Subgroups — C₅₉×#24280 #28436389

Order 15104.

1328 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#135×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₉×#473	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅₉×#472	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₉×#3005	C₂	1	2	yes	maximal, complemented
5	C₅₉×#2929	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₉×#2046	C₂	1	2	yes	maximal, complemented
7	C₅₉×#2019	C₂	1	2	yes	maximal, complemented
8	C₅₉×#1979	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₉×#1978	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₉×#1810	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₉×#1785	C₂	1	2	yes	maximal, complemented
12	C₂²×#41×C₅₉	C₂²	1	2	yes	—
13	C₂×#137×C₅₉	C₂²	1	2	yes	—
14	C₂×AGL(1,ℤ/8)×C₅₉	C₂²	1	2	yes	—
15	C₂×#135×C₅₉	—	2	4	no	—
16	C₂×#135×C₅₉	—	2	4	no	—
17	C₂×#135×C₅₉	—	2	2	no	—
18	C₂×#135×C₅₉	—	2	2	no	—
19	C₂²×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
20	C₂²×SD₁₆×C₅₉	C₂²	1	2	yes	—
21	C₂²×D₈×C₅₉	C₂²	1	1	yes	characteristic
22	C₂×#131×C₅₉	C₂²	1	1	yes	characteristic
23	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₅₉×#525	C₂²	1	1	yes	characteristic
25	C₅₉×#473	—	2	4	no	—
26	C₅₉×#472	—	2	4	no	—
27	C₅₉×#452	—	2	4	no	—
28	C₅₉×#449	—	2	4	no	—
29	C₅₉×#435	—	2	2	no	—
30	C₈×D₄×C₅₉	—	2	2	no	—
31	C₂×#108×C₅₉	C₂²	1	1	yes	characteristic
32	C₂×#108×C₅₉	C₂²	1	1	yes	characteristic
33	C₅₉×#424	C₂²	1	1	yes	characteristic
34	C₅₉×#419	C₂²	1	2	yes	—
35	C₅₉×#418	C₂²	1	2	yes	—
36	C₂×#101×C₅₉	C₂²	1	2	yes	—
37	C₂×#101×C₅₉	C₂²	1	2	yes	—
38	C₂×#100×C₅₉	C₂²	1	2	yes	—
39	C₂×#100×C₅₉	C₂²	1	2	yes	—
40	C₅₉×#408	C₂²	1	1	yes	characteristic
41	C₅₉×#407	C₂²	1	2	yes	—
42	C₅₉×#407	C₂²	1	1	yes	characteristic
43	C₅₉×#405	C₂²	1	1	yes	characteristic
44	C₅₉×#341	C₂²	1	2	yes	—
45	C₂×#41×C₅₉	—	2	4	no	—
46	C₂×#41×C₅₉	—	2	4	no	—
47	C₂×#41×C₅₉	—	2	4	no	—
48	C₂×#41×C₅₉	—	2	4	no	—
49	C₂×#41×C₅₉	—	4	2	no	—
50	C₂×#41×C₅₉	—	4	2	no	—
51	C₂×#41×C₅₉	C₂³	1	1	yes	characteristic
52	C₂²×Q₈×C₅₉	C₂³	1	2	yes	—
53	C₂²×D₄×C₅₉	—	2	2	no	—
54	C₂²×D₄×C₅₉	C₂³	1	2	yes	—
55	C₂³×C₄×C₅₉	—	2	2	no	—
56	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
