Subgroups — #41×C₅₉⋊C₁₆ #28742208

Order 15104.

602 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₅₉⋊C₁₆	C₂	1	2	yes	maximal, complemented
2	D₄×C₅₉⋊C₁₆	C₂	1	6	yes	maximal, complemented
3	#21425895	C₂	1	3	yes	maximal, complemented
4	C₂×C₄×C₅₉⋊C₁₆	C₂	1	3	yes	maximal, complemented
5	C₈×#41×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₁₆	C₂²	1	6	yes	—
7	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	6	yes	—
8	C₂²⋊(C₅₉⋊C₁₆)	C₂²	1	6	yes	—
9	C₄×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₄×C₅₉⋊C₁₆	C₂²	1	6	yes	—
11	C₄×C₅₉⋊C₁₆	C₂²	1	3	yes	—
12	C₄×#41×C₅₉	C₄	1	1	yes	characteristic
13	C₈×Q₈×C₅₉	C₄	1	1	yes	characteristic
14	C₈×Q₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₈×D₄×C₅₉	C₄	1	3	yes	—
16	C₈×D₄×C₅₉	C₂²	1	3	yes	—
17	C₅₉×#431	C₄	1	3	yes	—
18	C₂×C₄×C₈×C₅₉	C₂²	1	3	yes	—
19	C₂×C₅₉⋊C₁₆	—	2	12	no	—
20	C₂×C₅₉⋊C₁₆	C₂³	1	6	yes	—
21	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
22	C₂×#41×C₅₉	C₈	1	1	yes	characteristic
23	C₂²×C₈×C₅₉	C₂×C₄	1	3	yes	—
24	C₂²×C₈×C₅₉	C₂³	1	3	yes	—
25	C₄×Q₈×C₅₉	C₈	1	1	yes	characteristic
26	C₄×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
27	C₄×D₄×C₅₉	C₈	1	3	yes	—
28	C₄×D₄×C₅₉	C₂×C₄	1	3	yes	—
29	#117×C₅₉	C₈	1	3	yes	—
30	C₂×C₄²×C₅₉	C₂×C₄	1	3	yes	—
31	C₄⋊C₈×C₅₉	C₂×C₄	1	3	yes	—
32	C₄⋊C₈×C₅₉	C₂×C₄	1	3	yes	—
33	C₂²⋊C₈×C₅₉	C₂×C₄	1	6	yes	—
34	C₄×C₈×C₅₉	C₂×C₄	1	3	yes	—
35	C₄×C₈×C₅₉	C₂×C₄	1	3	yes	—
36	C₄×C₈×C₅₉	C₂³	1	3	yes	—
37	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
38	#41×C₅₉	C₁₆	1	8	yes	complemented
39	C₂×Q₈×C₅₉	C₂×C₈	1	1	yes	characteristic
40	C₂×D₄×C₅₉	C₂×C₈	1	3	yes	—
41	C₂²×C₄×C₅₉	C₂×C₈	1	3	yes	—
42	C₂²×C₄×C₅₉	C₂×C₈	1	3	yes	—
43	C₂²×C₄×C₅₉	C₂²×C₄	1	3	yes	—
44	C₂×C₈×C₅₉	—	2	6	no	—
45	C₂×C₈×C₅₉	—	2	6	no	—
46	C₂×C₈×C₅₉	C₂²×C₄	1	3	yes	—
47	C₂×C₈×C₅₉	C₂²×C₄	1	3	yes	—
48	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
50	C₄⋊C₄×C₅₉	C₂×C₈	1	3	yes	—
51	C₄⋊C₄×C₅₉	C₂×C₈	1	3	yes	—
52	C₂²⋊C₄×C₅₉	C₂×C₈	1	6	yes	—
53	C₄²×C₅₉	C₂×C₈	1	3	yes	—
54	C₄²×C₅₉	C₂×C₈	1	3	yes	—
55	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₅₉	C₂²×C₄	1	3	yes	—
57	C₅₉⋊C₁₆	—	2	6	no	—
58	C₅₉⋊C₁₆	#41	1	4	yes	direct product factor
59	C₂³×C₅₉	C₂²×C₈	1	3	yes	—
60	Q₈×C₅₉	C₂×C₁₆	1	4	yes	complemented
61	D₄×C₅₉	C₂×C₁₆	1	12	yes	complemented
62	C₂×C₄×C₅₉	—	2	3	no	—
63	C₂×C₄×C₅₉	—	2	3	no	—
64	C₂×C₄×C₅₉	—	2	6	no	—
65	C₂×C₄×C₅₉	C₂×C₁₆	1	12	yes	complemented
