Subgroups — C₁₆×#11745689 #28742207

Order 15104.

602 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#11745689	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₆×D₂₃₆	C₂	1	3	yes	maximal, complemented
4	#21425915	C₂	1	3	yes	maximal, complemented
5	#21425894	C₂	1	3	yes	maximal, complemented
6	C₄×C₁₆×D₅₉	C₂	1	3	yes	maximal, complemented
7	Q₈×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#11745689	C₄	1	1	yes	characteristic
9	C₂×C₁₆×D₅₉	C₂²	1	3	yes	—
10	C₂×C₁₆×D₅₉	C₂²	1	3	yes	—
11	Q₈×C₅₉⋊C₈	C₄	1	1	yes	characteristic
12	C₈×D₂₃₆	C₂²	1	3	yes	—
13	#16837467	C₄	1	3	yes	—
14	#16837454	C₄	1	3	yes	—
15	C₄×C₈×D₅₉	C₂²	1	3	yes	—
16	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	3	yes	—
17	(C₂×D₅₉)⋊C₁₆	C₂²	1	6	yes	—
18	C₄×C₅₉⋊C₁₆	C₂²	1	3	yes	—
19	C₁₆×Dic₅₉	C₂²	1	1	yes	characteristic
20	C₁₆×Dic₅₉	C₂²	1	3	yes	—
21	C₈×Q₈×C₅₉	C₂²	1	1	yes	characteristic
22	C₅₉×C₄⋊C₁₆	C₂²	1	3	yes	—
23	C₄×C₁₆×C₅₉	C₂²	1	3	yes	—
24	C₂×#11745689	C₈	1	1	yes	characteristic
25	C₂×C₈×D₅₉	C₂×C₄	1	3	yes	—
26	C₂×C₈×D₅₉	C₂³	1	3	yes	—
27	Q₈×Dic₅₉	C₈	1	1	yes	characteristic
28	C₄×D₂₃₆	C₂×C₄	1	3	yes	—
29	#14273689	C₈	1	3	yes	—
30	#14273685	C₈	1	3	yes	—
31	C₄²×D₅₉	C₂×C₄	1	3	yes	—
32	C₂×C₅₉⋊C₁₆	C₂³	1	3	yes	—
33	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
34	C₁₆×D₅₉	—	2	6	no	—
35	C₁₆×D₅₉	—	2	6	no	—
36	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	3	yes	—
37	(C₂×D₅₉)⋊C₈	C₂×C₄	1	6	yes	—
38	C₄×C₅₉⋊C₈	C₂×C₄	1	3	yes	—
39	C₈×Dic₅₉	C₂×C₄	1	3	yes	—
40	C₈×Dic₅₉	C₂³	1	1	yes	characteristic
41	C₄×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₁₆×C₅₉	C₂³	1	3	yes	—
43	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₅₉	C₂×C₄	1	3	yes	—
45	C₄×C₈×C₅₉	C₂³	1	3	yes	—
46	C₂×Q₈×C₅₉	C₂×C₈	1	1	yes	characteristic
47	#11745689	C₁₆	1	8	yes	direct product factor
48	C₂×D₂₃₆	C₂×C₈	1	3	yes	—
49	C₂×C₄×D₅₉	C₂×C₈	1	3	yes	—
50	C₂×C₄×D₅₉	C₂×C₈	1	3	yes	—
51	C₂×C₄×D₅₉	C₂²×C₄	1	3	yes	—
52	C₂×C₈×C₅₉	C₂²×C₄	1	3	yes	—
53	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
54	C₄⋊C₄×C₅₉	C₂×C₈	1	3	yes	—
55	C₄²×C₅₉	C₂²×C₄	1	3	yes	—
56	(C₂×D₅₉)⋊C₄	C₂×C₈	1	6	yes	—
57	C₄⋊Dic₅₉	C₂×C₈	1	3	yes	—
58	C₄×Dic₅₉	C₂×C₈	1	3	yes	—
59	C₄×Dic₅₉	C₂×C₈	1	3	yes	—
60	C₄×Dic₅₉	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₅₉⋊C₈	C₂²×C₄	1	3	yes	—
62	C₂×C₅₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
63	C₈×D₅₉	—	2	3	no	—
64	C₈×D₅₉	—	2	3	no	—
65	C₈×D₅₉	—	2	6	no	—
66	C₁₆×C₅₉	—	2	3	no	—
67	C₁₆×C₅₉	#41	1	2	yes	complemented
68	C₅₉⋊C₁₆	—	2	3	no	—
69	C₅₉⋊C₁₆	#41	1	2	yes	complemented
