Subgroups — C₅₉×#24225 #28436334

Order 15104.

1366 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#3497	C₂	1	1	yes	maximal, characteristic
2	C₅₉×#3037	C₂	1	2	yes	maximal, complemented
3	C₂×C₅₉×#456	C₂	1	2	yes	maximal, complemented
4	C₄×C₄⋊Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#2039	C₂	1	2	yes	maximal, complemented
6	C₅₉×#2007	C₂	1	2	yes	maximal, complemented
7	C₅₉×#1962	C₂	1	2	yes	maximal, complemented
8	C₅₉×#1801	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₉×#1793	C₂	1	2	yes	maximal, complemented
10	C₂×2₋¹⁺⁴×C₅₉	C₂²	1	2	yes	—
11	C₂×#137×C₅₉	C₂²	1	4	yes	—
12	C₅₉×#550	—	2	2	no	—
13	D₄×Q₈×C₅₉	—	2	2	no	—
14	C₅₉×#548	—	2	4	no	—
15	C₅₉×#537	—	2	2	no	—
16	C₅₉×#537	C₂²	1	2	yes	—
17	C₅₉×#532	—	2	2	no	—
18	C₅₉×#532	C₂²	1	2	yes	—
19	C₂×C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
22	C₅₉×#464	—	2	4	no	—
23	C₅₉×#462	—	2	2	no	—
24	C₅₉×#460	—	2	2	no	—
25	C₅₉×#456	—	2	8	no	—
26	C₅₉×#423	C₂²	1	2	yes	—
27	C₂×#104×C₅₉	C₂²	1	4	yes	—
28	C₂×#104×C₅₉	C₂²	1	4	yes	—
29	C₅₉×#417	C₂²	1	4	yes	—
30	C₂×#101×C₅₉	C₂²	1	2	yes	—
31	C₅₉×#391	C₂²	1	1	yes	characteristic
32	C₅₉×#391	C₂²	1	1	yes	characteristic
33	C₅₉×#389	—	2	2	no	—
34	C₅₉×#384	C₂²	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₅₉	—	2	2	no	—
36	C₄³×C₅₉	C₂²	1	1	yes	characteristic
37	C₅₉×#343	C₂²	1	2	yes	—
38	2₋¹⁺⁴×C₅₉	—	2	4	no	—
39	2₋¹⁺⁴×C₅₉	—	2	4	no	—
40	2₋¹⁺⁴×C₅₉	—	4	4	no	—
41	C₂×#41×C₅₉	—	2	4	no	—
42	C₂×#41×C₅₉	—	2	4	no	—
43	C₂×#41×C₅₉	C₂³	1	4	yes	—
44	C₂²×Q₈×C₅₉	—	2	4	no	—
45	C₂²×Q₈×C₅₉	C₂³	1	1	yes	characteristic
46	C₂²×Q₈×C₅₉	C₂³	1	1	yes	characteristic
47	#137×C₅₉	—	2	8	no	—
48	#137×C₅₉	—	4	4	no	—
49	C₂×Q₁₆×C₅₉	—	2	4	no	—
50	C₂×SD₁₆×C₅₉	—	2	4	no	—
51	C₂×M₁₆×C₅₉	C₂³	1	4	yes	—
52	C₄⋊Q₈×C₅₉	—	2	2	no	—
53	C₄⋊Q₈×C₅₉	—	2	2	no	—
54	C₄⋊Q₈×C₅₉	D₄	1	2	yes	—
55	C₄⋊Q₈×C₅₉	D₄	1	4	yes	complemented
56	#127×C₅₉	—	2	2	no	—
57	#127×C₅₉	D₄	1	2	yes	—
58	#124×C₅₉	—	2	4	no	—
59	#124×C₅₉	D₄	1	4	yes	—
60	C₂²⋊Q₈×C₅₉	—	2	8	no	—
61	C₂²⋊Q₈×C₅₉	—	2	4	no	—
62	#121×C₅₉	—	2	8	no	—
63	#121×C₅₉	—	2	4	no	—
