Subgroups — C₅₉×#26504 #28438613

Order 15104.

720 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#3163	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₉×#3081	C₂	1	2	yes	maximal, complemented
3	C₅₉×#2948	C₂	1	1	yes	maximal, characteristic
4	C₅₉×#2063	C₂	1	2	yes	maximal, complemented
5	C₅₉×#2027	C₂	1	2	yes	maximal, complemented
6	C₅₉×#1978	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#1972	C₂	1	1	yes	maximal, characteristic
8	C₅₉×#1951	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₉×#1946	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₉×#1856	C₂	1	2	yes	maximal, complemented
11	C₅₉×#1773	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₅₉×#544	C₂²	1	2	yes	—
14	C₅₉×#518	C₂²	1	2	yes	—
15	C₅₉×#501	—	2	2	no	—
16	C₅₉×#499	—	2	1	no	—
17	C₅₉×#498	—	2	1	no	—
18	C₅₉×#478	—	2	2	no	—
19	C₅₉×#478	—	2	2	no	—
20	C₅₉×#476	—	2	2	no	—
21	C₅₉×#474	—	2	2	no	—
22	C₅₉×#442	—	2	1	no	—
23	C₅₉×#442	—	2	1	no	—
24	C₄×D₈×C₅₉	—	2	2	no	—
25	C₂×#108×C₅₉	C₂²	1	1	yes	characteristic
26	C₂×#108×C₅₉	C₂²	1	1	yes	characteristic
27	C₅₉×#428	C₂²	1	2	yes	—
28	C₅₉×#428	C₂²	1	2	yes	—
29	C₅₉×#427	C₂²	1	1	yes	characteristic
30	C₅₉×#427	C₂²	1	1	yes	characteristic
31	C₂×#106×C₅₉	C₂²	1	1	yes	characteristic
32	C₅₉×#424	C₂²	1	2	yes	—
33	C₅₉×#418	C₂²	1	2	yes	—
34	C₅₉×#418	C₂²	1	2	yes	—
35	C₂×#102×C₅₉	C₂²	1	2	yes	—
36	C₂×#100×C₅₉	C₂²	1	2	yes	—
37	C₅₉×#405	C₂²	1	1	yes	characteristic
38	C₅₉×#405	C₂²	1	1	yes	characteristic
39	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
40	C₅₉×#390	C₂²	1	2	yes	—
41	C₅₉×#343	C₂²	1	2	yes	—
42	C₅₉×#339	C₂²	1	1	yes	characteristic
43	C₅₉×#339	C₂²	1	2	yes	—
44	C₅₉×#339	C₂²	1	1	yes	characteristic
45	C₂²×D₄×C₅₉	C₂³	1	2	yes	—
46	C₂×D₈×C₅₉	—	2	1	no	—
47	C₂×D₈×C₅₉	—	2	1	no	—
48	C₂×D₈×C₅₉	—	2	2	no	—
49	C₂×D₈×C₅₉	Q₈	1	4	yes	complemented
50	C₂×M₁₆×C₅₉	C₂³	1	1	yes	characteristic
51	C₂×M₁₆×C₅₉	C₂³	1	2	yes	—
52	C₂×M₁₆×C₅₉	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
55	C₄⋊Q₈×C₅₉	—	2	2	no	—
56	C₄⋊Q₈×C₅₉	—	2	2	no	—
57	#125×C₅₉	—	2	2	no	—
58	#125×C₅₉	—	2	2	no	—
59	C₂²⋊Q₈×C₅₉	—	4	2	no	—
60	C₄×D₄×C₅₉	—	2	2	no	—
61	C₄×D₄×C₅₉	—	2	2	no	—
62	C₄×D₄×C₅₉	—	2	2	no	—
63	C₄×D₄×C₅₉	—	2	2	no	—
