Subgroups — #36351479

Order 22144.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14583801	C₂	1	1	yes	maximal, characteristic
2	C₈×#13063436	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₆×C₁₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14583801	C₄	1	2	yes	—
5	C₄×#13063436	C₄	1	2	yes	—
6	C₂×C₁₆×C₁₇₃	C₄	1	2	yes	—
7	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
8	C₂×#13063436	C₈	1	4	yes	—
9	#14583801	C₈	1	4	yes	complemented
10	C₂×C₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
11	C₂×C₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
12	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
13	C₁₆×C₁₇₃	C₈	1	4	yes	complemented
14	#13063436	—	2	4	no	—
15	C₂×C₄×C₁₇₃	C₄²	1	1	yes	characteristic
16	C₂×C₄×C₁₇₃	C₂×C₈	1	2	yes	—
17	C₈×C₁₇₃	C₂×C₈	1	2	yes	—
18	C₈×C₁₇₃	C₂×C₈	1	2	yes	—
19	C₂²×C₁₇₃	C₄×C₈	1	1	yes	characteristic
20	C₄×C₁₇₃	C₄×C₈	1	1	yes	characteristic
21	C₄×C₁₇₃	C₄×C₈	1	1	yes	characteristic
22	C₄×C₁₇₃	M₃₂	1	4	yes	—
23	C₂×C₁₇₃	C₈²	1	1	yes	characteristic
24	C₂×C₁₇₃	#346	1	2	yes	—
25	C₁₇₃	#1311	1	1	yes	characteristic, complemented, Sylow
26	#1311	—	173	1	no	maximal, Sylow
27	C₄×C₁₆	—	173	1	no	—
28	C₄×C₁₆	D₁₇₃	1	1	yes	characteristic
29	C₈²	—	173	1	no	—
30	C₂×C₁₆	—	173	2	no	—
31	C₂×C₁₆	Dic₁₇₃	1	2	yes	—
32	C₄×C₈	—	173	2	no	—
33	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic
34	C₂×C₈	—	173	4	no	—
35	C₂×C₈	C₄×D₁₇₃	1	1	yes	characteristic
36	C₂×C₈	C₂×Dic₁₇₃	1	1	yes	characteristic
37	C₄²	C₄×D₁₇₃	1	1	yes	characteristic
38	C₁₆	—	173	4	no	—
39	C₁₆	#13063436	1	4	yes	complemented
40	C₂×C₄	C₄×Dic₁₇₃	1	1	yes	characteristic
41	C₂×C₄	C₈×D₁₇₃	1	2	yes	—
42	C₈	—	346	4	no	—
43	C₈	C₈×D₁₇₃	1	2	yes	—
44	C₈	C₂×#13063436	1	2	yes	—
45	C₂²	C₈×Dic₁₇₃	1	1	yes	characteristic
46	C₄	C₈×Dic₁₇₃	1	1	yes	characteristic
47	C₄	C₄×#13063436	1	1	yes	characteristic
48	C₄	#19149618	1	4	yes	—
49	C₂	C₈×#13063436	1	1	yes	characteristic
50	C₂	#26011266	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂×D₁₇₃
commutator	C₂×C₁₇₃	C₈²
Fitting	C₄×C₁₆×C₁₇₃	C₂
socle, prime radical	C₂²×C₁₇₃	C₄×C₈
cosocle	C₄×C₈×C₁₇₃	C₂²
hypercenter	C₄×C₁₆	D₁₇₃
hypocenter	C₁₇₃	#1311
LCS	G ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
9. #14583801	C₈	semidirect
13. C₁₆×C₁₇₃	C₈	semidirect
25. C₁₇₃	#1311	semidirect
39. C₁₆	#13063436	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14583801	2
C₈×#13063436	2
C₄×C₁₆×C₁₇₃	2
#1311	173