64	#121×C₅₉	—	2	2	no	—
65	#121×C₅₉	—	2	4	no	—
66	#121×C₅₉	—	2	2	no	—
67	C₄×Q₈×C₅₉	—	2	2	no	—
68	C₄×Q₈×C₅₉	—	2	2	no	—
69	C₄×Q₈×C₅₉	—	2	2	no	—
70	C₄×Q₈×C₅₉	—	2	2	no	—
71	C₄×D₄×C₅₉	—	2	4	no	—
72	C₄×D₄×C₅₉	—	2	4	no	—
73	C₄×D₄×C₅₉	—	2	2	no	—
74	C₄×D₄×C₅₉	—	2	2	no	—
75	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
76	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
77	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
78	C₂×C₄⋊C₄×C₅₉	D₄	1	2	yes	—
79	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
81	C₂×C₄²×C₅₉	—	2	1	no	—
82	C₂×C₄²×C₅₉	—	2	1	no	—
83	C₂×C₄²×C₅₉	—	2	1	no	—
84	C₂×C₄²×C₅₉	D₄	1	2	yes	—
85	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
86	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
87	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
88	C₄⋊C₈×C₅₉	—	2	2	no	—
89	C₄⋊C₈×C₅₉	—	2	2	no	—
90	#104×C₅₉	—	2	8	no	—
91	#104×C₅₉	—	4	4	no	—
92	#103×C₅₉	—	2	4	no	—
93	#102×C₅₉	—	2	4	no	—
94	#101×C₅₉	—	2	4	no	—
95	#95×C₅₉	—	2	2	no	—
96	#95×C₅₉	—	2	2	no	—
97	#41×C₅₉	—	2	4	no	—
98	#41×C₅₉	—	2	4	no	—
99	#41×C₅₉	—	2	4	no	—
100	#41×C₅₉	—	2	4	no	—
101	#41×C₅₉	—	4	8	no	—
102	#41×C₅₉	—	4	8	no	—
103	#41×C₅₉	—	4	8	no	—
104	#41×C₅₉	—	4	8	no	—
105	C₂×Q₈×C₅₉	—	2	4	no	—
106	C₂×Q₈×C₅₉	—	2	2	no	—
107	C₂×Q₈×C₅₉	—	2	1	no	—
108	C₂×Q₈×C₅₉	—	2	2	no	—
109	C₂×Q₈×C₅₉	—	2	2	no	—
110	C₂×Q₈×C₅₉	—	2	2	no	—
111	C₂×Q₈×C₅₉	—	4	8	no	—
112	C₂×Q₈×C₅₉	—	4	8	no	—
113	C₂×Q₈×C₅₉	C₂×D₄	1	4	yes	—
114	C₂×Q₈×C₅₉	C₂×D₄	1	2	yes	—
115	C₂×Q₈×C₅₉	C₂×D₄	1	2	yes	—
116	C₂×Q₈×C₅₉	#41	1	2	yes	—
117	C₂×D₄×C₅₉	—	2	4	no	—
118	C₂×D₄×C₅₉	—	2	4	no	—
119	C₂×D₄×C₅₉	—	2	4	no	—
120	C₂×D₄×C₅₉	—	2	2	no	—
121	C₂×D₄×C₅₉	—	2	2	no	—
122	C₂×D₄×C₅₉	C₂×D₄	1	4	yes	—
123	C₂²×C₄×C₅₉	—	2	1	no	—
124	C₂²×C₄×C₅₉	—	2	8	no	—
125	C₂²×C₄×C₅₉	—	2	1	no	—
126	C₂²×C₄×C₅₉	—	2	1	no	—
127	C₂²×C₄×C₅₉	—	2	4	no	—
128	C₂²×C₄×C₅₉	—	2	1	no	—
129	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
131	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
132	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
