Subgroups — #31774263

Order 17344.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22576157	C₂	1	1	yes	maximal, characteristic
2	#22576080	C₂	1	4	yes	maximal, complemented
3	#22576071	C₂	1	1	yes	maximal, characteristic, complemented
4	#125×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₀₈₄	—	2	2	no	—
6	C₂×D₁₀₈₄	C₂²	1	2	yes	—
7	C₂×C₂₇₁⋊C₈	C₂²	1	2	yes	—
8	C₄⋊C₄×C₂₇₁	—	2	2	no	—
9	C₄⋊C₄×C₂₇₁	C₂²	1	2	yes	—
10	C₄²×C₂₇₁	C₂²	1	1	yes	characteristic
11	C₂²×D₂₇₁	—	2	2	no	—
12	D₁₀₈₄	—	2	4	no	—
13	D₁₀₈₄	—	4	4	no	—
14	C₂₇₁⋊C₈	—	2	2	no	—
15	C₂×C₄×C₂₇₁	—	2	2	no	—
16	C₂×C₄×C₂₇₁	D₄	1	2	yes	—
17	C₂×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
18	C₂²×C₂₇₁	C₂×D₄	1	1	yes	characteristic
19	C₂×D₂₇₁	—	4	2	no	—
20	C₂×D₂₇₁	—	4	4	no	—
21	C₄×C₂₇₁	—	2	2	no	—
22	C₄×C₂₇₁	—	4	2	no	—
23	C₄×C₂₇₁	#41	1	2	yes	—
24	C₂×C₂₇₁	#124	1	1	yes	characteristic
25	C₂×C₂₇₁	AGL(1,ℤ/8)	1	2	yes	—
26	D₂₇₁	—	8	2	no	—
27	C₂₇₁	#490	1	1	yes	characteristic, complemented, Sylow
28	#490	—	271	1	no	maximal, Sylow
29	#127	—	271	1	no	—
30	#125	D₂₇₁	1	1	yes	characteristic, complemented
31	#102	—	271	4	no	—
32	#97	—	271	1	no	—
33	C₂×D₄	—	271	2	no	—
34	C₂×D₄	—	542	2	no	—
35	C₂×C₈	—	271	2	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	C₂×D₂₇₁	1	2	yes	—
38	C₄²	C₂×D₂₇₁	1	1	yes	characteristic
39	C₂³	—	542	2	no	—
40	D₄	—	542	4	no	—
41	D₄	—	1084	4	no	—
42	C₂×C₄	—	2	2	no	—
43	C₂×C₄	C₂²⋊D₂₇₁	1	2	yes	—
44	C₂×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
45	C₈	—	542	2	no	—
46	C₂²	—	1084	2	no	—
47	C₂²	—	1084	4	no	—
48	C₂²	C₂×C₂²⋊D₂₇₁	1	1	yes	characteristic
49	C₄	—	2	2	no	—
50	C₄	—	4	2	no	—
51	C₄	#16864655	1	2	yes	—
52	C₂	—	2168	2	no	—
53	C₂	#22576149	1	1	yes	characteristic
54	C₂	#22576185	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₇₁
commutator, cosocle	C₂×C₄×C₂₇₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₇₁
Fitting	#125×C₂₇₁	C₂
socle	C₂²×C₂₇₁	C₂×D₄
prime radical	#22576157	C₂
hypercenter	#125	D₂₇₁
hypocenter	C₂₇₁	#490
LCS	G ▷ C₂×C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
2. #22576080	C₂	semidirect
3. #22576071	C₂	semidirect
4. #125×C₂₇₁	C₂	semidirect
27. C₂₇₁	#490	semidirect
30. #125	D₂₇₁	semidirect

Maximal subgroups

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

subgroup	index
#22576157	2
#22576080	2
#22576071	2
#125×C₂₇₁	2
#490	271