Subgroups — C₅₉×#60799 #28472908

Order 15104.

2262 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#3444	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#124×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₉×#3215	C₂	1	4	yes	maximal, complemented
4	C₅₉×#3213	C₂	1	4	yes	maximal, complemented
5	C₅₉×#3211	C₂	1	8	yes	maximal, complemented
6	C₅₉×#3153	C₂	1	2	yes	maximal, complemented
7	C₅₉×#3140	C₂	1	2	yes	maximal, complemented
8	C₅₉×#3040	C₂	1	4	yes	maximal, complemented
9	C₅₉×#3004	C₂	1	2	yes	maximal, complemented
10	C₅₉×#3003	C₂	1	2	yes	maximal, complemented
11	C₅₉×#2920	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×Q₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₂³×D₄×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×#137×C₅₉	C₂²	1	4	yes	complemented
15	C₂×AGL(1,ℤ/8)×C₅₉	C₂²	1	4	yes	—
16	C₅₉×#533	C₂²	1	2	yes	complemented
17	C₅₉×#533	C₂²	1	1	yes	characteristic, complemented
18	C₅₉×#532	C₂²	1	1	yes	characteristic, complemented
19	C₅₉×#532	C₂²	1	2	yes	complemented
20	C₅₉×#529	—	2	2	no	—
21	C₅₉×#529	—	2	2	no	—
22	C₅₉×#529	C₂²	1	4	yes	complemented
23	C₂×#124×C₅₉	—	2	2	no	—
24	C₂×#124×C₅₉	—	2	2	no	—
25	C₂×#124×C₅₉	—	2	2	no	—
26	C₂×#124×C₅₉	C₂²	1	2	yes	—
27	C₂×#124×C₅₉	C₂²	1	2	yes	—
28	C₂×#124×C₅₉	C₂²	1	4	yes	—
29	C₂×#124×C₅₉	C₂²	1	4	yes	—
30	C₅₉×#525	—	2	2	no	—
31	C₅₉×#525	C₂²	1	2	yes	—
32	C₄×#41×C₅₉	—	2	2	no	—
33	C₄×#41×C₅₉	C₂²	1	2	yes	complemented
34	C₂²×C₂²⋊C₄×C₅₉	—	2	2	no	—
35	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
36	C₅₉×#497	C₂²	1	4	yes	complemented
37	C₅₉×#496	C₂²	1	4	yes	complemented
38	C₅₉×#495	C₂²	1	8	yes	complemented
39	C₅₉×#489	C₂²	1	8	yes	complemented
40	C₅₉×#488	C₂²	1	8	yes	complemented
41	C₅₉×#464	C₂²	1	16	yes	complemented
42	C₅₉×#463	C₂²	1	16	yes	complemented
43	C₅₉×#451	C₂²	1	8	yes	complemented
44	C₅₉×#450	C₂²	1	8	yes	—
45	C₅₉×#448	C₂²	1	8	yes	complemented
46	C₅₉×#447	C₂²	1	8	yes	—
47	C₅₉×#432	C₂²	1	4	yes	complemented
48	C₅₉×#431	C₂²	1	4	yes	complemented
49	C₅₉×#423	C₂²	1	2	yes	complemented
50	C₅₉×#419	C₂²	1	4	yes	complemented
51	C₅₉×#418	C₂²	1	4	yes	—
52	C₅₉×#407	C₂²	1	2	yes	—
53	C₄×M₁₆×C₅₉	C₂²	1	2	yes	complemented
54	C₂⁵×C₅₉	—	2	1	no	—
55	C₂×#41×C₅₉	—	2	2	no	—
56	C₂×#41×C₅₉	C₂³	1	2	yes	—
57	C₂²×Q₈×C₅₉	—	2	2	no	—
58	C₂²×Q₈×C₅₉	—	2	1	no	—
59	C₂²×Q₈×C₅₉	—	4	4	no	—
60	C₂²×Q₈×C₅₉	C₂³	1	2	yes	—
61	C₂²×Q₈×C₅₉	C₂³	1	4	yes	—
62	C₂²×D₄×C₅₉	—	2	2	no	—
63	C₂²×D₄×C₅₉	—	2	1	no	—
64	C₂²×D₄×C₅₉	—	4	4	no	—
65	C₂²×D₄×C₅₉	C₂³	1	2	yes	—
66	C₂²×D₄×C₅₉	C₂³	1	4	yes	—
67	C₂³×C₄×C₅₉	—	2	1	no	—
68	C₂³×C₄×C₅₉	D₄	1	2	yes	—
69	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
