Subgroups — C₅₉×#24224 #28436333

Order 15104.

1450 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#3496	C₂	1	1	yes	maximal, characteristic
2	C₅₉×#3037	C₂	1	2	yes	maximal, complemented
3	C₂×C₅₉×#455	C₂	1	2	yes	maximal, complemented
4	C₄×#124×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#2040	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×#1800	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₉×#1792	C₂	1	2	yes	maximal, complemented
10	C₂×H(5,𝔽₂)×C₅₉	C₂²	1	2	yes	—
11	C₂×#137×C₅₉	C₂²	1	4	yes	complemented
12	C₅₉×#547	—	2	2	no	—
13	C₅₉×#546	—	2	4	no	—
14	D₄²×C₅₉	—	2	2	no	—
15	C₅₉×#537	C₂²	1	2	yes	—
16	C₅₉×#534	—	2	2	no	—
17	C₅₉×#532	C₂²	1	2	yes	—
18	C₂×#124×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×#121×C₅₉	—	2	2	no	—
20	C₂×C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic, complemented
21	C₂×C₄×D₄×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×#455	—	2	8	no	—
26	C₅₉×#423	C₂²	1	2	yes	complemented
27	C₂×#104×C₅₉	C₂²	1	4	yes	complemented
28	C₂×#104×C₅₉	C₂²	1	4	yes	complemented
29	C₅₉×#417	C₂²	1	4	yes	complemented
30	C₂×#100×C₅₉	C₂²	1	2	yes	—
31	C₅₉×#391	C₂²	1	1	yes	characteristic, complemented
32	C₅₉×#390	C₂²	1	1	yes	characteristic
33	C₅₉×#388	—	2	2	no	—
34	C₅₉×#382	C₂²	1	1	yes	characteristic, complemented
35	C₄×C₂²⋊C₄×C₅₉	—	2	2	no	—
36	C₄³×C₅₉	C₂²	1	1	yes	characteristic, complemented
37	C₅₉×#343	C₂²	1	2	yes	complemented
38	H(5,𝔽₂)×C₅₉	—	2	4	no	—
39	H(5,𝔽₂)×C₅₉	—	2	4	no	—
40	H(5,𝔽₂)×C₅₉	—	4	4	no	—
41	C₂×#41×C₅₉	—	2	4	no	—
42	C₂×#41×C₅₉	C₂³	1	4	yes	—
43	C₂²×Q₈×C₅₉	C₂³	1	1	yes	characteristic
44	C₂²×D₄×C₅₉	—	2	4	no	—
45	C₂²×D₄×C₅₉	—	2	2	no	—
46	C₂²×D₄×C₅₉	—	2	4	no	—
47	C₂²×D₄×C₅₉	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₅₉	D₄	1	2	yes	—
49	#137×C₅₉	—	2	8	no	—
50	#137×C₅₉	—	4	4	no	—
51	C₂×Q₁₆×C₅₉	—	2	4	no	—
52	C₂×SD₁₆×C₅₉	—	2	4	no	—
53	C₂×M₁₆×C₅₉	C₂³	1	4	yes	—
54	C₄⋊Q₈×C₅₉	D₄	1	2	yes	—
55	#127×C₅₉	D₄	1	2	yes	—
56	#124×C₅₉	—	2	2	no	—
57	#124×C₅₉	D₄	1	4	yes	—
58	#124×C₅₉	D₄	1	4	yes	complemented
59	#123×C₅₉	—	2	8	no	—
60	C₂²⋊Q₈×C₅₉	—	2	4	no	—
61	#121×C₅₉	—	2	4	no	—
62	#121×C₅₉	—	2	4	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	#120×C₅₉	—	2	8	no	—
