Subgroups — #28741925

Order 15104.

646 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#504×D₅₉	C₂	1	2	yes	maximal, complemented
2	#21432994	C₂	1	1	yes	maximal, characteristic
3	#21425938	C₂	1	1	yes	maximal, characteristic, complemented
4	#21425937	C₂	1	1	yes	maximal, characteristic, complemented
5	#21425935	C₂	1	2	yes	maximal, complemented
6	#21425934	C₂	1	1	yes	maximal, characteristic, complemented
7	#21425931	C₂	1	2	yes	maximal, complemented
8	#21425860	C₂	1	1	yes	maximal, characteristic, complemented
9	#21425722	C₂	1	2	yes	maximal, complemented
10	#21425691	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₉×#2170	C₂	1	1	yes	maximal, characteristic, complemented
12	#16838199	C₄	1	1	yes	characteristic
13	#131×D₅₉	C₂²	1	2	yes	—
14	#16837820	—	2	2	no	—
15	#16837820	C₂²	1	2	yes	complemented
16	#16837819	C₂²	1	2	yes	complemented
17	#16837818	—	2	2	no	—
18	#16837818	C₂²	1	2	yes	complemented
19	C₂×#14273666	C₂²	1	1	yes	characteristic
20	M₃₂×D₅₉	—	2	2	no	—
21	M₃₂×D₅₉	C₂²	1	2	yes	—
22	C₂×C₁₆×D₅₉	—	2	2	no	—
23	C₂×C₁₆×D₅₉	C₂²	1	1	yes	characteristic
24	#16837524	C₄	1	1	yes	characteristic
25	#16837523	C₄	1	1	yes	characteristic
26	#16837521	C₄	1	2	yes	—
27	#16837520	C₂²	1	1	yes	characteristic
28	#16837517	C₂²	1	2	yes	—
29	#16837449	C₄	1	1	yes	characteristic
30	#16837417	C₄	1	2	yes	—
31	#16837356	C₂²	1	1	yes	characteristic
32	#16837223	C₂²	1	1	yes	characteristic, complemented
33	#16837222	C₂²	1	2	yes	complemented
34	#16837187	C₂²	1	2	yes	complemented
35	#16837186	C₂²	1	2	yes	complemented
36	#16837185	C₂²	1	2	yes	complemented
37	#16837178	C₂²	1	1	yes	characteristic, complemented
38	#16837177	C₂²	1	1	yes	characteristic, complemented
39	C₁₆×Dic₅₉	C₂²	1	1	yes	characteristic, complemented
40	C₅₉×#504	C₂²	1	2	yes	complemented
41	C₅₉×#443	C₂²	1	1	yes	characteristic
42	C₅₉×#364	C₂²	1	1	yes	characteristic, complemented
43	C₅₉×#350	C₂²	1	2	yes	complemented
44	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic, complemented
45	#41×D₅₉	C₂×C₄	1	2	yes	—
46	#14273759	C₈	1	2	yes	—
47	#14273756	C₈	1	2	yes	—
48	#14273754	C₈	1	2	yes	—
49	#14273751	C₈	1	2	yes	—
50	#14273749	C₂×C₄	1	1	yes	characteristic
51	#14273747	C₂×C₄	1	2	yes	—
52	#14273735	—	2	2	no	—
53	#14273735	C₂×C₄	1	2	yes	—
54	#14273734	C₂×C₄	1	2	yes	—
55	#14273733	—	2	2	no	—
56	#14273733	C₂³	1	2	yes	—
57	C₂×#11745659	C₂×C₄	1	1	yes	characteristic
58	M₁₆×D₅₉	—	2	2	no	—
59	M₁₆×D₅₉	C₂×C₄	1	2	yes	—
60	C₂×C₈×D₅₉	—	2	2	no	—
61	C₂×C₈×D₅₉	C₂³	1	1	yes	characteristic
62	#14273684	C₂×C₄	1	1	yes	characteristic
63	#14273667	—	2	2	no	—
64	#14273667	C₂³	1	2	yes	—
65	#14273666	—	2	2	no	—
66	#14273666	—	2	2	no	—
67	#14273666	—	2	2	no	—
