Subgroups — C₂³×#10621729 #72259331

Order 5408.

874 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10621729	C₂	1	14	yes	maximal, direct product factor
2	C₂³×C₁₃⋊D₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₁₃²	C₄	1	1	yes	characteristic, complemented
4	C₂²×C₁₃⋊D₁₃	C₄	1	7	yes	complemented
5	C₂²×C₁₃⋊D₁₃	C₂²	1	7	yes	—
6	C₂×#10621729	C₂²	1	28	yes	direct product factor
7	C₂²×C₁₃²	C₂×C₄	1	7	yes	complemented
8	C₂×C₁₃⋊D₁₃	C₂×C₄	1	21	yes	complemented
9	C₂×C₁₃⋊D₁₃	C₂³	1	7	yes	—
10	#10621729	C₂³	1	8	yes	direct product factor
11	C₂³×#261	—	13	2	no	maximal
12	C₂×C₁₃²	C₂²×C₄	1	7	yes	complemented
13	C₁₃⋊D₁₃	C₂²×C₄	1	7	yes	complemented
14	C₁₃⋊D₁₃	C₂⁴	1	1	yes	characteristic
15	C₂³×D₁₃	—	13	2	no	—
16	C₂³×D₁₃	—	26	6	no	—
17	C₂²×#261	—	13	28	no	—
18	C₁₃²	C₂³×C₄	1	1	yes	characteristic, complemented, Sylow
19	C₂³×C₁₃	—	2	6	no	—
20	C₂³×C₁₃	#261	1	2	yes	complemented
21	C₂²×D₁₃	—	13	14	no	—
22	C₂²×D₁₃	—	13	14	no	—
23	C₂²×D₁₃	—	26	42	no	—
24	C₂²×D₁₃	—	26	42	no	—
25	C₂×#261	—	13	56	no	—
26	C₂²×C₁₃	—	2	42	no	—
27	C₂²×C₁₃	C₂×#261	1	14	yes	complemented
28	C₂×D₁₃	—	13	42	no	—
29	C₂×D₁₃	—	13	14	no	—
30	C₂×D₁₃	—	26	126	no	—
31	C₂×D₁₃	—	26	42	no	—
32	#261	—	13	16	no	—
33	C₂³×C₄	—	169	1	no	Sylow
34	C₂×C₁₃	—	2	42	no	—
35	C₂×C₁₃	C₂²×#261	1	14	yes	complemented
36	D₁₃	—	13	2	no	—
37	D₁₃	—	13	14	no	—
38	D₁₃	—	26	42	no	—
39	D₁₃	—	26	6	no	—
40	C₂⁴	—	169	1	no	—
41	C₂²×C₄	—	169	14	no	—
42	C₁₃	—	2	6	no	—
43	C₁₃	C₂³×#261	1	2	yes	complemented
44	C₂³	—	169	7	no	—
45	C₂³	—	169	7	no	—
46	C₂³	#10621729	1	1	yes	characteristic, direct product factor
47	C₂×C₄	—	169	28	no	—
48	C₂²	—	169	7	no	—
49	C₂²	—	169	21	no	—
50	C₂²	C₂×#10621729	1	7	yes	direct product factor
51	C₄	—	169	8	no	—
52	C₂	—	169	1	no	—
53	C₂	—	169	7	no	—
54	C₂	C₂²×#10621729	1	7	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂³	#10621729
commutator, hypocenter	C₁₃²	C₂³×C₄
Frattini	C₁	G
Fitting, socle	C₂³×C₁₃²	C₄
prime radical	C₂³×C₁₃⋊D₁₃	C₂
cosocle	C₁₃⋊D₁₃	C₂⁴
LCS	G ▷ C₁₃²
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
1. C₂²×#10621729	C₂	direct (N × H)
3. C₂³×C₁₃²	C₄	semidirect
4. C₂²×C₁₃⋊D₁₃	C₄	semidirect
6. C₂×#10621729	C₂²	direct (N × H)
7. C₂²×C₁₃²	C₂×C₄	semidirect
8. C₂×C₁₃⋊D₁₃	C₂×C₄	semidirect
10. #10621729	C₂³	direct (N × H)
12. C₂×C₁₃²	C₂²×C₄	semidirect
13. C₁₃⋊D₁₃	C₂²×C₄	semidirect
18. C₁₃²	C₂³×C₄	semidirect
20. C₂³×C₁₃	#261	semidirect
27. C₂²×C₁₃	C₂×#261	semidirect
35. C₂×C₁₃	C₂²×#261	semidirect
43. C₁₃	C₂³×#261	semidirect
46. C₂³	#10621729	direct (N × H)
50. C₂²	C₂×#10621729	direct (N × H)
54. C₂	C₂²×#10621729	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×#10621729	2
C₂³×C₁₃⋊D₁₃	2
C₂³×#261	13

Subgroups — C23×#10621729 #72259331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#10621729 #72259331