Subgroups — #28479396

Order 15104.

554 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21433052	C₂	1	1	yes	maximal, characteristic
2	C₄×#14273653	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#21422479	C₂	1	1	yes	maximal, characteristic
5	M₁₆×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#21421928	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#11745685	C₂²	1	1	yes	characteristic
9	#16837526	C₄	1	2	yes	complemented
10	#16837467	C₄	1	2	yes	complemented
11	(C₂²⋊D₅₉)⋊C₈	—	2	2	no	—
12	#16837455	C₂²	1	1	yes	characteristic
13	#16837453	—	2	1	no	—
14	C₂×#14273653	C₄	1	2	yes	—
15	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₅₉	—	2	2	no	—
17	C₄×C₈×D₅₉	—	2	1	no	—
18	C₄²×Dic₅₉	C₂²	1	1	yes	characteristic
19	#16837163	C₄	1	2	yes	—
20	#16837134	C₄	1	2	yes	complemented
21	#16837131	C₄	1	2	yes	complemented
22	#16837123	—	2	1	no	—
23	C₈×C₅₉⋊C₈	—	2	1	no	—
24	C₅₉×#431	C₂²	1	1	yes	characteristic
25	C₄×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
27	C₂×#11745685	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₅₉⋊C₈	D₄	1	2	yes	—
29	C₂×C₈×D₅₉	—	2	2	no	—
30	C₄×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
32	C₄×D₂₃₆	C₂×C₄	1	1	yes	characteristic
33	C₄×D₂₃₆	C₂×C₄	1	1	yes	characteristic
34	C₄×C₂²⋊D₅₉	—	2	1	no	—
35	C₄×C₂²⋊D₅₉	—	2	1	no	—
36	#14273686	C₂×C₄	1	1	yes	characteristic
37	#14273684	—	2	1	no	—
38	C₂×C₄×Dic₅₉	—	2	1	no	—
39	C₂×C₄×Dic₅₉	—	2	1	no	—
40	C₂×C₄×Dic₅₉	C₂×C₄	1	2	yes	—
41	C₄²×D₅₉	—	2	1	no	—
42	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	2	yes	—
43	Dic₅₉⋊C₈	—	2	2	no	—
44	#14273653	C₈	1	8	yes	complemented
45	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
46	(C₂×D₅₉)⋊C₈	—	2	2	no	—
47	C₄×C₅₉⋊C₈	—	2	1	no	—
48	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
49	C₄×C₅₉⋊C₈	D₄	1	1	yes	characteristic
50	C₄×C₅₉⋊C₈	D₄	1	1	yes	characteristic
51	C₈×Dic₅₉	—	2	8	no	—
52	C₈×Dic₅₉	—	2	2	no	—
53	C₂×M₁₆×C₅₉	C₂×C₄	1	2	yes	—
54	C₂²×C₈×C₅₉	—	2	1	no	—
55	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₅₉	C₂×C₄	1	2	yes	—
57	C₂²⋊C₈×C₅₉	—	2	1	no	—
58	#97×C₅₉	—	2	1	no	—
59	C₄×C₈×C₅₉	—	2	1	no	—
60	C₄×C₈×C₅₉	—	2	1	no	—
61	C₄×C₈×C₅₉	—	2	1	no	—
62	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
63	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
66	C₂×C₂²⋊D₅₉	—	2	1	no	—
67	C₂²×Dic₅₉	—	2	2	no	—
68	#11745685	—	2	2	no	—
69	#11745685	C₂×C₈	1	4	yes	—
70	C₂×D₂₃₆	C₄²	1	1	yes	characteristic
71	C₂×C₄×D₅₉	—	2	1	no	—
72	C₂×C₄×D₅₉	—	2	1	no	—
73	C₂×C₄×D₅₉	—	2	1	no	—
74	C₂×Dic₁₁₈	C₄²	1	1	yes	characteristic
75	M₁₆×C₅₉	—	2	2	no	—
76	M₁₆×C₅₉	C₂×C₈	1	4	yes	complemented
77	C₂×C₈×C₅₉	—	2	4	no	—
78	C₂×C₈×C₅₉	—	2	2	no	—
79	C₂×C₈×C₅₉	—	2	2	no	—
80	C₂×C₈×C₅₉	—	2	2	no	—
81	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₅₉	#41	1	1	yes	characteristic
85	C₂²⋊Dic₅₉	—	2	1	no	—
86	(C₂×D₅₉)⋊C₄	—	2	1	no	—
87	(C₂×D₅₉)⋊C₄	—	2	1	no	—
88	C₄⋊Dic₅₉	C₄²	1	1	yes	characteristic
89	C₄⋊Dic₅₉	C₄²	1	1	yes	characteristic
90	Dic₅₉⋊C₄	—	2	1	no	—
91	Dic₅₉⋊C₄	—	2	1	no	—
92	C₄×Dic₅₉	—	2	4	no	—
93	C₄×Dic₅₉	—	2	4	no	—
94	C₄×Dic₅₉	—	2	1	no	—
