Subgroups — #30664010

Order 16768.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₁₃₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	#22553065	C₂	1	1	yes	maximal, characteristic
3	#494×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃₁⋊C₁₆	—	2	1	no	—
5	C₄⋊(C₁₃₁⋊C₈)	—	2	1	no	—
6	C₂×Q₁₆×C₁₃₁	—	2	1	no	—
7	C₂×Q₁₆×C₁₃₁	C₄	1	2	yes	—
8	C₄⋊Q₈×C₁₃₁	—	2	1	no	—
9	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₁₃₁⋊C₈	—	4	1	no	—
11	C₁₃₁⋊C₁₆	—	4	1	no	—
12	C₂×Q₈×C₁₃₁	—	2	2	no	—
13	Q₁₆×C₁₃₁	—	4	2	no	—
14	Q₁₆×C₁₃₁	—	4	1	no	—
15	Q₁₆×C₁₃₁	—	4	1	no	—
16	C₂×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₁₃₁	D₄	1	1	yes	characteristic
18	C₄⋊C₄×C₁₃₁	—	4	1	no	—
19	C₄⋊C₄×C₁₃₁	—	4	1	no	—
20	C₄²×C₁₃₁	D₄	1	1	yes	characteristic
21	C₁₃₁⋊C₈	—	8	1	no	—
22	Q₈×C₁₃₁	—	4	2	no	—
23	Q₈×C₁₃₁	—	4	2	no	—
24	C₂×C₄×C₁₃₁	—	4	2	no	—
25	C₂×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₁₃₁	D₈	1	1	yes	characteristic
27	C₂×C₄×C₁₃₁	SD₁₆	1	1	yes	characteristic
28	C₈×C₁₃₁	—	2	1	no	—
29	C₈×C₁₃₁	D₈	1	1	yes	characteristic
30	C₈×C₁₃₁	SD₁₆	1	1	yes	characteristic
31	C₂²×C₁₃₁	#102	1	1	yes	characteristic
32	C₄×C₁₃₁	—	2	1	no	—
33	C₄×C₁₃₁	—	8	2	no	—
34	C₄×C₁₃₁	#100	1	1	yes	characteristic
35	C₄×C₁₃₁	#102	1	1	yes	characteristic
36	C₄×C₁₃₁	SD₃₂	1	1	yes	characteristic
37	C₄×C₁₃₁	Q₃₂	1	1	yes	characteristic
38	C₂×C₁₃₁	#331	1	1	yes	characteristic
39	C₂×C₁₃₁	#358	1	1	yes	characteristic
40	C₂×C₁₃₁	#362	1	1	yes	characteristic
41	C₁₃₁	#1362	1	1	yes	characteristic, complemented, Sylow
42	#1362	—	131	1	no	maximal, Sylow
43	#494	D₁₃₁	1	1	yes	characteristic
44	C₄⋊C₁₆	—	131	1	no	—
45	#335	—	131	1	no	—
46	C₂×Q₁₆	—	2	1	no	—
47	C₂×Q₁₆	Dic₁₃₁	1	2	yes	—
48	C₄⋊Q₈	—	2	1	no	—
49	C₂×C₁₆	—	262	1	no	—
50	C₄⋊C₈	—	262	1	no	—
51	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
52	C₂×Q₈	—	2	2	no	—
53	Q₁₆	—	4	2	no	—
54	Q₁₆	—	4	1	no	—
55	Q₁₆	—	4	1	no	—
56	C₂×C₈	—	524	1	no	—
57	C₂×C₈	C₂×Dic₁₃₁	1	1	yes	characteristic
58	C₂×C₈	C₂²⋊D₁₃₁	1	1	yes	characteristic
59	C₄⋊C₄	—	4	1	no	—
60	C₄⋊C₄	—	4	1	no	—
61	C₄²	C₂²⋊D₁₃₁	1	1	yes	characteristic
62	C₁₆	—	524	1	no	—
63	Q₈	—	4	2	no	—
64	Q₈	—	4	2	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	C₂²⋊Dic₁₃₁	1	1	yes	characteristic
67	C₂×C₄	#14523498	1	1	yes	characteristic
68	C₂×C₄	#14523501	1	1	yes	characteristic
69	C₈	—	2	1	no	—
70	C₈	—	1048	1	no	—
71	C₈	#14523498	1	1	yes	characteristic
72	C₈	#14523501	1	1	yes	characteristic
73	C₂²	#16862167	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	8	2	no	—
76	C₄	#16862163	1	1	yes	characteristic
77	C₄	#16862167	1	1	yes	characteristic
78	C₄	#16862189	1	1	yes	characteristic
79	C₄	#16862192	1	1	yes	characteristic
80	C₂	#22553056	1	1	yes	characteristic
81	C₂	#22553116	1	1	yes	characteristic
82	C₂	#22553128	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16862167
commutator	C₂×C₈×C₁₃₁	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₃₁
Fitting	#494×C₁₃₁	C₂
socle, prime radical	C₂²×C₁₃₁	#102
cosocle	C₄×C₈×C₁₃₁	C₂²
hypercenter	#494	D₁₃₁
hypocenter	C₁₃₁	#1362
LCS	G ▷ C₂×C₈×C₁₃₁ ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #494

Complements

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

N	H ≅ G/N	type
41. C₁₃₁	#1362	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₁₃₁⋊C₁₆)	2
#22553065	2
#494×C₁₃₁	2
#1362	131