Subgroups — C₅₉×#33965 #28446074

Order 15104.

498 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#3395	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₉×#3393	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#2251	C₂	1	2	yes	maximal
4	C₅₉×#2247	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#2211	C₂	1	2	yes	maximal
6	C₅₉×#2209	C₂	1	2	yes	maximal, complemented
7	C₅₉×#2209	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₉×#2208	C₂	1	1	yes	maximal, characteristic
9	C₅₉×#2183	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₉×#2176	C₂	1	2	yes	maximal, complemented
11	C₄×SD₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic
12	C₅₉×#557	—	2	1	no	—
13	C₅₉×#556	—	2	1	no	—
14	C₂×Q₃₂×C₅₉	C₂²	1	2	yes	—
15	C₂×Q₃₂×C₅₉	C₂²	1	2	yes	—
16	C₂×Q₃₂×C₅₉	C₂²	1	2	yes	—
17	C₂×SD₃₂×C₅₉	C₂²	1	1	yes	characteristic
18	C₂×SD₃₂×C₅₉	C₂²	1	2	yes	—
19	C₅₉×#495	C₂²	1	2	yes	—
20	C₅₉×#494	C₂²	1	1	yes	characteristic
21	C₅₉×#494	C₂²	1	1	yes	characteristic
22	C₅₉×#494	C₂²	1	2	yes	—
23	C₅₉×#464	—	2	2	no	—
24	C₅₉×#462	—	2	1	no	—
25	C₅₉×#462	—	2	2	no	—
26	C₅₉×#461	—	2	1	no	—
27	C₈×Q₈×C₅₉	C₂²	1	1	yes	characteristic
28	C₄×Q₁₆×C₅₉	C₂²	1	2	yes	—
29	C₄×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
30	C₄×Q₁₆×C₅₉	C₂²	1	2	yes	—
31	C₄×D₈×C₅₉	C₂²	1	1	yes	characteristic
32	C₅₉×#367	C₂²	1	1	yes	characteristic
33	C₅₉×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₅₉×C₄⋊C₁₆	C₂²	1	2	yes	—
35	C₅₉×#358	C₂²	1	1	yes	characteristic
36	C₅₉×#358	C₂²	1	2	yes	—
37	C₅₉×#358	C₂²	1	2	yes	—
38	C₅₉×#357	C₂²	1	1	yes	characteristic
39	C₅₉×#346	C₂²	1	2	yes	—
40	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
41	C₂×Q₁₆×C₅₉	—	2	1	no	—
42	C₂×Q₁₆×C₅₉	—	2	2	no	—
43	C₂×Q₁₆×C₅₉	—	2	1	no	—
44	C₂×Q₁₆×C₅₉	C₂³	1	1	yes	characteristic
45	C₂×Q₁₆×C₅₉	C₂³	1	2	yes	—
46	C₂×Q₁₆×C₅₉	C₂³	1	2	yes	—
47	C₂×SD₁₆×C₅₉	—	2	2	no	—
48	C₂×D₈×C₅₉	C₂³	1	1	yes	characteristic
49	C₄⋊Q₈×C₅₉	—	2	1	no	—
50	C₄⋊Q₈×C₅₉	—	2	1	no	—
51	C₄⋊Q₈×C₅₉	—	2	2	no	—
52	#126×C₅₉	—	4	2	no	—
53	#125×C₅₉	—	4	1	no	—
54	#125×C₅₉	—	4	2	no	—
55	#124×C₅₉	—	2	2	no	—
56	C₂²⋊Q₈×C₅₉	—	4	1	no	—
57	C₄×Q₈×C₅₉	—	2	1	no	—
58	C₄×Q₈×C₅₉	—	2	2	no	—
59	C₄×Q₈×C₅₉	—	2	2	no	—
60	C₄×Q₈×C₅₉	—	4	1	no	—
61	C₄×Q₈×C₅₉	D₄	1	1	yes	characteristic
