Subgroups — #28481394

Order 15104.

492 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21426573	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₅₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#16837132	C₂	1	2	yes	maximal
4	#21421935	C₂	1	2	yes	maximal
5	C₅₉×#1842	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
8	C₂×C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
9	#16837308	C₄	1	1	yes	characteristic
10	#16837308	C₄	1	1	yes	characteristic
11	#16837300	C₄	1	1	yes	characteristic
12	#16837300	C₄	1	1	yes	characteristic
13	#16837286	—	2	1	no	—
14	#16837286	—	2	1	no	—
15	C₄×Dic₅₉⋊C₄	—	2	1	no	—
16	C₄×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
17	#16837248	—	2	1	no	—
18	#16837157	—	2	1	no	—
19	#16837132	—	2	4	no	—
20	C₂×C₄⋊C₈×C₅₉	—	2	1	no	—
21	C₂×C₄⋊C₈×C₅₉	—	2	1	no	—
22	C₂×C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
23	C₄³×C₅₉	C₂²	1	1	yes	characteristic
24	C₂²×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₅₉⋊C₈	—	2	2	no	—
26	C₄×Dic₁₁₈	—	2	1	no	—
27	C₄×Dic₁₁₈	—	2	1	no	—
28	C₄×Dic₁₁₈	D₄	1	2	yes	—
29	C₄×Dic₁₁₈	Q₈	1	2	yes	—
30	C₂×C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
33	C₂×Dic₅₉⋊C₄	—	2	1	no	—
34	C₂×Dic₅₉⋊C₄	—	2	1	no	—
35	C₂×Dic₅₉⋊C₄	—	2	1	no	—
36	C₂×C₄×Dic₅₉	—	2	1	no	—
37	C₂×C₄×Dic₅₉	—	2	1	no	—
38	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
39	C₄×C₅₉⋊C₈	—	2	2	no	—
40	C₄×C₅₉⋊C₈	—	2	2	no	—
41	C₅₉⋊#95	—	2	1	no	—
42	C₅₉⋊#95	—	2	1	no	—
43	C₅₉⋊#95	—	2	1	no	—
44	C₅₉⋊#95	—	2	1	no	—
45	C₅₉⋊#95	—	2	1	no	—
46	C₅₉⋊#95	—	2	1	no	—
47	C₅₉⋊#95	—	2	1	no	—
48	C₂²×C₈×C₅₉	—	2	2	no	—
49	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₅₉	—	2	2	no	—
57	C₄⋊C₈×C₅₉	—	2	2	no	—
58	C₄⋊C₈×C₅₉	—	4	2	no	—
59	C₄⋊C₈×C₅₉	—	4	2	no	—
60	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×Dic₅₉	—	2	1	no	—
68	C₂²×Dic₅₉	—	2	1	no	—
69	C₂²×Dic₅₉	—	2	1	no	—
70	C₂²×Dic₅₉	—	2	1	no	—
71	C₂×Dic₁₁₈	—	2	2	no	—
72	C₂×Dic₁₁₈	—	2	2	no	—
73	C₂×Dic₁₁₈	—	2	1	no	—
74	C₂×Dic₁₁₈	C₄⋊C₄	1	2	yes	—
75	C₂×C₈×C₅₉	—	4	2	no	—
76	C₂×C₈×C₅₉	—	4	2	no	—
77	C₂×C₈×C₅₉	—	4	2	no	—
78	C₄²×C₅₉	—	2	2	no	—
79	C₄²×C₅₉	—	2	2	no	—
80	C₄²×C₅₉	—	2	2	no	—
81	C₄²×C₅₉	—	2	1	no	—
82	C₄²×C₅₉	—	2	1	no	—
83	C₄²×C₅₉	—	4	1	no	—
84	C₄²×C₅₉	SD₁₆	1	2	yes	—
85	C₄²×C₅₉	Q₁₆	1	2	yes	—
86	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
88	C₄²×C₅₉	#41	1	1	yes	characteristic
89	C₄²×C₅₉	#41	1	1	yes	characteristic
90	C₄⋊Dic₅₉	—	2	1	no	—
91	C₄⋊Dic₅₉	—	2	2	no	—
92	C₄⋊Dic₅₉	—	2	2	no	—
93	C₄⋊Dic₅₉	C₄⋊C₄	1	2	yes	—
94	Dic₅₉⋊C₄	—	2	2	no	—
95	Dic₅₉⋊C₄	—	2	2	no	—
96	Dic₅₉⋊C₄	—	4	2	no	—
97	Dic₅₉⋊C₄	—	4	2	no	—
98	C₄×Dic₅₉	—	2	2	no	—
99	C₄×Dic₅₉	—	2	2	no	—
