Subgroups — #28478244

Order 15104.

798 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#21426503	C₂	1	1	yes	maximal, characteristic
3	#21421935	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₉×#423	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₅₉⋊C₈	—	2	1	no	—
6	C₂×#14273686	—	2	1	no	—
7	C₂×C₄×C₅₉⋊C₈	—	2	1	no	—
8	C₂×C₄×C₅₉⋊C₈	—	2	2	no	—
9	C₂×C₄×C₅₉⋊C₈	C₄	1	4	yes	complemented
10	C₂×C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
11	#16837272	C₄	1	2	yes	—
12	#16837268	—	2	1	no	—
13	#16837262	—	2	1	no	—
14	#16837262	C₄	1	2	yes	—
15	C₄×C₄⋊Dic₅₉	—	2	1	no	—
16	C₄×C₄⋊Dic₅₉	C₂²	1	2	yes	—
17	C₂²×M₁₆×C₅₉	—	2	1	no	—
18	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
19	C₅₉×#423	—	2	2	no	—
20	C₅₉×#423	C₄	1	4	yes	complemented
21	C₂×C₄⋊C₈×C₅₉	—	2	1	no	—
22	C₂×C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
23	C₂²×C₅₉⋊C₈	—	2	8	no	—
24	C₂²×C₅₉⋊C₈	—	2	4	no	—
25	C₂²×C₅₉⋊C₈	—	2	2	no	—
26	#14273686	—	4	2	no	—
27	#14273686	—	4	2	no	—
28	C₂×C₄⋊Dic₅₉	—	2	2	no	—
29	C₂×C₄⋊Dic₅₉	—	2	2	no	—
30	C₂×C₄⋊Dic₅₉	C₂×C₄	1	4	yes	—
31	C₂×C₂²⋊Dic₅₉	—	2	2	no	—
32	C₂×C₄×Dic₅₉	—	2	2	no	—
33	C₄×C₅₉⋊C₈	—	2	4	no	—
34	C₄×C₅₉⋊C₈	—	2	2	no	—
35	C₄×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
36	C₅₉⋊#95	—	2	2	no	—
37	C₅₉⋊#95	—	2	2	no	—
38	C₂³×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
41	C₂×M₁₆×C₅₉	—	2	4	no	—
42	C₂×M₁₆×C₅₉	—	4	2	no	—
43	C₂×M₁₆×C₅₉	—	4	2	no	—
44	C₂²×C₈×C₅₉	—	2	2	no	—
45	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₉	D₄	1	2	yes	—
51	C₂×C₄²×C₅₉	Q₈	1	2	yes	—
52	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₅₉	—	2	4	no	—
54	C₄⋊C₈×C₅₉	—	2	2	no	—
55	C₄⋊C₈×C₅₉	C₂×C₄	1	4	yes	—
56	C₂⁴×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₉	—	2	4	no	—
58	C₂²×C₄×C₅₉	—	2	4	no	—
59	C₂²×C₄×C₅₉	C₄²	1	2	yes	—
60	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₅₉	D₈	1	2	yes	—
69	C₂²×C₄×C₅₉	SD₁₆	1	2	yes	—
70	C₂²×C₄×C₅₉	SD₁₆	1	2	yes	—
71	C₂²×C₄×C₅₉	Q₁₆	1	2	yes	—
72	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
75	C₂²×Dic₅₉	—	2	4	no	—
76	M₁₆×C₅₉	—	4	8	no	—
77	C₂×C₈×C₅₉	—	2	4	no	—
78	C₂×C₈×C₅₉	—	4	2	no	—
79	C₂×C₈×C₅₉	—	4	2	no	—
80	C₄²×C₅₉	C₄²	1	2	yes	—
81	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
84	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
85	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
86	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
90	C₂²⋊Dic₅₉	—	4	4	no	—
91	C₄⋊Dic₅₉	—	2	8	no	—
92	C₄⋊Dic₅₉	—	4	4	no	—
93	C₄⋊Dic₅₉	—	4	4	no	—
94	C₄×Dic₅₉	—	4	2	no	—