68	#14273666	—	2	2	no	—
69	C₂×C₅₉⋊C₁₆	—	2	2	no	—
70	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
71	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
72	C₁₆×D₅₉	—	2	2	no	—
73	C₁₆×D₅₉	—	2	2	no	—
74	C₁₆×D₅₉	—	2	2	no	—
75	C₁₆×D₅₉	D₄	1	2	yes	—
76	C₁₆×D₅₉	D₄	1	2	yes	—
77	#14273663	C₂×C₄	1	1	yes	characteristic
78	#14273662	C₂×C₄	1	2	yes	—
79	#14273655	C₂×C₄	1	2	yes	—
80	#14273654	C₂×C₄	1	2	yes	—
81	#14273653	C₂×C₄	1	2	yes	—
82	#14273638	C₂×C₄	1	1	yes	characteristic
83	#14273637	C₂×C₄	1	1	yes	characteristic
84	C₈×Dic₅₉	C₂³	1	1	yes	characteristic
85	#135×C₅₉	C₂×C₄	1	1	yes	characteristic
86	#131×C₅₉	C₂³	1	2	yes	—
87	M₃₂×C₅₉	—	2	2	no	—
88	M₃₂×C₅₉	C₂³	1	2	yes	—
89	C₂×C₁₆×C₅₉	—	2	2	no	—
90	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
91	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
92	#108×C₅₉	C₂×C₄	1	1	yes	characteristic
93	#104×C₅₉	C₂×C₄	1	2	yes	—
94	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
95	#41×C₅₉	C₂²×C₄	1	2	yes	—
96	#11745689	C₂×C₈	1	2	yes	—
97	Q₈×D₅₉	C₂×C₈	1	2	yes	—
98	#11745687	—	2	2	no	—
99	#11745687	C₂×C₈	1	2	yes	—
100	D₄×D₅₉	—	2	2	no	—
101	D₄×D₅₉	C₂×C₈	1	2	yes	—
102	#11745685	—	2	2	no	—
103	#11745685	C₂²×C₄	1	2	yes	—
104	C₂×C₄×D₅₉	—	2	2	no	—
105	C₂×C₄×D₅₉	C₂²×C₄	1	1	yes	characteristic
106	Q₁₆×C₅₉	C₂×C₈	1	1	yes	characteristic
107	SD₁₆×C₅₉	C₂×C₈	1	2	yes	—
108	D₈×C₅₉	C₂×C₈	1	1	yes	characteristic
109	M₁₆×C₅₉	—	2	2	no	—
110	M₁₆×C₅₉	C₂²×C₄	1	2	yes	—
111	C₂×C₈×C₅₉	—	2	2	no	—
112	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
114	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
115	#11745672	C₂×C₈	1	2	yes	—
116	#11745671	C₂×C₈	1	2	yes	—
117	#11745670	C₂×C₈	1	2	yes	—
118	#11745669	C₂×C₈	1	2	yes	—
119	(C₂×D₅₉)⋊C₄	—	2	1	no	—
120	Dic₅₉⋊C₄	—	2	1	no	—
121	C₄×Dic₅₉	C₂²×C₄	1	1	yes	characteristic
122	#11745664	—	2	2	no	—
123	#11745664	C₂²×C₄	1	2	yes	—
124	C₂×C₅₉⋊C₈	—	2	2	no	—
125	C₂×C₅₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₅₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
127	Dic₂₃₆	C₂×C₈	1	1	yes	characteristic
128	D₄₇₂	C₂×C₈	1	1	yes	characteristic
129	#11745660	C₂×C₈	1	2	yes	—
130	#11745659	—	2	2	no	—
131	#11745659	—	2	2	no	—
132	#11745659	—	2	2	no	—
133	#11745659	C₂×C₈	1	2	yes	—
134	#11745659	C₂×C₈	1	2	yes	—
135	C₈×D₅₉	—	2	2	no	—
136	C₈×D₅₉	—	2	2	no	—
137	C₈×D₅₉	—	2	2	no	—
138	C₈×D₅₉	C₂×D₄	1	2	yes	—
139	C₈×D₅₉	#41	1	2	yes	—
140	C₁₆×C₅₉	—	2	2	no	—
141	C₁₆×C₅₉	—	2	1	no	—
142	C₁₆×C₅₉	C₂×D₄	1	1	yes	characteristic
143	C₁₆×C₅₉	C₂×D₄	1	1	yes	characteristic
144	C₅₉⋊C₁₆	—	2	2	no	—
145	C₅₉⋊C₁₆	—	2	1	no	—
146	C₅₉⋊C₁₆	C₂×D₄	1	2	yes	—
147	C₂²×D₅₉	—	2	2	no	—
148	C₂²×D₅₉	#131	1	1	yes	characteristic