70	#137×C₅₉	—	4	8	no	—
71	AGL(1,ℤ/8)×C₅₉	—	4	8	no	—
72	C₂×Q₁₆×C₅₉	C₂³	1	8	yes	complemented
73	C₂×SD₁₆×C₅₉	C₂³	1	8	yes	—
74	C₂×SD₁₆×C₅₉	C₂³	1	8	yes	complemented
75	C₂×D₈×C₅₉	C₂³	1	8	yes	—
76	C₂×M₁₆×C₅₉	C₂³	1	4	yes	—
77	C₄⋊Q₈×C₅₉	C₂³	1	1	yes	characteristic, complemented
78	C₄⋊Q₈×C₅₉	C₂³	1	2	yes	complemented
79	#127×C₅₉	C₂³	1	1	yes	characteristic, complemented
80	#126×C₅₉	—	2	4	no	—
81	#125×C₅₉	—	2	2	no	—
82	#125×C₅₉	C₂³	1	2	yes	complemented
83	#124×C₅₉	—	2	2	no	—
84	#124×C₅₉	—	2	2	no	—
85	#124×C₅₉	—	2	4	no	—
86	#124×C₅₉	—	2	4	no	—
87	#124×C₅₉	—	2	4	no	—
88	#124×C₅₉	—	2	4	no	—
89	#124×C₅₉	—	2	4	no	—
90	#124×C₅₉	—	2	2	no	—
91	#124×C₅₉	C₂³	1	8	yes	—
92	#124×C₅₉	C₂³	1	4	yes	—
93	#124×C₅₉	C₂³	1	4	yes	—
94	#124×C₅₉	C₂³	1	2	yes	complemented
95	#123×C₅₉	—	2	4	no	—
96	#123×C₅₉	—	2	2	no	—
97	C₂²⋊Q₈×C₅₉	—	2	2	no	—
98	C₂²⋊Q₈×C₅₉	—	2	2	no	—
99	C₂²⋊Q₈×C₅₉	C₂³	1	4	yes	—
100	#121×C₅₉	—	2	2	no	—
101	#121×C₅₉	—	2	2	no	—
102	#121×C₅₉	C₂³	1	4	yes	—
103	#120×C₅₉	—	2	2	no	—
104	C₄×Q₈×C₅₉	—	2	2	no	—
105	C₄×Q₈×C₅₉	—	2	2	no	—
106	C₄×Q₈×C₅₉	C₂³	1	4	yes	complemented
107	C₄×D₄×C₅₉	—	2	4	no	—
108	C₄×D₄×C₅₉	—	2	4	no	—
109	C₄×D₄×C₅₉	—	2	4	no	—
110	C₄×D₄×C₅₉	—	2	2	no	—
111	C₄×D₄×C₅₉	—	2	2	no	—
112	C₄×D₄×C₅₉	C₂³	1	4	yes	complemented
113	#117×C₅₉	—	2	2	no	—
114	#117×C₅₉	—	2	4	no	—
115	#117×C₅₉	—	2	2	no	—
116	#117×C₅₉	C₂³	1	2	yes	—
117	C₂×C₂²⋊C₄×C₅₉	—	2	4	no	—
118	C₂×C₂²⋊C₄×C₅₉	—	2	4	no	—
119	C₂×C₂²⋊C₄×C₅₉	—	2	4	no	—
120	C₂×C₂²⋊C₄×C₅₉	—	2	4	no	—
121	C₂×C₂²⋊C₄×C₅₉	—	2	4	no	—
122	C₂×C₂²⋊C₄×C₅₉	—	2	4	no	—
123	C₂×C₄²×C₅₉	—	2	2	no	—
124	C₂×C₄²×C₅₉	D₄	1	2	yes	—
125	C₂×C₄²×C₅₉	D₄	1	2	yes	—
126	C₂×C₄²×C₅₉	D₄	1	2	yes	—
127	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
128	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
129	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
130	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
131	C₄⋊C₈×C₅₉	C₂³	1	8	yes	complemented
132	#103×C₅₉	C₂³	1	16	yes	complemented
133	#102×C₅₉	C₂³	1	16	yes	—
134	C₂²⋊C₈×C₅₉	C₂³	1	8	yes	—
135	#97×C₅₉	C₂³	1	4	yes	complemented
136	C₄×C₈×C₅₉	C₂³	1	4	yes	complemented
137	C₂⁴×C₅₉	—	2	4	no	—
138	C₂⁴×C₅₉	—	2	2	no	—
139	C₂⁴×C₅₉	—	4	4	no	—
140	C₂⁴×C₅₉	—	4	8	no	—
141	C₂⁴×C₅₉	C₂×D₄	1	1	yes	characteristic
142	#41×C₅₉	—	2	8	no	—
143	#41×C₅₉	—	4	8	no	—
144	C₂×Q₈×C₅₉	—	2	2	no	—
145	C₂×Q₈×C₅₉	—	2	4	no	—
146	C₂×Q₈×C₅₉	—	2	2	no	—
147	C₂×Q₈×C₅₉	—	2	8	no	—
148	C₂×Q₈×C₅₉	—	4	8	no	—
149	C₂×Q₈×C₅₉	—	4	4	no	—
150	C₂×Q₈×C₅₉	—	4	8	no	—
151	C₂×Q₈×C₅₉	C₂⁴	1	4	yes	—
152	C₂×Q₈×C₅₉	C₂⁴	1	2	yes	—
153	C₂×D₄×C₅₉	—	2	2	no	—
154	C₂×D₄×C₅₉	—	2	4	no	—
155	C₂×D₄×C₅₉	—	2	2	no	—
