Subgroups — D₃×Q₈×C₂₇ #13047371

Order 1296.

170 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×Q₈×C₂₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₃×C₂₇	C₂	1	3	yes	maximal
3	Dic₆×C₂₇	C₂	1	3	yes	maximal, complemented
4	D₃×Q₈×C₉	C₃	1	1	yes	maximal, characteristic
5	C₂×Q₈×C₂₇	—	3	1	no	maximal
6	C₂×D₃×C₂₇	C₂²	1	1	yes	characteristic
7	C₃×C₄×C₂₇	C₂²	1	3	yes	—
8	Dic₃×C₂₇	C₂²	1	3	yes	—
9	C₃×Q₈×C₉	C₂×C₃	1	1	yes	characteristic
10	C₄×D₃×C₉	C₂×C₃	1	3	yes	—
11	C₉×Dic₆	C₂×C₃	1	3	yes	—
12	Q₈×C₂₇	—	2	1	no	—
13	Q₈×C₂₇	—	3	3	no	—
14	Q₈×C₂₇	D₃	1	1	yes	characteristic, direct product factor
15	C₂×C₄×C₂₇	—	3	3	no	—
16	C₂×C₃×C₂₇	C₂³	1	1	yes	characteristic
17	D₃×C₂₇	Q₈	1	2	yes	direct product factor
18	C₃×D₃×Q₈	C₉	1	1	yes	characteristic
19	C₂×Q₈×C₉	—	3	1	no	—
20	C₂×D₃×C₉	C₂²×C₃	1	1	yes	characteristic
21	C₃×C₄×C₉	C₂²×C₃	1	3	yes	—
22	C₉×Dic₃	C₂²×C₃	1	3	yes	—
23	C₂²×C₂₇	—	3	1	no	—
24	C₄×C₂₇	—	2	3	no	—
25	C₄×C₂₇	—	3	3	no	—
26	C₄×C₂₇	C₂×D₃	1	3	yes	—
27	C₃×C₂₇	C₂×Q₈	1	1	yes	characteristic, complemented, Sylow
28	C₃²×Q₈	C₂×C₉	1	1	yes	characteristic
29	C₃×C₄×D₃	C₂×C₉	1	3	yes	—
30	C₃×Dic₆	C₂×C₉	1	3	yes	—
31	Q₈×C₉	—	2	1	no	—
32	Q₈×C₉	—	3	3	no	—
33	Q₈×C₉	C₃×D₃	1	1	yes	characteristic
34	C₂×C₄×C₉	—	3	3	no	—
35	C₂×C₃×C₉	C₂³×C₃	1	1	yes	characteristic
36	D₃×C₉	C₃×Q₈	1	2	yes	—
37	C₂×C₂₇	—	2	1	no	—
38	C₂×C₂₇	—	3	2	no	—
39	C₂×C₂₇	C₂²×D₃	1	1	yes	characteristic
40	C₂×C₃×Q₈	—	3	1	no	—
41	D₃×Q₈	C₂₇	1	1	yes	characteristic, direct product factor
42	C₂×C₃×D₃	C₂²×C₉	1	1	yes	characteristic
43	C₃²×C₄	C₂²×C₉	1	3	yes	—
44	C₃×Dic₃	C₂²×C₉	1	3	yes	—
45	C₂²×C₉	—	3	1	no	—
46	C₄×C₉	—	2	3	no	—
47	C₄×C₉	—	3	3	no	—
48	C₄×C₉	C₂×C₃×D₃	1	3	yes	—
49	C₃×C₉	C₂×C₃×Q₈	1	1	yes	characteristic
50	C₂₇	—	2	1	no	—
51	C₂₇	D₃×Q₈	1	1	yes	characteristic, direct product factor
52	C₃×Q₈	—	2	1	no	—
53	C₃×Q₈	—	3	3	no	—
54	C₃×Q₈	C₂×C₂₇	1	1	yes	characteristic, complemented
55	C₃×Q₈	D₃×C₉	1	1	yes	characteristic
56	C₂×C₃×C₄	—	3	3	no	—
