Subgroups — C₅₉×#33518 #28445627

Order 15104.

1124 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	C₂³×Q₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₉×#510	C₂	1	2	yes	maximal, complemented
3	C₅₉×#3050	C₂	1	1	yes	maximal, characteristic
4	C₅₉×#2188	C₂	1	4	yes	maximal, complemented
5	C₅₉×#2185	C₂	1	4	yes	maximal, complemented
6	C₅₉×#2141	C₂	1	2	yes	maximal, complemented
7	C₅₉×#2112	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×Q₈×C₅₉	—	2	1	no	—
9	C₂×#135×C₅₉	C₂²	1	1	yes	characteristic
10	C₂²×Q₁₆×C₅₉	—	2	8	no	—
11	C₂²×Q₁₆×C₅₉	—	2	1	no	—
12	C₂²×Q₁₆×C₅₉	—	2	2	no	—
13	C₂²×Q₁₆×C₅₉	C₂²	1	4	yes	—
14	C₂²×Q₁₆×C₅₉	C₂²	1	2	yes	—
15	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₅₉×#525	—	2	1	no	—
17	C₅₉×#510	—	2	8	no	—
18	C₂×Q₃₂×C₅₉	C₂²	1	4	yes	complemented
19	C₂×SD₃₂×C₅₉	C₂²	1	4	yes	complemented
20	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
21	C₅₉×#467	C₂²	1	2	yes	—
22	C₅₉×#466	C₂²	1	2	yes	—
23	C₅₉×#465	—	2	2	no	—
24	C₅₉×#427	C₂²	1	1	yes	characteristic
25	C₅₉×#416	—	2	1	no	—
26	C₅₉×#358	C₂²	1	8	yes	complemented
27	C₅₉×C₂²⋊C₁₆	C₂²	1	4	yes	—
28	C₂×#41×C₅₉	—	2	1	no	—
29	C₂²×Q₈×C₅₉	—	2	8	no	—
30	C₂²×Q₈×C₅₉	—	2	4	no	—
31	C₂²×Q₈×C₅₉	—	2	2	no	—
32	C₂²×Q₈×C₅₉	—	4	4	no	—
33	C₂²×Q₈×C₅₉	—	4	2	no	—
34	C₂²×Q₈×C₅₉	—	4	1	no	—
35	C₂³×C₄×C₅₉	—	4	1	no	—
36	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
37	#135×C₅₉	—	2	4	no	—
38	C₂×Q₁₆×C₅₉	—	2	8	no	—
39	C₂×Q₁₆×C₅₉	—	2	16	no	—
40	C₂×Q₁₆×C₅₉	—	2	16	no	—
41	C₂×Q₁₆×C₅₉	—	2	4	no	—
42	C₂×Q₁₆×C₅₉	—	2	2	no	—
43	C₂×Q₁₆×C₅₉	—	2	4	no	—
44	C₂×Q₁₆×C₅₉	D₄	1	8	yes	complemented
45	C₂×Q₁₆×C₅₉	C₂³	1	1	yes	characteristic
46	C₂×Q₁₆×C₅₉	C₂³	1	4	yes	—
47	C₂×SD₁₆×C₅₉	—	2	1	no	—
48	C₂×D₈×C₅₉	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₅₉	—	2	4	no	—
50	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₅₉	D₄	1	2	yes	—
52	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
53	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
54	#123×C₅₉	—	4	1	no	—
55	C₂²⋊Q₈×C₅₉	—	2	2	no	—
56	#121×C₅₉	—	2	2	no	—
57	#120×C₅₉	—	4	1	no	—
58	C₄×D₄×C₅₉	—	4	2	no	—
59	#117×C₅₉	—	2	1	no	—
60	Q₃₂×C₅₉	—	2	8	no	—
61	SD₃₂×C₅₉	—	2	8	no	—
62	M₃₂×C₅₉	—	2	4	no	—
63	C₂×C₁₆×C₅₉	C₂³	1	4	yes	—
