Subgroups — C₅₉×#11048 #28423157

Order 15104.

850 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	C₅₉×#2905	C₂	1	1	yes	maximal, characteristic
2	C₂²×#101×C₅₉	C₂	1	2	yes	maximal
3	C₅₉×#2310	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₉×#344	C₂	1	2	yes	maximal
5	C₅₉×#1742	C₂	1	1	yes	maximal, characteristic
6	C₂³×Q₈×C₅₉	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₅₉	C₂²	1	2	yes	—
8	C₂²×M₁₆×C₅₉	C₂²	1	2	yes	—
9	C₂×#117×C₅₉	C₂²	1	1	yes	characteristic
10	C₂×#117×C₅₉	C₂²	1	1	yes	characteristic
11	C₅₉×#424	—	2	4	no	—
12	C₅₉×#423	—	2	2	no	—
13	C₂×#101×C₅₉	—	2	4	no	—
14	C₂×#101×C₅₉	—	2	4	no	—
15	C₂×#101×C₅₉	—	2	4	no	—
16	C₅₉×#391	—	2	2	no	—
17	C₅₉×#391	—	2	2	no	—
18	C₅₉×#381	C₄	1	4	yes	—
19	C₅₉×#344	—	2	4	no	—
20	C₅₉×#343	—	2	2	no	—
21	C₂²×Q₈×C₅₉	—	2	1	no	—
22	C₂²×Q₈×C₅₉	—	2	1	no	—
23	C₂²×Q₈×C₅₉	—	2	2	no	—
24	C₂²×Q₈×C₅₉	—	2	2	no	—
25	C₂²×Q₈×C₅₉	—	2	1	no	—
26	C₂²×Q₈×C₅₉	—	2	1	no	—
27	C₂²×Q₈×C₅₉	—	2	2	no	—
28	C₂²×Q₈×C₅₉	—	2	2	no	—
29	C₂²×Q₈×C₅₉	D₄	1	2	yes	—
30	C₂²×Q₈×C₅₉	Q₈	1	2	yes	—
31	C₂³×C₄×C₅₉	C₂×C₄	1	2	yes	—
32	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
33	C₂×M₁₆×C₅₉	—	2	2	no	—
34	C₂×M₁₆×C₅₉	—	2	2	no	—
35	C₂×M₁₆×C₅₉	—	2	2	no	—
36	C₂×M₁₆×C₅₉	—	2	2	no	—
37	C₂×M₁₆×C₅₉	—	2	2	no	—
38	C₂×M₁₆×C₅₉	—	2	2	no	—
39	C₂×M₁₆×C₅₉	—	2	2	no	—
40	C₂×M₁₆×C₅₉	—	2	2	no	—
41	C₂×M₁₆×C₅₉	—	2	2	no	—
42	C₂×M₁₆×C₅₉	—	2	2	no	—
43	C₂×M₁₆×C₅₉	D₄	1	4	yes	—
44	C₂×M₁₆×C₅₉	D₄	1	4	yes	—
45	C₂²×C₈×C₅₉	—	2	2	no	—
46	C₂²×C₈×C₅₉	—	2	2	no	—
47	#117×C₅₉	—	2	2	no	—
48	#117×C₅₉	—	2	2	no	—
49	#117×C₅₉	—	2	2	no	—
50	#117×C₅₉	—	2	2	no	—
51	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
52	C₂×C₄⋊C₄×C₅₉	D₄	1	2	yes	—
53	C₂×C₂²⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
54	C₂×C₂²⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₅₉	—	2	1	no	—
56	C₂×C₄²×C₅₉	D₄	1	2	yes	—
57	C₄⋊C₈×C₅₉	—	2	8	no	—
58	#101×C₅₉	—	2	16	no	—
59	#95×C₅₉	—	2	8	no	—
60	C₂⁴×C₅₉	C₂²×C₄	1	1	yes	characteristic
61	C₂×Q₈×C₅₉	—	2	4	no	—
62	C₂×Q₈×C₅₉	—	2	2	no	—
63	C₂×Q₈×C₅₉	—	2	2	no	—
64	C₂×Q₈×C₅₉	—	2	2	no	—
65	C₂×Q₈×C₅₉	—	2	2	no	—
66	C₂×Q₈×C₅₉	—	2	4	no	—
67	C₂×Q₈×C₅₉	—	2	4	no	—
68	C₂×Q₈×C₅₉	—	2	4	no	—
69	C₂×Q₈×C₅₉	—	4	1	no	—
70	C₂×Q₈×C₅₉	—	4	2	no	—
71	C₂×Q₈×C₅₉	—	4	2	no	—
72	C₂×Q₈×C₅₉	—	4	2	no	—