156	C₂×D₄×C₅₉	—	2	4	no	—
157	C₂×D₄×C₅₉	—	2	2	no	—
158	C₂×D₄×C₅₉	—	2	8	no	—
159	C₂×D₄×C₅₉	—	4	8	no	—
160	C₂×D₄×C₅₉	—	4	4	no	—
161	C₂×D₄×C₅₉	—	4	8	no	—
162	C₂×D₄×C₅₉	C₂⁴	1	4	yes	—
163	C₂×D₄×C₅₉	C₂⁴	1	2	yes	—
164	C₂²×C₄×C₅₉	—	2	4	no	—
165	C₂²×C₄×C₅₉	—	2	4	no	—
166	C₂²×C₄×C₅₉	—	2	2	no	—
167	C₂²×C₄×C₅₉	—	2	2	no	—
168	C₂²×C₄×C₅₉	—	2	4	no	—
169	C₂²×C₄×C₅₉	—	4	4	no	—
170	C₂²×C₄×C₅₉	—	4	4	no	—
171	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
172	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
174	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
175	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
176	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
177	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
178	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
179	C₂²×C₄×C₅₉	C₂⁴	1	2	yes	—
180	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
181	Q₁₆×C₅₉	—	4	8	no	—
182	SD₁₆×C₅₉	—	4	8	no	—
183	SD₁₆×C₅₉	—	4	8	no	—
184	D₈×C₅₉	—	4	8	no	—
185	M₁₆×C₅₉	—	2	8	no	—
186	C₂×C₈×C₅₉	C₂⁴	1	8	yes	—
187	C₄⋊C₄×C₅₉	—	2	2	no	—
188	C₄⋊C₄×C₅₉	—	2	2	no	—
189	C₄⋊C₄×C₅₉	—	2	4	no	—
190	C₄⋊C₄×C₅₉	—	2	4	no	—
191	C₄⋊C₄×C₅₉	—	2	4	no	—
192	C₄⋊C₄×C₅₉	C₂⁴	1	4	yes	—
193	C₂²⋊C₄×C₅₉	—	2	4	no	—
194	C₂²⋊C₄×C₅₉	—	2	2	no	—
195	C₂²⋊C₄×C₅₉	—	2	8	no	—
196	C₂²⋊C₄×C₅₉	—	2	8	no	—
197	C₂²⋊C₄×C₅₉	—	2	8	no	—
198	C₂²⋊C₄×C₅₉	—	2	8	no	—
199	C₂²⋊C₄×C₅₉	—	2	4	no	—
200	C₂²⋊C₄×C₅₉	—	2	4	no	—
201	C₂²⋊C₄×C₅₉	—	2	4	no	—
202	C₄²×C₅₉	—	2	4	no	—
203	C₄²×C₅₉	—	2	2	no	—
204	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
205	C₄²×C₅₉	C₂×D₄	1	2	yes	—
206	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
207	C₄²×C₅₉	C₂×D₄	1	2	yes	—
208	C₄²×C₅₉	C₂×D₄	1	2	yes	—
209	C₄²×C₅₉	C₂×D₄	1	2	yes	—
210	C₄²×C₅₉	C₂×D₄	1	2	yes	—
211	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
212	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
213	C₄²×C₅₉	C₂⁴	1	2	yes	—
214	C₂³×C₅₉	—	2	2	no	—
215	C₂³×C₅₉	—	2	4	no	—
216	C₂³×C₅₉	—	2	4	no	—
217	C₂³×C₅₉	—	2	2	no	—
218	C₂³×C₅₉	—	4	8	no	—
219	C₂³×C₅₉	—	4	8	no	—
220	C₂³×C₅₉	—	4	8	no	—
221	C₂³×C₅₉	—	4	4	no	—
222	C₂³×C₅₉	—	4	8	no	—
223	C₂³×C₅₉	—	8	16	no	—
224	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
225	C₂³×C₅₉	C₂²×D₄	1	2	yes	—
226	Q₈×C₅₉	—	2	4	no	—
227	Q₈×C₅₉	—	2	8	no	—
228	Q₈×C₅₉	—	4	8	no	—
229	Q₈×C₅₉	—	4	4	no	—
230	Q₈×C₅₉	—	4	4	no	—
231	D₄×C₅₉	—	2	4	no	—
232	D₄×C₅₉	—	2	8	no	—
233	D₄×C₅₉	—	4	8	no	—
234	D₄×C₅₉	—	4	4	no	—
235	D₄×C₅₉	—	4	4	no	—
236	D₄×C₅₉	—	4	8	no	—
237	D₄×C₅₉	—	4	4	no	—
238	C₂×C₄×C₅₉	—	2	2	no	—
239	C₂×C₄×C₅₉	—	2	4	no	—
240	C₂×C₄×C₅₉	—	2	4	no	—
241	C₂×C₄×C₅₉	—	2	4	no	—
