Subgroups — #30664008

Order 16768.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22553689	C₂	1	1	yes	maximal, characteristic
2	#22553065	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊C₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₀₄₈	—	2	1	no	—
5	C₂×D₁₀₄₈	C₄	1	2	yes	—
6	#16862243	—	2	1	no	—
7	C₄⋊(C₁₃₁⋊C₈)	—	2	1	no	—
8	C₂×C₁₆×C₁₃₁	—	2	1	no	—
9	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
10	C₂×D₅₂₄	—	2	2	no	—
11	C₂×D₅₂₄	—	4	1	no	—
12	C₂×D₅₂₄	—	4	1	no	—
13	D₁₀₄₈	—	4	2	no	—
14	D₁₀₄₈	—	4	1	no	—
15	D₁₀₄₈	—	4	1	no	—
16	C₂×C₁₃₁⋊C₈	—	4	1	no	—
17	C₂×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₁₃₁	D₄	1	1	yes	characteristic
19	C₄²×C₁₃₁	D₄	1	1	yes	characteristic
20	C₁₆×C₁₃₁	—	4	1	no	—
21	C₂²×D₁₃₁	—	4	2	no	—
22	D₅₂₄	—	4	2	no	—
23	D₅₂₄	—	4	2	no	—
24	D₅₂₄	—	8	2	no	—
25	D₅₂₄	—	8	1	no	—
26	D₅₂₄	—	8	1	no	—
27	C₁₃₁⋊C₈	—	8	1	no	—
28	C₂×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₃₁	D₈	1	1	yes	characteristic
30	C₂×C₄×C₁₃₁	SD₁₆	1	1	yes	characteristic
31	C₈×C₁₃₁	—	2	1	no	—
32	C₈×C₁₃₁	D₈	1	1	yes	characteristic
33	C₈×C₁₃₁	SD₁₆	1	1	yes	characteristic
34	C₂²×C₁₃₁	#102	1	1	yes	characteristic
35	C₂×D₁₃₁	—	8	2	no	—
36	C₂×D₁₃₁	—	8	2	no	—
37	C₂×D₁₃₁	—	8	2	no	—
38	C₄×C₁₃₁	—	2	1	no	—
39	C₄×C₁₃₁	#100	1	1	yes	characteristic
40	C₄×C₁₃₁	#102	1	1	yes	characteristic
41	C₄×C₁₃₁	D₁₆	1	1	yes	characteristic
42	C₄×C₁₃₁	SD₃₂	1	1	yes	characteristic
43	C₂×C₁₃₁	#331	1	1	yes	characteristic
44	C₂×C₁₃₁	#357	1	1	yes	characteristic
45	C₂×C₁₃₁	#361	1	1	yes	characteristic
46	D₁₃₁	—	16	2	no	—
47	C₁₃₁	#1361	1	1	yes	characteristic, complemented, Sylow
48	#1361	—	131	1	no	maximal, Sylow
49	#493	—	131	1	no	—
50	C₄⋊C₁₆	D₁₃₁	1	1	yes	characteristic, complemented
51	#335	—	131	1	no	—
52	C₂×D₈	—	262	1	no	—
53	C₂×D₈	—	131	2	no	—
54	#127	—	262	1	no	—
55	C₂×C₁₆	—	2	1	no	—
56	C₄⋊C₈	—	262	1	no	—
57	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
58	C₂×D₄	—	262	2	no	—
59	C₂×D₄	—	524	1	no	—
60	C₂×D₄	—	524	1	no	—
61	D₈	—	524	2	no	—
62	D₈	—	524	1	no	—
63	D₈	—	524	1	no	—
64	C₂×C₈	—	524	1	no	—
65	C₂×C₈	C₄×D₁₃₁	1	1	yes	characteristic
66	C₂×C₈	C₂²⋊D₁₃₁	1	1	yes	characteristic
67	C₄²	D₅₂₄	1	1	yes	characteristic
68	C₁₆	—	4	1	no	—
69	C₂³	—	524	2	no	—
70	D₄	—	524	2	no	—
71	D₄	—	524	2	no	—
72	D₄	—	1048	2	no	—
73	D₄	—	1048	1	no	—
74	D₄	—	1048	1	no	—
75	C₂×C₄	(C₂×D₁₃₁)⋊C₄	1	1	yes	characteristic
76	C₂×C₄	D₁₀₄₈	1	1	yes	characteristic
77	C₂×C₄	#14523502	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	1048	1	no	—
80	C₈	#14523498	1	1	yes	characteristic
81	C₈	#14523500	1	1	yes	characteristic
82	C₂²	—	1048	2	no	—
83	C₂²	—	1048	2	no	—
84	C₂²	—	1048	2	no	—
85	C₂²	#16862168	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	#16862162	1	1	yes	characteristic
88	C₄	#16862166	1	1	yes	characteristic
89	C₄	D₂₀₉₆	1	1	yes	characteristic
90	C₄	#16862191	1	1	yes	characteristic
91	C₂	—	2096	2	no	—
92	C₂	#22553055	1	1	yes	characteristic
93	C₂	#22553114	1	1	yes	characteristic
94	C₂	#22553126	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16862168
commutator	C₂×C₈×C₁₃₁	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₃₁
Fitting	C₄⋊C₁₆×C₁₃₁	C₂
socle	C₂²×C₁₃₁	#102
prime radical	#22553689	C₂
cosocle	C₄×C₈×C₁₃₁	C₂²
hypercenter	C₄⋊C₁₆	D₁₃₁
hypocenter	C₁₃₁	#1361
LCS	G ▷ C₂×C₈×C₁₃₁ ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. #22553065	C₂	semidirect
3. C₄⋊C₁₆×C₁₃₁	C₂	semidirect
47. C₁₃₁	#1361	semidirect
50. C₄⋊C₁₆	D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#22553689	2
#22553065	2
C₄⋊C₁₆×C₁₃₁	2
#1361	131