Subgroups — #28481408

Order 15104.

472 conjugacy classes of subgroups.

The subgroup lattice has 340 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	#21424213	C₂	1	1	yes	maximal, characteristic
3	C₂×#16837134	C₂	1	2	yes	maximal
4	#21421936	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
8	#16837308	C₄	1	1	yes	characteristic
9	#16837308	C₄	1	1	yes	characteristic
10	#16837300	C₄	1	1	yes	characteristic
11	#16837300	C₄	1	1	yes	characteristic
12	#16837286	—	2	1	no	—
13	#16837286	—	2	1	no	—
14	C₄×Dic₅₉⋊C₄	—	2	1	no	—
15	C₄×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
16	#16837248	—	2	1	no	—
17	#16837157	—	2	1	no	—
18	#16837157	—	2	1	no	—
19	#16837134	—	2	4	no	—
20	C₂×C₄⋊C₈×C₅₉	—	2	1	no	—
21	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
22	C₄³×C₅₉	C₂²	1	1	yes	characteristic
23	C₅₉×#336	—	2	1	no	—
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	2	no	—
39	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
40	C₅₉⋊#95	—	2	1	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	C₂²×C₈×C₅₉	—	2	2	no	—
48	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₉	D₄	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	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₅₉	—	2	4	no	—
56	C₄×C₈×C₅₉	—	2	2	no	—
57	C₄×C₈×C₅₉	—	2	2	no	—
58	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×Dic₅₉	—	2	1	no	—
66	C₂²×Dic₅₉	—	2	1	no	—
67	C₂²×Dic₅₉	—	2	1	no	—
68	C₂²×Dic₅₉	—	2	1	no	—
69	C₂×Dic₁₁₈	—	2	2	no	—
70	C₂×Dic₁₁₈	—	2	2	no	—
71	C₂×Dic₁₁₈	—	2	1	no	—
72	C₂×Dic₁₁₈	C₄⋊C₄	1	2	yes	—
73	C₂×C₈×C₅₉	—	2	4	no	—
74	C₂×C₈×C₅₉	—	4	2	no	—
75	C₂×C₈×C₅₉	—	4	2	no	—
76	C₄²×C₅₉	—	2	2	no	—
77	C₄²×C₅₉	—	2	2	no	—
78	C₄²×C₅₉	—	2	2	no	—
79	C₄²×C₅₉	—	2	2	no	—
80	C₄²×C₅₉	—	2	2	no	—
81	C₄²×C₅₉	—	4	1	no	—
82	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
84	C₄²×C₅₉	#41	1	1	yes	characteristic
85	C₄²×C₅₉	#41	1	1	yes	characteristic
86	C₄⋊Dic₅₉	—	2	1	no	—
87	C₄⋊Dic₅₉	—	2	2	no	—
88	C₄⋊Dic₅₉	—	2	2	no	—
89	C₄⋊Dic₅₉	C₄⋊C₄	1	2	yes	—
90	Dic₅₉⋊C₄	—	2	2	no	—
91	Dic₅₉⋊C₄	—	2	2	no	—
92	Dic₅₉⋊C₄	—	4	2	no	—
93	Dic₅₉⋊C₄	—	4	2	no	—
94	C₄×Dic₅₉	—	2	2	no	—
95	C₄×Dic₅₉	—	2	2	no	—
96	C₄×Dic₅₉	—	4	2	no	—
97	C₂×C₅₉⋊C₈	—	4	2	no	—
98	C₂×C₅₉⋊C₈	—	4	2	no	—
99	C₂×C₅₉⋊C₈	—	4	2	no	—
100	C₂³×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₉	—	2	1	no	—
102	C₂×C₄×C₅₉	—	2	1	no	—
103	C₂×C₄×C₅₉	—	2	1	no	—
104	C₂×C₄×C₅₉	—	2	1	no	—
105	C₂×C₄×C₅₉	—	2	1	no	—
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	#104	1	2	yes	—
115	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
117	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
118	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
119	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₄×C₅₉	#131	1	2	yes	—
122	C₂×C₄×C₅₉	#131	1	2	yes	—
123	C₂×C₄×C₅₉	#135	1	2	yes	—
124	C₂×C₄×C₅₉	#137	1	2	yes	—
125	C₂×Dic₅₉	—	2	2	no	—
126	C₂×Dic₅₉	—	2	2	no	—
127	C₂×Dic₅₉	—	4	1	no	—
128	C₂×Dic₅₉	—	4	1	no	—
129	C₂×Dic₅₉	—	4	1	no	—
130	C₂×Dic₅₉	—	4	1	no	—
