Subgroups — C₂²×#66419 #13049309

Order 1296.

251 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#4208	C₂	1	1	yes	maximal, characteristic
2	C₂×#66419	C₂	1	6	yes	maximal, direct product factor
3	C₂²×#1088	C₃	1	1	yes	maximal, characteristic
4	C₂²×#737	C₄	1	1	yes	characteristic, complemented
5	C₂×#4208	C₄	1	3	yes	complemented
6	C₂×#4208	C₂²	1	3	yes	—
7	#66419	C₂²	1	4	yes	direct product factor
8	C₂²×#272	C₂×C₃	1	1	yes	characteristic
9	C₂²×D₃×C₉	—	6	2	no	—
10	C₂×#1088	C₂×C₃	1	6	yes	—
11	C₂×#737	C₂×C₄	1	3	yes	complemented
12	#4208	C₂×C₄	1	3	yes	complemented
13	#4208	C₂³	1	1	yes	characteristic
14	C₂²×C₄×C₉	—	9	1	no	maximal
15	C₂²×3₋¹⁺²	—	4	2	no	—
16	C₂²×H(3,𝔽₃)	C₃×C₄	1	1	yes	characteristic
17	C₂²×C₃×C₉	—	2	2	no	—
18	C₂×#272	C₃×C₄	1	3	yes	—
19	C₂×#272	C₂²×C₃	1	3	yes	—
20	C₂×D₃×C₉	—	6	6	no	—
21	C₂×D₃×C₉	—	6	6	no	—
22	#1088	C₂²×C₃	1	4	yes	—
23	#737	C₂²×C₄	1	1	yes	characteristic, complemented, Sylow
24	C₂²×C₃×D₃	—	6	2	no	—
25	C₂³×C₉	—	9	1	no	—
26	C₂×C₄×C₉	—	9	6	no	—
27	C₂×3₋¹⁺²	—	4	6	no	—
28	C₂×H(3,𝔽₃)	C₂×C₃×C₄	1	3	yes	—
29	C₂×C₃×C₉	—	2	6	no	—
30	#272	C₂×C₃×C₄	1	3	yes	—
31	#272	C₂³×C₃	1	1	yes	characteristic
32	D₃×C₉	—	6	6	no	—
33	D₃×C₉	—	6	2	no	—
34	C₂²×C₃×C₄	—	9	1	no	—
35	C₂²×C₃²	—	2	2	no	—
36	C₂×C₃×D₃	—	6	6	no	—
37	C₂×C₃×D₃	—	6	6	no	—
38	C₂²×C₉	—	4	2	no	—
39	C₂²×C₉	—	9	3	no	—
40	C₂²×C₉	—	9	3	no	—
41	C₂²×C₉	#157	1	1	yes	characteristic
42	C₄×C₉	—	9	4	no	—
43	3₋¹⁺²	—	4	2	no	—
44	H(3,𝔽₃)	C₂²×C₃×C₄	1	1	yes	characteristic
45	C₃×C₉	—	2	2	no	—
46	C₂³×C₃	—	9	1	no	—
47	C₂²×D₃	—	6	2	no	—
48	C₂×C₃×C₄	—	9	6	no	—
49	C₂×C₃²	—	2	6	no	—
50	C₃×D₃	—	6	6	no	—
51	C₃×D₃	—	6	2	no	—
52	C₂×C₉	—	4	6	no	—
53	C₂×C₉	—	9	1	no	—
54	C₂×C₉	—	9	3	no	—
55	C₂×C₉	C₂×#157	1	3	yes	—
56	C₂²×C₄	—	9	1	no	Sylow
57	C₂²×C₃	—	6	2	no	—
58	C₂²×C₃	—	9	3	no	—
59	C₂²×C₃	—	9	3	no	—
60	C₂²×C₃	C₃×#157	1	1	yes	characteristic
61	C₂×D₃	—	6	6	no	—
62	C₂×D₃	—	6	6	no	—
63	C₃×C₄	—	9	4	no	—
64	C₃²	—	2	2	no	—
65	C₉	—	4	2	no	—
66	C₉	C₂²×#157	1	1	yes	characteristic
67	C₂³	—	9	1	no	—
68	C₂×C₄	—	9	6	no	—
69	C₂×C₃	—	6	6	no	—
70	C₂×C₃	—	9	1	no	—
71	C₂×C₃	—	9	3	no	—
72	C₂×C₃	C₂×C₃×#157	1	3	yes	—
73	D₃	—	6	2	no	—
74	D₃	—	6	6	no	—
75	C₂²	—	9	3	no	—
76	C₂²	—	9	3	no	—
77	C₂²	#66419	1	1	yes	characteristic, direct product factor
78	C₄	—	9	4	no	—
79	C₃	—	6	2	no	—
80	C₃	C₂²×C₃×#157	1	1	yes	characteristic
81	C₂	—	9	1	no	—
82	C₂	—	9	3	no	—
83	C₂	C₂×#66419	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₉	#157
commutator, hypocenter	H(3,𝔽₃)	C₂²×C₃×C₄
Frattini	C₃	C₂²×C₃×#157
Fitting	C₂²×#737	C₄
socle	C₂²×C₃	C₃×#157
prime radical	C₂²×#272	C₂×C₃
cosocle	#272	C₂³×C₃
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²×C₉

Complements

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

N	H ≅ G/N	type
2. C₂×#66419	C₂	direct (N × H)
4. C₂²×#737	C₄	semidirect
5. C₂×#4208	C₄	semidirect
7. #66419	C₂²	direct (N × H)
11. C₂×#737	C₂×C₄	semidirect
12. #4208	C₂×C₄	semidirect
23. #737	C₂²×C₄	semidirect
77. C₂²	#66419	direct (N × H)
83. C₂	C₂×#66419	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×#4208	2
C₂×#66419	2
C₂²×#1088	3
C₂²×C₄×C₉	9

Subgroups — C22×#66419 #13049309

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#66419 #13049309