Subgroups — #508×Dic₅₉ #28741972

Order 15104.

690 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#135×Dic₅₉	C₂	1	2	yes	maximal, complemented
2	Q₃₂×Dic₅₉	C₂	1	2	yes	maximal, complemented
3	SD₃₂×Dic₅₉	C₂	1	4	yes	maximal, complemented
4	D₁₆×Dic₅₉	C₂	1	2	yes	maximal, complemented
5	#21425880	C₂	1	1	yes	maximal, characteristic, complemented
6	#21425803	C₂	1	2	yes	maximal, complemented
7	C₂×C₁₆×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₉×#508	C₂	1	1	yes	maximal, characteristic
9	#41×Dic₅₉	—	2	2	no	—
10	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	Q₁₆×Dic₅₉	C₂²	1	4	yes	—
12	SD₁₆×Dic₅₉	—	2	4	no	—
13	D₈×Dic₅₉	C₂²	1	4	yes	—
14	#16837440	C₂²	1	2	yes	—
15	#16837398	—	2	2	no	—
16	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
17	#16837230	C₂²	1	2	yes	—
18	#16837228	C₂²	1	2	yes	—
19	#16837206	C₂²	1	4	yes	—
20	#16837203	C₂²	1	4	yes	—
21	C₁₆×Dic₅₉	C₂²	1	2	yes	—
22	C₁₆×Dic₅₉	C₂²	1	2	yes	—
23	C₂×#135×C₅₉	C₂²	1	2	yes	—
24	C₅₉×#508	C₄	1	8	yes	complemented
25	C₂×Q₃₂×C₅₉	C₂²	1	1	yes	characteristic
26	C₂×SD₃₂×C₅₉	C₂²	1	2	yes	—
27	C₂×D₁₆×C₅₉	C₂²	1	1	yes	characteristic
28	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
29	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
30	Q₈×Dic₅₉	—	2	4	no	—
31	D₄×Dic₅₉	—	2	4	no	—
32	D₄×Dic₅₉	—	4	4	no	—
33	#14273686	—	2	2	no	—
34	#14273686	—	4	2	no	—
35	C₂×C₄×Dic₅₉	—	4	2	no	—
36	C₂×C₄×Dic₅₉	D₄	1	1	yes	characteristic
37	C₂×C₅₉⋊C₁₆	—	2	2	no	—
38	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
39	#14273661	C₂³	1	4	yes	—
40	#14273659	—	2	2	no	—
41	#14273651	—	2	4	no	—
42	#14273648	—	2	4	no	—
43	C₈×Dic₅₉	D₄	1	2	yes	—
44	C₈×Dic₅₉	C₂³	1	2	yes	—
45	C₂×#41×C₅₉	—	2	2	no	—
46	#135×C₅₉	—	2	4	no	—
47	#135×C₅₉	C₂×C₄	1	8	yes	complemented
48	C₂×Q₁₆×C₅₉	C₂³	1	2	yes	—
49	C₂×SD₁₆×C₅₉	—	2	2	no	—
50	C₂×D₈×C₅₉	C₂³	1	2	yes	—
51	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
52	Q₃₂×C₅₉	C₂×C₄	1	4	yes	complemented
53	SD₃₂×C₅₉	C₂×C₄	1	8	yes	complemented
54	D₁₆×C₅₉	C₂×C₄	1	4	yes	complemented
55	C₂×C₁₆×C₅₉	C₂×C₄	1	4	yes	complemented
56	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
57	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
58	#41×C₅₉	—	2	8	no	—
59	#41×C₅₉	—	4	4	no	—
60	C₂×Q₈×C₅₉	—	2	2	no	—
61	C₂×D₄×C₅₉	—	2	2	no	—
62	C₂×D₄×C₅₉	—	4	2	no	—