134	Q₁₆×C₅₉	—	4	8	no	—
135	SD₁₆×C₅₉	—	4	8	no	—
136	M₁₆×C₅₉	—	2	4	no	—
137	M₁₆×C₅₉	—	2	4	no	—
138	C₂×C₈×C₅₉	—	2	4	no	—
139	C₄⋊C₄×C₅₉	—	2	4	no	—
140	C₄⋊C₄×C₅₉	—	2	4	no	—
141	C₄⋊C₄×C₅₉	—	2	2	no	—
142	C₄⋊C₄×C₅₉	—	2	2	no	—
143	C₄⋊C₄×C₅₉	—	2	8	no	—
144	C₄⋊C₄×C₅₉	—	2	2	no	—
145	C₄⋊C₄×C₅₉	—	2	2	no	—
146	C₄⋊C₄×C₅₉	—	4	4	no	—
147	C₄⋊C₄×C₅₉	—	4	2	no	—
148	C₂²⋊C₄×C₅₉	—	2	8	no	—
149	C₂²⋊C₄×C₅₉	—	2	4	no	—
150	C₂²⋊C₄×C₅₉	—	2	4	no	—
151	C₄²×C₅₉	—	2	4	no	—
152	C₄²×C₅₉	—	2	2	no	—
153	C₄²×C₅₉	—	2	2	no	—
154	C₄²×C₅₉	—	2	1	no	—
155	C₄²×C₅₉	—	2	2	no	—
156	C₄²×C₅₉	—	4	2	no	—
157	C₄²×C₅₉	—	4	1	no	—
158	C₄²×C₅₉	—	4	1	no	—
159	C₄²×C₅₉	C₂×D₄	1	2	yes	—
160	C₄²×C₅₉	C₂×D₄	1	2	yes	—
161	C₄²×C₅₉	C₂×D₄	1	2	yes	—
162	C₂³×C₅₉	—	2	4	no	—
163	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
164	Q₈×C₅₉	—	2	4	no	—
165	Q₈×C₅₉	—	2	4	no	—
166	Q₈×C₅₉	—	2	4	no	—
167	Q₈×C₅₉	—	2	4	no	—
168	Q₈×C₅₉	—	2	4	no	—
169	Q₈×C₅₉	—	4	8	no	—
170	Q₈×C₅₉	—	4	2	no	—
171	D₄×C₅₉	—	2	4	no	—
172	D₄×C₅₉	—	2	4	no	—
173	D₄×C₅₉	—	4	8	no	—
174	D₄×C₅₉	—	4	4	no	—
175	D₄×C₅₉	—	4	4	no	—
176	D₄×C₅₉	—	4	4	no	—
177	D₄×C₅₉	—	8	2	no	—
178	D₄×C₅₉	—	8	4	no	—
179	C₂×C₄×C₅₉	—	2	2	no	—
180	C₂×C₄×C₅₉	—	2	2	no	—
181	C₂×C₄×C₅₉	—	2	2	no	—
182	C₂×C₄×C₅₉	—	2	2	no	—
183	C₂×C₄×C₅₉	—	2	2	no	—
184	C₂×C₄×C₅₉	—	2	2	no	—
185	C₂×C₄×C₅₉	—	2	2	no	—
186	C₂×C₄×C₅₉	—	2	2	no	—
187	C₂×C₄×C₅₉	—	2	4	no	—
188	C₂×C₄×C₅₉	—	2	2	no	—
189	C₂×C₄×C₅₉	—	4	1	no	—
190	C₂×C₄×C₅₉	—	4	16	no	—
191	C₂×C₄×C₅₉	—	4	1	no	—
192	C₂×C₄×C₅₉	—	4	1	no	—
193	C₂×C₄×C₅₉	—	4	1	no	—
194	C₂×C₄×C₅₉	—	4	1	no	—
195	C₂×C₄×C₅₉	—	4	4	no	—
196	C₂×C₄×C₅₉	—	4	4	no	—
197	C₂×C₄×C₅₉	—	4	2	no	—
198	C₂×C₄×C₅₉	—	4	2	no	—
199	C₂×C₄×C₅₉	—	4	1	no	—
200	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
201	C₂×C₄×C₅₉	#120	1	2	yes	—
202	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
203	C₂×C₄×C₅₉	#121	1	4	yes	—
204	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
205	C₂×C₄×C₅₉	C₂²×D₄	1	2	yes	—
206	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
207	C₂×C₄×C₅₉	H(5,𝔽₂)	1	2	yes	—
