Subgroups — #28481521

Order 15104.

598 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈×D₅₉	C₂	1	1	yes	maximal, characteristic
2	#21424073	C₂	1	2	yes	maximal
3	C₂×#16837153	C₂	1	1	yes	maximal, characteristic, complemented
4	#21421930	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₉×#334	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₅₉	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₅₉	C₄	1	1	yes	characteristic
8	C₂×C₄²×D₅₉	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
10	C₂×(C₂×D₅₉)⋊C₈	—	2	2	no	—
11	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₅₉	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₅₉	C₄	1	1	yes	characteristic
14	C₄×C₈×D₅₉	—	2	2	no	—
15	C₄×C₈×D₅₉	—	2	2	no	—
16	#16837156	—	2	2	no	—
17	#16837153	—	2	2	no	—
18	C₂×C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₅₉×#334	—	2	2	no	—
21	C₂²×C₄×D₅₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×D₅₉	D₄	1	1	yes	characteristic
23	C₂²×C₄×D₅₉	Q₈	1	1	yes	characteristic
24	C₂²×C₅₉⋊C₈	—	2	2	no	—
25	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×D₅₉	—	2	2	no	—
28	C₂×C₈×D₅₉	—	2	2	no	—
29	C₂×C₈×D₅₉	—	2	2	no	—
30	C₂×C₈×D₅₉	—	2	2	no	—
31	C₂×C₈×D₅₉	—	2	2	no	—
32	C₂×C₈×D₅₉	C₈	1	4	yes	complemented
33	C₂×C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×Dic₅₉	D₄	1	1	yes	characteristic
35	C₂×C₄×Dic₅₉	Q₈	1	1	yes	characteristic
36	C₄²×D₅₉	—	2	1	no	—
37	C₄²×D₅₉	—	2	1	no	—
38	C₄²×D₅₉	—	2	1	no	—
39	C₄²×D₅₉	—	2	1	no	—
40	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
41	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
42	(C₂×D₅₉)⋊C₈	—	4	4	no	—
43	C₄×C₅₉⋊C₈	—	2	1	no	—
44	C₄×C₅₉⋊C₈	—	2	1	no	—
45	C₈×Dic₅₉	—	2	2	no	—
46	C₈×Dic₅₉	—	2	2	no	—
47	C₂²×C₈×C₅₉	—	2	2	no	—
48	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
51	C₄⋊C₈×C₅₉	—	2	2	no	—
52	C₄⋊C₈×C₅₉	—	2	2	no	—
53	C₄×C₈×C₅₉	D₄	1	2	yes	—
54	C₄×C₈×C₅₉	D₄	1	2	yes	—
55	C₂³×D₅₉	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
59	C₂²×Dic₅₉	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×Dic₅₉	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×Dic₅₉	C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×D₅₉	—	2	2	no	—
63	C₂×C₄×D₅₉	—	2	1	no	—
64	C₂×C₄×D₅₉	—	2	1	no	—
65	C₂×C₄×D₅₉	—	2	1	no	—
66	C₂×C₄×D₅₉	—	2	1	no	—
67	C₂×C₄×D₅₉	—	2	1	no	—
68	C₂×C₄×D₅₉	—	2	1	no	—
69	C₂×C₄×D₅₉	—	2	1	no	—
70	C₂×C₄×D₅₉	—	2	1	no	—