57	C₄×D₃	C₂×C₂₇	1	3	yes	—
58	Dic₆	C₂×C₂₇	1	3	yes	complemented
59	C₂×C₃²	C₂³×C₉	1	1	yes	characteristic
60	C₃×D₃	Q₈×C₉	1	2	yes	—
61	C₂×C₉	—	2	1	no	—
62	C₂×C₉	—	3	2	no	—
63	C₂×C₉	C₂²×C₃×D₃	1	1	yes	characteristic
64	C₂×Q₈	—	3	1	no	Sylow
65	C₂²×C₃	—	3	1	no	—
66	C₂×D₃	C₂²×C₂₇	1	1	yes	characteristic
67	C₃×C₄	—	2	3	no	—
68	C₃×C₄	—	3	3	no	—
69	C₃×C₄	C₂²×C₂₇	1	3	yes	—
70	C₃×C₄	C₂×D₃×C₉	1	3	yes	—
71	Dic₃	C₂²×C₂₇	1	3	yes	—
72	C₃²	C₂×Q₈×C₉	1	1	yes	characteristic
73	C₉	—	2	1	no	—
74	C₉	C₃×D₃×Q₈	1	1	yes	characteristic
75	Q₈	—	3	3	no	—
76	Q₈	D₃×C₂₇	1	1	yes	characteristic, direct product factor
77	C₂×C₄	—	3	3	no	—
78	C₂×C₃	—	2	1	no	—
79	C₂×C₃	—	3	2	no	—
80	C₂×C₃	C₂³×C₂₇	1	1	yes	characteristic
81	C₂×C₃	C₂²×D₃×C₉	1	1	yes	characteristic
82	D₃	Q₈×C₂₇	1	2	yes	direct product factor
83	C₂²	—	3	1	no	—
84	C₄	—	3	3	no	—
85	C₄	C₂×D₃×C₂₇	1	3	yes	—
86	C₃	—	2	1	no	—
87	C₃	C₂×Q₈×C₂₇	1	1	yes	characteristic, complemented
88	C₃	D₃×Q₈×C₉	1	1	yes	characteristic
89	C₂	—	3	2	no	—
90	C₂	C₂²×D₃×C₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₂₇	C₂²×D₃
commutator	C₂×C₃	C₂³×C₂₇
Frattini	C₂×C₉	C₂²×C₃×D₃
Fitting	C₃×Q₈×C₂₇	C₂
socle	C₂×C₃²	C₂³×C₉
prime radical	C₂×C₃×D₃	C₂²×C₉
cosocle	C₂×C₃×C₉	C₂³×C₃
hypercenter	Q₈×C₂₇	D₃
hypocenter	C₃	C₂×Q₈×C₂₇
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₂₇ ◁ Q₈×C₂₇

Complements

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

N	H ≅ G/N	type
1. C₃×Q₈×C₂₇	C₂	semidirect
3. Dic₆×C₂₇	C₂	semidirect
14. Q₈×C₂₇	D₃	direct (N × H)
17. D₃×C₂₇	Q₈	direct (N × H)
27. C₃×C₂₇	C₂×Q₈	semidirect
41. D₃×Q₈	C₂₇	direct (N × H)
51. C₂₇	D₃×Q₈	direct (N × H)
54. C₃×Q₈	C₂×C₂₇	semidirect
58. Dic₆	C₂×C₂₇	semidirect
76. Q₈	D₃×C₂₇	direct (N × H)
82. D₃	Q₈×C₂₇	direct (N × H)
87. C₃	C₂×Q₈×C₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₃×Q₈×C₂₇	2
C₄×D₃×C₂₇	2
Dic₆×C₂₇	2
D₃×Q₈×C₉	3
C₂×Q₈×C₂₇	3

Subgroups — D3×Q8×C27 #13047371

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×Q₈×C₂₇ #13047371