95	C₄×Dic₅₉	—	4	2	no	—
96	C₂×C₅₉⋊C₈	—	2	8	no	—
97	C₂×C₅₉⋊C₈	—	2	16	no	—
98	C₂×C₅₉⋊C₈	—	2	4	no	—
99	C₂³×C₅₉	—	2	2	no	—
100	C₂³×C₅₉	—	2	2	no	—
101	C₂³×C₅₉	#95	1	2	yes	—
102	C₂³×C₅₉	#102	1	2	yes	—
103	C₂³×C₅₉	#103	1	2	yes	—
104	C₂³×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅₉	—	2	2	no	—
106	C₂×C₄×C₅₉	—	2	2	no	—
107	C₂×C₄×C₅₉	—	2	4	no	—
108	C₂×C₄×C₅₉	—	2	4	no	—
109	C₂×C₄×C₅₉	—	2	4	no	—
110	C₂×C₄×C₅₉	—	2	4	no	—
111	C₂×C₄×C₅₉	#95	1	2	yes	—
112	C₂×C₄×C₅₉	#95	1	2	yes	—
113	C₂×C₄×C₅₉	#95	1	2	yes	—
114	C₂×C₄×C₅₉	#102	1	2	yes	—
115	C₂×C₄×C₅₉	#102	1	4	yes	—
116	C₂×C₄×C₅₉	#103	1	2	yes	—
117	C₂×C₄×C₅₉	#103	1	4	yes	—
118	C₂×C₄×C₅₉	#104	1	4	yes	—
119	C₂×C₄×C₅₉	#106	1	4	yes	—
120	C₂×C₄×C₅₉	#107	1	4	yes	—
121	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
122	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₅₉	#131	1	4	yes	—
128	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
131	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
132	C₂×Dic₅₉	—	4	4	no	—
133	C₂×Dic₅₉	—	4	4	no	—
134	C₂×Dic₅₉	—	4	4	no	—
135	C₈×C₅₉	—	4	4	no	—
136	C₅₉⋊C₈	—	2	8	no	—
137	#7897	—	59	1	no	maximal, Sylow
138	C₂²×C₅₉	—	2	2	no	—
139	C₂²×C₅₉	—	2	2	no	—
140	C₂²×C₅₉	—	2	4	no	—
141	C₂²×C₅₉	—	2	4	no	—
142	C₂²×C₅₉	#340	1	4	yes	—
143	C₂²×C₅₉	C₂×#95	1	1	yes	characteristic
144	C₂²×C₅₉	#408	1	2	yes	—
145	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
146	C₂²×C₅₉	C₂×#103	1	1	yes	characteristic
147	C₂²×C₅₉	C₂×#104	1	2	yes	—
148	C₄×C₅₉	—	2	2	no	—
149	C₄×C₅₉	—	2	1	no	—
150	C₄×C₅₉	—	2	1	no	—
151	C₄×C₅₉	#339	1	4	yes	—
152	C₄×C₅₉	#340	1	4	yes	—
153	C₄×C₅₉	#405	1	2	yes	—
154	C₄×C₅₉	C₂×#102	1	1	yes	characteristic
155	C₄×C₅₉	C₂×#103	1	1	yes	characteristic
156	C₄×C₅₉	#424	1	2	yes	—
157	C₄×C₅₉	C₂×#106	1	1	yes	characteristic
158	C₄×C₅₉	C₂×#107	1	1	yes	characteristic
159	Dic₅₉	—	8	4	no	—
160	C₂×#423	D₅₉	1	1	yes	characteristic
161	C₂²×C₄×C₈	—	59	1	no	—
162	#2285	—	59	1	no	—
163	#1279	—	59	4	no	—
164	C₂×C₅₉	—	2	4	no	—
165	C₂×C₅₉	#1492	1	2	yes	—
166	C₂×C₅₉	#1493	1	2	yes	—
167	C₂×C₅₉	#1729	1	1	yes	characteristic
168	C₂×C₅₉	C₂×#339	1	1	yes	characteristic
169	C₂×C₅₉	C₂×#340	1	1	yes	characteristic
170	C₂²×M₁₆	—	2	1	no	—
171	C₂³×C₈	—	118	1	no	—
172	C₂×#117	—	118	1	no	—
173	C₂²×C₄²	C₂×D₅₉	1	1	yes	characteristic
174	#423	—	2	2	no	—
175	#423	Dic₅₉	1	4	yes	complemented
176	C₂×C₄⋊C₈	—	2	1	no	—
177	C₂×C₄⋊C₈	C₂×D₅₉	1	2	yes	—
178	C₂×C₄×C₈	—	59	4	no	—
179	C₂×C₄×C₈	—	59	2	no	—
180	C₂×C₄×C₈	—	118	1	no	—
181	C₂×C₄×C₈	—	118	2	no	—
182	#384	—	59	2	no	—
183	#382	—	118	1	no	—
