Subgroups — #17985804

Order 4544.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₈₄	C₂	1	1	yes	maximal, characteristic
2	#14556062	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₁₄₂	—	2	1	no	—
5	Dic₂₈₄	—	2	1	no	—
6	Dic₂₈₄	C₄	1	2	yes	complemented
7	C₄⋊Dic₇₁	—	2	1	no	—
8	C₂×C₈×C₇₁	C₂²	1	1	yes	characteristic
9	C₁₆×C₇₁	—	2	1	no	—
10	C₂×C₄×C₇₁	D₄	1	1	yes	characteristic
11	C₂×Dic₇₁	—	4	1	no	—
12	C₂×Dic₇₁	—	4	1	no	—
13	Dic₁₄₂	—	2	2	no	—
14	Dic₁₄₂	—	4	1	no	—
15	C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
16	C₈×C₇₁	D₄	1	1	yes	characteristic
17	C₂²×C₇₁	D₈	1	1	yes	characteristic
18	C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
19	C₄×C₇₁	SD₁₆	1	1	yes	characteristic
20	Dic₇₁	—	4	2	no	—
21	Dic₇₁	—	8	1	no	—
22	C₂×C₇₁	#102	1	1	yes	characteristic
23	C₂×C₇₁	SD₃₂	1	1	yes	characteristic
24	C₂×C₇₁	Q₃₂	1	1	yes	characteristic
25	C₇₁	#358	1	1	yes	characteristic, complemented, Sylow
26	#358	—	71	1	no	maximal, Sylow
27	C₂×Q₁₆	—	71	1	no	—
28	C₂×C₁₆	D₇₁	1	1	yes	characteristic
29	#107	—	71	1	no	—
30	C₂×Q₈	—	142	1	no	—
31	Q₁₆	—	71	2	no	—
32	Q₁₆	—	142	1	no	—
33	C₂×C₈	C₂×D₇₁	1	1	yes	characteristic
34	C₄⋊C₄	—	142	1	no	—
35	C₁₆	—	2	1	no	—
36	Q₈	—	284	1	no	—
37	Q₈	—	142	2	no	—
38	C₂×C₄	—	284	1	no	—
39	C₂×C₄	—	284	1	no	—
40	C₂×C₄	D₂₈₄	1	1	yes	characteristic
41	C₈	C₄×D₇₁	1	1	yes	characteristic
42	C₈	C₂²⋊D₇₁	1	1	yes	characteristic
43	C₂²	D₅₆₈	1	1	yes	characteristic
44	C₄	—	284	2	no	—
45	C₄	—	568	1	no	—
46	C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
47	C₄	#11765854	1	1	yes	characteristic
48	C₂	#14556050	1	1	yes	characteristic
49	C₂	#14556073	1	1	yes	characteristic
50	C₂	Dic₅₆₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₅₆₈
commutator	C₈×C₇₁	C₂×C₄
Frattini	C₂×C₈	C₂×D₇₁
Fitting	C₂×C₁₆×C₇₁	C₂
socle, prime radical	C₂²×C₇₁	D₈
cosocle	C₂×C₈×C₇₁	C₂²
hypercenter	C₂×C₁₆	D₇₁
hypocenter	C₇₁	#358
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
6. Dic₂₈₄	C₄	semidirect
25. C₇₁	#358	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₈₄	2
#14556062	2
C₂×C₁₆×C₇₁	2
#358	71