149	Q₈×C₅₉	C₂²×C₈	1	2	yes	—
150	D₄×C₅₉	—	2	2	no	—
151	D₄×C₅₉	C₂²×C₈	1	2	yes	—
152	C₂×C₄×C₅₉	—	2	2	no	—
153	C₂×C₄×C₅₉	—	2	1	no	—
154	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
155	C₂²⋊D₅₉	—	2	2	no	—
156	C₂²⋊D₅₉	—	2	2	no	—
157	C₂²⋊D₅₉	—	2	2	no	—
158	C₂×Dic₅₉	—	2	2	no	—
159	C₂×Dic₅₉	—	2	1	no	—
160	C₂×Dic₅₉	#131	1	1	yes	characteristic
161	D₂₃₆	—	2	2	no	—
162	D₂₃₆	C₂²×C₈	1	2	yes	—
163	C₄×D₅₉	—	2	2	no	—
164	C₄×D₅₉	—	2	2	no	—
165	C₄×D₅₉	—	2	2	no	—
166	C₄×D₅₉	C₄×D₄	1	2	yes	—
167	C₄×D₅₉	C₂²×C₈	1	2	yes	—
168	Dic₁₁₈	—	2	2	no	—
169	Dic₁₁₈	C₂²×C₈	1	2	yes	—
170	C₈×C₅₉	—	2	2	no	—
171	C₈×C₅₉	C₂²×C₈	1	1	yes	characteristic
172	C₈×C₅₉	C₂²×C₈	1	1	yes	characteristic
173	C₈×C₅₉	C₂²×D₄	1	1	yes	characteristic
174	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
175	C₅₉⋊C₈	—	2	2	no	—
176	C₅₉⋊C₈	C₄×D₄	1	2	yes	—
177	C₅₉⋊C₈	C₂²×C₈	1	2	yes	—
178	#33441	—	59	1	no	maximal, Sylow
179	C₂²×C₅₉	—	2	2	no	—
180	C₂²×C₅₉	C₂×#131	1	1	yes	characteristic
181	C₂×D₅₉	—	2	2	no	—
182	C₂×D₅₉	—	4	1	no	—
183	C₂×D₅₉	—	4	4	no	—
184	C₂×D₅₉	C₈×D₄	1	2	yes	—
185	C₄×C₅₉	—	2	2	no	—
186	C₄×C₅₉	—	4	1	no	—
187	C₄×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
188	C₄×C₅₉	C₂³×C₈	1	1	yes	characteristic
189	Dic₅₉	—	2	2	no	—
190	Dic₅₉	—	4	1	no	—
191	Dic₅₉	C₈×D₄	1	2	yes	—
192	C₂×#504	—	59	2	no	—
193	#2955	—	59	1	no	—
194	#2171	—	59	1	no	—
195	#2171	—	59	1	no	—
196	#2171	—	59	2	no	—
197	#2170	—	59	2	no	—
198	#2170	—	59	1	no	—
199	#2170	D₅₉	1	1	yes	characteristic, complemented
200	#2153	—	59	1	no	—
201	#2117	—	59	2	no	—
202	#2108	—	59	1	no	—
203	C₂×C₅₉	—	2	1	no	—
204	C₂×C₅₉	—	4	2	no	—
205	C₂×C₅₉	C₂×C₈×D₄	1	1	yes	characteristic
206	D₅₉	—	2	2	no	—
207	D₅₉	—	4	2	no	—
208	#575	—	59	1	no	—
209	C₂×#131	—	59	2	no	—
210	#504	—	59	2	no	—
211	#504	—	59	2	no	—
212	#504	—	59	2	no	—
213	#504	—	118	2	no	—
214	#504	—	118	2	no	—
215	#504	C₂×D₅₉	1	2	yes	complemented
216	C₂×M₃₂	—	59	1	no	—
217	C₂×M₃₂	—	59	2	no	—
218	C₂×M₃₂	—	118	2	no	—
219	C₂²×C₁₆	—	59	1	no	—
220	C₂²×C₁₆	—	118	2	no	—
221	#444	—	59	1	no	—
222	#444	—	59	1	no	—
223	#444	—	59	2	no	—
224	#443	—	59	2	no	—
225	#443	—	59	1	no	—
226	#443	C₂×D₅₉	1	1	yes	characteristic
227	#430	—	59	1	no	—
228	#421	—	59	2	no	—
229	#405	—	59	1	no	—
230	#364	—	59	2	no	—
231	#364	—	59	1	no	—
232	#364	C₂×D₅₉	1	1	yes	characteristic, complemented
233	#350	—	59	2	no	—
234	#350	—	59	2	no	—
235	#350	—	59	2	no	—
236	#350	C₂×D₅₉	1	2	yes	complemented
237	#346	—	59	1	no	—
238	#346	—	59	1	no	—
239	C₄×C₁₆	—	59	1	no	—
240	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic, complemented
241	C₅₉	#33441	1	1	yes	characteristic, complemented, Sylow
242	C₂×#41	—	59	2	no	—
243	#137	—	59	2	no	—
244	#137	—	59	2	no	—
