Subgroups — C₅₉×#24728 #28436837

Order 15104.

1118 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₉×#460	C₂	1	2	yes	maximal, complemented
2	C₂×C₅₉×#460	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×SD₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₉×#2541	C₂	1	1	yes	maximal, characteristic
5	Q₈×C₄⋊C₄×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₉×#2065	C₂	1	2	yes	maximal, complemented
7	C₅₉×#2041	C₂	1	2	yes	maximal, complemented
8	C₅₉×#1957	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₉×#1939	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₉×#1880	C₂	1	2	yes	maximal, complemented
11	C₅₉×#1802	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₅₉	C₂²	1	2	yes	—
13	C₂²×SD₁₆×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×#127×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×#127×C₅₉	C₂²	1	2	yes	—
16	C₂×#121×C₅₉	—	2	1	no	—
17	C₂×#121×C₅₉	—	2	1	no	—
18	C₂×C₄×Q₈×C₅₉	C₂²	1	2	yes	—
19	C₂×C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
21	C₅₉×#460	—	2	8	no	—
22	C₅₉×#460	—	2	4	no	—
23	C₄×SD₁₆×C₅₉	—	2	4	no	—
24	C₂×#106×C₅₉	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
27	C₂×#103×C₅₉	C₂²	1	1	yes	characteristic
28	C₂×#102×C₅₉	C₂²	1	2	yes	—
29	C₂×#102×C₅₉	C₂²	1	2	yes	—
30	C₂×#102×C₅₉	C₂²	1	2	yes	—
31	C₂×#102×C₅₉	C₂²	1	1	yes	characteristic
32	C₂×#102×C₅₉	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
34	C₅₉×#392	—	2	2	no	—
35	C₅₉×#390	C₂²	1	2	yes	—
36	C₅₉×#389	—	2	2	no	—
37	C₅₉×#389	C₂²	1	2	yes	—
38	C₅₉×#384	—	2	1	no	—
39	C₅₉×#384	C₂²	1	1	yes	characteristic
40	C₄×C₄⋊C₄×C₅₉	—	2	1	no	—
41	C₄×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
42	C₅₉×#340	C₂²	1	2	yes	—
43	C₅₉×#336	C₂²	1	2	yes	—
44	C₂²×Q₈×C₅₉	C₂³	1	1	yes	characteristic
45	C₂²×D₄×C₅₉	—	2	1	no	—
46	C₂²×D₄×C₅₉	—	2	2	no	—
47	C₂²×D₄×C₅₉	—	2	2	no	—
48	C₂²×D₄×C₅₉	—	2	1	no	—
49	C₂²×D₄×C₅₉	C₂³	1	2	yes	—
50	C₂²×D₄×C₅₉	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₅₉	—	2	1	no	—
52	C₂×SD₁₆×C₅₉	—	2	2	no	—
53	C₂×SD₁₆×C₅₉	—	2	2	no	—
54	C₂×SD₁₆×C₅₉	—	2	4	no	—
55	C₂×SD₁₆×C₅₉	—	2	4	no	—
56	C₂×SD₁₆×C₅₉	—	2	4	no	—
57	C₂×SD₁₆×C₅₉	D₄	1	4	yes	complemented
58	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
60	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
