Subgroups — #10619607

Order 648.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×AΣL(1,𝔽₂₇)	C₂	1	1	yes	maximal, characteristic
2	#66321	C₂	1	2	yes	maximal
3	C₃²×Dic₆	—	3	1	no	maximal
4	#6229	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂×AΣL(1,𝔽₂₇)	C₂²	1	1	yes	characteristic
6	C₃³×C₄	D₃	1	1	yes	characteristic
7	C₃²×Dic₃	—	3	2	no	—
8	C₄×3₋¹⁺²	—	2	1	no	—
9	C₄×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic
10	#1084	C₂×C₃	1	2	yes	—
11	AΣL(1,𝔽₂₇)	Q₈	1	1	yes	characteristic, complemented, Sylow
12	C₃²×Q₈	—	9	1	no	—
13	C₃×Dic₆	—	3	1	no	—
14	C₃×Dic₆	—	3	1	no	—
15	C₃×Dic₆	—	3	1	no	—
16	C₃×Dic₆	—	3	1	no	—
17	C₃×Dic₆	—	9	1	no	—
18	C₂×C₃³	C₂×D₃	1	1	yes	characteristic
19	C₂×3₋¹⁺²	—	2	1	no	—
20	C₂×H(3,𝔽₃)	C₂²×C₃	1	1	yes	characteristic
21	C₃²×C₄	—	3	1	no	—
22	C₃²×C₄	—	3	1	no	—
23	C₃²×C₄	—	3	1	no	—
24	C₃²×C₄	—	6	1	no	—
25	C₃²×C₄	—	6	1	no	—
26	C₃²×C₄	—	9	2	no	—
27	C₃²×C₄	C₃×D₃	1	1	yes	characteristic
28	C₃×Dic₃	—	3	2	no	—
29	C₃×Dic₃	—	3	2	no	—
30	C₃×Dic₃	—	3	2	no	—
31	C₃×Dic₃	—	3	2	no	—
32	C₃×Dic₃	—	9	2	no	—
33	C₄×C₉	—	6	1	no	—
34	C₃³	Dic₆	1	1	yes	characteristic, complemented
35	3₋¹⁺²	—	2	1	no	—
36	H(3,𝔽₃)	C₃×Q₈	1	1	yes	characteristic, complemented
37	C₃×Q₈	—	9	1	no	—
38	C₃×Q₈	—	9	1	no	—
39	C₃×Q₈	—	9	1	no	—
40	C₃×Q₈	—	9	1	no	—
41	Dic₆	—	3	1	no	—
42	Dic₆	—	9	1	no	—
43	C₂×C₃²	—	3	1	no	—
44	C₂×C₃²	—	3	1	no	—
45	C₂×C₃²	—	3	1	no	—
46	C₂×C₃²	—	6	1	no	—
47	C₂×C₃²	—	6	1	no	—
48	C₂×C₃²	C₂×C₃×D₃	1	1	yes	characteristic
49	C₂×C₉	—	6	1	no	—
50	C₃×C₄	—	3	1	no	—
51	C₃×C₄	—	6	1	no	—
52	C₃×C₄	—	6	1	no	—
53	C₃×C₄	—	6	1	no	—
54	C₃×C₄	—	9	2	no	—
55	C₃×C₄	—	9	2	no	—
56	C₃×C₄	—	9	2	no	—
57	C₃×C₄	—	9	2	no	—
58	C₃×C₄	—	9	1	no	—
59	C₃×C₄	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
60	Dic₃	—	3	2	no	—
61	Dic₃	—	9	2	no	—
62	C₃²	—	3	1	no	—
63	C₃²	—	3	1	no	—
64	C₃²	—	3	1	no	—
65	C₃²	—	6	1	no	—
66	C₃²	—	6	1	no	—
67	C₃²	C₃×Dic₆	1	1	yes	characteristic
68	C₉	—	6	1	no	—
69	Q₈	—	9	1	no	Sylow
70	C₂×C₃	—	3	1	no	—
71	C₂×C₃	—	6	1	no	—
72	C₂×C₃	—	6	1	no	—
73	C₂×C₃	—	6	1	no	—
74	C₂×C₃	—	9	1	no	—
75	C₂×C₃	C₂×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
76	C₄	—	9	2	no	—
77	C₄	#4174	1	1	yes	characteristic
78	C₃	—	3	1	no	—
79	C₃	—	6	1	no	—
80	C₃	—	6	1	no	—
81	C₃	—	6	1	no	—
82	C₃	—	9	1	no	—
83	C₃	#634⋊C₃	1	1	yes	characteristic
84	C₂	C₂×#4174	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	C₂×(C₃⋊D₃)⋊C₃
commutator, cosocle	C₂×H(3,𝔽₃)	C₂²×C₃
Frattini	C₂×C₃²	C₂×C₃×D₃
Fitting	C₄×AΣL(1,𝔽₂₇)	C₂
prime radical	C₂×AΣL(1,𝔽₂₇)	C₂²
hypercenter	C₃×C₄	(C₃⋊D₃)⋊C₃
hypocenter	H(3,𝔽₃)	C₃×Q₈
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄

Complements

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

N	H ≅ G/N	type
4. #6229	C₃	semidirect
11. AΣL(1,𝔽₂₇)	Q₈	semidirect
34. C₃³	Dic₆	semidirect
36. H(3,𝔽₃)	C₃×Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×AΣL(1,𝔽₂₇)	2
#66321	2
C₃²×Dic₆	3
#6229	3