245	AGL(1,ℤ/8)	—	59	2	no	—
246	AGL(1,ℤ/8)	—	59	2	no	—
247	#135	—	59	2	no	—
248	#135	—	59	1	no	—
249	#135	C₄×D₅₉	1	1	yes	characteristic
250	#131	—	59	2	no	—
251	#131	—	59	2	no	—
252	#131	—	59	2	no	—
253	#131	—	118	2	no	—
254	#131	—	118	2	no	—
255	#131	C₂²×D₅₉	1	2	yes	—
256	C₂×M₁₆	—	59	1	no	—
257	C₂×M₁₆	—	59	2	no	—
258	C₂×M₁₆	—	118	2	no	—
259	C₂²×C₈	—	59	1	no	—
260	C₂²×C₈	—	118	2	no	—
261	#117	—	59	1	no	—
262	M₃₂	—	2	2	no	—
263	M₃₂	—	59	2	no	—
264	M₃₂	—	118	2	no	—
265	M₃₂	—	118	2	no	—
266	M₃₂	—	118	2	no	—
267	M₃₂	—	118	2	no	—
268	M₃₂	—	118	2	no	—
269	M₃₂	C₂²×D₅₉	1	2	yes	—
270	C₂×C₁₆	—	2	2	no	—
271	C₂×C₁₆	—	59	1	no	—
272	C₂×C₁₆	—	59	2	no	—
273	C₂×C₁₆	—	59	2	no	—
274	C₂×C₁₆	—	59	1	no	—
275	C₂×C₁₆	—	118	2	no	—
276	C₂×C₁₆	—	118	2	no	—
277	C₂×C₁₆	—	118	2	no	—
278	C₂×C₁₆	—	118	2	no	—
279	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
280	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
281	#108	—	59	2	no	—
282	#108	—	59	1	no	—
283	#108	C₄×D₅₉	1	1	yes	characteristic
284	#104	—	59	2	no	—
285	#104	—	59	2	no	—
286	#104	—	59	2	no	—
287	#104	C₄×D₅₉	1	2	yes	—
288	#97	—	59	1	no	—
289	#97	—	59	1	no	—
290	C₄×C₈	—	59	1	no	—
291	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
292	#41	—	59	2	no	—
293	#41	—	59	2	no	—
294	#41	—	59	2	no	—
295	#41	—	118	2	no	—
296	#41	—	118	2	no	—
297	#41	C₂×C₄×D₅₉	1	2	yes	—
298	C₂×Q₈	—	59	2	no	—
299	C₂×D₄	—	59	2	no	—
300	C₂×D₄	—	118	2	no	—
301	C₂²×C₄	—	59	1	no	—
302	C₂²×C₄	—	118	2	no	—
303	Q₁₆	—	59	2	no	—
304	Q₁₆	—	59	1	no	—
305	Q₁₆	C₈×D₅₉	1	1	yes	characteristic
306	SD₁₆	—	59	2	no	—
307	SD₁₆	—	59	2	no	—
308	SD₁₆	—	59	2	no	—
309	SD₁₆	C₈×D₅₉	1	2	yes	—
310	D₈	—	59	2	no	—
311	D₈	—	59	1	no	—
312	D₈	C₈×D₅₉	1	1	yes	characteristic
313	M₁₆	—	2	2	no	—
314	M₁₆	—	59	2	no	—
315	M₁₆	—	59	2	no	—
316	M₁₆	—	59	2	no	—
317	M₁₆	—	118	2	no	—
318	M₁₆	—	118	2	no	—
319	M₁₆	—	118	2	no	—
320	M₁₆	—	118	2	no	—
321	M₁₆	C₂×C₄×D₅₉	1	2	yes	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	59	1	no	—
324	C₂×C₈	—	59	1	no	—
325	C₂×C₈	—	59	2	no	—
326	C₂×C₈	—	59	2	no	—
327	C₂×C₈	—	118	2	no	—
328	C₂×C₈	—	118	2	no	—
329	C₂×C₈	—	118	2	no	—
330	C₂×C₈	—	118	2	no	—
331	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
332	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
333	C₄⋊C₄	—	118	1	no	—
334	C₂²⋊C₄	—	118	1	no	—
335	C₄²	—	59	1	no	—
336	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
337	C₁₆	—	2	2	no	—
338	C₁₆	—	2	1	no	—
339	C₁₆	—	59	2	no	—
340	C₁₆	—	118	2	no	—
341	C₁₆	—	118	1	no	—
342	C₁₆	D₄×D₅₉	1	1	yes	characteristic
343	C₁₆	D₄×D₅₉	1	1	yes	characteristic
344	C₂³	—	59	1	no	—
345	C₂³	—	118	2	no	—
346	Q₈	—	59	2	no	—
347	Q₈	—	118	2	no	—
348	Q₈	C₂×C₈×D₅₉	1	2	yes	—
