Subgroups — D₅₉×#1336 #28498478

Order 15104.

398 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#443×D₅₉	C₂	1	1	yes	maximal, characteristic
2	#364×D₅₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#322×D₅₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#21422073	C₂	1	1	yes	maximal, characteristic, complemented
5	#21422072	C₂	1	1	yes	maximal, characteristic, complemented
6	#21422071	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#1336	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×D₅₉	C₄	1	1	yes	characteristic
9	#131×D₅₉	—	2	1	no	—
10	M₃₂×D₅₉	—	2	1	no	—
11	#16837520	C₂²	1	1	yes	characteristic
12	#16837519	C₄	1	1	yes	characteristic
13	#16837518	C₄	1	1	yes	characteristic
14	#16837517	—	2	1	no	—
15	#16837516	—	2	1	no	—
16	#108×D₅₉	C₄	1	1	yes	characteristic
17	#104×D₅₉	—	2	1	no	—
18	C₄×C₈×D₅₉	—	2	1	no	—
19	C₄×C₈×D₅₉	C₂²	1	1	yes	characteristic
20	#16837223	C₂²	1	1	yes	characteristic, complemented
21	#16837222	—	2	1	no	—
22	#16837124	—	2	1	no	—
23	#16837123	C₂²	1	1	yes	characteristic, complemented
24	C₅₉×#443	C₂²	1	1	yes	characteristic
25	C₅₉×#364	C₂²	1	1	yes	characteristic, complemented
26	C₅₉×#322	C₂²	1	1	yes	characteristic, complemented
27	#41×D₅₉	—	2	1	no	—
28	#14273750	C₈	1	1	yes	characteristic, complemented
29	#14273749	C₂×C₄	1	1	yes	characteristic
30	#14273748	—	2	1	no	—
31	#14273747	—	2	1	no	—
32	#14273746	C₈	1	1	yes	characteristic, complemented
33	Q₁₆×D₅₉	C₈	1	1	yes	characteristic, complemented
34	SD₁₆×D₅₉	—	2	1	no	—
35	D₈×D₅₉	C₈	1	1	yes	characteristic, complemented
36	#14273735	—	2	1	no	—
37	#14273735	—	4	1	no	—
38	#14273734	—	2	1	no	—
39	#14273733	—	2	1	no	—
40	#14273733	—	4	1	no	—
41	M₁₆×D₅₉	—	2	1	no	—
42	M₁₆×D₅₉	—	4	1	no	—
43	C₂×C₈×D₅₉	—	2	2	no	—
44	C₂×C₈×D₅₉	—	4	1	no	—
45	C₂×C₈×D₅₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×D₅₉	D₄	1	1	yes	characteristic
47	C₄²×D₅₉	D₄	1	1	yes	characteristic
48	#14273667	—	2	1	no	—
49	#14273666	—	4	1	no	—
50	C₁₆×D₅₉	—	4	1	no	—
51	#14273663	C₂×C₄	1	1	yes	characteristic
52	#14273662	—	2	1	no	—
53	#14273655	—	2	1	no	—
54	#14273654	—	2	1	no	—
55	#14273653	—	2	1	no	—
56	C₄×C₅₉⋊C₈	—	2	1	no	—
57	C₄×C₅₉⋊C₈	D₄	1	1	yes	characteristic
58	C₈×Dic₅₉	—	2	2	no	—
59	C₈×Dic₅₉	C₂×C₄	1	1	yes	characteristic
60	C₈×Dic₅₉	D₄	1	1	yes	characteristic
61	#135×C₅₉	C₂×C₄	1	1	yes	characteristic
62	#131×C₅₉	—	2	1	no	—
63	M₃₂×C₅₉	—	2	1	no	—
64	#108×C₅₉	C₂×C₄	1	1	yes	characteristic
65	#104×C₅₉	—	2	1	no	—
66	C₄×C₈×C₅₉	—	2	1	no	—
67	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
68	#41×C₅₉	—	2	1	no	—
69	#11745689	—	2	1	no	—
70	Q₈×D₅₉	—	2	1	no	—
71	#11745687	—	2	1	no	—
72	#11745687	—	4	1	no	—
73	D₄×D₅₉	—	2	1	no	—
74	D₄×D₅₉	—	4	1	no	—
75	#11745685	—	2	1	no	—
