Subgroups — #17986396

Order 4544.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14556127	C₂	1	2	yes	maximal
2	#14556060	C₂	1	1	yes	maximal, characteristic
3	#14556059	C₂	1	2	yes	maximal
4	C₈×Dic₇₁	C₂	1	1	yes	maximal, characteristic
5	#106×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₄₂	—	2	2	no	—
7	C₂×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₇₁	C₂²	1	2	yes	—
9	Dic₇₁⋊C₄	—	2	2	no	—
10	C₄×Dic₇₁	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₇₁	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
13	C₂×C₄×C₇₁	—	2	2	no	—
14	C₂×C₄×C₇₁	C₂³	1	1	yes	characteristic
15	C₂×Dic₇₁	—	2	2	no	—
16	C₂×Dic₇₁	D₄	1	2	yes	—
17	Dic₁₄₂	—	4	4	no	—
18	C₈×C₇₁	Q₈	1	2	yes	complemented
19	C₇₁⋊C₈	Q₈	1	2	yes	complemented
20	C₂²×C₇₁	C₂×D₄	1	1	yes	characteristic
21	C₄×C₇₁	—	4	2	no	—
22	C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
23	C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
24	Dic₇₁	—	4	2	no	—
25	Dic₇₁	SD₁₆	1	4	yes	—
26	C₂×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
27	C₂×C₇₁	C₂×SD₁₆	1	2	yes	—
28	C₇₁	#498	1	1	yes	characteristic, complemented, Sylow
29	#498	—	71	1	no	maximal, Sylow
30	C₄⋊Q₈	—	71	2	no	—
31	#106	—	71	1	no	—
32	#106	—	71	2	no	—
33	#106	D₇₁	1	1	yes	characteristic
34	C₄×C₈	—	71	1	no	—
35	C₂×Q₈	—	142	2	no	—
36	C₂×C₈	—	71	1	no	—
37	C₂×C₈	C₂×D₇₁	1	1	yes	characteristic
38	C₄⋊C₄	—	71	2	no	—
39	C₄⋊C₄	—	142	2	no	—
40	C₄⋊C₄	C₂×D₇₁	1	2	yes	—
41	C₄²	—	71	1	no	—
42	Q₈	—	284	4	no	—
43	C₂×C₄	—	2	2	no	—
44	C₂×C₄	—	71	2	no	—
45	C₂×C₄	—	142	2	no	—
46	C₂×C₄	C₂²×D₇₁	1	1	yes	characteristic
47	C₈	—	71	2	no	—
48	C₈	Dic₁₄₂	1	2	yes	complemented
49	C₂²	D₄×D₇₁	1	1	yes	characteristic
50	C₄	—	4	2	no	—
51	C₄	—	284	2	no	—
52	C₄	—	71	4	no	—
53	C₄	Q₈×D₇₁	1	1	yes	characteristic
54	C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
55	C₂	#14556127	1	1	yes	characteristic
56	C₂	SD₁₆×D₇₁	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₇₁
commutator, cosocle	C₂×C₄×C₇₁	C₂³
Frattini	C₂×C₄	C₂²×D₇₁
Fitting	#106×C₇₁	C₂
socle, prime radical	C₂²×C₇₁	C₂×D₄
hypercenter	#106	D₇₁
hypocenter	C₇₁	#498
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
18. C₈×C₇₁	Q₈	semidirect
19. C₇₁⋊C₈	Q₈	semidirect
28. C₇₁	#498	semidirect
48. C₈	Dic₁₄₂	semidirect

Maximal subgroups

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

subgroup	index
#14556127	2
#14556060	2
#14556059	2
C₈×Dic₇₁	2
#106×C₇₁	2
#498	71