62	C₄×D₄×C₅₉	—	2	1	no	—
63	#117×C₅₉	—	4	1	no	—
64	Q₃₂×C₅₉	—	2	4	no	—
65	Q₃₂×C₅₉	—	4	2	no	—
66	Q₃₂×C₅₉	—	4	2	no	—
67	SD₃₂×C₅₉	—	2	2	no	—
68	SD₃₂×C₅₉	—	4	2	no	—
69	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
70	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
71	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
72	#107×C₅₉	C₂³	1	1	yes	characteristic
73	#107×C₅₉	C₂³	1	1	yes	characteristic
74	C₄⋊C₈×C₅₉	D₄	1	1	yes	characteristic
75	C₄⋊C₈×C₅₉	D₄	1	2	yes	—
76	#103×C₅₉	—	2	2	no	—
77	#103×C₅₉	—	2	1	no	—
78	#103×C₅₉	—	2	2	no	—
79	#102×C₅₉	—	2	1	no	—
80	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₅₉	C₂³	1	2	yes	—
82	C₂×Q₈×C₅₉	—	2	2	no	—
83	C₂×Q₈×C₅₉	—	2	1	no	—
84	C₂×Q₈×C₅₉	—	2	2	no	—
85	C₂×Q₈×C₅₉	D₈	1	1	yes	characteristic
86	C₂×D₄×C₅₉	—	2	1	no	—
87	C₂²×C₄×C₅₉	—	4	1	no	—
88	Q₁₆×C₅₉	—	2	1	no	—
89	Q₁₆×C₅₉	—	2	2	no	—
90	Q₁₆×C₅₉	—	2	2	no	—
91	Q₁₆×C₅₉	—	2	2	no	—
92	Q₁₆×C₅₉	—	2	1	no	—
93	Q₁₆×C₅₉	—	2	2	no	—
94	Q₁₆×C₅₉	—	8	2	no	—
95	Q₁₆×C₅₉	—	8	1	no	—
96	Q₁₆×C₅₉	—	8	1	no	—
97	SD₁₆×C₅₉	—	8	2	no	—
98	D₈×C₅₉	—	2	1	no	—
99	D₈×C₅₉	—	2	1	no	—
100	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄×C₅₉	—	2	1	no	—
104	C₄⋊C₄×C₅₉	—	2	1	no	—
105	C₄⋊C₄×C₅₉	—	4	1	no	—
106	C₄⋊C₄×C₅₉	—	4	2	no	—
107	C₄⋊C₄×C₅₉	—	4	2	no	—
108	C₄⋊C₄×C₅₉	—	4	1	no	—
109	C₄⋊C₄×C₅₉	—	4	2	no	—
110	C₄⋊C₄×C₅₉	—	4	1	no	—
111	C₄⋊C₄×C₅₉	—	4	2	no	—
112	C₄⋊C₄×C₅₉	—	4	1	no	—
113	C₄⋊C₄×C₅₉	—	4	1	no	—
114	C₄⋊C₄×C₅₉	D₈	1	1	yes	characteristic
115	C₄⋊C₄×C₅₉	D₈	1	2	yes	—
116	C₂²⋊C₄×C₅₉	—	4	2	no	—
117	C₂²⋊C₄×C₅₉	—	4	1	no	—
118	C₂²⋊C₄×C₅₉	—	4	2	no	—
119	C₄²×C₅₉	—	4	2	no	—
120	C₄²×C₅₉	—	4	2	no	—
121	C₄²×C₅₉	—	4	1	no	—
122	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₅₉	C₂×D₄	1	2	yes	—
124	C₁₆×C₅₉	—	2	2	no	—
125	C₁₆×C₅₉	—	2	1	no	—
126	C₁₆×C₅₉	#41	1	2	yes	—
127	C₂³×C₅₉	—	4	1	no	—
128	Q₈×C₅₉	—	2	2	no	—
129	Q₈×C₅₉	—	4	1	no	—
130	Q₈×C₅₉	—	4	2	no	—
131	Q₈×C₅₉	—	4	2	no	—
132	Q₈×C₅₉	—	4	2	no	—
133	Q₈×C₅₉	—	4	1	no	—
134	Q₈×C₅₉	—	4	2	no	—
135	D₄×C₅₉	—	4	1	no	—
136	D₄×C₅₉	—	4	1	no	—
137	C₂×C₄×C₅₉	—	4	1	no	—
138	C₂×C₄×C₅₉	—	4	1	no	—
139	C₂×C₄×C₅₉	—	4	2	no	—
140	C₂×C₄×C₅₉	—	4	2	no	—
141	C₂×C₄×C₅₉	—	4	1	no	—
142	C₂×C₄×C₅₉	—	8	2	no	—
