Subgroups — #30668254

Order 16768.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22553728	C₂	1	1	yes	maximal, characteristic
2	#22553146	C₂	1	1	yes	maximal, characteristic, complemented
3	#22553145	C₂	1	2	yes	maximal, complemented
4	C₃₂×D₁₃₁	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₂×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#16862252	C₄	1	1	yes	characteristic
7	#16862186	C₄	1	1	yes	characteristic
8	#16862185	C₄	1	2	yes	—
9	C₁₆×D₁₃₁	C₂²	1	2	yes	—
10	C₁₃₁⋊C₃₂	C₂²	1	2	yes	—
11	C₂×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
12	C₃₂×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₃₂×C₁₃₁	C₂²	1	1	yes	characteristic
14	#14523513	C₈	1	1	yes	characteristic
15	#14523497	C₈	1	1	yes	characteristic
16	#14523496	C₈	1	2	yes	—
17	C₈×D₁₃₁	C₂×C₄	1	2	yes	—
18	C₁₃₁⋊C₁₆	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
20	C₁₆×C₁₃₁	C₂×C₄	1	1	yes	characteristic
21	C₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
22	Dic₂₆₂	C₁₆	1	1	yes	characteristic
23	D₅₂₄	C₁₆	1	1	yes	characteristic
24	C₂²⋊D₁₃₁	C₁₆	1	2	yes	—
25	C₄×D₁₃₁	C₂×C₈	1	2	yes	—
26	C₁₃₁⋊C₈	C₂×C₈	1	2	yes	—
27	C₂×C₄×C₁₃₁	C₂×C₈	1	1	yes	characteristic
28	C₈×C₁₃₁	C₂×C₈	1	1	yes	characteristic
29	C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₁₃₁	C₂×C₁₆	1	1	yes	characteristic
31	C₂×D₁₃₁	C₂×C₁₆	1	2	yes	—
32	C₄×C₁₃₁	C₂×C₁₆	1	1	yes	characteristic
33	C₄×C₁₃₁	C₂²×C₈	1	1	yes	characteristic
34	Dic₁₃₁	C₂×C₁₆	1	2	yes	—
35	C₂×C₁₃₁	—	2	1	no	—
36	C₂×C₁₃₁	C₂²×C₁₆	1	1	yes	characteristic
37	D₁₃₁	—	2	2	no	—
38	C₁₃₁	#2258	1	1	yes	characteristic, complemented, Sylow
39	#2258	—	131	1	no	maximal, Sylow
40	#504	—	131	1	no	—
41	M₆₄	—	131	1	no	—
42	M₆₄	—	131	2	no	—
43	C₂×C₃₂	—	131	2	no	—
44	C₂×C₃₂	D₁₃₁	1	1	yes	characteristic, complemented
45	#131	—	131	1	no	—
46	M₃₂	—	131	1	no	—
47	M₃₂	—	131	2	no	—
48	C₂×C₁₆	—	131	2	no	—
49	C₂×C₁₆	C₂×D₁₃₁	1	1	yes	characteristic
50	C₃₂	—	131	2	no	—
51	C₃₂	C₂×D₁₃₁	1	1	yes	characteristic
52	C₃₂	C₂×D₁₃₁	1	1	yes	characteristic
53	#41	—	131	1	no	—
54	M₁₆	—	131	1	no	—
55	M₁₆	—	131	2	no	—
56	C₂×C₈	—	131	2	no	—
57	C₂×C₈	C₄×D₁₃₁	1	1	yes	characteristic
58	C₁₆	—	131	2	no	—
59	C₁₆	C₄×D₁₃₁	1	1	yes	characteristic
60	C₁₆	C₂²×D₁₃₁	1	1	yes	characteristic
61	Q₈	—	131	1	no	—
62	D₄	—	131	2	no	—
63	D₄	—	131	1	no	—
64	C₂×C₄	—	131	2	no	—
65	C₂×C₄	C₈×D₁₃₁	1	1	yes	characteristic
66	C₈	—	131	2	no	—
67	C₈	C₈×D₁₃₁	1	1	yes	characteristic
68	C₈	C₂×C₄×D₁₃₁	1	1	yes	characteristic
69	C₂²	—	131	2	no	—
70	C₂²	C₁₆×D₁₃₁	1	1	yes	characteristic
71	C₄	—	131	2	no	—
72	C₄	C₁₆×D₁₃₁	1	1	yes	characteristic
73	C₄	C₂×C₈×D₁₃₁	1	1	yes	characteristic
74	C₂	—	2	1	no	—
75	C₂	—	262	2	no	—
76	C₂	C₂×C₁₆×D₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₃₂	C₂×D₁₃₁
commutator, socle	C₂×C₁₃₁	C₂²×C₁₆
Frattini	C₁₆	C₂²×D₁₃₁
Fitting	C₂×C₃₂×C₁₃₁	C₂
prime radical	#14523513	C₈
cosocle	C₁₆×C₁₃₁	C₂³
hypercenter	C₂×C₃₂	D₁₃₁
hypocenter	C₁₃₁	#2258
LCS	G ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₃₂ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
2. #22553146	C₂	semidirect
3. #22553145	C₂	semidirect
4. C₃₂×D₁₃₁	C₂	semidirect
5. C₂×C₃₂×C₁₃₁	C₂	semidirect
38. C₁₃₁	#2258	semidirect
44. C₂×C₃₂	D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#22553728	2
#22553146	2
#22553145	2
C₃₂×D₁₃₁	2
C₂×C₃₂×C₁₃₁	2
#2258	131