Subgroups — C₅₉×#26368 #28438477

Order 15104.

968 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₉×#474	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₈×C₅₉	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#2452	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄×C₄⋊C₄×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#2093	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₉×#2083	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#2067	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₉×#2057	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₉×#1942	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₉×#1916	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₉×#1879	C₂	1	1	yes	maximal, characteristic
12	C₅₉×#1879	C₂	1	1	yes	maximal, characteristic
13	C₅₉×#1862	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅₉×#1862	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₉×#1862	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×D₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊Q₈×C₅₉	—	2	1	no	—
19	C₂×C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₅₉	—	2	1	no	—
24	C₅₉×#474	—	2	4	no	—
25	C₄×D₈×C₅₉	—	2	4	no	—
26	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
27	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
28	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
29	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
30	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
32	C₂×#102×C₅₉	C₂²	1	1	yes	characteristic
33	C₂×#102×C₅₉	C₂²	1	1	yes	characteristic
34	C₂×#102×C₅₉	C₂²	1	1	yes	characteristic
35	C₂×#102×C₅₉	C₂²	1	1	yes	characteristic
36	C₂×#102×C₅₉	C₂²	1	1	yes	characteristic
37	C₂×#102×C₅₉	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
39	C₅₉×#396	—	2	1	no	—
40	C₅₉×#393	—	2	1	no	—
41	C₅₉×#390	C₂²	1	1	yes	characteristic
42	C₅₉×#390	C₂²	1	1	yes	characteristic
43	C₅₉×#389	C₂²	1	1	yes	characteristic
44	C₅₉×#389	C₂²	1	1	yes	characteristic
45	C₅₉×#389	C₂²	1	1	yes	characteristic
46	C₅₉×#389	C₂²	1	1	yes	characteristic
47	C₅₉×#385	—	2	1	no	—
48	C₅₉×#385	—	2	1	no	—
49	C₅₉×#385	—	2	1	no	—
50	C₅₉×#384	C₂²	1	1	yes	characteristic
51	C₅₉×#380	—	2	1	no	—
52	C₄×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
53	C₅₉×#340	C₂²	1	1	yes	characteristic
54	C₅₉×#340	C₂²	1	1	yes	characteristic
55	C₅₉×#340	C₂²	1	1	yes	characteristic
56	C₅₉×#340	C₂²	1	1	yes	characteristic
57	C₅₉×#336	C₂²	1	1	yes	characteristic
58	C₅₉×#336	C₂²	1	1	yes	characteristic
59	C₅₉×#336	C₂²	1	1	yes	characteristic
60	C₅₉×#336	C₂²	1	1	yes	characteristic
61	C₂²×Q₈×C₅₉	—	2	1	no	—
62	C₂²×D₄×C₅₉	C₂³	1	1	yes	characteristic
63	C₂²×D₄×C₅₉	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₅₉	—	2	1	no	—
65	C₂³×C₄×C₅₉	—	2	1	no	—
66	C₂³×C₄×C₅₉	—	2	1	no	—
67	C₂³×C₄×C₅₉	—	2	1	no	—
68	C₂×D₈×C₅₉	—	2	2	no	—