131	C₂×Dic₅₉	—	4	1	no	—
132	C₂×Dic₅₉	—	4	1	no	—
133	C₂×Dic₅₉	—	4	1	no	—
134	C₂×Dic₅₉	—	4	1	no	—
135	C₂×Dic₅₉	—	4	1	no	—
136	C₂×Dic₅₉	—	4	1	no	—
137	Dic₁₁₈	—	2	4	no	—
138	Dic₁₁₈	—	4	2	no	—
139	C₈×C₅₉	—	4	4	no	—
140	C₅₉⋊C₈	—	8	2	no	—
141	#8570	—	59	1	no	maximal, Sylow
142	C₂²×C₅₉	#380	1	1	yes	characteristic
143	C₂²×C₅₉	#408	1	1	yes	characteristic
144	C₂²×C₅₉	#408	1	1	yes	characteristic
145	C₂²×C₅₉	#416	1	1	yes	characteristic
146	C₂²×C₅₉	#419	1	1	yes	characteristic
147	C₂²×C₅₉	C₂×#104	1	1	yes	characteristic
148	C₂²×C₅₉	#421	1	1	yes	characteristic
149	C₄×C₅₉	—	2	1	no	—
150	C₄×C₅₉	—	2	1	no	—
151	C₄×C₅₉	—	2	2	no	—
152	C₄×C₅₉	—	4	1	no	—
153	C₄×C₅₉	—	4	1	no	—
154	C₄×C₅₉	—	4	1	no	—
155	C₄×C₅₉	#424	1	2	yes	—
156	C₄×C₅₉	#433	1	2	yes	—
157	C₄×C₅₉	#464	1	2	yes	—
158	C₄×C₅₉	#479	1	2	yes	—
159	Dic₅₉	—	4	2	no	—
160	Dic₅₉	—	4	2	no	—
161	Dic₅₉	—	8	1	no	—
162	Dic₅₉	—	8	1	no	—
163	#2295	—	59	1	no	—
164	#1831	—	59	1	no	—
165	C₄×C₄⋊C₈	D₅₉	1	1	yes	characteristic
166	C₂×Q₈⋊C₈	—	59	2	no	—
167	#1279	—	59	2	no	—
168	C₂×C₅₉	#1482	1	1	yes	characteristic
169	C₂×C₅₉	#1485	1	1	yes	characteristic
170	C₂×C₅₉	#1494	1	1	yes	characteristic
171	C₂×C₅₉	#1497	1	1	yes	characteristic
172	C₂×C₅₉	#1798	1	1	yes	characteristic
173	C₂×C₅₉	#1805	1	1	yes	characteristic
174	C₂×C₅₉	#1806	1	1	yes	characteristic
175	C₂×C₄×Q₈	—	59	1	no	—
176	C₂×C₄⋊C₈	—	2	1	no	—
177	C₂×C₄⋊C₈	—	59	2	no	—
178	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
179	#391	—	59	1	no	—
180	#391	—	59	1	no	—
181	#389	—	59	1	no	—
182	#389	—	59	1	no	—
183	#387	—	118	1	no	—
184	#387	—	118	1	no	—
185	C₄×C₄⋊C₄	—	59	1	no	—
186	C₄×C₄⋊C₄	—	118	1	no	—
187	#376	—	118	1	no	—
188	C₄³	C₂×D₅₉	1	1	yes	characteristic
189	#336	—	2	1	no	—
190	#336	—	118	1	no	—
191	#336	—	118	1	no	—
192	Q₈⋊C₈	—	118	4	no	—
193	C₅₉	#8570	1	1	yes	characteristic, complemented, Sylow
194	C₂²×Q₈	—	59	1	no	—
195	C₂²×C₈	—	2	2	no	—
196	C₂²×C₈	—	118	2	no	—
197	C₄×Q₈	—	59	2	no	—
198	C₄×Q₈	—	59	2	no	—
199	C₄×Q₈	—	118	1	no	—
200	C₄×Q₈	—	118	1	no	—
201	C₂×C₄⋊C₄	—	59	1	no	—
202	C₂×C₄⋊C₄	—	59	1	no	—
203	C₂×C₄⋊C₄	—	59	1	no	—
204	C₂×C₄⋊C₄	—	118	1	no	—
205	C₂×C₄⋊C₄	—	118	1	no	—
206	C₂×C₄⋊C₄	—	118	1	no	—
207	C₂×C₄²	—	118	1	no	—
208	C₂×C₄²	—	118	1	no	—
209	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
210	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
211	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
212	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
213	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
214	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
215	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
216	C₄⋊C₈	—	2	4	no	—
217	C₄⋊C₈	—	118	2	no	—
218	C₄⋊C₈	—	118	2	no	—
219	C₄×C₈	—	2	2	no	—
220	C₄×C₈	—	2	2	no	—
221	#95	—	118	1	no	—
222	#95	—	118	1	no	—
223	#95	—	118	1	no	—
224	#95	—	118	1	no	—
225	#95	—	118	1	no	—
226	#95	—	118	1	no	—
227	#95	—	118	1	no	—
228	C₂×Q₈	—	59	2	no	—
229	C₂×Q₈	—	118	1	no	—
230	C₂×Q₈	—	118	2	no	—
231	C₂×Q₈	—	118	2	no	—
232	C₂²×C₄	—	118	1	no	—