68	C₄×Q₈×C₅₉	—	2	2	no	—
69	C₄×Q₈×C₅₉	—	2	2	no	—
70	C₄×D₄×C₅₉	—	2	2	no	—
71	C₄×D₄×C₅₉	—	2	2	no	—
72	C₄×D₄×C₅₉	—	2	2	no	—
73	C₄×D₄×C₅₉	—	2	2	no	—
74	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
75	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
76	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
78	C₂×C₂²⋊C₄×C₅₉	—	2	4	no	—
79	C₂×C₂²⋊C₄×C₅₉	—	2	2	no	—
80	C₂×C₂²⋊C₄×C₅₉	—	2	2	no	—
81	C₂×C₂²⋊C₄×C₅₉	D₄	1	2	yes	—
82	C₂×C₄²×C₅₉	—	2	1	no	—
83	C₂×C₄²×C₅₉	—	2	1	no	—
84	C₂×C₄²×C₅₉	—	2	1	no	—
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	#100×C₅₉	—	2	4	no	—
95	#95×C₅₉	—	2	2	no	—
96	#95×C₅₉	—	2	2	no	—
97	C₂⁴×C₅₉	—	2	4	no	—
98	C₂⁴×C₅₉	C₂×D₄	1	2	yes	—
99	#41×C₅₉	—	2	4	no	—
100	#41×C₅₉	—	2	4	no	—
101	#41×C₅₉	—	2	4	no	—
102	#41×C₅₉	—	2	4	no	—
103	#41×C₅₉	—	4	8	no	—
104	#41×C₅₉	—	4	8	no	—
105	C₂×Q₈×C₅₉	—	2	1	no	—
106	C₂×Q₈×C₅₉	—	2	2	no	—
107	C₂×Q₈×C₅₉	—	2	2	no	—
108	C₂×Q₈×C₅₉	C₂×D₄	1	4	yes	—
109	C₂×Q₈×C₅₉	C₂×D₄	1	2	yes	complemented
110	C₂×D₄×C₅₉	—	2	8	no	—
111	C₂×D₄×C₅₉	—	2	4	no	—
112	C₂×D₄×C₅₉	—	2	4	no	—
113	C₂×D₄×C₅₉	—	2	2	no	—
114	C₂×D₄×C₅₉	—	2	2	no	—
115	C₂×D₄×C₅₉	—	2	2	no	—
116	C₂×D₄×C₅₉	—	2	2	no	—
117	C₂×D₄×C₅₉	—	4	8	no	—
118	C₂×D₄×C₅₉	—	4	8	no	—
119	C₂×D₄×C₅₉	—	4	8	no	—
120	C₂×D₄×C₅₉	—	4	8	no	—
121	C₂×D₄×C₅₉	—	8	4	no	—
122	C₂×D₄×C₅₉	C₂×D₄	1	4	yes	—
123	C₂×D₄×C₅₉	C₂×D₄	1	2	yes	—
124	C₂×D₄×C₅₉	#41	1	2	yes	—
125	C₂²×C₄×C₅₉	—	2	4	no	—
126	C₂²×C₄×C₅₉	—	2	1	no	—
127	C₂²×C₄×C₅₉	—	2	1	no	—
128	C₂²×C₄×C₅₉	—	2	1	no	—
129	C₂²×C₄×C₅₉	—	2	2	no	—
130	C₂²×C₄×C₅₉	—	2	4	no	—
131	C₂²×C₄×C₅₉	—	2	2	no	—
132	C₂²×C₄×C₅₉	—	2	1	no	—
133	C₂²×C₄×C₅₉	—	4	4	no	—
134	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
136	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
138	Q₁₆×C₅₉	—	4	8	no	—
139	SD₁₆×C₅₉	—	4	8	no	—
140	M₁₆×C₅₉	—	2	4	no	—
141	M₁₆×C₅₉	—	2	4	no	—
142	C₂×C₈×C₅₉	—	2	4	no	—
143	C₄⋊C₄×C₅₉	—	2	4	no	—
144	C₄⋊C₄×C₅₉	—	2	2	no	—
145	C₄⋊C₄×C₅₉	—	2	2	no	—
146	C₄⋊C₄×C₅₉	—	2	2	no	—
147	C₄⋊C₄×C₅₉	—	2	2	no	—
148	C₄⋊C₄×C₅₉	—	4	2	no	—
149	C₂²⋊C₄×C₅₉	—	2	8	no	—
150	C₂²⋊C₄×C₅₉	—	2	8	no	—
151	C₂²⋊C₄×C₅₉	—	2	4	no	—
152	C₂²⋊C₄×C₅₉	—	2	4	no	—
153	C₂²⋊C₄×C₅₉	—	2	4	no	—
154	C₂²⋊C₄×C₅₉	—	4	4	no	—
155	C₄²×C₅₉	—	2	1	no	—
156	C₄²×C₅₉	—	2	2	no	—