76	#11745685	—	4	1	no	—
77	C₂×C₄×D₅₉	—	2	1	no	—
78	C₂×C₄×D₅₉	—	4	1	no	—
79	C₂×C₄×D₅₉	C₂²⋊C₄	1	1	yes	characteristic
80	Q₁₆×C₅₉	C₂×C₈	1	1	yes	characteristic, complemented
81	SD₁₆×C₅₉	—	2	1	no	—
82	D₈×C₅₉	C₂×C₈	1	1	yes	characteristic, complemented
83	M₁₆×C₅₉	—	2	1	no	—
84	M₁₆×C₅₉	—	4	1	no	—
85	C₂×C₈×C₅₉	—	2	2	no	—
86	C₂×C₈×C₅₉	—	4	1	no	—
87	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
90	#11745672	—	2	1	no	—
91	#11745671	—	2	1	no	—
92	#11745670	—	2	1	no	—
93	#11745669	—	2	1	no	—
94	C₄×Dic₅₉	—	2	1	no	—
95	C₄×Dic₅₉	C₂²⋊C₄	1	1	yes	characteristic
96	#11745664	—	2	1	no	—
97	#11745664	—	4	1	no	—
98	C₂×C₅₉⋊C₈	—	2	2	no	—
99	C₂×C₅₉⋊C₈	—	4	1	no	—
100	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
102	Dic₂₃₆	C₂×C₈	1	1	yes	characteristic, complemented
103	D₄₇₂	C₂×C₈	1	1	yes	characteristic, complemented
104	#11745660	—	2	1	no	—
105	#11745659	—	4	1	no	—
106	#11745659	—	4	1	no	—
107	#11745659	—	4	1	no	—
108	C₈×D₅₉	—	4	4	no	—
109	C₈×D₅₉	—	4	1	no	—
110	C₈×D₅₉	—	4	1	no	—
111	C₈×D₅₉	—	4	1	no	—
112	C₈×D₅₉	C₂×C₈	1	1	yes	characteristic
113	C₈×D₅₉	C₂×C₈	1	1	yes	characteristic
114	C₈×D₅₉	M₁₆	1	1	yes	characteristic
115	C₈×D₅₉	M₁₆	1	1	yes	characteristic
116	C₈×D₅₉	D₈	1	1	yes	characteristic
117	C₈×D₅₉	D₈	1	1	yes	characteristic
118	C₈×D₅₉	SD₁₆	1	1	yes	characteristic
119	C₈×D₅₉	SD₁₆	1	1	yes	characteristic
120	C₁₆×C₅₉	—	4	1	no	—
121	C₅₉⋊C₁₆	—	4	1	no	—
122	C₂²×D₅₉	—	4	1	no	—
123	C₂²×D₅₉	#104	1	1	yes	characteristic
124	Q₈×C₅₉	—	2	1	no	—
125	D₄×C₅₉	—	2	1	no	—
126	D₄×C₅₉	—	4	1	no	—
127	C₂×C₄×C₅₉	—	2	1	no	—
128	C₂×C₄×C₅₉	—	4	1	no	—
129	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂²⋊D₅₉	—	4	1	no	—
131	C₂²⋊D₅₉	—	4	1	no	—
132	C₂²⋊D₅₉	—	4	1	no	—
133	C₂×Dic₅₉	—	2	1	no	—
134	C₂×Dic₅₉	—	4	1	no	—
135	C₂×Dic₅₉	#104	1	1	yes	characteristic
136	D₂₃₆	—	2	1	no	—
137	D₂₃₆	—	4	1	no	—
138	C₄×D₅₉	—	2	4	no	—
139	C₄×D₅₉	—	4	1	no	—
140	C₄×D₅₉	—	4	1	no	—
141	C₄×D₅₉	—	4	1	no	—
142	C₄×D₅₉	C₂²⋊C₈	1	1	yes	characteristic
143	C₄×D₅₉	C₂²⋊C₈	1	1	yes	characteristic
144	C₄×D₅₉	#102	1	1	yes	characteristic
145	C₄×D₅₉	#102	1	1	yes	characteristic
146	Dic₁₁₈	—	2	1	no	—
147	Dic₁₁₈	—	4	1	no	—
148	C₈×C₅₉	—	4	2	no	—
149	C₈×C₅₉	—	4	1	no	—
150	C₈×C₅₉	C₂²×C₈	1	1	yes	characteristic
151	C₈×C₅₉	C₂×M₁₆	1	1	yes	characteristic
152	C₈×C₅₉	C₂×D₈	1	1	yes	characteristic
153	C₈×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
154	C₅₉⋊C₈	—	4	2	no	—
155	C₅₉⋊C₈	—	4	1	no	—
156	C₅₉⋊C₈	C₂²⋊C₈	1	1	yes	characteristic
157	C₅₉⋊C₈	C₂²⋊C₈	1	1	yes	characteristic
158	C₅₉⋊C₈	#102	1	1	yes	characteristic
159	C₅₉⋊C₈	#102	1	1	yes	characteristic
160	C₂×#1336	—	59	1	no	maximal, Sylow
161	C₂²×C₅₉	—	4	1	no	—
162	C₂²×C₅₉	C₂×#104	1	1	yes	characteristic
163	C₂×D₅₉	—	2	2	no	—
164	C₂×D₅₉	—	4	1	no	—
