Subgroups — C₅₉×#24721 #28436830

Order 15104.

706 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#3079	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₉×#476	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×AGL(1,ℤ/8)×C₅₉	C₂	1	1	yes	maximal, characteristic
4	C₅₉×#2925	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#2079	C₂	1	2	yes	maximal, complemented
6	C₅₉×#2051	C₂	1	2	yes	maximal
7	C₅₉×#1997	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₉×#1951	C₂	1	2	yes	maximal, complemented
9	C₅₉×#1950	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₉×#1931	C₂	1	2	yes	maximal, complemented
11	C₅₉×#1800	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₅₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₄×#41×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
16	C₅₉×#477	—	2	1	no	—
17	C₅₉×#476	—	2	1	no	—
18	C₅₉×#476	—	2	1	no	—
19	C₅₉×#476	—	2	1	no	—
20	C₅₉×#476	—	2	2	no	—
21	C₅₉×#475	—	2	1	no	—
22	C₅₉×#474	—	2	1	no	—
23	C₅₉×#442	—	2	1	no	—
24	C₅₉×#440	—	2	1	no	—
25	C₄×SD₁₆×C₅₉	—	2	1	no	—
26	C₄×D₈×C₅₉	—	2	1	no	—
27	C₅₉×#433	—	2	2	no	—
28	C₅₉×#432	—	2	1	no	—
29	C₅₉×#431	—	2	1	no	—
30	C₂×#108×C₅₉	C₂²	1	2	yes	—
31	C₅₉×#428	C₂²	1	2	yes	—
32	C₅₉×#428	C₂²	1	1	yes	characteristic
33	C₂×#106×C₅₉	C₂²	1	2	yes	—
34	C₅₉×#423	C₂²	1	1	yes	characteristic
35	C₅₉×#423	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
37	C₅₉×#418	C₂²	1	1	yes	characteristic
38	C₅₉×#417	C₂²	1	2	yes	—
39	C₅₉×#417	C₂²	1	1	yes	characteristic
40	C₂×#102×C₅₉	C₂²	1	2	yes	—
41	C₂×#100×C₅₉	C₂²	1	2	yes	—
42	C₅₉×#408	C₂²	1	2	yes	—
43	C₅₉×#405	C₂²	1	2	yes	—
44	C₄×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
45	C₅₉×#343	C₂²	1	2	yes	—
46	C₅₉×#343	C₂²	1	2	yes	—
47	C₅₉×#343	C₂²	1	2	yes	—
48	C₅₉×#341	C₂²	1	2	yes	—
49	C₂×#41×C₅₉	C₂³	1	1	yes	characteristic
50	C₂²×Q₈×C₅₉	—	2	1	no	—
51	C₂²×D₄×C₅₉	C₂³	1	1	yes	characteristic
52	C₂³×C₄×C₅₉	—	2	1	no	—
53	AGL(1,ℤ/8)×C₅₉	—	2	2	no	—
54	AGL(1,ℤ/8)×C₅₉	—	2	2	no	—
55	C₂×SD₁₆×C₅₉	—	2	1	no	—
56	C₂×D₈×C₅₉	—	2	1	no	—
57	C₂×M₁₆×C₅₉	C₂³	1	1	yes	characteristic
58	C₂×M₁₆×C₅₉	C₂³	1	2	yes	—
59	C₂×M₁₆×C₅₉	C₂³	1	2	yes	—
60	C₂×M₁₆×C₅₉	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
62	C₄⋊Q₈×C₅₉	—	2	1	no	—
63	C₄⋊Q₈×C₅₉	—	2	1	no	—
64	C₄⋊Q₈×C₅₉	—	2	1	no	—
65	C₄⋊Q₈×C₅₉	—	2	1	no	—
66	C₄×Q₈×C₅₉	D₄	1	1	yes	characteristic
67	C₄×Q₈×C₅₉	D₄	1	1	yes	characteristic
68	C₄×D₄×C₅₉	—	2	1	no	—
69	C₄×D₄×C₅₉	—	2	1	no	—