157	C₄²×C₅₉	—	4	2	no	—
158	C₄²×C₅₉	—	4	2	no	—
159	C₄²×C₅₉	—	4	1	no	—
160	C₄²×C₅₉	—	4	1	no	—
161	C₄²×C₅₉	C₂×D₄	1	2	yes	—
162	C₄²×C₅₉	C₂×D₄	1	2	yes	—
163	C₄²×C₅₉	C₂×D₄	1	2	yes	complemented
164	C₂³×C₅₉	—	2	2	no	—
165	C₂³×C₅₉	—	2	2	no	—
166	C₂³×C₅₉	—	2	4	no	—
167	C₂³×C₅₉	—	4	2	no	—
168	C₂³×C₅₉	—	4	2	no	—
169	C₂³×C₅₉	—	4	8	no	—
170	C₂³×C₅₉	—	8	4	no	—
171	C₂³×C₅₉	—	8	4	no	—
172	C₂³×C₅₉	#121	1	4	yes	—
173	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
174	Q₈×C₅₉	—	2	4	no	—
175	Q₈×C₅₉	—	2	4	no	—
176	Q₈×C₅₉	—	2	4	no	—
177	Q₈×C₅₉	—	4	2	no	—
178	D₄×C₅₉	—	2	4	no	—
179	D₄×C₅₉	—	2	4	no	—
180	D₄×C₅₉	—	2	4	no	—
181	D₄×C₅₉	—	2	4	no	—
182	D₄×C₅₉	—	4	16	no	—
183	D₄×C₅₉	—	4	4	no	—
184	D₄×C₅₉	—	4	4	no	—
185	D₄×C₅₉	—	4	4	no	—
186	D₄×C₅₉	—	8	2	no	—
187	D₄×C₅₉	—	8	4	no	—
188	C₂×C₄×C₅₉	—	2	2	no	—
189	C₂×C₄×C₅₉	—	2	2	no	—
190	C₂×C₄×C₅₉	—	2	2	no	—
191	C₂×C₄×C₅₉	—	2	4	no	—
192	C₂×C₄×C₅₉	—	2	2	no	—
193	C₂×C₄×C₅₉	—	2	2	no	—
194	C₂×C₄×C₅₉	—	4	1	no	—
195	C₂×C₄×C₅₉	—	4	1	no	—
196	C₂×C₄×C₅₉	—	4	1	no	—
197	C₂×C₄×C₅₉	—	4	1	no	—
198	C₂×C₄×C₅₉	—	4	8	no	—
199	C₂×C₄×C₅₉	—	4	1	no	—
200	C₂×C₄×C₅₉	—	4	1	no	—
201	C₂×C₄×C₅₉	—	4	1	no	—
202	C₂×C₄×C₅₉	—	4	4	no	—
203	C₂×C₄×C₅₉	—	4	4	no	—
204	C₂×C₄×C₅₉	—	4	4	no	—
205	C₂×C₄×C₅₉	—	4	2	no	—
206	C₂×C₄×C₅₉	—	4	2	no	—
207	C₂×C₄×C₅₉	—	4	1	no	—
208	C₂×C₄×C₅₉	—	8	2	no	—
209	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
210	C₂×C₄×C₅₉	#120	1	2	yes	—
211	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
212	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₄×C₅₉	C₂²×D₄	1	2	yes	—
214	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
215	C₂×C₄×C₅₉	H(5,𝔽₂)	1	2	yes	—
216	C₈×C₅₉	—	4	4	no	—
217	#24224	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
218	C₂²×C₅₉	—	2	1	no	—
219	C₂²×C₅₉	—	2	1	no	—
220	C₂²×C₅₉	—	2	4	no	—
221	C₂²×C₅₉	—	4	4	no	—
222	C₂²×C₅₉	—	4	2	no	—
223	C₂²×C₅₉	—	4	2	no	—
224	C₂²×C₅₉	—	4	4	no	—
225	C₂²×C₅₉	—	4	4	no	—
226	C₂²×C₅₉	—	4	4	no	—
227	C₂²×C₅₉	—	8	2	no	—
228	C₂²×C₅₉	—	16	8	no	—
229	C₂²×C₅₉	C₂×#120	1	1	yes	characteristic
230	C₂²×C₅₉	C₂×#121	1	1	yes	characteristic
231	C₂²×C₅₉	#535	1	1	yes	characteristic
232	C₄×C₅₉	—	2	1	no	—
233	C₄×C₅₉	—	2	2	no	—
234	C₄×C₅₉	—	4	2	no	—
