Subgroups — #31773681

Order 17344.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22576100	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#131×C₂₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂₇₁⋊C₁₆	—	2	1	no	—
5	C₂₇₁⋊C₁₆	—	2	1	no	—
6	#41×C₂₇₁	C₄	1	1	yes	characteristic
7	M₁₆×C₂₇₁	—	2	1	no	—
8	M₁₆×C₂₇₁	C₄	1	1	yes	characteristic
9	C₂×C₈×C₂₇₁	—	2	1	no	—
10	C₂×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
11	Q₈×C₂₇₁	C₈	1	1	yes	characteristic
12	D₄×C₂₇₁	—	2	1	no	—
13	D₄×C₂₇₁	C₈	1	1	yes	characteristic
14	C₂×C₄×C₂₇₁	—	2	1	no	—
15	C₂×C₄×C₂₇₁	C₂×C₄	1	1	yes	characteristic
16	C₈×C₂₇₁	—	2	1	no	—
17	C₈×C₂₇₁	D₄	1	1	yes	characteristic
18	C₈×C₂₇₁	D₄	1	1	yes	characteristic
19	C₂²×C₂₇₁	—	2	1	no	—
20	C₂²×C₂₇₁	M₁₆	1	1	yes	characteristic
21	C₄×C₂₇₁	—	2	1	no	—
22	C₄×C₂₇₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₂₇₁	C₂×C₈	1	1	yes	characteristic
24	C₂×C₂₇₁	—	2	1	no	—
25	C₂×C₂₇₁	—	4	1	no	—
26	C₂×C₂₇₁	C₂²⋊C₈	1	1	yes	characteristic
27	C₂₇₁	#350	1	1	yes	characteristic, complemented, Sylow
28	#350	—	271	1	no	maximal, Sylow
29	#131	D₂₇₁	1	1	yes	characteristic
30	M₃₂	—	271	1	no	—
31	C₂×C₁₆	—	271	1	no	—
32	#41	Dic₂₇₁	1	1	yes	characteristic
33	M₁₆	—	2	1	no	—
34	M₁₆	Dic₂₇₁	1	1	yes	characteristic
35	C₂×C₈	—	2	1	no	—
36	C₂×C₈	C₂×D₂₇₁	1	1	yes	characteristic
37	C₁₆	—	542	1	no	—
38	C₁₆	—	542	1	no	—
39	Q₈	C₂₇₁⋊C₈	1	1	yes	characteristic
40	D₄	—	2	1	no	—
41	D₄	C₂₇₁⋊C₈	1	1	yes	characteristic
42	C₂×C₄	—	2	1	no	—
43	C₂×C₄	C₂×Dic₂₇₁	1	1	yes	characteristic
44	C₈	—	2	1	no	—
45	C₈	C₂²⋊D₂₇₁	1	1	yes	characteristic
46	C₈	C₂²⋊D₂₇₁	1	1	yes	characteristic
47	C₂²	—	2	1	no	—
48	C₂²	#16864638	1	1	yes	characteristic
49	C₄	—	2	1	no	—
50	C₄	C₂²⋊Dic₂₇₁	1	1	yes	characteristic
51	C₄	C₂×C₂₇₁⋊C₈	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	4	1	no	—
54	C₂	C₂²⋊(C₂₇₁⋊C₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²⋊D₂₇₁
commutator	C₄×C₂₇₁	C₂×C₈
Frattini	C₂×C₈	C₂×D₂₇₁
Fitting	#131×C₂₇₁	C₂
socle	C₂×C₂₇₁	C₂²⋊C₈
prime radical	#41×C₂₇₁	C₄
cosocle	C₂×C₈×C₂₇₁	C₂²
hypercenter	#131	D₂₇₁
hypocenter	C₂₇₁	#350
LCS	G ▷ C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #131

Complements

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

N	H ≅ G/N	type
1. #22576100	C₂	semidirect
2. C₂×C₂₇₁⋊C₁₆	C₂	semidirect
27. C₂₇₁	#350	semidirect

Maximal subgroups

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

subgroup	index
#22576100	2
C₂×C₂₇₁⋊C₁₆	2
#131×C₂₇₁	2
#350	271