Subgroups — #26011301

Order 11072.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19149619	C₂	1	1	yes	maximal, characteristic, complemented
2	#19149611	C₂	1	1	yes	maximal, characteristic, complemented
3	#135×C₁₇₃	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Dic₁₇₃	—	2	1	no	—
5	#14583801	—	2	1	no	—
6	#41×C₁₇₃	—	2	1	no	—
7	Q₁₆×C₁₇₃	C₄	1	1	yes	characteristic, complemented
8	SD₁₆×C₁₇₃	—	2	1	no	—
9	D₈×C₁₇₃	C₄	1	1	yes	characteristic, complemented
10	C₂×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₇₃	—	4	1	no	—
12	Q₈×C₁₇₃	—	2	1	no	—
13	D₄×C₁₇₃	—	2	1	no	—
14	D₄×C₁₇₃	—	4	1	no	—
15	C₂×C₄×C₁₇₃	—	4	1	no	—
16	C₂×C₄×C₁₇₃	D₄	1	1	yes	characteristic
17	C₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
18	C₈×C₁₇₃	D₄	1	1	yes	characteristic
19	C₂²×C₁₇₃	—	4	1	no	—
20	C₂²×C₁₇₃	D₈	1	1	yes	characteristic
21	C₄×C₁₇₃	—	4	1	no	—
22	C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₁₇₃	SD₁₆	1	1	yes	characteristic
24	Dic₁₇₃	—	8	1	no	—
25	C₂×C₁₇₃	—	2	1	no	—
26	C₂×C₁₇₃	—	8	1	no	—
27	C₂×C₁₇₃	#102	1	1	yes	characteristic
28	C₁₇₃	#360	1	1	yes	characteristic, complemented, Sylow
29	#360	—	173	1	no	maximal, Sylow
30	#135	D₁₇₃	1	1	yes	characteristic
31	M₃₂	—	173	1	no	—
32	#106	—	173	1	no	—
33	#41	—	2	1	no	—
34	Q₁₆	Dic₁₇₃	1	1	yes	characteristic, complemented
35	SD₁₆	—	2	1	no	—
36	D₈	Dic₁₇₃	1	1	yes	characteristic, complemented
37	C₂×C₈	C₂×D₁₇₃	1	1	yes	characteristic
38	C₄⋊C₄	—	346	1	no	—
39	C₁₆	—	346	1	no	—
40	Q₈	—	2	1	no	—
41	D₄	—	2	1	no	—
42	D₄	—	4	1	no	—
43	C₂×C₄	—	4	1	no	—
44	C₂×C₄	—	692	1	no	—
45	C₂×C₄	C₂²⋊D₁₇₃	1	1	yes	characteristic
46	C₈	C₂×Dic₁₇₃	1	1	yes	characteristic
47	C₈	C₂²⋊D₁₇₃	1	1	yes	characteristic
48	C₂²	—	4	1	no	—
49	C₂²	#14583811	1	1	yes	characteristic
50	C₄	—	4	1	no	—
51	C₄	—	1384	1	no	—
52	C₄	C₂²⋊Dic₁₇₃	1	1	yes	characteristic
53	C₄	#14583814	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	8	1	no	—
56	C₂	#19149600	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#19149600
commutator	C₈×C₁₇₃	C₂×C₄
Frattini	C₂×C₈	C₂×D₁₇₃
Fitting, prime radical	#135×C₁₇₃	C₂
socle	C₂×C₁₇₃	#102
cosocle	C₂×C₈×C₁₇₃	C₂²
hypercenter	#135	D₁₇₃
hypocenter	C₁₇₃	#360
LCS	G ▷ C₈×C₁₇₃ ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #19149619	C₂	semidirect
2. #19149611	C₂	semidirect
7. Q₁₆×C₁₇₃	C₄	semidirect
9. D₈×C₁₇₃	C₄	semidirect
28. C₁₇₃	#360	semidirect
34. Q₁₆	Dic₁₇₃	semidirect
36. D₈	Dic₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
#19149619	2
#19149611	2
#135×C₁₇₃	2
#360	173