235	C₄×C₅₉	—	4	2	no	—
236	C₄×C₅₉	—	4	4	no	—
237	C₄×C₅₉	—	4	2	no	—
238	C₄×C₅₉	—	4	2	no	—
239	C₄×C₅₉	—	8	1	no	—
240	C₄×C₅₉	#455	1	2	yes	complemented
241	C₄×C₅₉	D₄²	1	2	yes	—
242	C₄×C₅₉	#546	1	2	yes	—
243	#3496	C₂×C₅₉	1	1	yes	characteristic
244	#3037	C₂×C₅₉	1	2	yes	complemented
245	C₂×#455	C₂×C₅₉	1	2	yes	complemented
246	C₄×#124	C₂×C₅₉	1	1	yes	characteristic, complemented
247	#2040	C₂×C₅₉	1	2	yes	complemented
248	#2007	C₂×C₅₉	1	2	yes	complemented
249	#1962	C₂×C₅₉	1	2	yes	complemented
250	#1800	C₂×C₅₉	1	1	yes	characteristic, complemented
251	#1792	C₂×C₅₉	1	2	yes	complemented
252	C₂×C₅₉	—	2	1	no	—
253	C₂×C₅₉	—	2	1	no	—
254	C₂×C₅₉	—	4	4	no	—
255	C₂×C₅₉	—	8	4	no	—
256	C₂×C₅₉	#2408	1	1	yes	characteristic
257	C₂×C₅₉	C₂×#455	1	1	yes	characteristic
258	C₂×C₅₉	#3018	1	1	yes	characteristic
259	C₂×H(5,𝔽₂)	C₂²×C₅₉	1	2	yes	—
260	C₂×#137	C₂²×C₅₉	1	4	yes	complemented
261	#547	—	2	2	no	—
262	#546	—	2	4	no	—
263	D₄²	—	2	2	no	—
264	#537	C₂²×C₅₉	1	2	yes	—
265	#534	—	2	2	no	—
266	#532	C₂²×C₅₉	1	2	yes	—
267	C₂×#124	C₂²×C₅₉	1	1	yes	characteristic
268	C₂×#121	—	2	2	no	—
269	C₂×C₄×Q₈	C₂²×C₅₉	1	1	yes	characteristic, complemented
270	C₂×C₄×D₄	C₂²×C₅₉	1	1	yes	characteristic
271	#464	—	2	4	no	—
272	#462	—	2	2	no	—
273	#460	—	2	2	no	—
274	#455	—	2	8	no	—
275	#423	C₂²×C₅₉	1	2	yes	complemented
276	C₂×#104	C₂²×C₅₉	1	4	yes	complemented
277	C₂×#104	C₂²×C₅₉	1	4	yes	complemented
278	#417	C₂²×C₅₉	1	4	yes	complemented
279	C₂×#100	C₂²×C₅₉	1	2	yes	—
280	#391	C₂²×C₅₉	1	1	yes	characteristic, complemented
281	#390	C₂²×C₅₉	1	1	yes	characteristic
282	#388	—	2	2	no	—
283	#382	C₂²×C₅₉	1	1	yes	characteristic, complemented
284	C₄×C₂²⋊C₄	—	2	2	no	—
285	C₄³	C₂²×C₅₉	1	1	yes	characteristic, complemented
286	#343	C₂²×C₅₉	1	2	yes	complemented
287	C₅₉	#24224	1	1	yes	characteristic, direct product factor, Sylow
288	H(5,𝔽₂)	—	2	4	no	—
289	H(5,𝔽₂)	—	2	4	no	—
290	H(5,𝔽₂)	—	4	4	no	—
291	C₂×#41	—	2	4	no	—
292	C₂×#41	C₂³×C₅₉	1	4	yes	—
293	C₂²×Q₈	C₂³×C₅₉	1	1	yes	characteristic
294	C₂²×D₄	—	2	4	no	—
295	C₂²×D₄	—	2	2	no	—
296	C₂²×D₄	—	2	4	no	—
297	C₂²×D₄	C₂³×C₅₉	1	1	yes	characteristic
298	C₂³×C₄	D₄×C₅₉	1	2	yes	—
299	#137	—	2	8	no	—
300	#137	—	4	4	no	—
301	C₂×Q₁₆	—	2	4	no	—
302	C₂×SD₁₆	—	2	4	no	—
