Subgroups — C₂³×#14268622 #28403667

Order 14464.

440 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14268622	C₂	1	14	yes	maximal, direct product factor
2	C₂³×C₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#14268622	C₂²	1	28	yes	direct product factor
4	C₂³×C₄×C₁₁₃	C₄	1	1	yes	characteristic
5	C₂²×C₈×C₁₁₃	C₄	1	7	yes	—
6	C₂²×C₈×C₁₁₃	C₂²	1	7	yes	—
7	#14268622	C₂³	1	8	yes	direct product factor
8	C₂⁴×C₁₁₃	C₈	1	1	yes	characteristic
9	C₂²×C₄×C₁₁₃	C₈	1	7	yes	—
10	C₂²×C₄×C₁₁₃	C₂×C₄	1	7	yes	—
11	C₂×C₈×C₁₁₃	C₂×C₄	1	21	yes	—
12	C₂×C₈×C₁₁₃	C₂³	1	7	yes	—
13	C₂³×C₁₁₃	C₁₆	1	8	yes	complemented
14	C₂³×C₁₁₃	C₂×C₈	1	7	yes	—
15	C₂×C₄×C₁₁₃	C₂×C₈	1	21	yes	—
16	C₂×C₄×C₁₁₃	C₂²×C₄	1	7	yes	—
17	C₈×C₁₁₃	C₂²×C₄	1	7	yes	—
18	C₈×C₁₁₃	C₂⁴	1	1	yes	characteristic
19	C₂²×C₁₁₃	C₂×C₁₆	1	28	yes	complemented
20	C₂²×C₁₁₃	C₂²×C₈	1	7	yes	—
21	C₄×C₁₁₃	C₂²×C₈	1	7	yes	—
22	C₄×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
23	C₂×C₁₁₃	C₂²×C₁₆	1	14	yes	complemented
24	C₂×C₁₁₃	C₂³×C₈	1	1	yes	characteristic
25	C₂³×C₁₆	—	113	1	no	maximal, Sylow
26	C₁₁₃	C₂³×C₁₆	1	1	yes	characteristic, complemented, Sylow
27	C₂³×C₈	D₁₁₃	1	1	yes	characteristic
28	C₂²×C₁₆	—	113	14	no	—
29	C₂³×C₄	Dic₁₁₃	1	1	yes	characteristic
30	C₂²×C₈	Dic₁₁₃	1	7	yes	—
31	C₂²×C₈	C₂×D₁₁₃	1	7	yes	—
32	C₂×C₁₆	—	113	28	no	—
33	C₂⁴	#11744751	1	1	yes	characteristic
34	C₂²×C₄	#11744751	1	7	yes	—
35	C₂²×C₄	C₂×Dic₁₁₃	1	7	yes	—
36	C₂×C₈	C₂×Dic₁₁₃	1	21	yes	—
37	C₂×C₈	C₂²×D₁₁₃	1	7	yes	—
38	C₁₆	—	113	8	no	—
39	C₂³	#14268622	1	8	yes	direct product factor
40	C₂³	C₂×#11744751	1	7	yes	—
41	C₂×C₄	C₂×#11744751	1	21	yes	—
42	C₂×C₄	C₂²×Dic₁₁₃	1	7	yes	—
43	C₈	C₂²×Dic₁₁₃	1	7	yes	—
44	C₈	C₂³×D₁₁₃	1	1	yes	characteristic
45	C₂²	C₂×#14268622	1	28	yes	direct product factor
46	C₂²	C₂²×#11744751	1	7	yes	—
47	C₄	C₂²×#11744751	1	7	yes	—
48	C₄	C₂³×Dic₁₁₃	1	1	yes	characteristic
49	C₂	C₂²×#14268622	1	14	yes	direct product factor
50	C₂	C₂³×#11744751	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₈	D₁₁₃
commutator, hypocenter	C₁₁₃	C₂³×C₁₆
Frattini	C₈	C₂³×D₁₁₃
Fitting	C₂³×C₈×C₁₁₃	C₂
socle, prime radical	C₂⁴×C₁₁₃	C₈
cosocle	C₈×C₁₁₃	C₂⁴
LCS	G ▷ C₁₁₃
UCS	C₁ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#14268622	C₂	direct (N × H)
3. C₂×#14268622	C₂²	direct (N × H)
7. #14268622	C₂³	direct (N × H)
13. C₂³×C₁₁₃	C₁₆	semidirect
19. C₂²×C₁₁₃	C₂×C₁₆	semidirect
23. C₂×C₁₁₃	C₂²×C₁₆	semidirect
26. C₁₁₃	C₂³×C₁₆	semidirect
39. C₂³	#14268622	direct (N × H)
45. C₂²	C₂×#14268622	direct (N × H)
49. C₂	C₂²×#14268622	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×#14268622	2
C₂³×C₈×C₁₁₃	2
C₂³×C₁₆	113

Subgroups — C23×#14268622 #28403667

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14268622 #28403667