Subgroups — C₅₉×#18599 #28430708

Order 15104.

1190 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#3496	C₂	1	1	yes	maximal, characteristic
2	C₅₉×#3235	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₉×#3182	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₉×#3101	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#1647	C₂	1	2	yes	maximal, complemented
6	C₅₉×#1639	C₂	1	2	yes	maximal, complemented
7	C₅₉×#1635	C₂	1	2	yes	maximal, complemented
8	C₅₉×#1629	C₂	1	2	yes	maximal, complemented
9	C₅₉×#1553	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₉×#1490	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₉×#1484	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×H(5,𝔽₂)×C₅₉	—	2	1	no	—
13	C₅₉×#547	—	2	1	no	—
14	C₅₉×#547	C₂²	1	2	yes	—
15	C₅₉×#546	—	2	2	no	—
16	C₅₉×#546	—	2	2	no	—
17	D₄²×C₅₉	—	2	1	no	—
18	D₄²×C₅₉	C₂²	1	2	yes	—
19	C₅₉×#537	—	2	1	no	—
20	C₅₉×#534	—	2	2	no	—
21	C₅₉×#533	C₂²	1	1	yes	characteristic, complemented
22	C₅₉×#532	C₂²	1	1	yes	characteristic
23	C₅₉×#532	C₂²	1	1	yes	characteristic
24	C₂×#121×C₅₉	—	2	2	no	—
25	C₂×C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
26	C₅₉×#501	—	2	1	no	—
27	C₅₉×#500	C₂²	1	1	yes	characteristic, complemented
28	C₅₉×#492	C₂²	1	1	yes	characteristic, complemented
29	C₅₉×#490	—	2	1	no	—
30	C₅₉×#487	C₂²	1	1	yes	characteristic, complemented
31	C₅₉×#486	C₂²	1	1	yes	characteristic, complemented
32	C₅₉×#483	—	2	1	no	—
33	C₅₉×#482	—	2	2	no	—
34	C₅₉×#476	C₂²	1	2	yes	complemented
35	C₅₉×#476	C₂²	1	2	yes	complemented
36	C₅₉×#474	C₂²	1	2	yes	complemented
37	C₅₉×#468	—	2	2	no	—
38	C₅₉×#461	C₂²	1	2	yes	complemented
39	C₅₉×#450	—	2	1	no	—
40	C₅₉×#440	—	2	2	no	—
41	C₄×SD₁₆×C₅₉	—	2	2	no	—
42	C₅₉×#432	C₂²	1	1	yes	characteristic, complemented
43	C₅₉×#431	C₂²	1	1	yes	characteristic, complemented
44	C₄×M₁₆×C₅₉	C₂²	1	1	yes	characteristic, complemented
45	C₅₉×#332	C₂²	1	2	yes	complemented
46	C₅₉×#331	C₂²	1	2	yes	complemented
47	C₅₉×D₄⋊C₈	C₂²	1	2	yes	complemented
48	C₅₉×D₄⋊C₈	C₂²	1	2	yes	complemented
49	C₅₉×D₄⋊C₈	C₂²	1	2	yes	complemented
50	C₅₉×D₄⋊C₈	C₂²	1	2	yes	complemented
51	H(5,𝔽₂)×C₅₉	—	4	8	no	—
52	C₂×#41×C₅₉	—	2	2	no	—
53	C₂×#41×C₅₉	—	2	2	no	—
54	C₂×#41×C₅₉	—	2	1	no	—
55	C₂×#41×C₅₉	—	2	1	no	—
56	C₂²×D₄×C₅₉	—	2	4	no	—
57	C₂²×D₄×C₅₉	—	2	2	no	—
58	C₂²×D₄×C₅₉	—	2	2	no	—
59	C₂²×D₄×C₅₉	—	2	2	no	—