69	C₂×D₈×C₅₉	—	2	2	no	—
70	C₂×D₈×C₅₉	Q₈	1	4	yes	complemented
71	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
75	C₄⋊Q₈×C₅₉	—	2	2	no	—
76	C₄⋊Q₈×C₅₉	—	4	1	no	—
77	C₂²⋊Q₈×C₅₉	—	4	4	no	—
78	C₄×D₄×C₅₉	—	2	2	no	—
79	C₄×D₄×C₅₉	—	2	2	no	—
80	C₄×D₄×C₅₉	—	2	2	no	—
81	C₄×D₄×C₅₉	—	2	2	no	—
82	C₄×D₄×C₅₉	—	2	2	no	—
83	C₄×D₄×C₅₉	—	4	1	no	—
84	C₄×D₄×C₅₉	—	4	1	no	—
85	C₄×D₄×C₅₉	—	4	1	no	—
86	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
87	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
88	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
89	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
90	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
91	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
92	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
93	C₂×C₄⋊C₄×C₅₉	—	4	4	no	—
94	C₂×C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
95	C₂×C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
96	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
97	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
101	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
102	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
103	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
104	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
105	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
106	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
107	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
108	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
109	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
110	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
111	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
112	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
113	#107×C₅₉	—	2	2	no	—
114	#107×C₅₉	—	2	2	no	—
115	#107×C₅₉	—	2	2	no	—
116	#107×C₅₉	—	2	2	no	—
117	#107×C₅₉	—	4	1	no	—
118	C₄⋊C₈×C₅₉	—	2	2	no	—
119	#102×C₅₉	—	2	2	no	—
120	#102×C₅₉	—	2	2	no	—
121	#102×C₅₉	—	2	2	no	—
122	#102×C₅₉	—	2	2	no	—
123	#102×C₅₉	—	4	1	no	—
124	#102×C₅₉	D₄	1	4	yes	—
125	C₄×C₈×C₅₉	—	2	2	no	—
126	#95×C₅₉	—	2	1	no	—
127	#95×C₅₉	—	2	1	no	—
128	#95×C₅₉	—	2	1	no	—
129	#95×C₅₉	—	2	1	no	—
130	#95×C₅₉	D₄	1	1	yes	characteristic
131	#95×C₅₉	D₄	1	1	yes	characteristic
132	C₂⁴×C₅₉	—	2	1	no	—
133	C₂⁴×C₅₉	—	2	1	no	—
134	C₂×Q₈×C₅₉	—	4	2	no	—
135	C₂×Q₈×C₅₉	—	4	2	no	—
136	C₂×Q₈×C₅₉	—	8	1	no	—
137	C₂×D₄×C₅₉	—	2	1	no	—
138	C₂×D₄×C₅₉	—	2	1	no	—
139	C₂×D₄×C₅₉	—	2	1	no	—
140	C₂×D₄×C₅₉	—	2	1	no	—
141	C₂×D₄×C₅₉	—	2	1	no	—
142	C₂×D₄×C₅₉	—	2	1	no	—
143	C₂×D₄×C₅₉	—	2	1	no	—
144	C₂×D₄×C₅₉	—	2	1	no	—
145	C₂×D₄×C₅₉	C₂×D₄	1	2	yes	—
146	C₂×D₄×C₅₉	C₂×Q₈	1	2	yes	—
147	C₂×D₄×C₅₉	C₂×Q₈	1	2	yes	—
148	C₂×D₄×C₅₉	#41	1	2	yes	—
149	C₂²×C₄×C₅₉	—	2	4	no	—
150	C₂²×C₄×C₅₉	—	2	1	no	—
151	C₂²×C₄×C₅₉	—	2	1	no	—
