Subgroups — C₂³×#14268648 #28405373

Order 14464.

6520 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14268648	C₂	1	56	yes	maximal, direct product factor
2	C₂³×D₄₅₂	C₂	1	3	yes	maximal, complemented
3	C₂³×C₄×D₁₁₃	C₂	1	3	yes	maximal, complemented
4	C₂³×Q₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×D₁₁₃	C₂²	1	3	yes	—
6	C₂×#14268648	C₂²	1	448	yes	direct product factor
7	C₂²×D₄₅₂	C₂²	1	84	yes	complemented
8	C₂³×Dic₁₁₃	C₂²	1	1	yes	characteristic
9	C₂²×C₄×D₁₁₃	C₂²	1	84	yes	complemented
10	C₂²×Q₈×C₁₁₃	C₂²	1	28	yes	complemented
11	C₂³×C₄×C₁₁₃	C₂²	1	3	yes	—
12	C₂³×D₁₁₃	—	2	24	no	—
13	C₂³×D₁₁₃	C₂³	1	42	yes	—
14	#14268648	C₂³	1	512	yes	direct product factor
15	C₂×D₄₅₂	C₂³	1	336	yes	complemented
16	C₂²×Dic₁₁₃	C₂³	1	14	yes	—
17	C₂×C₄×D₁₁₃	C₂³	1	336	yes	complemented
18	C₂⁴×C₁₁₃	C₂³	1	1	yes	characteristic
19	C₂×Q₈×C₁₁₃	C₂³	1	112	yes	complemented
20	C₂²×C₄×C₁₁₃	C₂³	1	42	yes	—
21	C₂³×C₁₁₃	#41	1	8	yes	complemented
22	C₂³×C₁₁₃	C₂⁴	1	7	yes	—
23	C₂²×D₁₁₃	—	2	168	no	—
24	C₂²×D₁₁₃	C₂⁴	1	84	yes	—
25	Q₈×C₁₁₃	C₂⁴	1	64	yes	complemented
26	C₂×C₄×C₁₁₃	C₂⁴	1	84	yes	—
27	C₂×Dic₁₁₃	C₂⁴	1	28	yes	—
28	D₄₅₂	C₂⁴	1	192	yes	complemented
29	C₄×D₁₁₃	C₂⁴	1	192	yes	complemented
30	C₂²×C₁₁₃	C₂×#41	1	28	yes	complemented
31	C₂²×C₁₁₃	C₂⁵	1	7	yes	—
32	C₂×D₁₁₃	—	2	168	no	—
33	C₂×D₁₁₃	C₂⁵	1	24	yes	—
34	C₄×C₁₁₃	C₂⁵	1	24	yes	—
35	Dic₁₁₃	C₂⁵	1	8	yes	—
36	C₂×C₁₁₃	C₂²×#41	1	14	yes	complemented
37	C₂×C₁₁₃	C₂⁶	1	1	yes	characteristic
38	D₁₁₃	—	2	24	no	—
39	C₂³×#41	—	113	1	no	maximal, Sylow
40	C₁₁₃	C₂³×#41	1	1	yes	characteristic, complemented, Sylow
41	C₂²×#41	—	113	56	no	—
42	C₂³×Q₈	D₁₁₃	1	1	yes	characteristic, complemented
43	C₂³×D₄	—	113	3	no	—
44	C₂⁴×C₄	—	113	3	no	—
45	C₂⁵	—	113	3	no	—
46	C₂×#41	—	113	448	no	—
47	C₂²×Q₈	C₂×D₁₁₃	1	28	yes	complemented
48	C₂²×D₄	—	113	84	no	—
49	C₂³×C₄	—	113	1	no	—
50	C₂³×C₄	—	113	84	no	—
51	C₂³×C₄	C₂×D₁₁₃	1	3	yes	—
52	C₂⁴	—	113	42	no	—
53	C₂⁴	—	226	24	no	—
54	C₂⁴	C₂²×D₁₁₃	1	1	yes	characteristic
55	#41	—	113	512	no	—
56	C₂×Q₈	C₂²×D₁₁₃	1	112	yes	complemented
57	C₂×D₄	—	113	336	no	—
58	C₂²×C₄	—	113	14	no	—
59	C₂²×C₄	—	113	336	no	—
60	C₂²×C₄	C₂²×D₁₁₃	1	42	yes	—
61	C₂³	—	113	84	no	—
62	C₂³	—	226	168	no	—
63	C₂³	#14268648	1	8	yes	direct product factor
64	C₂³	C₂³×D₁₁₃	1	7	yes	—
65	Q₈	C₂³×D₁₁₃	1	64	yes	complemented
66	D₄	—	113	192	no	—
67	C₂×C₄	—	113	28	no	—
68	C₂×C₄	—	113	192	no	—
69	C₂×C₄	C₂³×D₁₁₃	1	84	yes	—
70	C₂²	—	113	24	no	—
71	C₂²	—	226	168	no	—
72	C₂²	C₂×#14268648	1	28	yes	direct product factor
73	C₂²	C₂⁴×D₁₁₃	1	7	yes	—
74	C₄	—	113	8	no	—
75	C₄	C₂⁴×D₁₁₃	1	24	yes	—
76	C₂	—	226	24	no	—
77	C₂	C₂²×#14268648	1	14	yes	direct product factor
78	C₂	C₂⁵×D₁₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₁₁₃
commutator, cosocle	C₂×C₁₁₃	C₂⁶
Frattini	C₂	C₂⁵×D₁₁₃
Fitting	C₂³×Q₈×C₁₁₃	C₂
socle	C₂⁴×C₁₁₃	C₂³
hypercenter	C₂³×Q₈	D₁₁₃
hypocenter	C₁₁₃	C₂³×#41
LCS	G ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#14268648	C₂	direct (N × H)
2. C₂³×D₄₅₂	C₂	semidirect
3. C₂³×C₄×D₁₁₃	C₂	semidirect
4. C₂³×Q₈×C₁₁₃	C₂	semidirect
6. C₂×#14268648	C₂²	direct (N × H)
7. C₂²×D₄₅₂	C₂²	semidirect
9. C₂²×C₄×D₁₁₃	C₂²	semidirect
10. C₂²×Q₈×C₁₁₃	C₂²	semidirect
14. #14268648	C₂³	direct (N × H)
15. C₂×D₄₅₂	C₂³	semidirect
17. C₂×C₄×D₁₁₃	C₂³	semidirect
19. C₂×Q₈×C₁₁₃	C₂³	semidirect
21. C₂³×C₁₁₃	#41	semidirect
25. Q₈×C₁₁₃	C₂⁴	semidirect
28. D₄₅₂	C₂⁴	semidirect
29. C₄×D₁₁₃	C₂⁴	semidirect
30. C₂²×C₁₁₃	C₂×#41	semidirect
36. C₂×C₁₁₃	C₂²×#41	semidirect
40. C₁₁₃	C₂³×#41	semidirect
42. C₂³×Q₈	D₁₁₃	semidirect
47. C₂²×Q₈	C₂×D₁₁₃	semidirect
56. C₂×Q₈	C₂²×D₁₁₃	semidirect
63. C₂³	#14268648	direct (N × H)
65. Q₈	C₂³×D₁₁₃	semidirect
72. C₂²	C₂×#14268648	direct (N × H)
77. C₂	C₂²×#14268648	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×#14268648	2
C₂³×D₄₅₂	2
C₂³×C₄×D₁₁₃	2
C₂³×Q₈×C₁₁₃	2
C₂³×#41	113

Subgroups — C23×#14268648 #28405373

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14268648 #28405373