60	C₂²×D₄×C₅₉	D₄	1	1	yes	characteristic
61	C₂³×C₄×C₅₉	—	2	1	no	—
62	C₂×SD₁₆×C₅₉	—	2	2	no	—
63	C₂×SD₁₆×C₅₉	—	2	2	no	—
64	C₂×M₁₆×C₅₉	—	2	1	no	—
65	C₄⋊Q₈×C₅₉	D₄	1	1	yes	characteristic
66	C₄⋊Q₈×C₅₉	C₂³	1	1	yes	characteristic
67	C₄⋊Q₈×C₅₉	C₂³	1	1	yes	characteristic
68	C₄⋊Q₈×C₅₉	C₂³	1	1	yes	characteristic
69	#127×C₅₉	D₄	1	1	yes	characteristic
70	#127×C₅₉	C₂³	1	1	yes	characteristic
71	#125×C₅₉	—	2	1	no	—
72	#124×C₅₉	—	2	1	no	—
73	#124×C₅₉	—	2	1	no	—
74	#123×C₅₉	—	2	4	no	—
75	#123×C₅₉	—	2	4	no	—
76	C₂²⋊Q₈×C₅₉	—	2	2	no	—
77	C₂²⋊Q₈×C₅₉	—	2	2	no	—
78	C₂²⋊Q₈×C₅₉	—	2	1	no	—
79	C₂²⋊Q₈×C₅₉	—	2	1	no	—
80	#121×C₅₉	—	2	4	no	—
81	#121×C₅₉	—	2	4	no	—
82	#121×C₅₉	—	2	2	no	—
83	#121×C₅₉	—	2	2	no	—
84	#121×C₅₉	—	2	1	no	—
85	#121×C₅₉	—	2	1	no	—
86	#121×C₅₉	—	2	2	no	—
87	#121×C₅₉	—	2	2	no	—
88	#121×C₅₉	D₄	1	2	yes	—
89	#121×C₅₉	D₄	1	2	yes	—
90	#120×C₅₉	—	2	4	no	—
91	#120×C₅₉	—	2	4	no	—
92	C₄×Q₈×C₅₉	—	2	1	no	—
93	C₄×Q₈×C₅₉	—	2	1	no	—
94	C₄×D₄×C₅₉	—	2	1	no	—
95	C₄×D₄×C₅₉	—	2	1	no	—
96	C₄×D₄×C₅₉	—	2	1	no	—
97	C₄×D₄×C₅₉	—	2	1	no	—
98	C₄×D₄×C₅₉	D₄	1	2	yes	—
99	C₄×D₄×C₅₉	D₄	1	2	yes	—
100	C₄×D₄×C₅₉	C₂³	1	2	yes	—
101	C₄×D₄×C₅₉	C₂³	1	2	yes	—
102	#117×C₅₉	—	2	1	no	—
103	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
104	C₂×C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
105	C₂×C₂²⋊C₄×C₅₉	—	2	4	no	—
106	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
107	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
108	#107×C₅₉	—	2	2	no	—
109	#107×C₅₉	—	2	2	no	—
110	#106×C₅₉	—	2	2	no	—
111	#106×C₅₉	—	2	2	no	—
112	C₄⋊C₈×C₅₉	C₂³	1	2	yes	—
113	C₄⋊C₈×C₅₉	C₂³	1	2	yes	—
114	#103×C₅₉	—	2	2	no	—
115	#103×C₅₉	—	2	2	no	—
116	#102×C₅₉	—	2	2	no	—
117	#102×C₅₉	—	2	2	no	—
118	#102×C₅₉	—	2	2	no	—
119	#102×C₅₉	—	2	2	no	—
120	C₂²⋊C₈×C₅₉	—	2	1	no	—
121	C₂²⋊C₈×C₅₉	—	2	1	no	—
122	#97×C₅₉	—	2	1	no	—
123	#97×C₅₉	—	2	1	no	—
124	C₄×C₈×C₅₉	—	2	1	no	—
125	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
126	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
127	C₂⁴×C₅₉	—	2	1	no	—
128	C₂⁴×C₅₉	—	2	4	no	—
129	#41×C₅₉	—	4	4	no	—
130	#41×C₅₉	—	4	4	no	—
131	#41×C₅₉	—	4	8	no	—
132	#41×C₅₉	—	4	4	no	—
133	#41×C₅₉	—	4	4	no	—
134	C₂×Q₈×C₅₉	—	2	1	no	—
135	C₂×Q₈×C₅₉	—	2	1	no	—
136	C₂×Q₈×C₅₉	—	2	1	no	—
137	C₂×Q₈×C₅₉	—	2	1	no	—
138	C₂×D₄×C₅₉	—	2	4	no	—
139	C₂×D₄×C₅₉	—	2	4	no	—
140	C₂×D₄×C₅₉	—	2	4	no	—