242	C₂×C₄×C₅₉	—	2	4	no	—
243	C₂×C₄×C₅₉	—	2	2	no	—
244	C₂×C₄×C₅₉	—	2	4	no	—
245	C₂×C₄×C₅₉	—	2	4	no	—
246	C₂×C₄×C₅₉	—	4	2	no	—
247	C₂×C₄×C₅₉	—	4	2	no	—
248	C₂×C₄×C₅₉	—	4	2	no	—
249	C₂×C₄×C₅₉	—	4	2	no	—
250	C₂×C₄×C₅₉	—	4	8	no	—
251	C₂×C₄×C₅₉	—	4	8	no	—
252	C₂×C₄×C₅₉	—	4	4	no	—
253	C₂×C₄×C₅₉	—	4	4	no	—
254	C₂×C₄×C₅₉	C₂²×D₄	1	2	yes	—
255	C₂×C₄×C₅₉	C₂²×D₄	1	2	yes	—
256	C₂×C₄×C₅₉	C₂²×D₄	1	2	yes	—
257	C₂×C₄×C₅₉	C₂²×D₄	1	2	yes	—
258	C₂×C₄×C₅₉	C₂²×D₄	1	2	yes	—
259	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
260	C₂×C₄×C₅₉	H(5,𝔽₂)	1	4	yes	—
261	C₂×C₄×C₅₉	C₂⁵	1	1	yes	characteristic
262	C₈×C₅₉	—	2	8	no	—
263	#60799	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
264	C₂²×C₅₉	—	2	8	no	—
265	C₂²×C₅₉	—	2	2	no	—
266	C₂²×C₅₉	—	2	2	no	—
267	C₂²×C₅₉	—	2	2	no	—
268	C₂²×C₅₉	—	4	2	no	—
269	C₂²×C₅₉	—	4	4	no	—
270	C₂²×C₅₉	—	4	4	no	—
271	C₂²×C₅₉	—	4	8	no	—
272	C₂²×C₅₉	—	8	8	no	—
273	C₂²×C₅₉	—	8	16	no	—
274	C₂²×C₅₉	#535	1	2	yes	—
275	C₂²×C₅₉	#575	1	4	yes	—
276	C₂²×C₅₉	C₂³×D₄	1	1	yes	characteristic
277	C₄×C₅₉	—	2	2	no	—
278	C₄×C₅₉	—	2	2	no	—
279	C₄×C₅₉	—	2	2	no	—
280	C₄×C₅₉	—	2	2	no	—
281	C₄×C₅₉	—	2	2	no	—
282	C₄×C₅₉	—	4	4	no	—
283	C₄×C₅₉	—	4	2	no	—
284	C₄×C₅₉	—	4	4	no	—
285	C₄×C₅₉	#535	1	2	yes	—
286	C₄×C₅₉	C₂×H(5,𝔽₂)	1	2	yes	—
287	#3444	C₂×C₅₉	1	1	yes	characteristic, complemented
288	C₂²×#124	C₂×C₅₉	1	1	yes	characteristic, complemented
289	#3215	C₂×C₅₉	1	4	yes	complemented
290	#3213	C₂×C₅₉	1	4	yes	complemented
291	#3211	C₂×C₅₉	1	8	yes	complemented
292	#3153	C₂×C₅₉	1	2	yes	complemented
293	#3140	C₂×C₅₉	1	2	yes	complemented
294	#3040	C₂×C₅₉	1	4	yes	complemented
295	#3004	C₂×C₅₉	1	2	yes	complemented
296	#3003	C₂×C₅₉	1	2	yes	complemented
297	#2920	C₂×C₅₉	1	1	yes	characteristic, complemented
298	C₂×C₅₉	—	2	2	no	—
299	C₂×C₅₉	—	2	4	no	—
300	C₂×C₅₉	—	8	2	no	—
301	C₂×C₅₉	—	8	4	no	—
302	C₂×C₅₉	C₂×#535	1	1	yes	characteristic
303	C₂×C₅₉	C₂×#575	1	2	yes	—
304	C₂³×Q₈	C₂²×C₅₉	1	1	yes	characteristic
305	C₂³×D₄	C₂²×C₅₉	1	1	yes	characteristic
306	C₂×#137	C₂²×C₅₉	1	4	yes	complemented
307	C₂×AGL(1,ℤ/8)	C₂²×C₅₉	1	4	yes	—
308	#533	C₂²×C₅₉	1	2	yes	complemented
309	#533	C₂²×C₅₉	1	1	yes	characteristic, complemented
310	#532	C₂²×C₅₉	1	1	yes	characteristic, complemented
311	#532	C₂²×C₅₉	1	2	yes	complemented
312	#529	—	2	2	no	—
313	#529	—	2	2	no	—
314	#529	C₂²×C₅₉	1	4	yes	complemented
315	C₂×#124	—	2	2	no	—
316	C₂×#124	—	2	2	no	—
317	C₂×#124	—	2	2	no	—
318	C₂×#124	C₂²×C₅₉	1	2	yes	—
319	C₂×#124	C₂²×C₅₉	1	4	yes	—
320	C₂×#124	C₂²×C₅₉	1	4	yes	—
321	C₂×#124	C₂²×C₅₉	1	2	yes	—
322	#525	—	2	2	no	—
323	#525	C₂²×C₅₉	1	2	yes	—
324	C₄×#41	—	2	2	no	—