184	#380	—	59	2	no	—
185	#380	—	118	1	no	—
186	C₄×C₄⋊C₄	—	59	2	no	—
187	C₄×C₄⋊C₄	—	118	1	no	—
188	C₅₉	#7897	1	1	yes	characteristic, complemented, Sylow
189	C₂³×C₄	C₄×D₅₉	1	1	yes	characteristic
190	C₂³×C₄	D₂₃₆	1	1	yes	characteristic
191	C₂³×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
192	C₂×M₁₆	—	2	4	no	—
193	C₂×M₁₆	—	4	2	no	—
194	C₂×M₁₆	—	4	2	no	—
195	C₂²×C₈	—	2	2	no	—
196	C₂²×C₈	—	118	8	no	—
197	C₂²×C₈	—	118	4	no	—
198	C₂²×C₈	—	118	2	no	—
199	#117	—	236	2	no	—
200	#117	—	236	2	no	—
201	C₂×C₄⋊C₄	—	59	4	no	—
202	C₂×C₄⋊C₄	—	118	2	no	—
203	C₂×C₄⋊C₄	—	118	2	no	—
204	C₂×C₂²⋊C₄	—	118	2	no	—
205	C₂×C₄²	—	118	2	no	—
206	C₂×C₄²	Dic₁₁₈	1	2	yes	—
207	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
208	C₂×C₄²	C₄×D₅₉	1	2	yes	—
209	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
210	C₂×C₄²	C₂×Dic₅₉	1	2	yes	—
211	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
212	C₂×C₄²	C₂²⋊D₅₉	1	2	yes	—
213	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
214	C₄⋊C₈	—	2	4	no	—
215	C₄⋊C₈	—	2	2	no	—
216	C₄⋊C₈	C₂×Dic₅₉	1	4	yes	—
217	C₄×C₈	—	59	4	no	—
218	C₄×C₈	—	118	4	no	—
219	C₄×C₈	—	118	2	no	—
220	#95	—	118	2	no	—
221	#95	—	118	2	no	—
222	C₂⁴	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	118	4	no	—
226	C₂²×C₄	C₄×Dic₅₉	1	2	yes	—
227	C₂²×C₄	Dic₅₉⋊C₄	1	2	yes	—
228	C₂²×C₄	Dic₅₉⋊C₄	1	2	yes	—
229	C₂²×C₄	C₄⋊Dic₅₉	1	2	yes	—
230	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
231	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
232	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
233	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
234	C₂²×C₄	#11745669	1	2	yes	—
235	C₂²×C₄	#11745670	1	2	yes	—
236	C₂²×C₄	#11745671	1	2	yes	—
237	C₂²×C₄	#11745672	1	2	yes	—
238	C₂²×C₄	C₂²⋊Dic₅₉	1	2	yes	—
239	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
242	M₁₆	—	4	8	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	118	8	no	—
247	C₂×C₈	—	118	16	no	—
248	C₂×C₈	—	118	4	no	—
249	C₄⋊C₄	—	118	8	no	—
250	C₄⋊C₄	—	236	4	no	—
251	C₄⋊C₄	—	236	4	no	—
252	C₂²⋊C₄	—	236	4	no	—
253	C₄²	—	236	2	no	—
254	C₄²	—	236	2	no	—
255	C₄²	C₄×Dic₅₉	1	2	yes	—
256	C₄²	Dic₅₉⋊C₄	1	2	yes	—
257	C₄²	Dic₅₉⋊C₄	1	2	yes	—
258	C₄²	C₄⋊Dic₅₉	1	2	yes	—
259	C₄²	(C₂×D₅₉)⋊C₄	1	2	yes	—
260	C₄²	C₂²⋊Dic₅₉	1	2	yes	—
261	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
262	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
263	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
264	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
