Subgroups — #28560461

Order 15104.

586 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21434679	C₂	1	1	yes	maximal, characteristic
2	#21433038	C₂	1	2	yes	maximal
3	#21422994	C₂	1	1	yes	maximal, characteristic, complemented
4	#21422993	C₂	1	1	yes	maximal, characteristic, complemented
5	#21422949	C₂	1	2	yes	maximal, complemented
6	#21422946	C₂	1	2	yes	maximal, complemented
7	#21422902	C₂	1	1	yes	maximal, characteristic, complemented
8	#335×D₅₉	C₂	1	1	yes	maximal, characteristic
9	#21422497	C₂	1	2	yes	maximal
10	C₈×#11745659	C₂	1	1	yes	maximal, characteristic
11	C₈×Q₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837956	C₂²	1	2	yes	—
13	#16837806	C₄	1	1	yes	characteristic
14	#16837780	C₄	1	1	yes	characteristic
15	#16837751	C₄	1	2	yes	—
16	#16837644	C₄	1	1	yes	characteristic
17	#16837642	C₄	1	2	yes	—
18	#16837640	C₄	1	1	yes	characteristic
19	#16837533	C₂²	1	2	yes	—
20	#16837532	—	2	2	no	—
21	#16837532	C₂²	1	2	yes	—
22	#16837531	—	2	2	no	—
23	#16837526	C₂²	1	2	yes	—
24	C₄×Dic₂₃₆	C₂²	1	1	yes	characteristic
25	C₄×#11745672	C₂²	1	2	yes	—
26	#16837453	C₂²	1	1	yes	characteristic
27	#16837427	—	2	2	no	—
28	C₄⋊C₈×D₅₉	C₂²	1	2	yes	—
29	#16837356	—	2	2	no	—
30	C₄×#11745659	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₅₉	C₂²	1	1	yes	characteristic
32	#16837155	C₂²	1	1	yes	characteristic
33	#16837154	C₂²	1	2	yes	—
34	#16837134	C₂²	1	2	yes	—
35	Q₈⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
36	#16837132	C₂²	1	2	yes	—
37	#16837124	C₂²	1	1	yes	characteristic
38	#16837123	C₂²	1	1	yes	characteristic
39	C₈×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
40	C₈×Q₈×C₅₉	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
42	C₅₉×#335	C₂²	1	1	yes	characteristic
43	C₅₉×Q₈⋊C₈	C₂²	1	2	yes	—
44	C₈²×C₅₉	C₂²	1	1	yes	characteristic
45	C₂×Dic₂₃₆	C₂×C₄	1	1	yes	characteristic
46	C₂×#11745672	C₂×C₄	1	2	yes	—
47	C₂×#11745659	D₄	1	1	yes	characteristic
48	C₂×#11745659	D₄	1	1	yes	characteristic
49	C₂×C₈×D₅₉	—	2	2	no	—
50	C₂×C₈×D₅₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₅₉	C₂×C₄	1	1	yes	characteristic
52	#14273726	C₂×C₄	1	2	yes	—
53	#14273725	C₂×C₄	1	2	yes	—
54	#14273717	—	2	2	no	—
55	#14273707	C₂×C₄	1	2	yes	—
56	#14273705	—	2	2	no	—
57	#14273703	C₂×C₄	1	2	yes	—
58	C₄×Dic₁₁₈	—	2	2	no	—
59	C₄×Dic₁₁₈	C₂³	1	2	yes	—
60	#14273684	—	2	2	no	—
61	C₄²×D₅₉	C₂³	1	1	yes	characteristic
62	#14273661	C₂×C₄	1	1	yes	characteristic