152	C₂²×C₄×C₅₉	—	2	1	no	—
153	C₂²×C₄×C₅₉	—	2	1	no	—
154	C₂²×C₄×C₅₉	—	2	1	no	—
155	C₂²×C₄×C₅₉	—	2	1	no	—
156	C₂²×C₄×C₅₉	—	4	2	no	—
157	C₂²×C₄×C₅₉	—	4	2	no	—
158	C₂²×C₄×C₅₉	—	4	2	no	—
159	C₂²×C₄×C₅₉	—	4	2	no	—
160	C₂²×C₄×C₅₉	—	4	2	no	—
161	C₂²×C₄×C₅₉	—	4	2	no	—
162	C₂²×C₄×C₅₉	—	4	2	no	—
163	C₂²×C₄×C₅₉	—	4	2	no	—
164	C₂²×C₄×C₅₉	—	8	1	no	—
165	C₂²×C₄×C₅₉	—	8	1	no	—
166	C₂²×C₄×C₅₉	—	8	1	no	—
167	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
174	D₈×C₅₉	—	2	8	no	—
175	C₂×C₈×C₅₉	—	2	1	no	—
176	C₂×C₈×C₅₉	—	2	1	no	—
177	C₂×C₈×C₅₉	—	2	1	no	—
178	C₂×C₈×C₅₉	—	2	1	no	—
179	C₂×C₈×C₅₉	—	2	1	no	—
180	C₂×C₈×C₅₉	—	2	1	no	—
181	C₂×C₈×C₅₉	—	2	1	no	—
182	C₂×C₈×C₅₉	—	2	1	no	—
183	C₂×C₈×C₅₉	—	2	1	no	—
184	C₂×C₈×C₅₉	—	2	1	no	—
185	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
186	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
187	C₄⋊C₄×C₅₉	—	2	1	no	—
188	C₄⋊C₄×C₅₉	—	2	1	no	—
189	C₄⋊C₄×C₅₉	—	2	1	no	—
190	C₄⋊C₄×C₅₉	—	2	1	no	—
191	C₄⋊C₄×C₅₉	—	2	1	no	—
192	C₄⋊C₄×C₅₉	—	2	1	no	—
193	C₄⋊C₄×C₅₉	—	2	1	no	—
194	C₄⋊C₄×C₅₉	—	2	1	no	—
195	C₄⋊C₄×C₅₉	—	2	1	no	—
196	C₄⋊C₄×C₅₉	—	2	1	no	—
197	C₄⋊C₄×C₅₉	—	4	1	no	—
198	C₄⋊C₄×C₅₉	—	4	2	no	—
199	C₄⋊C₄×C₅₉	—	4	2	no	—
200	C₄⋊C₄×C₅₉	—	4	2	no	—
201	C₄⋊C₄×C₅₉	—	4	2	no	—
202	C₄⋊C₄×C₅₉	—	4	2	no	—
203	C₄⋊C₄×C₅₉	—	4	2	no	—
204	C₄⋊C₄×C₅₉	—	8	1	no	—
205	C₄⋊C₄×C₅₉	D₈	1	4	yes	complemented
206	C₄⋊C₄×C₅₉	C₂×D₄	1	2	yes	—
207	C₄⋊C₄×C₅₉	#41	1	2	yes	—
208	C₂²⋊C₄×C₅₉	—	2	4	no	—
209	C₂²⋊C₄×C₅₉	—	4	2	no	—
210	C₂²⋊C₄×C₅₉	—	4	2	no	—
211	C₂²⋊C₄×C₅₉	—	4	2	no	—
212	C₂²⋊C₄×C₅₉	—	4	2	no	—
213	C₂²⋊C₄×C₅₉	—	4	2	no	—
214	C₂²⋊C₄×C₅₉	—	4	2	no	—
215	C₂²⋊C₄×C₅₉	—	8	1	no	—
216	C₄²×C₅₉	—	2	1	no	—
217	C₄²×C₅₉	—	2	1	no	—
218	C₄²×C₅₉	—	2	1	no	—
219	C₄²×C₅₉	—	2	1	no	—
220	C₄²×C₅₉	—	2	1	no	—
221	C₄²×C₅₉	—	2	1	no	—
222	C₂³×C₅₉	—	2	2	no	—
223	C₂³×C₅₉	—	2	2	no	—
224	C₂³×C₅₉	—	2	2	no	—
225	C₂³×C₅₉	—	2	2	no	—
226	C₂³×C₅₉	—	4	1	no	—
227	C₂³×C₅₉	—	4	1	no	—
228	C₂³×C₅₉	—	4	1	no	—
229	C₂³×C₅₉	—	4	1	no	—
230	C₂³×C₅₉	—	4	1	no	—
231	C₂³×C₅₉	—	4	1	no	—
232	C₂³×C₅₉	—	4	1	no	—
233	C₂³×C₅₉	—	4	1	no	—
234	C₂³×C₅₉	—	4	1	no	—
235	C₂³×C₅₉	—	4	1	no	—
236	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
237	Q₈×C₅₉	—	8	4	no	—
238	D₄×C₅₉	—	2	2	no	—
239	D₄×C₅₉	—	2	2	no	—
240	D₄×C₅₉	—	2	2	no	—
241	D₄×C₅₉	—	2	2	no	—
242	D₄×C₅₉	—	2	2	no	—
243	D₄×C₅₉	—	4	1	no	—
244	D₄×C₅₉	—	4	1	no	—
245	D₄×C₅₉	C₂²⋊Q₈	1	4	yes	complemented
246	C₂×C₄×C₅₉	—	2	1	no	—
247	C₂×C₄×C₅₉	—	2	1	no	—
248	C₂×C₄×C₅₉	—	2	1	no	—
249	C₂×C₄×C₅₉	—	2	1	no	—
250	C₂×C₄×C₅₉	—	2	1	no	—
251	C₂×C₄×C₅₉	—	2	1	no	—
252	C₂×C₄×C₅₉	—	2	1	no	—
253	C₂×C₄×C₅₉	—	2	2	no	—
254	C₂×C₄×C₅₉	—	2	1	no	—
255	C₂×C₄×C₅₉	—	2	1	no	—
256	C₂×C₄×C₅₉	—	2	2	no	—
257	C₂×C₄×C₅₉	—	2	1	no	—
258	C₂×C₄×C₅₉	—	2	1	no	—
259	C₂×C₄×C₅₉	—	2	1	no	—
260	C₂×C₄×C₅₉	—	2	1	no	—
261	C₂×C₄×C₅₉	—	2	1	no	—
262	C₂×C₄×C₅₉	—	2	1	no	—
263	C₂×C₄×C₅₉	—	4	8	no	—
