Subgroups — #36351624

Order 22144.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011312	C₂	1	1	yes	maximal, characteristic
2	#26011312	C₂	1	1	yes	maximal, characteristic
3	#499×C₁₇₃	C₂	1	1	yes	maximal, characteristic
4	#19149619	—	2	1	no	—
5	#19149619	—	2	1	no	—
6	#125×C₁₇₃	C₄	1	2	yes	—
7	#107×C₁₇₃	—	2	1	no	—
8	#106×C₁₇₃	—	2	1	no	—
9	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
10	#14583801	—	4	1	no	—
11	#14583801	—	4	1	no	—
12	C₂×C₈×C₁₇₃	D₄	1	1	yes	characteristic
13	C₂×C₈×C₁₇₃	D₄	1	1	yes	characteristic
14	C₄⋊C₄×C₁₇₃	—	2	2	no	—
15	C₄⋊C₄×C₁₇₃	—	4	2	no	—
16	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₁₇₃	—	2	1	no	—
18	C₂×C₄×C₁₇₃	—	4	2	no	—
19	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
20	C₈×C₁₇₃	SD₁₆	1	1	yes	characteristic
21	C₈×C₁₇₃	SD₁₆	1	1	yes	characteristic
22	C₈×C₁₇₃	Q₁₆	1	1	yes	characteristic
23	C₈×C₁₇₃	Q₁₆	1	1	yes	characteristic
24	C₂²×C₁₇₃	#100	1	1	yes	characteristic
25	C₄×C₁₇₃	—	4	1	no	—
26	C₄×C₁₇₃	—	8	2	no	—
27	C₄×C₁₇₃	#103	1	1	yes	characteristic
28	C₄×C₁₇₃	#103	1	1	yes	characteristic
29	C₂×C₁₇₃	—	2	1	no	—
30	C₂×C₁₇₃	#333	1	1	yes	characteristic
31	C₁₇₃	#1366	1	1	yes	characteristic, complemented, Sylow
32	#1366	—	173	1	no	maximal, Sylow
33	#499	D₁₇₃	1	1	yes	characteristic
34	#364	—	173	1	no	—
35	#364	—	173	1	no	—
36	#125	Dic₁₇₃	1	2	yes	—
37	M₃₂	—	346	1	no	—
38	M₃₂	—	346	1	no	—
39	#107	—	2	1	no	—
40	#106	—	2	1	no	—
41	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic
42	C₂×C₈	C₂²⋊D₁₇₃	1	1	yes	characteristic
43	C₂×C₈	C₂²⋊D₁₇₃	1	1	yes	characteristic
44	C₄⋊C₄	—	2	2	no	—
45	C₄⋊C₄	—	4	2	no	—
46	C₄²	C₂×Dic₁₇₃	1	1	yes	characteristic
47	C₁₆	—	692	1	no	—
48	C₁₆	—	692	1	no	—
49	C₂×C₄	—	2	1	no	—
50	C₂×C₄	—	4	2	no	—
51	C₂×C₄	C₂²⋊Dic₁₇₃	1	1	yes	characteristic
52	C₈	#14583813	1	1	yes	characteristic
53	C₈	#14583813	1	1	yes	characteristic
54	C₈	#14583816	1	1	yes	characteristic
55	C₈	#14583816	1	1	yes	characteristic
56	C₂²	#19149596	1	1	yes	characteristic
57	C₄	—	4	1	no	—
58	C₄	—	8	2	no	—
59	C₄	#19149603	1	1	yes	characteristic
60	C₄	#19149603	1	1	yes	characteristic
61	C₂	—	2	1	no	—
62	C₂	#26011240	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26011240
commutator	C₄²×C₁₇₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₇₃
Fitting	#499×C₁₇₃	C₂
socle	C₂×C₁₇₃	#333
prime radical	C₂²×C₁₇₃	#100
cosocle	C₄×C₈×C₁₇₃	C₂²
hypercenter	#499	D₁₇₃
hypocenter	C₁₇₃	#1366
LCS	G ▷ C₄²×C₁₇₃ ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #499

Complements

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

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

Maximal subgroups

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

subgroup	index
#26011312	2
#499×C₁₇₃	2
#1366	173