61	#127×C₅₉	—	2	4	no	—
62	#127×C₅₉	—	2	2	no	—
63	#127×C₅₉	—	2	2	no	—
64	#127×C₅₉	—	4	2	no	—
65	#121×C₅₉	—	4	4	no	—
66	#121×C₅₉	—	4	4	no	—
67	C₄×Q₈×C₅₉	—	2	2	no	—
68	C₄×Q₈×C₅₉	—	2	4	no	—
69	C₄×Q₈×C₅₉	—	4	2	no	—
70	C₄×Q₈×C₅₉	D₄	1	4	yes	complemented
71	C₄×D₄×C₅₉	—	2	2	no	—
72	C₄×D₄×C₅₉	—	2	2	no	—
73	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
74	C₂×C₄⋊C₄×C₅₉	—	2	1	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	1	yes	characteristic
79	C₂×C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
82	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
83	C₂×C₂²⋊C₄×C₅₉	—	2	2	no	—
84	C₂×C₂²⋊C₄×C₅₉	—	2	2	no	—
85	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
86	C₂×C₄²×C₅₉	—	2	1	no	—
87	C₂×C₄²×C₅₉	—	2	2	no	—
88	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
89	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
90	#106×C₅₉	—	2	2	no	—
91	C₄⋊C₈×C₅₉	—	2	4	no	—
92	C₄⋊C₈×C₅₉	—	2	2	no	—
93	#103×C₅₉	—	2	2	no	—
94	#102×C₅₉	—	2	4	no	—
95	#102×C₅₉	—	2	4	no	—
96	#102×C₅₉	—	2	2	no	—
97	#102×C₅₉	—	2	4	no	—
98	#102×C₅₉	—	4	2	no	—
99	C₄×C₈×C₅₉	—	2	2	no	—
100	#95×C₅₉	—	2	2	no	—
101	#95×C₅₉	D₄	1	2	yes	—
102	C₂⁴×C₅₉	—	2	1	no	—
103	C₂⁴×C₅₉	—	2	2	no	—
104	C₂×Q₈×C₅₉	—	2	1	no	—
105	C₂×Q₈×C₅₉	—	2	1	no	—
106	C₂×Q₈×C₅₉	—	2	1	no	—
107	C₂×Q₈×C₅₉	C₂×D₄	1	2	yes	—
108	C₂×Q₈×C₅₉	C₂×D₄	1	2	yes	—
109	C₂×Q₈×C₅₉	#41	1	2	yes	—
110	C₂×D₄×C₅₉	—	2	1	no	—
111	C₂×D₄×C₅₉	—	2	2	no	—
112	C₂×D₄×C₅₉	—	2	2	no	—
113	C₂×D₄×C₅₉	—	2	1	no	—
114	C₂×D₄×C₅₉	—	2	1	no	—
115	C₂×D₄×C₅₉	—	2	2	no	—
116	C₂×D₄×C₅₉	—	2	2	no	—
117	C₂×D₄×C₅₉	—	2	1	no	—
118	C₂×D₄×C₅₉	—	2	2	no	—
119	C₂×D₄×C₅₉	—	2	1	no	—
120	C₂×D₄×C₅₉	—	2	2	no	—
121	C₂×D₄×C₅₉	—	4	4	no	—
122	C₂×D₄×C₅₉	—	4	4	no	—
123	C₂×D₄×C₅₉	—	4	4	no	—
124	C₂×D₄×C₅₉	—	4	2	no	—
125	C₂×D₄×C₅₉	—	4	2	no	—
126	C₂×D₄×C₅₉	—	4	4	no	—
127	C₂×D₄×C₅₉	—	4	2	no	—
128	C₂×D₄×C₅₉	—	4	2	no	—
129	C₂×D₄×C₅₉	—	8	1	no	—
130	C₂×D₄×C₅₉	—	8	2	no	—
131	C₂×D₄×C₅₉	—	8	1	no	—
132	C₂×D₄×C₅₉	—	8	2	no	—
133	C₂×D₄×C₅₉	C₂×D₄	1	2	yes	—
134	C₂²×C₄×C₅₉	—	2	1	no	—
135	C₂²×C₄×C₅₉	—	2	1	no	—
136	C₂²×C₄×C₅₉	—	2	2	no	—
137	C₂²×C₄×C₅₉	—	2	1	no	—
138	C₂²×C₄×C₅₉	—	4	2	no	—
139	C₂²×C₄×C₅₉	—	4	2	no	—
140	C₂²×C₄×C₅₉	—	4	2	no	—
141	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
142	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
