Subgroups — C₂²×C₁₆×Dic₅₉ #28741311

Order 15104.

1022 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×Dic₅₉	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₅₉⋊C₈	C₄	1	1	yes	characteristic
6	C₂²×C₄×Dic₅₉	C₄	1	1	yes	characteristic
7	C₂²×C₅₉⋊C₁₆	C₄	1	8	yes	complemented
8	C₂²×C₅₉⋊C₁₆	C₂²	1	6	yes	—
9	C₂×C₈×Dic₅₉	C₄	1	6	yes	—
10	C₂×C₈×Dic₅₉	C₂²	1	6	yes	—
11	C₁₆×Dic₅₉	C₂²	1	16	yes	direct product factor
12	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₅₉	C₄	1	8	yes	complemented
14	C₂²×C₁₆×C₅₉	C₂²	1	6	yes	—
15	C₂³×Dic₅₉	C₈	1	2	yes	—
16	C₂²×C₅₉⋊C₈	C₈	1	8	yes	—
17	C₂²×C₅₉⋊C₈	C₂×C₄	1	6	yes	—
18	C₂×C₄×Dic₅₉	C₈	1	6	yes	—
19	C₂×C₄×Dic₅₉	C₂×C₄	1	6	yes	—
20	C₂×C₅₉⋊C₁₆	C₂×C₄	1	24	yes	complemented
21	C₂×C₅₉⋊C₁₆	C₂³	1	4	yes	—
22	C₈×Dic₅₉	C₂×C₄	1	12	yes	—
23	C₈×Dic₅₉	C₂³	1	4	yes	—
24	C₂³×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₉	C₂×C₄	1	3	yes	—
26	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₅₉	C₂³	1	3	yes	—
29	C₂×C₁₆×C₅₉	C₂×C₄	1	24	yes	complemented
30	C₂×C₁₆×C₅₉	C₂³	1	4	yes	—
31	C₂⁴×C₅₉	C₂×C₈	1	1	yes	characteristic
32	C₂²×C₄×C₅₉	C₄²	1	4	yes	—
33	C₂²×C₄×C₅₉	C₂×C₈	1	3	yes	—
34	C₂²×C₄×C₅₉	C₂×C₈	1	4	yes	—
35	C₂²×C₄×C₅₉	C₂²×C₄	1	3	yes	—
36	C₂²×Dic₅₉	C₁₆	1	16	yes	direct product factor
37	C₂²×Dic₅₉	C₂×C₈	1	12	yes	—
38	C₂×C₈×C₅₉	C₄²	1	12	yes	—
39	C₂×C₈×C₅₉	C₂²×C₄	1	3	yes	—
40	C₂×C₈×C₅₉	C₂²×C₄	1	6	yes	—
41	C₂×C₈×C₅₉	C₂²×C₄	1	6	yes	—
42	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
43	C₄×Dic₅₉	C₂×C₈	1	12	yes	—
44	C₄×Dic₅₉	C₂²×C₄	1	4	yes	—
45	C₂×C₅₉⋊C₈	C₂×C₈	1	24	yes	—
46	C₂×C₅₉⋊C₈	C₂²×C₄	1	4	yes	—
47	C₁₆×C₅₉	C₂²×C₄	1	8	yes	complemented
48	C₅₉⋊C₁₆	C₂²×C₄	1	8	yes	complemented
49	C₂³×C₅₉	C₄×C₈	1	4	yes	—
50	C₂³×C₅₉	C₂×C₁₆	1	8	yes	—
51	C₂³×C₅₉	C₂²×C₈	1	3	yes	—
52	C₂×C₄×C₅₉	C₄×C₈	1	12	yes	—
53	C₂×C₄×C₅₉	C₂×C₄²	1	6	yes	—
54	C₂×C₄×C₅₉	C₂²×C₈	1	6	yes	—
55	C₂×C₄×C₅₉	C₂²×C₈	1	3	yes	—
56	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
57	C₂×Dic₅₉	C₂×C₁₆	1	48	yes	direct product factor
58	C₂×Dic₅₉	C₂²×C₈	1	8	yes	—
59	C₈×C₅₉	C₂×C₄²	1	6	yes	—