208	C₈×C₅₉	—	4	4	no	—
209	#24225	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
210	C₂²×C₅₉	—	2	1	no	—
211	C₂²×C₅₉	—	2	1	no	—
212	C₂²×C₅₉	—	4	4	no	—
213	C₂²×C₅₉	—	4	4	no	—
214	C₂²×C₅₉	—	4	4	no	—
215	C₂²×C₅₉	C₂×#120	1	1	yes	characteristic
216	C₂²×C₅₉	C₂×#121	1	1	yes	characteristic
217	C₂²×C₅₉	#535	1	1	yes	characteristic
218	C₄×C₅₉	—	2	4	no	—
219	C₄×C₅₉	—	2	4	no	—
220	C₄×C₅₉	—	2	1	no	—
221	C₄×C₅₉	—	2	2	no	—
222	C₄×C₅₉	—	4	2	no	—
223	C₄×C₅₉	—	4	4	no	—
224	C₄×C₅₉	—	4	2	no	—
225	C₄×C₅₉	—	4	2	no	—
226	C₄×C₅₉	—	8	1	no	—
227	C₄×C₅₉	#456	1	2	yes	complemented
228	C₄×C₅₉	D₄²	1	2	yes	—
229	C₄×C₅₉	#546	1	2	yes	—
230	#3497	C₂×C₅₉	1	1	yes	characteristic
231	#3037	C₂×C₅₉	1	2	yes	complemented
232	C₂×#456	C₂×C₅₉	1	2	yes	complemented
233	C₄×C₄⋊Q₈	C₂×C₅₉	1	1	yes	characteristic, complemented
234	#2039	C₂×C₅₉	1	2	yes	complemented
235	#2007	C₂×C₅₉	1	2	yes	complemented
236	#1962	C₂×C₅₉	1	2	yes	complemented
237	#1801	C₂×C₅₉	1	1	yes	characteristic, complemented
238	#1793	C₂×C₅₉	1	2	yes	complemented
239	C₂×C₅₉	—	2	1	no	—
240	C₂×C₅₉	—	2	1	no	—
241	C₂×C₅₉	—	8	4	no	—
242	C₂×C₅₉	#2408	1	1	yes	characteristic
243	C₂×C₅₉	C₂×#456	1	1	yes	characteristic
244	C₂×C₅₉	#3018	1	1	yes	characteristic
245	C₂×2₋¹⁺⁴	C₂²×C₅₉	1	2	yes	—
246	C₂×#137	C₂²×C₅₉	1	4	yes	—
247	#550	—	2	2	no	—
248	D₄×Q₈	—	2	2	no	—
249	#548	—	2	4	no	—
250	#537	—	2	2	no	—
251	#537	C₂²×C₅₉	1	2	yes	—
252	#532	—	2	2	no	—
253	#532	C₂²×C₅₉	1	2	yes	—
254	C₂×C₄⋊Q₈	C₂²×C₅₉	1	1	yes	characteristic
255	C₂×C₄×Q₈	C₂²×C₅₉	1	1	yes	characteristic
256	C₂×C₄×Q₈	C₂²×C₅₉	1	1	yes	characteristic
257	#464	—	2	4	no	—
258	#462	—	2	2	no	—
259	#460	—	2	2	no	—
260	#456	—	2	8	no	—
261	#423	C₂²×C₅₉	1	2	yes	—
262	C₂×#104	C₂²×C₅₉	1	4	yes	—
263	C₂×#104	C₂²×C₅₉	1	4	yes	—
264	#417	C₂²×C₅₉	1	4	yes	—
265	C₂×#101	C₂²×C₅₉	1	2	yes	—
266	#391	C₂²×C₅₉	1	1	yes	characteristic
267	#391	C₂²×C₅₉	1	1	yes	characteristic
268	#389	—	2	2	no	—
269	#384	C₂²×C₅₉	1	1	yes	characteristic
270	C₄×C₄⋊C₄	—	2	2	no	—
271	C₄³	C₂²×C₅₉	1	1	yes	characteristic
272	#343	C₂²×C₅₉	1	2	yes	—