70	C₄×D₄×C₅₉	—	2	1	no	—
71	C₄×D₄×C₅₉	—	2	1	no	—
72	C₄×D₄×C₅₉	—	2	1	no	—
73	C₄×D₄×C₅₉	—	2	1	no	—
74	C₄×D₄×C₅₉	D₄	1	1	yes	characteristic
75	C₄×D₄×C₅₉	D₄	1	1	yes	characteristic
76	#117×C₅₉	—	2	1	no	—
77	#117×C₅₉	C₂³	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
79	C₂×C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
80	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
81	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
82	C₂×C₄²×C₅₉	—	2	1	no	—
83	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
84	#108×C₅₉	—	2	4	no	—
85	#107×C₅₉	—	2	1	no	—
86	#107×C₅₉	—	2	2	no	—
87	#106×C₅₉	—	2	2	no	—
88	#106×C₅₉	—	2	2	no	—
89	#106×C₅₉	—	2	1	no	—
90	#106×C₅₉	—	2	2	no	—
91	C₄⋊C₈×C₅₉	—	2	1	no	—
92	C₄⋊C₈×C₅₉	—	2	1	no	—
93	C₄⋊C₈×C₅₉	—	2	1	no	—
94	C₄⋊C₈×C₅₉	—	2	1	no	—
95	C₄⋊C₈×C₅₉	D₄	1	1	yes	characteristic
96	C₄⋊C₈×C₅₉	D₄	1	1	yes	characteristic
97	C₄⋊C₈×C₅₉	D₄	1	2	yes	—
98	#103×C₅₉	—	2	1	no	—
99	#103×C₅₉	—	2	2	no	—
100	#102×C₅₉	—	2	1	no	—
101	#102×C₅₉	—	2	2	no	—
102	#102×C₅₉	—	2	2	no	—
103	#102×C₅₉	—	2	2	no	—
104	#102×C₅₉	—	2	1	no	—
105	#102×C₅₉	—	2	1	no	—
106	#100×C₅₉	—	2	4	no	—
107	C₂²⋊C₈×C₅₉	—	2	2	no	—
108	C₂²⋊C₈×C₅₉	—	2	2	no	—
109	#97×C₅₉	—	2	2	no	—
110	#97×C₅₉	—	2	1	no	—
111	C₄×C₈×C₅₉	—	2	2	no	—
112	C₄×C₈×C₅₉	—	2	1	no	—
113	C₂⁴×C₅₉	—	2	1	no	—
114	#41×C₅₉	—	2	1	no	—
115	#41×C₅₉	—	2	1	no	—
116	#41×C₅₉	—	2	1	no	—
117	#41×C₅₉	—	2	1	no	—
118	C₂×Q₈×C₅₉	—	4	1	no	—
119	C₂×Q₈×C₅₉	—	4	1	no	—
120	C₂×Q₈×C₅₉	—	8	2	no	—
121	C₂×Q₈×C₅₉	C₂×D₄	1	1	yes	characteristic
122	C₂×D₄×C₅₉	—	2	1	no	—
123	C₂×D₄×C₅₉	—	2	1	no	—
124	C₂×D₄×C₅₉	—	2	1	no	—
125	C₂×D₄×C₅₉	—	2	1	no	—
126	C₂×D₄×C₅₉	—	2	1	no	—
127	C₂×D₄×C₅₉	—	2	1	no	—
128	C₂×D₄×C₅₉	C₂×D₄	1	1	yes	characteristic
129	C₂×D₄×C₅₉	#41	1	2	yes	—
130	C₂²×C₄×C₅₉	—	2	1	no	—
131	C₂²×C₄×C₅₉	—	2	1	no	—
132	C₂²×C₄×C₅₉	—	2	1	no	—
133	C₂²×C₄×C₅₉	—	2	1	no	—
134	C₂²×C₄×C₅₉	—	2	2	no	—
135	C₂²×C₄×C₅₉	—	4	1	no	—
136	C₂²×C₄×C₅₉	—	4	1	no	—
137	C₂²×C₄×C₅₉	—	8	2	no	—
138	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
139	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
140	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
141	SD₁₆×C₅₉	—	2	4	no	—
142	D₈×C₅₉	—	2	4	no	—
143	M₁₆×C₅₉	—	2	1	no	—
144	M₁₆×C₅₉	—	2	2	no	—
145	M₁₆×C₅₉	—	2	2	no	—
146	M₁₆×C₅₉	—	2	2	no	—
147	M₁₆×C₅₉	—	2	1	no	—
148	M₁₆×C₅₉	—	2	1	no	—
149	M₁₆×C₅₉	—	2	2	no	—
150	M₁₆×C₅₉	#41	1	2	yes	—
151	C₂×C₈×C₅₉	—	2	1	no	—