57	#137×C₅₉	—	2	4	no	—
58	#137×C₅₉	—	4	2	no	—
59	AGL(1,ℤ/8)×C₅₉	—	2	4	no	—
60	AGL(1,ℤ/8)×C₅₉	—	4	2	no	—
61	#135×C₅₉	—	2	16	no	—
62	#135×C₅₉	—	2	16	no	—
63	C₂×Q₁₆×C₅₉	—	2	2	no	—
64	C₂×Q₁₆×C₅₉	—	2	1	no	—
65	C₂×Q₁₆×C₅₉	—	2	1	no	—
66	C₂×Q₁₆×C₅₉	—	2	2	no	—
67	C₂×Q₁₆×C₅₉	D₄	1	4	yes	complemented
68	C₂×SD₁₆×C₅₉	—	2	4	no	—
69	C₂×SD₁₆×C₅₉	—	2	4	no	—
70	C₂×SD₁₆×C₅₉	—	2	2	no	—
71	C₂×SD₁₆×C₅₉	—	2	2	no	—
72	C₂×SD₁₆×C₅₉	—	2	4	no	—
73	C₂×D₈×C₅₉	—	2	2	no	—
74	C₂×D₈×C₅₉	—	2	1	no	—
75	C₂×D₈×C₅₉	—	2	1	no	—
76	C₂×D₈×C₅₉	—	2	2	no	—
77	C₂×D₈×C₅₉	D₄	1	4	yes	complemented
78	#131×C₅₉	—	2	2	no	—
79	#131×C₅₉	—	2	2	no	—
80	C₂×M₁₆×C₅₉	—	2	1	no	—
81	C₂×M₁₆×C₅₉	C₂³	1	2	yes	—
82	C₂×M₁₆×C₅₉	C₂³	1	2	yes	—
83	C₂×M₁₆×C₅₉	C₂³	1	1	yes	characteristic
84	C₂×M₁₆×C₅₉	C₂³	1	1	yes	characteristic
85	C₂²×C₈×C₅₉	—	2	2	no	—
86	C₂²×C₈×C₅₉	—	2	2	no	—
87	C₂²×C₈×C₅₉	—	2	1	no	—
88	C₂²×C₈×C₅₉	D₄	1	2	yes	—
89	C₂²×C₈×C₅₉	D₄	1	2	yes	—
90	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
91	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
92	#123×C₅₉	—	2	1	no	—
93	C₂²⋊Q₈×C₅₉	D₄	1	2	yes	—
94	#121×C₅₉	D₄	1	2	yes	—
95	#120×C₅₉	—	2	1	no	—
96	C₄×D₄×C₅₉	—	2	2	no	—
97	#117×C₅₉	C₂³	1	1	yes	characteristic
98	#108×C₅₉	—	2	2	no	—
99	#108×C₅₉	—	2	2	no	—
100	C₄⋊C₈×C₅₉	—	2	2	no	—
101	#103×C₅₉	—	2	4	no	—
102	#102×C₅₉	—	2	4	no	—
103	#101×C₅₉	—	2	4	no	—
104	#101×C₅₉	—	4	2	no	—
105	#100×C₅₉	—	2	4	no	—
106	#100×C₅₉	—	4	2	no	—
107	C₂²⋊C₈×C₅₉	—	2	2	no	—
108	C₂²⋊C₈×C₅₉	—	2	4	no	—
109	C₂²⋊C₈×C₅₉	—	2	2	no	—
110	#97×C₅₉	—	2	1	no	—
111	C₄×C₈×C₅₉	—	2	1	no	—
112	C₂⁴×C₅₉	—	2	2	no	—
113	C₂⁴×C₅₉	#41	1	1	yes	characteristic
114	#41×C₅₉	—	2	16	no	—
115	#41×C₅₉	—	2	1	no	—
116	#41×C₅₉	—	2	1	no	—
117	#41×C₅₉	—	2	1	no	—
118	#41×C₅₉	—	2	1	no	—
119	#41×C₅₉	—	4	8	no	—
120	#41×C₅₉	—	4	8	no	—
121	#41×C₅₉	—	4	8	no	—
122	C₂×Q₈×C₅₉	—	2	2	no	—
123	C₂×Q₈×C₅₉	—	2	2	no	—
124	C₂×Q₈×C₅₉	—	2	2	no	—
125	C₂×Q₈×C₅₉	—	2	2	no	—
126	C₂×Q₈×C₅₉	—	2	2	no	—
127	C₂×Q₈×C₅₉	C₂×D₄	1	4	yes	—
128	C₂×Q₈×C₅₉	C₂×D₄	1	1	yes	characteristic
129	C₂×D₄×C₅₉	—	2	2	no	—
130	C₂×D₄×C₅₉	—	2	1	no	—
131	C₂×D₄×C₅₉	—	2	2	no	—
132	C₂×D₄×C₅₉	—	2	2	no	—
133	C₂×D₄×C₅₉	—	2	2	no	—