64	#117×C₅₉	C₂³	1	1	yes	characteristic
65	#117×C₅₉	C₂³	1	1	yes	characteristic
66	#117×C₅₉	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
68	C₂×C₂²⋊C₄×C₅₉	—	4	2	no	—
69	C₂×C₂²⋊C₄×C₅₉	—	4	2	no	—
70	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
71	#108×C₅₉	—	2	2	no	—
72	#108×C₅₉	—	2	2	no	—
73	#107×C₅₉	—	2	2	no	—
74	#107×C₅₉	—	2	2	no	—
75	#107×C₅₉	D₄	1	2	yes	—
76	#107×C₅₉	D₄	1	2	yes	—
77	#106×C₅₉	—	2	1	no	—
78	#106×C₅₉	—	2	1	no	—
79	#106×C₅₉	—	2	2	no	—
80	#106×C₅₉	—	2	2	no	—
81	#106×C₅₉	D₄	1	1	yes	characteristic
82	#106×C₅₉	D₄	1	1	yes	characteristic
83	#106×C₅₉	D₄	1	2	yes	—
84	C₄⋊C₈×C₅₉	—	2	2	no	—
85	C₄⋊C₈×C₅₉	—	2	2	no	—
86	#102×C₅₉	—	2	2	no	—
87	#102×C₅₉	—	2	2	no	—
88	#102×C₅₉	—	2	2	no	—
89	#102×C₅₉	—	2	2	no	—
90	#102×C₅₉	—	2	2	no	—
91	#102×C₅₉	—	2	2	no	—
92	#100×C₅₉	—	2	4	no	—
93	#97×C₅₉	—	2	1	no	—
94	#97×C₅₉	—	2	1	no	—
95	#97×C₅₉	—	2	2	no	—
96	C₄×C₈×C₅₉	—	2	1	no	—
97	C₄×C₈×C₅₉	—	2	1	no	—
98	C₂⁴×C₅₉	—	2	2	no	—
99	C₂×Q₈×C₅₉	—	4	2	no	—
100	C₂×D₄×C₅₉	—	2	2	no	—
101	C₂×D₄×C₅₉	—	2	2	no	—
102	C₂×D₄×C₅₉	—	2	2	no	—
103	C₂×D₄×C₅₉	—	2	2	no	—
104	C₂×D₄×C₅₉	—	2	2	no	—
105	C₂×D₄×C₅₉	C₂×Q₈	1	4	yes	—
106	C₂²×C₄×C₅₉	—	2	1	no	—
107	C₂²×C₄×C₅₉	—	2	2	no	—
108	C₂²×C₄×C₅₉	—	4	2	no	—
109	C₂²×C₄×C₅₉	—	4	2	no	—
110	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
111	D₈×C₅₉	—	2	4	no	—
112	D₈×C₅₉	—	2	4	no	—
113	M₁₆×C₅₉	—	2	1	no	—
114	M₁₆×C₅₉	—	2	2	no	—
115	M₁₆×C₅₉	—	2	2	no	—
116	M₁₆×C₅₉	—	2	1	no	—
117	M₁₆×C₅₉	—	2	1	no	—
118	M₁₆×C₅₉	#41	1	2	yes	—
119	C₂×C₈×C₅₉	—	2	1	no	—
120	C₂×C₈×C₅₉	—	2	1	no	—
121	C₂×C₈×C₅₉	—	2	1	no	—
122	C₂×C₈×C₅₉	—	2	1	no	—
123	C₂×C₈×C₅₉	—	2	2	no	—
124	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
129	C₂×C₈×C₅₉	#41	1	2	yes	—
130	C₄⋊C₄×C₅₉	—	2	2	no	—
131	C₄⋊C₄×C₅₉	—	2	2	no	—
132	C₄⋊C₄×C₅₉	—	4	2	no	—
133	C₄⋊C₄×C₅₉	—	4	2	no	—
134	C₄⋊C₄×C₅₉	—	4	2	no	—
135	C₄⋊C₄×C₅₉	C₂×D₄	1	2	yes	—
136	C₄⋊C₄×C₅₉	C₂×D₄	1	2	yes	—
137	C₄⋊C₄×C₅₉	C₂×D₄	1	2	yes	—
138	C₄⋊C₄×C₅₉	C₂×D₄	1	2	yes	—
139	C₂²⋊C₄×C₅₉	—	2	1	no	—
140	C₂²⋊C₄×C₅₉	—	2	2	no	—
141	C₂²⋊C₄×C₅₉	—	2	1	no	—
142	C₂²⋊C₄×C₅₉	—	4	2	no	—
143	C₂²⋊C₄×C₅₉	—	4	2	no	—
144	C₂²⋊C₄×C₅₉	—	8	4	no	—
145	C₄²×C₅₉	—	2	1	no	—
146	C₄²×C₅₉	—	2	1	no	—
147	C₄²×C₅₉	—	2	2	no	—