64	#107×C₅₉	C₂³	1	2	yes	—
65	#106×C₅₉	—	2	1	no	—
66	#103×C₅₉	—	2	2	no	—
67	#102×C₅₉	—	2	2	no	—
68	C₂⁴×C₅₉	D₈	1	1	yes	characteristic
69	#41×C₅₉	—	4	2	no	—
70	#41×C₅₉	—	4	2	no	—
71	C₂×Q₈×C₅₉	—	2	16	no	—
72	C₂×Q₈×C₅₉	—	2	8	no	—
73	C₂×Q₈×C₅₉	—	2	4	no	—
74	C₂×Q₈×C₅₉	—	2	1	no	—
75	C₂×Q₈×C₅₉	—	4	8	no	—
76	C₂×Q₈×C₅₉	—	4	8	no	—
77	C₂×Q₈×C₅₉	—	4	16	no	—
78	C₂×Q₈×C₅₉	—	4	4	no	—
79	C₂×Q₈×C₅₉	—	4	2	no	—
80	C₂×Q₈×C₅₉	—	4	4	no	—
81	C₂×D₄×C₅₉	—	2	1	no	—
82	C₂×D₄×C₅₉	—	4	2	no	—
83	C₂×D₄×C₅₉	—	4	1	no	—
84	C₂²×C₄×C₅₉	—	2	2	no	—
85	C₂²×C₄×C₅₉	—	2	2	no	—
86	C₂²×C₄×C₅₉	—	4	8	no	—
87	C₂²×C₄×C₅₉	—	4	4	no	—
88	C₂²×C₄×C₅₉	—	4	2	no	—
89	C₂²×C₄×C₅₉	—	4	1	no	—
90	C₂²×C₄×C₅₉	D₈	1	2	yes	—
91	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
92	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
94	Q₁₆×C₅₉	—	2	16	no	—
95	Q₁₆×C₅₉	—	2	8	no	—
96	Q₁₆×C₅₉	—	2	4	no	—
97	Q₁₆×C₅₉	—	2	2	no	—
98	Q₁₆×C₅₉	C₂×D₄	1	8	yes	—
99	SD₁₆×C₅₉	—	4	2	no	—
100	D₈×C₅₉	—	2	2	no	—
101	C₂×C₈×C₅₉	—	2	2	no	—
102	C₂×C₈×C₅₉	—	2	4	no	—
103	C₂×C₈×C₅₉	—	2	4	no	—
104	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₅₉	C₂×D₄	1	4	yes	—
107	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
108	C₄⋊C₄×C₅₉	—	2	2	no	—
109	C₄⋊C₄×C₅₉	—	4	2	no	—
110	C₂²⋊C₄×C₅₉	—	4	1	no	—
111	C₂²⋊C₄×C₅₉	—	4	2	no	—
112	C₂²⋊C₄×C₅₉	—	4	2	no	—
113	C₄²×C₅₉	—	4	1	no	—
114	C₁₆×C₅₉	—	2	4	no	—
115	C₂³×C₅₉	—	2	2	no	—
116	C₂³×C₅₉	—	2	4	no	—
117	C₂³×C₅₉	—	4	1	no	—
118	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
119	C₂³×C₅₉	C₂×D₈	1	2	yes	—
120	Q₈×C₅₉	—	2	8	no	—
121	Q₈×C₅₉	—	4	16	no	—
122	Q₈×C₅₉	—	4	4	no	—
123	Q₈×C₅₉	—	4	8	no	—
124	Q₈×C₅₉	—	4	1	no	—
125	Q₈×C₅₉	—	4	1	no	—
126	D₄×C₅₉	—	4	1	no	—
127	D₄×C₅₉	—	4	1	no	—
128	D₄×C₅₉	—	8	2	no	—
129	D₄×C₅₉	—	8	4	no	—
130	C₂×C₄×C₅₉	—	2	2	no	—
131	C₂×C₄×C₅₉	—	2	2	no	—
132	C₂×C₄×C₅₉	—	2	2	no	—
133	C₂×C₄×C₅₉	—	2	4	no	—
134	C₂×C₄×C₅₉	—	4	16	no	—
135	C₂×C₄×C₅₉	—	4	4	no	—
136	C₂×C₄×C₅₉	—	4	8	no	—
137	C₂×C₄×C₅₉	—	4	2	no	—
138	C₂×C₄×C₅₉	—	4	1	no	—
139	C₂×C₄×C₅₉	—	8	2	no	—
140	C₂×C₄×C₅₉	#120	1	2	yes	—
141	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
142	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
144	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
