Subgroups — #11772227

Order 1152.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10593253	C₂	1	2	yes	maximal
2	C₁₆×#157	C₂	1	1	yes	maximal, characteristic
3	C₄⋊AGL(1,𝔽₉)	—	2	2	no	—
4	C₈×#157	C₂²	1	1	yes	characteristic
5	#63764	C₄	1	1	yes	characteristic
6	C₁₆×C₃⋊D₃	C₄	1	1	yes	characteristic
7	C₂×AGL(1,𝔽₉)	—	4	2	no	—
8	C₄×#157	D₄	1	1	yes	characteristic
9	#3787	Q₈	1	1	yes	characteristic
10	C₈×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
11	#3708	—	2	1	no	—
12	C₃²×C₁₆	C₈	1	1	yes	characteristic, complemented
13	C₃⋊(C₃⋊C₁₆)	C₈	1	1	yes	characteristic, complemented
14	#1371	—	9	1	no	maximal, Sylow
15	D₃×C₁₆	—	12	1	no	—
16	C₂×#157	D₈	1	1	yes	characteristic
17	C₂×#157	Q₁₆	1	1	yes	characteristic
18	AGL(1,𝔽₉)	—	8	2	no	—
19	C₄×C₃⋊D₃	C₄⋊C₄	1	1	yes	characteristic
20	#622	—	2	1	no	—
21	C₃²×C₈	C₂×C₈	1	1	yes	characteristic
22	C₃⋊(C₃⋊C₈)	M₁₆	1	1	yes	characteristic
23	C₄×C₁₆	—	9	1	no	—
24	#335	—	9	2	no	—
25	D₃×C₈	—	12	1	no	—
26	C₃×C₁₆	—	4	1	no	—
27	C₃⋊C₁₆	—	12	1	no	—
28	C₂×C₃⋊D₃	#107	1	1	yes	characteristic
29	#157	—	2	1	no	—
30	#157	D₁₆	1	1	yes	characteristic
31	#157	Q₃₂	1	1	yes	characteristic
32	C₃²×C₄	C₄⋊C₈	1	1	yes	characteristic
33	C₃⋊Dic₃	#108	1	1	yes	characteristic
34	C₂×C₁₆	—	9	1	no	—
35	C₂×C₁₆	—	9	1	no	—
36	C₄⋊C₈	—	18	2	no	—
37	C₄×C₈	—	9	1	no	—
38	C₄×D₃	—	12	1	no	—
39	C₃×C₈	—	4	1	no	—
40	C₃⋊C₈	—	12	1	no	—
41	C₂×C₃²	#335	1	1	yes	characteristic
42	C₃⋊D₃	#366	1	1	yes	characteristic
43	C₃⋊D₃	#368	1	1	yes	characteristic
44	C₂×C₈	—	9	1	no	—
45	C₂×C₈	—	9	1	no	—
46	C₂×C₈	—	36	2	no	—
47	C₄²	—	9	1	no	—
48	C₁₆	—	9	1	no	—
49	C₁₆	—	18	1	no	—
50	C₁₆	AGL(1,𝔽₉)	1	1	yes	characteristic, complemented
51	C₂×D₃	—	12	1	no	—
52	C₃×C₄	—	4	1	no	—
53	Dic₃	—	12	1	no	—
54	C₃²	#1371	1	1	yes	characteristic, complemented, Sylow
55	C₂×C₄	—	9	1	no	—
56	C₂×C₄	—	9	1	no	—
57	C₂×C₄	—	9	1	no	—
58	C₈	—	9	1	no	—
59	C₈	—	18	1	no	—
60	C₈	—	72	2	no	—
61	C₈	C₂×AGL(1,𝔽₉)	1	1	yes	characteristic
62	C₂×C₃	—	4	1	no	—
63	D₃	—	12	1	no	—
64	D₃	—	12	1	no	—
65	C₂²	—	9	1	no	—
66	C₄	—	9	1	no	—
67	C₄	—	9	1	no	—
68	C₄	—	9	1	no	—
69	C₄	—	18	1	no	—
70	C₄	C₄⋊AGL(1,𝔽₉)	1	1	yes	characteristic
71	C₃	—	4	1	no	—
72	C₂	—	9	1	no	—
73	C₂	—	9	1	no	—
74	C₂	#10593253	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#10593253
commutator	C₃²×C₈	C₂×C₈
Frattini	C₈	C₂×AGL(1,𝔽₉)
Fitting	C₃²×C₁₆	C₈
socle	C₂×C₃²	#335
prime radical	C₂×C₃⋊D₃	#107
cosocle	C₈×#157	C₂²
hypercenter	C₁₆	AGL(1,𝔽₉)
hypocenter	C₃²	#1371
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

Complements

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

N	H ≅ G/N	type
12. C₃²×C₁₆	C₈	semidirect
13. C₃⋊(C₃⋊C₁₆)	C₈	semidirect
50. C₁₆	AGL(1,𝔽₉)	semidirect
54. C₃²	#1371	semidirect

Maximal subgroups

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

subgroup	index
#10593253	2
C₁₆×#157	2
#1371	9