264	C₂×C₄×C₅₉	—	4	1	no	—
265	C₂×C₄×C₅₉	—	4	1	no	—
266	C₂×C₄×C₅₉	—	4	1	no	—
267	C₂×C₄×C₅₉	—	4	1	no	—
268	C₂×C₄×C₅₉	—	4	1	no	—
269	C₂×C₄×C₅₉	—	4	1	no	—
270	C₂×C₄×C₅₉	—	4	1	no	—
271	C₂×C₄×C₅₉	—	4	1	no	—
272	C₂×C₄×C₅₉	—	4	1	no	—
273	C₂×C₄×C₅₉	—	4	1	no	—
274	C₂×C₄×C₅₉	—	4	1	no	—
275	C₂×C₄×C₅₉	—	4	1	no	—
276	C₂×C₄×C₅₉	—	4	1	no	—
277	C₂×C₄×C₅₉	—	4	1	no	—
278	C₂×C₄×C₅₉	—	4	1	no	—
279	C₂×C₄×C₅₉	—	4	1	no	—
280	C₂×C₄×C₅₉	—	8	4	no	—
281	C₂×C₄×C₅₉	—	8	4	no	—
282	C₂×C₄×C₅₉	—	8	4	no	—
283	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
284	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
285	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
286	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
287	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
288	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
289	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
290	C₂×C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
291	C₂×C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
292	C₈×C₅₉	—	2	2	no	—
293	C₈×C₅₉	—	2	2	no	—
294	C₈×C₅₉	—	4	1	no	—
295	C₈×C₅₉	—	4	1	no	—
296	#26368	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
297	C₂²×C₅₉	—	2	4	no	—
298	C₂²×C₅₉	—	4	2	no	—
299	C₂²×C₅₉	—	4	2	no	—
300	C₂²×C₅₉	—	4	2	no	—
301	C₂²×C₅₉	—	4	2	no	—
302	C₂²×C₅₉	—	4	2	no	—
303	C₂²×C₅₉	—	4	2	no	—
304	C₂²×C₅₉	—	4	2	no	—
305	C₂²×C₅₉	—	4	2	no	—
306	C₂²×C₅₉	—	4	2	no	—
307	C₂²×C₅₉	—	8	1	no	—
308	C₂²×C₅₉	—	8	1	no	—
309	C₂²×C₅₉	—	8	1	no	—
310	C₂²×C₅₉	—	8	1	no	—
311	C₂²×C₅₉	#525	1	1	yes	characteristic
312	C₂²×C₅₉	D₄²	1	1	yes	characteristic
313	C₂²×C₅₉	D₄×Q₈	1	1	yes	characteristic
314	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
315	C₂²×C₅₉	#575	1	1	yes	characteristic
316	C₂²×C₅₉	#576	1	1	yes	characteristic
317	C₂²×C₅₉	#578	1	1	yes	characteristic
318	C₄×C₅₉	—	2	2	no	—
319	C₄×C₅₉	—	2	2	no	—
320	C₄×C₅₉	—	2	2	no	—
321	C₄×C₅₉	—	4	2	no	—
322	C₄×C₅₉	—	4	1	no	—
323	C₄×C₅₉	—	4	1	no	—
324	C₄×C₅₉	—	4	1	no	—
325	C₄×C₅₉	—	4	2	no	—
326	C₄×C₅₉	—	8	1	no	—
327	C₄×C₅₉	—	8	1	no	—
328	C₄×C₅₉	—	8	1	no	—
329	C₄×C₅₉	—	8	1	no	—
330	C₄×C₅₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
331	C₄×C₅₉	#539	1	1	yes	characteristic
332	C₄×C₅₉	#547	1	1	yes	characteristic
333	C₄×C₅₉	#554	1	1	yes	characteristic
334	C₂×#474	C₂×C₅₉	1	1	yes	characteristic, complemented
335	C₂×C₄×D₈	C₂×C₅₉	1	1	yes	characteristic
336	#2452	C₂×C₅₉	1	1	yes	characteristic, complemented
337	D₄×C₄⋊C₄	C₂×C₅₉	1	1	yes	characteristic, complemented
338	#2093	C₂×C₅₉	1	1	yes	characteristic, complemented
339	#2083	C₂×C₅₉	1	1	yes	characteristic, complemented
340	#2067	C₂×C₅₉	1	1	yes	characteristic, complemented
341	#2057	C₂×C₅₉	1	1	yes	characteristic, complemented
342	#1942	C₂×C₅₉	1	1	yes	characteristic, complemented
343	#1916	C₂×C₅₉	1	1	yes	characteristic, complemented