141	C₂×D₄×C₅₉	—	2	2	no	—
142	C₂×D₄×C₅₉	—	2	2	no	—
143	C₂×D₄×C₅₉	—	2	4	no	—
144	C₂×D₄×C₅₉	—	2	1	no	—
145	C₂×D₄×C₅₉	—	2	1	no	—
146	C₂×D₄×C₅₉	—	2	1	no	—
147	C₂×D₄×C₅₉	—	2	1	no	—
148	C₂×D₄×C₅₉	—	2	1	no	—
149	C₂×D₄×C₅₉	—	4	8	no	—
150	C₂×D₄×C₅₉	—	4	4	no	—
151	C₂×D₄×C₅₉	—	4	16	no	—
152	C₂×D₄×C₅₉	—	4	4	no	—
153	C₂×D₄×C₅₉	—	8	4	no	—
154	C₂×D₄×C₅₉	C₂×D₄	1	2	yes	—
155	C₂×D₄×C₅₉	C₂×D₄	1	2	yes	—
156	C₂×D₄×C₅₉	C₂×D₄	1	2	yes	—
157	C₂²×C₄×C₅₉	—	2	4	no	—
158	C₂²×C₄×C₅₉	—	2	1	no	—
159	C₂²×C₄×C₅₉	—	2	2	no	—
160	C₂²×C₄×C₅₉	—	2	2	no	—
161	C₂²×C₄×C₅₉	—	2	2	no	—
162	C₂²×C₄×C₅₉	—	2	2	no	—
163	C₂²×C₄×C₅₉	—	8	2	no	—
164	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
167	SD₁₆×C₅₉	—	4	4	no	—
168	SD₁₆×C₅₉	—	4	4	no	—
169	M₁₆×C₅₉	—	4	2	no	—
170	C₂×C₈×C₅₉	—	2	1	no	—
171	C₂×C₈×C₅₉	—	2	2	no	—
172	C₂×C₈×C₅₉	—	2	1	no	—
173	C₂×C₈×C₅₉	—	2	2	no	—
174	C₄⋊C₄×C₅₉	—	2	4	no	—
175	C₄⋊C₄×C₅₉	—	2	1	no	—
176	C₄⋊C₄×C₅₉	—	2	1	no	—
177	C₄⋊C₄×C₅₉	—	2	1	no	—
178	C₄⋊C₄×C₅₉	—	2	1	no	—
179	C₄⋊C₄×C₅₉	—	2	1	no	—
180	C₄⋊C₄×C₅₉	—	2	1	no	—
181	C₄⋊C₄×C₅₉	—	2	1	no	—
182	C₄⋊C₄×C₅₉	—	4	1	no	—
183	C₄⋊C₄×C₅₉	—	4	1	no	—
184	C₄⋊C₄×C₅₉	C₂×D₄	1	2	yes	—
185	C₄⋊C₄×C₅₉	C₂×D₄	1	2	yes	—
186	C₄⋊C₄×C₅₉	C₂×D₄	1	2	yes	—
187	C₂²⋊C₄×C₅₉	—	2	4	no	—
188	C₂²⋊C₄×C₅₉	—	2	4	no	—
189	C₂²⋊C₄×C₅₉	—	2	2	no	—
190	C₂²⋊C₄×C₅₉	—	2	2	no	—
191	C₂²⋊C₄×C₅₉	—	2	2	no	—
192	C₂²⋊C₄×C₅₉	—	2	2	no	—
193	C₂²⋊C₄×C₅₉	—	2	2	no	—
194	C₂²⋊C₄×C₅₉	—	2	2	no	—
195	C₂²⋊C₄×C₅₉	—	4	1	no	—
196	C₄²×C₅₉	—	4	1	no	—
197	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
198	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
199	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
200	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
201	C₂³×C₅₉	—	2	4	no	—
202	C₂³×C₅₉	—	2	2	no	—
203	C₂³×C₅₉	—	2	2	no	—
204	C₂³×C₅₉	—	2	2	no	—
205	C₂³×C₅₉	—	4	8	no	—
206	C₂³×C₅₉	—	8	1	no	—
207	C₂³×C₅₉	—	8	1	no	—
208	C₂³×C₅₉	—	8	4	no	—
209	C₂³×C₅₉	—	8	4	no	—
210	C₂³×C₅₉	#120	1	1	yes	characteristic
211	Q₈×C₅₉	—	4	2	no	—
212	Q₈×C₅₉	—	4	2	no	—
213	Q₈×C₅₉	—	4	2	no	—
214	Q₈×C₅₉	—	4	2	no	—
215	D₄×C₅₉	—	2	4	no	—
216	D₄×C₅₉	—	2	2	no	—
217	D₄×C₅₉	—	2	4	no	—
218	D₄×C₅₉	—	4	2	no	—
219	D₄×C₅₉	—	4	2	no	—
220	D₄×C₅₉	—	4	4	no	—
221	D₄×C₅₉	—	4	4	no	—
222	D₄×C₅₉	—	4	8	no	—
223	D₄×C₅₉	—	4	8	no	—
224	D₄×C₅₉	—	4	2	no	—
225	D₄×C₅₉	—	4	2	no	—
226	D₄×C₅₉	—	8	1	no	—
227	D₄×C₅₉	—	8	4	no	—
228	D₄×C₅₉	AGL(1,ℤ/8)	1	4	yes	—