144	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
146	SD₁₆×C₅₉	—	2	8	no	—
147	SD₁₆×C₅₉	—	4	8	no	—
148	C₂×C₈×C₅₉	—	2	1	no	—
149	C₂×C₈×C₅₉	—	2	1	no	—
150	C₂×C₈×C₅₉	—	2	1	no	—
151	C₂×C₈×C₅₉	—	2	2	no	—
152	C₂×C₈×C₅₉	—	2	2	no	—
153	C₂×C₈×C₅₉	—	2	2	no	—
154	C₂×C₈×C₅₉	—	2	1	no	—
155	C₂×C₈×C₅₉	—	2	1	no	—
156	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
157	C₄⋊C₄×C₅₉	—	2	4	no	—
158	C₄⋊C₄×C₅₉	—	2	2	no	—
159	C₄⋊C₄×C₅₉	—	2	1	no	—
160	C₄⋊C₄×C₅₉	—	2	2	no	—
161	C₄⋊C₄×C₅₉	—	2	1	no	—
162	C₄⋊C₄×C₅₉	—	2	1	no	—
163	C₄⋊C₄×C₅₉	—	2	2	no	—
164	C₄⋊C₄×C₅₉	—	4	4	no	—
165	C₄⋊C₄×C₅₉	—	4	2	no	—
166	C₄⋊C₄×C₅₉	—	8	1	no	—
167	C₄⋊C₄×C₅₉	—	8	2	no	—
168	C₄⋊C₄×C₅₉	SD₁₆	1	4	yes	complemented
169	C₄⋊C₄×C₅₉	C₂×D₄	1	2	yes	—
170	C₂²⋊C₄×C₅₉	—	4	4	no	—
171	C₂²⋊C₄×C₅₉	—	4	2	no	—
172	C₂²⋊C₄×C₅₉	—	4	4	no	—
173	C₄²×C₅₉	—	2	4	no	—
174	C₄²×C₅₉	—	2	2	no	—
175	C₄²×C₅₉	—	2	2	no	—
176	C₄²×C₅₉	—	4	4	no	—
177	C₄²×C₅₉	C₂×D₄	1	2	yes	—
178	C₄²×C₅₉	#41	1	2	yes	—
179	C₂³×C₅₉	—	2	2	no	—
180	C₂³×C₅₉	—	4	1	no	—
181	C₂³×C₅₉	—	4	1	no	—
182	C₂³×C₅₉	—	4	1	no	—
183	C₂³×C₅₉	—	4	1	no	—
184	C₂³×C₅₉	—	4	1	no	—
185	C₂³×C₅₉	—	4	1	no	—
186	C₂³×C₅₉	—	4	2	no	—
187	C₂³×C₅₉	—	4	2	no	—
188	C₂³×C₅₉	—	4	2	no	—
189	C₂³×C₅₉	—	4	2	no	—
190	C₂³×C₅₉	—	4	2	no	—
191	C₂³×C₅₉	—	4	2	no	—
192	C₂³×C₅₉	—	4	2	no	—
193	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
194	Q₈×C₅₉	—	2	2	no	—
195	Q₈×C₅₉	—	2	2	no	—
196	Q₈×C₅₉	—	2	2	no	—
197	Q₈×C₅₉	#121	1	4	yes	complemented
198	D₄×C₅₉	—	2	2	no	—
199	D₄×C₅₉	—	2	2	no	—
200	D₄×C₅₉	—	4	2	no	—
201	D₄×C₅₉	—	4	1	no	—
202	D₄×C₅₉	—	4	2	no	—
203	D₄×C₅₉	—	4	2	no	—
204	D₄×C₅₉	—	4	1	no	—
205	D₄×C₅₉	—	4	2	no	—
206	D₄×C₅₉	—	8	8	no	—
207	D₄×C₅₉	—	8	4	no	—
208	D₄×C₅₉	—	8	8	no	—
209	D₄×C₅₉	—	8	4	no	—
210	C₂×C₄×C₅₉	—	2	2	no	—
211	C₂×C₄×C₅₉	—	2	2	no	—
212	C₂×C₄×C₅₉	—	2	2	no	—
213	C₂×C₄×C₅₉	—	2	2	no	—
214	C₂×C₄×C₅₉	—	2	1	no	—
215	C₂×C₄×C₅₉	—	2	2	no	—
216	C₂×C₄×C₅₉	—	2	1	no	—
217	C₂×C₄×C₅₉	—	2	2	no	—
218	C₂×C₄×C₅₉	—	2	2	no	—
219	C₂×C₄×C₅₉	—	2	2	no	—
220	C₂×C₄×C₅₉	—	2	2	no	—
221	C₂×C₄×C₅₉	—	4	1	no	—
222	C₂×C₄×C₅₉	—	4	1	no	—
223	C₂×C₄×C₅₉	—	4	1	no	—
224	C₂×C₄×C₅₉	—	4	2	no	—
225	C₂×C₄×C₅₉	—	4	1	no	—
226	C₂×C₄×C₅₉	—	4	1	no	—
227	C₂×C₄×C₅₉	—	4	2	no	—
228	C₂×C₄×C₅₉	—	4	2	no	—
229	C₂×C₄×C₅₉	—	8	4	no	—