143	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
144	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
145	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
148	C₈×C₅₉	—	2	2	no	—
149	C₈×C₅₉	—	2	1	no	—
150	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
151	C₈×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
152	#33965	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₂²×C₅₉	—	8	1	no	—
154	C₂²×C₅₉	—	8	1	no	—
155	C₂²×C₅₉	—	8	1	no	—
156	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
157	C₄×C₅₉	—	2	2	no	—
158	C₄×C₅₉	—	2	2	no	—
159	C₄×C₅₉	—	2	1	no	—
160	C₄×C₅₉	—	8	2	no	—
161	C₄×C₅₉	—	8	1	no	—
162	C₄×C₅₉	—	8	2	no	—
163	C₄×C₅₉	—	8	1	no	—
164	C₄×C₅₉	—	8	1	no	—
165	C₄×C₅₉	#510	1	2	yes	—
166	C₄×C₅₉	#550	1	1	yes	characteristic
167	C₄×C₅₉	#576	1	1	yes	characteristic
168	#3395	C₂×C₅₉	1	1	yes	characteristic, complemented
169	#3393	C₂×C₅₉	1	1	yes	characteristic
170	#2251	C₂×C₅₉	1	2	yes	—
171	#2247	C₂×C₅₉	1	1	yes	characteristic, complemented
172	#2211	C₂×C₅₉	1	2	yes	—
173	#2209	C₂×C₅₉	1	2	yes	complemented
174	#2209	C₂×C₅₉	1	1	yes	characteristic, complemented
175	#2208	C₂×C₅₉	1	1	yes	characteristic
176	#2183	C₂×C₅₉	1	1	yes	characteristic, complemented
177	#2176	C₂×C₅₉	1	2	yes	complemented
178	C₄×SD₃₂	C₂×C₅₉	1	1	yes	characteristic
179	C₂×C₅₉	—	16	1	no	—
180	C₂×C₅₉	#3391	1	1	yes	characteristic
181	C₂×C₅₉	C₂×#510	1	1	yes	characteristic
182	C₂×C₅₉	#3417	1	1	yes	characteristic
183	#557	—	2	1	no	—
184	#556	—	2	1	no	—
185	C₂×Q₃₂	C₂²×C₅₉	1	2	yes	—
186	C₂×Q₃₂	C₂²×C₅₉	1	2	yes	—
187	C₂×Q₃₂	C₂²×C₅₉	1	2	yes	—
188	C₂×SD₃₂	C₂²×C₅₉	1	1	yes	characteristic
189	C₂×SD₃₂	C₂²×C₅₉	1	2	yes	—
190	#495	C₂²×C₅₉	1	2	yes	—
191	#494	C₂²×C₅₉	1	1	yes	characteristic
192	#494	C₂²×C₅₉	1	1	yes	characteristic
193	#494	C₂²×C₅₉	1	2	yes	—
194	#464	—	2	2	no	—
195	#462	—	2	1	no	—
196	#462	—	2	2	no	—
197	#461	—	2	1	no	—
198	C₈×Q₈	C₂²×C₅₉	1	1	yes	characteristic
199	C₄×Q₁₆	C₂²×C₅₉	1	2	yes	—
200	C₄×Q₁₆	C₂²×C₅₉	1	1	yes	characteristic
201	C₄×Q₁₆	C₂²×C₅₉	1	2	yes	—
202	C₄×D₈	C₂²×C₅₉	1	1	yes	characteristic
203	#367	C₂²×C₅₉	1	1	yes	characteristic
204	C₄⋊C₁₆	C₂²×C₅₉	1	1	yes	characteristic
205	C₄⋊C₁₆	C₂²×C₅₉	1	2	yes	—
206	#358	C₂²×C₅₉	1	1	yes	characteristic
207	#358	C₂²×C₅₉	1	2	yes	—
