Subgroups — #36351760

Order 22144.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011325	C₂	1	1	yes	maximal, characteristic
2	C₁₆×Dic₁₇₃	C₂	1	1	yes	maximal, characteristic
3	M₆₄×C₁₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14583801	C₄	1	1	yes	characteristic
5	C₈×Dic₁₇₃	C₄	1	1	yes	characteristic
6	#19149585	—	2	1	no	—
7	C₂×C₁₆×C₁₇₃	C₂²	1	1	yes	characteristic
8	C₃₂×C₁₇₃	—	2	1	no	—
9	C₂×#13063436	C₈	1	1	yes	characteristic
10	C₄×Dic₁₇₃	C₈	1	1	yes	characteristic
11	#14583801	—	2	1	no	—
12	C₂×C₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
13	C₁₆×C₁₇₃	D₄	1	1	yes	characteristic
14	C₁₆×C₁₇₃	Q₈	1	1	yes	characteristic
15	C₂×Dic₁₇₃	—	2	1	no	—
16	#13063436	C₁₆	1	2	yes	—
17	C₂×C₄×C₁₇₃	C₂×C₈	1	1	yes	characteristic
18	C₈×C₁₇₃	C₄⋊C₄	1	1	yes	characteristic
19	C₈×C₁₇₃	M₁₆	1	1	yes	characteristic
20	C₂²×C₁₇₃	M₃₂	1	1	yes	characteristic
21	C₄×C₁₇₃	C₄⋊C₈	1	1	yes	characteristic
22	C₄×C₁₇₃	C₂×C₁₆	1	1	yes	characteristic
23	Dic₁₇₃	—	2	2	no	—
24	C₂×C₁₇₃	—	2	1	no	—
25	C₂×C₁₇₃	C₄⋊C₁₆	1	1	yes	characteristic
26	C₁₇₃	#1422	1	1	yes	characteristic, complemented, Sylow
27	#1422	—	173	1	no	maximal, Sylow
28	M₆₄	—	173	1	no	—
29	M₆₄	D₁₇₃	1	1	yes	characteristic
30	C₄×C₁₆	—	173	1	no	—
31	C₂×C₁₆	—	173	1	no	—
32	C₂×C₁₆	C₂×D₁₇₃	1	1	yes	characteristic
33	C₄×C₈	—	173	1	no	—
34	C₃₂	—	2	1	no	—
35	C₃₂	—	346	1	no	—
36	C₂×C₈	—	173	1	no	—
37	C₂×C₈	C₄×D₁₇₃	1	1	yes	characteristic
38	C₄²	—	173	1	no	—
39	C₁₆	—	346	1	no	—
40	C₁₆	C₂²⋊D₁₇₃	1	1	yes	characteristic
41	C₁₆	Dic₃₄₆	1	1	yes	characteristic
42	C₂×C₄	—	346	1	no	—
43	C₂×C₄	C₈×D₁₇₃	1	1	yes	characteristic
44	C₈	—	173	2	no	—
45	C₈	#14583805	1	1	yes	characteristic
46	C₈	#14583809	1	1	yes	characteristic
47	C₂²	#19149618	1	1	yes	characteristic
48	C₄	—	346	2	no	—
49	C₄	#19149608	1	1	yes	characteristic
50	C₄	C₁₆×D₁₇₃	1	1	yes	characteristic
51	C₂	—	2	1	no	—
52	C₂	#26011309	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂²⋊D₁₇₃
commutator	C₄×C₁₇₃	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₁₇₃
Fitting	M₆₄×C₁₇₃	C₂
socle	C₂×C₁₇₃	C₄⋊C₁₆
prime radical	C₂²×C₁₇₃	M₃₂
cosocle	C₂×C₁₆×C₁₇₃	C₂²
hypercenter	M₆₄	D₁₇₃
hypocenter	C₁₇₃	#1422
LCS	G ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
26. C₁₇₃	#1422	semidirect

Maximal subgroups

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

subgroup	index
#26011325	2
C₁₆×Dic₁₇₃	2
M₆₄×C₁₇₃	2
#1422	173