229	C₂×C₄×C₅₉	—	2	2	no	—
230	C₂×C₄×C₅₉	—	2	2	no	—
231	C₂×C₄×C₅₉	—	2	2	no	—
232	C₂×C₄×C₅₉	—	2	2	no	—
233	C₂×C₄×C₅₉	—	2	2	no	—
234	C₂×C₄×C₅₉	—	4	1	no	—
235	C₂×C₄×C₅₉	—	4	4	no	—
236	C₂×C₄×C₅₉	—	4	8	no	—
237	C₂×C₄×C₅₉	—	4	2	no	—
238	C₂×C₄×C₅₉	—	4	2	no	—
239	C₂×C₄×C₅₉	—	4	4	no	—
240	C₂×C₄×C₅₉	—	4	1	no	—
241	C₂×C₄×C₅₉	—	4	1	no	—
242	C₂×C₄×C₅₉	—	8	2	no	—
243	C₂×C₄×C₅₉	—	8	1	no	—
244	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
245	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
246	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
247	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
248	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
249	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
250	C₈×C₅₉	—	2	2	no	—
251	C₈×C₅₉	—	4	1	no	—
252	C₈×C₅₉	—	4	2	no	—
253	C₈×C₅₉	—	4	2	no	—
254	#18599	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
255	C₂²×C₅₉	—	2	1	no	—
256	C₂²×C₅₉	—	2	4	no	—
257	C₂²×C₅₉	—	2	1	no	—
258	C₂²×C₅₉	—	2	1	no	—
259	C₂²×C₅₉	—	4	2	no	—
260	C₂²×C₅₉	—	4	2	no	—
261	C₂²×C₅₉	—	4	2	no	—
262	C₂²×C₅₉	—	4	2	no	—
263	C₂²×C₅₉	—	4	2	no	—
264	C₂²×C₅₉	—	4	2	no	—
265	C₂²×C₅₉	—	4	2	no	—
266	C₂²×C₅₉	—	4	2	no	—
267	C₂²×C₅₉	—	8	4	no	—
268	C₂²×C₅₉	—	16	8	no	—
269	C₂²×C₅₉	C₂×#120	1	1	yes	characteristic
270	C₄×C₅₉	—	2	1	no	—
271	C₄×C₅₉	—	2	1	no	—
272	C₄×C₅₉	—	2	1	no	—
273	C₄×C₅₉	—	2	1	no	—
274	C₄×C₅₉	—	4	2	no	—
275	C₄×C₅₉	—	4	2	no	—
276	C₄×C₅₉	—	4	2	no	—
277	C₄×C₅₉	—	8	1	no	—
278	C₄×C₅₉	—	8	1	no	—
279	C₄×C₅₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
280	C₄×C₅₉	#577	1	2	yes	—
281	#3496	C₂×C₅₉	1	1	yes	characteristic
282	#3235	C₂×C₅₉	1	1	yes	characteristic, complemented
283	#3182	C₂×C₅₉	1	1	yes	characteristic, complemented
284	#3101	C₂×C₅₉	1	1	yes	characteristic, complemented
285	#1647	C₂×C₅₉	1	2	yes	complemented
286	#1639	C₂×C₅₉	1	2	yes	complemented
287	#1635	C₂×C₅₉	1	2	yes	complemented
288	#1629	C₂×C₅₉	1	2	yes	complemented
289	#1553	C₂×C₅₉	1	1	yes	characteristic, complemented
290	#1490	C₂×C₅₉	1	1	yes	characteristic, complemented
291	#1484	C₂×C₅₉	1	1	yes	characteristic, complemented
292	C₂×C₅₉	—	4	1	no	—
293	C₂×C₅₉	—	4	4	no	—
294	C₂×C₅₉	—	8	2	no	—
295	C₂×C₅₉	—	8	2	no	—
296	C₂×C₅₉	#3003	1	1	yes	characteristic
297	C₂×C₅₉	#3009	1	1	yes	characteristic
298	C₂×C₅₉	#3018	1	1	yes	characteristic
299	C₂×H(5,𝔽₂)	—	2	1	no	—