134	C₂×D₄×C₅₉	—	2	2	no	—
135	C₂×D₄×C₅₉	—	2	2	no	—
136	C₂×D₄×C₅₉	—	4	4	no	—
137	C₂×D₄×C₅₉	—	4	4	no	—
138	C₂×D₄×C₅₉	—	4	4	no	—
139	C₂×D₄×C₅₉	C₂×D₄	1	4	yes	—
140	C₂×D₄×C₅₉	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₅₉	—	2	1	no	—
142	C₂²×C₄×C₅₉	—	2	1	no	—
143	C₂²×C₄×C₅₉	—	2	1	no	—
144	C₂²×C₄×C₅₉	—	2	1	no	—
145	C₂²×C₄×C₅₉	—	2	2	no	—
146	C₂²×C₄×C₅₉	—	2	1	no	—
147	C₂²×C₄×C₅₉	—	4	4	no	—
148	C₂²×C₄×C₅₉	—	4	4	no	—
149	C₂²×C₄×C₅₉	—	4	4	no	—
150	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
151	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
152	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
153	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
154	Q₁₆×C₅₉	—	2	4	no	—
155	Q₁₆×C₅₉	—	2	4	no	—
156	Q₁₆×C₅₉	—	4	4	no	—
157	SD₁₆×C₅₉	—	2	16	no	—
158	SD₁₆×C₅₉	—	4	4	no	—
159	SD₁₆×C₅₉	—	4	4	no	—
160	D₈×C₅₉	—	2	4	no	—
161	D₈×C₅₉	—	2	4	no	—
162	D₈×C₅₉	—	4	4	no	—
163	M₁₆×C₅₉	—	2	1	no	—
164	M₁₆×C₅₉	—	2	2	no	—
165	M₁₆×C₅₉	—	2	1	no	—
166	M₁₆×C₅₉	—	2	2	no	—
167	M₁₆×C₅₉	—	2	1	no	—
168	M₁₆×C₅₉	—	2	2	no	—
169	M₁₆×C₅₉	—	2	1	no	—
170	M₁₆×C₅₉	—	2	2	no	—
171	M₁₆×C₅₉	—	4	2	no	—
172	C₂×C₈×C₅₉	—	2	1	no	—
173	C₂×C₈×C₅₉	—	2	1	no	—
174	C₂×C₈×C₅₉	—	2	4	no	—
175	C₂×C₈×C₅₉	—	2	4	no	—
176	C₂×C₈×C₅₉	—	2	1	no	—
177	C₂×C₈×C₅₉	—	2	2	no	—
178	C₂×C₈×C₅₉	—	2	2	no	—
179	C₂×C₈×C₅₉	—	2	1	no	—
180	C₂×C₈×C₅₉	—	4	2	no	—
181	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
182	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
183	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
184	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
185	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
186	C₄⋊C₄×C₅₉	—	2	2	no	—
187	C₄⋊C₄×C₅₉	C₂×D₄	1	2	yes	—
188	C₂²⋊C₄×C₅₉	—	2	2	no	—
189	C₂²⋊C₄×C₅₉	—	2	1	no	—
190	C₂²⋊C₄×C₅₉	—	2	2	no	—
191	C₄²×C₅₉	—	2	1	no	—
192	C₂³×C₅₉	—	2	1	no	—
193	C₂³×C₅₉	—	2	1	no	—
194	C₂³×C₅₉	—	2	1	no	—
195	C₂³×C₅₉	—	2	4	no	—
196	C₂³×C₅₉	—	4	2	no	—
197	C₂³×C₅₉	—	4	2	no	—
198	C₂³×C₅₉	—	4	2	no	—
199	C₂³×C₅₉	—	8	4	no	—
200	C₂³×C₅₉	#121	1	2	yes	—
201	C₂³×C₅₉	C₂×#41	1	1	yes	characteristic
202	Q₈×C₅₉	—	2	4	no	—
203	Q₈×C₅₉	—	2	1	no	—
204	Q₈×C₅₉	—	2	4	no	—
205	Q₈×C₅₉	—	2	1	no	—