230	C₂×C₄×C₅₉	#121	1	2	yes	—
231	C₂×C₄×C₅₉	#121	1	2	yes	—
232	C₂×C₄×C₅₉	C₂×SD₁₆	1	4	yes	—
233	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
234	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
235	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
236	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
238	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
239	C₂×C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
240	C₂×C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
241	C₈×C₅₉	—	2	2	no	—
242	C₈×C₅₉	—	4	2	no	—
243	C₈×C₅₉	—	4	1	no	—
244	#24728	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
245	C₂²×C₅₉	—	4	2	no	—
246	C₂²×C₅₉	—	4	2	no	—
247	C₂²×C₅₉	—	4	2	no	—
248	C₂²×C₅₉	—	8	1	no	—
249	C₂²×C₅₉	—	8	1	no	—
250	C₂²×C₅₉	—	8	1	no	—
251	C₂²×C₅₉	—	8	1	no	—
252	C₂²×C₅₉	—	8	2	no	—
253	C₂²×C₅₉	—	8	2	no	—
254	C₂²×C₅₉	—	8	2	no	—
255	C₂²×C₅₉	—	8	2	no	—
256	C₂²×C₅₉	—	8	2	no	—
257	C₂²×C₅₉	—	8	2	no	—
258	C₂²×C₅₉	—	8	2	no	—
259	C₂²×C₅₉	C₂×#121	1	1	yes	characteristic
260	C₂²×C₅₉	D₄²	1	1	yes	characteristic
261	C₂²×C₅₉	#548	1	1	yes	characteristic
262	C₂²×C₅₉	C₂²×SD₁₆	1	1	yes	characteristic
263	C₂²×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
264	C₂²×C₅₉	#576	1	1	yes	characteristic
265	C₂²×C₅₉	#577	1	1	yes	characteristic
266	C₄×C₅₉	—	2	4	no	—
267	C₄×C₅₉	—	2	4	no	—
268	C₄×C₅₉	—	2	2	no	—
269	C₄×C₅₉	—	4	2	no	—
270	C₄×C₅₉	—	4	2	no	—
271	C₄×C₅₉	—	8	2	no	—
272	C₄×C₅₉	—	8	1	no	—
273	C₄×C₅₉	#460	1	4	yes	complemented
274	C₄×C₅₉	C₂×#121	1	1	yes	characteristic
275	C₄×C₅₉	#537	1	1	yes	characteristic
276	C₄×C₅₉	#547	1	1	yes	characteristic
277	C₄×C₅₉	#553	1	1	yes	characteristic
278	C₂×#460	C₂×C₅₉	1	2	yes	complemented
279	C₂×#460	C₂×C₅₉	1	1	yes	characteristic, complemented
280	C₂×C₄×SD₁₆	C₂×C₅₉	1	1	yes	characteristic, complemented
281	#2541	C₂×C₅₉	1	1	yes	characteristic
282	Q₈×C₄⋊C₄	C₂×C₅₉	1	1	yes	characteristic, complemented
283	#2065	C₂×C₅₉	1	2	yes	complemented
284	#2041	C₂×C₅₉	1	2	yes	complemented
285	#1957	C₂×C₅₉	1	1	yes	characteristic, complemented
286	#1939	C₂×C₅₉	1	1	yes	characteristic, complemented
287	#1880	C₂×C₅₉	1	2	yes	complemented
288	#1802	C₂×C₅₉	1	1	yes	characteristic, complemented
289	C₂×C₅₉	—	8	2	no	—
290	C₂×C₅₉	—	16	2	no	—
291	C₂×C₅₉	#2416	1	1	yes	characteristic