300	#547	—	2	1	no	—
301	#547	C₂²×C₅₉	1	2	yes	—
302	#546	—	2	2	no	—
303	#546	—	2	2	no	—
304	D₄²	—	2	1	no	—
305	D₄²	C₂²×C₅₉	1	2	yes	—
306	#537	—	2	1	no	—
307	#534	—	2	2	no	—
308	#533	C₂²×C₅₉	1	1	yes	characteristic, complemented
309	#532	C₂²×C₅₉	1	1	yes	characteristic
310	#532	C₂²×C₅₉	1	1	yes	characteristic
311	C₂×#121	—	2	2	no	—
312	C₂×C₄×D₄	C₂²×C₅₉	1	1	yes	characteristic
313	#501	—	2	1	no	—
314	#500	C₂²×C₅₉	1	1	yes	characteristic, complemented
315	#492	C₂²×C₅₉	1	1	yes	characteristic, complemented
316	#490	—	2	1	no	—
317	#487	C₂²×C₅₉	1	1	yes	characteristic, complemented
318	#486	C₂²×C₅₉	1	1	yes	characteristic, complemented
319	#483	—	2	1	no	—
320	#482	—	2	2	no	—
321	#476	C₂²×C₅₉	1	2	yes	complemented
322	#476	C₂²×C₅₉	1	2	yes	complemented
323	#474	C₂²×C₅₉	1	2	yes	complemented
324	#468	—	2	2	no	—
325	#461	C₂²×C₅₉	1	2	yes	complemented
326	#450	—	2	1	no	—
327	#440	—	2	2	no	—
328	C₄×SD₁₆	—	2	2	no	—
329	#432	C₂²×C₅₉	1	1	yes	characteristic, complemented
330	#431	C₂²×C₅₉	1	1	yes	characteristic, complemented
331	C₄×M₁₆	C₂²×C₅₉	1	1	yes	characteristic, complemented
332	#332	C₂²×C₅₉	1	2	yes	complemented
333	#331	C₂²×C₅₉	1	2	yes	complemented
334	D₄⋊C₈	C₂²×C₅₉	1	2	yes	complemented
335	D₄⋊C₈	C₂²×C₅₉	1	2	yes	complemented
336	D₄⋊C₈	C₂²×C₅₉	1	2	yes	complemented
337	D₄⋊C₈	C₂²×C₅₉	1	2	yes	complemented
338	C₅₉	#18599	1	1	yes	characteristic, direct product factor, Sylow
339	H(5,𝔽₂)	—	4	8	no	—
340	C₂×#41	—	2	2	no	—
341	C₂×#41	—	2	2	no	—
342	C₂×#41	—	2	1	no	—
343	C₂×#41	—	2	1	no	—
344	C₂²×D₄	—	2	4	no	—
345	C₂²×D₄	—	2	2	no	—
346	C₂²×D₄	—	2	2	no	—
347	C₂²×D₄	—	2	2	no	—
348	C₂²×D₄	D₄×C₅₉	1	1	yes	characteristic
349	C₂³×C₄	—	2	1	no	—
350	C₂×SD₁₆	—	2	2	no	—
351	C₂×SD₁₆	—	2	2	no	—
352	C₂×M₁₆	—	2	1	no	—
353	C₄⋊Q₈	D₄×C₅₉	1	1	yes	characteristic
354	C₄⋊Q₈	C₂³×C₅₉	1	1	yes	characteristic
355	C₄⋊Q₈	C₂³×C₅₉	1	1	yes	characteristic
356	C₄⋊Q₈	C₂³×C₅₉	1	1	yes	characteristic
357	#127	D₄×C₅₉	1	1	yes	characteristic
358	#127	C₂³×C₅₉	1	1	yes	characteristic
359	#125	—	2	1	no	—
360	#124	—	2	1	no	—
361	#124	—	2	1	no	—
362	#123	—	2	4	no	—
363	#123	—	2	4	no	—
364	C₂²⋊Q₈	—	2	2	no	—
365	C₂²⋊Q₈	—	2	2	no	—
366	C₂²⋊Q₈	—	2	1	no	—
367	C₂²⋊Q₈	—	2	1	no	—
368	#121	—	2	4	no	—
369	#121	—	2	4	no	—
370	#121	—	2	2	no	—
371	#121	—	2	2	no	—
372	#121	—	2	1	no	—
373	#121	—	2	1	no	—
374	#121	—	2	2	no	—
375	#121	—	2	2	no	—