73	C₂×Q₈×C₅₉	—	4	1	no	—
74	C₂×Q₈×C₅₉	—	4	2	no	—
75	C₂×Q₈×C₅₉	—	4	2	no	—
76	C₂×Q₈×C₅₉	—	4	1	no	—
77	C₂×Q₈×C₅₉	—	4	2	no	—
78	C₂×Q₈×C₅₉	—	4	1	no	—
79	C₂²×C₄×C₅₉	—	2	2	no	—
80	C₂²×C₄×C₅₉	—	2	2	no	—
81	C₂²×C₄×C₅₉	—	2	4	no	—
82	C₂²×C₄×C₅₉	—	2	4	no	—
83	C₂²×C₄×C₅₉	—	2	2	no	—
84	C₂²×C₄×C₅₉	C₂²⋊C₄	1	4	yes	—
85	C₂²×C₄×C₅₉	C₄⋊C₄	1	4	yes	—
86	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
92	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
93	C₂²×C₄×C₅₉	#41	1	2	yes	—
94	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
95	C₂²×C₄×C₅₉	#41	1	2	yes	—
96	M₁₆×C₅₉	—	2	4	no	—
97	M₁₆×C₅₉	—	2	4	no	—
98	M₁₆×C₅₉	—	2	4	no	—
99	M₁₆×C₅₉	—	2	4	no	—
100	M₁₆×C₅₉	—	4	2	no	—
101	M₁₆×C₅₉	—	4	2	no	—
102	M₁₆×C₅₉	—	4	2	no	—
103	M₁₆×C₅₉	—	4	2	no	—
104	C₂×C₈×C₅₉	—	2	4	no	—
105	C₂×C₈×C₅₉	—	2	4	no	—
106	C₂×C₈×C₅₉	—	4	2	no	—
107	C₂×C₈×C₅₉	—	4	2	no	—
108	C₂×C₈×C₅₉	—	4	2	no	—
109	C₂×C₈×C₅₉	—	4	2	no	—
110	C₄⋊C₄×C₅₉	—	2	4	no	—
111	C₄⋊C₄×C₅₉	—	4	1	no	—
112	C₄⋊C₄×C₅₉	—	4	1	no	—
113	C₂²⋊C₄×C₅₉	—	2	4	no	—
114	C₂²⋊C₄×C₅₉	—	2	4	no	—
115	C₄²×C₅₉	—	2	2	no	—
116	C₄²×C₅₉	—	2	2	no	—
117	C₄²×C₅₉	—	4	2	no	—
118	C₂³×C₅₉	—	4	2	no	—
119	C₂³×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂³×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂³×C₅₉	#117	1	1	yes	characteristic
122	C₂³×C₅₉	C₄×D₄	1	2	yes	—
123	C₂³×C₅₉	C₄×D₄	1	2	yes	—
124	Q₈×C₅₉	—	4	2	no	—
125	Q₈×C₅₉	—	4	2	no	—
126	Q₈×C₅₉	—	4	4	no	—
127	Q₈×C₅₉	—	4	4	no	—
128	Q₈×C₅₉	—	4	2	no	—
129	Q₈×C₅₉	—	4	2	no	—
130	C₂×C₄×C₅₉	—	2	4	no	—
131	C₂×C₄×C₅₉	—	2	8	no	—
132	C₂×C₄×C₅₉	—	2	4	no	—
133	C₂×C₄×C₅₉	—	4	2	no	—
134	C₂×C₄×C₅₉	—	4	4	no	—
135	C₂×C₄×C₅₉	—	4	2	no	—
136	C₂×C₄×C₅₉	—	4	4	no	—
137	C₂×C₄×C₅₉	—	4	4	no	—
138	C₂×C₄×C₅₉	—	4	2	no	—
139	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
140	C₂×C₄×C₅₉	#120	1	2	yes	—
141	C₂×C₄×C₅₉	#121	1	2	yes	—
142	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
143	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
144	C₂×C₄×C₅₉	#121	1	2	yes	—
145	C₂×C₄×C₅₉	#121	1	2	yes	—
146	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
147	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
148	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
149	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