292	C₂×C₅₉	C₂×#460	1	1	yes	characteristic
293	C₂×C₅₉	#3045	1	1	yes	characteristic
294	C₂×C₅₉	#3274	1	1	yes	characteristic
295	C₂×C₅₉	D₄×SD₁₆	1	1	yes	characteristic
296	C₂×C₅₉	#3362	1	1	yes	characteristic
297	C₂×C₅₉	#3368	1	1	yes	characteristic
298	C₂²×SD₁₆	C₂²×C₅₉	1	2	yes	—
299	C₂²×SD₁₆	C₂²×C₅₉	1	1	yes	characteristic
300	C₂×#127	C₂²×C₅₉	1	1	yes	characteristic
301	C₂×#127	C₂²×C₅₉	1	2	yes	—
302	C₂×#121	—	2	1	no	—
303	C₂×#121	—	2	1	no	—
304	C₂×C₄×Q₈	C₂²×C₅₉	1	2	yes	—
305	C₂×C₄×Q₈	C₂²×C₅₉	1	1	yes	characteristic
306	C₂×C₄×D₄	C₂²×C₅₉	1	1	yes	characteristic
307	#460	—	2	8	no	—
308	#460	—	2	4	no	—
309	C₄×SD₁₆	—	2	4	no	—
310	C₂×#106	C₂²×C₅₉	1	1	yes	characteristic
311	C₂×C₄⋊C₈	C₂²×C₅₉	1	1	yes	characteristic
312	C₂×C₄⋊C₈	C₂²×C₅₉	1	2	yes	—
313	C₂×#103	C₂²×C₅₉	1	1	yes	characteristic
314	C₂×#102	C₂²×C₅₉	1	2	yes	—
315	C₂×#102	C₂²×C₅₉	1	2	yes	—
316	C₂×#102	C₂²×C₅₉	1	2	yes	—
317	C₂×#102	C₂²×C₅₉	1	1	yes	characteristic
318	C₂×#102	C₂²×C₅₉	1	2	yes	—
319	C₂×C₄×C₈	C₂²×C₅₉	1	1	yes	characteristic
320	#392	—	2	2	no	—
321	#390	C₂²×C₅₉	1	2	yes	—
322	#389	—	2	2	no	—
323	#389	C₂²×C₅₉	1	2	yes	—
324	#384	—	2	1	no	—
325	#384	C₂²×C₅₉	1	1	yes	characteristic
326	C₄×C₄⋊C₄	—	2	1	no	—
327	C₄×C₄⋊C₄	C₂²×C₅₉	1	1	yes	characteristic
328	#340	C₂²×C₅₉	1	2	yes	—
329	#336	C₂²×C₅₉	1	2	yes	—
330	C₅₉	#24728	1	1	yes	characteristic, direct product factor, Sylow
331	C₂²×Q₈	C₂³×C₅₉	1	1	yes	characteristic
332	C₂²×D₄	—	2	1	no	—
333	C₂²×D₄	—	2	2	no	—
334	C₂²×D₄	—	2	2	no	—
335	C₂²×D₄	—	2	1	no	—
336	C₂²×D₄	C₂³×C₅₉	1	2	yes	—
337	C₂²×D₄	C₂³×C₅₉	1	1	yes	characteristic
338	C₂³×C₄	—	2	1	no	—
339	C₂×SD₁₆	—	2	2	no	—
340	C₂×SD₁₆	—	2	2	no	—
341	C₂×SD₁₆	—	2	4	no	—
342	C₂×SD₁₆	—	2	4	no	—
343	C₂×SD₁₆	—	2	4	no	—
344	C₂×SD₁₆	D₄×C₅₉	1	4	yes	complemented
345	C₂²×C₈	C₂³×C₅₉	1	2	yes	—
346	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
347	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
348	#127	—	2	4	no	—
349	#127	—	2	2	no	—
350	#127	—	2	2	no	—
351	#127	—	4	2	no	—
352	#121	—	4	4	no	—
353	#121	—	4	4	no	—
354	C₄×Q₈	—	2	2	no	—
355	C₄×Q₈	—	2	4	no	—
356	C₄×Q₈	—	4	2	no	—
357	C₄×Q₈	D₄×C₅₉	1	4	yes	complemented
358	C₄×D₄	—	2	2	no	—
359	C₄×D₄	—	2	2	no	—
360	C₂×C₄⋊C₄	—	2	1	no	—
361	C₂×C₄⋊C₄	—	2	2	no	—
362	C₂×C₄⋊C₄	—	2	1	no	—
363	C₂×C₄⋊C₄	—	2	2	no	—
