Dic₅₃₈	—	2	1	no	—
23	C₂×Dic₂₆₉	—	2	1	no	—
24	C₂×Dic₂₆₉	—	2	1	no	—
25	C₂×Dic₂₆₉	D₄	1	1	yes	characteristic
26	C₂×Dic₂₆₉	D₄	1	1	yes	characteristic
27	#14524130	—	2	1	no	—
28	Q₈×C₂₆₉	—	2	1	no	—
29	Q₈×C₂₆₉	—	2	1	no	—
30	Q₈×C₂₆₉	—	2	1	no	—
31	Q₈×C₂₆₉	D₄	1	2	yes	—
32	C₂×C₄×C₂₆₉	—	2	1	no	—
33	C₂×C₄×C₂₆₉	—	2	1	no	—
34	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
35	C₈×C₂₆₉	—	2	1	no	—
36	C₂²×C₂₆₉	C₂×D₄	1	1	yes	characteristic
37	C₄×C₂₆₉	—	2	2	no	—
38	C₄×C₂₆₉	—	4	1	no	—
39	C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
40	C₄×C₂₆₉	#41	1	1	yes	characteristic
41	Dic₂₆₉	—	2	1	no	—
42	Dic₂₆₉	—	4	1	no	—
43	Dic₂₆₉	—	4	1	no	—
44	Dic₂₆₉	Q₁₆	1	2	yes	complemented
45	C₂×C₂₆₉	#121	1	1	yes	characteristic
46	C₂×C₂₆₉	C₂×Q₁₆	1	1	yes	characteristic
47	C₂×C₂₆₉	#137	1	1	yes	characteristic
48	C₂₆₉	#462	1	1	yes	characteristic, complemented, Sylow
49	#462	—	269	1	no	maximal, Sylow
50	C₂×Q₁₆	—	269	1	no	—
51	C₂×Q₁₆	D₂₆₉	1	1	yes	characteristic
52	C₄⋊Q₈	—	269	1	no	—
53	C₄×Q₈	—	269	1	no	—
54	C₄⋊C₈	—	269	1	no	—
55	#103	—	269	1	no	—
56	#103	—	269	1	no	—
57	C₂×Q₈	—	269	1	no	—
58	C₂×Q₈	C₂×D₂₆₉	1	1	yes	characteristic
59	C₂×Q₈	C₂×D₂₆₉	1	1	yes	characteristic
60	Q₁₆	—	2	2	no	—
61	Q₁₆	—	538	2	no	—
62	C₂×C₈	—	269	1	no	—
63	C₂×C₈	C₂×D₂₆₉	1	1	yes	characteristic
64	C₄⋊C₄	—	269	1	no	—
65	C₄⋊C₄	—	538	1	no	—
66	C₄⋊C₄	—	538	1	no	—
67	C₄⋊C₄	—	538	1	no	—
68	C₄²	—	269	1	no	—
69	C₄²	—	538	1	no	—
70	Q₈	—	2	1	no	—
71	Q₈	—	2	1	no	—
72	Q₈	—	2	1	no	—
73	Q₈	—	538	1	no	—
74	Q₈	—	538	1	no	—
75	Q₈	C₂²⋊D₂₆₉	1	2	yes	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	269	1	no	—
79	C₂×C₄	—	269	1	no	—
80	C₂×C₄	—	538	1	no	—
81	C₂×C₄	—	538	1	no	—
82	C₂×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
83	C₈	—	2	1	no	—
84	C₈	—	538	1	no	—
85	C₂²	D₄×D₂₆₉	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	4	1	no	—
88	C₄	—	269	2	no	—
89	C₄	—	538	1	no	—
90	C₄	—	1076	1	no	—
91	C₄	—	1076	1	no	—
92	C₄	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
93	C₄	#16864401	1	1	yes	characteristic
94	C₂	#22575425	1	1	yes	characteristic
95	C₂	Q₁₆×D₂₆₉	1	1	yes	characteristic
96	C₂	#22575485	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₆₉
commutator, cosocle	C₂×C₄×C₂₆₉	C₂³
Frattini	C₂×C₄	C₂²×D₂₆₉
Fitting	C₂×Q₁₆×C₂₆₉	C₂
socle, prime radical	C₂²×C₂₆₉	C₂×D₄
hypercenter	C₂×Q₁₆	D₂₆₉
hypocenter	C₂₆₉	#462
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
44. Dic₂₆₉	Q₁₆	semidirect
48. C₂₆₉	#462	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16864392	2
#22575451	2
Q₈×Dic₂₆₉	2
#22575382	2
#22575378	2
#22575377	2
C₂×Q₁₆×C₂₆₉	2
#462	269