152	C₂×C₈×C₅₉	—	2	2	no	—
153	C₂×C₈×C₅₉	—	2	2	no	—
154	C₂×C₈×C₅₉	—	2	2	no	—
155	C₂×C₈×C₅₉	—	2	1	no	—
156	C₂×C₈×C₅₉	—	2	2	no	—
157	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
158	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
159	C₄⋊C₄×C₅₉	—	2	1	no	—
160	C₄⋊C₄×C₅₉	—	2	1	no	—
161	C₄⋊C₄×C₅₉	—	2	2	no	—
162	C₄⋊C₄×C₅₉	—	2	1	no	—
163	C₄⋊C₄×C₅₉	—	2	2	no	—
164	C₄⋊C₄×C₅₉	—	2	1	no	—
165	C₄⋊C₄×C₅₉	—	4	1	no	—
166	C₄⋊C₄×C₅₉	—	4	1	no	—
167	C₄⋊C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
168	C₄⋊C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
169	C₂²⋊C₄×C₅₉	—	2	1	no	—
170	C₂²⋊C₄×C₅₉	—	2	1	no	—
171	C₂²⋊C₄×C₅₉	—	2	1	no	—
172	C₂²⋊C₄×C₅₉	—	4	1	no	—
173	C₂²⋊C₄×C₅₉	—	4	1	no	—
174	C₄²×C₅₉	—	2	1	no	—
175	C₄²×C₅₉	—	2	1	no	—
176	C₄²×C₅₉	—	2	1	no	—
177	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
178	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
179	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
180	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
181	C₂³×C₅₉	—	2	1	no	—
182	C₂³×C₅₉	—	2	2	no	—
183	C₂³×C₅₉	—	4	1	no	—
184	C₂³×C₅₉	—	4	1	no	—
185	C₂³×C₅₉	—	8	1	no	—
186	C₂³×C₅₉	—	8	1	no	—
187	C₂³×C₅₉	C₂×#41	1	1	yes	characteristic
188	Q₈×C₅₉	—	2	1	no	—
189	Q₈×C₅₉	—	2	1	no	—
190	Q₈×C₅₉	—	8	2	no	—
191	Q₈×C₅₉	—	8	2	no	—
192	D₄×C₅₉	—	2	1	no	—
193	D₄×C₅₉	—	2	2	no	—
194	D₄×C₅₉	—	2	1	no	—
195	D₄×C₅₉	—	2	2	no	—
196	D₄×C₅₉	—	4	1	no	—
197	D₄×C₅₉	—	4	1	no	—
198	D₄×C₅₉	—	4	1	no	—
199	D₄×C₅₉	—	4	1	no	—
200	C₂×C₄×C₅₉	—	2	2	no	—
201	C₂×C₄×C₅₉	—	2	1	no	—
202	C₂×C₄×C₅₉	—	2	1	no	—
203	C₂×C₄×C₅₉	—	2	1	no	—
204	C₂×C₄×C₅₉	—	4	1	no	—
205	C₂×C₄×C₅₉	—	4	1	no	—
206	C₂×C₄×C₅₉	—	4	1	no	—
207	C₂×C₄×C₅₉	—	4	2	no	—
208	C₂×C₄×C₅₉	—	4	1	no	—
209	C₂×C₄×C₅₉	—	4	2	no	—
210	C₂×C₄×C₅₉	—	8	1	no	—
211	C₂×C₄×C₅₉	—	8	2	no	—
212	C₂×C₄×C₅₉	—	8	2	no	—
213	C₂×C₄×C₅₉	—	8	1	no	—
214	C₂×C₄×C₅₉	—	8	2	no	—
215	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
216	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
217	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
218	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
219	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
220	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
222	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
223	C₂×C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
224	C₂×C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
