Subgroups — C₂×(C₂₇⋊C₃)⋊C₈ #13046621

Order 1296.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₂₇⋊C₃	C₂	1	1	yes	maximal, characteristic
2	(C₂₇⋊C₃)⋊C₈	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×C₉⋊C₈	—	3	1	no	maximal
4	C₂×C₂₇⋊C₈	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₂₇⋊C₃	C₄	1	1	yes	characteristic
6	C₄×C₂₇⋊C₃	C₄	1	1	yes	characteristic
7	C₄×C₂₇⋊C₃	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄×C₉	D₃	1	1	yes	characteristic
9	C₃×C₉⋊C₈	—	3	2	no	—
10	C₂×C₄×C₂₇	—	2	1	no	—
11	C₂×C₄×C₂₇	C₂×C₃	1	1	yes	characteristic
12	C₂₇⋊C₈	C₂×C₃	1	2	yes	complemented
13	C₂×C₂₇⋊C₃	C₈	1	2	yes	complemented
14	C₂×C₂₇⋊C₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₃×C₃⋊C₈	—	9	1	no	—
16	C₂×C₉⋊C₈	—	3	1	no	—
17	C₂²×C₃×C₉	Dic₃	1	1	yes	characteristic
18	C₃×C₄×C₉	Dic₃	1	1	yes	characteristic
19	C₃×C₄×C₉	C₂×D₃	1	1	yes	characteristic
20	C₂²×C₂₇	—	2	1	no	—
21	C₂²×C₂₇	C₃×C₄	1	1	yes	characteristic
22	C₄×C₂₇	—	2	1	no	—
23	C₄×C₂₇	—	2	1	no	—
24	C₄×C₂₇	C₃×C₄	1	1	yes	characteristic
25	C₄×C₂₇	C₂²×C₃	1	1	yes	characteristic
26	C₂₇⋊C₃	C₂×C₈	1	1	yes	characteristic, complemented, Sylow
27	C₂×C₃²×C₄	D₉	1	1	yes	characteristic
28	C₃×C₃⋊C₈	—	9	2	no	—
29	C₂×C₄×C₉	—	2	1	no	—
30	C₂×C₄×C₉	C₃×D₃	1	1	yes	characteristic
31	C₉⋊C₈	—	3	2	no	—
32	C₂×C₃×C₉	C₃⋊C₈	1	2	yes	—
33	C₂×C₃×C₉	C₂×Dic₃	1	1	yes	characteristic
34	C₂×C₂₇	—	2	2	no	—
35	C₂×C₂₇	—	2	1	no	—
36	C₂×C₂₇	C₃×C₈	1	2	yes	complemented
37	C₂×C₂₇	C₂×C₃×C₄	1	1	yes	characteristic
38	C₂×C₃×C₈	—	27	1	no	—
39	C₂×C₃⋊C₈	—	9	1	no	—
40	C₂²×C₃²	Dic₉	1	1	yes	characteristic
41	C₃²×C₄	Dic₉	1	1	yes	characteristic
42	C₃²×C₄	C₂×D₉	1	1	yes	characteristic
43	C₂²×C₉	—	2	1	no	—
44	C₂²×C₉	C₃×Dic₃	1	1	yes	characteristic
45	C₄×C₉	—	2	1	no	—
46	C₄×C₉	—	2	1	no	—
47	C₄×C₉	C₃×Dic₃	1	1	yes	characteristic
48	C₄×C₉	C₂×C₃×D₃	1	1	yes	characteristic
49	C₃×C₉	C₂×C₃⋊C₈	1	1	yes	characteristic
50	C₂₇	—	2	1	no	—
51	C₂₇	C₂×C₃×C₈	1	1	yes	characteristic, complemented
52	C₂×C₃×C₄	—	3	1	no	—
53	C₂×C₃×C₄	C₃×D₉	1	1	yes	characteristic
54	C₃×C₈	—	27	2	no	—
55	C₃⋊C₈	—	9	2	no	—
56	C₂×C₃²	C₉⋊C₈	1	2	yes	—
57	C₂×C₃²	C₂×Dic₉	1	1	yes	characteristic
58	C₂×C₉	—	2	2	no	—
59	C₂×C₉	—	2	1	no	—
60	C₂×C₉	C₃×C₃⋊C₈	1	2	yes	—
61	C₂×C₉	C₂×C₃×Dic₃	1	1	yes	characteristic
62	C₂×C₈	—	27	1	no	Sylow
63	C₂²×C₃	—	3	1	no	—
64	C₂²×C₃	C₃×Dic₉	1	1	yes	characteristic
65	C₃×C₄	—	3	1	no	—
66	C₃×C₄	—	3	1	no	—
67	C₃×C₄	C₃×Dic₉	1	1	yes	characteristic
68	C₃×C₄	C₂×C₃×D₉	1	1	yes	characteristic
69	C₃²	C₂×C₉⋊C₈	1	1	yes	characteristic
70	C₉	—	2	1	no	—
71	C₉	C₂×C₃×C₃⋊C₈	1	1	yes	characteristic
72	C₂×C₄	D₂₇⋊C₃	1	1	yes	characteristic
73	C₈	—	27	2	no	—
74	C₂×C₃	—	3	2	no	—
75	C₂×C₃	—	3	1	no	—
76	C₂×C₃	C₃×C₉⋊C₈	1	2	yes	—
77	C₂×C₃	C₂×C₃×Dic₉	1	1	yes	characteristic
78	C₂²	(C₂₇⋊C₃)⋊C₄	1	1	yes	characteristic
79	C₄	(C₂₇⋊C₃)⋊C₄	1	1	yes	characteristic
80	C₄	C₂×D₂₇⋊C₃	1	1	yes	characteristic
81	C₃	—	3	1	no	—
82	C₃	C₂×C₃×C₉⋊C₈	1	1	yes	characteristic
83	C₂	(C₂₇⋊C₃)⋊C₈	1	2	yes	direct product factor
84	C₂	C₂×(C₂₇⋊C₃)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄	D₂₇⋊C₃
commutator, hypocenter	C₂₇	C₂×C₃×C₈
Frattini	C₄×C₉	C₂×C₃×D₃
Fitting	C₂×C₄×C₂₇⋊C₃	C₂
socle	C₂²×C₃	C₃×Dic₉
prime radical	C₂²×C₃²	Dic₉
cosocle	C₄×C₂₇	C₂²×C₃
LCS	G ▷ C₂₇
UCS	C₁ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
2. (C₂₇⋊C₃)⋊C₈	C₂	direct (N × H)
4. C₂×C₂₇⋊C₈	C₃	semidirect
12. C₂₇⋊C₈	C₂×C₃	semidirect
13. C₂×C₂₇⋊C₃	C₈	semidirect
26. C₂₇⋊C₃	C₂×C₈	semidirect
36. C₂×C₂₇	C₃×C₈	semidirect
51. C₂₇	C₂×C₃×C₈	semidirect
83. C₂	(C₂₇⋊C₃)⋊C₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₂₇⋊C₃	2
(C₂₇⋊C₃)⋊C₈	2
C₂×C₃×C₉⋊C₈	3
C₂×C₂₇⋊C₈	3

Subgroups — C2×(C27⋊C3)⋊C8 #13046621

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×(C₂₇⋊C₃)⋊C₈ #13046621