Subgroups — #36351635

Order 22144.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011244	C₂	1	2	yes	maximal
2	C₄×C₁₆×C₁₇₃	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#13063436	—	2	2	no	—
4	C₂×C₁₆×C₁₇₃	C₄	1	1	yes	characteristic
5	C₂×C₁₆×C₁₇₃	C₄	1	1	yes	characteristic
6	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
7	C₂×#13063436	—	4	2	no	—
8	C₂×C₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
9	C₂×C₈×C₁₇₃	Q₈	1	1	yes	characteristic
10	C₄²×C₁₇₃	D₄	1	1	yes	characteristic
11	C₁₆×C₁₇₃	—	2	1	no	—
12	C₁₆×C₁₇₃	C₈	1	2	yes	complemented
13	#13063436	—	8	2	no	—
14	C₂×C₄×C₁₇₃	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₄×C₁₇₃	D₈	1	1	yes	characteristic
16	C₂×C₄×C₁₇₃	Q₁₆	1	1	yes	characteristic
17	C₈×C₁₇₃	—	2	1	no	—
18	C₈×C₁₇₃	C₂×C₈	1	1	yes	characteristic
19	C₈×C₁₇₃	M₁₆	1	1	yes	characteristic
20	C₂²×C₁₇₃	#107	1	1	yes	characteristic
21	C₄×C₁₇₃	—	2	1	no	—
22	C₄×C₁₇₃	C₄⋊C₈	1	1	yes	characteristic
23	C₄×C₁₇₃	#108	1	1	yes	characteristic
24	C₄×C₁₇₃	D₁₆	1	1	yes	characteristic
25	C₄×C₁₇₃	Q₃₂	1	1	yes	characteristic
26	C₂×C₁₇₃	#335	1	1	yes	characteristic
27	C₂×C₁₇₃	#366	1	1	yes	characteristic
28	C₂×C₁₇₃	#368	1	1	yes	characteristic
29	C₁₇₃	#1371	1	1	yes	characteristic, complemented, Sylow
30	#1371	—	173	1	no	maximal, Sylow
31	C₄×C₁₆	D₁₇₃	1	1	yes	characteristic
32	#335	—	173	2	no	—
33	C₂×C₁₆	Dic₁₇₃	1	1	yes	characteristic
34	C₂×C₁₆	Dic₁₇₃	1	1	yes	characteristic
35	C₄⋊C₈	—	346	2	no	—
36	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic
37	C₂×C₈	—	692	2	no	—
38	C₂×C₈	C₂×Dic₁₇₃	1	1	yes	characteristic
39	C₂×C₈	Dic₃₄₆	1	1	yes	characteristic
40	C₄²	D₆₉₂	1	1	yes	characteristic
41	C₁₆	—	2	1	no	—
42	C₁₆	#13063436	1	2	yes	complemented
43	C₂×C₄	C₄⋊Dic₁₇₃	1	1	yes	characteristic
44	C₂×C₄	D₁₃₈₄	1	1	yes	characteristic
45	C₂×C₄	Dic₆₉₂	1	1	yes	characteristic
46	C₈	—	2	1	no	—
47	C₈	—	1384	2	no	—
48	C₈	C₂×#13063436	1	1	yes	characteristic
49	C₈	#14583810	1	1	yes	characteristic
50	C₂²	#19149613	1	1	yes	characteristic
51	C₄	—	2	1	no	—
52	C₄	C₄⋊#13063436	1	1	yes	characteristic
53	C₄	#19149615	1	1	yes	characteristic
54	C₄	D₂₇₆₈	1	1	yes	characteristic
55	C₄	Dic₁₃₈₄	1	1	yes	characteristic
56	C₂	#26011244	1	1	yes	characteristic
57	C₂	#26011317	1	1	yes	characteristic
58	C₂	#26011321	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₁₃₈₄
commutator	C₈×C₁₇₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₇₃
Fitting	C₄×C₁₆×C₁₇₃	C₂
socle, prime radical	C₂²×C₁₇₃	#107
cosocle	C₄×C₈×C₁₇₃	C₂²
hypercenter	C₄×C₁₆	D₁₇₃
hypocenter	C₁₇₃	#1371
LCS	G ▷ C₈×C₁₇₃ ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
12. C₁₆×C₁₇₃	C₈	semidirect
29. C₁₇₃	#1371	semidirect
42. C₁₆	#13063436	semidirect

Maximal subgroups

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

subgroup	index
#26011244	2
C₄×C₁₆×C₁₇₃	2
#1371	173