Subgroups — C₂²×Dic₅₉⋊C₁₆ #28741663

Order 15104.

862 conjugacy classes of subgroups.

The subgroup lattice has 168 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×Dic₅₉⋊C₁₆	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	—	2	4	no	—
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	Dic₅₉⋊C₁₆	C₂²	1	16	yes	direct product factor
12	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₅₉	—	2	4	no	—
14	C₂²×C₁₆×C₅₉	C₂²	1	6	yes	—
15	C₂³×Dic₅₉	C₈	1	1	yes	characteristic
16	C₂³×Dic₅₉	C₈	1	1	yes	characteristic
17	C₂²×C₅₉⋊C₈	—	2	4	no	—
18	C₂²×C₅₉⋊C₈	C₂×C₄	1	6	yes	—
19	C₂×C₄×Dic₅₉	C₈	1	6	yes	—
20	C₂×C₄×Dic₅₉	C₂×C₄	1	6	yes	—
21	C₂×C₅₉⋊C₁₆	—	2	12	no	—
22	C₂×C₅₉⋊C₁₆	C₂³	1	4	yes	—
23	C₈×Dic₅₉	C₂×C₄	1	12	yes	—
24	C₈×Dic₅₉	C₂³	1	4	yes	—
25	C₂³×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₉	C₂×C₄	1	3	yes	—
27	C₂²×C₈×C₅₉	D₄	1	4	yes	—
28	C₂²×C₈×C₅₉	Q₈	1	4	yes	—
29	C₂²×C₈×C₅₉	C₂³	1	3	yes	—
30	C₂×C₁₆×C₅₉	—	2	12	no	—
31	C₂×C₁₆×C₅₉	C₂³	1	4	yes	—
32	C₂⁴×C₅₉	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄×C₅₉	C₄⋊C₄	1	4	yes	—
34	C₂²×C₄×C₅₉	C₂×C₈	1	3	yes	—
35	C₂²×C₄×C₅₉	M₁₆	1	4	yes	—
36	C₂²×C₄×C₅₉	C₂²×C₄	1	3	yes	—
37	C₂²×Dic₅₉	—	2	4	no	—
38	C₂²×Dic₅₉	C₁₆	1	8	yes	complemented
39	C₂²×Dic₅₉	C₂×C₈	1	6	yes	—
40	C₂²×Dic₅₉	C₂×C₈	1	6	yes	—
41	C₂×C₈×C₅₉	C₄⋊C₄	1	12	yes	—
42	C₂×C₈×C₅₉	C₂²×C₄	1	3	yes	—
43	C₂×C₈×C₅₉	C₂×D₄	1	6	yes	—
44	C₂×C₈×C₅₉	C₂×Q₈	1	6	yes	—
45	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
46	C₄×Dic₅₉	C₂×C₈	1	12	yes	—
47	C₄×Dic₅₉	C₂²×C₄	1	4	yes	—
48	C₂×C₅₉⋊C₈	—	2	12	no	—
49	C₂×C₅₉⋊C₈	C₂²×C₄	1	4	yes	—
50	C₁₆×C₅₉	—	2	4	no	—
51	C₅₉⋊C₁₆	—	2	4	no	—
52	C₂³×C₅₉	C₄⋊C₈	1	4	yes	—
53	C₂³×C₅₉	C₂×C₁₆	1	4	yes	—
54	C₂³×C₅₉	M₃₂	1	4	yes	—
55	C₂³×C₅₉	C₂²×C₈	1	3	yes	—
56	C₂×C₄×C₅₉	C₄⋊C₈	1	12	yes	—
57	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	6	yes	—
58	C₂×C₄×C₅₉	C₂²×C₈	1	3	yes	—
59	C₂×C₄×C₅₉	C₂×M₁₆	1	6	yes	—
60	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
61	C₂×Dic₅₉	—	2	12	no	—
