Subgroups — #46834505

Order 34688.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31774313	C₂	1	1	yes	maximal, characteristic
2	#31773731	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₂×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#22576166	C₄	1	1	yes	characteristic
5	#22576100	C₄	1	1	yes	characteristic
6	#22576099	—	2	1	no	—
7	C₁₆×D₂₇₁	—	2	1	no	—
8	C₂₇₁⋊C₃₂	—	2	1	no	—
9	C₂×C₁₆×C₂₇₁	C₂²	1	1	yes	characteristic
10	C₃₂×C₂₇₁	—	2	1	no	—
11	#16864654	C₈	1	1	yes	characteristic
12	#16864638	C₈	1	1	yes	characteristic
13	#16864637	—	2	1	no	—
14	C₈×D₂₇₁	—	2	1	no	—
15	C₂₇₁⋊C₁₆	—	2	1	no	—
16	C₂×C₈×C₂₇₁	C₂×C₄	1	1	yes	characteristic
17	C₁₆×C₂₇₁	D₄	1	1	yes	characteristic
18	C₁₆×C₂₇₁	D₄	1	1	yes	characteristic
19	Dic₅₄₂	C₁₆	1	1	yes	characteristic
20	D₁₀₈₄	C₁₆	1	1	yes	characteristic
21	C₂²⋊D₂₇₁	—	2	1	no	—
22	C₄×D₂₇₁	—	2	1	no	—
23	C₂₇₁⋊C₈	—	2	1	no	—
24	C₂×C₄×C₂₇₁	C₂×C₈	1	1	yes	characteristic
25	C₈×C₂₇₁	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₂₇₁	M₁₆	1	1	yes	characteristic
27	C₂²×C₂₇₁	M₃₂	1	1	yes	characteristic
28	C₂×D₂₇₁	—	2	1	no	—
29	C₄×C₂₇₁	C₂²⋊C₈	1	1	yes	characteristic
30	C₄×C₂₇₁	C₂×C₁₆	1	1	yes	characteristic
31	Dic₂₇₁	—	2	1	no	—
32	C₂×C₂₇₁	—	2	1	no	—
33	C₂×C₂₇₁	C₂²⋊C₁₆	1	1	yes	characteristic
34	D₂₇₁	—	4	1	no	—
35	C₂₇₁	#1401	1	1	yes	characteristic, complemented, Sylow
36	#1401	—	271	1	no	maximal, Sylow
37	#504	—	271	1	no	—
38	M₆₄	—	271	1	no	—
39	C₂×C₃₂	D₂₇₁	1	1	yes	characteristic, complemented
40	#131	—	271	1	no	—
41	M₃₂	—	271	1	no	—
42	M₃₂	—	542	1	no	—
43	C₂×C₁₆	—	542	1	no	—
44	C₂×C₁₆	C₂×D₂₇₁	1	1	yes	characteristic
45	C₃₂	—	2	1	no	—
46	C₃₂	—	542	1	no	—
47	#41	—	271	1	no	—
48	M₁₆	—	271	1	no	—
49	M₁₆	—	542	1	no	—
50	C₂×C₈	—	542	1	no	—
51	C₂×C₈	C₄×D₂₇₁	1	1	yes	characteristic
52	C₁₆	—	542	1	no	—
53	C₁₆	C₂²⋊D₂₇₁	1	1	yes	characteristic
54	C₁₆	D₁₀₈₄	1	1	yes	characteristic
55	Q₈	—	271	1	no	—
56	D₄	—	271	1	no	—
57	D₄	—	542	1	no	—
58	C₂×C₄	—	542	1	no	—
59	C₂×C₄	C₈×D₂₇₁	1	1	yes	characteristic
60	C₈	—	542	1	no	—
61	C₈	(C₂×D₂₇₁)⋊C₄	1	1	yes	characteristic
62	C₈	#16864637	1	1	yes	characteristic
63	C₂²	—	542	1	no	—
64	C₂²	#22576099	1	1	yes	characteristic
65	C₄	—	542	1	no	—
66	C₄	(C₂×D₂₇₁)⋊C₈	1	1	yes	characteristic
67	C₄	C₁₆×D₂₇₁	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	1084	1	no	—
70	C₂	(C₂×D₂₇₁)⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	D₁₀₈₄
commutator	C₄×C₂₇₁	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₂₇₁
Fitting	C₂×C₃₂×C₂₇₁	C₂
socle	C₂×C₂₇₁	C₂²⋊C₁₆
prime radical	#16864654	C₈
cosocle	C₂×C₁₆×C₂₇₁	C₂²
hypercenter	C₂×C₃₂	D₂₇₁
hypocenter	C₂₇₁	#1401
LCS	G ▷ C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
2. #31773731	C₂	semidirect
3. C₂×C₃₂×C₂₇₁	C₂	semidirect
35. C₂₇₁	#1401	semidirect
39. C₂×C₃₂	D₂₇₁	semidirect

Maximal subgroups

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

subgroup	index
#31774313	2
#31773731	2
C₂×C₃₂×C₂₇₁	2
#1401	271