Subgroups — #31770812

Order 17216.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16864392	C₂	1	2	yes	maximal
2	C₄×Dic₅₃₈	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#14524130	C₂	1	1	yes	maximal, characteristic
4	#22575377	C₂	1	2	yes	maximal
5	C₄⋊Q₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₃₈	C₂²	1	1	yes	characteristic
7	#16864392	—	2	4	no	—
8	C₂×#14524130	C₂²	1	2	yes	—
9	C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
10	#16864381	—	2	1	no	—
11	C₄×Dic₂₆₉	—	2	1	no	—
12	C₂×Q₈×C₂₆₉	C₂²	1	2	yes	—
13	C₄⋊C₄×C₂₆₉	—	2	1	no	—
14	C₄⋊C₄×C₂₆₉	—	2	1	no	—
15	C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
16	Dic₅₃₈	—	2	1	no	—
17	Dic₅₃₈	D₄	1	2	yes	—
18	C₂×Dic₂₆₉	—	2	1	no	—
19	C₂×Dic₂₆₉	—	2	1	no	—
20	#14524130	—	2	2	no	—
21	Q₈×C₂₆₉	—	2	2	no	—
22	Q₈×C₂₆₉	—	2	2	no	—
23	C₂×C₄×C₂₆₉	—	2	2	no	—
24	C₂×C₄×C₂₆₉	D₄	1	1	yes	characteristic
25	C₂×C₄×C₂₆₉	D₄	1	1	yes	characteristic
26	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
27	C₂²×C₂₆₉	C₂×D₄	1	1	yes	characteristic
28	C₄×C₂₆₉	—	2	1	no	—
29	C₄×C₂₆₉	—	4	2	no	—
30	C₄×C₂₆₉	Q₁₆	1	2	yes	complemented
31	C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
32	C₄×C₂₆₉	#41	1	1	yes	characteristic
33	Dic₂₆₉	—	2	2	no	—
34	Dic₂₆₉	—	4	1	no	—
35	C₂×C₂₆₉	#121	1	1	yes	characteristic
36	C₂×C₂₆₉	C₂×Q₁₆	1	1	yes	characteristic
37	C₂×C₂₆₉	#137	1	1	yes	characteristic
38	C₂₆₉	#462	1	1	yes	characteristic, complemented, Sylow
39	#462	—	269	1	no	maximal, Sylow
40	C₂×Q₁₆	—	269	2	no	—
41	C₄⋊Q₈	D₂₆₉	1	1	yes	characteristic
42	C₄×Q₈	—	269	1	no	—
43	C₄⋊C₈	—	269	1	no	—
44	#103	—	269	2	no	—
45	C₂×Q₈	—	269	1	no	—
46	C₂×Q₈	C₂×D₂₆₉	1	2	yes	—
47	Q₁₆	—	538	4	no	—
48	C₂×C₈	—	269	2	no	—
49	C₄⋊C₄	—	2	1	no	—
50	C₄⋊C₄	—	2	1	no	—
51	C₄⋊C₄	—	269	1	no	—
52	C₄⋊C₄	—	538	1	no	—
53	C₄²	—	538	1	no	—
54	C₄²	C₂×D₂₆₉	1	1	yes	characteristic
55	Q₈	—	2	2	no	—
56	Q₈	—	2	2	no	—
57	Q₈	—	269	2	no	—
58	Q₈	—	538	1	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	538	1	no	—
61	C₂×C₄	—	538	1	no	—
62	C₂×C₄	C₂²⋊D₂₆₉	1	1	yes	characteristic
63	C₂×C₄	C₂²⋊D₂₆₉	1	1	yes	characteristic
64	C₂×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
65	C₈	—	538	2	no	—
66	C₂²	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	4	2	no	—
69	C₄	—	538	2	no	—
70	C₄	—	1076	1	no	—
71	C₄	#16864392	1	2	yes	complemented
72	C₄	D₄×D₂₆₉	1	1	yes	characteristic
73	C₄	#16864401	1	1	yes	characteristic
74	C₂	#22575424	1	1	yes	characteristic
75	C₂	C₂×#16864392	1	1	yes	characteristic
76	C₂	#22575484	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₆₉
commutator, cosocle	C₂×C₄×C₂₆₉	C₂³
Frattini	C₂×C₄	C₂²×D₂₆₉
Fitting	C₄⋊Q₈×C₂₆₉	C₂
socle, prime radical	C₂²×C₂₆₉	C₂×D₄
hypercenter	C₄⋊Q₈	D₂₆₉
hypocenter	C₂₆₉	#462
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
30. C₄×C₂₆₉	Q₁₆	semidirect
38. C₂₆₉	#462	semidirect
71. C₄	#16864392	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
C₄×Dic₅₃₈	2
C₄⋊#14524130	2
#22575377	2
C₄⋊Q₈×C₂₆₉	2
#462	269