Subgroups — C₂₇×#239 #13047372

Order 1296.

178 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	D₁₂×C₂₇	C₂	1	3	yes	maximal, complemented
3	C₄×D₃×C₂₇	C₂	1	3	yes	maximal, complemented
4	C₉×#239	C₃	1	1	yes	maximal, characteristic
5	#41×C₂₇	—	3	1	no	maximal
6	C₂×D₃×C₂₇	C₂²	1	3	yes	—
7	C₃×C₄×C₂₇	C₂²	1	3	yes	—
8	Dic₃×C₂₇	C₂²	1	1	yes	characteristic
9	C₃×Q₈×C₉	C₂×C₃	1	1	yes	characteristic
10	C₉×D₁₂	C₂×C₃	1	3	yes	—
11	C₄×D₃×C₉	C₂×C₃	1	3	yes	—
12	Q₈×C₂₇	—	2	1	no	—
13	Q₈×C₂₇	D₃	1	1	yes	characteristic, complemented
14	D₄×C₂₇	—	3	3	no	—
15	C₂×C₄×C₂₇	—	3	3	no	—
16	C₂×C₃×C₂₇	C₂³	1	1	yes	characteristic
17	D₃×C₂₇	—	2	3	no	—
18	C₃×#239	C₉	1	1	yes	characteristic
19	C₉×#41	—	3	1	no	—
20	C₂×D₃×C₉	C₂²×C₃	1	3	yes	—
21	C₃×C₄×C₉	C₂²×C₃	1	3	yes	—
22	C₉×Dic₃	C₂²×C₃	1	1	yes	characteristic
23	C₂²×C₂₇	—	3	3	no	—
24	C₄×C₂₇	—	2	3	no	—
25	C₄×C₂₇	—	3	1	no	—
26	C₄×C₂₇	C₂×D₃	1	3	yes	—
27	C₃×C₂₇	#41	1	1	yes	characteristic, complemented, Sylow
28	C₃²×Q₈	C₂×C₉	1	1	yes	characteristic
29	C₃×D₁₂	C₂×C₉	1	3	yes	—
30	C₃×C₄×D₃	C₂×C₉	1	3	yes	—
31	Q₈×C₉	—	2	1	no	—
32	Q₈×C₉	C₃×D₃	1	1	yes	characteristic
33	D₄×C₉	—	3	3	no	—
34	C₂×C₄×C₉	—	3	3	no	—
35	C₂×C₃×C₉	C₂³×C₃	1	1	yes	characteristic
36	D₃×C₉	—	2	3	no	—
37	C₂×C₂₇	—	2	1	no	—
38	C₂×C₂₇	—	6	3	no	—
39	C₂×C₂₇	C₂²×D₃	1	1	yes	characteristic
40	C₃×#41	—	3	1	no	—
41	#239	C₂₇	1	1	yes	characteristic, direct product factor
42	C₂×C₃×D₃	C₂²×C₉	1	3	yes	—
43	C₃²×C₄	C₂²×C₉	1	3	yes	—
44	C₃×Dic₃	C₂²×C₉	1	1	yes	characteristic
45	C₂²×C₉	—	3	3	no	—
46	C₄×C₉	—	2	3	no	—
47	C₄×C₉	—	3	1	no	—
48	C₄×C₉	C₂×C₃×D₃	1	3	yes	—
49	C₃×C₉	C₃×#41	1	1	yes	characteristic
50	C₂₇	—	2	1	no	—
51	C₂₇	#239	1	1	yes	characteristic, direct product factor
52	C₃×Q₈	—	2	1	no	—
53	C₃×Q₈	C₂×C₂₇	1	1	yes	characteristic, complemented
54	C₃×Q₈	D₃×C₉	1	1	yes	characteristic
55	C₃×D₄	—	3	3	no	—
56	C₂×C₃×C₄	—	3	3	no	—
57	D₁₂	C₂×C₂₇	1	3	yes	complemented
58	C₄×D₃	C₂×C₂₇	1	3	yes	complemented
59	C₂×C₃²	C₂³×C₉	1	1	yes	characteristic
60	C₃×D₃	—	2	3	no	—
61	C₂×C₉	—	2	1	no	—
62	C₂×C₉	—	6	3	no	—
63	C₂×C₉	C₂²×C₃×D₃	1	1	yes	characteristic
64	#41	—	3	1	no	Sylow
65	C₂²×C₃	—	3	3	no	—
66	C₂×D₃	C₂²×C₂₇	1	3	yes	—
67	C₃×C₄	—	2	3	no	—
68	C₃×C₄	—	3	1	no	—
69	C₃×C₄	C₂²×C₂₇	1	3	yes	—
70	C₃×C₄	C₂×D₃×C₉	1	3	yes	—
71	Dic₃	C₂²×C₂₇	1	1	yes	characteristic
72	C₃²	C₉×#41	1	1	yes	characteristic
73	C₉	—	2	1	no	—
74	C₉	C₃×#239	1	1	yes	characteristic
75	Q₈	D₃×C₂₇	1	1	yes	characteristic, complemented
76	D₄	—	3	3	no	—
77	C₂×C₄	—	3	3	no	—
78	C₂×C₃	—	2	1	no	—
79	C₂×C₃	—	6	3	no	—
80	C₂×C₃	C₂³×C₂₇	1	1	yes	characteristic
81	C₂×C₃	C₂²×D₃×C₉	1	1	yes	characteristic
82	D₃	—	2	3	no	—
83	C₂²	—	3	3	no	—
84	C₄	—	3	1	no	—
85	C₄	C₂×D₃×C₂₇	1	3	yes	—
86	C₃	—	2	1	no	—
87	C₃	#41×C₂₇	1	1	yes	characteristic, complemented
88	C₃	C₉×#239	1	1	yes	characteristic
89	C₂	—	6	3	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	C₃×#239	C₉
cosocle	C₂×C₃×C₉	C₂³×C₃
hypercenter	Q₈×C₂₇	D₃
hypocenter	C₃	#41×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
2. D₁₂×C₂₇	C₂	semidirect
3. C₄×D₃×C₂₇	C₂	semidirect
13. Q₈×C₂₇	D₃	semidirect
27. C₃×C₂₇	#41	semidirect
41. #239	C₂₇	direct (N × H)
51. C₂₇	#239	direct (N × H)
53. C₃×Q₈	C₂×C₂₇	semidirect
57. D₁₂	C₂×C₂₇	semidirect
58. C₄×D₃	C₂×C₂₇	semidirect
75. Q₈	D₃×C₂₇	semidirect
87. C₃	#41×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
D₁₂×C₂₇	2
C₄×D₃×C₂₇	2
C₉×#239	3
#41×C₂₇	3

Subgroups — C27×#239 #13047372

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₇×#239 #13047372