303	C₂×M₁₆	C₂³×C₅₉	1	4	yes	—
304	C₄⋊Q₈	D₄×C₅₉	1	2	yes	—
305	#127	D₄×C₅₉	1	2	yes	—
306	#124	—	2	2	no	—
307	#124	D₄×C₅₉	1	4	yes	—
308	#124	D₄×C₅₉	1	4	yes	complemented
309	#123	—	2	8	no	—
310	C₂²⋊Q₈	—	2	4	no	—
311	#121	—	2	4	no	—
312	#121	—	2	4	no	—
313	#121	—	2	4	no	—
314	#121	—	2	2	no	—
315	#121	—	2	4	no	—
316	#121	—	2	2	no	—
317	#120	—	2	8	no	—
318	C₄×Q₈	—	2	2	no	—
319	C₄×Q₈	—	2	2	no	—
320	C₄×D₄	—	2	2	no	—
321	C₄×D₄	—	2	2	no	—
322	C₄×D₄	—	2	2	no	—
323	C₄×D₄	—	2	2	no	—
324	C₂×C₄⋊C₄	—	2	2	no	—
325	C₂×C₄⋊C₄	—	2	1	no	—
326	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
327	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
328	C₂×C₂²⋊C₄	—	2	4	no	—
329	C₂×C₂²⋊C₄	—	2	2	no	—
330	C₂×C₂²⋊C₄	—	2	2	no	—
331	C₂×C₂²⋊C₄	D₄×C₅₉	1	2	yes	—
332	C₂×C₄²	—	2	1	no	—
333	C₂×C₄²	—	2	1	no	—
334	C₂×C₄²	—	2	1	no	—
335	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
336	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
337	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
338	C₄⋊C₈	—	2	2	no	—
339	C₄⋊C₈	—	2	2	no	—
340	#104	—	2	8	no	—
341	#104	—	4	4	no	—
342	#103	—	2	4	no	—
343	#102	—	2	4	no	—
344	#100	—	2	4	no	—
345	#95	—	2	2	no	—
346	#95	—	2	2	no	—
347	C₂⁴	—	2	4	no	—
348	C₂⁴	C₂×D₄×C₅₉	1	2	yes	—
349	#41	—	2	4	no	—
350	#41	—	2	4	no	—
351	#41	—	2	4	no	—
352	#41	—	2	4	no	—
353	#41	—	4	8	no	—
354	#41	—	4	8	no	—
355	C₂×Q₈	—	2	1	no	—
356	C₂×Q₈	—	2	2	no	—
357	C₂×Q₈	—	2	2	no	—
358	C₂×Q₈	C₂×D₄×C₅₉	1	4	yes	—
359	C₂×Q₈	C₂×D₄×C₅₉	1	2	yes	complemented
360	C₂×D₄	—	2	8	no	—
361	C₂×D₄	—	2	4	no	—
362	C₂×D₄	—	2	4	no	—
363	C₂×D₄	—	2	2	no	—
364	C₂×D₄	—	2	2	no	—
365	C₂×D₄	—	2	2	no	—
366	C₂×D₄	—	2	2	no	—
367	C₂×D₄	—	4	8	no	—
368	C₂×D₄	—	4	8	no	—
369	C₂×D₄	—	4	8	no	—
370	C₂×D₄	—	4	8	no	—
371	C₂×D₄	—	8	4	no	—
372	C₂×D₄	C₂×D₄×C₅₉	1	4	yes	—
373	C₂×D₄	C₂×D₄×C₅₉	1	2	yes	—
374	C₂×D₄	#41×C₅₉	1	2	yes	—
375	C₂²×C₄	—	2	4	no	—
376	C₂²×C₄	—	2	1	no	—
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	4	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	1	no	—
383	C₂²×C₄	—	4	4	no	—
384	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
385	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
386	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
