Subgroups — #46834419

Order 34688.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₂₇₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	#31773650	C₂	1	1	yes	maximal, characteristic
3	#500×C₂₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₇₁⋊C₁₆	—	2	1	no	—
5	C₄⋊(C₂₇₁⋊C₈)	—	2	1	no	—
6	C₄⋊Q₈×C₂₇₁	—	2	1	no	—
7	#107×C₂₇₁	—	2	1	no	—
8	#107×C₂₇₁	C₄	1	2	yes	—
9	C₄×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
10	C₂×C₂₇₁⋊C₈	—	4	1	no	—
11	C₂₇₁⋊C₁₆	—	4	1	no	—
12	C₂×Q₈×C₂₇₁	—	4	1	no	—
13	C₂×C₈×C₂₇₁	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₂₇₁	D₄	1	1	yes	characteristic
15	C₄⋊C₄×C₂₇₁	—	2	2	no	—
16	C₄⋊C₄×C₂₇₁	—	4	1	no	—
17	C₄²×C₂₇₁	D₄	1	1	yes	characteristic
18	C₂₇₁⋊C₈	—	8	1	no	—
19	Q₈×C₂₇₁	—	8	1	no	—
20	Q₈×C₂₇₁	—	8	1	no	—
21	C₂×C₄×C₂₇₁	—	4	2	no	—
22	C₂×C₄×C₂₇₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₄×C₂₇₁	D₈	1	1	yes	characteristic
24	C₂×C₄×C₂₇₁	SD₁₆	1	1	yes	characteristic
25	C₈×C₂₇₁	—	2	1	no	—
26	C₈×C₂₇₁	SD₁₆	1	1	yes	characteristic
27	C₈×C₂₇₁	Q₁₆	1	1	yes	characteristic
28	C₂²×C₂₇₁	#102	1	1	yes	characteristic
29	C₄×C₂₇₁	—	2	1	no	—
30	C₄×C₂₇₁	—	8	2	no	—
31	C₄×C₂₇₁	#101	1	1	yes	characteristic
32	C₄×C₂₇₁	#103	1	1	yes	characteristic
33	C₄×C₂₇₁	D₁₆	1	1	yes	characteristic
34	C₄×C₂₇₁	SD₃₂	1	1	yes	characteristic
35	C₂×C₂₇₁	#332	1	1	yes	characteristic
36	C₂×C₂₇₁	#357	1	1	yes	characteristic
37	C₂×C₂₇₁	#362	1	1	yes	characteristic
38	C₂₇₁	#1364	1	1	yes	characteristic, complemented, Sylow
39	#1364	—	271	1	no	maximal, Sylow
40	#500	D₂₇₁	1	1	yes	characteristic
41	C₄⋊C₁₆	—	271	1	no	—
42	#335	—	271	1	no	—
43	C₄⋊Q₈	—	2	1	no	—
44	C₂×C₁₆	—	542	1	no	—
45	#107	—	2	1	no	—
46	#107	Dic₂₇₁	1	2	yes	—
47	C₄⋊C₈	—	542	1	no	—
48	C₄×C₈	C₂×D₂₇₁	1	1	yes	characteristic
49	C₂×Q₈	—	4	1	no	—
50	C₂×C₈	—	1084	1	no	—
51	C₂×C₈	C₂×Dic₂₇₁	1	1	yes	characteristic
52	C₂×C₈	C₂²⋊D₂₇₁	1	1	yes	characteristic
53	C₄⋊C₄	—	2	2	no	—
54	C₄⋊C₄	—	4	1	no	—
55	C₄²	C₂²⋊D₂₇₁	1	1	yes	characteristic
56	C₁₆	—	1084	1	no	—
57	Q₈	—	8	1	no	—
58	Q₈	—	8	1	no	—
59	C₂×C₄	—	4	2	no	—
60	C₂×C₄	C₂²⋊Dic₂₇₁	1	1	yes	characteristic
61	C₂×C₄	#16864639	1	1	yes	characteristic
62	C₂×C₄	#16864642	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	2168	1	no	—
65	C₈	#16864641	1	1	yes	characteristic
66	C₈	#16864644	1	1	yes	characteristic
67	C₂²	#22576081	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	8	2	no	—
70	C₄	#22576079	1	1	yes	characteristic
71	C₄	#22576084	1	1	yes	characteristic
72	C₄	#22576101	1	1	yes	characteristic
73	C₄	#22576104	1	1	yes	characteristic
74	C₂	#31773643	1	1	yes	characteristic
75	C₂	#31773698	1	1	yes	characteristic
76	C₂	#31773713	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#22576081
commutator	C₂×C₈×C₂₇₁	C₂×C₄
Frattini	C₄×C₈	C₂×D₂₇₁
Fitting	#500×C₂₇₁	C₂
socle, prime radical	C₂²×C₂₇₁	#102
cosocle	C₄×C₈×C₂₇₁	C₂²
hypercenter	#500	D₂₇₁
hypocenter	C₂₇₁	#1364
LCS	G ▷ C₂×C₈×C₂₇₁ ▷ C₂×C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

Complements

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

N	H ≅ G/N	type
38. C₂₇₁	#1364	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₂₇₁⋊C₁₆)	2
#31773650	2
#500×C₂₇₁	2
#1364	271