62	C₂×Dic₅₉	C₂×C₁₆	1	24	yes	complemented
63	C₂×Dic₅₉	C₂²×C₈	1	4	yes	—
64	C₂×Dic₅₉	C₂²×C₈	1	4	yes	—
65	C₈×C₅₉	C₂×C₄⋊C₄	1	6	yes	—
66	C₈×C₅₉	C₂²×D₄	1	1	yes	characteristic
67	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
68	C₅₉⋊C₈	—	2	4	no	—
69	C₂²×C₄⋊C₁₆	—	59	1	no	maximal, Sylow
70	C₂²×C₅₉	C₄⋊C₁₆	1	16	yes	complemented
71	C₂²×C₅₉	C₂×C₄⋊C₈	1	6	yes	—
72	C₂²×C₅₉	C₂²×C₁₆	1	6	yes	—
73	C₂²×C₅₉	C₂×M₃₂	1	6	yes	—
74	C₂²×C₅₉	C₂³×C₈	1	1	yes	characteristic
75	C₄×C₅₉	C₂×C₄⋊C₈	1	6	yes	—
76	C₄×C₅₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
77	C₄×C₅₉	C₂²×M₁₆	1	1	yes	characteristic
78	Dic₅₉	—	2	4	no	—
79	Dic₅₉	C₂²×C₁₆	1	8	yes	complemented
80	C₂³×C₁₆	—	59	1	no	—
81	C₂³×C₁₆	D₅₉	1	1	yes	characteristic
82	C₂²×C₄×C₈	—	59	1	no	—
83	C₂×C₄⋊C₁₆	—	59	12	no	—
84	C₂×C₅₉	C₂×C₄⋊C₁₆	1	12	yes	complemented
85	C₂×C₅₉	C₂²×C₄⋊C₈	1	1	yes	characteristic
86	C₂×C₅₉	C₂³×C₁₆	1	1	yes	characteristic
87	C₂×C₅₉	C₂²×M₃₂	1	1	yes	characteristic
88	C₂³×C₈	—	59	1	no	—
89	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
90	C₂²×C₄²	—	59	1	no	—
91	C₂²×C₁₆	—	2	4	no	—
92	C₂²×C₁₆	—	59	6	no	—
93	C₂²×C₁₆	—	118	4	no	—
94	C₂²×C₁₆	C₂×D₅₉	1	6	yes	—
95	C₂×C₄×C₈	—	59	6	no	—
96	C₂×C₄×C₈	—	59	6	no	—
97	C₄⋊C₁₆	—	59	16	no	—
98	C₅₉	C₂²×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
99	C₂³×C₄	—	59	1	no	—
100	C₂³×C₄	—	59	1	no	—
101	C₂³×C₄	C₄×D₅₉	1	1	yes	characteristic
102	C₂²×C₈	—	59	6	no	—
103	C₂²×C₈	—	118	4	no	—
104	C₂²×C₈	Dic₁₁₈	1	4	yes	—
105	C₂²×C₈	C₄×D₅₉	1	3	yes	—
106	C₂²×C₈	C₂²⋊D₅₉	1	4	yes	—
107	C₂²×C₈	C₂²×D₅₉	1	3	yes	—
108	C₂×C₄²	—	59	6	no	—
109	C₂×C₄²	—	59	6	no	—
110	C₂×C₁₆	—	2	12	no	—
111	C₂×C₁₆	—	59	4	no	—
112	C₂×C₁₆	—	118	12	no	—
113	C₂×C₁₆	C₂²×D₅₉	1	4	yes	—
114	C₄×C₈	—	59	12	no	—
115	C₄×C₈	—	59	4	no	—
116	C₂⁴	C₈×D₅₉	1	1	yes	characteristic
117	C₂²×C₄	—	59	8	no	—
118	C₂²×C₄	—	59	6	no	—
119	C₂²×C₄	—	59	6	no	—
120	C₂²×C₄	—	118	4	no	—
121	C₂²×C₄	C₈×D₅₉	1	3	yes	—
122	C₂²×C₄	#11745659	1	4	yes	—
123	C₂²×C₄	Dic₅₉⋊C₄	1	4	yes	—