66	C₂×C₄×C₅₉	C₂²×C₈	1	3	yes	—
67	C₂×C₄×C₅₉	C₂²×C₈	1	3	yes	—
68	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₅₉	C₂²×C₈	1	3	yes	—
70	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
72	C₈×C₅₉	—	2	3	no	—
73	C₈×C₅₉	—	2	3	no	—
74	C₈×C₅₉	#131	1	2	yes	—
75	C₈×C₅₉	C₂×#41	1	2	yes	—
76	C₁₆×#41	—	59	1	no	maximal, Sylow
77	C₂²×C₅₉	—	2	3	no	—
78	C₂²×C₅₉	—	2	3	no	—
79	C₂²×C₅₉	C₂²×C₁₆	1	6	yes	—
80	C₂²×C₅₉	C₂³×C₈	1	1	yes	characteristic
81	C₄×C₅₉	—	2	3	no	—
82	C₄×C₅₉	—	2	3	no	—
83	C₄×C₅₉	C₂²×C₁₆	1	2	yes	—
84	C₄×C₅₉	C₂²×C₁₆	1	6	yes	—
85	C₄×C₅₉	#504	1	2	yes	—
86	C₄×C₅₉	C₄×#41	1	1	yes	characteristic
87	C₄×C₅₉	C₂×#131	1	1	yes	characteristic
88	C₈×#41	D₅₉	1	1	yes	characteristic
89	Q₈×C₁₆	—	59	2	no	—
90	D₄×C₁₆	—	59	6	no	—
91	#2162	—	59	3	no	—
92	C₂×C₄×C₁₆	—	59	3	no	—
93	C₂×C₅₉	—	2	6	no	—
94	C₂×C₅₉	C₈×#41	1	1	yes	characteristic
95	C₂×C₅₉	C₂³×C₁₆	1	1	yes	characteristic
96	C₂×C₅₉	C₂×#504	1	1	yes	characteristic
97	C₄×#41	Dic₅₉	1	1	yes	characteristic
98	C₂²×C₁₆	—	59	6	no	—
99	C₈×Q₈	Dic₅₉	1	1	yes	characteristic
100	C₈×Q₈	C₂×D₅₉	1	1	yes	characteristic
101	C₈×D₄	Dic₅₉	1	3	yes	—
102	C₈×D₄	C₂×D₅₉	1	3	yes	—
103	#431	Dic₅₉	1	3	yes	—
104	C₂×C₄×C₈	C₂×D₅₉	1	3	yes	—
105	C₄⋊C₁₆	—	59	6	no	—
106	C₂²⋊C₁₆	—	59	6	no	—
107	C₄×C₁₆	—	59	6	no	—
108	C₄×C₁₆	—	59	1	no	—
109	C₄×C₁₆	—	59	3	no	—
110	C₅₉	C₁₆×#41	1	1	yes	characteristic, complemented, Sylow
111	C₂×#41	C₅₉⋊C₈	1	1	yes	characteristic
112	C₂²×C₈	C₂×Dic₅₉	1	3	yes	—
113	C₂²×C₈	C₂²×D₅₉	1	3	yes	—
114	C₄×Q₈	C₅₉⋊C₈	1	1	yes	characteristic
115	C₄×Q₈	C₂×Dic₅₉	1	1	yes	characteristic
116	C₄×D₄	C₅₉⋊C₈	1	3	yes	—
117	C₄×D₄	C₂×Dic₅₉	1	3	yes	—
118	#117	C₅₉⋊C₈	1	3	yes	—
119	C₂×C₄²	C₂×Dic₅₉	1	3	yes	—
120	C₂×C₁₆	—	59	2	no	—
121	C₂×C₁₆	—	59	6	no	—
122	C₂×C₁₆	—	118	12	no	—
123	C₄⋊C₈	C₂×Dic₅₉	1	3	yes	—
124	C₄⋊C₈	C₂×Dic₅₉	1	3	yes	—
125	C₂²⋊C₈	C₂×Dic₅₉	1	6	yes	—
126	C₄×C₈	C₂×Dic₅₉	1	3	yes	—
127	C₄×C₈	C₂×Dic₅₉	1	3	yes	—
128	C₄×C₈	C₂²×D₅₉	1	3	yes	—
129	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
130	#41	C₅₉⋊C₁₆	1	8	yes	direct product factor
131	C₂×Q₈	C₂×C₅₉⋊C₈	1	1	yes	characteristic
132	C₂×D₄	C₂×C₅₉⋊C₈	1	3	yes	—
133	C₂²×C₄	C₂×C₅₉⋊C₈	1	3	yes	—
134	C₂²×C₄	C₂×C₅₉⋊C₈	1	3	yes	—
135	C₂²×C₄	C₂²×Dic₅₉	1	3	yes	—
136	C₂×C₈	—	2	6	no	—
137	C₂×C₈	—	2	6	no	—
138	C₂×C₈	C₂²×Dic₅₉	1	3	yes	—
139	C₂×C₈	C₂²×Dic₅₉	1	3	yes	—
140	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
141	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