148	C₄²×C₅₉	—	2	1	no	—
149	C₄²×C₅₉	—	2	1	no	—
150	C₂³×C₅₉	—	2	4	no	—
151	C₂³×C₅₉	—	4	2	no	—
152	C₂³×C₅₉	—	4	2	no	—
153	C₂³×C₅₉	—	8	2	no	—
154	C₂³×C₅₉	—	8	2	no	—
155	C₂³×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
156	Q₈×C₅₉	—	16	2	no	—
157	D₄×C₅₉	—	2	4	no	—
158	D₄×C₅₉	—	2	4	no	—
159	D₄×C₅₉	—	4	2	no	—
160	D₄×C₅₉	—	4	2	no	—
161	C₂×C₄×C₅₉	—	2	2	no	—
162	C₂×C₄×C₅₉	—	2	2	no	—
163	C₂×C₄×C₅₉	—	2	2	no	—
164	C₂×C₄×C₅₉	—	2	2	no	—
165	C₂×C₄×C₅₉	—	4	2	no	—
166	C₂×C₄×C₅₉	—	4	1	no	—
167	C₂×C₄×C₅₉	—	4	2	no	—
168	C₂×C₄×C₅₉	—	8	2	no	—
169	C₂×C₄×C₅₉	—	16	2	no	—
170	C₂×C₄×C₅₉	—	16	2	no	—
171	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
174	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
175	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
176	C₂×C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
177	C₈×C₅₉	—	2	1	no	—
178	C₈×C₅₉	—	2	1	no	—
179	C₈×C₅₉	—	2	2	no	—
180	C₈×C₅₉	—	4	2	no	—
181	C₈×C₅₉	—	4	1	no	—
182	C₈×C₅₉	C₂²⋊Q₈	1	2	yes	—
183	C₈×C₅₉	C₂²⋊Q₈	1	2	yes	—
184	#26504	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
185	C₂²×C₅₉	—	2	2	no	—
186	C₂²×C₅₉	—	4	4	no	—
187	C₂²×C₅₉	—	8	4	no	—
188	C₂²×C₅₉	—	8	2	no	—
189	C₂²×C₅₉	—	8	2	no	—
190	C₂²×C₅₉	—	16	2	no	—
191	C₂²×C₅₉	#542	1	1	yes	characteristic
192	C₂²×C₅₉	D₄²	1	1	yes	characteristic
193	C₂²×C₅₉	#554	1	1	yes	characteristic
194	C₄×C₅₉	—	4	2	no	—
195	C₄×C₅₉	—	4	2	no	—
196	C₄×C₅₉	—	4	2	no	—
197	C₄×C₅₉	—	4	2	no	—
198	C₄×C₅₉	—	8	1	no	—
199	C₄×C₅₉	—	8	2	no	—
200	C₄×C₅₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
201	C₄×C₅₉	#532	1	1	yes	characteristic
202	C₄×C₅₉	#548	1	1	yes	characteristic
203	C₄×C₅₉	D₄×Q₈	1	1	yes	characteristic
204	#3163	C₂×C₅₉	1	1	yes	characteristic, complemented
205	#3081	C₂×C₅₉	1	2	yes	complemented
206	#2948	C₂×C₅₉	1	1	yes	characteristic
207	#2063	C₂×C₅₉	1	2	yes	complemented
208	#2027	C₂×C₅₉	1	2	yes	complemented
209	#1978	C₂×C₅₉	1	1	yes	characteristic
210	#1972	C₂×C₅₉	1	1	yes	characteristic
211	#1951	C₂×C₅₉	1	1	yes	characteristic, complemented
212	#1946	C₂×C₅₉	1	1	yes	characteristic, complemented
213	#1856	C₂×C₅₉	1	2	yes	complemented
214	#1773	C₂×C₅₉	1	1	yes	characteristic, complemented
215	C₂×C₅₉	—	2	1	no	—
