Subgroups — C₂²×#6864⋊C₂ #14516646

Order 1944.

237 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, complemented
2	C₂×#6864⋊C₂	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃×D₂₇	C₃	1	1	yes	maximal, characteristic
4	C₂²×C₉×D₉	—	3	1	no	maximal
5	C₂×#6864	C₂²	1	3	yes	complemented
6	#6864⋊C₂	C₂²	1	4	yes	direct product factor
7	C₂²×C₃×C₂₇	—	2	1	no	—
8	C₂²×C₃×C₂₇	C₂×C₃	1	1	yes	characteristic
9	C₂²×C₉²	D₃	1	1	yes	characteristic
10	C₂×C₃×D₂₇	C₂×C₃	1	6	yes	—
11	C₂×C₉×D₉	—	3	6	no	—
12	#6864	C₂³	1	1	yes	characteristic, complemented, Sylow
13	C₂²×D₃×C₉	—	9	1	no	—
14	C₂²×C₃×D₉	—	3	1	no	—
15	C₂²×D₂₇	C₉	1	1	yes	characteristic, complemented
16	C₂×C₃×C₂₇	—	2	3	no	—
17	C₂×C₃×C₂₇	C₂²×C₃	1	3	yes	—
18	C₂×C₉²	C₂×D₃	1	3	yes	—
19	C₃×D₂₇	C₂²×C₃	1	4	yes	—
20	C₉×D₉	—	3	4	no	—
21	C₂²×C₃×C₉	—	2	1	no	—
22	C₂²×C₃×C₉	D₉	1	1	yes	characteristic
23	C₂²×C₃×C₉	C₃×D₃	1	1	yes	characteristic
24	C₂×D₃×C₉	—	9	6	no	—
25	C₂×C₃×D₉	—	3	6	no	—
26	C₂²×C₂₇	—	2	3	no	—
27	C₂²×C₂₇	—	2	1	no	—
28	C₂²×C₂₇	C₂×C₉	1	1	yes	characteristic, complemented
29	C₂×D₂₇	C₂×C₉	1	6	yes	complemented
30	C₃×C₂₇	—	2	1	no	—
31	C₃×C₂₇	C₂³×C₃	1	1	yes	characteristic
32	C₉²	C₂²×D₃	1	1	yes	characteristic
33	C₂²×C₃×D₃	—	9	1	no	—
34	C₂³×C₉	—	27	1	no	—
35	C₂²×D₉	—	3	1	no	—
36	C₂×C₃×C₉	—	2	3	no	—
37	C₂×C₃×C₉	C₂×D₉	1	3	yes	—
38	C₂×C₃×C₉	C₂×C₃×D₃	1	3	yes	—
39	D₃×C₉	—	9	4	no	—
40	C₃×D₉	—	3	4	no	—
41	C₂×C₂₇	—	2	9	no	—
42	C₂×C₂₇	—	2	3	no	—
43	C₂×C₂₇	C₂²×C₉	1	3	yes	complemented
44	D₂₇	C₂²×C₉	1	4	yes	complemented
45	C₂²×C₃²	C₃×D₉	1	1	yes	characteristic
46	C₂×C₃×D₃	—	9	6	no	—
47	C₂²×C₉	—	2	1	no	—
48	C₂²×C₉	—	3	1	no	—
49	C₂²×C₉	—	6	1	no	—
50	C₂²×C₉	—	27	6	no	—
51	C₂²×C₉	D₃×C₉	1	1	yes	characteristic
52	C₂×D₉	—	3	6	no	—
53	C₃×C₉	—	2	1	no	—
54	C₃×C₉	C₂²×D₉	1	1	yes	characteristic
55	C₃×C₉	C₂²×C₃×D₃	1	1	yes	characteristic
56	C₂₇	—	2	3	no	—
57	C₂₇	—	2	1	no	—
58	C₂₇	C₂³×C₉	1	1	yes	characteristic, complemented
59	C₂³×C₃	—	27	1	no	—
60	C₂²×D₃	—	9	1	no	—
61	C₂×C₃²	C₂×C₃×D₉	1	3	yes	—
62	C₃×D₃	—	9	4	no	—
63	C₂×C₉	—	2	3	no	—
64	C₂×C₉	—	3	3	no	—
65	C₂×C₉	—	6	3	no	—
66	C₂×C₉	—	27	4	no	—
67	C₂×C₉	C₂×D₃×C₉	1	3	yes	—
68	D₉	—	3	4	no	—
69	C₂²×C₃	—	2	1	no	—
70	C₂²×C₃	—	27	6	no	—
71	C₂²×C₃	C₉×D₉	1	1	yes	characteristic
72	C₂²×C₃	D₂₇⋊C₃	1	1	yes	characteristic
73	C₂×D₃	—	9	6	no	—
74	C₃²	C₂²×C₃×D₉	1	1	yes	characteristic
75	C₉	—	2	1	no	—
76	C₉	—	3	1	no	—
77	C₉	—	6	1	no	—
78	C₉	C₂²×D₃×C₉	1	1	yes	characteristic
79	C₂³	—	27	1	no	Sylow
80	C₂×C₃	—	2	3	no	—
81	C₂×C₃	—	27	4	no	—
82	C₂×C₃	C₂×C₉×D₉	1	3	yes	—
83	C₂×C₃	C₂×D₂₇⋊C₃	1	3	yes	—
84	D₃	—	9	4	no	—
85	C₂²	—	27	6	no	—
86	C₂²	#6864⋊C₂	1	1	yes	characteristic, direct product factor
87	C₃	—	2	1	no	—
88	C₃	C₂²×C₉×D₉	1	1	yes	characteristic
89	C₃	C₂²×D₂₇⋊C₃	1	1	yes	characteristic
90	C₂	—	27	4	no	—
91	C₂	C₂×#6864⋊C₂	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₃	D₂₇⋊C₃
commutator, hypocenter	C₂₇	C₂³×C₉
Frattini	C₃×C₉	C₂²×C₃×D₃
Fitting	C₂²×#6864	C₂
socle	C₂²×C₃²	C₃×D₉
prime radical	C₂²×C₃×D₂₇	C₃
cosocle	C₃×C₂₇	C₂³×C₃
LCS	G ▷ C₂₇
UCS	C₁ ◁ C₂²×C₃

Complements

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

N	H ≅ G/N	type
1. C₂²×#6864	C₂	semidirect
2. C₂×#6864⋊C₂	C₂	direct (N × H)
5. C₂×#6864	C₂²	semidirect
6. #6864⋊C₂	C₂²	direct (N × H)
12. #6864	C₂³	semidirect
15. C₂²×D₂₇	C₉	semidirect
28. C₂²×C₂₇	C₂×C₉	semidirect
29. C₂×D₂₇	C₂×C₉	semidirect
43. C₂×C₂₇	C₂²×C₉	semidirect
44. D₂₇	C₂²×C₉	semidirect
58. C₂₇	C₂³×C₉	semidirect
86. C₂²	#6864⋊C₂	direct (N × H)
91. C₂	C₂×#6864⋊C₂	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
C₂×#6864⋊C₂	2
C₂²×C₃×D₂₇	3
C₂²×C₉×D₉	3

Subgroups — C22×#6864⋊C2 #14516646

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#6864⋊C₂ #14516646