Subgroups — #46834423

Order 34688.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31774302	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₂₇₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#335×C₂₇₁	C₂	1	1	yes	maximal, characteristic
4	#22576160	—	2	1	no	—
5	C₂×C₂₇₁⋊C₁₆	—	2	1	no	—
6	#22576094	—	2	1	no	—
7	#22576094	C₄	1	2	yes	—
8	C₄⋊C₈×C₂₇₁	—	2	1	no	—
9	C₄×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
10	C₂×Dic₅₄₂	—	4	1	no	—
11	C₄⋊Dic₂₇₁	—	2	2	no	—
12	C₄⋊Dic₂₇₁	—	4	1	no	—
13	C₂₇₁⋊C₁₆	—	4	1	no	—
14	C₂×C₈×C₂₇₁	—	4	1	no	—
15	C₂×C₈×C₂₇₁	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₂₇₁	D₄	1	1	yes	characteristic
17	C₄²×C₂₇₁	D₄	1	1	yes	characteristic
18	Dic₅₄₂	—	8	2	no	—
19	C₂×Dic₂₇₁	—	4	2	no	—
20	C₂×C₄×C₂₇₁	C₂²⋊C₄	1	1	yes	characteristic
21	C₂×C₄×C₂₇₁	D₈	1	1	yes	characteristic
22	C₂×C₄×C₂₇₁	SD₁₆	1	1	yes	characteristic
23	C₈×C₂₇₁	—	2	1	no	—
24	C₈×C₂₇₁	—	8	1	no	—
25	C₈×C₂₇₁	SD₁₆	1	1	yes	characteristic
26	C₈×C₂₇₁	Q₁₆	1	1	yes	characteristic
27	C₂²×C₂₇₁	#102	1	1	yes	characteristic
28	C₄×C₂₇₁	—	2	1	no	—
29	C₄×C₂₇₁	#101	1	1	yes	characteristic
30	C₄×C₂₇₁	#103	1	1	yes	characteristic
31	C₄×C₂₇₁	SD₃₂	1	1	yes	characteristic
32	C₄×C₂₇₁	Q₃₂	1	1	yes	characteristic
33	Dic₂₇₁	—	8	2	no	—
34	C₂×C₂₇₁	#332	1	1	yes	characteristic
35	C₂×C₂₇₁	#358	1	1	yes	characteristic
36	C₂×C₂₇₁	#361	1	1	yes	characteristic
37	C₂₇₁	#1365	1	1	yes	characteristic, complemented, Sylow
38	#1365	—	271	1	no	maximal, Sylow
39	#500	—	271	1	no	—
40	C₄⋊C₁₆	—	271	1	no	—
41	#335	D₂₇₁	1	1	yes	characteristic
42	C₄⋊Q₈	—	542	1	no	—
43	C₂×C₁₆	—	542	1	no	—
44	#107	—	271	2	no	—
45	#107	—	542	1	no	—
46	C₄⋊C₈	—	2	1	no	—
47	C₄×C₈	C₂×D₂₇₁	1	1	yes	characteristic
48	C₂×Q₈	—	1084	1	no	—
49	C₂×C₈	—	4	1	no	—
50	C₂×C₈	C₄×D₂₇₁	1	1	yes	characteristic
51	C₂×C₈	D₁₀₈₄	1	1	yes	characteristic
52	C₄⋊C₄	—	542	2	no	—
53	C₄⋊C₄	—	1084	1	no	—
54	C₄²	C₂²⋊D₂₇₁	1	1	yes	characteristic
55	C₁₆	—	1084	1	no	—
56	Q₈	—	2168	2	no	—
57	C₂×C₄	—	1084	2	no	—
58	C₂×C₄	(C₂×D₂₇₁)⋊C₄	1	1	yes	characteristic
59	C₂×C₄	#16864639	1	1	yes	characteristic
60	C₂×C₄	#16864641	1	1	yes	characteristic
61	C₈	—	2	1	no	—
62	C₈	—	8	1	no	—
63	C₈	#16864643	1	1	yes	characteristic
64	C₈	Dic₁₀₈₄	1	1	yes	characteristic
65	C₂²	#22576080	1	1	yes	characteristic
66	C₄	—	2	1	no	—
67	C₄	—	2168	2	no	—
68	C₄	#22576078	1	1	yes	characteristic
69	C₄	#22576085	1	1	yes	characteristic
70	C₄	#22576104	1	1	yes	characteristic
71	C₄	#22576106	1	1	yes	characteristic
72	C₂	#31773644	1	1	yes	characteristic
73	C₂	#31773700	1	1	yes	characteristic
74	C₂	#31773709	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#22576080
commutator	C₂×C₈×C₂₇₁	C₂×C₄
Frattini	C₄×C₈	C₂×D₂₇₁
Fitting	#335×C₂₇₁	C₂
socle, prime radical	C₂²×C₂₇₁	#102
cosocle	C₄×C₈×C₂₇₁	C₂²
hypercenter	#335	D₂₇₁
hypocenter	C₂₇₁	#1365
LCS	G ▷ C₂×C₈×C₂₇₁ ▷ C₂×C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #335

Complements

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

N	H ≅ G/N	type
37. C₂₇₁	#1365	semidirect

Maximal subgroups

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

subgroup	index
#31774302	2
C₄⋊(C₂₇₁⋊C₁₆)	2
#335×C₂₇₁	2
#1365	271