150	C₂×C₄×C₅₉	#123	1	2	yes	—
151	C₂×C₄×C₅₉	#123	1	2	yes	—
152	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
153	C₂×C₄×C₅₉	#124	1	2	yes	—
154	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
155	C₂×C₄×C₅₉	#126	1	2	yes	—
156	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
157	C₈×C₅₉	—	4	4	no	—
158	C₈×C₅₉	—	4	4	no	—
159	#11048	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂²×C₅₉	—	2	2	no	—
161	C₂²×C₅₉	—	2	4	no	—
162	C₂²×C₅₉	—	4	1	no	—
163	C₂²×C₅₉	—	4	1	no	—
164	C₂²×C₅₉	—	4	2	no	—
165	C₂²×C₅₉	#380	1	1	yes	characteristic
166	C₂²×C₅₉	#385	1	2	yes	—
167	C₂²×C₅₉	#386	1	1	yes	characteristic
168	C₂²×C₅₉	#388	1	2	yes	—
169	C₂²×C₅₉	#390	1	1	yes	characteristic
170	C₄×C₅₉	—	4	4	no	—
171	C₄×C₅₉	—	4	4	no	—
172	C₄×C₅₉	—	8	2	no	—
173	C₄×C₅₉	#392	1	1	yes	characteristic
174	C₄×C₅₉	#393	1	1	yes	characteristic
175	C₄×C₅₉	#394	1	2	yes	—
176	C₄×C₅₉	#396	1	2	yes	—
177	C₄×C₅₉	#397	1	1	yes	characteristic
178	C₄×C₅₉	#399	1	1	yes	characteristic
179	#2905	C₂×C₅₉	1	1	yes	characteristic
180	C₂²×#101	C₂×C₅₉	1	2	yes	—
181	#2310	C₂×C₅₉	1	1	yes	characteristic
182	C₂×#344	C₂×C₅₉	1	2	yes	—
183	#1742	C₂×C₅₉	1	1	yes	characteristic
184	C₂×C₅₉	—	2	1	no	—
185	C₂×C₅₉	—	2	1	no	—
186	C₂×C₅₉	—	2	2	no	—
187	C₂×C₅₉	—	2	2	no	—
188	C₂×C₅₉	#1439	1	1	yes	characteristic
189	C₂×C₅₉	#1915	1	2	yes	—
190	C₂³×Q₈	C₂²×C₅₉	1	1	yes	characteristic
191	C₂²×M₁₆	C₂²×C₅₉	1	2	yes	—
192	C₂²×M₁₆	C₂²×C₅₉	1	2	yes	—
193	C₂×#117	C₂²×C₅₉	1	1	yes	characteristic
194	C₂×#117	C₂²×C₅₉	1	1	yes	characteristic
195	#424	—	2	4	no	—
196	#423	—	2	2	no	—
197	C₂×#101	—	2	4	no	—
198	C₂×#101	—	2	4	no	—
199	C₂×#101	—	2	4	no	—
200	#391	—	2	2	no	—
201	#391	—	2	2	no	—
202	#381	C₄×C₅₉	1	4	yes	—
203	#344	—	2	4	no	—
204	#343	—	2	2	no	—
205	C₅₉	#11048	1	1	yes	characteristic, direct product factor, Sylow
206	C₂²×Q₈	—	2	1	no	—
207	C₂²×Q₈	—	2	1	no	—
208	C₂²×Q₈	—	2	2	no	—
209	C₂²×Q₈	—	2	2	no	—
210	C₂²×Q₈	—	2	1	no	—
211	C₂²×Q₈	—	2	1	no	—
212	C₂²×Q₈	—	2	2	no	—
213	C₂²×Q₈	—	2	2	no	—
214	C₂²×Q₈	D₄×C₅₉	1	2	yes	—
215	C₂²×Q₈	Q₈×C₅₉	1	2	yes	—
216	C₂³×C₄	C₂×C₄×C₅₉	1	2	yes	—
217	C₂³×C₄	C₂³×C₅₉	1	1	yes	characteristic
218	C₂×M₁₆	—	2	2	no	—
219	C₂×M₁₆	—	2	2	no	—
220	C₂×M₁₆	—	2	2	no	—
221	C₂×M₁₆	—	2	2	no	—
222	C₂×M₁₆	—	2	2	no	—
223	C₂×M₁₆	—	2	2	no	—