206	Q₈×C₅₉	—	4	2	no	—
207	Q₈×C₅₉	—	4	2	no	—
208	D₄×C₅₉	—	2	4	no	—
209	D₄×C₅₉	—	2	1	no	—
210	D₄×C₅₉	—	2	4	no	—
211	D₄×C₅₉	—	2	1	no	—
212	D₄×C₅₉	—	4	4	no	—
213	D₄×C₅₉	—	4	16	no	—
214	D₄×C₅₉	—	4	1	no	—
215	D₄×C₅₉	—	4	1	no	—
216	D₄×C₅₉	—	4	2	no	—
217	D₄×C₅₉	—	4	2	no	—
218	C₂×C₄×C₅₉	—	2	1	no	—
219	C₂×C₄×C₅₉	—	2	1	no	—
220	C₂×C₄×C₅₉	—	2	2	no	—
221	C₂×C₄×C₅₉	—	2	2	no	—
222	C₂×C₄×C₅₉	—	2	2	no	—
223	C₂×C₄×C₅₉	—	2	2	no	—
224	C₂×C₄×C₅₉	—	2	2	no	—
225	C₂×C₄×C₅₉	—	2	1	no	—
226	C₂×C₄×C₅₉	—	2	4	no	—
227	C₂×C₄×C₅₉	—	4	16	no	—
228	C₂×C₄×C₅₉	—	4	2	no	—
229	C₂×C₄×C₅₉	—	4	2	no	—
230	C₂×C₄×C₅₉	—	4	1	no	—
231	C₂×C₄×C₅₉	—	4	1	no	—
232	C₂×C₄×C₅₉	#121	1	2	yes	—
233	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
234	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
235	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
236	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
238	C₂×C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
239	C₈×C₅₉	—	2	2	no	—
240	C₈×C₅₉	—	4	1	no	—
241	C₈×C₅₉	—	4	1	no	—
242	C₈×C₅₉	—	4	2	no	—
243	C₈×C₅₉	—	4	2	no	—
244	C₈×C₅₉	#120	1	2	yes	—
245	C₈×C₅₉	#120	1	2	yes	—
246	#24280	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
247	C₂²×C₅₉	—	2	2	no	—
248	C₂²×C₅₉	—	2	2	no	—
249	C₂²×C₅₉	—	2	2	no	—
250	C₂²×C₅₉	—	2	2	no	—
251	C₂²×C₅₉	—	4	2	no	—
252	C₂²×C₅₉	—	4	2	no	—
253	C₂²×C₅₉	—	4	4	no	—
254	C₂²×C₅₉	—	4	1	no	—
255	C₂²×C₅₉	—	4	2	no	—
256	C₂²×C₅₉	—	8	4	no	—
257	C₂²×C₅₉	—	8	4	no	—
258	C₂²×C₅₉	—	8	4	no	—
259	C₂²×C₅₉	C₂×#121	1	1	yes	characteristic
260	C₂²×C₅₉	#546	1	1	yes	characteristic
261	C₂²×C₅₉	#546	1	1	yes	characteristic
262	C₄×C₅₉	—	2	2	no	—
263	C₄×C₅₉	—	4	4	no	—
264	C₄×C₅₉	—	4	1	no	—
265	C₄×C₅₉	—	4	2	no	—
266	C₄×C₅₉	C₂×#120	1	1	yes	characteristic
267	C₄×C₅₉	#535	1	1	yes	characteristic
268	C₄×C₅₉	D₄²	1	1	yes	characteristic
269	C₄×C₅₉	D₄²	1	1	yes	characteristic
270	C₂²×#135	C₂×C₅₉	1	1	yes	characteristic, complemented
271	C₂×#473	C₂×C₅₉	1	1	yes	characteristic, complemented
272	C₂×#472	C₂×C₅₉	1	1	yes	characteristic, complemented
273	#3005	C₂×C₅₉	1	2	yes	complemented
274	#2929	C₂×C₅₉	1	1	yes	characteristic, complemented
275	#2046	C₂×C₅₉	1	2	yes	complemented
