Subgroups — #36368712

Order 22144.

82 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, complemented
2	#26012417	C₂	1	1	yes	maximal, characteristic, complemented
3	D₁₆×D₁₇₃	C₂	1	1	yes	maximal, characteristic
4	#19149748	—	2	1	no	—
5	#19149729	—	2	1	no	—
6	D₈×D₁₇₃	—	2	1	no	—
7	D₂₇₆₈	C₄	1	1	yes	characteristic, complemented
8	#19149620	—	2	1	no	—
9	C₁₆×D₁₇₃	C₂²	1	1	yes	characteristic
10	D₁₆×C₁₇₃	C₄	1	1	yes	characteristic, complemented
11	C₄⋊#10621924	—	4	1	no	—
12	D₄×D₁₇₃	—	4	1	no	—
13	D₁₃₈₄	—	2	1	no	—
14	#14583811	—	4	1	no	—
15	C₈×D₁₇₃	D₄	1	1	yes	characteristic
16	#14583801	D₄	1	1	yes	characteristic
17	D₈×C₁₇₃	—	2	1	no	—
18	C₁₆×C₁₇₃	C₂×C₄	1	1	yes	characteristic
19	C₂×#10621924	—	8	1	no	—
20	C₂²×D₁₇₃	—	8	1	no	—
21	D₆₉₂	—	4	1	no	—
22	C₂²⋊D₁₇₃	—	8	1	no	—
23	C₄×D₁₇₃	D₈	1	1	yes	characteristic
24	#13063436	SD₁₆	1	1	yes	characteristic
25	D₄×C₁₇₃	—	4	1	no	—
26	C₈×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₁₇₃	—	8	1	no	—
28	C₂×D₁₇₃	—	8	1	no	—
29	C₂×D₁₇₃	—	16	1	no	—
30	C₂×D₁₇₃	—	16	1	no	—
31	C₂×D₁₇₃	D₁₆	1	1	yes	characteristic
32	#10621924	—	16	1	no	—
33	C₄×C₁₇₃	#102	1	1	yes	characteristic
34	Dic₁₇₃	SD₃₂	1	1	yes	characteristic
35	C₂×C₁₇₃	—	16	1	no	—
36	C₂×C₁₇₃	#357	1	1	yes	characteristic
37	D₁₇₃	—	16	1	no	—
38	D₁₇₃	D₃₂	1	1	yes	characteristic
39	D₁₇₃	SD₆₄	1	1	yes	characteristic
40	C₁₇₃	#1415	1	1	yes	characteristic, complemented, Sylow
41	#1415	—	173	1	no	maximal, Sylow
42	C₂×D₁₆	—	173	1	no	—
43	C₂×C₃₂	—	173	1	no	—
44	#366	—	173	1	no	—
45	C₂×D₈	—	346	1	no	—
46	D₁₆	—	346	1	no	—
47	D₁₆	—	173	1	no	—
48	D₁₆	#10621924	1	1	yes	characteristic, complemented
49	C₂×C₁₆	—	173	1	no	—
50	#107	—	346	1	no	—
51	C₃₂	—	346	1	no	—
52	C₂×D₄	—	692	1	no	—
53	D₈	—	2	1	no	—
54	D₈	—	346	1	no	—
55	D₈	—	692	1	no	—
56	C₂×C₈	—	173	1	no	—
57	C₄⋊C₄	—	692	1	no	—
58	C₁₆	—	173	1	no	—
59	C₁₆	C₂×#10621924	1	1	yes	characteristic
60	C₂³	—	1384	1	no	—
61	D₄	—	4	1	no	—
62	D₄	—	1384	1	no	—
63	D₄	—	692	1	no	—
64	C₂×C₄	—	1384	1	no	—
65	C₂×C₄	—	173	1	no	—
66	C₈	—	173	1	no	—
67	C₈	C₂²⋊#10621924	1	1	yes	characteristic
68	C₂²	—	8	1	no	—
69	C₂²	—	1384	1	no	—
70	C₂²	—	173	1	no	—
71	C₂²	—	2768	1	no	—
72	C₂²	—	2768	1	no	—
73	C₄	—	173	1	no	—
74	C₄	—	2768	1	no	—
75	C₄	#19149741	1	1	yes	characteristic
76	C₂	—	16	1	no	—
77	C₂	—	173	1	no	—
78	C₂	—	173	1	no	—
79	C₂	—	2768	1	no	—
80	C₂	#26012403	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26012403
commutator	C₁₆×C₁₇₃	C₂×C₄
Frattini	C₁₆	C₂×#10621924
Fitting	D₁₆×C₁₇₃	C₄
socle	C₂×C₁₇₃	#357
prime radical	D₁₆×D₁₇₃	C₂
cosocle	C₁₆×D₁₇₃	C₂²
hypercenter	D₁₆	#10621924
hypocenter	C₁₇₃	#1415
LCS	G ▷ C₁₆×C₁₇₃ ▷ C₈×C₁₇₃ ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ D₁₆

Complements

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

N	H ≅ G/N	type
1. D₁₇₃⋊C₃₂	C₂	semidirect
2. #26012417	C₂	semidirect
7. D₂₇₆₈	C₄	semidirect
10. D₁₆×C₁₇₃	C₄	semidirect
40. C₁₇₃	#1415	semidirect
48. D₁₆	#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
D₁₆×D₁₇₃	2
#1415	173