Subgroups — #31770710

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	C₄×Dic₅₃₈	C₂	1	1	yes	maximal, characteristic
3	#22575384	C₂	1	1	yes	maximal, characteristic
4	#22575377	C₂	1	1	yes	maximal, characteristic
5	#22575376	C₂	1	1	yes	maximal, characteristic
6	#22575364	C₂	1	1	yes	maximal, characteristic
7	C₄×Q₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₅₃₈	C₂²	1	1	yes	characteristic
9	#16864392	C₄	1	4	yes	complemented
10	C₂×#14524130	C₂²	1	1	yes	characteristic
11	C₂×#14524130	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
13	#16864381	—	2	1	no	—
14	C₄×Dic₂₆₉	—	2	1	no	—
15	C₂×Q₈×C₂₆₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×C₂₆₉	—	2	1	no	—
17	C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
18	C₄²×C₂₆₉	—	2	1	no	—
19	C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
20	Dic₅₃₈	—	2	1	no	—
21	Dic₅₃₈	C₂×C₄	1	2	yes	—
22	C₂×Dic₂₆₉	—	2	1	no	—
23	C₂×Dic₂₆₉	—	2	1	no	—
24	#14524130	—	2	1	no	—
25	#14524130	C₂×C₄	1	2	yes	—
26	Q₈×C₂₆₉	—	2	1	no	—
27	Q₈×C₂₆₉	C₂×C₄	1	2	yes	—
28	C₂×C₄×C₂₆₉	—	2	1	no	—
29	C₂×C₄×C₂₆₉	—	2	1	no	—
30	C₂×C₄×C₂₆₉	D₄	1	1	yes	characteristic
31	C₂×C₄×C₂₆₉	D₄	1	1	yes	characteristic
32	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
33	C₂²×C₂₆₉	C₂×D₄	1	1	yes	characteristic
34	C₄×C₂₆₉	—	2	2	no	—
35	C₄×C₂₆₉	—	2	1	no	—
36	C₄×C₂₆₉	—	2	1	no	—
37	C₄×C₂₆₉	—	4	1	no	—
38	C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
39	C₄×C₂₆₉	#41	1	1	yes	characteristic
40	Dic₂₆₉	—	2	2	no	—
41	Dic₂₆₉	—	4	1	no	—
42	C₂×C₂₆₉	C₄×D₄	1	1	yes	characteristic
43	C₂×C₂₆₉	#137	1	1	yes	characteristic
44	C₂×C₂₆₉	#137	1	1	yes	characteristic
45	C₂₆₉	#441	1	1	yes	characteristic, complemented, Sylow
46	#441	—	269	1	no	maximal, Sylow
47	C₂×Q₁₆	—	269	1	no	—
48	C₄×Q₈	—	269	1	no	—
49	C₄×Q₈	D₂₆₉	1	1	yes	characteristic
50	#106	—	269	1	no	—
51	#103	—	269	1	no	—
52	#103	—	269	1	no	—
53	#97	—	269	1	no	—
54	C₂×Q₈	—	269	1	no	—
55	C₂×Q₈	C₂×D₂₆₉	1	1	yes	characteristic
56	Q₁₆	—	269	4	no	—
57	C₂×C₈	—	269	1	no	—
58	C₂×C₈	—	269	1	no	—
59	C₄⋊C₄	—	2	1	no	—
60	C₄⋊C₄	—	269	1	no	—
61	C₄⋊C₄	—	538	1	no	—
62	C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
63	C₄²	—	2	1	no	—
64	C₄²	—	538	1	no	—
65	C₄²	C₂×D₂₆₉	1	1	yes	characteristic
66	Q₈	—	2	1	no	—
67	Q₈	—	269	2	no	—
68	Q₈	—	538	1	no	—
69	Q₈	C₄×D₂₆₉	1	2	yes	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	538	1	no	—
73	C₂×C₄	—	538	1	no	—
74	C₂×C₄	C₂²⋊D₂₆₉	1	1	yes	characteristic
75	C₂×C₄	C₂²⋊D₂₆₉	1	1	yes	characteristic
76	C₂×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
77	C₈	—	269	2	no	—
78	C₈	—	538	1	no	—
79	C₂²	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
80	C₄	—	2	2	no	—
81	C₄	—	2	1	no	—
82	C₄	—	2	1	no	—
83	C₄	—	4	1	no	—
84	C₄	—	538	2	no	—
85	C₄	—	1076	1	no	—
86	C₄	C₂×C₄×D₂₆₉	1	1	yes	characteristic
87	C₄	#16864402	1	1	yes	characteristic
88	C₂	C₄×C₂²⋊D₂₆₉	1	1	yes	characteristic
89	C₂	#22575483	1	1	yes	characteristic
90	C₂	#22575484	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₆₉
commutator	C₄×C₂₆₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₆₉
Fitting	C₄×Q₈×C₂₆₉	C₂
socle, prime radical	C₂²×C₂₆₉	C₂×D₄
cosocle	C₂×C₄×C₂₆₉	C₂³
hypercenter	C₄×Q₈	D₂₆₉
hypocenter	C₂₆₉	#441
LCS	G ▷ 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
9. #16864392	C₄	semidirect
45. C₂₆₉	#441	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
#22575384	2
#22575377	2
#22575376	2
#22575364	2
C₄×Q₈×C₂₆₉	2
#441	269