208	#358	C₂²×C₅₉	1	2	yes	—
209	#357	C₂²×C₅₉	1	1	yes	characteristic
210	#346	C₂²×C₅₉	1	2	yes	—
211	C₄×C₁₆	C₂²×C₅₉	1	1	yes	characteristic
212	C₅₉	#33965	1	1	yes	characteristic, direct product factor, Sylow
213	C₂×Q₁₆	—	2	1	no	—
214	C₂×Q₁₆	—	2	2	no	—
215	C₂×Q₁₆	—	2	1	no	—
216	C₂×Q₁₆	C₂³×C₅₉	1	1	yes	characteristic
217	C₂×Q₁₆	C₂³×C₅₉	1	2	yes	—
218	C₂×Q₁₆	C₂³×C₅₉	1	2	yes	—
219	C₂×SD₁₆	—	2	2	no	—
220	C₂×D₈	C₂³×C₅₉	1	1	yes	characteristic
221	C₄⋊Q₈	—	2	1	no	—
222	C₄⋊Q₈	—	2	1	no	—
223	C₄⋊Q₈	—	2	2	no	—
224	#126	—	4	2	no	—
225	#125	—	4	1	no	—
226	#125	—	4	2	no	—
227	#124	—	2	2	no	—
228	C₂²⋊Q₈	—	4	1	no	—
229	C₄×Q₈	—	2	2	no	—
230	C₄×Q₈	—	2	1	no	—
231	C₄×Q₈	—	2	2	no	—
232	C₄×Q₈	—	4	1	no	—
233	C₄×Q₈	D₄×C₅₉	1	1	yes	characteristic
234	C₄×D₄	—	2	1	no	—
235	#117	—	4	1	no	—
236	Q₃₂	—	2	4	no	—
237	Q₃₂	—	4	2	no	—
238	Q₃₂	—	4	2	no	—
239	SD₃₂	—	2	2	no	—
240	SD₃₂	—	4	2	no	—
241	C₂×C₁₆	C₂³×C₅₉	1	1	yes	characteristic
242	C₂×C₁₆	C₂³×C₅₉	1	2	yes	—
243	C₂×C₁₆	C₂³×C₅₉	1	1	yes	characteristic
244	#107	C₂³×C₅₉	1	1	yes	characteristic
245	#107	C₂³×C₅₉	1	1	yes	characteristic
246	C₄⋊C₈	D₄×C₅₉	1	1	yes	characteristic
247	C₄⋊C₈	D₄×C₅₉	1	2	yes	—
248	#103	—	2	2	no	—
249	#103	—	2	1	no	—
250	#103	—	2	2	no	—
251	#102	—	2	1	no	—
252	C₄×C₈	C₂³×C₅₉	1	1	yes	characteristic
253	C₄×C₈	C₂³×C₅₉	1	2	yes	—
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	—	2	1	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	D₈×C₅₉	1	1	yes	characteristic
258	C₂×D₄	—	2	1	no	—
259	C₂²×C₄	—	4	1	no	—
260	Q₁₆	—	2	1	no	—
261	Q₁₆	—	2	2	no	—
262	Q₁₆	—	2	2	no	—
263	Q₁₆	—	2	2	no	—
264	Q₁₆	—	2	1	no	—
265	Q₁₆	—	2	2	no	—
266	Q₁₆	—	8	2	no	—
267	Q₁₆	—	8	1	no	—
268	Q₁₆	—	8	1	no	—
269	SD₁₆	—	8	2	no	—
270	D₈	—	2	1	no	—
271	D₈	—	2	1	no	—
272	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
273	C₂×C₈	C₂×D₄×C₅₉	1	2	yes	—
274	C₂×C₈	C₂⁴×C₅₉	1	1	yes	characteristic
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	2	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	2	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄⋊C₄	—	4	2	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₄⋊C₄	D₈×C₅₉	1	1	yes	characteristic
287	C₄⋊C₄	D₈×C₅₉	1	2	yes	—
288	C₂²⋊C₄	—	4	2	no	—
289	C₂²⋊C₄	—	4	1	no	—
290	C₂²⋊C₄	—	4	2	no	—
