Subgroups — #31773685

Order 17344.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22576128	C₂	1	1	yes	maximal, characteristic
2	#22576075	C₂	1	1	yes	maximal, characteristic, complemented
3	#100×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₂₇₁	C₄	1	1	yes	characteristic, complemented
5	C₂×C₂²⋊D₂₇₁	—	2	1	no	—
6	C₂²⋊Dic₂₇₁	—	2	1	no	—
7	C₂²⋊Dic₂₇₁	C₄	1	1	yes	characteristic, complemented
8	(C₂×D₂₇₁)⋊C₄	—	2	1	no	—
9	C₂×D₄×C₂₇₁	C₂²	1	1	yes	characteristic
10	M₁₆×C₂₇₁	—	2	1	no	—
11	C₂²×D₂₇₁	—	2	1	no	—
12	C₂²×D₂₇₁	—	4	2	no	—
13	C₂²×D₂₇₁	—	4	1	no	—
14	C₂²⋊D₂₇₁	—	4	2	no	—
15	C₂×Dic₂₇₁	—	2	1	no	—
16	C₂×Dic₂₇₁	—	4	1	no	—
17	C₂³×C₂₇₁	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₂₇₁	D₄	1	1	yes	characteristic
19	D₄×C₂₇₁	—	4	1	no	—
20	C₂×C₄×C₂₇₁	D₄	1	1	yes	characteristic
21	C₈×C₂₇₁	—	4	1	no	—
22	C₂²×C₂₇₁	—	2	1	no	—
23	C₂²×C₂₇₁	—	2	1	no	—
24	C₂²×C₂₇₁	—	4	1	no	—
25	C₂²×C₂₇₁	—	4	1	no	—
26	C₂²×C₂₇₁	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×D₂₇₁	—	2	2	no	—
28	C₂×D₂₇₁	—	4	1	no	—
29	C₂×D₂₇₁	—	4	2	no	—
30	C₂×D₂₇₁	—	4	1	no	—
31	C₂×D₂₇₁	—	4	2	no	—
32	C₄×C₂₇₁	—	2	1	no	—
33	Dic₂₇₁	—	4	1	no	—
34	Dic₂₇₁	—	8	1	no	—
35	C₂×C₂₇₁	—	2	1	no	—
36	C₂×C₂₇₁	—	4	1	no	—
37	C₂×C₂₇₁	—	4	1	no	—
38	C₂×C₂₇₁	#99	1	1	yes	characteristic
39	D₂₇₁	—	4	2	no	—
40	C₂₇₁	AΣL(1,𝔽₁₆)	1	1	yes	characteristic, complemented, Sylow
41	AΣL(1,𝔽₁₆)	—	271	1	no	maximal, Sylow
42	#120	—	271	1	no	—
43	#100	D₂₇₁	1	1	yes	characteristic, complemented
44	#99	—	271	1	no	—
45	C₂⁴	—	271	1	no	—
46	C₂×D₄	—	542	1	no	—
47	C₂×D₄	C₂×D₂₇₁	1	1	yes	characteristic
48	M₁₆	—	2	1	no	—
49	C₂²⋊C₄	—	271	1	no	—
50	C₂²⋊C₄	—	542	1	no	—
51	C₂²⋊C₄	—	542	1	no	—
52	C₂³	—	542	1	no	—
53	C₂³	—	1084	1	no	—
54	C₂³	—	1084	2	no	—
55	C₂³	C₄×D₂₇₁	1	1	yes	characteristic
56	C₂³	D₁₀₈₄	1	1	yes	characteristic
57	D₄	—	4	1	no	—
58	D₄	—	1084	2	no	—
59	C₂×C₄	—	542	1	no	—
60	C₂×C₄	—	1084	1	no	—
61	C₂×C₄	C₂²⋊D₂₇₁	1	1	yes	characteristic
62	C₈	—	4	1	no	—
63	C₂²	—	2	1	no	—
64	C₂²	—	2	1	no	—
65	C₂²	—	4	1	no	—
66	C₂²	—	4	1	no	—
67	C₂²	—	542	2	no	—
68	C₂²	—	1084	1	no	—
69	C₂²	—	1084	2	no	—
70	C₂²	—	1084	2	no	—
71	C₂²	—	1084	1	no	—
72	C₂²	(C₂×D₂₇₁)⋊C₄	1	1	yes	characteristic
73	C₄	—	2	1	no	—
74	C₄	—	1084	1	no	—
75	C₄	—	2168	1	no	—
76	C₂	—	2	1	no	—
77	C₂	—	4	1	no	—
78	C₂	—	4	1	no	—
79	C₂	—	1084	2	no	—
80	C₂	#22576074	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22576074
commutator	C₂³×C₂₇₁	C₂×C₄
Frattini	C₂×D₄	C₂×D₂₇₁
Fitting	#100×C₂₇₁	C₂
socle	C₂×C₂₇₁	#99
prime radical	#22576128	C₂
cosocle	C₂×D₄×C₂₇₁	C₂²
hypercenter	#100	D₂₇₁
hypocenter	C₂₇₁	AΣL(1,𝔽₁₆)
LCS	G ▷ C₂³×C₂₇₁ ▷ C₂²×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #100

Complements

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

N	H ≅ G/N	type
2. #22576075	C₂	semidirect
3. #100×C₂₇₁	C₂	semidirect
4. C₂³×D₂₇₁	C₄	semidirect
7. C₂²⋊Dic₂₇₁	C₄	semidirect
40. C₂₇₁	AΣL(1,𝔽₁₆)	semidirect
43. #100	D₂₇₁	semidirect

Maximal subgroups

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

subgroup	index
#22576128	2
#22576075	2
#100×C₂₇₁	2
AΣL(1,𝔽₁₆)	271