273	C₅₉	#24225	1	1	yes	characteristic, direct product factor, Sylow
274	2₋¹⁺⁴	—	2	4	no	—
275	2₋¹⁺⁴	—	2	4	no	—
276	2₋¹⁺⁴	—	4	4	no	—
277	C₂×#41	—	2	4	no	—
278	C₂×#41	—	2	4	no	—
279	C₂×#41	C₂³×C₅₉	1	4	yes	—
280	C₂²×Q₈	—	2	4	no	—
281	C₂²×Q₈	C₂³×C₅₉	1	1	yes	characteristic
282	C₂²×Q₈	C₂³×C₅₉	1	1	yes	characteristic
283	#137	—	2	8	no	—
284	#137	—	4	4	no	—
285	C₂×Q₁₆	—	2	4	no	—
286	C₂×SD₁₆	—	2	4	no	—
287	C₂×M₁₆	C₂³×C₅₉	1	4	yes	—
288	C₄⋊Q₈	—	2	2	no	—
289	C₄⋊Q₈	—	2	2	no	—
290	C₄⋊Q₈	D₄×C₅₉	1	2	yes	—
291	C₄⋊Q₈	D₄×C₅₉	1	4	yes	complemented
292	#127	—	2	2	no	—
293	#127	D₄×C₅₉	1	2	yes	—
294	#124	—	2	4	no	—
295	#124	D₄×C₅₉	1	4	yes	—
296	C₂²⋊Q₈	—	2	8	no	—
297	C₂²⋊Q₈	—	2	4	no	—
298	#121	—	2	8	no	—
299	#121	—	2	4	no	—
300	#121	—	2	2	no	—
301	#121	—	2	4	no	—
302	#121	—	2	2	no	—
303	C₄×Q₈	—	2	2	no	—
304	C₄×Q₈	—	2	2	no	—
305	C₄×Q₈	—	2	2	no	—
306	C₄×Q₈	—	2	2	no	—
307	C₄×D₄	—	2	4	no	—
308	C₄×D₄	—	2	4	no	—
309	C₄×D₄	—	2	2	no	—
310	C₄×D₄	—	2	2	no	—
311	C₂×C₄⋊C₄	—	2	2	no	—
312	C₂×C₄⋊C₄	—	2	1	no	—
313	C₂×C₄⋊C₄	—	2	2	no	—
314	C₂×C₄⋊C₄	D₄×C₅₉	1	2	yes	—
315	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
316	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
317	C₂×C₄²	—	2	1	no	—
318	C₂×C₄²	—	2	1	no	—
319	C₂×C₄²	—	2	1	no	—
320	C₂×C₄²	D₄×C₅₉	1	2	yes	—
321	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
322	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
323	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
324	C₄⋊C₈	—	2	2	no	—
325	C₄⋊C₈	—	2	2	no	—
326	#104	—	2	8	no	—
327	#104	—	4	4	no	—
328	#103	—	2	4	no	—
329	#102	—	2	4	no	—
330	#101	—	2	4	no	—
331	#95	—	2	2	no	—
332	#95	—	2	2	no	—
333	#41	—	2	4	no	—
334	#41	—	2	4	no	—
335	#41	—	2	4	no	—
336	#41	—	2	4	no	—
337	#41	—	4	8	no	—
338	#41	—	4	8	no	—
339	#41	—	4	8	no	—
340	#41	—	4	8	no	—
341	C₂×Q₈	—	2	4	no	—
342	C₂×Q₈	—	2	2	no	—
343	C₂×Q₈	—	2	1	no	—
344	C₂×Q₈	—	2	2	no	—
345	C₂×Q₈	—	2	2	no	—
346	C₂×Q₈	—	2	2	no	—
347	C₂×Q₈	—	4	8	no	—
348	C₂×Q₈	—	4	8	no	—
349	C₂×Q₈	C₂×D₄×C₅₉	1	4	yes	—
350	C₂×Q₈	C₂×D₄×C₅₉	1	2	yes	—
