Subgroups — #46838638

Order 34688.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31774301	C₂	1	2	yes	maximal
2	#31773723	C₂	1	1	yes	maximal, characteristic
3	#31773722	C₂	1	2	yes	maximal
4	C₁₆×Dic₂₇₁	C₂	1	1	yes	maximal, characteristic
5	#366×C₂₇₁	C₂	1	1	yes	maximal, characteristic
6	#22576159	—	2	2	no	—
7	C₂×C₂₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
8	#22576094	C₂²	1	2	yes	—
9	#22576093	—	2	2	no	—
10	C₈×Dic₂₇₁	C₂²	1	1	yes	characteristic
11	C₂×C₁₆×C₂₇₁	C₂²	1	1	yes	characteristic
12	#107×C₂₇₁	C₂²	1	2	yes	—
13	C₂×Dic₅₄₂	—	4	2	no	—
14	C₂×C₂₇₁⋊C₈	D₄	1	1	yes	characteristic
15	C₄⋊Dic₂₇₁	—	2	2	no	—
16	Dic₂₇₁⋊C₄	—	4	2	no	—
17	C₄×Dic₂₇₁	D₄	1	1	yes	characteristic
18	C₂₇₁⋊C₁₆	Q₈	1	2	yes	complemented
19	C₂×C₈×C₂₇₁	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₂₇₁	—	2	2	no	—
21	C₁₆×C₂₇₁	Q₈	1	2	yes	complemented
22	Dic₅₄₂	—	8	4	no	—
23	C₂×Dic₂₇₁	—	4	2	no	—
24	C₂×Dic₂₇₁	D₈	1	1	yes	characteristic
25	C₂×Dic₂₇₁	D₈	1	1	yes	characteristic
26	C₂₇₁⋊C₈	Q₁₆	1	2	yes	—
27	C₂×C₄×C₂₇₁	—	4	2	no	—
28	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
29	C₈×C₂₇₁	C₂×Q₈	1	1	yes	characteristic
30	C₈×C₂₇₁	C₂×Q₈	1	1	yes	characteristic
31	C₂²×C₂₇₁	C₂×D₈	1	1	yes	characteristic
32	C₄×C₂₇₁	—	8	2	no	—
33	C₄×C₂₇₁	C₄⋊Q₈	1	1	yes	characteristic
34	C₄×C₂₇₁	C₂×Q₁₆	1	1	yes	characteristic
35	Dic₂₇₁	—	8	2	no	—
36	Dic₂₇₁	D₁₆	1	2	yes	—
37	Dic₂₇₁	Q₃₂	1	2	yes	—
38	C₂×C₂₇₁	#500	1	1	yes	characteristic
39	C₂×C₂₇₁	C₂×D₁₆	1	1	yes	characteristic
40	C₂×C₂₇₁	C₂×Q₃₂	1	1	yes	characteristic
41	C₂₇₁	#2252	1	1	yes	characteristic, complemented, Sylow
42	#2252	—	271	1	no	maximal, Sylow
43	#500	—	271	2	no	—
44	#366	—	271	1	no	—
45	#366	—	271	2	no	—
46	#366	D₂₇₁	1	1	yes	characteristic
47	C₄×C₁₆	—	271	1	no	—
48	C₄⋊Q₈	—	542	2	no	—
49	C₂×C₁₆	—	271	1	no	—
50	C₂×C₁₆	C₂×D₂₇₁	1	1	yes	characteristic
51	#107	—	271	2	no	—
52	#107	—	542	2	no	—
53	#107	C₂×D₂₇₁	1	2	yes	—
54	C₄×C₈	—	271	1	no	—
55	C₂×Q₈	—	1084	2	no	—
56	C₂×C₈	—	271	1	no	—
57	C₂×C₈	C₂²×D₂₇₁	1	1	yes	characteristic
58	C₄⋊C₄	—	2	2	no	—
59	C₄⋊C₄	—	542	2	no	—
60	C₄⋊C₄	—	1084	2	no	—
61	C₄²	—	271	1	no	—
62	C₁₆	—	271	2	no	—
63	C₁₆	Dic₅₄₂	1	2	yes	complemented
64	Q₈	—	2168	4	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	271	1	no	—
67	C₂×C₄	—	271	1	no	—
68	C₂×C₄	—	1084	2	no	—
69	C₂×C₄	D₄×D₂₇₁	1	1	yes	characteristic
70	C₈	—	271	2	no	—
71	C₈	Q₈×D₂₇₁	1	1	yes	characteristic
72	C₈	C₂×Dic₅₄₂	1	1	yes	characteristic
73	C₂²	D₈×D₂₇₁	1	1	yes	characteristic
74	C₄	—	8	2	no	—
75	C₄	—	271	2	no	—
76	C₄	—	271	2	no	—
77	C₄	—	2168	2	no	—
78	C₄	#22576159	1	1	yes	characteristic
79	C₄	Q₁₆×D₂₇₁	1	1	yes	characteristic
80	C₂	#31774301	1	1	yes	characteristic
81	C₂	D₁₆×D₂₇₁	1	1	yes	characteristic
82	C₂	Q₃₂×D₂₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₂₇₁
commutator, cosocle	C₂×C₈×C₂₇₁	C₂³
Frattini	C₂×C₈	C₂²×D₂₇₁
Fitting	#366×C₂₇₁	C₂
socle, prime radical	C₂²×C₂₇₁	C₂×D₈
hypercenter	#366	D₂₇₁
hypocenter	C₂₇₁	#2252
LCS	G ▷ C₂×C₈×C₂₇₁ ▷ C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

Complements

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

N	H ≅ G/N	type
18. C₂₇₁⋊C₁₆	Q₈	semidirect
21. C₁₆×C₂₇₁	Q₈	semidirect
41. C₂₇₁	#2252	semidirect
63. C₁₆	Dic₅₄₂	semidirect

Maximal subgroups

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

subgroup	index
#31774301	2
#31773723	2
#31773722	2
C₁₆×Dic₂₇₁	2
#366×C₂₇₁	2
#2252	271