165	C₂×D₅₉	—	8	2	no	—
166	C₂×D₅₉	D₄⋊C₈	1	1	yes	characteristic
167	C₂×D₅₉	D₄⋊C₈	1	1	yes	characteristic
168	C₄×C₅₉	—	2	2	no	—
169	C₄×C₅₉	—	4	1	no	—
170	C₄×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
171	C₄×C₅₉	C₂×#102	1	1	yes	characteristic
172	Dic₅₉	—	2	2	no	—
173	Dic₅₉	—	4	1	no	—
174	Dic₅₉	D₄⋊C₈	1	1	yes	characteristic
175	Dic₅₉	D₄⋊C₈	1	1	yes	characteristic
176	C₂×#443	—	59	1	no	—
177	C₂×#364	—	59	1	no	—
178	C₂×#322	—	59	1	no	—
179	#1336	—	59	1	no	—
180	#1336	—	59	1	no	—
181	#1336	—	59	1	no	—
182	#1336	D₅₉	1	1	yes	characteristic, direct product factor
183	C₂×C₅₉	—	2	1	no	—
184	C₂×C₅₉	—	8	1	no	—
185	C₂×C₅₉	C₂×D₄⋊C₈	1	1	yes	characteristic
186	D₅₉	—	2	1	no	—
187	D₅₉	—	8	1	no	—
188	D₅₉	#1336	1	2	yes	direct product factor
189	C₂×#135	—	59	1	no	—
190	C₂×#131	—	118	1	no	—
191	C₂×M₃₂	—	118	1	no	—
192	#443	—	59	1	no	—
193	#443	—	59	1	no	—
194	#443	—	59	1	no	—
195	#443	—	118	1	no	—
196	#443	—	118	1	no	—
197	#443	C₂×D₅₉	1	1	yes	characteristic
198	C₂×#108	—	59	1	no	—
199	C₂×#104	—	118	1	no	—
200	C₂×C₄×C₈	—	59	1	no	—
201	C₂×C₄×C₈	—	118	1	no	—
202	#364	—	59	1	no	—
203	#364	—	118	1	no	—
204	#364	C₂×D₅₉	1	1	yes	characteristic, complemented
205	#322	—	59	1	no	—
206	#322	—	118	1	no	—
207	#322	C₂×D₅₉	1	1	yes	characteristic, complemented
208	C₅₉	C₂×#1336	1	1	yes	characteristic, complemented, Sylow
209	C₂×#41	—	118	1	no	—
210	#135	—	59	1	no	—
211	#135	—	59	1	no	—
212	#135	—	59	1	no	—
213	#135	—	118	1	no	—
214	#135	—	118	1	no	—
215	#135	C₄×D₅₉	1	1	yes	characteristic
216	C₂×Q₁₆	—	59	1	no	—
217	C₂×SD₁₆	—	118	1	no	—
218	C₂×D₈	—	59	1	no	—
219	#131	—	2	1	no	—
220	#131	—	118	1	no	—
221	#131	—	118	1	no	—
222	#131	—	118	1	no	—
223	#131	—	236	1	no	—
224	#131	—	236	1	no	—
225	C₂×M₁₆	—	118	1	no	—
226	C₂×M₁₆	—	236	1	no	—
227	C₂²×C₈	—	59	1	no	—
228	C₂²×C₈	—	59	1	no	—
229	C₂²×C₈	—	118	2	no	—
230	C₂²×C₈	—	236	1	no	—
231	C₂×C₄²	—	59	1	no	—
232	M₃₂	—	2	1	no	—
233	M₃₂	—	118	1	no	—
234	M₃₂	—	236	1	no	—
235	C₂×C₁₆	—	236	1	no	—
236	#108	—	59	1	no	—
237	#108	—	118	1	no	—
238	#108	C₄×D₅₉	1	1	yes	characteristic
239	#104	—	2	1	no	—
240	#104	—	118	1	no	—
241	#104	—	118	1	no	—
242	#104	—	118	1	no	—
243	C₄×C₈	—	2	1	no	—
244	C₄×C₈	—	59	1	no	—
245	C₄×C₈	—	59	1	no	—
246	C₄×C₈	—	59	1	no	—
247	C₄×C₈	—	118	2	no	—
248	C₄×C₈	—	118	1	no	—
249	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
250	#41	—	2	1	no	—
251	#41	—	118	1	no	—
252	#41	—	118	1	no	—
253	#41	—	118	1	no	—
254	#41	—	236	1	no	—
255	#41	—	236	1	no	—
256	C₂×Q₈	—	118	1	no	—
257	C₂×D₄	—	118	1	no	—
258	C₂×D₄	—	236	1	no	—
259	C₂²×C₄	—	59	1	no	—
260	C₂²×C₄	—	118	1	no	—
261	C₂²×C₄	—	236	1	no	—
262	Q₁₆	—	59	1	no	—
263	Q₁₆	—	118	1	no	—
264	Q₁₆	C₈×D₅₉	1	1	yes	characteristic, complemented