376	#121	D₄×C₅₉	1	2	yes	—
377	#121	D₄×C₅₉	1	2	yes	—
378	#120	—	2	4	no	—
379	#120	—	2	4	no	—
380	C₄×Q₈	—	2	1	no	—
381	C₄×Q₈	—	2	1	no	—
382	C₄×D₄	—	2	1	no	—
383	C₄×D₄	—	2	1	no	—
384	C₄×D₄	—	2	1	no	—
385	C₄×D₄	—	2	1	no	—
386	C₄×D₄	D₄×C₅₉	1	2	yes	—
387	C₄×D₄	D₄×C₅₉	1	2	yes	—
388	C₄×D₄	C₂³×C₅₉	1	2	yes	—
389	C₄×D₄	C₂³×C₅₉	1	2	yes	—
390	#117	—	2	1	no	—
391	C₂×C₄⋊C₄	—	2	2	no	—
392	C₂×C₄⋊C₄	D₄×C₅₉	1	1	yes	characteristic
393	C₂×C₂²⋊C₄	—	2	4	no	—
394	C₂×C₂²⋊C₄	—	2	1	no	—
395	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
396	#107	—	2	2	no	—
397	#107	—	2	2	no	—
398	#106	—	2	2	no	—
399	#106	—	2	2	no	—
400	C₄⋊C₈	C₂³×C₅₉	1	2	yes	—
401	C₄⋊C₈	C₂³×C₅₉	1	2	yes	—
402	#103	—	2	2	no	—
403	#103	—	2	2	no	—
404	#102	—	2	2	no	—
405	#102	—	2	2	no	—
406	#102	—	2	2	no	—
407	#102	—	2	2	no	—
408	C₂²⋊C₈	—	2	1	no	—
409	C₂²⋊C₈	—	2	1	no	—
410	#97	—	2	1	no	—
411	#97	—	2	1	no	—
412	C₄×C₈	—	2	1	no	—
413	C₄×C₈	C₂³×C₅₉	1	1	yes	characteristic
414	C₄×C₈	C₂³×C₅₉	1	1	yes	characteristic
415	C₂⁴	—	2	1	no	—
416	C₂⁴	—	2	4	no	—
417	#41	—	4	4	no	—
418	#41	—	4	4	no	—
419	#41	—	4	8	no	—
420	#41	—	4	4	no	—
421	#41	—	4	4	no	—
422	C₂×Q₈	—	2	1	no	—
423	C₂×Q₈	—	2	1	no	—
424	C₂×Q₈	—	2	1	no	—
425	C₂×Q₈	—	2	1	no	—
426	C₂×D₄	—	2	4	no	—
427	C₂×D₄	—	2	4	no	—
428	C₂×D₄	—	2	4	no	—
429	C₂×D₄	—	2	2	no	—
430	C₂×D₄	—	2	2	no	—
431	C₂×D₄	—	2	4	no	—
432	C₂×D₄	—	2	1	no	—
433	C₂×D₄	—	2	1	no	—
434	C₂×D₄	—	2	1	no	—
435	C₂×D₄	—	2	1	no	—
436	C₂×D₄	—	2	1	no	—
437	C₂×D₄	—	4	8	no	—
438	C₂×D₄	—	4	4	no	—
439	C₂×D₄	—	4	16	no	—
440	C₂×D₄	—	4	4	no	—
441	C₂×D₄	—	8	4	no	—
442	C₂×D₄	C₂×D₄×C₅₉	1	2	yes	—
443	C₂×D₄	C₂×D₄×C₅₉	1	2	yes	—
444	C₂×D₄	C₂×D₄×C₅₉	1	2	yes	—
445	C₂²×C₄	—	2	4	no	—
446	C₂²×C₄	—	2	1	no	—
447	C₂²×C₄	—	2	2	no	—
448	C₂²×C₄	—	2	2	no	—
449	C₂²×C₄	—	2	2	no	—
450	C₂²×C₄	—	2	2	no	—
451	C₂²×C₄	—	8	2	no	—
452	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
453	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
454	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
455	SD₁₆	—	4	4	no	—
456	SD₁₆	—	4	4	no	—
457	M₁₆	—	4	2	no	—
458	C₂×C₈	—	2	1	no	—
459	C₂×C₈	—	2	2	no	—
460	C₂×C₈	—	2	1	no	—
461	C₂×C₈	—	2	2	no	—
462	C₄⋊C₄	—	2	4	no	—
463	C₄⋊C₄	—	2	1	no	—
464	C₄⋊C₄	—	2	1	no	—
465	C₄⋊C₄	—	2	1	no	—
466	C₄⋊C₄	—	2	1	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	4	1	no	—
471	C₄⋊C₄	—	4	1	no	—
472	C₄⋊C₄	C₂×D₄×C₅₉	1	2	yes	—