225	C₈×C₅₉	—	2	2	no	—
226	C₈×C₅₉	—	2	2	no	—
227	C₈×C₅₉	—	2	2	no	—
228	C₈×C₅₉	—	4	2	no	—
229	C₈×C₅₉	—	4	1	no	—
230	C₈×C₅₉	—	4	2	no	—
231	#24721	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
232	C₂²×C₅₉	—	2	1	no	—
233	C₂²×C₅₉	—	2	1	no	—
234	C₂²×C₅₉	—	4	1	no	—
235	C₂²×C₅₉	—	4	2	no	—
236	C₂²×C₅₉	—	4	1	no	—
237	C₂²×C₅₉	—	4	1	no	—
238	C₂²×C₅₉	—	8	2	no	—
239	C₂²×C₅₉	—	8	2	no	—
240	C₂²×C₅₉	—	8	1	no	—
241	C₂²×C₅₉	—	8	1	no	—
242	C₂²×C₅₉	C₂×#121	1	1	yes	characteristic
243	C₂²×C₅₉	#552	1	1	yes	characteristic
244	C₂²×C₅₉	#556	1	1	yes	characteristic
245	C₄×C₅₉	—	2	1	no	—
246	C₄×C₅₉	—	2	1	no	—
247	C₄×C₅₉	—	2	1	no	—
248	C₄×C₅₉	—	4	1	no	—
249	C₄×C₅₉	—	4	1	no	—
250	C₄×C₅₉	—	4	1	no	—
251	C₄×C₅₉	—	8	1	no	—
252	C₄×C₅₉	—	8	2	no	—
253	C₄×C₅₉	#471	1	2	yes	—
254	C₄×C₅₉	#525	1	1	yes	characteristic
255	C₄×C₅₉	#536	1	1	yes	characteristic
256	C₄×C₅₉	#547	1	1	yes	characteristic
257	C₄×C₅₉	#548	1	1	yes	characteristic
258	#3079	C₂×C₅₉	1	1	yes	characteristic, complemented
259	C₂×#476	C₂×C₅₉	1	1	yes	characteristic, complemented
260	C₄×AGL(1,ℤ/8)	C₂×C₅₉	1	1	yes	characteristic
261	#2925	C₂×C₅₉	1	1	yes	characteristic, complemented
262	#2079	C₂×C₅₉	1	2	yes	complemented
263	#2051	C₂×C₅₉	1	2	yes	—
264	#1997	C₂×C₅₉	1	1	yes	characteristic, complemented
265	#1951	C₂×C₅₉	1	2	yes	complemented
266	#1950	C₂×C₅₉	1	1	yes	characteristic, complemented
267	#1931	C₂×C₅₉	1	2	yes	complemented
268	#1800	C₂×C₅₉	1	1	yes	characteristic, complemented
269	C₂×C₅₉	—	2	1	no	—
270	C₂×C₅₉	—	2	1	no	—
271	C₂×C₅₉	—	8	1	no	—
272	C₂×C₅₉	—	8	2	no	—
273	C₂×C₅₉	#2415	1	1	yes	characteristic
274	C₂×C₅₉	C₂×#471	1	1	yes	characteristic
275	C₂×C₅₉	#3068	1	1	yes	characteristic
276	C₂×AGL(1,ℤ/8)	C₂²×C₅₉	1	1	yes	characteristic
277	C₂×C₄⋊Q₈	C₂²×C₅₉	1	1	yes	characteristic
278	C₄×#41	C₂²×C₅₉	1	1	yes	characteristic
279	C₂×C₄×D₄	C₂²×C₅₉	1	1	yes	characteristic
280	#477	—	2	1	no	—
281	#476	—	2	1	no	—
282	#476	—	2	1	no	—
283	#476	—	2	1	no	—
284	#476	—	2	2	no	—
285	#475	—	2	1	no	—
286	#474	—	2	1	no	—
287	#442	—	2	1	no	—
288	#440	—	2	1	no	—
289	C₄×SD₁₆	—	2	1	no	—
290	C₄×D₈	—	2	1	no	—
291	#433	—	2	2	no	—
292	#432	—	2	1	no	—
293	#431	—	2	1	no	—
294	C₂×#108	C₂²×C₅₉	1	2	yes	—
295	#428	C₂²×C₅₉	1	2	yes	—
296	#428	C₂²×C₅₉	1	1	yes	characteristic
297	C₂×#106	C₂²×C₅₉	1	2	yes	—
298	#423	C₂²×C₅₉	1	1	yes	characteristic
299	#423	C₂²×C₅₉	1	1	yes	characteristic
300	C₂×C₄⋊C₈	C₂²×C₅₉	1	1	yes	characteristic
