Subgroups — #14517202

Order 1944.

179 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10619912	C₃	1	1	yes	maximal, characteristic
2	C₃×Q₈×C₂₇	C₃	1	1	yes	maximal, characteristic, complemented
3	#10619570	C₃	1	2	yes	maximal, complemented
4	C₂₇⋊SL(2,𝔽₃)	C₃	1	6	yes	maximal, complemented
5	SL(2,𝔽₃)×C₂₇	C₃	1	3	yes	maximal, complemented
6	C₂×#6903	—	4	1	no	maximal
7	C₃×C₄×C₂₇	—	3	1	no	—
8	#6903	—	4	1	no	Sylow
9	C₃×Q₈×C₉	C₃²	1	1	yes	characteristic
10	Q₈⋊H(3,𝔽₃)	C₉	1	1	yes	characteristic
11	#6204	C₉	1	2	yes	—
12	C₉⋊SL(2,𝔽₃)	C₉	1	6	yes	—
13	C₉×SL(2,𝔽₃)	C₃²	1	3	yes	—
14	Q₈×C₂₇	C₃²	1	1	yes	characteristic
15	Q₈×C₂₇	C₃²	1	2	yes	—
16	Q₈⋊C₂₇	C₃²	1	6	yes	—
17	C₂×#737	—	4	1	no	—
18	C₂×C₂₇⋊C₃	—	4	2	no	—
19	C₂×C₂₇⋊C₃	—	4	6	no	—
20	C₂×C₃×C₂₇	—	4	3	no	—
21	C₂×C₃×C₂₇	A₄	1	1	yes	characteristic
22	C₃×C₄×C₉	—	3	1	no	—
23	C₄×C₂₇	—	3	2	no	—
24	C₄×C₂₇	—	3	1	no	—
25	#737	—	4	1	no	—
26	C₂₇⋊C₃	—	4	2	no	—
27	C₂₇⋊C₃	—	4	6	no	—
28	C₃×C₂₇	—	4	3	no	—
29	C₃×C₂₇	SL(2,𝔽₃)	1	1	yes	characteristic, complemented
30	C₃²×Q₈	C₃×C₉	1	1	yes	characteristic
31	C₃×SL(2,𝔽₃)	C₃×C₉	1	3	yes	—
32	Q₈×C₉	C₃×C₉	1	2	yes	—
33	Q₈×C₉	C₃³	1	1	yes	characteristic
34	Q₈⋊C₉	C₃×C₉	1	6	yes	—
35	C₂×3₋¹⁺²	—	4	2	no	—
36	C₂×3₋¹⁺²	—	4	6	no	—
37	C₂×H(3,𝔽₃)	—	4	1	no	—
38	C₂×C₃×C₉	—	4	3	no	—
39	C₂×C₃×C₉	C₃×A₄	1	1	yes	characteristic
40	C₂×C₂₇	—	4	6	no	—
41	C₂×C₂₇	C₃×A₄	1	2	yes	—
42	C₂×C₂₇	C₃×A₄	1	1	yes	characteristic
43	C₃²×C₄	—	3	1	no	—
44	C₄×C₉	—	3	2	no	—
45	C₄×C₉	—	3	1	no	—
46	3₋¹⁺²	—	4	2	no	—
47	3₋¹⁺²	—	4	6	no	—
48	H(3,𝔽₃)	—	4	1	no	—
49	C₃×C₉	—	4	3	no	—
50	C₃×C₉	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
51	C₂₇	—	4	6	no	—
52	C₂₇	C₃×SL(2,𝔽₃)	1	2	yes	—
53	C₂₇	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
54	C₃×Q₈	—	3	1	no	—
55	C₃×Q₈	C₃²×C₉	1	1	yes	characteristic
56	SL(2,𝔽₃)	—	3	3	no	—
57	C₂×C₃²	—	4	3	no	—
58	C₂×C₃²	C₉×A₄	1	1	yes	characteristic
59	C₂×C₉	—	4	6	no	—
60	C₂×C₉	C₉×A₄	1	2	yes	—
61	C₂×C₉	C₃²×A₄	1	1	yes	characteristic
62	C₃×C₄	—	3	3	no	—
63	C₃×C₄	—	3	1	no	—
64	C₃²	—	4	3	no	—
65	C₃²	C₉×SL(2,𝔽₃)	1	1	yes	characteristic
66	C₉	—	4	6	no	—
67	C₉	C₉×SL(2,𝔽₃)	1	2	yes	—
68	C₉	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
69	Q₈	#6903	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₃	—	3	1	no	—
71	C₂×C₃	—	12	3	no	—
72	C₂×C₃	C₃×C₉×A₄	1	1	yes	characteristic
73	C₄	—	3	1	no	—
74	C₃	—	3	1	no	—
75	C₃	—	12	3	no	—
76	C₃	C₃×C₉×SL(2,𝔽₃)	1	1	yes	characteristic
77	C₂	#11757795	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₂₇	C₃×A₄
commutator	C₃×Q₈	C₃²×C₉
Frattini	C₂×C₉	C₃²×A₄
Fitting	C₃×Q₈×C₂₇	C₃
socle	C₂×C₃	C₃×C₉×A₄
prime radical	Q₈⋊H(3,𝔽₃)	C₉
cosocle	Q₈×C₉	C₃³
hypercenter	C₂×C₃×C₂₇	A₄
hypocenter	Q₈	#6903
LCS	G ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₂₇ ◁ C₂×C₃×C₂₇

Complements

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

N	H ≅ G/N	type
2. C₃×Q₈×C₂₇	C₃	semidirect
3. #10619570	C₃	semidirect
4. C₂₇⋊SL(2,𝔽₃)	C₃	semidirect
5. SL(2,𝔽₃)×C₂₇	C₃	semidirect
29. C₃×C₂₇	SL(2,𝔽₃)	semidirect
69. Q₈	#6903	semidirect

Maximal subgroups

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

subgroup	index
#10619912	3
C₃×Q₈×C₂₇	3
#10619570	3
C₂₇⋊SL(2,𝔽₃)	3
SL(2,𝔽₃)×C₂₇	3
C₂×#6903	4