Subgroups — C₃×Q₈×D₂₇ #13047369

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	C₃×C₄×D₂₇	C₂	1	3	yes	maximal
3	C₃×Dic₅₄	C₂	1	3	yes	maximal, complemented
4	C₃×Q₈×D₉	—	3	1	no	maximal
5	Q₈×D₂₇	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂×C₃×D₂₇	C₂²	1	1	yes	characteristic
7	C₃×C₄×C₂₇	C₂²	1	3	yes	—
8	C₃×Dic₂₇	C₂²	1	3	yes	—
9	C₃×Q₈×C₉	D₃	1	1	yes	characteristic
10	C₃×C₄×D₉	—	3	3	no	—
11	C₃×Dic₁₈	—	3	3	no	—
12	Q₈×C₂₇	—	2	1	no	—
13	Q₈×C₂₇	C₂×C₃	1	1	yes	characteristic, complemented
14	C₄×D₂₇	C₂×C₃	1	3	yes	—
15	Dic₅₄	C₂×C₃	1	3	yes	complemented
16	C₂×C₃×C₂₇	C₂³	1	1	yes	characteristic
17	C₃×D₂₇	Q₈	1	2	yes	direct product factor
18	C₃×D₃×Q₈	—	9	1	no	—
19	Q₈×D₉	—	3	1	no	—
20	C₂×C₃×D₉	—	3	1	no	—
21	C₃×C₄×C₉	C₂×D₃	1	3	yes	—
22	C₃×Dic₉	—	3	3	no	—
23	C₂×D₂₇	C₂²×C₃	1	1	yes	characteristic
24	C₄×C₂₇	—	2	3	no	—
25	C₄×C₂₇	C₂²×C₃	1	3	yes	—
26	Dic₂₇	C₂²×C₃	1	3	yes	—
27	C₃×C₂₇	C₂×Q₈	1	1	yes	characteristic, complemented, Sylow
28	C₃²×Q₈	D₉	1	1	yes	characteristic
29	C₃×C₄×D₃	—	9	3	no	—
30	C₃×Dic₆	—	9	3	no	—
31	Q₈×C₉	—	2	1	no	—
32	Q₈×C₉	C₃×D₃	1	1	yes	characteristic
33	C₄×D₉	—	3	3	no	—
34	Dic₁₈	—	3	3	no	—
35	C₂×C₃×C₉	C₂²×D₃	1	1	yes	characteristic
36	C₃×D₉	—	3	2	no	—
37	C₂×C₂₇	—	2	1	no	—
38	C₂×C₂₇	C₂³×C₃	1	1	yes	characteristic
39	D₂₇	C₃×Q₈	1	2	yes	direct product factor
40	C₂×C₃×Q₈	—	27	1	no	—
41	D₃×Q₈	—	9	1	no	—
42	C₂×C₃×D₃	—	9	1	no	—
43	C₃²×C₄	C₂×D₉	1	3	yes	—
44	C₃×Dic₃	—	9	3	no	—
45	C₂×D₉	—	3	1	no	—
46	C₄×C₉	—	2	3	no	—
47	C₄×C₉	C₂×C₃×D₃	1	3	yes	—
48	Dic₉	—	3	3	no	—
49	C₃×C₉	D₃×Q₈	1	1	yes	characteristic
50	C₂₇	—	2	1	no	—
51	C₂₇	C₂×C₃×Q₈	1	1	yes	characteristic, complemented
52	C₃×Q₈	—	2	1	no	—
53	C₃×Q₈	—	27	3	no	—
54	C₃×Q₈	D₂₇	1	1	yes	characteristic, direct product factor
55	C₃×Q₈	C₃×D₉	1	1	yes	characteristic
56	C₂×C₃×C₄	—	27	3	no	—
57	C₄×D₃	—	9	3	no	—
58	Dic₆	—	9	3	no	—
59	C₂×C₃²	C₂²×D₉	1	1	yes	characteristic
60	C₃×D₃	—	9	2	no	—
61	C₂×C₉	—	2	1	no	—
62	C₂×C₉	C₂²×C₃×D₃	1	1	yes	characteristic
63	D₉	—	3	2	no	—
64	C₂×Q₈	—	27	1	no	Sylow
65	C₂²×C₃	—	27	1	no	—
66	C₂×D₃	—	9	1	no	—
67	C₃×C₄	—	2	3	no	—
68	C₃×C₄	—	27	3	no	—
69	C₃×C₄	C₂×D₂₇	1	3	yes	—
70	C₃×C₄	C₂×C₃×D₉	1	3	yes	—
71	Dic₃	—	9	3	no	—
72	C₃²	Q₈×D₉	1	1	yes	characteristic
73	C₉	—	2	1	no	—
74	C₉	C₃×D₃×Q₈	1	1	yes	characteristic
75	Q₈	—	27	3	no	—
76	Q₈	C₃×D₂₇	1	1	yes	characteristic, direct product factor
77	C₂×C₄	—	27	3	no	—
78	C₂×C₃	—	2	1	no	—
79	C₂×C₃	—	27	2	no	—
80	C₂×C₃	C₂²×D₂₇	1	1	yes	characteristic
81	C₂×C₃	C₂²×C₃×D₉	1	1	yes	characteristic
82	D₃	—	9	2	no	—
83	C₂²	—	27	1	no	—
84	C₄	—	27	3	no	—
85	C₄	C₂×C₃×D₂₇	1	3	yes	—
86	C₃	—	2	1	no	—
87	C₃	Q₈×D₂₇	1	1	yes	characteristic, direct product factor
88	C₃	C₃×Q₈×D₉	1	1	yes	characteristic
89	C₂	—	27	2	no	—
90	C₂	C₂²×C₃×D₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂²×D₂₇
commutator, cosocle	C₂×C₂₇	C₂³×C₃
Frattini	C₂×C₉	C₂²×C₃×D₃
Fitting	C₃×Q₈×C₂₇	C₂
socle	C₂×C₃²	C₂²×D₉
prime radical	C₂×C₃×D₂₇	C₂²
hypercenter	C₃×Q₈	D₂₇
hypocenter	C₂₇	C₂×C₃×Q₈
LCS	G ▷ C₂×C₂₇ ▷ C₂₇
UCS	C₁ ◁ C₂×C₃ ◁ C₃×Q₈

Complements

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

N	H ≅ G/N	type
1. C₃×Q₈×C₂₇	C₂	semidirect
3. C₃×Dic₅₄	C₂	semidirect
5. Q₈×D₂₇	C₃	direct (N × H)
13. Q₈×C₂₇	C₂×C₃	semidirect
15. Dic₅₄	C₂×C₃	semidirect
17. C₃×D₂₇	Q₈	direct (N × H)
27. C₃×C₂₇	C₂×Q₈	semidirect
39. D₂₇	C₃×Q₈	direct (N × H)
51. C₂₇	C₂×C₃×Q₈	semidirect
54. C₃×Q₈	D₂₇	direct (N × H)
76. Q₈	C₃×D₂₇	direct (N × H)
87. C₃	Q₈×D₂₇	direct (N × H)

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
C₃×C₄×D₂₇	2
C₃×Dic₅₄	2
C₃×Q₈×D₉	3
Q₈×D₂₇	3

Subgroups — C3×Q8×D27 #13047369

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×Q₈×D₂₇ #13047369