301	#418	C₂²×C₅₉	1	1	yes	characteristic
302	#417	C₂²×C₅₉	1	2	yes	—
303	#417	C₂²×C₅₉	1	1	yes	characteristic
304	C₂×#102	C₂²×C₅₉	1	2	yes	—
305	C₂×#100	C₂²×C₅₉	1	2	yes	—
306	#408	C₂²×C₅₉	1	2	yes	—
307	#405	C₂²×C₅₉	1	2	yes	—
308	C₄×M₁₆	C₂²×C₅₉	1	1	yes	characteristic
309	#343	C₂²×C₅₉	1	2	yes	—
310	#343	C₂²×C₅₉	1	2	yes	—
311	#343	C₂²×C₅₉	1	2	yes	—
312	#341	C₂²×C₅₉	1	2	yes	—
313	C₅₉	#24721	1	1	yes	characteristic, direct product factor, Sylow
314	C₂×#41	C₂³×C₅₉	1	1	yes	characteristic
315	C₂²×Q₈	—	2	1	no	—
316	C₂²×D₄	C₂³×C₅₉	1	1	yes	characteristic
317	C₂³×C₄	—	2	1	no	—
318	AGL(1,ℤ/8)	—	2	2	no	—
319	AGL(1,ℤ/8)	—	2	2	no	—
320	C₂×SD₁₆	—	2	1	no	—
321	C₂×D₈	—	2	1	no	—
322	C₂×M₁₆	C₂³×C₅₉	1	1	yes	characteristic
323	C₂×M₁₆	C₂³×C₅₉	1	2	yes	—
324	C₂×M₁₆	C₂³×C₅₉	1	2	yes	—
325	C₂×M₁₆	C₂³×C₅₉	1	1	yes	characteristic
326	C₂²×C₈	C₂³×C₅₉	1	2	yes	—
327	C₄⋊Q₈	—	2	1	no	—
328	C₄⋊Q₈	—	2	1	no	—
329	C₄⋊Q₈	—	2	1	no	—
330	C₄⋊Q₈	—	2	1	no	—
331	C₄×Q₈	D₄×C₅₉	1	1	yes	characteristic
332	C₄×Q₈	D₄×C₅₉	1	1	yes	characteristic
333	C₄×D₄	—	2	1	no	—
334	C₄×D₄	—	2	1	no	—
335	C₄×D₄	—	2	1	no	—
336	C₄×D₄	—	2	1	no	—
337	C₄×D₄	—	2	1	no	—
338	C₄×D₄	—	2	1	no	—
339	C₄×D₄	D₄×C₅₉	1	1	yes	characteristic
340	C₄×D₄	D₄×C₅₉	1	1	yes	characteristic
341	#117	—	2	1	no	—
342	#117	C₂³×C₅₉	1	1	yes	characteristic
343	C₂×C₄⋊C₄	—	2	1	no	—
344	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
345	C₂×C₄⋊C₄	C₂³×C₅₉	1	2	yes	—
346	C₂×C₂²⋊C₄	—	2	1	no	—
347	C₂×C₄²	—	2	1	no	—
348	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
349	#108	—	2	4	no	—
350	#107	—	2	1	no	—
351	#107	—	2	2	no	—
352	#106	—	2	2	no	—
353	#106	—	2	2	no	—
354	#106	—	2	1	no	—
355	#106	—	2	2	no	—
356	C₄⋊C₈	—	2	1	no	—
357	C₄⋊C₈	—	2	1	no	—
358	C₄⋊C₈	—	2	1	no	—
359	C₄⋊C₈	—	2	1	no	—
360	C₄⋊C₈	D₄×C₅₉	1	1	yes	characteristic
361	C₄⋊C₈	D₄×C₅₉	1	1	yes	characteristic
362	C₄⋊C₈	D₄×C₅₉	1	2	yes	—
363	#103	—	2	1	no	—
364	#103	—	2	2	no	—
365	#102	—	2	1	no	—
366	#102	—	2	2	no	—
367	#102	—	2	2	no	—
368	#102	—	2	2	no	—
369	#102	—	2	1	no	—
370	#102	—	2	1	no	—
371	#100	—	2	4	no	—
372	C₂²⋊C₈	—	2	2	no	—
373	C₂²⋊C₈	—	2	2	no	—
374	#97	—	2	2	no	—
375	#97	—	2	1	no	—
376	C₄×C₈	—	2	2	no	—
377	C₄×C₈	—	2	1	no	—
378	C₂⁴	—	2	1	no	—
379	#41	—	2	1	no	—
380	#41	—	2	1	no	—
381	#41	—	2	1	no	—
382	#41	—	2	1	no	—
383	C₂×Q₈	—	4	1	no	—
384	C₂×Q₈	—	4	1	no	—
385	C₂×Q₈	—	8	2	no	—
386	C₂×Q₈	C₂×D₄×C₅₉	1	1	yes	characteristic
387	C₂×D₄	—	2	1	no	—
388	C₂×D₄	—	2	1	no	—