344	#1879	C₂×C₅₉	1	1	yes	characteristic
345	#1879	C₂×C₅₉	1	1	yes	characteristic
346	#1862	C₂×C₅₉	1	1	yes	characteristic, complemented
347	#1862	C₂×C₅₉	1	1	yes	characteristic, complemented
348	#1862	C₂×C₅₉	1	1	yes	characteristic, complemented
349	C₂×C₅₉	—	4	4	no	—
350	C₂×C₅₉	—	8	2	no	—
351	C₂×C₅₉	#2449	1	1	yes	characteristic
352	C₂×C₅₉	#3100	1	1	yes	characteristic
353	C₂×C₅₉	#3124	1	1	yes	characteristic
354	C₂×C₅₉	#3269	1	1	yes	characteristic
355	C₂×C₅₉	D₄×D₈	1	1	yes	characteristic
356	C₂×C₅₉	Q₈×D₈	1	1	yes	characteristic
357	C₂×C₅₉	#3389	1	1	yes	characteristic
358	C₂²×D₈	C₂²×C₅₉	1	1	yes	characteristic
359	C₂×C₄⋊Q₈	C₂²×C₅₉	1	1	yes	characteristic
360	C₂×C₂²⋊Q₈	—	2	1	no	—
361	C₂×C₄×D₄	C₂²×C₅₉	1	1	yes	characteristic
362	C₂×C₄×D₄	C₂²×C₅₉	1	1	yes	characteristic
363	C₂×C₄×D₄	C₂²×C₅₉	1	1	yes	characteristic
364	C₂×C₄×D₄	C₂²×C₅₉	1	1	yes	characteristic
365	C₂²×C₄⋊C₄	—	2	1	no	—
366	#474	—	2	4	no	—
367	C₄×D₈	—	2	4	no	—
368	C₂×#107	C₂²×C₅₉	1	1	yes	characteristic
369	C₂×#107	C₂²×C₅₉	1	1	yes	characteristic
370	C₂×#107	C₂²×C₅₉	1	1	yes	characteristic
371	C₂×#107	C₂²×C₅₉	1	1	yes	characteristic
372	C₂×#107	C₂²×C₅₉	1	1	yes	characteristic
373	C₂×C₄⋊C₈	C₂²×C₅₉	1	1	yes	characteristic
374	C₂×#102	C₂²×C₅₉	1	1	yes	characteristic
375	C₂×#102	C₂²×C₅₉	1	1	yes	characteristic
376	C₂×#102	C₂²×C₅₉	1	1	yes	characteristic
377	C₂×#102	C₂²×C₅₉	1	1	yes	characteristic
378	C₂×#102	C₂²×C₅₉	1	1	yes	characteristic
379	C₂×#102	C₂²×C₅₉	1	1	yes	characteristic
380	C₂×C₄×C₈	C₂²×C₅₉	1	1	yes	characteristic
381	#396	—	2	1	no	—
382	#393	—	2	1	no	—
383	#390	C₂²×C₅₉	1	1	yes	characteristic
384	#390	C₂²×C₅₉	1	1	yes	characteristic
385	#389	C₂²×C₅₉	1	1	yes	characteristic
386	#389	C₂²×C₅₉	1	1	yes	characteristic
387	#389	C₂²×C₅₉	1	1	yes	characteristic
388	#389	C₂²×C₅₉	1	1	yes	characteristic
389	#385	—	2	1	no	—
390	#385	—	2	1	no	—
391	#385	—	2	1	no	—
392	#384	C₂²×C₅₉	1	1	yes	characteristic
393	#380	—	2	1	no	—
394	C₄×C₄⋊C₄	C₂²×C₅₉	1	1	yes	characteristic
395	#340	C₂²×C₅₉	1	1	yes	characteristic
396	#340	C₂²×C₅₉	1	1	yes	characteristic
397	#340	C₂²×C₅₉	1	1	yes	characteristic
398	#340	C₂²×C₅₉	1	1	yes	characteristic
399	#336	C₂²×C₅₉	1	1	yes	characteristic
400	#336	C₂²×C₅₉	1	1	yes	characteristic
401	#336	C₂²×C₅₉	1	1	yes	characteristic
402	#336	C₂²×C₅₉	1	1	yes	characteristic
403	C₅₉	#26368	1	1	yes	characteristic, direct product factor, Sylow
404	C₂²×Q₈	—	2	1	no	—
405	C₂²×D₄	C₂³×C₅₉	1	1	yes	characteristic
406	C₂²×D₄	C₂³×C₅₉	1	1	yes	characteristic
407	C₂³×C₄	—	2	1	no	—
408	C₂³×C₄	—	2	1	no	—
409	C₂³×C₄	—	2	1	no	—
410	C₂³×C₄	—	2	1	no	—
411	C₂×D₈	—	2	2	no	—
412	C₂×D₈	—	2	2	no	—
413	C₂×D₈	Q₈×C₅₉	1	4	yes	complemented
414	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
415	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
416	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