100	C₄×Dic₅₉	—	4	1	no	—
101	C₄×Dic₅₉	—	4	1	no	—
102	C₂×C₅₉⋊C₈	—	2	4	no	—
103	C₂×C₅₉⋊C₈	—	4	2	no	—
104	C₂×C₅₉⋊C₈	—	4	2	no	—
105	C₂³×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₅₉	—	2	1	no	—
107	C₂×C₄×C₅₉	—	2	1	no	—
108	C₂×C₄×C₅₉	—	2	1	no	—
109	C₂×C₄×C₅₉	—	2	1	no	—
110	C₂×C₄×C₅₉	—	2	1	no	—
111	C₂×C₄×C₅₉	—	2	1	no	—
112	C₂×C₄×C₅₉	—	2	1	no	—
113	C₂×C₄×C₅₉	—	2	1	no	—
114	C₂×C₄×C₅₉	—	2	1	no	—
115	C₂×C₄×C₅₉	—	2	1	no	—
116	C₂×C₄×C₅₉	—	2	1	no	—
117	C₂×C₄×C₅₉	#103	1	2	yes	—
118	C₂×C₄×C₅₉	#103	1	2	yes	—
119	C₂×C₄×C₅₉	#104	1	2	yes	—
120	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
122	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
123	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
124	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
126	C₂×C₄×C₅₉	#131	1	2	yes	—
127	C₂×C₄×C₅₉	#131	1	2	yes	—
128	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
130	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
132	C₂×Dic₅₉	—	2	2	no	—
133	C₂×Dic₅₉	—	2	2	no	—
134	C₂×Dic₅₉	—	4	1	no	—
135	C₂×Dic₅₉	—	4	1	no	—
136	C₂×Dic₅₉	—	4	1	no	—
137	C₂×Dic₅₉	—	4	1	no	—
138	C₂×Dic₅₉	—	4	1	no	—
139	C₂×Dic₅₉	—	4	1	no	—
140	C₂×Dic₅₉	—	4	1	no	—
141	C₂×Dic₅₉	—	4	1	no	—
142	C₂×Dic₅₉	—	4	1	no	—
143	C₂×Dic₅₉	—	4	1	no	—
144	Dic₁₁₈	—	2	4	no	—
145	Dic₁₁₈	—	4	2	no	—
146	C₈×C₅₉	—	8	2	no	—
147	C₅₉⋊C₈	—	4	4	no	—
148	#8568	—	59	1	no	maximal, Sylow
149	C₂²×C₅₉	#380	1	1	yes	characteristic
150	C₂²×C₅₉	#408	1	1	yes	characteristic
151	C₂²×C₅₉	#408	1	1	yes	characteristic
152	C₂²×C₅₉	C₂×#103	1	1	yes	characteristic
153	C₂²×C₅₉	#417	1	1	yes	characteristic
154	C₂²×C₅₉	C₂×#104	1	1	yes	characteristic
155	C₂²×C₅₉	#421	1	1	yes	characteristic
156	C₄×C₅₉	—	2	1	no	—
157	C₄×C₅₉	—	2	1	no	—
158	C₄×C₅₉	—	2	2	no	—
159	C₄×C₅₉	—	2	2	no	—
160	C₄×C₅₉	—	2	2	no	—
161	C₄×C₅₉	—	4	1	no	—
162	C₄×C₅₉	#424	1	2	yes	—
163	C₄×C₅₉	#433	1	2	yes	—
164	C₄×C₅₉	#460	1	1	yes	characteristic
165	C₄×C₅₉	#462	1	1	yes	characteristic
166	C₄×C₅₉	#475	1	1	yes	characteristic
167	C₄×C₅₉	#477	1	1	yes	characteristic
168	Dic₅₉	—	4	2	no	—
169	Dic₅₉	—	4	2	no	—
170	Dic₅₉	—	8	1	no	—
171	Dic₅₉	—	8	1	no	—
172	#2295	—	59	1	no	—
173	#1842	D₅₉	1	1	yes	characteristic
174	C₄×C₄⋊C₈	—	59	1	no	—
175	C₂×Q₈⋊C₈	—	59	2	no	—
176	#1279	—	59	2	no	—
177	C₂×C₅₉	#1481	1	1	yes	characteristic
178	C₂×C₅₉	#1483	1	1	yes	characteristic
179	C₂×C₅₉	#1493	1	1	yes	characteristic
180	C₂×C₅₉	#1495	1	1	yes	characteristic
181	C₂×C₅₉	#1798	1	1	yes	characteristic
182	C₂×C₅₉	#1803	1	1	yes	characteristic
183	C₂×C₅₉	#1806	1	1	yes	characteristic
184	C₂×C₄×Q₈	—	59	1	no	—
185	C₂×C₄⋊C₈	—	2	1	no	—
186	C₂×C₄⋊C₈	—	2	1	no	—
187	C₂×C₄⋊C₈	—	118	1	no	—