63	#14273660	C₂×C₄	1	2	yes	—
64	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
65	C₄⋊(C₅₉⋊C₈)	C₂³	1	2	yes	—
66	Dic₅₉⋊C₈	—	2	2	no	—
67	Dic₅₉⋊C₈	—	2	2	no	—
68	Dic₅₉⋊C₈	—	2	1	no	—
69	Dic₅₉⋊C₈	—	2	2	no	—
70	#14273652	C₂×C₄	1	2	yes	—
71	#14273651	C₂×C₄	1	2	yes	—
72	#14273650	C₂×C₄	1	2	yes	—
73	(C₂×D₅₉)⋊C₈	—	2	1	no	—
74	#14273638	—	2	2	no	—
75	#14273637	—	2	2	no	—
76	#14273637	D₄	1	1	yes	characteristic
77	#14273637	D₄	1	1	yes	characteristic
78	C₄×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
79	C₄×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
80	C₄×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
81	C₈×Dic₅₉	—	2	2	no	—
82	C₈×Dic₅₉	C₂×C₄	1	1	yes	characteristic
83	C₈×Dic₅₉	C₂×C₄	1	1	yes	characteristic
84	C₂×Q₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
85	C₄×Q₈×C₅₉	C₂³	1	2	yes	—
86	#107×C₅₉	C₂×C₄	1	1	yes	characteristic
87	C₄⋊C₈×C₅₉	—	2	2	no	—
88	C₄⋊C₈×C₅₉	C₂³	1	2	yes	—
89	#103×C₅₉	C₂×C₄	1	2	yes	—
90	C₄×C₈×C₅₉	—	2	2	no	—
91	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₅₉	C₂²×C₄	1	2	yes	—
95	C₂×C₄×D₅₉	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×D₅₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×D₅₉	#41	1	1	yes	characteristic
98	C₂×Dic₁₁₈	—	2	2	no	—
99	C₂×Dic₁₁₈	C₂²×C₄	1	2	yes	—
100	Q₁₆×C₅₉	—	2	2	no	—
101	C₂×C₈×C₅₉	—	2	2	no	—
102	C₂×C₈×C₅₉	—	2	1	no	—
103	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
107	C₄⋊C₄×C₅₉	—	2	2	no	—
108	C₄⋊C₄×C₅₉	C₂²×C₄	1	2	yes	—
109	C₄²×C₅₉	—	2	2	no	—
110	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
111	#11745672	—	4	2	no	—
112	(C₂×D₅₉)⋊C₄	—	2	2	no	—
113	(C₂×D₅₉)⋊C₄	—	2	2	no	—
114	C₄⋊Dic₅₉	—	2	2	no	—
115	C₄⋊Dic₅₉	C₂²×C₄	1	2	yes	—
116	Dic₅₉⋊C₄	—	2	2	no	—
117	Dic₅₉⋊C₄	—	2	2	no	—
118	Dic₅₉⋊C₄	—	2	2	no	—
119	Dic₅₉⋊C₄	—	2	2	no	—
120	Dic₅₉⋊C₄	—	2	2	no	—
121	C₄×Dic₅₉	—	2	2	no	—
122	C₄×Dic₅₉	C₂²×C₄	1	1	yes	characteristic
123	C₄×Dic₅₉	C₂×D₄	1	1	yes	characteristic
124	C₄×Dic₅₉	#41	1	1	yes	characteristic
125	C₂×C₅₉⋊C₈	—	2	2	no	—
126	C₂×C₅₉⋊C₈	—	2	1	no	—
127	C₂×C₅₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₅₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₅₉⋊C₈	C₂×D₄	1	1	yes	characteristic
130	C₂×C₅₉⋊C₈	C₂×D₄	1	1	yes	characteristic
131	Dic₂₃₆	—	2	2	no	—
132	#11745659	—	2	2	no	—
133	#11745659	Q₁₆	1	4	yes	complemented
134	C₈×D₅₉	—	2	2	no	—
135	C₈×D₅₉	—	4	1	no	—
136	C₈×D₅₉	—	4	4	no	—
