Subgroups — C₂³×#10625354 #20258187

Order 6016.

6520 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10625354	C₂	1	56	yes	maximal, direct product factor
2	C₂³×D₁₈₈	C₂	1	3	yes	maximal, complemented
3	C₂³×C₄×D₄₇	C₂	1	3	yes	maximal, complemented
4	C₂³×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×D₄₇	C₂²	1	3	yes	—
6	C₂×#10625354	C₂²	1	448	yes	direct product factor
7	C₂²×D₁₈₈	C₂²	1	84	yes	complemented
8	C₂³×Dic₄₇	C₂²	1	1	yes	characteristic
9	C₂²×C₄×D₄₇	C₂²	1	84	yes	complemented
10	C₂²×Q₈×C₄₇	C₂²	1	28	yes	complemented
11	C₂³×C₄×C₄₇	C₂²	1	3	yes	—
12	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
13	C₂³×D₄₇	—	2	24	no	—
14	C₂³×D₄₇	C₂³	1	42	yes	—
15	C₂×Q₈×C₄₇	C₂³	1	112	yes	complemented
16	C₂²×C₄×C₄₇	C₂³	1	42	yes	—
17	C₂²×Dic₄₇	C₂³	1	14	yes	—
18	#10625354	C₂³	1	512	yes	direct product factor
19	C₂×D₁₈₈	C₂³	1	336	yes	complemented
20	C₂×C₄×D₄₇	C₂³	1	336	yes	complemented
21	C₂³×C₄₇	#41	1	8	yes	complemented
22	C₂³×C₄₇	C₂⁴	1	7	yes	—
23	C₂²×D₄₇	—	2	168	no	—
24	C₂²×D₄₇	C₂⁴	1	84	yes	—
25	Q₈×C₄₇	C₂⁴	1	64	yes	complemented
26	C₂×C₄×C₄₇	C₂⁴	1	84	yes	—
27	C₂×Dic₄₇	C₂⁴	1	28	yes	—
28	D₁₈₈	C₂⁴	1	192	yes	complemented
29	C₄×D₄₇	C₂⁴	1	192	yes	complemented
30	C₂²×C₄₇	C₂×#41	1	28	yes	complemented
31	C₂²×C₄₇	C₂⁵	1	7	yes	—
32	C₂×D₄₇	—	2	168	no	—
33	C₂×D₄₇	C₂⁵	1	24	yes	—
34	C₄×C₄₇	C₂⁵	1	24	yes	—
35	Dic₄₇	C₂⁵	1	8	yes	—
36	C₂³×#41	—	47	1	no	maximal, Sylow
37	C₂×C₄₇	C₂²×#41	1	14	yes	complemented
38	C₂×C₄₇	C₂⁶	1	1	yes	characteristic
39	D₄₇	—	2	24	no	—
40	C₂²×#41	—	47	56	no	—
41	C₂³×Q₈	D₄₇	1	1	yes	characteristic, complemented
42	C₂³×D₄	—	47	3	no	—
43	C₂⁴×C₄	—	47	3	no	—
44	C₄₇	C₂³×#41	1	1	yes	characteristic, complemented, Sylow
45	C₂⁵	—	47	3	no	—
46	C₂×#41	—	47	448	no	—
47	C₂²×Q₈	C₂×D₄₇	1	28	yes	complemented
48	C₂²×D₄	—	47	84	no	—
49	C₂³×C₄	—	47	1	no	—
50	C₂³×C₄	—	47	84	no	—
51	C₂³×C₄	C₂×D₄₇	1	3	yes	—
52	C₂⁴	—	47	42	no	—
53	C₂⁴	—	94	24	no	—
54	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
55	#41	—	47	512	no	—
56	C₂×Q₈	C₂²×D₄₇	1	112	yes	complemented
57	C₂×D₄	—	47	336	no	—
58	C₂²×C₄	—	47	14	no	—
59	C₂²×C₄	—	47	336	no	—
60	C₂²×C₄	C₂²×D₄₇	1	42	yes	—
61	C₂³	—	47	84	no	—
62	C₂³	—	94	168	no	—
63	C₂³	#10625354	1	8	yes	direct product factor
64	C₂³	C₂³×D₄₇	1	7	yes	—
65	Q₈	C₂³×D₄₇	1	64	yes	complemented
66	D₄	—	47	192	no	—
67	C₂×C₄	—	47	28	no	—
68	C₂×C₄	—	47	192	no	—
69	C₂×C₄	C₂³×D₄₇	1	84	yes	—
70	C₂²	—	47	24	no	—
71	C₂²	—	94	168	no	—
72	C₂²	C₂×#10625354	1	28	yes	direct product factor
73	C₂²	C₂⁴×D₄₇	1	7	yes	—
74	C₄	—	47	8	no	—
75	C₄	C₂⁴×D₄₇	1	24	yes	—
76	C₂	—	94	24	no	—
77	C₂	C₂²×#10625354	1	14	yes	direct product factor
78	C₂	C₂⁵×D₄₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₄₇
commutator	C₂×C₄₇	C₂⁶
Frattini	C₂	C₂⁵×D₄₇
Fitting	C₂³×Q₈×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂³
hypercenter	C₂³×Q₈	D₄₇
hypocenter	C₄₇	C₂³×#41
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#10625354	C₂	direct (N × H)
2. C₂³×D₁₈₈	C₂	semidirect
3. C₂³×C₄×D₄₇	C₂	semidirect
4. C₂³×Q₈×C₄₇	C₂	semidirect
6. C₂×#10625354	C₂²	direct (N × H)
7. C₂²×D₁₈₈	C₂²	semidirect
9. C₂²×C₄×D₄₇	C₂²	semidirect
10. C₂²×Q₈×C₄₇	C₂²	semidirect
15. C₂×Q₈×C₄₇	C₂³	semidirect
18. #10625354	C₂³	direct (N × H)
19. C₂×D₁₈₈	C₂³	semidirect
20. C₂×C₄×D₄₇	C₂³	semidirect
21. C₂³×C₄₇	#41	semidirect
25. Q₈×C₄₇	C₂⁴	semidirect
28. D₁₈₈	C₂⁴	semidirect
29. C₄×D₄₇	C₂⁴	semidirect
30. C₂²×C₄₇	C₂×#41	semidirect
37. C₂×C₄₇	C₂²×#41	semidirect
41. C₂³×Q₈	D₄₇	semidirect
44. C₄₇	C₂³×#41	semidirect
47. C₂²×Q₈	C₂×D₄₇	semidirect
56. C₂×Q₈	C₂²×D₄₇	semidirect
63. C₂³	#10625354	direct (N × H)
65. Q₈	C₂³×D₄₇	semidirect
72. C₂²	C₂×#10625354	direct (N × H)
77. C₂	C₂²×#10625354	direct (N × H)

Subgroups — C23×#10625354 #20258187

Conjugacy classes of subgroups

Complements

Subgroups — C₂³×#10625354 #20258187