216	C₂×C₅₉	—	2	1	no	—
217	C₂×C₅₉	—	8	4	no	—
218	C₂×C₅₉	#2452	1	1	yes	characteristic
219	C₂×C₅₉	#3288	1	2	yes	—
220	C₂²×D₈	C₂²×C₅₉	1	1	yes	characteristic
221	#544	C₂²×C₅₉	1	2	yes	—
222	#518	C₂²×C₅₉	1	2	yes	—
223	#501	—	2	2	no	—
224	#499	—	2	1	no	—
225	#498	—	2	1	no	—
226	#478	—	2	2	no	—
227	#478	—	2	2	no	—
228	#476	—	2	2	no	—
229	#474	—	2	2	no	—
230	#442	—	2	1	no	—
231	#442	—	2	1	no	—
232	C₄×D₈	—	2	2	no	—
233	C₂×#108	C₂²×C₅₉	1	1	yes	characteristic
234	C₂×#108	C₂²×C₅₉	1	1	yes	characteristic
235	#428	C₂²×C₅₉	1	2	yes	—
236	#428	C₂²×C₅₉	1	2	yes	—
237	#427	C₂²×C₅₉	1	1	yes	characteristic
238	#427	C₂²×C₅₉	1	1	yes	characteristic
239	C₂×#106	C₂²×C₅₉	1	1	yes	characteristic
240	#424	C₂²×C₅₉	1	2	yes	—
241	#418	C₂²×C₅₉	1	2	yes	—
242	#418	C₂²×C₅₉	1	2	yes	—
243	C₂×#102	C₂²×C₅₉	1	2	yes	—
244	C₂×#100	C₂²×C₅₉	1	2	yes	—
245	#405	C₂²×C₅₉	1	1	yes	characteristic
246	#405	C₂²×C₅₉	1	1	yes	characteristic
247	C₂×#97	C₂²×C₅₉	1	1	yes	characteristic
248	#390	C₂²×C₅₉	1	2	yes	—
249	#343	C₂²×C₅₉	1	2	yes	—
250	#339	C₂²×C₅₉	1	1	yes	characteristic
251	#339	C₂²×C₅₉	1	2	yes	—
252	#339	C₂²×C₅₉	1	1	yes	characteristic
253	C₅₉	#26504	1	1	yes	characteristic, direct product factor, Sylow
254	C₂²×D₄	C₂³×C₅₉	1	2	yes	—
255	C₂×D₈	—	2	1	no	—
256	C₂×D₈	—	2	1	no	—
257	C₂×D₈	—	2	2	no	—
258	C₂×D₈	Q₈×C₅₉	1	4	yes	complemented
259	C₂×M₁₆	C₂³×C₅₉	1	1	yes	characteristic
260	C₂×M₁₆	C₂³×C₅₉	1	2	yes	—
261	C₂×M₁₆	C₂³×C₅₉	1	1	yes	characteristic
262	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
263	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
264	C₄⋊Q₈	—	2	2	no	—
265	C₄⋊Q₈	—	2	2	no	—
266	#125	—	2	2	no	—
267	#125	—	2	2	no	—
268	C₂²⋊Q₈	—	4	2	no	—
269	C₄×D₄	—	2	2	no	—
270	C₄×D₄	—	2	2	no	—
271	C₄×D₄	—	2	2	no	—
272	C₄×D₄	—	2	2	no	—
273	#117	C₂³×C₅₉	1	1	yes	characteristic
274	#117	C₂³×C₅₉	1	2	yes	—
275	#117	C₂³×C₅₉	1	1	yes	characteristic
276	C₂×C₄⋊C₄	C₂³×C₅₉	1	2	yes	—
277	C₂×C₂²⋊C₄	—	4	2	no	—
278	C₂×C₂²⋊C₄	—	4	2	no	—
279	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
280	#108	—	2	2	no	—
281	#108	—	2	2	no	—
282	#107	—	2	2	no	—
283	#107	—	2	2	no	—
284	#107	D₄×C₅₉	1	2	yes	—
285	#107	D₄×C₅₉	1	2	yes	—
286	#106	—	2	1	no	—
287	#106	—	2	1	no	—
288	#106	—	2	2	no	—