389	C₂×D₄	—	2	1	no	—
390	C₂×D₄	—	2	1	no	—
391	C₂×D₄	—	2	1	no	—
392	C₂×D₄	—	2	1	no	—
393	C₂×D₄	C₂×D₄×C₅₉	1	1	yes	characteristic
394	C₂×D₄	#41×C₅₉	1	2	yes	—
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	2	1	no	—
398	C₂²×C₄	—	2	1	no	—
399	C₂²×C₄	—	2	2	no	—
400	C₂²×C₄	—	4	1	no	—
401	C₂²×C₄	—	4	1	no	—
402	C₂²×C₄	—	8	2	no	—
403	C₂²×C₄	#41×C₅₉	1	1	yes	characteristic
404	C₂²×C₄	#41×C₅₉	1	1	yes	characteristic
405	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
406	SD₁₆	—	2	4	no	—
407	D₈	—	2	4	no	—
408	M₁₆	—	2	1	no	—
409	M₁₆	—	2	2	no	—
410	M₁₆	—	2	2	no	—
411	M₁₆	—	2	2	no	—
412	M₁₆	—	2	1	no	—
413	M₁₆	—	2	1	no	—
414	M₁₆	—	2	2	no	—
415	M₁₆	#41×C₅₉	1	2	yes	—
416	C₂×C₈	—	2	1	no	—
417	C₂×C₈	—	2	2	no	—
418	C₂×C₈	—	2	2	no	—
419	C₂×C₈	—	2	2	no	—
420	C₂×C₈	—	2	1	no	—
421	C₂×C₈	—	2	2	no	—
422	C₂×C₈	C₂×D₄×C₅₉	1	2	yes	—
423	C₂×C₈	C₂×D₄×C₅₉	1	2	yes	—
424	C₄⋊C₄	—	2	1	no	—
425	C₄⋊C₄	—	2	1	no	—
426	C₄⋊C₄	—	2	2	no	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	—	2	2	no	—
429	C₄⋊C₄	—	2	1	no	—
430	C₄⋊C₄	—	4	1	no	—
431	C₄⋊C₄	—	4	1	no	—
432	C₄⋊C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
433	C₄⋊C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
434	C₂²⋊C₄	—	2	1	no	—
435	C₂²⋊C₄	—	2	1	no	—
436	C₂²⋊C₄	—	2	1	no	—
437	C₂²⋊C₄	—	4	1	no	—
438	C₂²⋊C₄	—	4	1	no	—
439	C₄²	—	2	1	no	—
440	C₄²	—	2	1	no	—
441	C₄²	—	2	1	no	—
442	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
443	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
444	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
445	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
446	C₂³	—	2	1	no	—
447	C₂³	—	2	2	no	—
448	C₂³	—	4	1	no	—
449	C₂³	—	4	1	no	—
450	C₂³	—	8	1	no	—
451	C₂³	—	8	1	no	—
452	C₂³	C₂×#41×C₅₉	1	1	yes	characteristic
453	Q₈	—	2	1	no	—
454	Q₈	—	2	1	no	—
455	Q₈	—	8	2	no	—
456	Q₈	—	8	2	no	—
457	D₄	—	2	1	no	—
458	D₄	—	2	2	no	—
459	D₄	—	2	1	no	—
460	D₄	—	2	2	no	—
461	D₄	—	4	1	no	—
462	D₄	—	4	1	no	—
463	D₄	—	4	1	no	—
464	D₄	—	4	1	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	4	1	no	—
470	C₂×C₄	—	4	2	no	—
471	C₂×C₄	—	4	1	no	—
472	C₂×C₄	—	4	1	no	—
473	C₂×C₄	—	4	1	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	8	1	no	—
476	C₂×C₄	—	8	2	no	—
477	C₂×C₄	—	8	2	no	—
478	C₂×C₄	—	8	1	no	—
479	C₂×C₄	—	8	2	no	—
480	C₂×C₄	#121×C₅₉	1	1	yes	characteristic
481	C₂×C₄	#121×C₅₉	1	1	yes	characteristic
482	C₂×C₄	#121×C₅₉	1	1	yes	characteristic