142	C₄⋊C₄	C₂×C₅₉⋊C₈	1	3	yes	—
143	C₄⋊C₄	C₂×C₅₉⋊C₈	1	3	yes	—
144	C₂²⋊C₄	C₂×C₅₉⋊C₈	1	6	yes	—
145	C₄²	C₂×C₅₉⋊C₈	1	3	yes	—
146	C₄²	C₂×C₅₉⋊C₈	1	3	yes	—
147	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
148	C₄²	C₂²×Dic₅₉	1	3	yes	—
149	C₁₆	—	59	4	no	—
150	C₁₆	—	118	6	no	—
151	C₂³	C₂²×C₅₉⋊C₈	1	3	yes	—
152	Q₈	C₂×C₅₉⋊C₁₆	1	4	yes	complemented
153	D₄	C₂×C₅₉⋊C₁₆	1	12	yes	complemented
154	C₂×C₄	—	2	3	no	—
155	C₂×C₄	—	2	6	no	—
156	C₂×C₄	—	2	3	no	—
157	C₂×C₄	C₂×C₅₉⋊C₁₆	1	12	yes	complemented
158	C₂×C₄	C₂²×C₅₉⋊C₈	1	3	yes	—
159	C₂×C₄	C₂²×C₅₉⋊C₈	1	3	yes	—
160	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
161	C₂×C₄	C₂²×C₅₉⋊C₈	1	3	yes	—
162	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
163	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
164	C₈	—	2	3	no	—
165	C₈	—	2	3	no	—
166	C₈	#14273734	1	2	yes	—
167	C₈	#41×D₅₉	1	2	yes	—
168	C₂²	—	2	3	no	—
169	C₂²	—	2	3	no	—
170	C₂²	C₂²×C₅₉⋊C₁₆	1	6	yes	—
171	C₂²	C₂³×C₅₉⋊C₈	1	1	yes	characteristic
172	C₄	—	2	3	no	—
173	C₄	—	2	3	no	—
174	C₄	C₂²×C₅₉⋊C₁₆	1	2	yes	—
175	C₄	C₂²×C₅₉⋊C₁₆	1	6	yes	—
176	C₄	#16837819	1	2	yes	—
177	C₄	#41×Dic₅₉	1	1	yes	characteristic
178	C₄	C₂×#14273734	1	1	yes	characteristic
179	C₂	—	2	6	no	—
180	C₂	#41×C₅₉⋊C₈	1	1	yes	characteristic
181	C₂	C₂³×C₅₉⋊C₁₆	1	1	yes	characteristic
182	C₂	C₂×#16837819	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂²×D₅₉
commutator	C₂×C₅₉	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₈×#41×C₅₉	C₂
socle	C₂²×C₅₉	C₂³×C₈
prime radical	C₂×#41×C₅₉	C₈
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	C₈×#41	D₅₉
hypocenter	C₅₉	C₁₆×#41
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄×C₈ ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
1. Q₈×C₅₉⋊C₁₆	C₂	semidirect
2. D₄×C₅₉⋊C₁₆	C₂	semidirect
3. #21425895	C₂	semidirect
4. C₂×C₄×C₅₉⋊C₁₆	C₂	semidirect
38. #41×C₅₉	C₁₆	semidirect
58. C₅₉⋊C₁₆	#41	direct (N × H)
60. Q₈×C₅₉	C₂×C₁₆	semidirect
61. D₄×C₅₉	C₂×C₁₆	semidirect
65. C₂×C₄×C₅₉	C₂×C₁₆	semidirect
110. C₅₉	C₁₆×#41	semidirect
130. #41	C₅₉⋊C₁₆	direct (N × H)
152. Q₈	C₂×C₅₉⋊C₁₆	semidirect
153. D₄	C₂×C₅₉⋊C₁₆	semidirect
157. C₂×C₄	C₂×C₅₉⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₅₉⋊C₁₆	2
D₄×C₅₉⋊C₁₆	2
#21425895	2
C₂×C₄×C₅₉⋊C₁₆	2
C₈×#41×C₅₉	2
C₁₆×#41	59

Subgroups — #41×C59⋊C16 #28742208

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #41×C₅₉⋊C₁₆ #28742208