188	C₂×C₄⋊C₈	C₂×D₅₉	1	2	yes	—
189	C₂×C₄×C₈	—	59	2	no	—
190	#391	—	59	1	no	—
191	#391	—	59	1	no	—
192	#389	—	59	1	no	—
193	#389	—	59	1	no	—
194	#387	—	118	1	no	—
195	#387	—	118	1	no	—
196	C₄×C₄⋊C₄	—	59	1	no	—
197	C₄×C₄⋊C₄	—	118	1	no	—
198	#376	—	118	1	no	—
199	C₄³	C₂×D₅₉	1	1	yes	characteristic
200	#336	—	118	1	no	—
201	Q₈⋊C₈	—	118	4	no	—
202	C₅₉	#8568	1	1	yes	characteristic, complemented, Sylow
203	C₂²×Q₈	—	59	1	no	—
204	C₂²×C₈	—	2	2	no	—
205	C₂²×C₈	—	118	2	no	—
206	C₄×Q₈	—	59	2	no	—
207	C₄×Q₈	—	59	2	no	—
208	C₄×Q₈	—	118	1	no	—
209	C₄×Q₈	—	118	1	no	—
210	C₂×C₄⋊C₄	—	59	1	no	—
211	C₂×C₄⋊C₄	—	59	1	no	—
212	C₂×C₄⋊C₄	—	59	1	no	—
213	C₂×C₄⋊C₄	—	118	1	no	—
214	C₂×C₄⋊C₄	—	118	1	no	—
215	C₂×C₄⋊C₄	—	118	1	no	—
216	C₂×C₄²	—	118	1	no	—
217	C₂×C₄²	—	118	1	no	—
218	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
219	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
220	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
221	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
222	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
223	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
224	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
225	C₄⋊C₈	—	2	2	no	—
226	C₄⋊C₈	—	2	2	no	—
227	C₄⋊C₈	—	4	2	no	—
228	C₄⋊C₈	—	4	2	no	—
229	C₄⋊C₈	—	118	4	no	—
230	C₄×C₈	—	118	2	no	—
231	C₄×C₈	—	118	2	no	—
232	#95	—	118	1	no	—
233	#95	—	118	1	no	—
234	#95	—	118	1	no	—
235	#95	—	118	1	no	—
236	#95	—	118	1	no	—
237	#95	—	118	1	no	—
238	#95	—	118	1	no	—
239	C₂×Q₈	—	59	2	no	—
240	C₂×Q₈	—	118	1	no	—
241	C₂×Q₈	—	118	2	no	—
242	C₂×Q₈	—	118	2	no	—
243	C₂²×C₄	—	118	1	no	—
244	C₂²×C₄	—	118	1	no	—
245	C₂²×C₄	—	118	1	no	—
246	C₂²×C₄	—	118	1	no	—
247	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
248	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
249	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
250	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
251	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
254	C₂×C₈	—	4	2	no	—
255	C₂×C₈	—	4	2	no	—
256	C₂×C₈	—	4	2	no	—
257	C₂×C₈	—	118	4	no	—
258	C₂×C₈	—	236	2	no	—
259	C₂×C₈	—	236	2	no	—
260	C₄⋊C₄	—	59	2	no	—
261	C₄⋊C₄	—	118	1	no	—
262	C₄⋊C₄	—	118	2	no	—
263	C₄⋊C₄	—	118	2	no	—
264	C₄⋊C₄	—	118	2	no	—
265	C₄⋊C₄	—	118	2	no	—
266	C₄⋊C₄	—	236	2	no	—
267	C₄⋊C₄	—	236	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	2	1	no	—
273	C₄²	—	4	1	no	—
274	C₄²	—	118	2	no	—
275	C₄²	—	118	2	no	—
276	C₄²	—	236	1	no	—
277	C₄²	—	236	1	no	—
278	C₄²	#11745670	1	2	yes	—
279	C₄²	#11745672	1	2	yes	—
280	C₄²	D₄×D₅₉	1	1	yes	characteristic
281	C₄²	#11745687	1	1	yes	characteristic
