Subgroups — #36368589

Order 22144.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26012352	C₂	1	1	yes	maximal, characteristic
2	#26012352	C₂	1	1	yes	maximal, characteristic
3	#26011266	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#13063443	—	2	1	no	—
5	C₄⋊#13063443	—	2	1	no	—
6	C₂×#14583801	C₄	1	1	yes	characteristic
7	C₈×Dic₁₇₃	C₂²	1	1	yes	characteristic
8	C₂×C₁₆×C₁₇₃	C₄	1	1	yes	characteristic
9	C₂×#13063443	—	4	1	no	—
10	C₂×#13063443	—	4	1	no	—
11	C₂×#13063436	Q₈	1	1	yes	characteristic
12	C₄×Dic₁₇₃	D₄	1	1	yes	characteristic
13	#14583801	C₈	1	2	yes	complemented
14	C₂×C₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
15	C₁₆×C₁₇₃	—	2	1	no	—
16	#13063443	—	8	1	no	—
17	#13063443	—	8	1	no	—
18	C₂×Dic₁₇₃	SD₁₆	1	1	yes	characteristic
19	C₂×Dic₁₇₃	SD₁₆	1	1	yes	characteristic
20	#13063436	—	2	1	no	—
21	C₂×C₄×C₁₇₃	C₄⋊C₄	1	1	yes	characteristic
22	C₈×C₁₇₃	C₂×C₈	1	1	yes	characteristic
23	C₈×C₁₇₃	M₁₆	1	1	yes	characteristic
24	C₂²×C₁₇₃	#106	1	1	yes	characteristic
25	C₄×C₁₇₃	C₄⋊C₈	1	1	yes	characteristic
26	C₄×C₁₇₃	#108	1	1	yes	characteristic
27	Dic₁₇₃	—	2	1	no	—
28	Dic₁₇₃	—	2	1	no	—
29	C₂×C₁₇₃	#334	1	1	yes	characteristic
30	C₂×C₁₇₃	#365	1	1	yes	characteristic
31	C₂×C₁₇₃	#365	1	1	yes	characteristic
32	C₁₇₃	#1368	1	1	yes	characteristic, complemented, Sylow
33	#1368	—	173	1	no	maximal, Sylow
34	#346	—	173	1	no	—
35	#334	—	173	1	no	—
36	#334	—	173	1	no	—
37	C₂×C₁₆	—	173	1	no	—
38	C₂×C₁₆	#10621924	1	1	yes	characteristic
39	C₄⋊C₈	—	346	1	no	—
40	C₄⋊C₈	—	346	1	no	—
41	C₄×C₈	—	173	1	no	—
42	C₂×C₈	—	173	1	no	—
43	C₂×C₈	—	692	1	no	—
44	C₂×C₈	—	692	1	no	—
45	C₂×C₈	C₂×#10621924	1	1	yes	characteristic
46	C₄²	—	173	1	no	—
47	C₁₆	—	2	1	no	—
48	C₁₆	—	173	2	no	—
49	C₂×C₄	—	173	1	no	—
50	C₂×C₄	—	173	1	no	—
51	C₂×C₄	C₄⋊#10621924	1	1	yes	characteristic
52	C₈	—	346	1	no	—
53	C₈	—	1384	1	no	—
54	C₈	—	1384	1	no	—
55	C₈	D₁₇₃⋊C₈	1	1	yes	characteristic
56	C₈	#14583835	1	1	yes	characteristic
57	C₂²	#19149747	1	1	yes	characteristic
58	C₄	—	346	1	no	—
59	C₄	—	346	1	no	—
60	C₄	#19149746	1	1	yes	characteristic
61	C₄	#19149750	1	1	yes	characteristic
62	C₂	#26012353	1	1	yes	characteristic
63	C₂	#26012415	1	1	yes	characteristic
64	C₂	#26012416	1	1	yes	characteristic

	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
13. #14583801	C₈	semidirect
32. C₁₇₃	#1368	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