63	C₂²×C₄×C₅₉	—	4	2	no	—
64	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×Dic₅₉	—	4	4	no	—
66	C₂²×Dic₅₉	D₈	1	2	yes	—
67	Q₁₆×C₅₉	—	2	2	no	—
68	Q₁₆×C₅₉	C₂²×C₄	1	4	yes	—
69	SD₁₆×C₅₉	—	2	8	no	—
70	D₈×C₅₉	—	2	2	no	—
71	D₈×C₅₉	C₂²×C₄	1	4	yes	—
72	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₅₉	#41	1	2	yes	—
75	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
76	C₂²⋊Dic₅₉	—	4	2	no	—
77	C₂²⋊Dic₅₉	—	4	4	no	—
78	C₄⋊Dic₅₉	—	2	4	no	—
79	C₄⋊Dic₅₉	—	4	4	no	—
80	C₄×Dic₅₉	—	4	2	no	—
81	C₄×Dic₅₉	—	4	4	no	—
82	C₄×Dic₅₉	D₈	1	1	yes	characteristic
83	C₄×Dic₅₉	D₈	1	1	yes	characteristic
84	C₄×Dic₅₉	C₂×D₄	1	2	yes	—
85	C₂×C₅₉⋊C₈	—	2	2	no	—
86	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
87	C₁₆×C₅₉	C₂²×C₄	1	2	yes	—
88	C₁₆×C₅₉	C₂²×C₄	1	2	yes	—
89	C₅₉⋊C₁₆	—	2	2	no	—
90	C₂³×C₅₉	—	4	2	no	—
91	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
92	Q₈×C₅₉	—	2	4	no	—
93	Q₈×C₅₉	—	4	2	no	—
94	D₄×C₅₉	—	2	4	no	—
95	D₄×C₅₉	—	4	8	no	—
96	D₄×C₅₉	—	4	2	no	—
97	C₂×C₄×C₅₉	—	4	2	no	—
98	C₂×C₄×C₅₉	—	4	8	no	—
99	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₅₉	#135	1	2	yes	—
102	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
103	C₂×Dic₅₉	—	2	4	no	—
104	C₂×Dic₅₉	—	4	4	no	—
105	C₂×Dic₅₉	—	8	8	no	—
106	C₂×Dic₅₉	C₂×D₈	1	2	yes	—
107	C₂×Dic₅₉	C₂×D₈	1	2	yes	—
108	C₈×C₅₉	C₄×D₄	1	2	yes	—
109	C₈×C₅₉	C₂³×C₄	1	1	yes	characteristic
110	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
111	C₅₉⋊C₈	—	2	2	no	—
112	C₄×#508	—	59	1	no	maximal, Sylow
113	C₂²×C₅₉	—	2	2	no	—
114	C₂²×C₅₉	—	4	4	no	—
115	C₂²×C₅₉	—	8	4	no	—
116	C₂²×C₅₉	C₄×D₈	1	2	yes	—
117	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
118	C₄×C₅₉	—	4	4	no	—
119	C₄×C₅₉	C₄×D₈	1	2	yes	—
120	C₄×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
121	C₄×C₅₉	C₂×#135	1	1	yes	characteristic
122	Dic₅₉	—	2	2	no	—
123	Dic₅₉	—	8	4	no	—
124	Dic₅₉	#508	1	4	yes	direct product factor
125	C₂×#508	D₅₉	1	1	yes	characteristic
126	C₄×#135	—	59	2	no	—
127	C₄×Q₃₂	—	59	2	no	—
128	C₄×SD₃₂	—	59	4	no	—
129	C₄×D₁₆	—	59	2	no	—
130	#2158	—	59	1	no	—
131	#2138	—	59	2	no	—
132	C₂×C₄×C₁₆	—	59	1	no	—
133	C₂×C₅₉	—	2	2	no	—
134	C₂×C₅₉	—	8	4	no	—
135	C₂×C₅₉	C₂×C₄×D₈	1	1	yes	characteristic
136	C₂×C₅₉	C₂×#508	1	2	yes	—
137	C₂×#135	C₂×D₅₉	1	2	yes	—
138	C₄×#41	—	118	2	no	—
139	#508	Dic₅₉	1	8	yes	direct product factor