265	C₂³	—	2	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	C₅₉⋊#95	1	2	yes	—
268	C₂³	#14273647	1	2	yes	—
269	C₂³	#14273650	1	2	yes	—
270	C₂³	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	236	4	no	—
278	C₂×C₄	—	236	4	no	—
279	C₂×C₄	—	236	4	no	—
280	C₂×C₄	C₅₉⋊#95	1	2	yes	—
281	C₂×C₄	C₅₉⋊#95	1	2	yes	—
282	C₂×C₄	C₅₉⋊#95	1	2	yes	—
283	C₂×C₄	#14273647	1	2	yes	—
284	C₂×C₄	#14273648	1	4	yes	—
285	C₂×C₄	#14273650	1	2	yes	—
286	C₂×C₄	#14273651	1	4	yes	—
287	C₂×C₄	#14273653	1	4	yes	—
288	C₂×C₄	#14273658	1	4	yes	—
289	C₂×C₄	#14273660	1	4	yes	—
290	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
291	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
292	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
293	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
294	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
295	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
296	C₂×C₄	#14273735	1	4	yes	—
297	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
298	C₂×C₄	C₂×#11745671	1	1	yes	characteristic
299	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
300	C₂×C₄	C₂×#11745672	1	1	yes	characteristic
301	C₈	—	4	4	no	—
302	C₈	—	118	8	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	2	4	no	—
307	C₂²	#16837164	1	4	yes	—
308	C₂²	C₂×C₅₉⋊#95	1	1	yes	characteristic
309	C₂²	#16837370	1	2	yes	—
310	C₂²	C₂×#14273647	1	1	yes	characteristic
311	C₂²	C₂×#14273650	1	1	yes	characteristic
312	C₂²	C₂×#14273653	1	2	yes	—
313	C₄	—	2	2	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	472	4	no	—
317	C₄	#16837163	1	4	yes	—
318	C₄	#16837164	1	4	yes	—
319	C₄	#16837360	1	2	yes	—
320	C₄	C₂×#14273648	1	1	yes	characteristic
321	C₄	C₂×#14273651	1	1	yes	characteristic
322	C₄	#16837430	1	2	yes	—
323	C₄	C₂×#14273658	1	1	yes	characteristic
324	C₄	C₂×#14273660	1	1	yes	characteristic
325	C₂	—	2	4	no	—
326	C₂	#21422546	1	2	yes	—
327	C₂	#21422550	1	2	yes	—
328	C₂	#21423794	1	1	yes	characteristic
329	C₂	C₂×#16837163	1	1	yes	characteristic
330	C₂	C₂×#16837164	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₅₉×#423	C₂
socle	C₂³×C₅₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₅₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×#423	D₅₉
hypocenter	C₅₉	#7897
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
3. #21421935	C₂	semidirect
9. C₂×C₄×C₅₉⋊C₈	C₄	semidirect
20. C₅₉×#423	C₄	semidirect
175. #423	Dic₅₉	semidirect
188. C₅₉	#7897	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₅₉⋊C₈	2
#21426503	2
#21421935	2
C₂×C₅₉×#423	2
#7897	59