Subgroups — #30668274

Order 16768.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22553744	C₂	1	1	yes	maximal, characteristic, complemented
2	#22553150	C₂	1	2	yes	maximal, complemented
3	#22553147	C₂	1	2	yes	maximal, complemented
4	#22553146	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#16862268	—	2	1	no	—
7	Dic₁₀₄₈	C₂²	1	1	yes	characteristic, complemented
8	#16862191	—	2	1	no	—
9	D₂₀₉₆	C₂²	1	1	yes	characteristic, complemented
10	C₁₃₁⋊C₃₂	C₂²	1	2	yes	complemented
11	C₂×D₈×C₁₃₁	—	2	1	no	—
12	D₁₆×C₁₃₁	—	2	1	no	—
13	D₁₆×C₁₃₁	C₂²	1	2	yes	—
14	C₂×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
15	#14523513	—	4	1	no	—
16	Dic₅₂₄	—	2	1	no	—
17	#14523502	—	4	1	no	—
18	D₁₀₄₈	—	2	1	no	—
19	C₂×D₄×C₁₃₁	—	4	1	no	—
20	D₈×C₁₃₁	—	2	2	no	—
21	D₈×C₁₃₁	—	4	1	no	—
22	C₂×C₈×C₁₃₁	D₄	1	1	yes	characteristic
23	C₁₆×C₁₃₁	D₄	1	1	yes	characteristic
24	C₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
25	Dic₂₆₂	—	4	1	no	—
26	D₅₂₄	—	4	1	no	—
27	C₂²⋊D₁₃₁	—	8	1	no	—
28	C₄×D₁₃₁	—	8	1	no	—
29	C₂³×C₁₃₁	—	8	1	no	—
30	D₄×C₁₃₁	—	4	2	no	—
31	D₄×C₁₃₁	—	8	1	no	—
32	C₂×C₄×C₁₃₁	D₈	1	1	yes	characteristic
33	C₈×C₁₃₁	D₈	1	1	yes	characteristic
34	C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₁₃₁	—	8	2	no	—
36	C₂²×C₁₃₁	—	16	2	no	—
37	C₂²×C₁₃₁	D₁₆	1	1	yes	characteristic
38	C₂×D₁₃₁	—	8	1	no	—
39	C₄×C₁₃₁	D₁₆	1	1	yes	characteristic
40	C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
41	Dic₁₃₁	—	8	1	no	—
42	C₂×C₁₃₁	—	2	1	no	—
43	C₂×C₁₃₁	—	16	2	no	—
44	C₂×C₁₃₁	C₂×D₁₆	1	1	yes	characteristic
45	D₁₃₁	—	16	1	no	—
46	C₁₃₁	#2263	1	1	yes	characteristic, complemented, Sylow
47	#2263	—	131	1	no	maximal, Sylow
48	#508	—	131	1	no	—
49	C₂×D₁₆	D₁₃₁	1	1	yes	characteristic, complemented
50	SD₆₄	—	131	2	no	—
51	D₃₂	—	131	2	no	—
52	M₆₄	—	131	1	no	—
53	#135	—	262	1	no	—
54	C₂×D₈	—	2	1	no	—
55	Q₃₂	—	131	1	no	—
56	SD₃₂	—	262	1	no	—
57	D₁₆	—	2	1	no	—
58	D₁₆	—	131	1	no	—
59	D₁₆	C₂×D₁₃₁	1	2	yes	—
60	C₂×C₁₆	C₂×D₁₃₁	1	1	yes	characteristic
61	C₃₂	—	131	2	no	—
62	#41	—	524	1	no	—
63	C₂×D₄	—	4	1	no	—
64	Q₁₆	—	262	1	no	—
65	SD₁₆	—	524	1	no	—
66	D₈	—	2	2	no	—
67	D₈	—	4	1	no	—
68	D₈	—	262	1	no	—
69	C₂×C₈	C₂²⋊D₁₃₁	1	1	yes	characteristic
70	C₁₆	C₂²⋊D₁₃₁	1	1	yes	characteristic
71	C₁₆	C₂²×D₁₃₁	1	1	yes	characteristic
72	C₂³	—	8	1	no	—
73	Q₈	—	524	1	no	—
74	D₄	—	4	2	no	—
75	D₄	—	8	1	no	—
76	D₄	—	524	1	no	—
77	D₄	—	1048	1	no	—
78	C₂×C₄	—	1048	1	no	—
79	C₂×C₄	#14523498	1	1	yes	characteristic
80	C₈	#14523498	1	1	yes	characteristic
81	C₈	C₂×C₂²⋊D₁₃₁	1	1	yes	characteristic
82	C₂²	—	8	2	no	—
83	C₂²	—	16	2	no	—
84	C₂²	—	1048	1	no	—
85	C₂²	#16862187	1	1	yes	characteristic
86	C₄	—	1048	1	no	—
87	C₄	#16862187	1	1	yes	characteristic
88	C₄	C₂×#14523498	1	1	yes	characteristic
89	C₂	—	2	1	no	—
90	C₂	—	16	2	no	—
91	C₂	—	2096	1	no	—
92	C₂	C₂×#16862187	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#16862187
commutator, cosocle	C₁₆×C₁₃₁	C₂³
Frattini	C₁₆	C₂²×D₁₃₁
Fitting	C₂×D₁₆×C₁₃₁	C₂
socle	C₂×C₁₃₁	C₂×D₁₆
hypercenter	C₂×D₁₆	D₁₃₁
hypocenter	C₁₃₁	#2263
LCS	G ▷ C₁₆×C₁₃₁ ▷ C₈×C₁₃₁ ▷ C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×D₁₆

Complements

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

N	H ≅ G/N	type
1. #22553744	C₂	semidirect
2. #22553150	C₂	semidirect
3. #22553147	C₂	semidirect
4. #22553146	C₂	semidirect
5. C₂×D₁₆×C₁₃₁	C₂	semidirect
7. Dic₁₀₄₈	C₂²	semidirect
9. D₂₀₉₆	C₂²	semidirect
10. C₁₃₁⋊C₃₂	C₂²	semidirect
46. C₁₃₁	#2263	semidirect
49. C₂×D₁₆	D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#22553744	2
#22553150	2
#22553147	2
#22553146	2
C₂×D₁₆×C₁₃₁	2
#2263	131