60	C₈×C₅₉	C₂³×C₄	1	1	yes	characteristic
61	C₈×C₅₉	C₂³×C₄	1	1	yes	characteristic
62	C₅₉⋊C₈	C₂²×C₈	1	8	yes	—
63	C₂²×C₄×C₁₆	—	59	1	no	maximal, Sylow
64	C₂²×C₅₉	C₄×C₁₆	1	16	yes	complemented
65	C₂²×C₅₉	C₂×C₄×C₈	1	6	yes	—
66	C₂²×C₅₉	C₂²×C₁₆	1	12	yes	—
67	C₂²×C₅₉	C₂³×C₈	1	1	yes	characteristic
68	C₄×C₅₉	C₂×C₄×C₈	1	6	yes	—
69	C₄×C₅₉	C₂²×C₄²	1	1	yes	characteristic
70	C₄×C₅₉	C₂³×C₈	1	1	yes	characteristic
71	Dic₅₉	C₂²×C₁₆	1	16	yes	direct product factor
72	C₂³×C₁₆	—	59	1	no	—
73	C₂³×C₁₆	D₅₉	1	1	yes	characteristic
74	C₂²×C₄×C₈	—	59	1	no	—
75	C₂×C₄×C₁₆	—	59	12	no	—
76	C₂×C₅₉	C₂×C₄×C₁₆	1	12	yes	complemented
77	C₂×C₅₉	C₂²×C₄×C₈	1	1	yes	characteristic
78	C₂×C₅₉	C₂³×C₁₆	1	2	yes	—
79	C₂³×C₈	—	59	1	no	—
80	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
81	C₂²×C₄²	—	59	1	no	—
82	C₂²×C₁₆	—	59	8	no	—
83	C₂²×C₁₆	—	59	6	no	—
84	C₂²×C₁₆	Dic₅₉	1	8	yes	direct product factor
85	C₂²×C₁₆	C₂×D₅₉	1	6	yes	—
86	C₂×C₄×C₈	—	59	6	no	—
87	C₂×C₄×C₈	—	59	6	no	—
88	C₄×C₁₆	—	59	16	no	—
89	C₅₉	C₂²×C₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
90	C₂³×C₄	—	59	2	no	—
91	C₂³×C₄	C₄×D₅₉	1	1	yes	characteristic
92	C₂²×C₈	—	59	8	no	—
93	C₂²×C₈	—	59	6	no	—
94	C₂²×C₈	C₄×D₅₉	1	3	yes	—
95	C₂²×C₈	C₄×D₅₉	1	4	yes	—
96	C₂²×C₈	C₂×Dic₅₉	1	4	yes	—
97	C₂²×C₈	C₂²×D₅₉	1	3	yes	—
98	C₂×C₄²	—	59	6	no	—
99	C₂×C₄²	—	59	6	no	—
100	C₂×C₁₆	—	59	24	no	—
101	C₂×C₁₆	—	59	4	no	—
102	C₂×C₁₆	C₂×Dic₅₉	1	24	yes	direct product factor
103	C₂×C₁₆	C₂²×D₅₉	1	4	yes	—
104	C₄×C₈	—	59	12	no	—
105	C₄×C₈	—	59	4	no	—
106	C₂⁴	C₈×D₅₉	1	1	yes	characteristic
107	C₂²×C₄	—	59	16	no	—
108	C₂²×C₄	—	59	12	no	—
109	C₂²×C₄	C₈×D₅₉	1	3	yes	—
110	C₂²×C₄	C₈×D₅₉	1	4	yes	—
111	C₂²×C₄	C₄×Dic₅₉	1	4	yes	—
112	C₂²×C₄	C₂×C₄×D₅₉	1	3	yes	—
113	C₂×C₈	—	59	24	no	—
114	C₂×C₈	—	59	4	no	—
115	C₂×C₈	C₄×Dic₅₉	1	12	yes	—
116	C₂×C₈	C₂×C₄×D₅₉	1	3	yes	—
117	C₂×C₈	C₂×C₄×D₅₉	1	6	yes	—
118	C₂×C₈	C₂²×Dic₅₉	1	6	yes	—
119	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
120	C₄²	—	59	12	no	—
121	C₄²	—	59	4	no	—
122	C₁₆	—	59	8	no	—
