Subgroups — C₂×C₄×#6864 #14516201

Order 1944.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#6864	C₂	1	1	yes	maximal, characteristic
2	C₄×#6864	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×C₄×C₂₇	C₃	1	3	yes	maximal
4	C₂×C₄×C₉²	C₃	1	1	yes	maximal, characteristic
5	C₂×#6864	C₄	1	2	yes	direct product factor
6	C₂×#6864	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₂₇	C₂×C₃	1	3	yes	—
8	C₂²×C₉²	C₂×C₃	1	1	yes	characteristic
9	C₃×C₄×C₂₇	C₂×C₃	1	6	yes	—
10	C₄×C₉²	C₂×C₃	1	2	yes	—
11	#6864	C₂×C₄	1	1	yes	characteristic, direct product factor, Sylow
12	C₂×C₃×C₄×C₉	C₉	1	3	yes	—
13	C₂×C₃×C₄×C₉	C₃²	1	1	yes	characteristic
14	C₂×C₄×C₂₇	C₉	1	9	yes	complemented
15	C₂×C₃×C₂₇	C₃×C₄	1	6	yes	—
16	C₂×C₃×C₂₇	C₂²×C₃	1	3	yes	—
17	C₂×C₉²	C₃×C₄	1	2	yes	—
18	C₂×C₉²	C₂²×C₃	1	1	yes	characteristic
19	C₂²×C₃×C₉	C₂×C₉	1	3	yes	—
20	C₂²×C₃×C₉	C₂×C₃²	1	1	yes	characteristic
21	C₃×C₄×C₉	C₂×C₉	1	6	yes	—
22	C₃×C₄×C₉	C₂×C₃²	1	2	yes	—
23	C₂²×C₂₇	C₂×C₉	1	9	yes	—
24	C₄×C₂₇	C₂×C₉	1	18	yes	complemented
25	C₃×C₂₇	C₂×C₃×C₄	1	3	yes	—
26	C₉²	C₂×C₃×C₄	1	1	yes	characteristic
27	C₂×C₃²×C₄	C₃×C₉	1	1	yes	characteristic
28	C₂×C₄×C₉	—	3	3	no	—
29	C₂×C₄×C₉	C₃×C₉	1	3	yes	—
30	C₂×C₃×C₉	C₄×C₉	1	6	yes	—
31	C₂×C₃×C₉	C₂²×C₉	1	3	yes	—
32	C₂×C₃×C₉	C₃²×C₄	1	2	yes	—
33	C₂×C₃×C₉	C₂²×C₃²	1	1	yes	characteristic
34	C₂×C₂₇	C₄×C₉	1	18	yes	complemented
35	C₂×C₂₇	C₂²×C₉	1	9	yes	—
36	C₂²×C₃²	C₂×C₃×C₉	1	1	yes	characteristic
37	C₃²×C₄	C₂×C₃×C₉	1	2	yes	—
38	C₂²×C₉	—	3	3	no	—
39	C₂²×C₉	C₂×C₃×C₉	1	3	yes	—
40	C₄×C₉	—	3	6	no	—
41	C₄×C₉	C₂×C₃×C₉	1	6	yes	—
42	C₃×C₉	C₂×C₄×C₉	1	3	yes	—
43	C₃×C₉	C₂×C₃²×C₄	1	1	yes	characteristic
44	C₂₇	C₂×C₄×C₉	1	9	yes	complemented
45	C₂×C₃×C₄	C₉²	1	1	yes	characteristic
46	C₂×C₃×C₄	C₂₇⋊C₃	1	3	yes	—
47	C₂×C₃²	C₃×C₄×C₉	1	2	yes	—
48	C₂×C₃²	C₂²×C₃×C₉	1	1	yes	characteristic
49	C₂×C₉	—	3	6	no	—
50	C₂×C₉	—	3	3	no	—
51	C₂×C₉	C₃×C₄×C₉	1	6	yes	—
52	C₂×C₉	C₂²×C₃×C₉	1	3	yes	—
53	C₂²×C₃	C₂×C₉²	1	1	yes	characteristic
54	C₂²×C₃	C₂×C₂₇⋊C₃	1	3	yes	—
55	C₃×C₄	C₂×C₉²	1	2	yes	—
56	C₃×C₄	C₂×C₂₇⋊C₃	1	6	yes	—
57	C₃²	C₂×C₃×C₄×C₉	1	1	yes	characteristic
58	C₉	—	3	3	no	—
59	C₉	C₂×C₃×C₄×C₉	1	3	yes	—
60	C₂×C₄	#6864	1	1	yes	characteristic, direct product factor, Sylow
61	C₂×C₃	C₄×C₉²	1	2	yes	—
62	C₂×C₃	C₄×C₂₇⋊C₃	1	6	yes	—
63	C₂×C₃	C₂²×C₉²	1	1	yes	characteristic
64	C₂×C₃	C₂²×C₂₇⋊C₃	1	3	yes	—
65	C₂²	C₂×#6864	1	1	yes	characteristic
66	C₄	C₂×#6864	1	2	yes	direct product factor
67	C₃	C₂×C₄×C₉²	1	1	yes	characteristic
68	C₃	C₂×C₄×C₂₇⋊C₃	1	3	yes	—
69	C₂	C₄×#6864	1	2	yes	direct product factor
70	C₂	C₂²×#6864	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄×C₉	C₃²
commutator	C₃	C₂×C₄×C₉²
Frattini, cosocle	C₂×C₃×C₉	C₂²×C₃²
socle, prime radical	C₂²×C₃²	C₂×C₃×C₉
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×#6864	C₂	direct (N × H)
5. C₂×#6864	C₄	direct (N × H)
11. #6864	C₂×C₄	direct (N × H)
14. C₂×C₄×C₂₇	C₉	semidirect
24. C₄×C₂₇	C₂×C₉	semidirect
34. C₂×C₂₇	C₄×C₉	semidirect
44. C₂₇	C₂×C₄×C₉	semidirect
60. C₂×C₄	#6864	direct (N × H)
66. C₄	C₂×#6864	direct (N × H)
69. C₂	C₄×#6864	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#6864	2
C₄×#6864	2
C₂×C₃×C₄×C₂₇	3
C₂×C₄×C₉²	3

Subgroups — C2×C4×#6864 #14516201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#6864 #14516201