473	C₄⋊C₄	C₂×D₄×C₅₉	1	2	yes	—
474	C₄⋊C₄	C₂×D₄×C₅₉	1	2	yes	—
475	C₂²⋊C₄	—	2	4	no	—
476	C₂²⋊C₄	—	2	4	no	—
477	C₂²⋊C₄	—	2	2	no	—
478	C₂²⋊C₄	—	2	2	no	—
479	C₂²⋊C₄	—	2	2	no	—
480	C₂²⋊C₄	—	2	2	no	—
481	C₂²⋊C₄	—	2	2	no	—
482	C₂²⋊C₄	—	2	2	no	—
483	C₂²⋊C₄	—	4	1	no	—
484	C₄²	—	4	1	no	—
485	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
486	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
487	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
488	C₄²	C₂⁴×C₅₉	1	1	yes	characteristic
489	C₂³	—	2	4	no	—
490	C₂³	—	2	2	no	—
491	C₂³	—	2	2	no	—
492	C₂³	—	2	2	no	—
493	C₂³	—	4	8	no	—
494	C₂³	—	8	1	no	—
495	C₂³	—	8	1	no	—
496	C₂³	—	8	4	no	—
497	C₂³	—	8	4	no	—
498	C₂³	#120×C₅₉	1	1	yes	characteristic
499	Q₈	—	4	2	no	—
500	Q₈	—	4	2	no	—
501	Q₈	—	4	2	no	—
502	Q₈	—	4	2	no	—
503	D₄	—	2	4	no	—
504	D₄	—	2	2	no	—
505	D₄	—	2	4	no	—
506	D₄	—	4	2	no	—
507	D₄	—	4	2	no	—
508	D₄	—	4	4	no	—
509	D₄	—	4	4	no	—
510	D₄	—	4	8	no	—
511	D₄	—	4	8	no	—
512	D₄	—	4	2	no	—
513	D₄	—	4	2	no	—
514	D₄	—	8	1	no	—
515	D₄	—	8	4	no	—
516	D₄	AGL(1,ℤ/8)×C₅₉	1	4	yes	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	2	no	—
522	C₂×C₄	—	4	1	no	—
523	C₂×C₄	—	4	4	no	—
524	C₂×C₄	—	4	8	no	—
525	C₂×C₄	—	4	2	no	—
526	C₂×C₄	—	4	2	no	—
527	C₂×C₄	—	4	4	no	—
528	C₂×C₄	—	4	1	no	—
529	C₂×C₄	—	4	1	no	—
530	C₂×C₄	—	8	2	no	—
531	C₂×C₄	—	8	1	no	—
532	C₂×C₄	#120×C₅₉	1	1	yes	characteristic
533	C₂×C₄	#120×C₅₉	1	1	yes	characteristic
534	C₂×C₄	#120×C₅₉	1	1	yes	characteristic
535	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
536	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
537	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
538	C₈	—	2	2	no	—
539	C₈	—	4	1	no	—
540	C₈	—	4	2	no	—
541	C₈	—	4	2	no	—
542	C₂²	—	2	1	no	—
543	C₂²	—	2	4	no	—
544	C₂²	—	2	1	no	—
545	C₂²	—	2	1	no	—
546	C₂²	—	4	2	no	—
547	C₂²	—	4	2	no	—
548	C₂²	—	4	2	no	—
549	C₂²	—	4	2	no	—
550	C₂²	—	4	2	no	—
551	C₂²	—	4	2	no	—
552	C₂²	—	4	2	no	—
553	C₂²	—	4	2	no	—
554	C₂²	—	8	4	no	—
555	C₂²	—	16	8	no	—
556	C₂²	C₂×#120×C₅₉	1	1	yes	characteristic
557	C₄	—	2	1	no	—
558	C₄	—	2	1	no	—
559	C₄	—	2	1	no	—
560	C₄	—	2	1	no	—
561	C₄	—	4	2	no	—
562	C₄	—	4	2	no	—
563	C₄	—	4	2	no	—
564	C₄	—	8	1	no	—
565	C₄	—	8	1	no	—
566	C₄	C₂×AGL(1,ℤ/8)×C₅₉	1	2	yes	—
567	C₄	C₅₉×#577	1	2	yes	—
568	C₂	—	4	1	no	—
569	C₂	—	4	4	no	—
570	C₂	—	8	2	no	—