71	C₂×C₄×D₅₉	—	2	2	no	—
72	C₂×C₄×D₅₉	—	2	2	no	—
73	C₂×C₄×D₅₉	C₂×C₈	1	2	yes	—
74	C₂×C₄×D₅₉	M₁₆	1	2	yes	—
75	C₂×C₄×D₅₉	SD₁₆	1	4	yes	—
76	C₂×C₈×C₅₉	—	2	1	no	—
77	C₂×C₈×C₅₉	—	2	1	no	—
78	C₂×C₈×C₅₉	—	4	2	no	—
79	C₂×C₈×C₅₉	—	4	2	no	—
80	C₂×C₈×C₅₉	—	4	2	no	—
81	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
83	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
84	C₂×C₈×C₅₉	M₁₆	1	2	yes	—
85	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₅₉	#41	1	1	yes	characteristic
88	C₄²×C₅₉	#41	1	1	yes	characteristic
89	C₄×Dic₅₉	—	2	1	no	—
90	C₄×Dic₅₉	—	2	1	no	—
91	C₄×Dic₅₉	—	2	1	no	—
92	C₄×Dic₅₉	—	2	1	no	—
93	C₄×Dic₅₉	—	2	1	no	—
94	C₄×Dic₅₉	—	2	1	no	—
95	C₂×C₅₉⋊C₈	—	2	1	no	—
96	C₂×C₅₉⋊C₈	—	2	1	no	—
97	C₂×C₅₉⋊C₈	—	2	1	no	—
98	C₂×C₅₉⋊C₈	—	2	1	no	—
99	C₂×C₅₉⋊C₈	—	4	2	no	—
100	C₂×C₅₉⋊C₈	—	4	2	no	—
101	C₂×C₅₉⋊C₈	—	4	2	no	—
102	C₂×C₅₉⋊C₈	C₂×C₈	1	2	yes	—
103	C₂×C₅₉⋊C₈	M₁₆	1	2	yes	—
104	C₈×D₅₉	—	2	8	no	—
105	C₈×D₅₉	—	2	8	no	—
106	C₂³×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂²×D₅₉	—	2	1	no	—
108	C₂²×D₅₉	—	2	1	no	—
109	C₂²×D₅₉	—	2	1	no	—
110	C₂²×D₅₉	—	2	1	no	—
111	C₂²×D₅₉	C₄⋊C₈	1	2	yes	—
112	C₂²×D₅₉	#106	1	2	yes	—
113	C₂²×D₅₉	#108	1	2	yes	—
114	C₂×C₄×C₅₉	—	2	1	no	—
115	C₂×C₄×C₅₉	—	2	1	no	—
116	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
118	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
119	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
120	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
124	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
125	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
126	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
127	C₂×C₄×C₅₉	#135	1	2	yes	—
128	C₂×Dic₅₉	—	2	1	no	—
129	C₂×Dic₅₉	—	2	1	no	—
130	C₂×Dic₅₉	—	2	1	no	—
131	C₂×Dic₅₉	—	2	1	no	—
132	C₂×Dic₅₉	—	2	1	no	—
133	C₂×Dic₅₉	—	2	1	no	—
134	C₂×Dic₅₉	C₄⋊C₈	1	2	yes	—
135	C₂×Dic₅₉	#106	1	2	yes	—
136	C₂×Dic₅₉	#108	1	2	yes	—
137	C₄×D₅₉	—	2	4	no	—
138	C₄×D₅₉	—	2	4	no	—
139	C₄×D₅₉	—	2	2	no	—
140	C₄×D₅₉	—	2	2	no	—
141	C₄×D₅₉	—	2	2	no	—
142	C₄×D₅₉	—	2	2	no	—
143	C₄×D₅₉	—	2	4	no	—
144	C₄×D₅₉	—	2	4	no	—
145	C₈×C₅₉	—	2	2	no	—
146	C₈×C₅₉	—	8	2	no	—
147	C₈×C₅₉	C₂²⋊C₈	1	4	yes	complemented
148	C₅₉⋊C₈	—	2	2	no	—
149	C₅₉⋊C₈	—	2	2	no	—
150	C₅₉⋊C₈	—	8	2	no	—
151	#8595	—	59	1	no	maximal, Sylow
152	C₂²×C₅₉	#380	1	1	yes	characteristic