276	#2019	C₂×C₅₉	1	2	yes	complemented
277	#1979	C₂×C₅₉	1	1	yes	characteristic, complemented
278	#1978	C₂×C₅₉	1	1	yes	characteristic, complemented
279	#1810	C₂×C₅₉	1	1	yes	characteristic, complemented
280	#1785	C₂×C₅₉	1	2	yes	complemented
281	C₂×C₅₉	—	2	1	no	—
282	C₂×C₅₉	—	2	1	no	—
283	C₂×C₅₉	—	4	2	no	—
284	C₂×C₅₉	—	8	1	no	—
285	C₂×C₅₉	—	8	4	no	—
286	C₂×C₅₉	#2408	1	1	yes	characteristic
287	C₂×C₅₉	#3067	1	2	yes	—
288	C₂²×#41	C₂²×C₅₉	1	2	yes	—
289	C₂×#137	C₂²×C₅₉	1	2	yes	—
290	C₂×AGL(1,ℤ/8)	C₂²×C₅₉	1	2	yes	—
291	C₂×#135	—	2	4	no	—
292	C₂×#135	—	2	4	no	—
293	C₂×#135	—	2	2	no	—
294	C₂×#135	—	2	2	no	—
295	C₂²×Q₁₆	C₂²×C₅₉	1	1	yes	characteristic
296	C₂²×SD₁₆	C₂²×C₅₉	1	2	yes	—
297	C₂²×D₈	C₂²×C₅₉	1	1	yes	characteristic
298	C₂×#131	C₂²×C₅₉	1	1	yes	characteristic
299	C₂³×C₈	C₂²×C₅₉	1	1	yes	characteristic
300	#525	C₂²×C₅₉	1	1	yes	characteristic
301	#473	—	2	4	no	—
302	#472	—	2	4	no	—
303	#452	—	2	4	no	—
304	#449	—	2	4	no	—
305	#435	—	2	2	no	—
306	C₈×D₄	—	2	2	no	—
307	C₂×#108	C₂²×C₅₉	1	1	yes	characteristic
308	C₂×#108	C₂²×C₅₉	1	1	yes	characteristic
309	#424	C₂²×C₅₉	1	1	yes	characteristic
310	#419	C₂²×C₅₉	1	2	yes	—
311	#418	C₂²×C₅₉	1	2	yes	—
312	C₂×#101	C₂²×C₅₉	1	2	yes	—
313	C₂×#101	C₂²×C₅₉	1	2	yes	—
314	C₂×#100	C₂²×C₅₉	1	2	yes	—
315	C₂×#100	C₂²×C₅₉	1	2	yes	—
316	#408	C₂²×C₅₉	1	1	yes	characteristic
317	#407	C₂²×C₅₉	1	2	yes	—
318	#407	C₂²×C₅₉	1	1	yes	characteristic
319	#405	C₂²×C₅₉	1	1	yes	characteristic
320	#341	C₂²×C₅₉	1	2	yes	—
321	C₅₉	#24280	1	1	yes	characteristic, direct product factor, Sylow
322	C₂×#41	—	2	4	no	—
323	C₂×#41	—	2	4	no	—
324	C₂×#41	—	2	4	no	—
325	C₂×#41	—	2	4	no	—
326	C₂×#41	—	4	2	no	—
327	C₂×#41	—	4	2	no	—
328	C₂×#41	C₂³×C₅₉	1	1	yes	characteristic
329	C₂²×Q₈	C₂³×C₅₉	1	2	yes	—
330	C₂²×D₄	—	2	2	no	—
331	C₂²×D₄	C₂³×C₅₉	1	2	yes	—
332	C₂³×C₄	—	2	2	no	—
333	C₂³×C₄	C₂³×C₅₉	1	1	yes	characteristic
334	#137	—	2	4	no	—
335	#137	—	4	2	no	—
336	AGL(1,ℤ/8)	—	2	4	no	—
337	AGL(1,ℤ/8)	—	4	2	no	—
338	#135	—	2	16	no	—
339	#135	—	2	16	no	—
340	C₂×Q₁₆	—	2	2	no	—
341	C₂×Q₁₆	—	2	1	no	—
342	C₂×Q₁₆	—	2	1	no	—
343	C₂×Q₁₆	—	2	2	no	—
344	C₂×Q₁₆	D₄×C₅₉	1	4	yes	complemented
345	C₂×SD₁₆	—	2	4	no	—