224	C₂×M₁₆	—	2	2	no	—
225	C₂×M₁₆	—	2	2	no	—
226	C₂×M₁₆	—	2	2	no	—
227	C₂×M₁₆	—	2	2	no	—
228	C₂×M₁₆	D₄×C₅₉	1	4	yes	—
229	C₂×M₁₆	D₄×C₅₉	1	4	yes	—
230	C₂²×C₈	—	2	2	no	—
231	C₂²×C₈	—	2	2	no	—
232	#117	—	2	2	no	—
233	#117	—	2	2	no	—
234	#117	—	2	2	no	—
235	#117	—	2	2	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	D₄×C₅₉	1	2	yes	—
238	C₂×C₂²⋊C₄	C₂×C₄×C₅₉	1	2	yes	—
239	C₂×C₂²⋊C₄	C₂×C₄×C₅₉	1	2	yes	—
240	C₂×C₄²	—	2	1	no	—
241	C₂×C₄²	D₄×C₅₉	1	2	yes	—
242	C₄⋊C₈	—	2	8	no	—
243	#101	—	2	16	no	—
244	#95	—	2	8	no	—
245	C₂⁴	C₂²×C₄×C₅₉	1	1	yes	characteristic
246	C₂×Q₈	—	2	4	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	2	4	no	—
252	C₂×Q₈	—	2	4	no	—
253	C₂×Q₈	—	2	4	no	—
254	C₂×Q₈	—	4	1	no	—
255	C₂×Q₈	—	4	2	no	—
256	C₂×Q₈	—	4	2	no	—
257	C₂×Q₈	—	4	2	no	—
258	C₂×Q₈	—	4	1	no	—
259	C₂×Q₈	—	4	2	no	—
260	C₂×Q₈	—	4	2	no	—
261	C₂×Q₈	—	4	1	no	—
262	C₂×Q₈	—	4	2	no	—
263	C₂×Q₈	—	4	1	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	—	2	4	no	—
267	C₂²×C₄	—	2	4	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	C₂²⋊C₄×C₅₉	1	4	yes	—
270	C₂²×C₄	C₄⋊C₄×C₅₉	1	4	yes	—
271	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
272	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
273	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
274	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
275	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
276	C₂²×C₄	C₂×Q₈×C₅₉	1	1	yes	characteristic
277	C₂²×C₄	#41×C₅₉	1	1	yes	characteristic
278	C₂²×C₄	#41×C₅₉	1	2	yes	—
279	C₂²×C₄	#41×C₅₉	1	1	yes	characteristic
280	C₂²×C₄	#41×C₅₉	1	2	yes	—
281	M₁₆	—	2	4	no	—
282	M₁₆	—	2	4	no	—
283	M₁₆	—	2	4	no	—
284	M₁₆	—	2	4	no	—
285	M₁₆	—	4	2	no	—
286	M₁₆	—	4	2	no	—
287	M₁₆	—	4	2	no	—
288	M₁₆	—	4	2	no	—
289	C₂×C₈	—	2	4	no	—
290	C₂×C₈	—	2	4	no	—
291	C₂×C₈	—	4	2	no	—
292	C₂×C₈	—	4	2	no	—
293	C₂×C₈	—	4	2	no	—
294	C₂×C₈	—	4	2	no	—
295	C₄⋊C₄	—	2	4	no	—
296	C₄⋊C₄	—	4	1	no	—
297	C₄⋊C₄	—	4	1	no	—
298	C₂²⋊C₄	—	2	4	no	—
299	C₂²⋊C₄	—	2	4	no	—
300	C₄²	—	2	2	no	—
301	C₄²	—	2	2	no	—
302	C₄²	—	4	2	no	—
303	C₂³	—	4	2	no	—
304	C₂³	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
305	C₂³	C₂×C₄⋊C₄×C₅₉	1	1	yes	characteristic
306	C₂³	#117×C₅₉	1	1	yes	characteristic
307	C₂³	C₄×D₄×C₅₉	1	2	yes	—
308	C₂³	C₄×D₄×C₅₉	1	2	yes	—