145	C₈×C₅₉	—	2	2	no	—
146	C₈×C₅₉	#120	1	2	yes	—
147	C₈×C₅₉	C₂²×D₄	1	2	yes	—
148	#33518	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂²×C₅₉	—	2	8	no	—
150	C₂²×C₅₉	—	2	2	no	—
151	C₂²×C₅₉	—	2	2	no	—
152	C₂²×C₅₉	—	2	2	no	—
153	C₂²×C₅₉	—	8	2	no	—
154	C₂²×C₅₉	—	8	1	no	—
155	C₂²×C₅₉	#447	1	2	yes	—
156	C₂²×C₅₉	#510	1	4	yes	complemented
157	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
158	C₄×C₅₉	—	2	2	no	—
159	C₄×C₅₉	—	4	8	no	—
160	C₄×C₅₉	—	8	2	no	—
161	C₄×C₅₉	—	8	1	no	—
162	C₄×C₅₉	#447	1	2	yes	—
163	C₄×C₅₉	C₂×#120	1	1	yes	characteristic
164	C₄×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
165	C₂³×Q₁₆	C₂×C₅₉	1	1	yes	characteristic, complemented
166	C₂×#510	C₂×C₅₉	1	2	yes	complemented
167	#3050	C₂×C₅₉	1	1	yes	characteristic
168	#2188	C₂×C₅₉	1	4	yes	complemented
169	#2185	C₂×C₅₉	1	4	yes	complemented
170	#2141	C₂×C₅₉	1	2	yes	complemented
171	#2112	C₂×C₅₉	1	1	yes	characteristic, complemented
172	C₂×C₅₉	—	2	2	no	—
173	C₂×C₅₉	—	2	4	no	—
174	C₂×C₅₉	—	16	1	no	—
175	C₂×C₅₉	C₂×#447	1	1	yes	characteristic
176	C₂×C₅₉	C₂×#510	1	2	yes	—
177	C₂³×Q₈	—	2	1	no	—
178	C₂×#135	C₂²×C₅₉	1	1	yes	characteristic
179	C₂²×Q₁₆	—	2	8	no	—
180	C₂²×Q₁₆	—	2	1	no	—
181	C₂²×Q₁₆	—	2	2	no	—
182	C₂²×Q₁₆	C₂²×C₅₉	1	4	yes	—
183	C₂²×Q₁₆	C₂²×C₅₉	1	2	yes	—
184	C₂³×C₈	C₂²×C₅₉	1	1	yes	characteristic
185	#525	—	2	1	no	—
186	#510	—	2	8	no	—
187	C₂×Q₃₂	C₂²×C₅₉	1	4	yes	complemented
188	C₂×SD₃₂	C₂²×C₅₉	1	4	yes	complemented
189	C₂×M₃₂	C₂²×C₅₉	1	2	yes	—
190	#467	C₂²×C₅₉	1	2	yes	—
191	#466	C₂²×C₅₉	1	2	yes	—
192	#465	—	2	2	no	—
193	#427	C₂²×C₅₉	1	1	yes	characteristic
194	#416	—	2	1	no	—
195	#358	C₂²×C₅₉	1	8	yes	complemented
196	C₂²⋊C₁₆	C₂²×C₅₉	1	4	yes	—
197	C₅₉	#33518	1	1	yes	characteristic, direct product factor, Sylow
198	C₂×#41	—	2	1	no	—
199	C₂²×Q₈	—	2	8	no	—
200	C₂²×Q₈	—	2	4	no	—
201	C₂²×Q₈	—	2	2	no	—
202	C₂²×Q₈	—	4	4	no	—
203	C₂²×Q₈	—	4	2	no	—
204	C₂²×Q₈	—	4	1	no	—
205	C₂³×C₄	—	4	1	no	—
206	C₂³×C₄	D₄×C₅₉	1	1	yes	characteristic
207	#135	—	2	4	no	—
208	C₂×Q₁₆	—	2	8	no	—
209	C₂×Q₁₆	—	2	16	no	—
210	C₂×Q₁₆	—	2	16	no	—
211	C₂×Q₁₆	—	2	4	no	—
212	C₂×Q₁₆	—	2	2	no	—
213	C₂×Q₁₆	—	2	4	no	—
214	C₂×Q₁₆	D₄×C₅₉	1	8	yes	complemented
215	C₂×Q₁₆	C₂³×C₅₉	1	1	yes	characteristic
