Subgroups — #36368584

Order 22144.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26012414	C₂	1	1	yes	maximal, characteristic
2	#26012414	C₂	1	1	yes	maximal, characteristic
3	#26011890	C₂	1	1	yes	maximal, characteristic
4	#19149753	—	2	1	no	—
5	#19149753	—	2	1	no	—
6	#19149671	C₄	1	1	yes	characteristic
7	#19149612	—	2	1	no	—
8	#19149610	—	2	1	no	—
9	C₄×#13063436	C₂²	1	1	yes	characteristic
10	#125×C₁₇₃	C₄	1	1	yes	characteristic
11	#14583829	—	4	1	no	—
12	#14583829	—	4	1	no	—
13	C₂×#13063436	D₄	1	1	yes	characteristic
14	C₂×#13063436	D₄	1	1	yes	characteristic
15	C₄⋊Dic₁₇₃	—	2	1	no	—
16	#14583805	—	4	1	no	—
17	C₄⋊C₄×C₁₇₃	—	2	1	no	—
18	C₄⋊C₄×C₁₇₃	—	4	1	no	—
19	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₁₇₃	—	4	1	no	—
21	#13063436	SD₁₆	1	1	yes	characteristic
22	#13063436	SD₁₆	1	1	yes	characteristic
23	#13063436	Q₁₆	1	1	yes	characteristic
24	#13063436	Q₁₆	1	1	yes	characteristic
25	C₂×C₄×C₁₇₃	—	2	1	no	—
26	C₂×C₄×C₁₇₃	—	4	1	no	—
27	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂²×C₁₇₃	#100	1	1	yes	characteristic
29	C₄×C₁₇₃	—	4	1	no	—
30	C₄×C₁₇₃	—	8	1	no	—
31	C₄×C₁₇₃	#103	1	1	yes	characteristic
32	C₄×C₁₇₃	#103	1	1	yes	characteristic
33	Dic₁₇₃	—	8	1	no	—
34	C₂×C₁₇₃	—	2	1	no	—
35	C₂×C₁₇₃	#333	1	1	yes	characteristic
36	C₁₇₃	#1366	1	1	yes	characteristic, complemented, Sylow
37	#1366	—	173	1	no	maximal, Sylow
38	#499	—	173	1	no	—
39	#364	—	173	1	no	—
40	#364	—	173	1	no	—
41	#125	—	173	1	no	—
42	#125	#10621924	1	1	yes	characteristic
43	M₃₂	—	346	1	no	—
44	M₃₂	—	346	1	no	—
45	#107	—	346	1	no	—
46	#106	—	346	1	no	—
47	C₄×C₈	—	173	1	no	—
48	C₂×C₈	—	173	1	no	—
49	C₂×C₈	—	173	1	no	—
50	C₄⋊C₄	—	2	1	no	—
51	C₄⋊C₄	—	4	1	no	—
52	C₄⋊C₄	—	346	1	no	—
53	C₄⋊C₄	—	692	1	no	—
54	C₄²	C₂×#10621924	1	1	yes	characteristic
55	C₁₆	—	692	1	no	—
56	C₁₆	—	692	1	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	4	1	no	—
59	C₂×C₄	—	692	1	no	—
60	C₂×C₄	C₂²⋊#10621924	1	1	yes	characteristic
61	C₈	—	173	1	no	—
62	C₈	—	173	1	no	—
63	C₈	—	173	1	no	—
64	C₈	—	173	1	no	—
65	C₂²	#19149739	1	1	yes	characteristic
66	C₄	—	4	1	no	—
67	C₄	—	8	1	no	—
68	C₄	—	1384	1	no	—
69	C₄	#19149742	1	1	yes	characteristic
70	C₄	#19149742	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	#26012351	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26012351
commutator	C₄²×C₁₇₃	C₂×C₄
Frattini	C₄²	C₂×#10621924
Fitting	#125×C₁₇₃	C₄
socle	C₂×C₁₇₃	#333
prime radical	C₂²×C₁₇₃	#100
cosocle	C₄×#13063436	C₂²
hypercenter	#125	#10621924
hypocenter	C₁₇₃	#1366
LCS	G ▷ C₄²×C₁₇₃ ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
36. C₁₇₃	#1366	semidirect

Maximal subgroups

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

subgroup	index
#26012414	2
#26011890	2
#1366	173