417	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
418	C₄⋊Q₈	—	2	2	no	—
419	C₄⋊Q₈	—	4	1	no	—
420	C₂²⋊Q₈	—	4	4	no	—
421	C₄×D₄	—	2	2	no	—
422	C₄×D₄	—	2	2	no	—
423	C₄×D₄	—	2	2	no	—
424	C₄×D₄	—	2	2	no	—
425	C₄×D₄	—	2	2	no	—
426	C₄×D₄	—	4	1	no	—
427	C₄×D₄	—	4	1	no	—
428	C₄×D₄	—	4	1	no	—
429	C₂×C₄⋊C₄	—	2	1	no	—
430	C₂×C₄⋊C₄	—	2	1	no	—
431	C₂×C₄⋊C₄	—	2	1	no	—
432	C₂×C₄⋊C₄	—	2	1	no	—
433	C₂×C₄⋊C₄	—	2	1	no	—
434	C₂×C₄⋊C₄	—	2	1	no	—
435	C₂×C₄⋊C₄	—	2	1	no	—
436	C₂×C₄⋊C₄	—	4	4	no	—
437	C₂×C₄⋊C₄	D₄×C₅₉	1	1	yes	characteristic
438	C₂×C₄⋊C₄	D₄×C₅₉	1	1	yes	characteristic
439	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
440	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
441	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
442	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
443	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
444	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
445	C₂×C₂²⋊C₄	—	2	1	no	—
446	C₂×C₂²⋊C₄	—	2	1	no	—
447	C₂×C₂²⋊C₄	—	2	1	no	—
448	C₂×C₂²⋊C₄	—	2	1	no	—
449	C₂×C₂²⋊C₄	—	2	1	no	—
450	C₂×C₂²⋊C₄	—	2	1	no	—
451	C₂×C₂²⋊C₄	—	2	1	no	—
452	C₂×C₂²⋊C₄	—	2	1	no	—
453	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
454	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
455	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
456	#107	—	2	2	no	—
457	#107	—	2	2	no	—
458	#107	—	2	2	no	—
459	#107	—	2	2	no	—
460	#107	—	4	1	no	—
461	C₄⋊C₈	—	2	2	no	—
462	#102	—	2	2	no	—
463	#102	—	2	2	no	—
464	#102	—	2	2	no	—
465	#102	—	2	2	no	—
466	#102	—	4	1	no	—
467	#102	D₄×C₅₉	1	4	yes	—
468	C₄×C₈	—	2	2	no	—
469	#95	—	2	1	no	—
470	#95	—	2	1	no	—
471	#95	—	2	1	no	—
472	#95	—	2	1	no	—
473	#95	D₄×C₅₉	1	1	yes	characteristic
474	#95	D₄×C₅₉	1	1	yes	characteristic
475	C₂⁴	—	2	1	no	—
476	C₂⁴	—	2	1	no	—
477	C₂×Q₈	—	4	2	no	—
478	C₂×Q₈	—	4	2	no	—
479	C₂×Q₈	—	8	1	no	—
480	C₂×D₄	—	2	1	no	—
481	C₂×D₄	—	2	1	no	—
482	C₂×D₄	—	2	1	no	—
483	C₂×D₄	—	2	1	no	—
484	C₂×D₄	—	2	1	no	—
485	C₂×D₄	—	2	1	no	—
486	C₂×D₄	—	2	1	no	—
487	C₂×D₄	—	2	1	no	—
488	C₂×D₄	C₂×D₄×C₅₉	1	2	yes	—
489	C₂×D₄	C₂×Q₈×C₅₉	1	2	yes	—
490	C₂×D₄	C₂×Q₈×C₅₉	1	2	yes	—
491	C₂×D₄	#41×C₅₉	1	2	yes	—
492	C₂²×C₄	—	2	4	no	—
493	C₂²×C₄	—	2	1	no	—
494	C₂²×C₄	—	2	1	no	—
495	C₂²×C₄	—	2	1	no	—
496	C₂²×C₄	—	2	1	no	—
497	C₂²×C₄	—	2	1	no	—
498	C₂²×C₄	—	2	1	no	—
499	C₂²×C₄	—	4	2	no	—
500	C₂²×C₄	—	4	2	no	—
501	C₂²×C₄	—	4	2	no	—
502	C₂²×C₄	—	4	2	no	—
503	C₂²×C₄	—	4	2	no	—
504	C₂²×C₄	—	4	2	no	—
505	C₂²×C₄	—	4	2	no	—
506	C₂²×C₄	—	4	2	no	—
507	C₂²×C₄	—	8	1	no	—
508	C₂²×C₄	—	8	1	no	—
509	C₂²×C₄	—	8	1	no	—
510	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
511	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
512	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
513	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
514	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