483	C₂×C₄	#121×C₅₉	1	1	yes	characteristic
484	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
485	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
486	C₂×C₄	C₂×#41×C₅₉	1	1	yes	characteristic
487	C₂×C₄	C₂×#41×C₅₉	1	1	yes	characteristic
488	C₂×C₄	2₋¹⁺⁴×C₅₉	1	1	yes	characteristic
489	C₂×C₄	2₋¹⁺⁴×C₅₉	1	1	yes	characteristic
490	C₈	—	2	2	no	—
491	C₈	—	2	2	no	—
492	C₈	—	2	2	no	—
493	C₈	—	4	2	no	—
494	C₈	—	4	1	no	—
495	C₈	—	4	2	no	—
496	C₂²	—	2	1	no	—
497	C₂²	—	2	1	no	—
498	C₂²	—	4	1	no	—
499	C₂²	—	4	1	no	—
500	C₂²	—	4	1	no	—
501	C₂²	—	4	2	no	—
502	C₂²	—	8	2	no	—
503	C₂²	—	8	2	no	—
504	C₂²	—	8	1	no	—
505	C₂²	—	8	1	no	—
506	C₂²	C₂×#121×C₅₉	1	1	yes	characteristic
507	C₂²	C₅₉×#552	1	1	yes	characteristic
508	C₂²	C₅₉×#556	1	1	yes	characteristic
509	C₄	—	2	1	no	—
510	C₄	—	2	1	no	—
511	C₄	—	2	1	no	—
512	C₄	—	4	1	no	—
513	C₄	—	4	1	no	—
514	C₄	—	4	1	no	—
515	C₄	—	8	1	no	—
516	C₄	—	8	2	no	—
517	C₄	C₅₉×#471	1	2	yes	—
518	C₄	C₅₉×#525	1	1	yes	characteristic
519	C₄	C₅₉×#536	1	1	yes	characteristic
520	C₄	C₅₉×#547	1	1	yes	characteristic
521	C₄	C₅₉×#548	1	1	yes	characteristic
522	C₂	—	2	1	no	—
523	C₂	—	2	1	no	—
524	C₂	—	8	1	no	—
525	C₂	—	8	2	no	—
526	C₂	C₅₉×#2415	1	1	yes	characteristic
527	C₂	C₂×C₅₉×#471	1	1	yes	characteristic
528	C₂	C₅₉×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₉	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₉
prime radical	C₂×AGL(1,ℤ/8)×C₅₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ C₂×C₄²×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₉×#3079	C₂	semidirect
2. C₂×C₅₉×#476	C₂	semidirect
4. C₅₉×#2925	C₂	semidirect
5. C₅₉×#2079	C₂	semidirect
7. C₅₉×#1997	C₂	semidirect
8. C₅₉×#1951	C₂	semidirect
9. C₅₉×#1950	C₂	semidirect
10. C₅₉×#1931	C₂	semidirect
11. C₅₉×#1800	C₂	semidirect
231. #24721	C₅₉	direct (N × H)
258. #3079	C₂×C₅₉	semidirect
259. C₂×#476	C₂×C₅₉	semidirect
261. #2925	C₂×C₅₉	semidirect
262. #2079	C₂×C₅₉	semidirect
264. #1997	C₂×C₅₉	semidirect
265. #1951	C₂×C₅₉	semidirect
266. #1950	C₂×C₅₉	semidirect
267. #1931	C₂×C₅₉	semidirect
268. #1800	C₂×C₅₉	semidirect
313. C₅₉	#24721	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×#3079	2
C₂×C₅₉×#476	2
C₄×AGL(1,ℤ/8)×C₅₉	2
C₅₉×#2925	2
C₅₉×#2079	2
C₅₉×#2051	2
C₅₉×#1997	2
C₅₉×#1951	2
C₅₉×#1950	2
C₅₉×#1931	2
C₅₉×#1800	2
#24721	59

Subgroups — C59×#24721 #28436830

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#24721 #28436830