153	C₂²×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
154	C₂²×C₅₉	#423	1	1	yes	characteristic
155	C₂²×C₅₉	C₂×#106	1	1	yes	characteristic
156	C₂²×C₅₉	#427	1	1	yes	characteristic
157	C₂²×C₅₉	C₂×#108	1	1	yes	characteristic
158	C₂²×C₅₉	C₂×#108	1	1	yes	characteristic
159	C₂×D₅₉	—	2	2	no	—
160	C₂×D₅₉	—	2	2	no	—
161	C₂×D₅₉	—	2	2	no	—
162	C₂×D₅₉	—	2	2	no	—
163	C₂×D₅₉	—	2	2	no	—
164	C₂×D₅₉	—	2	2	no	—
165	C₂×D₅₉	#334	1	4	yes	complemented
166	C₄×C₅₉	—	2	1	no	—
167	C₄×C₅₉	—	2	1	no	—
168	C₄×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
169	C₄×C₅₉	#407	1	1	yes	characteristic
170	C₄×C₅₉	#431	1	1	yes	characteristic
171	C₄×C₅₉	#432	1	1	yes	characteristic
172	C₄×C₅₉	#465	1	2	yes	—
173	C₄×C₅₉	#465	1	2	yes	—
174	Dic₅₉	—	2	2	no	—
175	Dic₅₉	—	2	2	no	—
176	Dic₅₉	—	2	2	no	—
177	C₂²×C₄×C₈	—	59	1	no	—
178	#1797	—	59	2	no	—
179	C₂×#334	—	59	1	no	—
180	C₂×#334	D₅₉	1	1	yes	characteristic, complemented
181	#1277	—	59	2	no	—
182	C₂×C₅₉	C₂×#334	1	1	yes	characteristic
183	C₂×C₅₉	#1564	1	1	yes	characteristic
184	C₂×C₅₉	#1566	1	1	yes	characteristic
185	C₂×C₅₉	#1568	1	1	yes	characteristic
186	C₂×C₅₉	#1797	1	1	yes	characteristic
187	C₂×C₅₉	#1807	1	1	yes	characteristic
188	C₂×C₅₉	#1810	1	1	yes	characteristic
189	D₅₉	—	2	4	no	—
190	C₂³×C₈	—	59	1	no	—
191	C₂³×C₈	—	59	1	no	—
192	C₂²×C₄²	—	59	1	no	—
193	C₂×C₄⋊C₈	—	59	2	no	—
194	C₂×C₄⋊C₈	C₂×D₅₉	1	2	yes	—
195	C₂×C₂²⋊C₈	—	118	2	no	—
196	C₂×C₄×C₈	—	59	1	no	—
197	C₂×C₄×C₈	—	59	1	no	—
198	C₂×C₄×C₈	—	59	1	no	—
199	C₂×C₄×C₈	—	118	2	no	—
200	C₂×C₄×C₈	—	118	2	no	—
201	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
202	#336	—	118	2	no	—
203	#334	—	2	2	no	—
204	#334	—	118	2	no	—
205	C₅₉	#8595	1	1	yes	characteristic, complemented, Sylow
206	C₂³×C₄	—	59	1	no	—
207	C₂³×C₄	—	59	1	no	—
208	C₂³×C₄	—	59	1	no	—
209	C₂²×C₈	—	2	2	no	—
210	C₂²×C₈	—	59	1	no	—
211	C₂²×C₈	—	59	1	no	—
212	C₂²×C₈	—	59	4	no	—
213	C₂²×C₈	—	118	2	no	—
214	C₂²×C₈	—	118	2	no	—
215	C₂²×C₈	—	118	2	no	—
216	C₂²×C₈	—	118	2	no	—
217	C₂²×C₈	—	118	2	no	—
218	C₂²×C₈	—	118	2	no	—
219	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
220	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
221	C₂×C₄²	—	59	1	no	—
222	C₂×C₄²	—	59	1	no	—
223	C₂×C₄²	—	59	1	no	—
224	C₂×C₄²	—	118	1	no	—
225	C₂×C₄²	—	118	1	no	—
226	C₂×C₄²	—	118	1	no	—
227	C₂×C₄²	—	118	1	no	—
228	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
