Subgroups — C₂²×#11745671 #16838174

Order 3776.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₂₃₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#11745671	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₅₉	—	2	1	no	—
6	C₂×Q₈×C₅₉	—	2	3	no	—
7	C₂×Q₈×C₅₉	C₂²	1	6	yes	complemented
8	C₂²×C₄×C₅₉	—	2	1	no	—
9	C₂²×C₄×C₅₉	C₂²	1	1	yes	characteristic
10	C₂×D₂₃₆	—	2	3	no	—
11	C₂×D₂₃₆	C₂²	1	6	yes	—
12	#11745671	C₂²	1	16	yes	direct product factor
13	C₂×C₅₉⋊C₈	C₂²	1	6	yes	complemented
14	C₂³×C₅₉	D₄	1	1	yes	characteristic
15	C₂²×D₅₉	—	2	6	no	—
16	C₂²×D₅₉	—	4	4	no	—
17	Q₈×C₅₉	—	2	6	no	—
18	Q₈×C₅₉	C₂³	1	4	yes	complemented
19	C₂×C₄×C₅₉	—	2	6	no	—
20	C₂×C₄×C₅₉	D₄	1	3	yes	—
21	C₂×C₄×C₅₉	C₂³	1	3	yes	—
22	D₂₃₆	—	2	6	no	—
23	D₂₃₆	C₂³	1	4	yes	—
24	C₅₉⋊C₈	C₂³	1	4	yes	complemented
25	C₂²×C₅₉	SD₁₆	1	4	yes	complemented
26	C₂²×C₅₉	C₂×D₄	1	3	yes	—
27	C₂×D₅₉	—	2	4	no	—
28	C₂×D₅₉	—	4	12	no	—
29	C₄×C₅₉	—	2	4	no	—
30	C₄×C₅₉	C₂×D₄	1	3	yes	—
31	C₄×C₅₉	C₂⁴	1	1	yes	characteristic
32	C₂×C₅₉	C₂×SD₁₆	1	6	yes	complemented
33	C₂×C₅₉	C₂²×D₄	1	1	yes	characteristic
34	D₅₉	—	4	4	no	—
35	C₂²×SD₁₆	—	59	1	no	maximal, Sylow
36	C₅₉	C₂²×SD₁₆	1	1	yes	characteristic, complemented, Sylow
37	C₂²×Q₈	D₅₉	1	1	yes	characteristic, complemented
38	C₂²×D₄	—	59	1	no	—
39	C₂×SD₁₆	—	59	12	no	—
40	C₂²×C₈	—	59	1	no	—
41	C₂⁴	—	118	1	no	—
42	C₂×Q₈	—	2	3	no	—
43	C₂×Q₈	C₂×D₅₉	1	6	yes	complemented
44	C₂×D₄	—	59	6	no	—
45	C₂×D₄	—	118	3	no	—
46	C₂²×C₄	—	2	1	no	—
47	C₂²×C₄	C₂×D₅₉	1	1	yes	characteristic
48	SD₁₆	—	59	16	no	—
49	C₂×C₈	—	59	6	no	—
50	C₂³	—	118	6	no	—
51	C₂³	—	236	4	no	—
52	C₂³	C₂²⋊D₅₉	1	1	yes	characteristic
53	Q₈	—	2	6	no	—
54	Q₈	C₂²×D₅₉	1	4	yes	complemented
55	D₄	—	59	4	no	—
56	D₄	—	118	6	no	—
57	C₂×C₄	—	2	6	no	—
58	C₂×C₄	C₂²⋊D₅₉	1	3	yes	—
59	C₂×C₄	C₂²×D₅₉	1	3	yes	—
60	C₈	—	59	4	no	—
61	C₂²	—	118	4	no	—
62	C₂²	—	236	12	no	—
63	C₂²	#11745671	1	4	yes	direct product factor
64	C₂²	C₂×C₂²⋊D₅₉	1	3	yes	—
65	C₄	—	2	4	no	—
66	C₄	C₂×C₂²⋊D₅₉	1	3	yes	—
67	C₄	C₂³×D₅₉	1	1	yes	characteristic
68	C₂	—	236	4	no	—
69	C₂	C₂×#11745671	1	6	yes	direct product factor
70	C₂	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊D₅₉
commutator, cosocle	C₄×C₅₉	C₂⁴
Frattini	C₄	C₂³×D₅₉
Fitting	C₂²×Q₈×C₅₉	C₂
socle	C₂³×C₅₉	D₄
prime radical	C₂²×D₂₃₆	C₂
hypercenter	C₂²×Q₈	D₅₉
hypocenter	C₅₉	C₂²×SD₁₆
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
2. C₂×#11745671	C₂	direct (N × H)
3. C₂²×C₅₉⋊C₈	C₂	semidirect
4. C₂²×Q₈×C₅₉	C₂	semidirect
7. C₂×Q₈×C₅₉	C₂²	semidirect
12. #11745671	C₂²	direct (N × H)
13. C₂×C₅₉⋊C₈	C₂²	semidirect
18. Q₈×C₅₉	C₂³	semidirect
24. C₅₉⋊C₈	C₂³	semidirect
25. C₂²×C₅₉	SD₁₆	semidirect
32. C₂×C₅₉	C₂×SD₁₆	semidirect
36. C₅₉	C₂²×SD₁₆	semidirect
37. C₂²×Q₈	D₅₉	semidirect
43. C₂×Q₈	C₂×D₅₉	semidirect
54. Q₈	C₂²×D₅₉	semidirect
63. C₂²	#11745671	direct (N × H)
69. C₂	C₂×#11745671	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×D₂₃₆	2
C₂×#11745671	2
C₂²×C₅₉⋊C₈	2
C₂²×Q₈×C₅₉	2
C₂²×SD₁₆	59

Subgroups — C22×#11745671 #16838174

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11745671 #16838174