291	C₄²	—	4	2	no	—
292	C₄²	—	4	2	no	—
293	C₄²	—	4	1	no	—
294	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
295	C₄²	C₂×D₄×C₅₉	1	2	yes	—
296	C₁₆	—	2	2	no	—
297	C₁₆	—	2	1	no	—
298	C₁₆	#41×C₅₉	1	2	yes	—
299	C₂³	—	4	1	no	—
300	Q₈	—	2	2	no	—
301	Q₈	—	4	1	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	4	2	no	—
304	Q₈	—	4	2	no	—
305	Q₈	—	4	1	no	—
306	Q₈	—	4	2	no	—
307	D₄	—	4	1	no	—
308	D₄	—	4	1	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	8	2	no	—
315	C₂×C₄	C₂×D₈×C₅₉	1	2	yes	—
316	C₂×C₄	C₂×D₈×C₅₉	1	2	yes	—
317	C₂×C₄	C₂×D₈×C₅₉	1	1	yes	characteristic
318	C₂×C₄	C₂×D₈×C₅₉	1	1	yes	characteristic
319	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
320	C₈	—	2	2	no	—
321	C₈	—	2	1	no	—
322	C₈	C₂×#41×C₅₉	1	1	yes	characteristic
323	C₈	H(5,𝔽₂)×C₅₉	1	1	yes	characteristic
324	C₂²	—	8	1	no	—
325	C₂²	—	8	1	no	—
326	C₂²	—	8	1	no	—
327	C₂²	C₂²×D₈×C₅₉	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	1	no	—
331	C₄	—	8	2	no	—
332	C₄	—	8	1	no	—
333	C₄	—	8	2	no	—
334	C₄	—	8	1	no	—
335	C₄	—	8	1	no	—
336	C₄	C₅₉×#510	1	2	yes	—
337	C₄	C₅₉×#550	1	1	yes	characteristic
338	C₄	C₅₉×#576	1	1	yes	characteristic
339	C₂	—	16	1	no	—
340	C₂	C₅₉×#3391	1	1	yes	characteristic
341	C₂	C₂×C₅₉×#510	1	1	yes	characteristic
342	C₂	C₅₉×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₉	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅₉
prime radical	C₂×D₈×C₅₉	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ C₄×Q₈×C₅₉ ◁ C₈×Q₈×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₉×#3395	C₂	semidirect
4. C₅₉×#2247	C₂	semidirect
6. C₅₉×#2209	C₂	semidirect
7. C₅₉×#2209	C₂	semidirect
9. C₅₉×#2183	C₂	semidirect
10. C₅₉×#2176	C₂	semidirect
152. #33965	C₅₉	direct (N × H)
168. #3395	C₂×C₅₉	semidirect
171. #2247	C₂×C₅₉	semidirect
173. #2209	C₂×C₅₉	semidirect
174. #2209	C₂×C₅₉	semidirect
176. #2183	C₂×C₅₉	semidirect
177. #2176	C₂×C₅₉	semidirect
212. C₅₉	#33965	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₉×#3395	2
C₅₉×#3393	2
C₅₉×#2251	2
C₅₉×#2247	2
C₅₉×#2211	2
C₅₉×#2209	2
C₅₉×#2208	2
C₅₉×#2183	2
C₅₉×#2176	2
C₄×SD₃₂×C₅₉	2
#33965	59

Subgroups — C59×#33965 #28446074

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#33965 #28446074