325	C₄×#41	C₂²×C₅₉	1	2	yes	complemented
326	C₂²×C₂²⋊C₄	—	2	2	no	—
327	C₂²×C₄²	C₂²×C₅₉	1	1	yes	characteristic
328	#497	C₂²×C₅₉	1	4	yes	complemented
329	#496	C₂²×C₅₉	1	4	yes	complemented
330	#495	C₂²×C₅₉	1	8	yes	complemented
331	#489	C₂²×C₅₉	1	8	yes	complemented
332	#488	C₂²×C₅₉	1	8	yes	complemented
333	#464	C₂²×C₅₉	1	16	yes	complemented
334	#463	C₂²×C₅₉	1	16	yes	complemented
335	#451	C₂²×C₅₉	1	8	yes	complemented
336	#450	C₂²×C₅₉	1	8	yes	—
337	#448	C₂²×C₅₉	1	8	yes	complemented
338	#447	C₂²×C₅₉	1	8	yes	—
339	#432	C₂²×C₅₉	1	4	yes	complemented
340	#431	C₂²×C₅₉	1	4	yes	complemented
341	#423	C₂²×C₅₉	1	2	yes	complemented
342	#419	C₂²×C₅₉	1	4	yes	complemented
343	#418	C₂²×C₅₉	1	4	yes	—
344	#407	C₂²×C₅₉	1	2	yes	—
345	C₄×M₁₆	C₂²×C₅₉	1	2	yes	complemented
346	C₅₉	#60799	1	1	yes	characteristic, direct product factor, Sylow
347	C₂⁵	—	2	1	no	—
348	C₂×#41	—	2	2	no	—
349	C₂×#41	C₂³×C₅₉	1	2	yes	—
350	C₂²×Q₈	—	2	2	no	—
351	C₂²×Q₈	—	2	1	no	—
352	C₂²×Q₈	—	4	4	no	—
353	C₂²×Q₈	C₂³×C₅₉	1	2	yes	—
354	C₂²×Q₈	C₂³×C₅₉	1	4	yes	—
355	C₂²×D₄	—	2	2	no	—
356	C₂²×D₄	—	2	1	no	—
357	C₂²×D₄	—	4	4	no	—
358	C₂²×D₄	C₂³×C₅₉	1	2	yes	—
359	C₂²×D₄	C₂³×C₅₉	1	4	yes	—
360	C₂³×C₄	—	2	1	no	—
361	C₂³×C₄	D₄×C₅₉	1	2	yes	—
362	C₂³×C₄	C₂³×C₅₉	1	1	yes	characteristic
363	#137	—	4	8	no	—
364	AGL(1,ℤ/8)	—	4	8	no	—
365	C₂×Q₁₆	C₂³×C₅₉	1	8	yes	complemented
366	C₂×SD₁₆	C₂³×C₅₉	1	8	yes	—
367	C₂×SD₁₆	C₂³×C₅₉	1	8	yes	complemented
368	C₂×D₈	C₂³×C₅₉	1	8	yes	—
369	C₂×M₁₆	C₂³×C₅₉	1	4	yes	—
370	C₄⋊Q₈	C₂³×C₅₉	1	1	yes	characteristic, complemented
371	C₄⋊Q₈	C₂³×C₅₉	1	2	yes	complemented
372	#127	C₂³×C₅₉	1	1	yes	characteristic, complemented
373	#126	—	2	4	no	—
374	#125	—	2	2	no	—
375	#125	C₂³×C₅₉	1	2	yes	complemented
376	#124	—	2	2	no	—
377	#124	—	2	4	no	—
378	#124	—	2	2	no	—
379	#124	—	2	4	no	—
380	#124	—	2	4	no	—
381	#124	—	2	4	no	—
382	#124	—	2	4	no	—
383	#124	—	2	2	no	—
384	#124	C₂³×C₅₉	1	8	yes	—
385	#124	C₂³×C₅₉	1	4	yes	—
386	#124	C₂³×C₅₉	1	4	yes	—
387	#124	C₂³×C₅₉	1	2	yes	complemented
388	#123	—	2	4	no	—
389	#123	—	2	2	no	—
390	C₂²⋊Q₈	—	2	2	no	—
391	C₂²⋊Q₈	—	2	2	no	—
392	C₂²⋊Q₈	C₂³×C₅₉	1	4	yes	—
393	#121	—	2	2	no	—
394	#121	—	2	2	no	—
395	#121	C₂³×C₅₉	1	4	yes	—
396	#120	—	2	2	no	—
397	C₄×Q₈	—	2	2	no	—
398	C₄×Q₈	—	2	2	no	—
399	C₄×Q₈	C₂³×C₅₉	1	4	yes	complemented
400	C₄×D₄	—	2	4	no	—
401	C₄×D₄	—	2	4	no	—
402	C₄×D₄	—	2	4	no	—
403	C₄×D₄	—	2	2	no	—
404	C₄×D₄	—	2	2	no	—
405	C₄×D₄	C₂³×C₅₉	1	4	yes	complemented
406	#117	—	2	2	no	—
407	#117	—	2	4	no	—
408	#117	—	2	2	no	—
409	#117	C₂³×C₅₉	1	2	yes	—
410	C₂×C₂²⋊C₄	—	2	4	no	—
411	C₂×C₂²⋊C₄	—	2	4	no	—
412	C₂×C₂²⋊C₄	—	2	4	no	—