364	C₂×C₄⋊C₄	—	2	1	no	—
365	C₂×C₄⋊C₄	D₄×C₅₉	1	1	yes	characteristic
366	C₂×C₄⋊C₄	D₄×C₅₉	1	1	yes	characteristic
367	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
368	C₂×C₄⋊C₄	C₂³×C₅₉	1	2	yes	—
369	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
370	C₂×C₂²⋊C₄	—	2	1	no	—
371	C₂×C₂²⋊C₄	—	2	2	no	—
372	C₂×C₂²⋊C₄	—	2	2	no	—
373	C₂×C₄²	—	2	1	no	—
374	C₂×C₄²	—	2	2	no	—
375	C₂×C₄²	C₂³×C₅₉	1	2	yes	—
376	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
377	#106	—	2	2	no	—
378	C₄⋊C₈	—	2	4	no	—
379	C₄⋊C₈	—	2	2	no	—
380	#103	—	2	2	no	—
381	#102	—	2	4	no	—
382	#102	—	2	4	no	—
383	#102	—	2	2	no	—
384	#102	—	2	4	no	—
385	#102	—	4	2	no	—
386	C₄×C₈	—	2	2	no	—
387	#95	—	2	2	no	—
388	#95	D₄×C₅₉	1	2	yes	—
389	C₂⁴	—	2	1	no	—
390	C₂⁴	—	2	2	no	—
391	C₂×Q₈	—	2	1	no	—
392	C₂×Q₈	—	2	1	no	—
393	C₂×Q₈	—	2	1	no	—
394	C₂×Q₈	C₂×D₄×C₅₉	1	2	yes	—
395	C₂×Q₈	C₂×D₄×C₅₉	1	2	yes	—
396	C₂×Q₈	#41×C₅₉	1	2	yes	—
397	C₂×D₄	—	2	1	no	—
398	C₂×D₄	—	2	2	no	—
399	C₂×D₄	—	2	2	no	—
400	C₂×D₄	—	2	1	no	—
401	C₂×D₄	—	2	1	no	—
402	C₂×D₄	—	2	2	no	—
403	C₂×D₄	—	2	2	no	—
404	C₂×D₄	—	2	1	no	—
405	C₂×D₄	—	2	2	no	—
406	C₂×D₄	—	2	1	no	—
407	C₂×D₄	—	2	2	no	—
408	C₂×D₄	—	4	4	no	—
409	C₂×D₄	—	4	4	no	—
410	C₂×D₄	—	4	4	no	—
411	C₂×D₄	—	4	2	no	—
412	C₂×D₄	—	4	2	no	—
413	C₂×D₄	—	4	4	no	—
414	C₂×D₄	—	4	2	no	—
415	C₂×D₄	—	4	2	no	—
416	C₂×D₄	—	8	1	no	—
417	C₂×D₄	—	8	2	no	—
418	C₂×D₄	—	8	1	no	—
419	C₂×D₄	—	8	2	no	—
420	C₂×D₄	C₂×D₄×C₅₉	1	2	yes	—
421	C₂²×C₄	—	2	1	no	—
422	C₂²×C₄	—	2	1	no	—
423	C₂²×C₄	—	2	2	no	—
424	C₂²×C₄	—	2	1	no	—
425	C₂²×C₄	—	4	2	no	—
426	C₂²×C₄	—	4	2	no	—
427	C₂²×C₄	—	4	2	no	—
428	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
429	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
430	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
431	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
432	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
433	SD₁₆	—	2	8	no	—
434	SD₁₆	—	4	8	no	—
435	C₂×C₈	—	2	1	no	—
436	C₂×C₈	—	2	1	no	—
437	C₂×C₈	—	2	1	no	—
438	C₂×C₈	—	2	2	no	—
439	C₂×C₈	—	2	2	no	—
440	C₂×C₈	—	2	2	no	—
441	C₂×C₈	—	2	1	no	—
442	C₂×C₈	—	2	1	no	—
443	C₂×C₈	C₂×D₄×C₅₉	1	2	yes	—
444	C₄⋊C₄	—	2	4	no	—
445	C₄⋊C₄	—	2	2	no	—
446	C₄⋊C₄	—	2	1	no	—
447	C₄⋊C₄	—	2	2	no	—
448	C₄⋊C₄	—	2	1	no	—
