Subgroups — #30664124

Order 16768.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22553105	C₂	1	2	yes	maximal, complemented
2	#493×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#16862161	—	2	2	no	—
4	C₂×D₈×C₁₃₁	—	2	2	no	—
5	#127×C₁₃₁	C₄	1	1	yes	characteristic, complemented
6	#127×C₁₃₁	C₂²	1	1	yes	characteristic
7	C₄×C₈×C₁₃₁	C₄	1	1	yes	characteristic, complemented
8	C₂²⋊Dic₁₃₁	—	4	2	no	—
9	C₂×D₄×C₁₃₁	—	2	1	no	—
10	C₂×D₄×C₁₃₁	—	4	1	no	—
11	C₂×D₄×C₁₃₁	—	4	1	no	—
12	C₂×D₄×C₁₃₁	D₄	1	2	yes	—
13	D₈×C₁₃₁	—	4	4	no	—
14	C₂×C₈×C₁₃₁	—	2	1	no	—
15	C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
16	C₂×Dic₁₃₁	—	8	2	no	—
17	C₂³×C₁₃₁	—	2	2	no	—
18	C₂³×C₁₃₁	—	4	1	no	—
19	D₄×C₁₃₁	—	4	2	no	—
20	D₄×C₁₃₁	—	4	2	no	—
21	D₄×C₁₃₁	—	8	2	no	—
22	D₄×C₁₃₁	—	8	2	no	—
23	C₂×C₄×C₁₃₁	—	2	1	no	—
24	C₂×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
25	C₈×C₁₃₁	—	2	2	no	—
26	C₂²×C₁₃₁	—	4	2	no	—
27	C₂²×C₁₃₁	—	8	2	no	—
28	C₂²×C₁₃₁	—	8	2	no	—
29	C₂²×C₁₃₁	—	8	1	no	—
30	C₂²×C₁₃₁	#99	1	1	yes	characteristic
31	C₄×C₁₃₁	—	2	2	no	—
32	C₄×C₁₃₁	—	2	1	no	—
33	Dic₁₃₁	—	16	2	no	—
34	C₂×C₁₃₁	—	2	1	no	—
35	C₂×C₁₃₁	—	8	2	no	—
36	C₂×C₁₃₁	—	16	1	no	—
37	C₂×C₁₃₁	#353	1	1	yes	characteristic
38	C₁₃₁	#1408	1	1	yes	characteristic, complemented, Sylow
39	#1408	—	131	1	no	maximal, Sylow
40	#493	D₁₃₁	1	1	yes	characteristic
41	#353	—	131	2	no	—
42	C₂×D₈	—	2	2	no	—
43	#127	Dic₁₃₁	1	1	yes	characteristic, complemented
44	#127	C₂×D₁₃₁	1	1	yes	characteristic
45	#99	—	262	2	no	—
46	C₄×C₈	Dic₁₃₁	1	1	yes	characteristic, complemented
47	C₂×D₄	—	2	1	no	—
48	C₂×D₄	—	4	1	no	—
49	C₂×D₄	—	4	1	no	—
50	C₂×D₄	C₂²⋊D₁₃₁	1	2	yes	—
51	D₈	—	4	4	no	—
52	C₂×C₈	—	2	1	no	—
53	C₂²⋊C₄	—	524	2	no	—
54	C₄²	C₂×Dic₁₃₁	1	1	yes	characteristic
55	C₂³	—	2	2	no	—
56	C₂³	—	4	1	no	—
57	D₄	—	4	2	no	—
58	D₄	—	4	2	no	—
59	D₄	—	8	2	no	—
60	D₄	—	8	2	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	1048	2	no	—
63	C₂×C₄	C₂²⋊Dic₁₃₁	1	1	yes	characteristic
64	C₈	—	2	2	no	—
65	C₂²	—	4	2	no	—
66	C₂²	—	8	2	no	—
67	C₂²	—	8	2	no	—
68	C₂²	—	8	1	no	—
69	C₂²	#16862161	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	2	1	no	—
72	C₄	—	2096	2	no	—
73	C₂	—	2	1	no	—
74	C₂	—	8	2	no	—
75	C₂	—	16	1	no	—
76	C₂	#22553105	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22553105
commutator	C₄²×C₁₃₁	C₂×C₄
Frattini	#127	C₂×D₁₃₁
Fitting, prime radical	#493×C₁₃₁	C₂
socle	C₂×C₁₃₁	#353
cosocle	#127×C₁₃₁	C₂²
hypercenter	#493	D₁₃₁
hypocenter	C₁₃₁	#1408
LCS	G ▷ C₄²×C₁₃₁ ▷ C₂×C₄×C₁₃₁ ▷ C₂²×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #127 ◁ #493

Complements

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

N	H ≅ G/N	type
1. #22553105	C₂	semidirect
5. #127×C₁₃₁	C₄	semidirect
7. C₄×C₈×C₁₃₁	C₄	semidirect
38. C₁₃₁	#1408	semidirect
43. #127	Dic₁₃₁	semidirect
46. 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
#22553105	2
#493×C₁₃₁	2
#1408	131