Subgroups — #31774264

Order 17344.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22576158	C₂	1	1	yes	maximal, characteristic
2	#22576151	C₂	1	1	yes	maximal, characteristic
3	#22576085	C₂	1	1	yes	maximal, characteristic
4	#22576084	C₂	1	1	yes	maximal, characteristic
5	#22576083	C₂	1	1	yes	maximal, characteristic
6	#22576070	C₂	1	1	yes	maximal, characteristic
7	#103×C₂₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₅₄₂	C₂²	1	1	yes	characteristic
9	C₂×C₂₇₁⋊C₈	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₂₇₁	—	2	1	no	—
11	C₄⋊Dic₂₇₁	C₂²	1	1	yes	characteristic
12	Dic₂₇₁⋊C₄	—	2	1	no	—
13	Dic₂₇₁⋊C₄	—	2	1	no	—
14	Dic₂₇₁⋊C₄	—	2	1	no	—
15	C₄×Dic₂₇₁	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₂₇₁	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₂₇₁	C₂²	1	1	yes	characteristic
19	Dic₅₄₂	—	2	1	no	—
20	Dic₅₄₂	—	2	1	no	—
21	C₂×Dic₂₇₁	—	2	1	no	—
22	C₂×Dic₂₇₁	—	2	1	no	—
23	C₂×Dic₂₇₁	D₄	1	1	yes	characteristic
24	C₂×Dic₂₇₁	D₄	1	1	yes	characteristic
25	C₂₇₁⋊C₈	—	2	1	no	—
26	Q₈×C₂₇₁	—	2	1	no	—
27	Q₈×C₂₇₁	—	2	1	no	—
28	C₂×C₄×C₂₇₁	—	2	1	no	—
29	C₂×C₄×C₂₇₁	—	2	1	no	—
30	C₂×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
31	C₈×C₂₇₁	—	2	1	no	—
32	C₂²×C₂₇₁	C₂×D₄	1	1	yes	characteristic
33	C₄×C₂₇₁	—	4	1	no	—
34	C₄×C₂₇₁	—	4	1	no	—
35	C₄×C₂₇₁	#41	1	1	yes	characteristic
36	C₄×C₂₇₁	#41	1	1	yes	characteristic
37	Dic₂₇₁	—	2	1	no	—
38	Dic₂₇₁	—	2	1	no	—
39	Dic₂₇₁	—	4	1	no	—
40	Dic₂₇₁	—	4	1	no	—
41	C₂×C₂₇₁	#124	1	1	yes	characteristic
42	C₂×C₂₇₁	#137	1	1	yes	characteristic
43	C₂×C₂₇₁	#137	1	1	yes	characteristic
44	C₂₇₁	#491	1	1	yes	characteristic, complemented, Sylow
45	#491	—	271	1	no	maximal, Sylow
46	C₄⋊Q₈	—	271	1	no	—
47	#125	—	271	1	no	—
48	#103	—	271	1	no	—
49	#103	—	271	1	no	—
50	#103	—	271	1	no	—
51	#103	D₂₇₁	1	1	yes	characteristic
52	#97	—	271	1	no	—
53	C₂×Q₈	—	271	1	no	—
54	C₂×Q₈	C₂×D₂₇₁	1	1	yes	characteristic
55	C₂×C₈	—	271	1	no	—
56	C₂×C₈	C₂×D₂₇₁	1	1	yes	characteristic
57	C₄⋊C₄	—	271	1	no	—
58	C₄⋊C₄	—	542	1	no	—
59	C₄⋊C₄	—	542	1	no	—
60	C₄⋊C₄	—	542	1	no	—
61	C₄⋊C₄	—	542	1	no	—
62	C₄⋊C₄	C₂×D₂₇₁	1	1	yes	characteristic
63	C₄²	—	271	1	no	—
64	Q₈	—	2	1	no	—
65	Q₈	—	2	1	no	—
66	Q₈	—	542	1	no	—
67	Q₈	—	542	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	271	1	no	—
71	C₂×C₄	—	271	1	no	—
72	C₂×C₄	—	542	1	no	—
73	C₂×C₄	—	542	1	no	—
74	C₂×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
75	C₈	—	2	1	no	—
76	C₈	—	542	1	no	—
77	C₂²	D₄×D₂₇₁	1	1	yes	characteristic
78	C₄	—	4	1	no	—
79	C₄	—	4	1	no	—
80	C₄	—	542	1	no	—
81	C₄	—	542	1	no	—
82	C₄	—	1084	1	no	—
83	C₄	—	1084	1	no	—
84	C₄	#16864653	1	1	yes	characteristic
85	C₄	#16864654	1	1	yes	characteristic
86	C₂	#22576146	1	1	yes	characteristic
87	C₂	#22576189	1	1	yes	characteristic
88	C₂	#22576192	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₇₁
commutator, cosocle	C₂×C₄×C₂₇₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₇₁
Fitting	#103×C₂₇₁	C₂
socle, prime radical	C₂²×C₂₇₁	C₂×D₄
hypercenter	#103	D₂₇₁
hypocenter	C₂₇₁	#491
LCS	G ▷ C₂×C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
44. C₂₇₁	#491	semidirect

Maximal subgroups

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

subgroup	index
#22576158	2
#22576151	2
#22576085	2
#22576084	2
#22576083	2
#22576070	2
#103×C₂₇₁	2
#491	271