515	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
516	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
517	D₈	—	2	8	no	—
518	C₂×C₈	—	2	1	no	—
519	C₂×C₈	—	2	1	no	—
520	C₂×C₈	—	2	1	no	—
521	C₂×C₈	—	2	1	no	—
522	C₂×C₈	—	2	1	no	—
523	C₂×C₈	—	2	1	no	—
524	C₂×C₈	—	2	1	no	—
525	C₂×C₈	—	2	1	no	—
526	C₂×C₈	—	2	1	no	—
527	C₂×C₈	—	2	1	no	—
528	C₂×C₈	C₂×D₄×C₅₉	1	2	yes	—
529	C₂×C₈	C₂×Q₈×C₅₉	1	2	yes	—
530	C₄⋊C₄	—	2	1	no	—
531	C₄⋊C₄	—	2	1	no	—
532	C₄⋊C₄	—	2	1	no	—
533	C₄⋊C₄	—	2	1	no	—
534	C₄⋊C₄	—	2	1	no	—
535	C₄⋊C₄	—	2	1	no	—
536	C₄⋊C₄	—	2	1	no	—
537	C₄⋊C₄	—	2	1	no	—
538	C₄⋊C₄	—	2	1	no	—
539	C₄⋊C₄	—	2	1	no	—
540	C₄⋊C₄	—	4	1	no	—
541	C₄⋊C₄	—	4	2	no	—
542	C₄⋊C₄	—	4	2	no	—
543	C₄⋊C₄	—	4	2	no	—
544	C₄⋊C₄	—	4	2	no	—
545	C₄⋊C₄	—	4	2	no	—
546	C₄⋊C₄	—	4	2	no	—
547	C₄⋊C₄	—	8	1	no	—
548	C₄⋊C₄	D₈×C₅₉	1	4	yes	complemented
549	C₄⋊C₄	C₂×D₄×C₅₉	1	2	yes	—
550	C₄⋊C₄	#41×C₅₉	1	2	yes	—
551	C₂²⋊C₄	—	2	4	no	—
552	C₂²⋊C₄	—	4	2	no	—
553	C₂²⋊C₄	—	4	2	no	—
554	C₂²⋊C₄	—	4	2	no	—
555	C₂²⋊C₄	—	4	2	no	—
556	C₂²⋊C₄	—	4	2	no	—
557	C₂²⋊C₄	—	4	2	no	—
558	C₂²⋊C₄	—	8	1	no	—
559	C₄²	—	2	1	no	—
560	C₄²	—	2	1	no	—
561	C₄²	—	2	1	no	—
562	C₄²	—	2	1	no	—
563	C₄²	—	2	1	no	—
564	C₄²	—	2	1	no	—
565	C₂³	—	2	2	no	—
566	C₂³	—	2	2	no	—
567	C₂³	—	2	2	no	—
568	C₂³	—	2	2	no	—
569	C₂³	—	4	1	no	—
570	C₂³	—	4	1	no	—
571	C₂³	—	4	1	no	—
572	C₂³	—	4	1	no	—
573	C₂³	—	4	1	no	—
574	C₂³	—	4	1	no	—
575	C₂³	—	4	1	no	—
576	C₂³	—	4	1	no	—
577	C₂³	—	4	1	no	—
578	C₂³	—	4	1	no	—
579	C₂³	C₂²×D₄×C₅₉	1	1	yes	characteristic
580	Q₈	—	8	4	no	—
581	D₄	—	2	2	no	—
582	D₄	—	2	2	no	—
583	D₄	—	2	2	no	—
584	D₄	—	2	2	no	—
585	D₄	—	2	2	no	—
586	D₄	—	4	1	no	—
587	D₄	—	4	1	no	—
588	D₄	C₂²⋊Q₈×C₅₉	1	4	yes	complemented
589	C₂×C₄	—	2	1	no	—
590	C₂×C₄	—	2	1	no	—
591	C₂×C₄	—	2	1	no	—
592	C₂×C₄	—	2	1	no	—
593	C₂×C₄	—	2	1	no	—
594	C₂×C₄	—	2	1	no	—
595	C₂×C₄	—	2	1	no	—
596	C₂×C₄	—	2	2	no	—
597	C₂×C₄	—	2	1	no	—
598	C₂×C₄	—	2	1	no	—
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	2	1	no	—
601	C₂×C₄	—	2	1	no	—
602	C₂×C₄	—	2	1	no	—
603	C₂×C₄	—	2	1	no	—
604	C₂×C₄	—	2	1	no	—
605	C₂×C₄	—	2	1	no	—
606	C₂×C₄	—	4	8	no	—
607	C₂×C₄	—	4	1	no	—
608	C₂×C₄	—	4	1	no	—
609	C₂×C₄	—	4	1	no	—
610	C₂×C₄	—	4	1	no	—
611	C₂×C₄	—	4	1	no	—
612	C₂×C₄	—	4	1	no	—
613	C₂×C₄	—	4	1	no	—
614	C₂×C₄	—	4	1	no	—
615	C₂×C₄	—	4	1	no	—
616	C₂×C₄	—	4	1	no	—
617	C₂×C₄	—	4	1	no	—
618	C₂×C₄	—	4	1	no	—
619	C₂×C₄	—	4	1	no	—
620	C₂×C₄	—	4	1	no	—
621	C₂×C₄	—	4	1	no	—
622	C₂×C₄	—	4	1	no	—
623	C₂×C₄	—	8	4	no	—
624	C₂×C₄	—	8	4	no	—
625	C₂×C₄	—	8	4	no	—
626	C₂×C₄	C₂×D₈×C₅₉	1	2	yes	—
627	C₂×C₄	C₂×D₈×C₅₉	1	2	yes	—
628	C₂×C₄	C₂×D₈×C₅₉	1	2	yes	—
629	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
630	C₂×C₄	C₂²×Q₈×C₅₉	1	1	yes	characteristic
631	C₂×C₄	C₂×#41×C₅₉	1	1	yes	characteristic
632	C₂×C₄	C₂×#41×C₅₉	1	1	yes	characteristic
633	C₂×C₄	H(5,𝔽₂)×C₅₉	1	1	yes	characteristic