123	C₁₆	C₂²×Dic₅₉	1	8	yes	direct product factor
124	C₂³	C₈×Dic₅₉	1	4	yes	—
125	C₂³	C₁₆×D₅₉	1	8	yes	—
126	C₂³	C₂×C₈×D₅₉	1	3	yes	—
127	C₂×C₄	—	59	8	no	—
128	C₂×C₄	—	59	48	no	—
129	C₂×C₄	C₈×Dic₅₉	1	12	yes	—
130	C₂×C₄	C₂×C₄×Dic₅₉	1	6	yes	—
131	C₂×C₄	C₂×C₈×D₅₉	1	6	yes	—
132	C₂×C₄	C₂×C₈×D₅₉	1	3	yes	—
133	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
134	C₈	—	59	8	no	—
135	C₈	C₂×C₄×Dic₅₉	1	6	yes	—
136	C₈	C₂²×C₄×D₅₉	1	1	yes	characteristic
137	C₈	C₂³×Dic₅₉	1	1	yes	characteristic
138	C₂²	C₁₆×Dic₅₉	1	16	yes	direct product factor
139	C₂²	C₂×C₈×Dic₅₉	1	6	yes	—
140	C₂²	C₂×C₁₆×D₅₉	1	12	yes	—
141	C₂²	C₂²×C₈×D₅₉	1	1	yes	characteristic
142	C₄	—	59	16	no	—
143	C₄	C₂×C₈×Dic₅₉	1	6	yes	—
144	C₄	C₂²×C₄×Dic₅₉	1	1	yes	characteristic
145	C₄	C₂²×C₈×D₅₉	1	1	yes	characteristic
146	C₂	C₂×C₁₆×Dic₅₉	1	12	yes	direct product factor
147	C₂	C₂²×C₈×Dic₅₉	1	1	yes	characteristic
148	C₂	C₂²×C₁₆×D₅₉	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂³×C₁₆	D₅₉
commutator, hypocenter	C₅₉	C₂²×C₄×C₁₆
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂³×C₁₆×C₅₉	C₂
socle, prime radical	C₂⁴×C₅₉	C₂×C₈
cosocle	C₂×C₈×C₅₉	C₂⁴
LCS	G ▷ C₅₉
UCS	C₁ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×C₁₆×Dic₅₉	C₂	direct (N × H)
7. C₂²×C₅₉⋊C₁₆	C₄	semidirect
11. C₁₆×Dic₅₉	C₂²	direct (N × H)
13. C₂²×C₁₆×C₅₉	C₄	semidirect
20. C₂×C₅₉⋊C₁₆	C₂×C₄	semidirect
29. C₂×C₁₆×C₅₉	C₂×C₄	semidirect
36. C₂²×Dic₅₉	C₁₆	direct (N × H)
47. C₁₆×C₅₉	C₂²×C₄	semidirect
48. C₅₉⋊C₁₆	C₂²×C₄	semidirect
57. C₂×Dic₅₉	C₂×C₁₆	direct (N × H)
64. C₂²×C₅₉	C₄×C₁₆	semidirect
71. Dic₅₉	C₂²×C₁₆	direct (N × H)
76. C₂×C₅₉	C₂×C₄×C₁₆	semidirect
84. C₂²×C₁₆	Dic₅₉	direct (N × H)
89. C₅₉	C₂²×C₄×C₁₆	semidirect
102. C₂×C₁₆	C₂×Dic₅₉	direct (N × H)
123. C₁₆	C₂²×Dic₅₉	direct (N × H)
138. C₂²	C₁₆×Dic₅₉	direct (N × H)
146. C₂	C₂×C₁₆×Dic₅₉	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₅₉⋊C₁₆	2
C₂²×C₈×Dic₅₉	2
C₂×C₁₆×Dic₅₉	2
C₂³×C₁₆×C₅₉	2
C₂²×C₄×C₁₆	59

Subgroups — C22×C16×Dic59 #28741311

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₆×Dic₅₉ #28741311