140	C₂×Q₃₂	C₂×D₅₉	1	1	yes	characteristic
141	C₂×SD₃₂	C₂×D₅₉	1	2	yes	—
142	C₂×D₁₆	C₂×D₅₉	1	1	yes	characteristic
143	C₂²×C₁₆	—	59	1	no	—
144	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
145	C₄×Q₁₆	—	59	4	no	—
146	C₄×SD₁₆	—	118	4	no	—
147	C₄×D₈	—	59	4	no	—
148	#427	—	59	2	no	—
149	#416	—	118	2	no	—
150	C₂×C₄×C₈	—	59	1	no	—
151	#367	—	59	2	no	—
152	#366	—	59	2	no	—
153	#358	—	59	4	no	—
154	#357	—	59	4	no	—
155	C₄×C₁₆	—	59	2	no	—
156	C₄×C₁₆	—	59	2	no	—
157	C₅₉	C₄×#508	1	1	yes	characteristic, complemented, Sylow
158	C₂×#41	—	2	2	no	—
159	#135	—	2	4	no	—
160	#135	C₂×Dic₅₉	1	8	yes	complemented
161	C₂×Q₁₆	C₂²×D₅₉	1	2	yes	—
162	C₂×SD₁₆	—	2	2	no	—
163	C₂×D₈	C₂²×D₅₉	1	2	yes	—
164	C₂²×C₈	—	59	1	no	—
165	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
166	C₄×Q₈	—	118	4	no	—
167	C₄×D₄	—	118	4	no	—
168	C₄×D₄	—	236	4	no	—
169	#117	—	118	2	no	—
170	#117	—	236	2	no	—
171	C₂×C₄²	—	59	1	no	—
172	C₂×C₄²	—	236	2	no	—
173	Q₃₂	C₂×Dic₅₉	1	4	yes	complemented
174	SD₃₂	C₂×Dic₅₉	1	8	yes	complemented
175	D₁₆	C₂×Dic₅₉	1	4	yes	complemented
176	C₂×C₁₆	—	59	2	no	—
177	C₂×C₁₆	—	118	2	no	—
178	C₂×C₁₆	C₂×Dic₅₉	1	4	yes	complemented
179	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
180	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
181	#107	—	59	4	no	—
182	#106	—	118	2	no	—
183	#103	—	118	4	no	—
184	#102	—	118	4	no	—
185	C₄×C₈	—	59	2	no	—
186	C₄×C₈	—	59	2	no	—
187	#41	—	2	8	no	—
188	#41	—	4	4	no	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	4	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	—	59	2	no	—
194	C₂²×C₄	—	236	4	no	—
195	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
196	Q₁₆	—	2	2	no	—
197	Q₁₆	C₂²×Dic₅₉	1	4	yes	—
198	SD₁₆	—	2	8	no	—
199	D₈	—	2	2	no	—
200	D₈	C₂²×Dic₅₉	1	4	yes	—
201	C₂×C₈	—	59	2	no	—
202	C₂×C₈	—	118	2	no	—
203	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
204	C₂×C₈	#11745687	1	2	yes	—
205	C₂×C₈	C₂²×Dic₅₉	1	2	yes	—
206	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
207	C₄⋊C₄	—	118	4	no	—
208	C₄⋊C₄	—	236	4	no	—
209	C₂²⋊C₄	—	236	4	no	—
210	C₂²⋊C₄	—	236	2	no	—
211	C₄²	—	59	1	no	—
212	C₄²	—	59	1	no	—
213	C₄²	—	59	2	no	—
214	C₄²	—	236	4	no	—
215	C₄²	—	236	2	no	—
216	C₁₆	—	118	2	no	—
217	C₁₆	C₂²×Dic₅₉	1	2	yes	—
218	C₁₆	C₂²×Dic₅₉	1	2	yes	—
219	C₂³	—	4	2	no	—