216	C₂×Q₁₆	C₂³×C₅₉	1	4	yes	—
217	C₂×SD₁₆	—	2	1	no	—
218	C₂×D₈	C₂³×C₅₉	1	1	yes	characteristic
219	C₂²×C₈	—	2	4	no	—
220	C₂²×C₈	D₄×C₅₉	1	1	yes	characteristic
221	C₂²×C₈	D₄×C₅₉	1	2	yes	—
222	C₂²×C₈	C₂³×C₅₉	1	2	yes	—
223	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
224	#123	—	4	1	no	—
225	C₂²⋊Q₈	—	2	2	no	—
226	#121	—	2	2	no	—
227	#120	—	4	1	no	—
228	C₄×D₄	—	4	2	no	—
229	#117	—	2	1	no	—
230	Q₃₂	—	2	8	no	—
231	SD₃₂	—	2	8	no	—
232	M₃₂	—	2	4	no	—
233	C₂×C₁₆	C₂³×C₅₉	1	4	yes	—
234	#107	C₂³×C₅₉	1	2	yes	—
235	#106	—	2	1	no	—
236	#103	—	2	2	no	—
237	#102	—	2	2	no	—
238	C₂⁴	D₈×C₅₉	1	1	yes	characteristic
239	#41	—	4	2	no	—
240	#41	—	4	2	no	—
241	C₂×Q₈	—	2	16	no	—
242	C₂×Q₈	—	2	8	no	—
243	C₂×Q₈	—	2	4	no	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	—	4	8	no	—
246	C₂×Q₈	—	4	16	no	—
247	C₂×Q₈	—	4	8	no	—
248	C₂×Q₈	—	4	4	no	—
249	C₂×Q₈	—	4	2	no	—
250	C₂×Q₈	—	4	4	no	—
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	—	4	2	no	—
253	C₂×D₄	—	4	1	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	4	8	no	—
257	C₂²×C₄	—	4	4	no	—
258	C₂²×C₄	—	4	2	no	—
259	C₂²×C₄	—	4	1	no	—
260	C₂²×C₄	D₈×C₅₉	1	2	yes	—
261	C₂²×C₄	D₈×C₅₉	1	1	yes	characteristic
262	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
263	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
264	Q₁₆	—	2	16	no	—
265	Q₁₆	—	2	8	no	—
266	Q₁₆	—	2	4	no	—
267	Q₁₆	—	2	2	no	—
268	Q₁₆	C₂×D₄×C₅₉	1	8	yes	—
269	SD₁₆	—	4	2	no	—
270	D₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	4	no	—
273	C₂×C₈	—	2	4	no	—
274	C₂×C₈	C₂×D₄×C₅₉	1	2	yes	—
275	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
276	C₂×C₈	C₂×D₄×C₅₉	1	4	yes	—
277	C₂×C₈	C₂⁴×C₅₉	1	1	yes	characteristic
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	4	2	no	—
280	C₂²⋊C₄	—	4	1	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₄²	—	4	1	no	—
284	C₁₆	—	2	4	no	—
285	C₂³	—	2	2	no	—
286	C₂³	—	2	4	no	—
287	C₂³	—	4	1	no	—
288	C₂³	C₂×D₈×C₅₉	1	1	yes	characteristic
289	C₂³	C₂×D₈×C₅₉	1	2	yes	—
290	Q₈	—	2	8	no	—
291	Q₈	—	4	16	no	—
292	Q₈	—	4	8	no	—
293	Q₈	—	4	4	no	—
294	Q₈	—	4	1	no	—
295	Q₈	—	4	1	no	—
296	D₄	—	4	1	no	—
297	D₄	—	4	1	no	—
298	D₄	—	8	2	no	—