265	SD₁₆	—	2	1	no	—
266	SD₁₆	—	118	1	no	—
267	SD₁₆	—	118	1	no	—
268	SD₁₆	—	118	1	no	—
269	D₈	—	59	1	no	—
270	D₈	—	118	1	no	—
271	D₈	C₈×D₅₉	1	1	yes	characteristic, complemented
272	M₁₆	—	2	1	no	—
273	M₁₆	—	4	1	no	—
274	M₁₆	—	118	1	no	—
275	M₁₆	—	236	1	no	—
276	M₁₆	—	236	1	no	—
277	M₁₆	—	236	1	no	—
278	M₁₆	—	236	1	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	4	1	no	—
281	C₂×C₈	—	59	1	no	—
282	C₂×C₈	—	59	1	no	—
283	C₂×C₈	—	59	1	no	—
284	C₂×C₈	—	59	1	no	—
285	C₂×C₈	—	59	1	no	—
286	C₂×C₈	—	59	1	no	—
287	C₂×C₈	—	59	1	no	—
288	C₂×C₈	—	59	1	no	—
289	C₂×C₈	—	59	1	no	—
290	C₂×C₈	—	59	1	no	—
291	C₂×C₈	—	118	2	no	—
292	C₂×C₈	—	236	1	no	—
293	C₂×C₈	—	236	1	no	—
294	C₂×C₈	—	236	4	no	—
295	C₂×C₈	—	236	1	no	—
296	C₂×C₈	—	236	1	no	—
297	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
298	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
299	C₄²	—	59	1	no	—
300	C₄²	—	118	1	no	—
301	C₄²	D₄×D₅₉	1	1	yes	characteristic
302	C₁₆	—	4	1	no	—
303	C₁₆	—	236	1	no	—
304	C₂³	—	59	1	no	—
305	C₂³	—	236	1	no	—
306	Q₈	—	2	1	no	—
307	Q₈	—	118	1	no	—
308	Q₈	—	236	1	no	—
309	D₄	—	2	1	no	—
310	D₄	—	4	1	no	—
311	D₄	—	118	1	no	—
312	D₄	—	236	1	no	—
313	D₄	—	236	1	no	—
314	D₄	—	236	1	no	—
315	D₄	—	236	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	59	1	no	—
319	C₂×C₄	—	59	1	no	—
320	C₂×C₄	—	59	1	no	—
321	C₂×C₄	—	59	1	no	—
322	C₂×C₄	—	59	1	no	—
323	C₂×C₄	—	118	4	no	—
324	C₂×C₄	—	118	1	no	—
325	C₂×C₄	—	236	1	no	—
326	C₂×C₄	—	236	1	no	—
327	C₂×C₄	—	236	1	no	—
328	C₂×C₄	—	236	1	no	—
329	C₂×C₄	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
330	C₈	—	4	2	no	—
331	C₈	—	4	1	no	—
332	C₈	—	59	1	no	—
333	C₈	—	59	1	no	—
334	C₈	—	59	1	no	—
335	C₈	—	59	1	no	—
336	C₈	—	236	2	no	—
337	C₈	—	236	1	no	—
338	C₈	C₂×C₈×D₅₉	1	1	yes	characteristic
339	C₈	M₁₆×D₅₉	1	1	yes	characteristic
340	C₈	D₈×D₅₉	1	1	yes	characteristic
341	C₈	SD₁₆×D₅₉	1	1	yes	characteristic
342	C₂²	—	4	1	no	—
343	C₂²	—	59	1	no	—
344	C₂²	—	59	1	no	—
345	C₂²	—	118	2	no	—
346	C₂²	—	472	2	no	—
347	C₂²	—	236	1	no	—
348	C₂²	#104×D₅₉	1	1	yes	characteristic
349	C₄	—	2	2	no	—
350	C₄	—	4	1	no	—
351	C₄	—	59	1	no	—
352	C₄	—	59	1	no	—
353	C₄	—	118	2	no	—
354	C₄	—	236	1	no	—
355	C₄	C₂²⋊C₈×D₅₉	1	1	yes	characteristic
356	C₄	#102×D₅₉	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	8	1	no	—
359	C₂	—	59	2	no	—
360	C₂	—	118	1	no	—
361	C₂	—	472	1	no	—
362	C₂	D₄⋊C₈×D₅₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#102×D₅₉
commutator	C₈×C₅₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#1336	C₂
socle	C₂×C₅₉	C₂×D₄⋊C₈
prime radical	#135×D₅₉	C₄
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	#1336	D₅₉
hypocenter	C₅₉	C₂×#1336
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1336