449	C₄⋊C₄	—	2	1	no	—
450	C₄⋊C₄	—	2	2	no	—
451	C₄⋊C₄	—	4	4	no	—
452	C₄⋊C₄	—	4	2	no	—
453	C₄⋊C₄	—	8	1	no	—
454	C₄⋊C₄	—	8	2	no	—
455	C₄⋊C₄	SD₁₆×C₅₉	1	4	yes	complemented
456	C₄⋊C₄	C₂×D₄×C₅₉	1	2	yes	—
457	C₂²⋊C₄	—	4	4	no	—
458	C₂²⋊C₄	—	4	2	no	—
459	C₂²⋊C₄	—	4	4	no	—
460	C₄²	—	2	4	no	—
461	C₄²	—	2	2	no	—
462	C₄²	—	2	2	no	—
463	C₄²	—	4	4	no	—
464	C₄²	C₂×D₄×C₅₉	1	2	yes	—
465	C₄²	#41×C₅₉	1	2	yes	—
466	C₂³	—	2	2	no	—
467	C₂³	—	4	1	no	—
468	C₂³	—	4	1	no	—
469	C₂³	—	4	1	no	—
470	C₂³	—	4	1	no	—
471	C₂³	—	4	1	no	—
472	C₂³	—	4	1	no	—
473	C₂³	—	4	2	no	—
474	C₂³	—	4	2	no	—
475	C₂³	—	4	2	no	—
476	C₂³	—	4	2	no	—
477	C₂³	—	4	2	no	—
478	C₂³	—	4	2	no	—
479	C₂³	—	4	2	no	—
480	C₂³	C₂²×D₄×C₅₉	1	1	yes	characteristic
481	Q₈	—	2	2	no	—
482	Q₈	—	2	2	no	—
483	Q₈	—	2	2	no	—
484	Q₈	#121×C₅₉	1	4	yes	complemented
485	D₄	—	2	2	no	—
486	D₄	—	2	2	no	—
487	D₄	—	4	2	no	—
488	D₄	—	4	1	no	—
489	D₄	—	4	2	no	—
490	D₄	—	4	2	no	—
491	D₄	—	4	1	no	—
492	D₄	—	4	2	no	—
493	D₄	—	8	8	no	—
494	D₄	—	8	4	no	—
495	D₄	—	8	8	no	—
496	D₄	—	8	4	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	1	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	1	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	2	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	4	1	no	—
509	C₂×C₄	—	4	1	no	—
510	C₂×C₄	—	4	1	no	—
511	C₂×C₄	—	4	2	no	—
512	C₂×C₄	—	4	1	no	—
513	C₂×C₄	—	4	1	no	—
514	C₂×C₄	—	4	2	no	—
515	C₂×C₄	—	4	2	no	—
516	C₂×C₄	—	8	4	no	—
517	C₂×C₄	#121×C₅₉	1	2	yes	—
518	C₂×C₄	#121×C₅₉	1	2	yes	—
519	C₂×C₄	C₂×SD₁₆×C₅₉	1	4	yes	—
520	C₂×C₄	C₂×SD₁₆×C₅₉	1	2	yes	—
521	C₂×C₄	AGL(1,ℤ/8)×C₅₉	1	2	yes	—
522	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
523	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
524	C₂×C₄	C₂×#41×C₅₉	1	1	yes	characteristic
525	C₂×C₄	C₂×#41×C₅₉	1	1	yes	characteristic
526	C₂×C₄	H(5,𝔽₂)×C₅₉	1	1	yes	characteristic
527	C₂×C₄	2₋¹⁺⁴×C₅₉	1	1	yes	characteristic
528	C₈	—	2	2	no	—
529	C₈	—	4	2	no	—
530	C₈	—	4	1	no	—
531	C₂²	—	4	2	no	—
532	C₂²	—	4	2	no	—
533	C₂²	—	4	2	no	—
534	C₂²	—	8	1	no	—
535	C₂²	—	8	1	no	—
536	C₂²	—	8	1	no	—
537	C₂²	—	8	1	no	—
538	C₂²	—	8	2	no	—
539	C₂²	—	8	2	no	—
540	C₂²	—	8	2	no	—
541	C₂²	—	8	2	no	—
542	C₂²	—	8	2	no	—
543	C₂²	—	8	2	no	—
544	C₂²	—	8	2	no	—
545	C₂²	C₂×#121×C₅₉	1	1	yes	characteristic