289	#106	—	2	2	no	—
290	#106	D₄×C₅₉	1	1	yes	characteristic
291	#106	D₄×C₅₉	1	1	yes	characteristic
292	#106	D₄×C₅₉	1	2	yes	—
293	C₄⋊C₈	—	2	2	no	—
294	C₄⋊C₈	—	2	2	no	—
295	#102	—	2	2	no	—
296	#102	—	2	2	no	—
297	#102	—	2	2	no	—
298	#102	—	2	2	no	—
299	#102	—	2	2	no	—
300	#102	—	2	2	no	—
301	#100	—	2	4	no	—
302	#97	—	2	1	no	—
303	#97	—	2	1	no	—
304	#97	—	2	2	no	—
305	C₄×C₈	—	2	1	no	—
306	C₄×C₈	—	2	1	no	—
307	C₂⁴	—	2	2	no	—
308	C₂×Q₈	—	4	2	no	—
309	C₂×D₄	—	2	2	no	—
310	C₂×D₄	—	2	2	no	—
311	C₂×D₄	—	2	2	no	—
312	C₂×D₄	—	2	2	no	—
313	C₂×D₄	—	2	2	no	—
314	C₂×D₄	C₂×Q₈×C₅₉	1	4	yes	—
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	2	no	—
317	C₂²×C₄	—	4	2	no	—
318	C₂²×C₄	—	4	2	no	—
319	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
320	D₈	—	2	4	no	—
321	D₈	—	2	4	no	—
322	M₁₆	—	2	1	no	—
323	M₁₆	—	2	2	no	—
324	M₁₆	—	2	2	no	—
325	M₁₆	—	2	1	no	—
326	M₁₆	—	2	1	no	—
327	M₁₆	#41×C₅₉	1	2	yes	—
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	2	1	no	—
332	C₂×C₈	—	2	2	no	—
333	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
334	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
335	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
336	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
337	C₂×C₈	C₂×Q₈×C₅₉	1	2	yes	—
338	C₂×C₈	#41×C₅₉	1	2	yes	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	4	2	no	—
342	C₄⋊C₄	—	4	2	no	—
343	C₄⋊C₄	—	4	2	no	—
344	C₄⋊C₄	C₂×D₄×C₅₉	1	2	yes	—
345	C₄⋊C₄	C₂×D₄×C₅₉	1	2	yes	—
346	C₄⋊C₄	C₂×D₄×C₅₉	1	2	yes	—
347	C₄⋊C₄	C₂×D₄×C₅₉	1	2	yes	—
348	C₂²⋊C₄	—	2	1	no	—
349	C₂²⋊C₄	—	2	2	no	—
350	C₂²⋊C₄	—	2	1	no	—
351	C₂²⋊C₄	—	4	2	no	—
352	C₂²⋊C₄	—	4	2	no	—
353	C₂²⋊C₄	—	8	4	no	—
354	C₄²	—	2	1	no	—
355	C₄²	—	2	1	no	—
356	C₄²	—	2	2	no	—
357	C₄²	—	2	1	no	—
358	C₄²	—	2	1	no	—
359	C₂³	—	2	4	no	—
360	C₂³	—	4	2	no	—
361	C₂³	—	4	2	no	—
362	C₂³	—	8	2	no	—
363	C₂³	—	8	2	no	—
364	C₂³	H(5,𝔽₂)×C₅₉	1	1	yes	characteristic
365	Q₈	—	16	2	no	—
366	D₄	—	2	4	no	—
367	D₄	—	2	4	no	—
368	D₄	—	4	2	no	—
369	D₄	—	4	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	4	2	no	—
375	C₂×C₄	—	4	1	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	—	8	2	no	—
378	C₂×C₄	—	16	2	no	—
379	C₂×C₄	—	16	2	no	—
