Subgroups — #36368588

Order 22144.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26012352	C₂	1	2	yes	maximal
2	#26011266	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#13063443	—	2	2	no	—
4	C₂×#14583801	C₄	1	1	yes	characteristic
5	C₈×Dic₁₇₃	C₂²	1	1	yes	characteristic
6	C₂×C₁₆×C₁₇₃	C₄	1	1	yes	characteristic
7	C₂×#13063443	—	4	2	no	—
8	C₂×#13063436	Q₈	1	1	yes	characteristic
9	C₄×Dic₁₇₃	D₄	1	1	yes	characteristic
10	#14583801	—	2	1	no	—
11	C₂×C₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₁₇₃	C₈	1	2	yes	complemented
13	#13063443	—	8	2	no	—
14	C₂×Dic₁₇₃	SD₁₆	1	2	yes	—
15	#13063436	—	2	1	no	—
16	C₂×C₄×C₁₇₃	C₄⋊C₄	1	1	yes	characteristic
17	C₈×C₁₇₃	C₂×C₈	1	1	yes	characteristic
18	C₈×C₁₇₃	M₁₆	1	1	yes	characteristic
19	C₂²×C₁₇₃	#106	1	1	yes	characteristic
20	C₄×C₁₇₃	C₄⋊C₈	1	1	yes	characteristic
21	C₄×C₁₇₃	#108	1	1	yes	characteristic
22	Dic₁₇₃	—	2	2	no	—
23	C₂×C₁₇₃	#334	1	1	yes	characteristic
24	C₂×C₁₇₃	#365	1	2	yes	—
25	C₁₇₃	#1368	1	1	yes	characteristic, complemented, Sylow
26	#1368	—	173	1	no	maximal, Sylow
27	#346	—	173	1	no	—
28	#334	—	173	2	no	—
29	C₂×C₁₆	—	173	1	no	—
30	C₂×C₁₆	#10621924	1	1	yes	characteristic
31	C₄⋊C₈	—	346	2	no	—
32	C₄×C₈	—	173	1	no	—
33	C₂×C₈	—	173	1	no	—
34	C₂×C₈	—	692	2	no	—
35	C₂×C₈	C₂×#10621924	1	1	yes	characteristic
36	C₄²	—	173	1	no	—
37	C₁₆	—	346	1	no	—
38	C₁₆	#13063443	1	2	yes	complemented
39	C₂×C₄	—	173	2	no	—
40	C₂×C₄	C₄⋊#10621924	1	1	yes	characteristic
41	C₈	—	346	1	no	—
42	C₈	—	1384	2	no	—
43	C₈	C₂×#13063443	1	1	yes	characteristic
44	C₈	#14583836	1	1	yes	characteristic
45	C₂²	#19149747	1	1	yes	characteristic
46	C₄	—	346	2	no	—
47	C₄	C₄⋊#13063443	1	1	yes	characteristic
48	C₄	#19149749	1	1	yes	characteristic
49	C₂	#26012352	1	1	yes	characteristic
50	C₂	#26012416	1	2	yes	—

	subgroup	G/H
center	C₂²	#19149747
commutator	C₈×C₁₇₃	C₂×C₈
Frattini	C₂×C₈	C₂×#10621924
Fitting	C₂×C₁₆×C₁₇₃	C₄
socle, prime radical	C₂²×C₁₇₃	#106
cosocle	C₈×Dic₁₇₃	C₂²
hypercenter	C₂×C₁₆	#10621924
hypocenter	C₁₇₃	#1368
LCS	G ▷ C₈×C₁₇₃ ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
12. C₁₆×C₁₇₃	C₈	semidirect
25. C₁₇₃	#1368	semidirect
38. C₁₆	#13063443	semidirect

Maximal subgroups

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

subgroup	index
#26012352	2
#26011266	2
#1368	173