571	C₂	—	8	2	no	—
572	C₂	C₅₉×#3003	1	1	yes	characteristic
573	C₂	C₅₉×#3009	1	1	yes	characteristic
574	C₂	C₅₉×#3018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₉	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₉
prime radical	C₅₉×#3496	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ 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
2. C₅₉×#3235	C₂	semidirect
3. C₅₉×#3182	C₂	semidirect
4. C₅₉×#3101	C₂	semidirect
5. C₅₉×#1647	C₂	semidirect
6. C₅₉×#1639	C₂	semidirect
7. C₅₉×#1635	C₂	semidirect
8. C₅₉×#1629	C₂	semidirect
9. C₅₉×#1553	C₂	semidirect
10. C₅₉×#1490	C₂	semidirect
11. C₅₉×#1484	C₂	semidirect
21. C₅₉×#533	C₂²	semidirect
27. C₅₉×#500	C₂²	semidirect
28. C₅₉×#492	C₂²	semidirect
30. C₅₉×#487	C₂²	semidirect
31. C₅₉×#486	C₂²	semidirect
34. C₅₉×#476	C₂²	semidirect
35. C₅₉×#476	C₂²	semidirect
36. C₅₉×#474	C₂²	semidirect
38. C₅₉×#461	C₂²	semidirect
42. C₅₉×#432	C₂²	semidirect
43. C₅₉×#431	C₂²	semidirect
44. C₄×M₁₆×C₅₉	C₂²	semidirect
45. C₅₉×#332	C₂²	semidirect
46. C₅₉×#331	C₂²	semidirect
47. C₅₉×D₄⋊C₈	C₂²	semidirect
48. C₅₉×D₄⋊C₈	C₂²	semidirect
49. C₅₉×D₄⋊C₈	C₂²	semidirect
50. C₅₉×D₄⋊C₈	C₂²	semidirect
254. #18599	C₅₉	direct (N × H)
282. #3235	C₂×C₅₉	semidirect
283. #3182	C₂×C₅₉	semidirect
284. #3101	C₂×C₅₉	semidirect
285. #1647	C₂×C₅₉	semidirect
286. #1639	C₂×C₅₉	semidirect
287. #1635	C₂×C₅₉	semidirect
288. #1629	C₂×C₅₉	semidirect
289. #1553	C₂×C₅₉	semidirect
290. #1490	C₂×C₅₉	semidirect
291. #1484	C₂×C₅₉	semidirect
308. #533	C₂²×C₅₉	semidirect
314. #500	C₂²×C₅₉	semidirect
315. #492	C₂²×C₅₉	semidirect
317. #487	C₂²×C₅₉	semidirect
318. #486	C₂²×C₅₉	semidirect
321. #476	C₂²×C₅₉	semidirect
322. #476	C₂²×C₅₉	semidirect
323. #474	C₂²×C₅₉	semidirect
325. #461	C₂²×C₅₉	semidirect
329. #432	C₂²×C₅₉	semidirect
330. #431	C₂²×C₅₉	semidirect
331. C₄×M₁₆	C₂²×C₅₉	semidirect
332. #332	C₂²×C₅₉	semidirect
333. #331	C₂²×C₅₉	semidirect
334. D₄⋊C₈	C₂²×C₅₉	semidirect
335. D₄⋊C₈	C₂²×C₅₉	semidirect
336. D₄⋊C₈	C₂²×C₅₉	semidirect
337. D₄⋊C₈	C₂²×C₅₉	semidirect
338. C₅₉	#18599	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×#3496	2
C₅₉×#3235	2
C₅₉×#3182	2
C₅₉×#3101	2
C₅₉×#1647	2
C₅₉×#1639	2
C₅₉×#1635	2
C₅₉×#1629	2
C₅₉×#1553	2
C₅₉×#1490	2
C₅₉×#1484	2
#18599	59

Subgroups — C59×#18599 #28430708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#18599 #28430708