309	Q₈	—	4	2	no	—
310	Q₈	—	4	2	no	—
311	Q₈	—	4	4	no	—
312	Q₈	—	4	4	no	—
313	Q₈	—	4	2	no	—
314	Q₈	—	4	2	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	2	8	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	4	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	4	no	—
322	C₂×C₄	—	4	4	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	#120×C₅₉	1	1	yes	characteristic
325	C₂×C₄	#120×C₅₉	1	2	yes	—
326	C₂×C₄	#121×C₅₉	1	2	yes	—
327	C₂×C₄	#121×C₅₉	1	1	yes	characteristic
328	C₂×C₄	#121×C₅₉	1	1	yes	characteristic
329	C₂×C₄	#121×C₅₉	1	2	yes	—
330	C₂×C₄	#121×C₅₉	1	2	yes	—
331	C₂×C₄	C₂²⋊Q₈×C₅₉	1	2	yes	—
332	C₂×C₄	C₂²⋊Q₈×C₅₉	1	1	yes	characteristic
333	C₂×C₄	C₂²⋊Q₈×C₅₉	1	1	yes	characteristic
334	C₂×C₄	C₂²⋊Q₈×C₅₉	1	2	yes	—
335	C₂×C₄	#123×C₅₉	1	2	yes	—
336	C₂×C₄	#123×C₅₉	1	2	yes	—
337	C₂×C₄	#123×C₅₉	1	1	yes	characteristic
338	C₂×C₄	#124×C₅₉	1	2	yes	—
339	C₂×C₄	#124×C₅₉	1	1	yes	characteristic
340	C₂×C₄	#126×C₅₉	1	2	yes	—
341	C₂×C₄	#127×C₅₉	1	1	yes	characteristic
342	C₈	—	4	4	no	—
343	C₈	—	4	4	no	—
344	C₂²	—	2	2	no	—
345	C₂²	—	2	4	no	—
346	C₂²	—	4	1	no	—
347	C₂²	—	4	1	no	—
348	C₂²	—	4	2	no	—
349	C₂²	C₅₉×#380	1	1	yes	characteristic
350	C₂²	C₅₉×#385	1	2	yes	—
351	C₂²	C₅₉×#386	1	1	yes	characteristic
352	C₂²	C₅₉×#388	1	2	yes	—
353	C₂²	C₅₉×#390	1	1	yes	characteristic
354	C₄	—	4	4	no	—
355	C₄	—	4	4	no	—
356	C₄	—	8	2	no	—
357	C₄	C₅₉×#392	1	1	yes	characteristic
358	C₄	C₅₉×#393	1	1	yes	characteristic
359	C₄	C₅₉×#394	1	2	yes	—
360	C₄	C₅₉×#396	1	2	yes	—
361	C₄	C₅₉×#397	1	1	yes	characteristic
362	C₄	C₅₉×#399	1	1	yes	characteristic
363	C₂	—	2	1	no	—
364	C₂	—	2	1	no	—
365	C₂	—	2	2	no	—
366	C₂	—	2	2	no	—
367	C₂	C₅₉×#1439	1	1	yes	characteristic
368	C₂	C₅₉×#1915	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₉	#386
commutator	C₂⁴	C₂²×C₄×C₅₉
Frattini, cosocle	C₂³×C₄	C₂³×C₅₉
prime radical	C₂⁴×C₅₉	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ C₂³×C₄×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
159. #11048	C₅₉	direct (N × H)
205. C₅₉	#11048	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×#2905	2
C₂²×#101×C₅₉	2
C₅₉×#2310	2
C₂×C₅₉×#344	2
C₅₉×#1742	2
#11048	59

Subgroups — C59×#11048 #28423157

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#11048 #28423157