220	C₂³	D₈×D₅₉	1	1	yes	characteristic
221	Q₈	—	2	4	no	—
222	Q₈	—	4	2	no	—
223	D₄	—	2	4	no	—
224	D₄	—	4	8	no	—
225	D₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	8	no	—
228	C₂×C₄	—	59	2	no	—
229	C₂×C₄	—	59	2	no	—
230	C₂×C₄	—	118	4	no	—
231	C₂×C₄	—	472	8	no	—
232	C₂×C₄	—	236	4	no	—
233	C₂×C₄	D₄×Dic₅₉	1	2	yes	—
234	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
235	C₂×C₄	#14273746	1	2	yes	—
236	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
237	C₈	—	118	2	no	—
238	C₈	D₄×Dic₅₉	1	2	yes	—
239	C₈	C₂³×Dic₅₉	1	1	yes	characteristic
240	C₈	C₂×#11745687	1	1	yes	characteristic
241	C₂²	—	2	2	no	—
242	C₂²	—	4	4	no	—
243	C₂²	—	8	4	no	—
244	C₂²	D₈×Dic₅₉	1	2	yes	—
245	C₂²	C₂×D₈×D₅₉	1	1	yes	characteristic
246	C₄	—	4	4	no	—
247	C₄	—	59	4	no	—
248	C₄	—	118	2	no	—
249	C₄	—	472	4	no	—
250	C₄	D₈×Dic₅₉	1	2	yes	—
251	C₄	C₂×D₄×Dic₅₉	1	1	yes	characteristic
252	C₄	C₂×#14273746	1	1	yes	characteristic
253	C₂	—	2	2	no	—
254	C₂	—	8	4	no	—
255	C₂	C₂×D₈×Dic₅₉	1	1	yes	characteristic
256	C₂	#508×D₅₉	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₅₉
commutator	C₈×C₅₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₉
Fitting, prime radical	C₂×C₅₉×#508	C₂
socle	C₂²×C₅₉	C₂²×D₈
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	C₂×#508	D₅₉
hypocenter	C₅₉	C₄×#508
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. #135×Dic₅₉	C₂	semidirect
2. Q₃₂×Dic₅₉	C₂	semidirect
3. SD₃₂×Dic₅₉	C₂	semidirect
4. D₁₆×Dic₅₉	C₂	semidirect
5. #21425880	C₂	semidirect
6. #21425803	C₂	semidirect
7. C₂×C₁₆×Dic₅₉	C₂	semidirect
24. C₅₉×#508	C₄	semidirect
47. #135×C₅₉	C₂×C₄	semidirect
52. Q₃₂×C₅₉	C₂×C₄	semidirect
53. SD₃₂×C₅₉	C₂×C₄	semidirect
54. D₁₆×C₅₉	C₂×C₄	semidirect
55. C₂×C₁₆×C₅₉	C₂×C₄	semidirect
124. Dic₅₉	#508	direct (N × H)
139. #508	Dic₅₉	direct (N × H)
157. C₅₉	C₄×#508	semidirect
160. #135	C₂×Dic₅₉	semidirect
173. Q₃₂	C₂×Dic₅₉	semidirect
174. SD₃₂	C₂×Dic₅₉	semidirect
175. D₁₆	C₂×Dic₅₉	semidirect
178. C₂×C₁₆	C₂×Dic₅₉	semidirect

Maximal subgroups

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

subgroup	index
#135×Dic₅₉	2
Q₃₂×Dic₅₉	2
SD₃₂×Dic₅₉	2
D₁₆×Dic₅₉	2
#21425880	2
#21425803	2
C₂×C₁₆×Dic₅₉	2
C₂×C₅₉×#508	2
C₄×#508	59

Subgroups — #508×Dic59 #28741972

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #508×Dic₅₉ #28741972