124	C₂²×C₄	C₂×C₄×D₅₉	1	3	yes	—
125	C₂×C₈	—	59	4	no	—
126	C₂×C₈	—	118	12	no	—
127	C₂×C₈	Dic₅₉⋊C₄	1	12	yes	—
128	C₂×C₈	C₂×Dic₁₁₈	1	6	yes	—
129	C₂×C₈	C₂×C₄×D₅₉	1	3	yes	—
130	C₂×C₈	C₂×C₂²⋊D₅₉	1	6	yes	—
131	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
132	C₄²	—	59	12	no	—
133	C₄²	—	59	4	no	—
134	C₁₆	—	2	4	no	—
135	C₁₆	—	118	4	no	—
136	C₂³	Dic₅₉⋊C₈	1	4	yes	—
137	C₂³	C₁₆×D₅₉	1	4	yes	—
138	C₂³	#14273666	1	4	yes	—
139	C₂³	C₂×C₈×D₅₉	1	3	yes	—
140	C₂×C₄	—	59	4	no	—
141	C₂×C₄	—	59	24	no	—
142	C₂×C₄	—	59	4	no	—
143	C₂×C₄	—	118	12	no	—
144	C₂×C₄	Dic₅₉⋊C₈	1	12	yes	—
145	C₂×C₄	C₂×Dic₅₉⋊C₄	1	6	yes	—
146	C₂×C₄	C₂×C₈×D₅₉	1	3	yes	—
147	C₂×C₄	C₂×#11745659	1	6	yes	—
148	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
149	C₈	—	118	4	no	—
150	C₈	C₂×Dic₅₉⋊C₄	1	6	yes	—
151	C₈	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
152	C₈	C₂²×Dic₁₁₈	1	1	yes	characteristic
153	C₂²	Dic₅₉⋊C₁₆	1	16	yes	direct product factor
154	C₂²	C₂×Dic₅₉⋊C₈	1	6	yes	—
155	C₂²	C₂×C₁₆×D₅₉	1	6	yes	—
156	C₂²	C₂×#14273666	1	6	yes	—
157	C₂²	C₂²×C₈×D₅₉	1	1	yes	characteristic
158	C₄	—	59	8	no	—
159	C₄	—	118	4	no	—
160	C₄	C₂×Dic₅₉⋊C₈	1	6	yes	—
161	C₄	C₂²×Dic₅₉⋊C₄	1	1	yes	characteristic
162	C₄	C₂²×#11745659	1	1	yes	characteristic
163	C₂	C₂×Dic₅₉⋊C₁₆	1	12	yes	direct product factor
164	C₂	C₂²×Dic₅₉⋊C₈	1	1	yes	characteristic
165	C₂	C₂²×C₁₆×D₅₉	1	1	yes	characteristic
166	C₂	C₂²×#14273666	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₂×Dic₅₉⋊C₁₆	C₂	direct (N × H)
11. Dic₅₉⋊C₁₆	C₂²	direct (N × H)
38. C₂²×Dic₅₉	C₁₆	semidirect
62. C₂×Dic₅₉	C₂×C₁₆	semidirect
70. C₂²×C₅₉	C₄⋊C₁₆	semidirect
79. Dic₅₉	C₂²×C₁₆	semidirect
84. C₂×C₅₉	C₂×C₄⋊C₁₆	semidirect
98. C₅₉	C₂²×C₄⋊C₁₆	semidirect
153. C₂²	Dic₅₉⋊C₁₆	direct (N × H)
163. C₂	C₂×Dic₅₉⋊C₁₆	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×Dic₅₉⋊C₁₆	2
C₂³×C₁₆×C₅₉	2
C₂²×C₄⋊C₁₆	59

Subgroups — C22×Dic59⋊C16 #28741663

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Dic₅₉⋊C₁₆ #28741663