351	C₂×Q₈	C₂×D₄×C₅₉	1	2	yes	—
352	C₂×Q₈	#41×C₅₉	1	2	yes	—
353	C₂×D₄	—	2	4	no	—
354	C₂×D₄	—	2	4	no	—
355	C₂×D₄	—	2	4	no	—
356	C₂×D₄	—	2	2	no	—
357	C₂×D₄	—	2	2	no	—
358	C₂×D₄	C₂×D₄×C₅₉	1	4	yes	—
359	C₂²×C₄	—	2	1	no	—
360	C₂²×C₄	—	2	8	no	—
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	4	no	—
364	C₂²×C₄	—	2	1	no	—
365	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
366	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
367	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
368	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
369	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
370	Q₁₆	—	4	8	no	—
371	SD₁₆	—	4	8	no	—
372	M₁₆	—	2	4	no	—
373	M₁₆	—	2	4	no	—
374	C₂×C₈	—	2	4	no	—
375	C₄⋊C₄	—	2	4	no	—
376	C₄⋊C₄	—	2	4	no	—
377	C₄⋊C₄	—	2	2	no	—
378	C₄⋊C₄	—	2	2	no	—
379	C₄⋊C₄	—	2	8	no	—
380	C₄⋊C₄	—	2	2	no	—
381	C₄⋊C₄	—	2	2	no	—
382	C₄⋊C₄	—	4	4	no	—
383	C₄⋊C₄	—	4	2	no	—
384	C₂²⋊C₄	—	2	8	no	—
385	C₂²⋊C₄	—	2	4	no	—
386	C₂²⋊C₄	—	2	4	no	—
387	C₄²	—	2	4	no	—
388	C₄²	—	2	2	no	—
389	C₄²	—	2	2	no	—
390	C₄²	—	2	1	no	—
391	C₄²	—	2	2	no	—
392	C₄²	—	4	2	no	—
393	C₄²	—	4	1	no	—
394	C₄²	—	4	1	no	—
395	C₄²	C₂×D₄×C₅₉	1	2	yes	—
396	C₄²	C₂×D₄×C₅₉	1	2	yes	—
397	C₄²	C₂×D₄×C₅₉	1	2	yes	—
398	C₂³	—	2	4	no	—
399	C₂³	C₂²×D₄×C₅₉	1	1	yes	characteristic
400	Q₈	—	2	4	no	—
401	Q₈	—	2	4	no	—
402	Q₈	—	2	4	no	—
403	Q₈	—	2	4	no	—
404	Q₈	—	2	4	no	—
405	Q₈	—	4	8	no	—
406	Q₈	—	4	2	no	—
407	D₄	—	2	4	no	—
408	D₄	—	2	4	no	—
409	D₄	—	4	8	no	—
410	D₄	—	4	4	no	—
411	D₄	—	4	4	no	—
412	D₄	—	4	4	no	—
413	D₄	—	8	2	no	—
414	D₄	—	8	4	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	4	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	4	1	no	—
426	C₂×C₄	—	4	16	no	—
427	C₂×C₄	—	4	1	no	—
428	C₂×C₄	—	4	1	no	—
429	C₂×C₄	—	4	1	no	—
430	C₂×C₄	—	4	1	no	—
431	C₂×C₄	—	4	4	no	—
432	C₂×C₄	—	4	4	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	2	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	#120×C₅₉	1	1	yes	characteristic
437	C₂×C₄	#120×C₅₉	1	2	yes	—
438	C₂×C₄	#120×C₅₉	1	1	yes	characteristic