233	C₂²×C₄	—	118	1	no	—
234	C₂²×C₄	—	118	1	no	—
235	C₂²×C₄	—	118	1	no	—
236	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
237	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
238	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
239	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	236	2	no	—
247	C₂×C₈	—	236	2	no	—
248	C₂×C₈	—	236	2	no	—
249	C₄⋊C₄	—	59	2	no	—
250	C₄⋊C₄	—	118	1	no	—
251	C₄⋊C₄	—	118	2	no	—
252	C₄⋊C₄	—	118	2	no	—
253	C₄⋊C₄	—	118	2	no	—
254	C₄⋊C₄	—	118	2	no	—
255	C₄⋊C₄	—	236	2	no	—
256	C₄⋊C₄	—	236	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	4	1	no	—
263	C₄²	—	118	2	no	—
264	C₄²	—	118	2	no	—
265	C₄²	—	236	2	no	—
266	C₄²	D₄×D₅₉	1	1	yes	characteristic
267	C₄²	#11745687	1	1	yes	characteristic
268	C₄²	Q₈×D₅₉	1	1	yes	characteristic
269	C₄²	#11745689	1	1	yes	characteristic
270	C₂³	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
271	Q₈	—	118	4	no	—
272	Q₈	—	236	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	118	2	no	—
287	C₂×C₄	—	118	2	no	—
288	C₂×C₄	—	236	1	no	—
289	C₂×C₄	—	236	1	no	—
290	C₂×C₄	—	236	1	no	—
291	C₂×C₄	—	236	1	no	—
292	C₂×C₄	—	236	1	no	—
293	C₂×C₄	—	236	1	no	—
294	C₂×C₄	—	236	1	no	—
295	C₂×C₄	—	236	1	no	—
296	C₂×C₄	—	236	1	no	—
297	C₂×C₄	—	236	1	no	—
298	C₂×C₄	#14273655	1	2	yes	—
299	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
300	C₂×C₄	#14273685	1	1	yes	characteristic
301	C₂×C₄	#14273698	1	1	yes	characteristic
302	C₂×C₄	#14273702	1	1	yes	characteristic
303	C₂×C₄	#14273703	1	1	yes	characteristic
304	C₂×C₄	#14273707	1	1	yes	characteristic
305	C₂×C₄	#14273733	1	2	yes	—
306	C₂×C₄	#14273735	1	2	yes	—
307	C₂×C₄	#14273749	1	2	yes	—
308	C₂×C₄	#14273760	1	2	yes	—
309	C₈	—	4	4	no	—
310	C₈	—	472	2	no	—
311	C₂²	#16837259	1	1	yes	characteristic
312	C₂²	#16837368	1	1	yes	characteristic
313	C₂²	#16837370	1	1	yes	characteristic
314	C₂²	#16837399	1	1	yes	characteristic
315	C₂²	#16837412	1	1	yes	characteristic
316	C₂²	C₂×#14273655	1	1	yes	characteristic
317	C₂²	#16837420	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	472	1	no	—
325	C₄	—	472	1	no	—
326	C₄	—	236	2	no	—
327	C₄	—	236	2	no	—
328	C₄	#16837427	1	2	yes	—
329	C₄	#16837462	1	2	yes	—
330	C₄	#16837625	1	2	yes	—
331	C₄	#16837710	1	2	yes	—
332	C₂	#21422505	1	1	yes	characteristic
333	C₂	#21422518	1	1	yes	characteristic
334	C₂	#21422557	1	1	yes	characteristic
335	C₂	#21422570	1	1	yes	characteristic
336	C₂	#21424077	1	1	yes	characteristic
337	C₂	#21424112	1	1	yes	characteristic
338	C₂	#21424117	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₃₆
commutator	C₂²×C₄×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₄×C₄⋊C₈×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₄×C₄⋊C₈	D₅₉
hypocenter	C₅₉	#8570
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
193. C₅₉	#8570	semidirect

Maximal subgroups

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

subgroup	index
#21426573	2
#21424213	2
C₂×#16837134	2
#21421936	2
C₄×C₄⋊C₈×C₅₉	2
#8570	59