546	C₂²	D₄²×C₅₉	1	1	yes	characteristic
547	C₂²	C₅₉×#548	1	1	yes	characteristic
548	C₂²	C₂²×SD₁₆×C₅₉	1	1	yes	characteristic
549	C₂²	C₂×AGL(1,ℤ/8)×C₅₉	1	1	yes	characteristic
550	C₂²	C₅₉×#576	1	1	yes	characteristic
551	C₂²	C₅₉×#577	1	1	yes	characteristic
552	C₄	—	2	4	no	—
553	C₄	—	2	2	no	—
554	C₄	—	2	4	no	—
555	C₄	—	4	2	no	—
556	C₄	—	4	2	no	—
557	C₄	—	8	2	no	—
558	C₄	—	8	1	no	—
559	C₄	C₅₉×#460	1	4	yes	complemented
560	C₄	C₂×#121×C₅₉	1	1	yes	characteristic
561	C₄	C₅₉×#537	1	1	yes	characteristic
562	C₄	C₅₉×#547	1	1	yes	characteristic
563	C₄	C₅₉×#553	1	1	yes	characteristic
564	C₂	—	8	2	no	—
565	C₂	—	16	2	no	—
566	C₂	C₅₉×#2416	1	1	yes	characteristic
567	C₂	C₂×C₅₉×#460	1	1	yes	characteristic
568	C₂	C₅₉×#3045	1	1	yes	characteristic
569	C₂	C₅₉×#3274	1	1	yes	characteristic
570	C₂	D₄×SD₁₆×C₅₉	1	1	yes	characteristic
571	C₂	C₅₉×#3362	1	1	yes	characteristic
572	C₂	C₅₉×#3368	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₉
prime radical	C₅₉×#2541	C₂
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×#460	C₂	semidirect
2. C₂×C₅₉×#460	C₂	semidirect
3. C₂×C₄×SD₁₆×C₅₉	C₂	semidirect
5. Q₈×C₄⋊C₄×C₅₉	C₂	semidirect
6. C₅₉×#2065	C₂	semidirect
7. C₅₉×#2041	C₂	semidirect
8. C₅₉×#1957	C₂	semidirect
9. C₅₉×#1939	C₂	semidirect
10. C₅₉×#1880	C₂	semidirect
11. C₅₉×#1802	C₂	semidirect
57. C₂×SD₁₆×C₅₉	D₄	semidirect
70. C₄×Q₈×C₅₉	D₄	semidirect
168. C₄⋊C₄×C₅₉	SD₁₆	semidirect
197. Q₈×C₅₉	#121	semidirect
244. #24728	C₅₉	direct (N × H)
273. C₄×C₅₉	#460	semidirect
278. C₂×#460	C₂×C₅₉	semidirect
279. C₂×#460	C₂×C₅₉	semidirect
280. C₂×C₄×SD₁₆	C₂×C₅₉	semidirect
282. Q₈×C₄⋊C₄	C₂×C₅₉	semidirect
283. #2065	C₂×C₅₉	semidirect
284. #2041	C₂×C₅₉	semidirect
285. #1957	C₂×C₅₉	semidirect
286. #1939	C₂×C₅₉	semidirect
287. #1880	C₂×C₅₉	semidirect
288. #1802	C₂×C₅₉	semidirect
330. C₅₉	#24728	direct (N × H)
344. C₂×SD₁₆	D₄×C₅₉	semidirect
357. C₄×Q₈	D₄×C₅₉	semidirect
455. C₄⋊C₄	SD₁₆×C₅₉	semidirect
484. Q₈	#121×C₅₉	semidirect
559. C₄	C₅₉×#460	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₉×#460	2
C₂×C₄×SD₁₆×C₅₉	2
C₅₉×#2541	2
Q₈×C₄⋊C₄×C₅₉	2
C₅₉×#2065	2
C₅₉×#2041	2
C₅₉×#1957	2
C₅₉×#1939	2
C₅₉×#1880	2
C₅₉×#1802	2
#24728	59

Subgroups — C59×#24728 #28436837

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#24728 #28436837