Subgroups — #36368713

Order 22144.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₇₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic
2	#26012417	C₂	1	1	yes	maximal, characteristic
3	Q₃₂×D₁₇₃	C₂	1	1	yes	maximal, characteristic
4	#19149748	—	2	1	no	—
5	#19149729	—	2	1	no	—
6	Q₁₆×D₁₇₃	—	2	1	no	—
7	Dic₁₃₈₄	C₄	1	1	yes	characteristic, complemented
8	#19149625	—	2	1	no	—
9	C₁₆×D₁₇₃	C₂²	1	1	yes	characteristic
10	Q₃₂×C₁₇₃	C₄	1	1	yes	characteristic, complemented
11	C₄⋊#10621924	—	4	1	no	—
12	Q₈×D₁₇₃	—	4	1	no	—
13	Dic₆₉₂	—	2	1	no	—
14	#14583816	—	4	1	no	—
15	C₈×D₁₇₃	D₄	1	1	yes	characteristic
16	#14583801	D₄	1	1	yes	characteristic
17	Q₁₆×C₁₇₃	—	2	1	no	—
18	C₁₆×C₁₇₃	C₂×C₄	1	1	yes	characteristic
19	C₂×#10621924	—	8	1	no	—
20	Dic₃₄₆	—	4	1	no	—
21	Dic₃₄₆	—	8	1	no	—
22	C₄×D₁₇₃	—	8	1	no	—
23	C₄×D₁₇₃	D₈	1	1	yes	characteristic
24	#13063436	SD₁₆	1	1	yes	characteristic
25	Q₈×C₁₇₃	—	4	1	no	—
26	C₈×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×D₁₇₃	D₁₆	1	1	yes	characteristic
28	#10621924	—	16	1	no	—
29	C₄×C₁₇₃	—	8	1	no	—
30	C₄×C₁₇₃	#102	1	1	yes	characteristic
31	Dic₁₇₃	—	8	1	no	—
32	Dic₁₇₃	SD₃₂	1	1	yes	characteristic
33	C₂×C₁₇₃	#357	1	1	yes	characteristic
34	D₁₇₃	SD₆₄	1	1	yes	characteristic
35	D₁₇₃	Q₆₄	1	1	yes	characteristic
36	C₁₇₃	#1416	1	1	yes	characteristic, complemented, Sylow
37	#1416	—	173	1	no	maximal, Sylow
38	C₂×Q₃₂	—	173	1	no	—
39	C₂×C₃₂	—	173	1	no	—
40	#366	—	173	1	no	—
41	C₂×Q₁₆	—	346	1	no	—
42	Q₃₂	—	346	1	no	—
43	Q₃₂	—	173	1	no	—
44	Q₃₂	#10621924	1	1	yes	characteristic, complemented
45	C₂×C₁₆	—	173	1	no	—
46	#107	—	346	1	no	—
47	C₃₂	—	346	1	no	—
48	C₂×Q₈	—	692	1	no	—
49	Q₁₆	—	2	1	no	—
50	Q₁₆	—	346	1	no	—
51	Q₁₆	—	692	1	no	—
52	C₂×C₈	—	173	1	no	—
53	C₄⋊C₄	—	692	1	no	—
54	C₁₆	—	173	1	no	—
55	C₁₆	C₂×#10621924	1	1	yes	characteristic
56	Q₈	—	4	1	no	—
57	Q₈	—	1384	1	no	—
58	Q₈	—	692	1	no	—
59	C₂×C₄	—	1384	1	no	—
60	C₂×C₄	—	1384	1	no	—
61	C₂×C₄	—	173	1	no	—
62	C₈	—	173	1	no	—
63	C₈	C₂²⋊#10621924	1	1	yes	characteristic
64	C₂²	—	173	1	no	—
65	C₄	—	8	1	no	—
66	C₄	—	1384	1	no	—
67	C₄	—	173	1	no	—
68	C₄	—	2768	1	no	—
69	C₄	#19149741	1	1	yes	characteristic
70	C₂	—	173	1	no	—
71	C₂	—	173	1	no	—
72	C₂	#26012403	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26012403
commutator	C₁₆×C₁₇₃	C₂×C₄
Frattini	C₁₆	C₂×#10621924
Fitting	Q₃₂×C₁₇₃	C₄
socle	C₂×C₁₇₃	#357
prime radical	C₂×D₁₇₃	D₁₆
cosocle	C₁₆×D₁₇₃	C₂²
hypercenter	Q₃₂	#10621924
hypocenter	C₁₇₃	#1416
LCS	G ▷ C₁₆×C₁₇₃ ▷ C₈×C₁₇₃ ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
7. Dic₁₃₈₄	C₄	semidirect
10. Q₃₂×C₁₇₃	C₄	semidirect
36. C₁₇₃	#1416	semidirect
44. Q₃₂	#10621924	semidirect

Maximal subgroups

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

subgroup	index
D₁₇₃⋊C₃₂	2
#26012417	2
Q₃₂×D₁₇₃	2
#1416	173