380	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
381	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
382	C₂×C₄	C₂²×Q₈×C₅₉	1	1	yes	characteristic
383	C₂×C₄	C₂×#41×C₅₉	1	1	yes	characteristic
384	C₂×C₄	C₂×#41×C₅₉	1	1	yes	characteristic
385	C₂×C₄	2₋¹⁺⁴×C₅₉	1	1	yes	characteristic
386	C₈	—	2	1	no	—
387	C₈	—	2	1	no	—
388	C₈	—	2	2	no	—
389	C₈	—	4	2	no	—
390	C₈	—	4	1	no	—
391	C₈	C₂²⋊Q₈×C₅₉	1	2	yes	—
392	C₈	C₂²⋊Q₈×C₅₉	1	2	yes	—
393	C₂²	—	2	2	no	—
394	C₂²	—	4	4	no	—
395	C₂²	—	8	4	no	—
396	C₂²	—	8	2	no	—
397	C₂²	—	8	2	no	—
398	C₂²	—	16	2	no	—
399	C₂²	C₅₉×#542	1	1	yes	characteristic
400	C₂²	D₄²×C₅₉	1	1	yes	characteristic
401	C₂²	C₅₉×#554	1	1	yes	characteristic
402	C₄	—	4	2	no	—
403	C₄	—	4	2	no	—
404	C₄	—	4	2	no	—
405	C₄	—	4	2	no	—
406	C₄	—	8	1	no	—
407	C₄	—	8	2	no	—
408	C₄	C₂×C₂²⋊Q₈×C₅₉	1	1	yes	characteristic
409	C₄	C₅₉×#532	1	1	yes	characteristic
410	C₄	C₅₉×#548	1	1	yes	characteristic
411	C₄	D₄×Q₈×C₅₉	1	1	yes	characteristic
412	C₂	—	2	1	no	—
413	C₂	—	2	1	no	—
414	C₂	—	8	4	no	—
415	C₂	C₅₉×#2452	1	1	yes	characteristic
416	C₂	C₅₉×#3288	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₉	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₉
prime radical	C₂²×D₈×C₅₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ C₂²×C₄×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₉×#3163	C₂	semidirect
2. C₅₉×#3081	C₂	semidirect
4. C₅₉×#2063	C₂	semidirect
5. C₅₉×#2027	C₂	semidirect
8. C₅₉×#1951	C₂	semidirect
9. C₅₉×#1946	C₂	semidirect
10. C₅₉×#1856	C₂	semidirect
11. C₅₉×#1773	C₂	semidirect
49. C₂×D₈×C₅₉	Q₈	semidirect
184. #26504	C₅₉	direct (N × H)
204. #3163	C₂×C₅₉	semidirect
205. #3081	C₂×C₅₉	semidirect
207. #2063	C₂×C₅₉	semidirect
208. #2027	C₂×C₅₉	semidirect
211. #1951	C₂×C₅₉	semidirect
212. #1946	C₂×C₅₉	semidirect
213. #1856	C₂×C₅₉	semidirect
214. #1773	C₂×C₅₉	semidirect
253. C₅₉	#26504	direct (N × H)
258. C₂×D₈	Q₈×C₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₅₉×#3163	2
C₅₉×#3081	2
C₅₉×#2948	2
C₅₉×#2063	2
C₅₉×#2027	2
C₅₉×#1978	2
C₅₉×#1972	2
C₅₉×#1951	2
C₅₉×#1946	2
C₅₉×#1856	2
C₅₉×#1773	2
#26504	59

Subgroups — C59×#26504 #28438613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#26504 #28438613