137	C₂²×D₅₉	C₄×D₄	1	1	yes	characteristic
138	Q₈×C₅₉	—	2	2	no	—
139	Q₈×C₅₉	—	2	2	no	—
140	C₂×C₄×C₅₉	—	2	2	no	—
141	C₂×C₄×C₅₉	—	2	2	no	—
142	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
145	C₂×Dic₅₉	—	2	2	no	—
146	C₂×Dic₅₉	—	2	2	no	—
147	C₂×Dic₅₉	C₄×D₄	1	1	yes	characteristic
148	C₂×Dic₅₉	C₄×D₄	1	1	yes	characteristic
149	C₂×Dic₅₉	C₄×D₄	1	1	yes	characteristic
150	C₄×D₅₉	—	2	1	no	—
151	C₄×D₅₉	—	2	1	no	—
152	C₄×D₅₉	—	4	1	no	—
153	C₄×D₅₉	C₂×Q₁₆	1	2	yes	—
154	C₄×D₅₉	#135	1	2	yes	—
155	Dic₁₁₈	—	2	2	no	—
156	Dic₁₁₈	—	2	2	no	—
157	Dic₁₁₈	—	8	2	no	—
158	C₈×C₅₉	—	2	1	no	—
159	C₈×C₅₉	—	2	1	no	—
160	C₈×C₅₉	—	4	1	no	—
161	C₈×C₅₉	—	4	2	no	—
162	C₈×C₅₉	#131	1	2	yes	—
163	C₈×C₅₉	C₂×Q₁₆	1	2	yes	—
164	C₅₉⋊C₈	—	2	1	no	—
165	C₅₉⋊C₈	—	2	1	no	—
166	C₅₉⋊C₈	—	2	1	no	—
167	C₅₉⋊C₈	—	4	1	no	—
168	C₅₉⋊C₈	—	4	2	no	—
169	C₅₉⋊C₈	C₂×Q₁₆	1	2	yes	—
170	#18068	—	59	1	no	maximal, Sylow
171	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
172	C₂×D₅₉	—	2	1	no	—
173	C₂×D₅₉	—	2	1	no	—
174	C₂×D₅₉	C₄×Q₁₆	1	2	yes	—
175	C₄×C₅₉	—	2	1	no	—
176	C₄×C₅₉	—	4	2	no	—
177	C₄×C₅₉	—	4	2	no	—
178	C₄×C₅₉	C₂×#131	1	1	yes	characteristic
179	C₄×C₅₉	#568	1	1	yes	characteristic
180	C₄×C₅₉	C₂²×Q₁₆	1	1	yes	characteristic
181	C₄×C₅₉	C₂×#135	1	1	yes	characteristic
182	Dic₅₉	—	2	1	no	—
183	Dic₅₉	—	2	1	no	—
184	Dic₅₉	—	4	2	no	—
185	Dic₅₉	—	4	2	no	—
186	Dic₅₉	C₄×Q₁₆	1	2	yes	—
187	#3170	—	59	1	no	—
188	#2962	—	59	2	no	—
189	#1591	—	59	1	no	—
190	#1591	—	59	1	no	—
191	#1584	—	59	2	no	—
192	#1584	—	59	2	no	—
193	C₈×Q₁₆	—	59	1	no	—
194	C₈×Q₁₆	D₅₉	1	1	yes	characteristic, complemented
195	C₂×#335	—	59	1	no	—
196	#1481	—	59	2	no	—
197	C₈×M₁₆	—	59	1	no	—
198	C₂×C₅₉	#2935	1	1	yes	characteristic
199	C₂×C₅₉	C₂×C₄×Q₁₆	1	1	yes	characteristic
200	C₂×C₅₉	#2957	1	1	yes	characteristic
201	D₅₉	—	2	2	no	—
202	#533	—	59	2	no	—
203	#500	—	59	1	no	—
204	#494	—	59	1	no	—
205	#487	—	59	2	no	—
206	#467	—	59	1	no	—
207	#467	—	59	1	no	—
208	#467	—	59	2	no	—
209	#446	—	59	2	no	—
210	#446	—	59	2	no	—
211	#446	—	118	2	no	—
212	#446	—	118	2	no	—
213	C₈×Q₈	—	59	2	no	—
214	C₈×Q₈	C₂×D₅₉	1	2	yes	—
215	C₄×Q₁₆	—	59	2	no	—
216	C₄×Q₁₆	—	59	1	no	—
217	C₄×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
218	#431	—	59	1	no	—
219	#424	—	118	2	no	—
220	C₂×C₄⋊C₈	—	59	2	no	—
221	#405	—	118	2	no	—
222	C₄×M₁₆	—	59	1	no	—
223	C₂×C₄×C₈	—	59	1	no	—
224	#335	—	59	2	no	—
225	#335	—	59	1	no	—
226	#335	C₂×D₅₉	1	1	yes	characteristic