Complements

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

N	H ≅ G/N	type
2. #364×D₅₉	C₂	semidirect
3. #322×D₅₉	C₂	semidirect
4. #21422073	C₂	semidirect
5. #21422072	C₂	semidirect
6. #21422071	C₂	semidirect
7. C₅₉×#1336	C₂	semidirect
20. #16837223	C₂²	semidirect
23. #16837123	C₂²	semidirect
25. C₅₉×#364	C₂²	semidirect
26. C₅₉×#322	C₂²	semidirect
28. #14273750	C₈	semidirect
32. #14273746	C₈	semidirect
33. Q₁₆×D₅₉	C₈	semidirect
35. D₈×D₅₉	C₈	semidirect
80. Q₁₆×C₅₉	C₂×C₈	semidirect
82. D₈×C₅₉	C₂×C₈	semidirect
102. Dic₂₃₆	C₂×C₈	semidirect
103. D₄₇₂	C₂×C₈	semidirect
182. #1336	D₅₉	direct (N × H)
188. D₅₉	#1336	direct (N × H)
204. #364	C₂×D₅₉	semidirect
207. #322	C₂×D₅₉	semidirect
208. C₅₉	C₂×#1336	semidirect
264. Q₁₆	C₈×D₅₉	semidirect
271. D₈	C₈×D₅₉	semidirect

Maximal subgroups

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

subgroup	index
#443×D₅₉	2
#364×D₅₉	2
#322×D₅₉	2
#21422073	2
#21422072	2
#21422071	2
C₅₉×#1336	2
C₂×#1336	59

Subgroups — D59×#1336 #28498478

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₉×#1336 #28498478