Subgroups — #36355845

Order 22144.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011896	C₂	1	2	yes	maximal
2	#26011319	C₂	1	4	yes	maximal
3	C₄×C₁₆×C₁₇₃	C₂	1	1	yes	maximal, characteristic
4	#19149679	—	2	2	no	—
5	#19149613	—	2	2	no	—
6	#19149613	C₂²	1	4	yes	—
7	C₂×C₁₆×C₁₇₃	C₂²	1	2	yes	—
8	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
9	C₂×Dic₃₄₆	—	4	2	no	—
10	C₄⋊Dic₁₇₃	—	2	4	no	—
11	C₄⋊Dic₁₇₃	—	4	2	no	—
12	C₂×C₈×C₁₇₃	D₄	1	1	yes	characteristic
13	C₂×C₈×C₁₇₃	C₂³	1	1	yes	characteristic
14	C₄²×C₁₇₃	D₄	1	1	yes	characteristic
15	C₁₆×C₁₇₃	Q₈	1	4	yes	complemented
16	Dic₃₄₆	—	8	4	no	—
17	C₂×Dic₁₇₃	—	4	4	no	—
18	C₂×C₄×C₁₇₃	D₈	1	2	yes	—
19	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
20	C₈×C₁₇₃	Q₁₆	1	2	yes	—
21	C₈×C₁₇₃	C₂×Q₈	1	2	yes	—
22	C₂²×C₁₇₃	C₂×D₈	1	1	yes	characteristic
23	C₄×C₁₇₃	SD₃₂	1	4	yes	—
24	C₄×C₁₇₃	C₄⋊Q₈	1	1	yes	characteristic
25	C₄×C₁₇₃	C₂×Q₁₆	1	1	yes	characteristic
26	Dic₁₇₃	—	8	4	no	—
27	C₂×C₁₇₃	#500	1	1	yes	characteristic
28	C₂×C₁₇₃	C₂×SD₃₂	1	2	yes	—
29	C₁₇₃	#2254	1	1	yes	characteristic, complemented, Sylow
30	#2254	—	173	1	no	maximal, Sylow
31	#500	—	173	2	no	—
32	#367	—	173	4	no	—
33	C₄×C₁₆	D₁₇₃	1	1	yes	characteristic
34	C₄⋊Q₈	—	346	2	no	—
35	C₂×C₁₆	C₂×D₁₇₃	1	2	yes	—
36	#107	—	346	2	no	—
37	#107	—	173	4	no	—
38	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic
39	C₂×Q₈	—	692	2	no	—
40	C₂×C₈	D₆₉₂	1	1	yes	characteristic
41	C₂×C₈	C₂²×D₁₇₃	1	1	yes	characteristic
42	C₄⋊C₄	—	346	4	no	—
43	C₄⋊C₄	—	692	2	no	—
44	C₄²	D₆₉₂	1	1	yes	characteristic
45	C₁₆	Dic₃₄₆	1	4	yes	complemented
46	Q₈	—	1384	4	no	—
47	C₂×C₄	—	692	4	no	—
48	C₂×C₄	D₁₃₈₄	1	2	yes	—
49	C₂×C₄	C₂×D₆₉₂	1	1	yes	characteristic
50	C₈	Dic₆₉₂	1	2	yes	—
51	C₈	C₂×Dic₃₄₆	1	2	yes	—
52	C₂²	C₂×D₁₃₈₄	1	1	yes	characteristic
53	C₄	—	1384	4	no	—
54	C₄	#19149624	1	4	yes	—
55	C₄	#19149679	1	1	yes	characteristic
56	C₄	C₂×Dic₆₉₂	1	1	yes	characteristic
57	C₂	#26011896	1	1	yes	characteristic
58	C₂	C₂×#19149624	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₃₈₄
commutator, cosocle	C₂×C₈×C₁₇₃	C₂³
Frattini	C₂×C₈	C₂²×D₁₇₃
Fitting	C₄×C₁₆×C₁₇₃	C₂
socle, prime radical	C₂²×C₁₇₃	C₂×D₈
hypercenter	C₄×C₁₆	D₁₇₃
hypocenter	C₁₇₃	#2254
LCS	G ▷ C₂×C₈×C₁₇₃ ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
15. C₁₆×C₁₇₃	Q₈	semidirect
29. C₁₇₃	#2254	semidirect
45. C₁₆	Dic₃₄₆	semidirect

Maximal subgroups

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

subgroup	index
#26011896	2
#26011319	2
C₄×C₁₆×C₁₇₃	2
#2254	173