Subgroups — #31770367

Order 17216.

114 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	C₂×C₄⋊C₄×C₂₆₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₂₆₉	—	2	1	no	—
5	C₂×#14524130	—	2	1	no	—
6	C₂×#14524130	—	2	1	no	—
7	C₂×#14524130	C₄	1	2	yes	complemented
8	C₄⋊Dic₂₆₉	—	2	1	no	—
9	C₄⋊Dic₂₆₉	C₄	1	2	yes	complemented
10	C₂²×C₄×C₂₆₉	—	2	1	no	—
11	C₂²×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₂₆₉	—	2	1	no	—
13	C₄⋊C₄×C₂₆₉	C₄	1	2	yes	complemented
14	C₂×Dic₂₆₉	—	2	2	no	—
15	C₂×Dic₂₆₉	—	4	1	no	—
16	C₂×Dic₂₆₉	—	4	1	no	—
17	#14524130	—	2	2	no	—
18	C₂³×C₂₆₉	D₄	1	1	yes	characteristic
19	C₂×C₄×C₂₆₉	—	2	2	no	—
20	C₂×C₄×C₂₆₉	—	4	1	no	—
21	C₂×C₄×C₂₆₉	—	4	1	no	—
22	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₂₆₉	D₄	1	1	yes	characteristic
26	C₂×C₄×C₂₆₉	D₄	1	1	yes	characteristic
27	C₂×C₄×C₂₆₉	Q₈	1	1	yes	characteristic
28	C₂²×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₂₆₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₂₆₉	D₈	1	1	yes	characteristic
32	C₂²×C₂₆₉	SD₁₆	1	1	yes	characteristic
33	C₂²×C₂₆₉	SD₁₆	1	1	yes	characteristic
34	C₂²×C₂₆₉	Q₁₆	1	1	yes	characteristic
35	C₄×C₂₆₉	—	4	2	no	—
36	C₄×C₂₆₉	C₄²	1	1	yes	characteristic
37	C₄×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
38	C₄×C₂₆₉	C₄⋊C₄	1	1	yes	characteristic
39	C₄×C₂₆₉	C₄⋊C₄	1	1	yes	characteristic
40	Dic₂₆₉	—	4	2	no	—
41	C₂×C₂₆₉	#95	1	1	yes	characteristic
42	C₂×C₂₆₉	#102	1	1	yes	characteristic
43	C₂×C₂₆₉	#102	1	1	yes	characteristic
44	C₂×C₂₆₉	#103	1	1	yes	characteristic
45	C₂×C₂₆₉	#103	1	1	yes	characteristic
46	C₂×C₂₆₉	#106	1	1	yes	characteristic
47	C₂×C₂₆₉	#107	1	1	yes	characteristic
48	C₂₆₉	#340	1	1	yes	characteristic, complemented, Sylow
49	#340	—	269	1	no	maximal, Sylow
50	C₂²×C₈	—	269	1	no	—
51	C₂×C₄⋊C₄	—	269	1	no	—
52	C₂×C₄⋊C₄	D₂₆₉	1	1	yes	characteristic
53	C₂²×C₄	—	2	1	no	—
54	C₂²×C₄	—	538	1	no	—
55	C₂²×C₄	C₂×D₂₆₉	1	1	yes	characteristic
56	C₂×C₈	—	269	2	no	—
57	C₂×C₈	—	538	1	no	—
58	C₂×C₈	—	538	1	no	—
59	C₄⋊C₄	—	2	1	no	—
60	C₄⋊C₄	—	269	2	no	—
61	C₄⋊C₄	—	538	1	no	—
62	C₄⋊C₄	Dic₂₆₉	1	2	yes	complemented
63	C₂³	C₂²⋊D₂₆₉	1	1	yes	characteristic
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	4	1	no	—
66	C₂×C₄	—	4	1	no	—
67	C₂×C₄	—	538	2	no	—
68	C₂×C₄	—	1076	1	no	—
69	C₂×C₄	—	1076	1	no	—
70	C₂×C₄	C₄×D₂₆₉	1	1	yes	characteristic
71	C₂×C₄	C₄×D₂₆₉	1	1	yes	characteristic
72	C₂×C₄	C₂×Dic₂₆₉	1	1	yes	characteristic
73	C₂×C₄	C₂²⋊D₂₆₉	1	1	yes	characteristic
74	C₂×C₄	D₁₀₇₆	1	1	yes	characteristic
75	C₂×C₄	Dic₅₃₈	1	1	yes	characteristic
76	C₈	—	538	2	no	—
77	C₂²	#16864379	1	1	yes	characteristic
78	C₂²	C₂²⋊Dic₂₆₉	1	1	yes	characteristic
79	C₂²	#16864381	1	1	yes	characteristic
80	C₂²	#16864387	1	1	yes	characteristic
81	C₂²	#16864389	1	1	yes	characteristic
82	C₂²	#16864390	1	1	yes	characteristic
83	C₂²	#16864392	1	1	yes	characteristic
84	C₄	—	4	2	no	—
85	C₄	—	1076	2	no	—
86	C₄	C₄×Dic₂₆₉	1	1	yes	characteristic
87	C₄	#16864379	1	1	yes	characteristic
88	C₄	#16864381	1	1	yes	characteristic
89	C₄	C₄⋊Dic₂₆₉	1	1	yes	characteristic
90	C₂	#22575360	1	1	yes	characteristic
91	C₂	#22575373	1	1	yes	characteristic
92	C₂	#22575374	1	1	yes	characteristic
93	C₂	#22575376	1	1	yes	characteristic
94	C₂	#22575377	1	1	yes	characteristic
95	C₂	#22575384	1	1	yes	characteristic
96	C₂	#22575386	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊D₂₆₉
commutator	C₄×C₂₆₉	C₄²
Frattini	C₂²×C₄	C₂×D₂₆₉
Fitting	C₂×C₄⋊C₄×C₂₆₉	C₂
socle, prime radical	C₂³×C₂₆₉	D₄
cosocle	C₂²×C₄×C₂₆₉	C₂²
hypercenter	C₂×C₄⋊C₄	D₂₆₉
hypocenter	C₂₆₉	#340
LCS	G ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
7. C₂×#14524130	C₄	semidirect
9. C₄⋊Dic₂₆₉	C₄	semidirect
13. C₄⋊C₄×C₂₆₉	C₄	semidirect
48. C₂₆₉	#340	semidirect
62. C₄⋊C₄	Dic₂₆₉	semidirect

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
C₂×C₄⋊C₄×C₂₆₉	2
#340	269