387	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
388	Q₁₆	—	4	8	no	—
389	SD₁₆	—	4	8	no	—
390	M₁₆	—	2	4	no	—
391	M₁₆	—	2	4	no	—
392	C₂×C₈	—	2	4	no	—
393	C₄⋊C₄	—	2	4	no	—
394	C₄⋊C₄	—	2	2	no	—
395	C₄⋊C₄	—	2	2	no	—
396	C₄⋊C₄	—	2	2	no	—
397	C₄⋊C₄	—	2	2	no	—
398	C₄⋊C₄	—	4	2	no	—
399	C₂²⋊C₄	—	2	8	no	—
400	C₂²⋊C₄	—	2	8	no	—
401	C₂²⋊C₄	—	2	4	no	—
402	C₂²⋊C₄	—	2	4	no	—
403	C₂²⋊C₄	—	2	4	no	—
404	C₂²⋊C₄	—	4	4	no	—
405	C₄²	—	2	1	no	—
406	C₄²	—	2	2	no	—
407	C₄²	—	4	2	no	—
408	C₄²	—	4	2	no	—
409	C₄²	—	4	1	no	—
410	C₄²	—	4	1	no	—
411	C₄²	C₂×D₄×C₅₉	1	2	yes	—
412	C₄²	C₂×D₄×C₅₉	1	2	yes	—
413	C₄²	C₂×D₄×C₅₉	1	2	yes	complemented
414	C₂³	—	2	2	no	—
415	C₂³	—	2	2	no	—
416	C₂³	—	2	4	no	—
417	C₂³	—	4	2	no	—
418	C₂³	—	4	2	no	—
419	C₂³	—	4	8	no	—
420	C₂³	—	8	4	no	—
421	C₂³	—	8	4	no	—
422	C₂³	#121×C₅₉	1	4	yes	—
423	C₂³	C₂²×D₄×C₅₉	1	1	yes	characteristic
424	Q₈	—	2	4	no	—
425	Q₈	—	2	4	no	—
426	Q₈	—	2	4	no	—
427	Q₈	—	4	2	no	—
428	D₄	—	2	4	no	—
429	D₄	—	2	4	no	—
430	D₄	—	2	4	no	—
431	D₄	—	2	4	no	—
432	D₄	—	4	16	no	—
433	D₄	—	4	4	no	—
434	D₄	—	4	4	no	—
435	D₄	—	4	4	no	—
436	D₄	—	8	2	no	—
437	D₄	—	8	4	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	4	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	4	1	no	—
445	C₂×C₄	—	4	1	no	—
446	C₂×C₄	—	4	1	no	—
447	C₂×C₄	—	4	1	no	—
448	C₂×C₄	—	4	8	no	—
449	C₂×C₄	—	4	1	no	—
450	C₂×C₄	—	4	1	no	—
451	C₂×C₄	—	4	1	no	—
452	C₂×C₄	—	4	4	no	—
453	C₂×C₄	—	4	4	no	—
454	C₂×C₄	—	4	4	no	—
455	C₂×C₄	—	4	2	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	—	4	1	no	—
458	C₂×C₄	—	8	2	no	—
459	C₂×C₄	#120×C₅₉	1	1	yes	characteristic
460	C₂×C₄	#120×C₅₉	1	2	yes	—
461	C₂×C₄	#120×C₅₉	1	1	yes	characteristic
462	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
463	C₂×C₄	C₂²×D₄×C₅₉	1	2	yes	—
464	C₂×C₄	C₂×#41×C₅₉	1	1	yes	characteristic
465	C₂×C₄	H(5,𝔽₂)×C₅₉	1	2	yes	—
466	C₈	—	4	4	no	—
467	C₂²	—	2	1	no	—
468	C₂²	—	2	1	no	—
469	C₂²	—	2	4	no	—
470	C₂²	—	4	4	no	—
471	C₂²	—	4	2	no	—
472	C₂²	—	4	2	no	—
473	C₂²	—	4	4	no	—
474	C₂²	—	4	4	no	—
475	C₂²	—	4	4	no	—
476	C₂²	—	8	2	no	—
477	C₂²	—	16	8	no	—
478	C₂²	C₂×#120×C₅₉	1	1	yes	characteristic
479	C₂²	C₂×#121×C₅₉	1	1	yes	characteristic
480	C₂²	C₅₉×#535	1	1	yes	characteristic