346	C₂×SD₁₆	—	2	4	no	—
347	C₂×SD₁₆	—	2	2	no	—
348	C₂×SD₁₆	—	2	2	no	—
349	C₂×SD₁₆	—	2	4	no	—
350	C₂×D₈	—	2	2	no	—
351	C₂×D₈	—	2	1	no	—
352	C₂×D₈	—	2	1	no	—
353	C₂×D₈	—	2	2	no	—
354	C₂×D₈	D₄×C₅₉	1	4	yes	complemented
355	#131	—	2	2	no	—
356	#131	—	2	2	no	—
357	C₂×M₁₆	—	2	1	no	—
358	C₂×M₁₆	C₂³×C₅₉	1	1	yes	characteristic
359	C₂×M₁₆	C₂³×C₅₉	1	2	yes	—
360	C₂×M₁₆	C₂³×C₅₉	1	2	yes	—
361	C₂×M₁₆	C₂³×C₅₉	1	1	yes	characteristic
362	C₂²×C₈	—	2	2	no	—
363	C₂²×C₈	—	2	2	no	—
364	C₂²×C₈	—	2	1	no	—
365	C₂²×C₈	D₄×C₅₉	1	2	yes	—
366	C₂²×C₈	D₄×C₅₉	1	2	yes	—
367	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
368	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
369	#123	—	2	1	no	—
370	C₂²⋊Q₈	D₄×C₅₉	1	2	yes	—
371	#121	D₄×C₅₉	1	2	yes	—
372	#120	—	2	1	no	—
373	C₄×D₄	—	2	2	no	—
374	#117	C₂³×C₅₉	1	1	yes	characteristic
375	#108	—	2	2	no	—
376	#108	—	2	2	no	—
377	C₄⋊C₈	—	2	2	no	—
378	#103	—	2	4	no	—
379	#102	—	2	4	no	—
380	#101	—	2	4	no	—
381	#101	—	4	2	no	—
382	#100	—	2	4	no	—
383	#100	—	4	2	no	—
384	C₂²⋊C₈	—	2	2	no	—
385	C₂²⋊C₈	—	2	4	no	—
386	C₂²⋊C₈	—	2	2	no	—
387	#97	—	2	1	no	—
388	C₄×C₈	—	2	1	no	—
389	C₂⁴	—	2	2	no	—
390	C₂⁴	#41×C₅₉	1	1	yes	characteristic
391	#41	—	2	16	no	—
392	#41	—	2	1	no	—
393	#41	—	2	1	no	—
394	#41	—	2	1	no	—
395	#41	—	2	1	no	—
396	#41	—	4	8	no	—
397	#41	—	4	8	no	—
398	#41	—	4	8	no	—
399	C₂×Q₈	—	2	2	no	—
400	C₂×Q₈	—	2	2	no	—
401	C₂×Q₈	—	2	2	no	—
402	C₂×Q₈	—	2	2	no	—
403	C₂×Q₈	—	2	2	no	—
404	C₂×Q₈	C₂×D₄×C₅₉	1	4	yes	—
405	C₂×Q₈	C₂×D₄×C₅₉	1	1	yes	characteristic
406	C₂×D₄	—	2	2	no	—
407	C₂×D₄	—	2	1	no	—
408	C₂×D₄	—	2	2	no	—
409	C₂×D₄	—	2	2	no	—
410	C₂×D₄	—	2	2	no	—
411	C₂×D₄	—	2	2	no	—
412	C₂×D₄	—	2	2	no	—
413	C₂×D₄	—	4	4	no	—
414	C₂×D₄	—	4	4	no	—
415	C₂×D₄	—	4	4	no	—
416	C₂×D₄	C₂×D₄×C₅₉	1	4	yes	—
417	C₂×D₄	C₂×D₄×C₅₉	1	1	yes	characteristic
418	C₂²×C₄	—	2	1	no	—
419	C₂²×C₄	—	2	1	no	—
420	C₂²×C₄	—	2	1	no	—
421	C₂²×C₄	—	2	1	no	—
422	C₂²×C₄	—	2	2	no	—
423	C₂²×C₄	—	2	1	no	—
424	C₂²×C₄	—	4	4	no	—
425	C₂²×C₄	—	4	4	no	—
426	C₂²×C₄	—	4	4	no	—
427	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
428	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