282	C₄²	Q₈×D₅₉	1	1	yes	characteristic
283	C₄²	#11745689	1	1	yes	characteristic
284	C₂³	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
285	Q₈	—	118	4	no	—
286	Q₈	—	236	2	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	118	2	no	—
299	C₂×C₄	—	118	2	no	—
300	C₂×C₄	—	236	1	no	—
301	C₂×C₄	—	236	1	no	—
302	C₂×C₄	—	236	1	no	—
303	C₂×C₄	—	236	1	no	—
304	C₂×C₄	—	236	1	no	—
305	C₂×C₄	—	236	1	no	—
306	C₂×C₄	—	236	1	no	—
307	C₂×C₄	—	236	1	no	—
308	C₂×C₄	—	236	1	no	—
309	C₂×C₄	—	236	1	no	—
310	C₂×C₄	#14273650	1	2	yes	—
311	C₂×C₄	#14273650	1	2	yes	—
312	C₂×C₄	#14273653	1	2	yes	—
313	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
314	C₂×C₄	#14273685	1	1	yes	characteristic
315	C₂×C₄	#14273698	1	1	yes	characteristic
316	C₂×C₄	#14273702	1	1	yes	characteristic
317	C₂×C₄	#14273703	1	1	yes	characteristic
318	C₂×C₄	#14273707	1	1	yes	characteristic
319	C₂×C₄	#14273733	1	2	yes	—
320	C₂×C₄	#14273735	1	2	yes	—
321	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
322	C₂×C₄	C₂×#11745672	1	1	yes	characteristic
323	C₂×C₄	#14273753	1	1	yes	characteristic
324	C₂×C₄	#14273758	1	1	yes	characteristic
325	C₈	—	8	2	no	—
326	C₈	—	236	4	no	—
327	C₂²	#16837259	1	1	yes	characteristic
328	C₂²	#16837368	1	1	yes	characteristic
329	C₂²	#16837370	1	1	yes	characteristic
330	C₂²	C₂×#14273650	1	1	yes	characteristic
331	C₂²	#16837401	1	1	yes	characteristic
332	C₂²	C₂×#14273653	1	1	yes	characteristic
333	C₂²	#16837418	1	1	yes	characteristic
334	C₄	—	2	2	no	—
335	C₄	—	2	2	no	—
336	C₄	—	2	2	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	472	1	no	—
341	C₄	—	472	1	no	—
342	C₄	—	236	2	no	—
343	C₄	—	236	2	no	—
344	C₄	#16837427	1	2	yes	—
345	C₄	#16837462	1	2	yes	—
346	C₄	#16837599	1	1	yes	characteristic
347	C₄	#16837609	1	1	yes	characteristic
348	C₄	#16837684	1	1	yes	characteristic
349	C₄	#16837694	1	1	yes	characteristic
350	C₂	#21422498	1	1	yes	characteristic
351	C₂	#21422507	1	1	yes	characteristic
352	C₂	#21422550	1	1	yes	characteristic
353	C₂	#21422559	1	1	yes	characteristic
354	C₂	#21424077	1	1	yes	characteristic
355	C₂	#21424098	1	1	yes	characteristic
356	C₂	#21424113	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₉
commutator	C₂²×C₄×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1842	C₂
socle, prime radical	C₂³×C₅₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1842	D₅₉
hypocenter	C₅₉	#8568
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
202. C₅₉	#8568	semidirect

Maximal subgroups

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

subgroup	index
#21426573	2
C₄×C₄⋊(C₅₉⋊C₈)	2
C₂×#16837132	2
#21421935	2
C₅₉×#1842	2
#8568	59