Subgroups — #30668207

Order 16768.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22553714	C₂	1	1	yes	maximal, characteristic
2	#22553692	C₂	1	1	yes	maximal, characteristic
3	#22553117	C₂	1	4	yes	maximal
4	#346×C₁₃₁	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₅₂₄	—	2	1	no	—
6	C₂×Dic₅₂₄	C₂²	1	2	yes	—
7	#16862246	—	2	1	no	—
8	#16862238	—	2	1	no	—
9	#16862180	C₂²	1	2	yes	—
10	#16862178	—	2	1	no	—
11	C₂×C₁₆×C₁₃₁	C₂²	1	2	yes	—
12	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₂×Dic₂₆₂	—	2	2	no	—
14	Dic₅₂₄	—	2	4	no	—
15	Dic₅₂₄	—	4	2	no	—
16	C₄⋊Dic₁₃₁	—	2	2	no	—
17	C₄⋊Dic₁₃₁	—	4	2	no	—
18	Dic₁₃₁⋊C₄	—	4	2	no	—
19	C₂×C₈×C₁₃₁	D₄	1	1	yes	characteristic
20	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
21	C₄²×C₁₃₁	D₄	1	1	yes	characteristic
22	C₁₆×C₁₃₁	—	2	2	no	—
23	Dic₂₆₂	—	4	2	no	—
24	Dic₂₆₂	—	4	2	no	—
25	C₂×Dic₁₃₁	—	4	2	no	—
26	C₂×Dic₁₃₁	—	4	2	no	—
27	C₂×C₄×C₁₃₁	D₈	1	2	yes	—
28	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
29	C₈×C₁₃₁	SD₁₆	1	2	yes	—
30	C₈×C₁₃₁	#41	1	2	yes	—
31	C₂²×C₁₃₁	C₂×D₈	1	1	yes	characteristic
32	C₄×C₁₃₁	—	2	2	no	—
33	C₄×C₁₃₁	#124	1	1	yes	characteristic
34	C₄×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
35	Dic₁₃₁	—	8	2	no	—
36	Dic₁₃₁	—	8	2	no	—
37	C₂×C₁₃₁	#486	1	1	yes	characteristic
38	C₂×C₁₃₁	#510	1	2	yes	—
39	C₁₃₁	#2245	1	1	yes	characteristic, complemented, Sylow
40	#2245	—	131	1	no	maximal, Sylow
41	#499	—	131	1	no	—
42	#494	—	131	1	no	—
43	#358	—	131	4	no	—
44	#346	D₁₃₁	1	1	yes	characteristic
45	C₂×Q₁₆	—	262	1	no	—
46	C₂×Q₁₆	—	131	2	no	—
47	C₄⋊Q₈	—	262	1	no	—
48	#125	—	262	1	no	—
49	C₂×C₁₆	C₂×D₁₃₁	1	2	yes	—
50	#107	—	131	2	no	—
51	#106	—	262	1	no	—
52	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
53	C₂×Q₈	—	262	2	no	—
54	Q₁₆	—	262	4	no	—
55	Q₁₆	—	524	2	no	—
56	C₂×C₈	D₅₂₄	1	1	yes	characteristic
57	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
58	C₄⋊C₄	—	262	2	no	—
59	C₄⋊C₄	—	524	2	no	—
60	C₄⋊C₄	—	524	2	no	—
61	C₄²	D₅₂₄	1	1	yes	characteristic
62	C₁₆	—	2	2	no	—
63	Q₈	—	524	2	no	—
64	Q₈	—	524	2	no	—
65	C₂×C₄	—	524	2	no	—
66	C₂×C₄	—	524	2	no	—
67	C₂×C₄	D₁₀₄₈	1	2	yes	—
68	C₂×C₄	C₂×D₅₂₄	1	1	yes	characteristic
69	C₈	#14523502	1	2	yes	—
70	C₈	#14523513	1	2	yes	—
71	C₂²	C₂×D₁₀₄₈	1	1	yes	characteristic
72	C₄	—	2	2	no	—
73	C₄	—	1048	2	no	—
74	C₄	—	1048	2	no	—
75	C₄	#16862234	1	1	yes	characteristic
76	C₄	C₂×#14523502	1	1	yes	characteristic
77	C₂	#22553659	1	1	yes	characteristic
78	C₂	#22553755	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₀₄₈
commutator, cosocle	C₂×C₈×C₁₃₁	C₂³
Frattini	C₂×C₈	C₂²×D₁₃₁
Fitting	#346×C₁₃₁	C₂
socle, prime radical	C₂²×C₁₃₁	C₂×D₈
hypercenter	#346	D₁₃₁
hypocenter	C₁₃₁	#2245
LCS	G ▷ C₂×C₈×C₁₃₁ ▷ C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
39. C₁₃₁	#2245	semidirect

Maximal subgroups

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

subgroup	index
#22553714	2
#22553692	2
#22553117	2
#346×C₁₃₁	2
#2245	131