229	C₄⋊C₈	—	2	2	no	—
230	C₄⋊C₈	—	2	2	no	—
231	C₄⋊C₈	—	118	2	no	—
232	C₄⋊C₈	—	118	2	no	—
233	C₂²⋊C₈	—	236	4	no	—
234	C₄×C₈	—	118	2	no	—
235	C₄×C₈	—	118	2	no	—
236	C₄×C₈	—	118	1	no	—
237	C₄×C₈	—	118	1	no	—
238	C₄×C₈	D₂₃₆	1	2	yes	—
239	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
240	C₂⁴	—	59	1	no	—
241	C₂²×C₄	—	59	1	no	—
242	C₂²×C₄	—	59	1	no	—
243	C₂²×C₄	—	59	1	no	—
244	C₂²×C₄	—	59	2	no	—
245	C₂²×C₄	—	59	2	no	—
246	C₂²×C₄	—	59	4	no	—
247	C₂²×C₄	—	118	2	no	—
248	C₂²×C₄	—	118	2	no	—
249	C₂²×C₄	—	118	1	no	—
250	C₂²×C₄	—	118	1	no	—
251	C₂²×C₄	—	118	1	no	—
252	C₂²×C₄	—	118	1	no	—
253	C₂²×C₄	—	118	1	no	—
254	C₂²×C₄	—	118	1	no	—
255	C₂²×C₄	—	118	1	no	—
256	C₂²×C₄	—	118	1	no	—
257	C₂²×C₄	—	118	2	no	—
258	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
259	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
260	C₂²×C₄	Q₈×D₅₉	1	1	yes	characteristic
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	4	2	no	—
264	C₂×C₈	—	4	2	no	—
265	C₂×C₈	—	4	2	no	—
266	C₂×C₈	—	59	2	no	—
267	C₂×C₈	—	59	2	no	—
268	C₂×C₈	—	118	8	no	—
269	C₂×C₈	—	118	1	no	—
270	C₂×C₈	—	118	1	no	—
271	C₂×C₈	—	118	1	no	—
272	C₂×C₈	—	118	1	no	—
273	C₂×C₈	—	118	8	no	—
274	C₂×C₈	—	236	2	no	—
275	C₂×C₈	—	236	2	no	—
276	C₂×C₈	—	236	2	no	—
277	C₂×C₈	C₈×D₅₉	1	2	yes	—
278	C₂×C₈	#11745659	1	2	yes	—
279	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
280	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
281	C₄²	—	118	1	no	—
282	C₄²	—	118	1	no	—
283	C₄²	—	118	1	no	—
284	C₄²	—	118	1	no	—
285	C₄²	—	118	1	no	—
286	C₄²	—	118	1	no	—
287	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
288	C₄²	#11745687	1	1	yes	characteristic
289	C₄²	#11745689	1	1	yes	characteristic
290	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
291	C₂³	—	59	2	no	—
292	C₂³	—	59	2	no	—
293	C₂³	—	59	2	no	—
294	C₂³	—	118	1	no	—
295	C₂³	—	118	1	no	—
296	C₂³	—	118	1	no	—
297	C₂³	—	118	1	no	—
298	C₂³	C₄⋊C₄×D₅₉	1	1	yes	characteristic
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	59	2	no	—
302	C₂×C₄	—	59	2	no	—
303	C₂×C₄	—	59	2	no	—
304	C₂×C₄	—	118	4	no	—
305	C₂×C₄	—	118	4	no	—
306	C₂×C₄	—	118	1	no	—
307	C₂×C₄	—	118	1	no	—
308	C₂×C₄	—	118	1	no	—
309	C₂×C₄	—	118	1	no	—
310	C₂×C₄	—	118	1	no	—
311	C₂×C₄	—	118	1	no	—
312	C₂×C₄	—	118	4	no	—
313	C₂×C₄	—	118	2	no	—
314	C₂×C₄	—	118	2	no	—
315	C₂×C₄	—	118	2	no	—
316	C₂×C₄	—	118	2	no	—