227	Q₈⋊C₈	—	59	2	no	—
228	Q₈⋊C₈	—	59	2	no	—
229	Q₈⋊C₈	—	59	2	no	—
230	Q₈⋊C₈	C₂×D₅₉	1	2	yes	—
231	#322	—	59	1	no	—
232	#322	—	59	1	no	—
233	C₈²	—	59	1	no	—
234	C₈²	C₂×D₅₉	1	1	yes	characteristic
235	C₅₉	#18068	1	1	yes	characteristic, complemented, Sylow
236	C₂×Q₁₆	—	59	2	no	—
237	C₂×Q₁₆	—	59	1	no	—
238	C₂×Q₁₆	C₄×D₅₉	1	1	yes	characteristic
239	C₂×M₁₆	—	59	1	no	—
240	C₂×M₁₆	—	59	1	no	—
241	C₂²×C₈	—	59	1	no	—
242	C₂²×C₈	—	59	1	no	—
243	C₂²×C₈	—	118	2	no	—
244	C₄⋊Q₈	—	59	2	no	—
245	C₄⋊Q₈	—	59	2	no	—
246	#125	—	118	2	no	—
247	C₂²⋊Q₈	—	59	2	no	—
248	C₂²⋊Q₈	—	59	2	no	—
249	C₂²⋊Q₈	—	118	2	no	—
250	C₄×Q₈	—	59	2	no	—
251	C₄×Q₈	—	118	2	no	—
252	C₄×Q₈	C₂²×D₅₉	1	2	yes	—
253	#117	—	118	2	no	—
254	C₂×C₄²	—	59	1	no	—
255	#107	—	59	1	no	—
256	#107	—	59	2	no	—
257	#107	C₄×D₅₉	1	1	yes	characteristic
258	C₄⋊C₈	—	2	2	no	—
259	C₄⋊C₈	—	59	2	no	—
260	C₄⋊C₈	—	118	2	no	—
261	C₄⋊C₈	—	118	2	no	—
262	C₄⋊C₈	—	118	1	no	—
263	C₄⋊C₈	—	118	2	no	—
264	C₄⋊C₈	—	118	2	no	—
265	C₄⋊C₈	C₂²×D₅₉	1	2	yes	—
266	#103	—	59	2	no	—
267	#103	—	59	2	no	—
268	#103	—	59	2	no	—
269	#103	C₄×D₅₉	1	2	yes	—
270	C₂²⋊C₈	—	118	1	no	—
271	#97	—	59	1	no	—
272	#97	—	59	1	no	—
273	#97	—	118	2	no	—
274	#97	—	118	2	no	—
275	C₄×C₈	—	2	2	no	—
276	C₄×C₈	—	59	1	no	—
277	C₄×C₈	—	59	1	no	—
278	C₄×C₈	—	59	1	no	—
279	C₄×C₈	—	59	1	no	—
280	C₄×C₈	—	59	1	no	—
281	C₄×C₈	—	118	2	no	—
282	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
283	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
284	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
285	C₂×Q₈	—	59	2	no	—
286	C₂×Q₈	—	118	2	no	—
287	C₂×Q₈	C₂×C₄×D₅₉	1	2	yes	—
288	C₂²×C₄	—	59	1	no	—
289	C₂²×C₄	—	59	1	no	—
290	C₂²×C₄	—	59	1	no	—
291	Q₁₆	—	2	2	no	—
292	Q₁₆	—	118	2	no	—
293	Q₁₆	—	236	2	no	—
294	M₁₆	—	59	4	no	—
295	M₁₆	—	118	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	2	1	no	—
298	C₂×C₈	—	59	1	no	—
299	C₂×C₈	—	59	1	no	—
300	C₂×C₈	—	59	1	no	—
301	C₂×C₈	—	59	1	no	—
302	C₂×C₈	—	118	2	no	—
303	C₂×C₈	—	118	2	no	—
304	C₂×C₈	—	118	1	no	—
305	C₂×C₈	—	236	1	no	—
306	C₂×C₈	—	236	4	no	—
307	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
308	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
309	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
310	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	59	2	no	—
313	C₄⋊C₄	—	118	2	no	—
314	C₄⋊C₄	—	118	2	no	—
315	C₄⋊C₄	—	118	2	no	—
316	C₄⋊C₄	—	118	2	no	—