634	C₂×C₄	2₋¹⁺⁴×C₅₉	1	1	yes	characteristic
635	C₈	—	2	2	no	—
636	C₈	—	2	2	no	—
637	C₈	—	4	1	no	—
638	C₈	—	4	1	no	—
639	C₂²	—	2	4	no	—
640	C₂²	—	4	2	no	—
641	C₂²	—	4	2	no	—
642	C₂²	—	4	2	no	—
643	C₂²	—	4	2	no	—
644	C₂²	—	4	2	no	—
645	C₂²	—	4	2	no	—
646	C₂²	—	4	2	no	—
647	C₂²	—	4	2	no	—
648	C₂²	—	4	2	no	—
649	C₂²	—	8	1	no	—
650	C₂²	—	8	1	no	—
651	C₂²	—	8	1	no	—
652	C₂²	—	8	1	no	—
653	C₂²	C₅₉×#525	1	1	yes	characteristic
654	C₂²	D₄²×C₅₉	1	1	yes	characteristic
655	C₂²	D₄×Q₈×C₅₉	1	1	yes	characteristic
656	C₂²	C₂²×D₈×C₅₉	1	1	yes	characteristic
657	C₂²	C₅₉×#575	1	1	yes	characteristic
658	C₂²	C₅₉×#576	1	1	yes	characteristic
659	C₂²	C₅₉×#578	1	1	yes	characteristic
660	C₄	—	2	2	no	—
661	C₄	—	2	2	no	—
662	C₄	—	2	2	no	—
663	C₄	—	4	1	no	—
664	C₄	—	4	1	no	—
665	C₄	—	4	2	no	—
666	C₄	—	4	2	no	—
667	C₄	—	4	1	no	—
668	C₄	—	8	1	no	—
669	C₄	—	8	1	no	—
670	C₄	—	8	1	no	—
671	C₄	—	8	1	no	—
672	C₄	C₂×C₂²⋊Q₈×C₅₉	1	1	yes	characteristic
673	C₄	C₅₉×#539	1	1	yes	characteristic
674	C₄	C₅₉×#547	1	1	yes	characteristic
675	C₄	C₅₉×#554	1	1	yes	characteristic
676	C₂	—	4	4	no	—
677	C₂	—	8	2	no	—
678	C₂	C₅₉×#2449	1	1	yes	characteristic
679	C₂	C₅₉×#3100	1	1	yes	characteristic
680	C₂	C₅₉×#3124	1	1	yes	characteristic
681	C₂	C₅₉×#3269	1	1	yes	characteristic
682	C₂	D₄×D₈×C₅₉	1	1	yes	characteristic
683	C₂	Q₈×D₈×C₅₉	1	1	yes	characteristic
684	C₂	C₅₉×#3389	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₉
prime radical	C₂²×D₈×C₅₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₉ ◁ C₄×C₄⋊C₄×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₉×#474	C₂	semidirect
3. C₅₉×#2452	C₂	semidirect
4. D₄×C₄⋊C₄×C₅₉	C₂	semidirect
5. C₅₉×#2093	C₂	semidirect
6. C₅₉×#2083	C₂	semidirect
7. C₅₉×#2067	C₂	semidirect
8. C₅₉×#2057	C₂	semidirect
9. C₅₉×#1942	C₂	semidirect
10. C₅₉×#1916	C₂	semidirect
13. C₅₉×#1862	C₂	semidirect
14. C₅₉×#1862	C₂	semidirect
15. C₅₉×#1862	C₂	semidirect
70. C₂×D₈×C₅₉	Q₈	semidirect
205. C₄⋊C₄×C₅₉	D₈	semidirect
245. D₄×C₅₉	C₂²⋊Q₈	semidirect
296. #26368	C₅₉	direct (N × H)
334. C₂×#474	C₂×C₅₉	semidirect
336. #2452	C₂×C₅₉	semidirect
337. D₄×C₄⋊C₄	C₂×C₅₉	semidirect
338. #2093	C₂×C₅₉	semidirect
339. #2083	C₂×C₅₉	semidirect
340. #2067	C₂×C₅₉	semidirect
341. #2057	C₂×C₅₉	semidirect
342. #1942	C₂×C₅₉	semidirect
343. #1916	C₂×C₅₉	semidirect
346. #1862	C₂×C₅₉	semidirect
347. #1862	C₂×C₅₉	semidirect
348. #1862	C₂×C₅₉	semidirect
403. C₅₉	#26368	direct (N × H)
413. C₂×D₈	Q₈×C₅₉	semidirect
548. C₄⋊C₄	D₈×C₅₉	semidirect
588. D₄	C₂²⋊Q₈×C₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₉×#474	2
C₂×C₄×D₈×C₅₉	2
C₅₉×#2452	2
D₄×C₄⋊C₄×C₅₉	2
C₅₉×#2093	2
C₅₉×#2083	2
C₅₉×#2067	2
C₅₉×#2057	2
C₅₉×#1942	2
C₅₉×#1916	2
C₅₉×#1879	2
C₅₉×#1862	2
#26368	59

Subgroups — C59×#26368 #28438477

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#26368 #28438477