349	D₄	—	2	2	no	—
350	D₄	—	59	2	no	—
351	D₄	—	118	2	no	—
352	D₄	—	118	2	no	—
353	D₄	—	118	2	no	—
354	D₄	—	118	2	no	—
355	D₄	C₂×C₈×D₅₉	1	2	yes	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	59	1	no	—
359	C₂×C₄	—	59	2	no	—
360	C₂×C₄	—	59	2	no	—
361	C₂×C₄	—	118	2	no	—
362	C₂×C₄	—	118	2	no	—
363	C₂×C₄	—	118	2	no	—
364	C₂×C₄	—	118	2	no	—
365	C₂×C₄	—	118	1	no	—
366	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
367	C₈	—	2	2	no	—
368	C₈	—	59	2	no	—
369	C₈	—	59	2	no	—
370	C₈	—	118	2	no	—
371	C₈	C₂×C₈×D₅₉	1	1	yes	characteristic
372	C₈	C₂×C₈×D₅₉	1	1	yes	characteristic
373	C₈	C₂×D₄×D₅₉	1	1	yes	characteristic
374	C₈	#41×D₅₉	1	1	yes	characteristic
375	C₂²	—	2	2	no	—
376	C₂²	—	59	2	no	—
377	C₂²	—	118	2	no	—
378	C₂²	—	236	1	no	—
379	C₂²	—	236	4	no	—
380	C₂²	#131×D₅₉	1	1	yes	characteristic
381	C₄	—	2	2	no	—
382	C₄	—	4	1	no	—
383	C₄	—	59	2	no	—
384	C₄	—	118	2	no	—
385	C₄	—	236	1	no	—
386	C₄	C₄×D₄×D₅₉	1	1	yes	characteristic
387	C₄	C₂²×C₈×D₅₉	1	1	yes	characteristic
388	C₂	—	2	1	no	—
389	C₂	—	4	2	no	—
390	C₂	—	118	2	no	—
391	C₂	—	236	2	no	—
392	C₂	C₈×D₄×D₅₉	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	D₄×D₅₉
commutator	C₄×C₅₉	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₅₉×#2170	C₂
socle	C₂×C₅₉	C₂×C₈×D₄
prime radical	#16838199	C₄
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	#2170	D₅₉
hypocenter	C₅₉	#33441
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ #2170

Complements

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

N	H ≅ G/N	type
1. #504×D₅₉	C₂	semidirect
3. #21425938	C₂	semidirect
4. #21425937	C₂	semidirect
5. #21425935	C₂	semidirect
6. #21425934	C₂	semidirect
7. #21425931	C₂	semidirect
8. #21425860	C₂	semidirect
9. #21425722	C₂	semidirect
10. #21425691	C₂	semidirect
11. C₅₉×#2170	C₂	semidirect
15. #16837820	C₂²	semidirect
16. #16837819	C₂²	semidirect
18. #16837818	C₂²	semidirect
32. #16837223	C₂²	semidirect
33. #16837222	C₂²	semidirect
34. #16837187	C₂²	semidirect
35. #16837186	C₂²	semidirect
36. #16837185	C₂²	semidirect
37. #16837178	C₂²	semidirect
38. #16837177	C₂²	semidirect
39. C₁₆×Dic₅₉	C₂²	semidirect
40. C₅₉×#504	C₂²	semidirect
42. C₅₉×#364	C₂²	semidirect
43. C₅₉×#350	C₂²	semidirect
44. C₄×C₁₆×C₅₉	C₂²	semidirect
199. #2170	D₅₉	semidirect
215. #504	C₂×D₅₉	semidirect
232. #364	C₂×D₅₉	semidirect
236. #350	C₂×D₅₉	semidirect
240. C₄×C₁₆	C₂×D₅₉	semidirect
241. C₅₉	#33441	semidirect

Maximal subgroups

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

subgroup	index
#504×D₅₉	2
#21432994	2
#21425938	2
#21425937	2
#21425935	2
#21425934	2
#21425931	2
#21425860	2
#21425722	2
#21425691	2
C₅₉×#2170	2
#33441	59