Subgroups — #46838235

Order 34688.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₁₆×Dic₂₇₁	C₂	1	2	yes	maximal
2	#31773725	C₂	1	1	yes	maximal, characteristic
3	#31773701	C₂	1	2	yes	maximal
4	#31773672	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₃₂×C₂₇₁	C₂	1	1	yes	maximal, characteristic
6	Q₈×Dic₂₇₁	—	2	2	no	—
7	C₂×C₂₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
8	#22576094	C₂²	1	2	yes	—
9	#22576084	—	2	2	no	—
10	C₈×Dic₂₇₁	C₂²	1	1	yes	characteristic
11	C₂×Q₁₆×C₂₇₁	C₂²	1	2	yes	—
12	Q₃₂×C₂₇₁	C₄	1	4	yes	complemented
13	C₂×C₁₆×C₂₇₁	C₂²	1	1	yes	characteristic
14	C₂×C₂₇₁⋊C₈	D₄	1	1	yes	characteristic
15	C₄⋊Dic₂₇₁	—	2	2	no	—
16	C₄⋊Dic₂₇₁	—	4	2	no	—
17	C₄×Dic₂₇₁	—	4	2	no	—
18	C₄×Dic₂₇₁	D₄	1	1	yes	characteristic
19	C₂₇₁⋊C₁₆	—	2	1	no	—
20	C₂×Q₈×C₂₇₁	—	2	2	no	—
21	Q₁₆×C₂₇₁	—	2	2	no	—
22	Q₁₆×C₂₇₁	C₂×C₄	1	4	yes	—
23	C₂×C₈×C₂₇₁	C₂³	1	1	yes	characteristic
24	C₁₆×C₂₇₁	C₂×C₄	1	2	yes	—
25	C₂×Dic₂₇₁	—	4	2	no	—
26	C₂×Dic₂₇₁	D₈	1	2	yes	—
27	C₂₇₁⋊C₈	—	2	1	no	—
28	Q₈×C₂₇₁	—	2	4	no	—
29	Q₈×C₂₇₁	—	4	2	no	—
30	C₂×C₄×C₂₇₁	—	4	2	no	—
31	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
32	C₈×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₂₇₁	#41	1	1	yes	characteristic
34	C₂²×C₂₇₁	C₂×D₈	1	1	yes	characteristic
35	C₄×C₂₇₁	—	4	4	no	—
36	C₄×C₂₇₁	C₄×D₄	1	1	yes	characteristic
37	C₄×C₂₇₁	#135	1	1	yes	characteristic
38	Dic₂₇₁	—	2	2	no	—
39	Dic₂₇₁	—	8	2	no	—
40	C₂×C₂₇₁	C₄×D₈	1	1	yes	characteristic
41	C₂×C₂₇₁	#510	1	2	yes	—
42	C₂₇₁	#2176	1	1	yes	characteristic, complemented, Sylow
43	#2176	—	271	1	no	maximal, Sylow
44	C₂×Q₃₂	D₂₇₁	1	1	yes	characteristic
45	C₄×Q₁₆	—	271	2	no	—
46	#367	—	271	1	no	—
47	#358	—	271	2	no	—
48	#346	—	271	1	no	—
49	C₂×Q₁₆	C₂×D₂₇₁	1	2	yes	—
50	C₄×Q₈	—	542	2	no	—
51	Q₃₂	Dic₂₇₁	1	4	yes	complemented
52	C₂×C₁₆	—	271	1	no	—
53	C₂×C₁₆	C₂×D₂₇₁	1	1	yes	characteristic
54	#107	—	271	2	no	—
55	#103	—	542	2	no	—
56	C₄×C₈	—	271	1	no	—
57	C₂×Q₈	—	2	2	no	—
58	Q₁₆	—	2	2	no	—
59	Q₁₆	C₂×Dic₂₇₁	1	4	yes	—
60	C₂×C₈	—	271	1	no	—
61	C₂×C₈	C₂²×D₂₇₁	1	1	yes	characteristic
62	C₄⋊C₄	—	542	2	no	—
63	C₄⋊C₄	—	1084	2	no	—
64	C₄²	—	271	1	no	—
65	C₄²	—	1084	2	no	—
66	C₁₆	—	542	1	no	—
67	C₁₆	C₂×Dic₂₇₁	1	2	yes	—
68	Q₈	—	2	4	no	—
69	Q₈	—	4	2	no	—
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	—	271	2	no	—
72	C₂×C₄	—	1084	2	no	—
73	C₂×C₄	D₄×D₂₇₁	1	1	yes	characteristic
74	C₈	—	542	1	no	—
75	C₈	C₂²×Dic₂₇₁	1	1	yes	characteristic
76	C₈	#16864653	1	1	yes	characteristic
77	C₂²	D₈×D₂₇₁	1	1	yes	characteristic
78	C₄	—	4	4	no	—
79	C₄	—	542	2	no	—
80	C₄	—	2168	2	no	—
81	C₄	D₄×Dic₂₇₁	1	1	yes	characteristic
82	C₄	#22576179	1	1	yes	characteristic
83	C₂	D₈×Dic₂₇₁	1	1	yes	characteristic
84	C₂	#31774339	1	2	yes	—

	subgroup	G/H
center	C₂²	D₈×D₂₇₁
commutator	C₈×C₂₇₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₂₇₁
Fitting	C₂×Q₃₂×C₂₇₁	C₂
socle, prime radical	C₂²×C₂₇₁	C₂×D₈
cosocle	C₂×C₈×C₂₇₁	C₂³
hypercenter	C₂×Q₃₂	D₂₇₁
hypocenter	C₂₇₁	#2176
LCS	G ▷ C₈×C₂₇₁ ▷ C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
12. Q₃₂×C₂₇₁	C₄	semidirect
42. C₂₇₁	#2176	semidirect
51. Q₃₂	Dic₂₇₁	semidirect

Maximal subgroups

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

subgroup	index
Q₁₆×Dic₂₇₁	2
#31773725	2
#31773701	2
#31773672	2
C₂×Q₃₂×C₂₇₁	2
#2176	271