70	C₂²×D₅₉	C₂²×C₈	1	3	yes	—
71	Q₈×C₅₉	C₂×C₁₆	1	4	yes	complemented
72	C₂×C₄×C₅₉	C₂²×C₈	1	3	yes	—
73	C₂×C₄×C₅₉	C₂²×C₈	1	3	yes	—
74	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
75	C₂×Dic₅₉	C₂²×C₈	1	3	yes	—
76	C₂×Dic₅₉	C₂²×C₈	1	1	yes	characteristic
77	C₂×Dic₅₉	C₂²×C₈	1	1	yes	characteristic
78	D₂₃₆	C₂×C₁₆	1	12	yes	complemented
79	C₄×D₅₉	—	2	3	no	—
80	C₄×D₅₉	—	2	3	no	—
81	C₄×D₅₉	—	2	6	no	—
82	C₄×D₅₉	C₂×C₁₆	1	12	yes	complemented
83	C₈×C₅₉	—	2	3	no	—
84	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
85	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
86	C₅₉⋊C₈	—	2	3	no	—
87	C₅₉⋊C₈	#131	1	2	yes	—
88	C₁₆×#41	—	59	1	no	maximal, Sylow
89	C₂²×C₅₉	C₂³×C₈	1	1	yes	characteristic
90	C₂×D₅₉	—	2	3	no	—
91	C₂×D₅₉	—	2	3	no	—
92	C₂×D₅₉	C₂²×C₁₆	1	6	yes	—
93	C₄×C₅₉	—	2	3	no	—
94	C₄×C₅₉	C₂²×C₁₆	1	6	yes	—
95	C₄×C₅₉	C₄×#41	1	1	yes	characteristic
96	C₄×C₅₉	C₂×#131	1	1	yes	characteristic
97	Dic₅₉	—	2	3	no	—
98	Dic₅₉	C₂²×C₁₆	1	2	yes	—
99	Dic₅₉	#504	1	2	yes	—
100	C₈×#41	—	59	1	no	—
101	Q₈×C₁₆	—	59	1	no	—
102	Q₈×C₁₆	D₅₉	1	1	yes	characteristic, complemented
103	D₄×C₁₆	—	59	3	no	—
104	D₄×C₁₆	—	59	3	no	—
105	#2162	—	59	3	no	—
106	C₂×C₄×C₁₆	—	59	3	no	—
107	C₂×C₅₉	C₈×#41	1	1	yes	characteristic
108	C₂×C₅₉	C₂³×C₁₆	1	1	yes	characteristic
109	C₂×C₅₉	C₂×#504	1	1	yes	characteristic
110	D₅₉	—	2	6	no	—
111	C₄×#41	—	59	1	no	—
112	C₂²×C₁₆	—	59	3	no	—
113	C₂²×C₁₆	—	59	3	no	—
114	C₈×Q₈	—	59	1	no	—
115	C₈×Q₈	C₂×D₅₉	1	1	yes	characteristic
116	C₈×D₄	—	59	3	no	—
117	C₈×D₄	—	59	3	no	—
118	#431	—	59	3	no	—
119	C₂×C₄×C₈	—	59	3	no	—
120	C₄⋊C₁₆	—	59	3	no	—
121	C₄⋊C₁₆	C₂×D₅₉	1	3	yes	—
122	C₂²⋊C₁₆	—	59	6	no	—
123	C₄×C₁₆	—	59	3	no	—
124	C₄×C₁₆	—	59	1	no	—
125	C₄×C₁₆	—	59	3	no	—
126	C₄×C₁₆	C₂×D₅₉	1	3	yes	—
127	C₅₉	C₁₆×#41	1	1	yes	characteristic, complemented, Sylow
128	C₂×#41	—	59	1	no	—
129	C₂²×C₈	—	59	3	no	—
130	C₂²×C₈	—	59	3	no	—
131	C₄×Q₈	—	59	1	no	—
132	C₄×Q₈	C₄×D₅₉	1	1	yes	characteristic
133	C₄×D₄	—	59	3	no	—
134	C₄×D₄	—	59	3	no	—
135	#117	—	59	3	no	—
136	C₂×C₄²	—	59	3	no	—
137	C₂×C₁₆	—	59	1	no	—
138	C₂×C₁₆	—	59	3	no	—
139	C₂×C₁₆	—	118	6	no	—
140	C₂×C₁₆	—	118	6	no	—
141	C₂×C₁₆	C₂²×D₅₉	1	3	yes	—
142	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
143	C₄⋊C₈	—	59	3	no	—
144	C₄⋊C₈	C₄×D₅₉	1	3	yes	—
145	C₂²⋊C₈	—	59	6	no	—
146	C₄×C₈	—	59	3	no	—
147	C₄×C₈	—	59	1	no	—
148	C₄×C₈	—	59	3	no	—
149	C₄×C₈	C₂²×D₅₉	1	3	yes	—
150	#41	—	59	8	no	—
151	C₂×Q₈	C₈×D₅₉	1	1	yes	characteristic
152	C₂×D₄	—	59	3	no	—
153	C₂²×C₄	—	59	3	no	—
154	C₂²×C₄	—	59	3	no	—
155	C₂²×C₄	—	59	3	no	—
156	C₂×C₈	—	59	1	no	—