299	D₄	—	8	4	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	4	16	no	—
305	C₂×C₄	—	4	8	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	8	2	no	—
310	C₂×C₄	#120×C₅₉	1	2	yes	—
311	C₂×C₄	C₂×D₈×C₅₉	1	2	yes	—
312	C₂×C₄	C₂×D₈×C₅₉	1	1	yes	characteristic
313	C₂×C₄	AGL(1,ℤ/8)×C₅₉	1	2	yes	—
314	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
315	C₈	—	2	2	no	—
316	C₈	#120×C₅₉	1	2	yes	—
317	C₈	C₂²×D₄×C₅₉	1	2	yes	—
318	C₂²	—	2	8	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	8	1	no	—
323	C₂²	—	8	2	no	—
324	C₂²	C₅₉×#447	1	2	yes	—
325	C₂²	C₅₉×#510	1	4	yes	complemented
326	C₂²	C₂²×D₈×C₅₉	1	1	yes	characteristic
327	C₄	—	2	2	no	—
328	C₄	—	4	8	no	—
329	C₄	—	8	1	no	—
330	C₄	—	8	2	no	—
331	C₄	C₅₉×#447	1	2	yes	—
332	C₄	C₂×#120×C₅₉	1	1	yes	characteristic
333	C₄	C₂×AGL(1,ℤ/8)×C₅₉	1	1	yes	characteristic
334	C₂	—	2	2	no	—
335	C₂	—	2	4	no	—
336	C₂	—	16	1	no	—
337	C₂	C₂×C₅₉×#447	1	1	yes	characteristic
338	C₂	C₂×C₅₉×#510	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₉	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅₉
prime radical	C₅₉×#3050	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ C₂³×C₄×C₅₉ ◁ C₂³×C₈×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×Q₁₆×C₅₉	C₂	semidirect
2. C₂×C₅₉×#510	C₂	semidirect
4. C₅₉×#2188	C₂	semidirect
5. C₅₉×#2185	C₂	semidirect
6. C₅₉×#2141	C₂	semidirect
7. C₅₉×#2112	C₂	semidirect
18. C₂×Q₃₂×C₅₉	C₂²	semidirect
19. C₂×SD₃₂×C₅₉	C₂²	semidirect
26. C₅₉×#358	C₂²	semidirect
44. C₂×Q₁₆×C₅₉	D₄	semidirect
148. #33518	C₅₉	direct (N × H)
156. C₂²×C₅₉	#510	semidirect
165. C₂³×Q₁₆	C₂×C₅₉	semidirect
166. C₂×#510	C₂×C₅₉	semidirect
168. #2188	C₂×C₅₉	semidirect
169. #2185	C₂×C₅₉	semidirect
170. #2141	C₂×C₅₉	semidirect
171. #2112	C₂×C₅₉	semidirect
187. C₂×Q₃₂	C₂²×C₅₉	semidirect
188. C₂×SD₃₂	C₂²×C₅₉	semidirect
195. #358	C₂²×C₅₉	semidirect
197. C₅₉	#33518	direct (N × H)
214. C₂×Q₁₆	D₄×C₅₉	semidirect
325. C₂²	C₅₉×#510	semidirect

Maximal subgroups

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

subgroup	index
C₂³×Q₁₆×C₅₉	2
C₂×C₅₉×#510	2
C₅₉×#3050	2
C₅₉×#2188	2
C₅₉×#2185	2
C₅₉×#2141	2
C₅₉×#2112	2
#33518	59

Subgroups — C59×#33518 #28445627

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#33518 #28445627