Subgroups — C₂³×#13182004 #28376125

Order 13952.

440 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13182004	C₂	1	14	yes	maximal, direct product factor
2	C₂³×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#13182004	C₂²	1	28	yes	direct product factor
4	C₂³×C₄×C₁₀₉	C₄	1	1	yes	characteristic
5	C₂²×C₈×C₁₀₉	C₄	1	7	yes	—
6	C₂²×C₈×C₁₀₉	C₂²	1	7	yes	—
7	#13182004	C₂³	1	8	yes	direct product factor
8	C₂⁴×C₁₀₉	C₈	1	1	yes	characteristic
9	C₂²×C₄×C₁₀₉	C₈	1	7	yes	—
10	C₂²×C₄×C₁₀₉	C₂×C₄	1	7	yes	—
11	C₂×C₈×C₁₀₉	C₂×C₄	1	21	yes	—
12	C₂×C₈×C₁₀₉	C₂³	1	7	yes	—
13	C₂³×C₁₀₉	C₁₆	1	8	yes	complemented
14	C₂³×C₁₀₉	C₂×C₈	1	7	yes	—
15	C₂×C₄×C₁₀₉	C₂×C₈	1	21	yes	—
16	C₂×C₄×C₁₀₉	C₂²×C₄	1	7	yes	—
17	C₈×C₁₀₉	C₂²×C₄	1	7	yes	—
18	C₈×C₁₀₉	C₂⁴	1	1	yes	characteristic
19	C₂²×C₁₀₉	C₂×C₁₆	1	28	yes	complemented
20	C₂²×C₁₀₉	C₂²×C₈	1	7	yes	—
21	C₄×C₁₀₉	C₂²×C₈	1	7	yes	—
22	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
23	C₂×C₁₀₉	C₂²×C₁₆	1	14	yes	complemented
24	C₂×C₁₀₉	C₂³×C₈	1	1	yes	characteristic
25	C₂³×C₁₆	—	109	1	no	maximal, Sylow
26	C₁₀₉	C₂³×C₁₆	1	1	yes	characteristic, complemented, Sylow
27	C₂³×C₈	D₁₀₉	1	1	yes	characteristic
28	C₂²×C₁₆	—	109	14	no	—
29	C₂³×C₄	Dic₁₀₉	1	1	yes	characteristic
30	C₂²×C₈	Dic₁₀₉	1	7	yes	—
31	C₂²×C₈	C₂×D₁₀₉	1	7	yes	—
32	C₂×C₁₆	—	109	28	no	—
33	C₂⁴	#11724784	1	1	yes	characteristic
34	C₂²×C₄	#11724784	1	7	yes	—
35	C₂²×C₄	C₂×Dic₁₀₉	1	7	yes	—
36	C₂×C₈	C₂×Dic₁₀₉	1	21	yes	—
37	C₂×C₈	C₂²×D₁₀₉	1	7	yes	—
38	C₁₆	—	109	8	no	—
39	C₂³	#13182004	1	8	yes	direct product factor
40	C₂³	C₂×#11724784	1	7	yes	—
41	C₂×C₄	C₂×#11724784	1	21	yes	—
42	C₂×C₄	C₂²×Dic₁₀₉	1	7	yes	—
43	C₈	C₂²×Dic₁₀₉	1	7	yes	—
44	C₈	C₂³×D₁₀₉	1	1	yes	characteristic
45	C₂²	C₂×#13182004	1	28	yes	direct product factor
46	C₂²	C₂²×#11724784	1	7	yes	—
47	C₄	C₂²×#11724784	1	7	yes	—
48	C₄	C₂³×Dic₁₀₉	1	1	yes	characteristic
49	C₂	C₂²×#13182004	1	14	yes	direct product factor
50	C₂	C₂³×#11724784	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₈	D₁₀₉
commutator, hypocenter	C₁₀₉	C₂³×C₁₆
Frattini	C₈	C₂³×D₁₀₉
Fitting	C₂³×C₈×C₁₀₉	C₂
socle, prime radical	C₂⁴×C₁₀₉	C₈
cosocle	C₈×C₁₀₉	C₂⁴
LCS	G ▷ C₁₀₉
UCS	C₁ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#13182004	C₂	direct (N × H)
3. C₂×#13182004	C₂²	direct (N × H)
7. #13182004	C₂³	direct (N × H)
13. C₂³×C₁₀₉	C₁₆	semidirect
19. C₂²×C₁₀₉	C₂×C₁₆	semidirect
23. C₂×C₁₀₉	C₂²×C₁₆	semidirect
26. C₁₀₉	C₂³×C₁₆	semidirect
39. C₂³	#13182004	direct (N × H)
45. C₂²	C₂×#13182004	direct (N × H)
49. C₂	C₂²×#13182004	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×#13182004	2
C₂³×C₈×C₁₀₉	2
C₂³×C₁₆	109

Subgroups — C23×#13182004 #28376125

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#13182004 #28376125