157	C₂×C₈	—	59	3	no	—
158	C₂×C₈	—	118	6	no	—
159	C₂×C₈	—	118	3	no	—
160	C₂×C₈	—	118	3	no	—
161	C₂×C₈	C₂×C₄×D₅₉	1	3	yes	—
162	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
163	C₄⋊C₄	—	59	3	no	—
164	C₄⋊C₄	C₈×D₅₉	1	3	yes	—
165	C₂²⋊C₄	—	59	6	no	—
166	C₄²	—	59	3	no	—
167	C₄²	—	59	3	no	—
168	C₄²	—	59	1	no	—
169	C₄²	C₂×C₄×D₅₉	1	3	yes	—
170	C₁₆	—	2	3	no	—
171	C₁₆	—	59	2	no	—
172	C₁₆	—	118	3	no	—
173	C₁₆	#11745689	1	2	yes	direct product factor
174	C₂³	—	59	3	no	—
175	Q₈	C₁₆×D₅₉	1	4	yes	complemented
176	D₄	—	59	12	no	—
177	C₂×C₄	—	59	1	no	—
178	C₂×C₄	—	59	1	no	—
179	C₂×C₄	—	59	12	no	—
180	C₂×C₄	—	59	3	no	—
181	C₂×C₄	—	118	6	no	—
182	C₂×C₄	—	118	3	no	—
183	C₂×C₄	—	118	3	no	—
184	C₂×C₄	C₂×C₈×D₅₉	1	3	yes	—
185	C₂×C₄	C₂×C₈×D₅₉	1	3	yes	—
186	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
187	C₈	—	2	3	no	—
188	C₈	—	59	2	no	—
189	C₈	—	118	3	no	—
190	C₈	#41×D₅₉	1	1	yes	characteristic
191	C₈	C₂×#11745689	1	1	yes	characteristic
192	C₂²	—	59	6	no	—
193	C₂²	—	118	3	no	—
194	C₂²	—	118	3	no	—
195	C₂²	C₂²×C₈×D₅₉	1	1	yes	characteristic
196	C₄	—	2	3	no	—
197	C₄	—	59	2	no	—
198	C₄	—	59	2	no	—
199	C₄	—	118	3	no	—
200	C₄	C₂×C₁₆×D₅₉	1	6	yes	—
201	C₄	C₄×#11745689	1	1	yes	characteristic
202	C₄	#131×D₅₉	1	1	yes	characteristic
203	C₂	—	118	6	no	—
204	C₂	C₈×#11745689	1	1	yes	characteristic
205	C₂	C₂²×C₁₆×D₅₉	1	1	yes	characteristic
206	C₂	#504×D₅₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₅₉
commutator	C₂×C₅₉	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	Q₈×C₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₂³×C₈
prime radical	C₂×#11745689	C₈
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	Q₈×C₁₆	D₅₉
hypocenter	C₅₉	C₁₆×#41
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₁₆ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
2. Q₈×C₅₉⋊C₁₆	C₂	semidirect
3. C₁₆×D₂₃₆	C₂	semidirect
4. #21425915	C₂	semidirect
5. #21425894	C₂	semidirect
6. C₄×C₁₆×D₅₉	C₂	semidirect
7. Q₈×C₁₆×C₅₉	C₂	semidirect
47. #11745689	C₁₆	direct (N × H)
67. C₁₆×C₅₉	#41	semidirect
69. C₅₉⋊C₁₆	#41	semidirect
71. Q₈×C₅₉	C₂×C₁₆	semidirect
78. D₂₃₆	C₂×C₁₆	semidirect
82. C₄×D₅₉	C₂×C₁₆	semidirect
102. Q₈×C₁₆	D₅₉	semidirect
127. C₅₉	C₁₆×#41	semidirect
173. C₁₆	#11745689	direct (N × H)
175. Q₈	C₁₆×D₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₈×#11745689	2
Q₈×C₅₉⋊C₁₆	2
C₁₆×D₂₃₆	2
#21425915	2
#21425894	2
C₄×C₁₆×D₅₉	2
Q₈×C₁₆×C₅₉	2
C₁₆×#41	59

Subgroups — C16×#11745689 #28742207

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₆×#11745689 #28742207