429	C₂²×C₄	#41×C₅₉	1	1	yes	characteristic
430	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
431	Q₁₆	—	2	4	no	—
432	Q₁₆	—	2	4	no	—
433	Q₁₆	—	4	4	no	—
434	SD₁₆	—	2	16	no	—
435	SD₁₆	—	4	4	no	—
436	SD₁₆	—	4	4	no	—
437	D₈	—	2	4	no	—
438	D₈	—	2	4	no	—
439	D₈	—	4	4	no	—
440	M₁₆	—	2	1	no	—
441	M₁₆	—	2	2	no	—
442	M₁₆	—	2	1	no	—
443	M₁₆	—	2	2	no	—
444	M₁₆	—	2	1	no	—
445	M₁₆	—	2	2	no	—
446	M₁₆	—	2	1	no	—
447	M₁₆	—	2	2	no	—
448	M₁₆	—	4	2	no	—
449	C₂×C₈	—	2	1	no	—
450	C₂×C₈	—	2	1	no	—
451	C₂×C₈	—	2	4	no	—
452	C₂×C₈	—	2	4	no	—
453	C₂×C₈	—	2	1	no	—
454	C₂×C₈	—	2	2	no	—
455	C₂×C₈	—	2	2	no	—
456	C₂×C₈	—	2	1	no	—
457	C₂×C₈	—	4	2	no	—
458	C₂×C₈	C₂×D₄×C₅₉	1	2	yes	—
459	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
460	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
461	C₂×C₈	C₂×D₄×C₅₉	1	2	yes	—
462	C₂×C₈	C₂×D₄×C₅₉	1	2	yes	—
463	C₄⋊C₄	—	2	2	no	—
464	C₄⋊C₄	C₂×D₄×C₅₉	1	2	yes	—
465	C₂²⋊C₄	—	2	2	no	—
466	C₂²⋊C₄	—	2	1	no	—
467	C₂²⋊C₄	—	2	2	no	—
468	C₄²	—	2	1	no	—
469	C₂³	—	2	1	no	—
470	C₂³	—	2	1	no	—
471	C₂³	—	2	1	no	—
472	C₂³	—	2	4	no	—
473	C₂³	—	4	2	no	—
474	C₂³	—	4	2	no	—
475	C₂³	—	4	2	no	—
476	C₂³	—	8	4	no	—
477	C₂³	#121×C₅₉	1	2	yes	—
478	C₂³	C₂×#41×C₅₉	1	1	yes	characteristic
479	Q₈	—	2	4	no	—
480	Q₈	—	2	1	no	—
481	Q₈	—	2	4	no	—
482	Q₈	—	2	1	no	—
483	Q₈	—	4	2	no	—
484	Q₈	—	4	2	no	—
485	D₄	—	2	4	no	—
486	D₄	—	2	1	no	—
487	D₄	—	2	4	no	—
488	D₄	—	2	1	no	—
489	D₄	—	4	4	no	—
490	D₄	—	4	16	no	—
491	D₄	—	4	1	no	—
492	D₄	—	4	1	no	—
493	D₄	—	4	2	no	—
494	D₄	—	4	2	no	—
495	C₂×C₄	—	2	1	no	—
496	C₂×C₄	—	2	1	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	2	no	—
502	C₂×C₄	—	2	1	no	—
503	C₂×C₄	—	2	4	no	—
504	C₂×C₄	—	4	16	no	—
505	C₂×C₄	—	4	2	no	—
506	C₂×C₄	—	4	2	no	—
507	C₂×C₄	—	4	1	no	—
508	C₂×C₄	—	4	1	no	—
509	C₂×C₄	#121×C₅₉	1	2	yes	—
510	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
511	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
512	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
513	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
514	C₂×C₄	H(5,𝔽₂)×C₅₉	1	1	yes	characteristic
515	C₂×C₄	H(5,𝔽₂)×C₅₉	1	1	yes	characteristic
516	C₈	—	2	2	no	—
517	C₈	—	4	1	no	—