413	C₂×C₂²⋊C₄	—	2	4	no	—
414	C₂×C₂²⋊C₄	—	2	4	no	—
415	C₂×C₂²⋊C₄	—	2	4	no	—
416	C₂×C₄²	—	2	2	no	—
417	C₂×C₄²	D₄×C₅₉	1	2	yes	—
418	C₂×C₄²	D₄×C₅₉	1	2	yes	—
419	C₂×C₄²	D₄×C₅₉	1	2	yes	—
420	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
421	C₂×C₄²	C₂³×C₅₉	1	2	yes	—
422	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
423	C₂×C₄²	C₂³×C₅₉	1	2	yes	—
424	C₄⋊C₈	C₂³×C₅₉	1	8	yes	complemented
425	#103	C₂³×C₅₉	1	16	yes	complemented
426	#102	C₂³×C₅₉	1	16	yes	—
427	C₂²⋊C₈	C₂³×C₅₉	1	8	yes	—
428	#97	C₂³×C₅₉	1	4	yes	complemented
429	C₄×C₈	C₂³×C₅₉	1	4	yes	complemented
430	C₂⁴	—	2	4	no	—
431	C₂⁴	—	2	2	no	—
432	C₂⁴	—	4	4	no	—
433	C₂⁴	—	4	8	no	—
434	C₂⁴	C₂×D₄×C₅₉	1	1	yes	characteristic
435	#41	—	2	8	no	—
436	#41	—	4	8	no	—
437	C₂×Q₈	—	2	2	no	—
438	C₂×Q₈	—	2	4	no	—
439	C₂×Q₈	—	2	2	no	—
440	C₂×Q₈	—	2	8	no	—
441	C₂×Q₈	—	4	8	no	—
442	C₂×Q₈	—	4	4	no	—
443	C₂×Q₈	—	4	8	no	—
444	C₂×Q₈	C₂⁴×C₅₉	1	4	yes	—
445	C₂×Q₈	C₂⁴×C₅₉	1	2	yes	—
446	C₂×D₄	—	2	2	no	—
447	C₂×D₄	—	2	4	no	—
448	C₂×D₄	—	2	2	no	—
449	C₂×D₄	—	2	4	no	—
450	C₂×D₄	—	2	2	no	—
451	C₂×D₄	—	2	8	no	—
452	C₂×D₄	—	4	8	no	—
453	C₂×D₄	—	4	4	no	—
454	C₂×D₄	—	4	8	no	—
455	C₂×D₄	C₂⁴×C₅₉	1	4	yes	—
456	C₂×D₄	C₂⁴×C₅₉	1	2	yes	—
457	C₂²×C₄	—	2	4	no	—
458	C₂²×C₄	—	2	4	no	—
459	C₂²×C₄	—	2	2	no	—
460	C₂²×C₄	—	2	2	no	—
461	C₂²×C₄	—	2	4	no	—
462	C₂²×C₄	—	4	4	no	—
463	C₂²×C₄	—	4	4	no	—
464	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
465	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
466	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
467	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
468	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
469	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
470	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
471	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
472	C₂²×C₄	C₂⁴×C₅₉	1	2	yes	—
473	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
474	Q₁₆	—	4	8	no	—
475	SD₁₆	—	4	8	no	—
476	SD₁₆	—	4	8	no	—
477	D₈	—	4	8	no	—
478	M₁₆	—	2	8	no	—
479	C₂×C₈	C₂⁴×C₅₉	1	8	yes	—
480	C₄⋊C₄	—	2	2	no	—
481	C₄⋊C₄	—	2	2	no	—
482	C₄⋊C₄	—	2	4	no	—
483	C₄⋊C₄	—	2	4	no	—
484	C₄⋊C₄	—	2	4	no	—
485	C₄⋊C₄	C₂⁴×C₅₉	1	4	yes	—
486	C₂²⋊C₄	—	2	4	no	—
487	C₂²⋊C₄	—	2	2	no	—
488	C₂²⋊C₄	—	2	8	no	—
489	C₂²⋊C₄	—	2	8	no	—
490	C₂²⋊C₄	—	2	8	no	—
491	C₂²⋊C₄	—	2	8	no	—
492	C₂²⋊C₄	—	2	4	no	—
493	C₂²⋊C₄	—	2	4	no	—
494	C₂²⋊C₄	—	2	4	no	—
495	C₄²	—	2	4	no	—
496	C₄²	—	2	2	no	—
497	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
498	C₄²	C₂×D₄×C₅₉	1	2	yes	—
499	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
500	C₄²	C₂×D₄×C₅₉	1	2	yes	—