317	C₂×C₄	—	118	4	no	—
318	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
319	C₂×C₄	#14273685	1	1	yes	characteristic
320	C₂×C₄	#14273698	1	1	yes	characteristic
321	C₂×C₄	#14273702	1	1	yes	characteristic
322	C₂×C₄	#14273703	1	1	yes	characteristic
323	C₂×C₄	#14273707	1	1	yes	characteristic
324	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
325	C₂×C₄	M₁₆×D₅₉	1	1	yes	characteristic
326	C₂×C₄	M₁₆×D₅₉	1	1	yes	characteristic
327	C₂×C₄	C₂×#11745659	1	1	yes	characteristic
328	C₂×C₄	SD₁₆×D₅₉	1	2	yes	—
329	C₂×C₄	#14273748	1	2	yes	—
330	C₈	—	2	2	no	—
331	C₈	—	8	2	no	—
332	C₈	—	118	2	no	—
333	C₈	—	118	2	no	—
334	C₈	—	472	2	no	—
335	C₈	(C₂×D₅₉)⋊C₈	1	4	yes	complemented
336	C₂²	—	59	4	no	—
337	C₂²	—	118	2	no	—
338	C₂²	—	118	2	no	—
339	C₂²	—	118	2	no	—
340	C₂²	—	118	2	no	—
341	C₂²	—	118	2	no	—
342	C₂²	—	118	2	no	—
343	C₂²	#16837259	1	1	yes	characteristic
344	C₂²	C₄⋊C₈×D₅₉	1	1	yes	characteristic
345	C₂²	#16837424	1	1	yes	characteristic
346	C₂²	#106×D₅₉	1	1	yes	characteristic
347	C₂²	#16837437	1	1	yes	characteristic
348	C₂²	#108×D₅₉	1	1	yes	characteristic
349	C₂²	#108×D₅₉	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	118	2	no	—
353	C₄	—	118	2	no	—
354	C₄	—	118	2	no	—
355	C₄	C₂×(C₂×D₅₉)⋊C₈	1	1	yes	characteristic
356	C₄	#16837365	1	1	yes	characteristic
357	C₄	#16837454	1	1	yes	characteristic
358	C₄	#16837458	1	1	yes	characteristic
359	C₄	#16837628	1	2	yes	—
360	C₄	#16837632	1	2	yes	—
361	C₂	—	118	4	no	—
362	C₂	#334×D₅₉	1	1	yes	characteristic
363	C₂	#21422841	1	1	yes	characteristic
364	C₂	#21422850	1	1	yes	characteristic
365	C₂	#21422856	1	1	yes	characteristic
366	C₂	#21424073	1	1	yes	characteristic
367	C₂	#21424118	1	1	yes	characteristic
368	C₂	#21424131	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₅₉×#334	C₂
socle	C₂³×C₅₉	C₂×C₄⋊C₄
prime radical	C₂³×D₅₉	C₄⋊C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×#334	D₅₉
hypocenter	C₅₉	#8595
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
3. C₂×#16837153	C₂	semidirect
4. #21421930	C₂	semidirect
5. C₂×C₅₉×#334	C₂	semidirect
32. C₂×C₈×D₅₉	C₈	semidirect
147. C₈×C₅₉	C₂²⋊C₈	semidirect
165. C₂×D₅₉	#334	semidirect
180. C₂×#334	D₅₉	semidirect
205. C₅₉	#8595	semidirect
335. C₈	(C₂×D₅₉)⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₈×D₅₉	2
#21424073	2
C₂×#16837153	2
#21421930	2
C₂×C₅₉×#334	2
#8595	59