518	C₈	—	4	1	no	—
519	C₈	—	4	2	no	—
520	C₈	—	4	2	no	—
521	C₈	#120×C₅₉	1	2	yes	—
522	C₈	#120×C₅₉	1	2	yes	—
523	C₂²	—	2	2	no	—
524	C₂²	—	2	2	no	—
525	C₂²	—	2	2	no	—
526	C₂²	—	2	2	no	—
527	C₂²	—	4	2	no	—
528	C₂²	—	4	2	no	—
529	C₂²	—	4	2	no	—
530	C₂²	—	4	4	no	—
531	C₂²	—	4	1	no	—
532	C₂²	—	8	4	no	—
533	C₂²	—	8	4	no	—
534	C₂²	—	8	4	no	—
535	C₂²	C₂×#121×C₅₉	1	1	yes	characteristic
536	C₂²	C₅₉×#546	1	1	yes	characteristic
537	C₂²	C₅₉×#546	1	1	yes	characteristic
538	C₄	—	2	2	no	—
539	C₄	—	4	2	no	—
540	C₄	—	4	4	no	—
541	C₄	—	4	1	no	—
542	C₄	C₂×#120×C₅₉	1	1	yes	characteristic
543	C₄	C₅₉×#535	1	1	yes	characteristic
544	C₄	D₄²×C₅₉	1	1	yes	characteristic
545	C₄	D₄²×C₅₉	1	1	yes	characteristic
546	C₂	—	2	1	no	—
547	C₂	—	2	1	no	—
548	C₂	—	4	2	no	—
549	C₂	—	8	1	no	—
550	C₂	—	8	4	no	—
551	C₂	C₅₉×#2408	1	1	yes	characteristic
552	C₂	C₅₉×#3067	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₉	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ C₂³×C₄×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#135×C₅₉	C₂	semidirect
2. C₂×C₅₉×#473	C₂	semidirect
3. C₂×C₅₉×#472	C₂	semidirect
4. C₅₉×#3005	C₂	semidirect
5. C₅₉×#2929	C₂	semidirect
6. C₅₉×#2046	C₂	semidirect
7. C₅₉×#2019	C₂	semidirect
8. C₅₉×#1979	C₂	semidirect
9. C₅₉×#1978	C₂	semidirect
10. C₅₉×#1810	C₂	semidirect
11. C₅₉×#1785	C₂	semidirect
67. C₂×Q₁₆×C₅₉	D₄	semidirect
77. C₂×D₈×C₅₉	D₄	semidirect
246. #24280	C₅₉	direct (N × H)
270. C₂²×#135	C₂×C₅₉	semidirect
271. C₂×#473	C₂×C₅₉	semidirect
272. C₂×#472	C₂×C₅₉	semidirect
273. #3005	C₂×C₅₉	semidirect
274. #2929	C₂×C₅₉	semidirect
275. #2046	C₂×C₅₉	semidirect
276. #2019	C₂×C₅₉	semidirect
277. #1979	C₂×C₅₉	semidirect
278. #1978	C₂×C₅₉	semidirect
279. #1810	C₂×C₅₉	semidirect
280. #1785	C₂×C₅₉	semidirect
321. C₅₉	#24280	direct (N × H)
344. C₂×Q₁₆	D₄×C₅₉	semidirect
354. C₂×D₈	D₄×C₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#135×C₅₉	2
C₂×C₅₉×#473	2
C₂×C₅₉×#472	2
C₅₉×#3005	2
C₅₉×#2929	2
C₅₉×#2046	2
C₅₉×#2019	2
C₅₉×#1979	2
C₅₉×#1978	2
C₅₉×#1810	2
C₅₉×#1785	2
#24280	59

Subgroups — C59×#24280 #28436389

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#24280 #28436389