439	C₂×C₄	#121×C₅₉	1	4	yes	—
440	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
441	C₂×C₄	C₂²×D₄×C₅₉	1	2	yes	—
442	C₂×C₄	C₂×#41×C₅₉	1	1	yes	characteristic
443	C₂×C₄	H(5,𝔽₂)×C₅₉	1	2	yes	—
444	C₈	—	4	4	no	—
445	C₂²	—	2	1	no	—
446	C₂²	—	2	1	no	—
447	C₂²	—	4	4	no	—
448	C₂²	—	4	4	no	—
449	C₂²	—	4	4	no	—
450	C₂²	C₂×#120×C₅₉	1	1	yes	characteristic
451	C₂²	C₂×#121×C₅₉	1	1	yes	characteristic
452	C₂²	C₅₉×#535	1	1	yes	characteristic
453	C₄	—	2	4	no	—
454	C₄	—	2	1	no	—
455	C₄	—	2	2	no	—
456	C₄	—	2	4	no	—
457	C₄	—	4	2	no	—
458	C₄	—	4	2	no	—
459	C₄	—	4	4	no	—
460	C₄	—	4	2	no	—
461	C₄	—	8	1	no	—
462	C₄	C₅₉×#456	1	2	yes	complemented
463	C₄	D₄²×C₅₉	1	2	yes	—
464	C₄	C₅₉×#546	1	2	yes	—
465	C₂	—	2	1	no	—
466	C₂	—	2	1	no	—
467	C₂	—	8	4	no	—
468	C₂	C₅₉×#2408	1	1	yes	characteristic
469	C₂	C₂×C₅₉×#456	1	1	yes	characteristic
470	C₂	C₅₉×#3018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₉	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₉
prime radical	C₅₉×#3497	C₂
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
2. C₅₉×#3037	C₂	semidirect
3. C₂×C₅₉×#456	C₂	semidirect
4. C₄×C₄⋊Q₈×C₅₉	C₂	semidirect
5. C₅₉×#2039	C₂	semidirect
6. C₅₉×#2007	C₂	semidirect
7. C₅₉×#1962	C₂	semidirect
8. C₅₉×#1801	C₂	semidirect
9. C₅₉×#1793	C₂	semidirect
55. C₄⋊Q₈×C₅₉	D₄	semidirect
209. #24225	C₅₉	direct (N × H)
227. C₄×C₅₉	#456	semidirect
231. #3037	C₂×C₅₉	semidirect
232. C₂×#456	C₂×C₅₉	semidirect
233. C₄×C₄⋊Q₈	C₂×C₅₉	semidirect
234. #2039	C₂×C₅₉	semidirect
235. #2007	C₂×C₅₉	semidirect
236. #1962	C₂×C₅₉	semidirect
237. #1801	C₂×C₅₉	semidirect
238. #1793	C₂×C₅₉	semidirect
273. C₅₉	#24225	direct (N × H)
291. C₄⋊Q₈	D₄×C₅₉	semidirect
462. C₄	C₅₉×#456	semidirect

Maximal subgroups

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

subgroup	index
C₅₉×#3497	2
C₅₉×#3037	2
C₂×C₅₉×#456	2
C₄×C₄⋊Q₈×C₅₉	2
C₅₉×#2039	2
C₅₉×#2007	2
C₅₉×#1962	2
C₅₉×#1801	2
C₅₉×#1793	2
#24225	59

Subgroups — C59×#24225 #28436334

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#24225 #28436334