Subgroups — #36351483

Order 22144.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011267	C₂	1	1	yes	maximal, characteristic
2	#26011212	C₂	1	1	yes	maximal, characteristic
3	#346×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	—	2	2	no	—
9	#14583801	—	2	2	no	—
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	—	4	2	no	—
15	C₂×C₄×C₁₇₃	C₄²	1	1	yes	characteristic
16	C₂×C₄×C₁₇₃	M₁₆	1	2	yes	—
17	C₈×C₁₇₃	C₂×C₈	1	2	yes	—
18	C₈×C₁₇₃	M₁₆	1	2	yes	—
19	C₂²×C₁₇₃	#97	1	1	yes	characteristic
20	C₄×C₁₇₃	—	2	2	no	—
21	C₄×C₁₇₃	C₄×C₈	1	1	yes	characteristic
22	C₄×C₁₇₃	#97	1	1	yes	characteristic
23	C₂×C₁₇₃	#322	1	1	yes	characteristic
24	C₂×C₁₇₃	#347	1	2	yes	—
25	C₁₇₃	#1313	1	1	yes	characteristic, complemented, Sylow
26	#1313	—	173	1	no	maximal, Sylow
27	#346	—	173	1	no	—
28	#346	D₁₇₃	1	1	yes	characteristic
29	#322	—	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	—	346	2	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	—	346	2	no	—
39	C₁₆	#13063436	1	4	yes	complemented
40	C₂×C₄	C₄×Dic₁₇₃	1	1	yes	characteristic
41	C₂×C₄	#14583809	1	2	yes	—
42	C₈	—	692	2	no	—
43	C₈	C₂×#13063436	1	2	yes	—
44	C₈	#14583809	1	2	yes	—
45	C₂²	#19149589	1	1	yes	characteristic
46	C₄	—	2	2	no	—
47	C₄	C₄×#13063436	1	1	yes	characteristic
48	C₄	#19149589	1	1	yes	characteristic
49	C₂	#26011212	1	1	yes	characteristic
50	C₂	#26011268	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄×Dic₁₇₃
commutator	C₄×C₁₇₃	C₄×C₈
Frattini	C₄×C₈	C₂×D₁₇₃
Fitting	#346×C₁₇₃	C₂
socle, prime radical	C₂²×C₁₇₃	#97
cosocle	C₄×C₈×C₁₇₃	C₂²
hypercenter	#346	D₁₇₃
hypocenter	C₁₇₃	#1313
LCS	G ▷ C₄×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #346

Complements

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

N	H ≅ G/N	type
13. C₁₆×C₁₇₃	C₈	semidirect
25. C₁₇₃	#1313	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
#26011267	2
#26011212	2
#346×C₁₇₃	2
#1313	173