481	C₄	—	2	1	no	—
482	C₄	—	2	2	no	—
483	C₄	—	4	2	no	—
484	C₄	—	4	2	no	—
485	C₄	—	4	2	no	—
486	C₄	—	4	4	no	—
487	C₄	—	4	2	no	—
488	C₄	—	8	1	no	—
489	C₄	C₅₉×#455	1	2	yes	complemented
490	C₄	D₄²×C₅₉	1	2	yes	—
491	C₄	C₅₉×#546	1	2	yes	—
492	C₂	—	2	1	no	—
493	C₂	—	2	1	no	—
494	C₂	—	4	4	no	—
495	C₂	—	8	4	no	—
496	C₂	C₅₉×#2408	1	1	yes	characteristic
497	C₂	C₂×C₅₉×#455	1	1	yes	characteristic
498	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×#3496	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×#455	C₂	semidirect
4. C₄×#124×C₅₉	C₂	semidirect
5. C₅₉×#2040	C₂	semidirect
6. C₅₉×#2007	C₂	semidirect
7. C₅₉×#1962	C₂	semidirect
8. C₅₉×#1800	C₂	semidirect
9. C₅₉×#1792	C₂	semidirect
11. C₂×#137×C₅₉	C₂²	semidirect
20. C₂×C₄×Q₈×C₅₉	C₂²	semidirect
26. C₅₉×#423	C₂²	semidirect
27. C₂×#104×C₅₉	C₂²	semidirect
28. C₂×#104×C₅₉	C₂²	semidirect
29. C₅₉×#417	C₂²	semidirect
31. C₅₉×#391	C₂²	semidirect
34. C₅₉×#382	C₂²	semidirect
36. C₄³×C₅₉	C₂²	semidirect
37. C₅₉×#343	C₂²	semidirect
58. #124×C₅₉	D₄	semidirect
109. C₂×Q₈×C₅₉	C₂×D₄	semidirect
163. C₄²×C₅₉	C₂×D₄	semidirect
217. #24224	C₅₉	direct (N × H)
240. C₄×C₅₉	#455	semidirect
244. #3037	C₂×C₅₉	semidirect
245. C₂×#455	C₂×C₅₉	semidirect
246. C₄×#124	C₂×C₅₉	semidirect
247. #2040	C₂×C₅₉	semidirect
248. #2007	C₂×C₅₉	semidirect
249. #1962	C₂×C₅₉	semidirect
250. #1800	C₂×C₅₉	semidirect
251. #1792	C₂×C₅₉	semidirect
260. C₂×#137	C₂²×C₅₉	semidirect
269. C₂×C₄×Q₈	C₂²×C₅₉	semidirect
275. #423	C₂²×C₅₉	semidirect
276. C₂×#104	C₂²×C₅₉	semidirect
277. C₂×#104	C₂²×C₅₉	semidirect
278. #417	C₂²×C₅₉	semidirect
280. #391	C₂²×C₅₉	semidirect
283. #382	C₂²×C₅₉	semidirect
285. C₄³	C₂²×C₅₉	semidirect
286. #343	C₂²×C₅₉	semidirect
287. C₅₉	#24224	direct (N × H)
308. #124	D₄×C₅₉	semidirect
359. C₂×Q₈	C₂×D₄×C₅₉	semidirect
413. C₄²	C₂×D₄×C₅₉	semidirect
489. C₄	C₅₉×#455	semidirect

Maximal subgroups

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

subgroup	index
C₅₉×#3496	2
C₅₉×#3037	2
C₂×C₅₉×#455	2
C₄×#124×C₅₉	2
C₅₉×#2040	2
C₅₉×#2007	2
C₅₉×#1962	2
C₅₉×#1800	2
C₅₉×#1792	2
#24224	59

Subgroups — C59×#24224 #28436333

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#24224 #28436333