Subgroups — #26012308

Order 11072.

248 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19149726	C₂	1	2	yes	maximal, complemented
2	#19149712	C₂	1	6	yes	maximal, complemented
3	#19149701	C₂	1	3	yes	maximal, complemented
4	C₂×Dic₆₉₂	C₂	1	3	yes	maximal, complemented
5	#131×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583826	C₂²	1	6	yes	—
7	#14583825	—	2	6	no	—
8	C₂×Dic₃₄₆	C₂²	1	6	yes	—
9	Q₈×D₁₇₃	—	2	2	no	—
10	Dic₆₉₂	C₂²	1	3	yes	—
11	Dic₆₉₂	C₂²	1	6	yes	—
12	#14583815	C₂²	1	6	yes	—
13	D₁₃₈₄	C₂²	1	1	yes	characteristic
14	#41×C₁₇₃	C₂²	1	1	yes	characteristic
15	M₁₆×C₁₇₃	C₂²	1	3	yes	—
16	C₂×C₈×C₁₇₃	C₂²	1	3	yes	—
17	Dic₃₄₆	—	2	6	no	—
18	Dic₃₄₆	C₂³	1	6	yes	—
19	D₆₉₂	C₂³	1	2	yes	—
20	C₂²⋊D₁₇₃	—	2	6	no	—
21	C₂×Dic₁₇₃	—	2	6	no	—
22	C₄×D₁₇₃	—	2	6	no	—
23	Q₈×C₁₇₃	D₄	1	1	yes	characteristic
24	D₄×C₁₇₃	D₄	1	3	yes	—
25	C₂×C₄×C₁₇₃	C₂³	1	3	yes	—
26	C₈×C₁₇₃	C₂³	1	1	yes	characteristic
27	C₈×C₁₇₃	C₂³	1	3	yes	—
28	C₂²×C₁₇₃	C₂×D₄	1	3	yes	—
29	C₂×D₁₇₃	—	2	2	no	—
30	C₄×C₁₇₃	C₂×D₄	1	3	yes	—
31	C₄×C₁₇₃	C₂⁴	1	1	yes	characteristic
32	Dic₁₇₃	—	2	6	no	—
33	C₂×C₁₇₃	—	2	3	no	—
34	C₂×C₁₇₃	C₂²×D₄	1	1	yes	characteristic
35	D₁₇₃	—	4	2	no	—
36	C₁₇₃	#578	1	1	yes	characteristic, complemented, Sylow
37	#578	—	173	1	no	maximal, Sylow
38	2₋¹⁺⁴	—	173	2	no	—
39	#137	—	173	6	no	—
40	#135	—	173	3	no	—
41	C₂×Q₁₆	—	173	3	no	—
42	#131	D₁₇₃	1	1	yes	characteristic, complemented
43	#41	—	346	6	no	—
44	#41	—	173	6	no	—
45	#41	C₂×D₁₇₃	1	1	yes	characteristic
46	C₂×Q₈	—	346	2	no	—
47	C₂×Q₈	—	173	6	no	—
48	Q₁₆	—	173	3	no	—
49	Q₁₆	—	173	6	no	—
50	SD₁₆	—	173	6	no	—
51	D₈	—	173	1	no	—
52	M₁₆	C₂×D₁₇₃	1	3	yes	—
53	C₂×C₈	C₂×D₁₇₃	1	3	yes	—
54	Q₈	—	346	6	no	—
55	Q₈	—	173	6	no	—
56	Q₈	D₆₉₂	1	1	yes	characteristic
57	D₄	—	346	6	no	—
58	D₄	—	173	2	no	—
59	D₄	D₆₉₂	1	3	yes	—
60	C₂×C₄	—	346	6	no	—
61	C₂×C₄	—	346	6	no	—
62	C₂×C₄	C₂²×D₁₇₃	1	3	yes	—
63	C₈	C₂²×D₁₇₃	1	1	yes	characteristic
64	C₈	C₂²×D₁₇₃	1	3	yes	—
65	C₂²	—	346	2	no	—
66	C₂²	C₂×D₆₉₂	1	3	yes	—
67	C₄	—	346	6	no	—
68	C₄	C₂×D₆₉₂	1	3	yes	—
69	C₄	C₂³×D₁₇₃	1	1	yes	characteristic
70	C₂	—	2	3	no	—
71	C₂	—	692	2	no	—
72	C₂	C₂²×D₆₉₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₆₉₂
commutator, cosocle	C₄×C₁₇₃	C₂⁴
Frattini	C₄	C₂³×D₁₇₃
Fitting	#131×C₁₇₃	C₂
socle	C₂×C₁₇₃	C₂²×D₄
hypercenter	#131	D₁₇₃
hypocenter	C₁₇₃	#578
LCS	G ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
1. #19149726	C₂	semidirect
2. #19149712	C₂	semidirect
3. #19149701	C₂	semidirect
4. C₂×Dic₆₉₂	C₂	semidirect
5. #131×C₁₇₃	C₂	semidirect
36. C₁₇₃	#578	semidirect
42. #131	D₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
#19149726	2
#19149712	2
#19149701	2
C₂×Dic₆₉₂	2
#131×C₁₇₃	2
#578	173