501	C₄²	C₂×D₄×C₅₉	1	2	yes	—
502	C₄²	C₂×D₄×C₅₉	1	2	yes	—
503	C₄²	C₂×D₄×C₅₉	1	2	yes	—
504	C₄²	C₂⁴×C₅₉	1	1	yes	characteristic
505	C₄²	C₂⁴×C₅₉	1	1	yes	characteristic
506	C₄²	C₂⁴×C₅₉	1	2	yes	—
507	C₂³	—	2	2	no	—
508	C₂³	—	2	4	no	—
509	C₂³	—	2	4	no	—
510	C₂³	—	2	2	no	—
511	C₂³	—	4	8	no	—
512	C₂³	—	4	8	no	—
513	C₂³	—	4	8	no	—
514	C₂³	—	4	4	no	—
515	C₂³	—	4	8	no	—
516	C₂³	—	8	16	no	—
517	C₂³	C₂²×D₄×C₅₉	1	1	yes	characteristic
518	C₂³	C₂²×D₄×C₅₉	1	2	yes	—
519	Q₈	—	2	4	no	—
520	Q₈	—	2	8	no	—
521	Q₈	—	4	8	no	—
522	Q₈	—	4	4	no	—
523	Q₈	—	4	4	no	—
524	D₄	—	2	4	no	—
525	D₄	—	2	8	no	—
526	D₄	—	4	8	no	—
527	D₄	—	4	4	no	—
528	D₄	—	4	4	no	—
529	D₄	—	4	8	no	—
530	D₄	—	4	4	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	4	no	—
533	C₂×C₄	—	2	4	no	—
534	C₂×C₄	—	2	4	no	—
535	C₂×C₄	—	2	4	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	4	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	4	2	no	—
540	C₂×C₄	—	4	2	no	—
541	C₂×C₄	—	4	2	no	—
542	C₂×C₄	—	4	2	no	—
543	C₂×C₄	—	4	8	no	—
544	C₂×C₄	—	4	8	no	—
545	C₂×C₄	—	4	4	no	—
546	C₂×C₄	—	4	4	no	—
547	C₂×C₄	C₂²×D₄×C₅₉	1	2	yes	—
548	C₂×C₄	C₂²×D₄×C₅₉	1	2	yes	—
549	C₂×C₄	C₂²×D₄×C₅₉	1	2	yes	—
550	C₂×C₄	C₂²×D₄×C₅₉	1	2	yes	—
551	C₂×C₄	C₂²×D₄×C₅₉	1	2	yes	—
552	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
553	C₂×C₄	H(5,𝔽₂)×C₅₉	1	4	yes	—
554	C₂×C₄	C₂⁵×C₅₉	1	1	yes	characteristic
555	C₈	—	2	8	no	—
556	C₂²	—	2	8	no	—
557	C₂²	—	2	2	no	—
558	C₂²	—	2	2	no	—
559	C₂²	—	2	2	no	—
560	C₂²	—	4	4	no	—
561	C₂²	—	4	8	no	—
562	C₂²	—	4	2	no	—
563	C₂²	—	4	4	no	—
564	C₂²	—	8	8	no	—
565	C₂²	—	8	16	no	—
566	C₂²	C₅₉×#535	1	2	yes	—
567	C₂²	C₅₉×#575	1	4	yes	—
568	C₂²	C₂³×D₄×C₅₉	1	1	yes	characteristic
569	C₄	—	2	2	no	—
570	C₄	—	2	2	no	—
571	C₄	—	2	2	no	—
572	C₄	—	2	2	no	—
573	C₄	—	2	2	no	—
574	C₄	—	4	4	no	—
575	C₄	—	4	4	no	—
576	C₄	—	4	2	no	—
577	C₄	C₅₉×#535	1	2	yes	—
578	C₄	C₂×H(5,𝔽₂)×C₅₉	1	2	yes	—
579	C₂	—	2	2	no	—
580	C₂	—	2	4	no	—
581	C₂	—	8	4	no	—
582	C₂	—	8	2	no	—
583	C₂	C₂×C₅₉×#535	1	1	yes	characteristic
584	C₂	C₂×C₅₉×#575	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₉	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ C₂²×C₄²×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₉×#3444	C₂	semidirect
2. C₂²×#124×C₅₉	C₂	semidirect
3. C₅₉×#3215	C₂	semidirect
4. C₅₉×#3213	C₂	semidirect
5. C₅₉×#3211	C₂	semidirect
6. C₅₉×#3153	C₂	semidirect
7. C₅₉×#3140	C₂	semidirect
8. C₅₉×#3040	C₂	semidirect
9. C₅₉×#3004	C₂	semidirect
10. C₅₉×#3003	C₂	semidirect
11. C₅₉×#2920	C₂	semidirect
14. C₂×#137×C₅₉	C₂²	semidirect
16. C₅₉×#533	C₂²	semidirect
17. C₅₉×#533	C₂²	semidirect
18. C₅₉×#532	C₂²	semidirect