317	C₄⋊C₄	—	118	2	no	—
318	C₄⋊C₄	—	118	2	no	—
319	C₄⋊C₄	C₂×C₄×D₅₉	1	2	yes	—
320	C₂²⋊C₄	—	118	2	no	—
321	C₂²⋊C₄	—	118	2	no	—
322	C₄²	—	2	2	no	—
323	C₄²	—	59	1	no	—
324	C₄²	—	59	1	no	—
325	C₄²	—	59	1	no	—
326	C₄²	—	118	2	no	—
327	C₄²	C₂³×D₅₉	1	1	yes	characteristic
328	C₂³	—	59	1	no	—
329	Q₈	—	2	2	no	—
330	Q₈	—	2	2	no	—
331	Q₈	—	118	2	no	—
332	Q₈	—	118	2	no	—
333	Q₈	—	472	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	59	1	no	—
337	C₂×C₄	—	59	2	no	—
338	C₂×C₄	—	59	2	no	—
339	C₂×C₄	—	59	1	no	—
340	C₂×C₄	—	59	1	no	—
341	C₂×C₄	—	118	1	no	—
342	C₂×C₄	—	118	1	no	—
343	C₂×C₄	—	118	2	no	—
344	C₂×C₄	—	118	2	no	—
345	C₂×C₄	—	236	1	no	—
346	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
347	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
348	C₂×C₄	#41×D₅₉	1	1	yes	characteristic
349	C₈	—	2	1	no	—
350	C₈	—	2	1	no	—
351	C₈	—	4	1	no	—
352	C₈	—	4	2	no	—
353	C₈	—	59	2	no	—
354	C₈	—	118	1	no	—
355	C₈	—	118	1	no	—
356	C₈	—	118	1	no	—
357	C₈	—	236	1	no	—
358	C₈	—	236	2	no	—
359	C₈	#14273733	1	2	yes	—
360	C₈	Q₁₆×D₅₉	1	2	yes	—
361	C₂²	—	59	2	no	—
362	C₂²	—	118	1	no	—
363	C₂²	—	118	1	no	—
364	C₂²	C₄×D₄×D₅₉	1	1	yes	characteristic
365	C₄	—	2	1	no	—
366	C₄	—	4	2	no	—
367	C₄	—	4	2	no	—
368	C₄	—	59	2	no	—
369	C₄	—	118	1	no	—
370	C₄	—	118	1	no	—
371	C₄	—	236	2	no	—
372	C₄	—	236	2	no	—
373	C₄	C₂×#14273733	1	1	yes	characteristic
374	C₄	#16838167	1	1	yes	characteristic
375	C₄	C₂×Q₁₆×D₅₉	1	1	yes	characteristic
376	C₄	#135×D₅₉	1	1	yes	characteristic
377	C₂	—	118	2	no	—
378	C₂	#21432835	1	1	yes	characteristic
379	C₂	C₄×Q₁₆×D₅₉	1	1	yes	characteristic
380	C₂	#21433008	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₄²	C₂³×D₅₉
Fitting	C₈×Q₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₂×C₄×D₄
prime radical	C₂²×D₅₉	C₄×D₄
cosocle	C₄²×C₅₉	C₂⁴
hypercenter	C₈×Q₁₆	D₅₉
hypocenter	C₅₉	#18068
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #21422994	C₂	semidirect
4. #21422993	C₂	semidirect
5. #21422949	C₂	semidirect
6. #21422946	C₂	semidirect
7. #21422902	C₂	semidirect
11. C₈×Q₁₆×C₅₉	C₂	semidirect
133. #11745659	Q₁₆	semidirect
194. C₈×Q₁₆	D₅₉	semidirect
235. C₅₉	#18068	semidirect

Maximal subgroups

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

subgroup	index
#21434679	2
#21433038	2
#21422994	2
#21422993	2
#21422949	2
#21422946	2
#21422902	2
#335×D₅₉	2
#21422497	2
C₈×#11745659	2
C₈×Q₁₆×C₅₉	2
#18068	59