Subgroups — #30664136

Order 16768.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22553109	C₂	1	1	yes	maximal, characteristic
2	#22553107	C₂	1	1	yes	maximal, characteristic
3	#488×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#16862165	—	2	1	no	—
5	#16862161	—	2	1	no	—
6	#125×C₁₃₁	C₄	1	1	yes	characteristic, complemented
7	#124×C₁₃₁	C₂²	1	1	yes	characteristic
8	#103×C₁₃₁	—	2	1	no	—
9	#102×C₁₃₁	—	2	1	no	—
10	C₄×C₈×C₁₃₁	C₄	1	1	yes	characteristic, complemented
11	#14523497	—	4	1	no	—
12	C₂²⋊Dic₁₃₁	—	4	1	no	—
13	C₂×Q₈×C₁₃₁	D₄	1	1	yes	characteristic
14	C₂×D₄×C₁₃₁	D₄	1	1	yes	characteristic
15	C₂×C₈×C₁₃₁	—	2	1	no	—
16	C₄⋊C₄×C₁₃₁	—	2	1	no	—
17	C₄⋊C₄×C₁₃₁	—	4	1	no	—
18	C₂²⋊C₄×C₁₃₁	—	4	1	no	—
19	C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₁₃₁	—	8	1	no	—
21	C₁₃₁⋊C₈	—	8	1	no	—
22	C₂³×C₁₃₁	—	2	1	no	—
23	Q₈×C₁₃₁	—	4	1	no	—
24	D₄×C₁₃₁	—	4	1	no	—
25	C₂×C₄×C₁₃₁	—	2	1	no	—
26	C₂×C₄×C₁₃₁	—	2	1	no	—
27	C₂×C₄×C₁₃₁	—	4	1	no	—
28	C₂×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
29	C₈×C₁₃₁	—	4	1	no	—
30	C₂²×C₁₃₁	—	4	1	no	—
31	C₂²×C₁₃₁	—	8	1	no	—
32	C₂²×C₁₃₁	#99	1	1	yes	characteristic
33	C₄×C₁₃₁	—	2	1	no	—
34	C₄×C₁₃₁	—	4	1	no	—
35	C₄×C₁₃₁	—	4	1	no	—
36	C₄×C₁₃₁	—	8	1	no	—
37	Dic₁₃₁	—	16	1	no	—
38	C₂×C₁₃₁	—	2	1	no	—
39	C₂×C₁₃₁	—	8	1	no	—
40	C₂×C₁₃₁	#354	1	1	yes	characteristic
41	C₁₃₁	#1414	1	1	yes	characteristic, complemented, Sylow
42	#1414	—	131	1	no	maximal, Sylow
43	#488	D₁₃₁	1	1	yes	characteristic
44	#355	—	131	1	no	—
45	#354	—	131	1	no	—
46	#125	Dic₁₃₁	1	1	yes	characteristic, complemented
47	#124	C₂×D₁₃₁	1	1	yes	characteristic
48	#103	—	2	1	no	—
49	#102	—	2	1	no	—
50	#101	—	262	1	no	—
51	#99	—	262	1	no	—
52	C₄×C₈	Dic₁₃₁	1	1	yes	characteristic, complemented
53	C₂×Q₈	C₂²⋊D₁₃₁	1	1	yes	characteristic
54	C₂×D₄	C₂²⋊D₁₃₁	1	1	yes	characteristic
55	M₁₆	—	524	1	no	—
56	C₂×C₈	—	2	1	no	—
57	C₄⋊C₄	—	2	1	no	—
58	C₄⋊C₄	—	4	1	no	—
59	C₂²⋊C₄	—	4	1	no	—
60	C₂²⋊C₄	—	524	1	no	—
61	C₄²	C₂×Dic₁₃₁	1	1	yes	characteristic
62	C₂³	—	2	1	no	—
63	Q₈	—	4	1	no	—
64	D₄	—	4	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	4	1	no	—
68	C₂×C₄	—	1048	1	no	—
69	C₂×C₄	C₂²⋊Dic₁₃₁	1	1	yes	characteristic
70	C₈	—	4	1	no	—
71	C₈	—	1048	1	no	—
72	C₂²	—	4	1	no	—
73	C₂²	—	8	1	no	—
74	C₂²	#16862161	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	4	1	no	—
78	C₄	—	8	1	no	—
79	C₄	—	2096	1	no	—
80	C₂	—	2	1	no	—
81	C₂	—	8	1	no	—
82	C₂	#22553107	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22553107
commutator	C₄²×C₁₃₁	C₂×C₄
Frattini	#124	C₂×D₁₃₁
Fitting	#488×C₁₃₁	C₂
socle	C₂×C₁₃₁	#354
prime radical	C₂×D₄×C₁₃₁	D₄
cosocle	#124×C₁₃₁	C₂²
hypercenter	#488	D₁₃₁
hypocenter	C₁₃₁	#1414
LCS	G ▷ C₄²×C₁₃₁ ▷ C₂×C₄×C₁₃₁ ▷ C₂²×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #488

Complements

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

N	H ≅ G/N	type
6. #125×C₁₃₁	C₄	semidirect
10. C₄×C₈×C₁₃₁	C₄	semidirect
41. C₁₃₁	#1414	semidirect
46. #125	Dic₁₃₁	semidirect
52. C₄×C₈	Dic₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#22553109	2
#22553107	2
#488×C₁₃₁	2
#1414	131