19. C₅₉×#532	C₂²	semidirect
22. C₅₉×#529	C₂²	semidirect
33. C₄×#41×C₅₉	C₂²	semidirect
36. C₅₉×#497	C₂²	semidirect
37. C₅₉×#496	C₂²	semidirect
38. C₅₉×#495	C₂²	semidirect
39. C₅₉×#489	C₂²	semidirect
40. C₅₉×#488	C₂²	semidirect
41. C₅₉×#464	C₂²	semidirect
42. C₅₉×#463	C₂²	semidirect
43. C₅₉×#451	C₂²	semidirect
45. C₅₉×#448	C₂²	semidirect
47. C₅₉×#432	C₂²	semidirect
48. C₅₉×#431	C₂²	semidirect
49. C₅₉×#423	C₂²	semidirect
50. C₅₉×#419	C₂²	semidirect
53. C₄×M₁₆×C₅₉	C₂²	semidirect
72. C₂×Q₁₆×C₅₉	C₂³	semidirect
74. C₂×SD₁₆×C₅₉	C₂³	semidirect
77. C₄⋊Q₈×C₅₉	C₂³	semidirect
78. C₄⋊Q₈×C₅₉	C₂³	semidirect
79. #127×C₅₉	C₂³	semidirect
82. #125×C₅₉	C₂³	semidirect
94. #124×C₅₉	C₂³	semidirect
106. C₄×Q₈×C₅₉	C₂³	semidirect
112. C₄×D₄×C₅₉	C₂³	semidirect
131. C₄⋊C₈×C₅₉	C₂³	semidirect
132. #103×C₅₉	C₂³	semidirect
135. #97×C₅₉	C₂³	semidirect
136. C₄×C₈×C₅₉	C₂³	semidirect
263. #60799	C₅₉	direct (N × H)
287. #3444	C₂×C₅₉	semidirect
288. C₂²×#124	C₂×C₅₉	semidirect
289. #3215	C₂×C₅₉	semidirect
290. #3213	C₂×C₅₉	semidirect
291. #3211	C₂×C₅₉	semidirect
292. #3153	C₂×C₅₉	semidirect
293. #3140	C₂×C₅₉	semidirect
294. #3040	C₂×C₅₉	semidirect
295. #3004	C₂×C₅₉	semidirect
296. #3003	C₂×C₅₉	semidirect
297. #2920	C₂×C₅₉	semidirect
306. C₂×#137	C₂²×C₅₉	semidirect
308. #533	C₂²×C₅₉	semidirect
309. #533	C₂²×C₅₉	semidirect
310. #532	C₂²×C₅₉	semidirect
311. #532	C₂²×C₅₉	semidirect
314. #529	C₂²×C₅₉	semidirect
325. C₄×#41	C₂²×C₅₉	semidirect
328. #497	C₂²×C₅₉	semidirect
329. #496	C₂²×C₅₉	semidirect
330. #495	C₂²×C₅₉	semidirect
331. #489	C₂²×C₅₉	semidirect
332. #488	C₂²×C₅₉	semidirect
333. #464	C₂²×C₅₉	semidirect
334. #463	C₂²×C₅₉	semidirect
335. #451	C₂²×C₅₉	semidirect
337. #448	C₂²×C₅₉	semidirect
339. #432	C₂²×C₅₉	semidirect
340. #431	C₂²×C₅₉	semidirect
341. #423	C₂²×C₅₉	semidirect
342. #419	C₂²×C₅₉	semidirect
345. C₄×M₁₆	C₂²×C₅₉	semidirect
346. C₅₉	#60799	direct (N × H)
365. C₂×Q₁₆	C₂³×C₅₉	semidirect
367. C₂×SD₁₆	C₂³×C₅₉	semidirect
370. C₄⋊Q₈	C₂³×C₅₉	semidirect
371. C₄⋊Q₈	C₂³×C₅₉	semidirect
372. #127	C₂³×C₅₉	semidirect
375. #125	C₂³×C₅₉	semidirect
387. #124	C₂³×C₅₉	semidirect
399. C₄×Q₈	C₂³×C₅₉	semidirect
405. C₄×D₄	C₂³×C₅₉	semidirect
424. C₄⋊C₈	C₂³×C₅₉	semidirect
425. #103	C₂³×C₅₉	semidirect
428. #97	C₂³×C₅₉	semidirect
429. C₄×C₈	C₂³×C₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₅₉×#3444	2
C₂²×#124×C₅₉	2
C₅₉×#3215	2
C₅₉×#3213	2
C₅₉×#3211	2
C₅₉×#3153	2
C₅₉×#3140	2
C₅₉×#3040	2
C₅₉×#3004	2
C₅₉×#3003	2
C₅₉×#2920	2
#60799	59

Subgroups — C59×#60799 #28472908

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#60799 #28472908