95	C₄×Dic₅₉	—	2	1	no	—
96	C₄×Dic₅₉	—	2	1	no	—
97	C₄×Dic₅₉	—	2	2	no	—
98	C₄×Dic₅₉	C₂×C₈	1	4	yes	complemented
99	C₂×C₅₉⋊C₈	—	2	4	no	—
100	C₂×C₅₉⋊C₈	—	2	2	no	—
101	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	2	yes	—
102	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	2	yes	—
103	C₈×D₅₉	—	4	4	no	—
104	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
105	C₂²×D₅₉	—	2	1	no	—
106	C₂×C₄×C₅₉	—	2	2	no	—
107	C₂×C₄×C₅₉	—	2	2	no	—
108	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
111	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
115	C₂×C₄×C₅₉	#131	1	2	yes	—
116	C₂²⋊D₅₉	—	2	4	no	—
117	C₂×Dic₅₉	—	2	4	no	—
118	C₂×Dic₅₉	—	2	4	no	—
119	C₂×Dic₅₉	—	2	1	no	—
120	C₂×Dic₅₉	—	2	4	no	—
121	C₂×Dic₅₉	—	2	1	no	—
122	C₂×Dic₅₉	—	2	1	no	—
123	D₂₃₆	—	2	1	no	—
124	D₂₃₆	C₄×C₈	1	2	yes	complemented
125	C₄×D₅₉	—	2	4	no	—
126	C₄×D₅₉	—	4	2	no	—
127	C₄×D₅₉	—	4	2	no	—
128	Dic₁₁₈	—	2	1	no	—
129	Dic₁₁₈	C₄×C₈	1	2	yes	complemented
130	C₈×C₅₉	—	2	4	no	—
131	C₈×C₅₉	—	2	4	no	—
132	C₈×C₅₉	—	4	2	no	—
133	C₅₉⋊C₈	—	2	2	no	—
134	C₅₉⋊C₈	—	4	2	no	—
135	C₅₉⋊C₈	#104	1	4	yes	complemented
136	C₈×#104	—	59	1	no	maximal, Sylow
137	C₂²×C₅₉	—	2	1	no	—
138	C₂²×C₅₉	—	2	1	no	—
139	C₂²×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
140	C₂²×C₅₉	C₈×D₄	1	2	yes	—
141	C₂×D₅₉	—	2	2	no	—
142	C₂×D₅₉	—	4	1	no	—
143	C₂×D₅₉	—	4	1	no	—
144	C₄×C₅₉	—	2	1	no	—
145	C₄×C₅₉	—	2	1	no	—
146	C₄×C₅₉	C₂×C₄×C₈	1	1	yes	characteristic
147	C₄×C₅₉	#405	1	1	yes	characteristic
148	C₄×C₅₉	C₂×#104	1	2	yes	—
149	C₄×C₅₉	C₈×D₄	1	2	yes	—
150	C₄×C₅₉	#443	1	2	yes	—
151	Dic₅₉	—	2	8	no	—
152	Dic₅₉	—	2	2	no	—
153	Dic₅₉	—	4	1	no	—
154	Dic₅₉	—	4	1	no	—
155	C₈×#41	—	59	1	no	—
156	C₄×#104	—	59	1	no	—
157	C₄²×C₈	—	59	1	no	—
158	#1476	—	59	1	no	—
159	C₈×M₁₆	—	59	1	no	—
160	#1278	D₅₉	1	1	yes	characteristic, complemented
161	#1276	—	59	1	no	—
162	C₂×C₅₉	—	2	2	no	—
163	C₂×C₅₉	C₈×C₂²⋊C₄	1	1	yes	characteristic
164	C₂×C₅₉	C₄×#104	1	1	yes	characteristic
165	C₂×C₅₉	#1764	1	1	yes	characteristic
166	D₅₉	—	4	2	no	—
167	C₄×#41	—	59	1	no	—
168	C₈×Q₈	—	59	2	no	—
169	C₈×D₄	—	59	2	no	—
170	C₈×D₄	—	118	2	no	—
171	#431	—	59	1	no	—
172	#431	—	118	1	no	—
173	#431	C₂×D₅₉	1	1	yes	characteristic
174	C₂×#104	—	59	2	no	—
175	C₄×M₁₆	C₂×D₅₉	1	1	yes	characteristic
176	C₂×C₄×C₈	—	59	1	no	—
177	C₂×C₄×C₈	—	118	1	no	—
178	C₂×C₄×C₈	—	118	2	no	—
179	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
180	C₄³	—	59	1	no	—
181	#339	—	59	2	no	—
182	Q₈⋊C₈	—	59	2	no	—
183	D₄⋊C₈	—	59	2	no	—
184	#322	—	118	1	no	—
185	C₈²	—	118	1	no	—
186	C₅₉	C₈×#104	1	1	yes	characteristic, complemented, Sylow
187	C₂×#41	—	59	1	no	—
188	C₂×M₁₆	C₄×D₅₉	1	2	yes	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	—	59	2	no	—
191	C₂²×C₈	—	118	2	no	—
192	C₄×Q₈	—	59	1	no	—
193	C₄×Q₈	—	59	1	no	—
194	C₄×D₄	—	59	1	no	—
195	C₄×D₄	—	59	1	no	—
196	C₄×D₄	—	118	1	no	—
197	C₄×D₄	—	118	1	no	—
198	#117	—	59	1	no	—
199	#117	—	118	1	no	—
200	C₂×C₄²	—	59	2	no	—
201	C₂×C₄²	—	118	1	no	—
202	C₂×C₄²	—	118	1	no	—
203	C₂×C₄²	—	118	1	no	—
204	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
205	C₄⋊C₈	—	59	2	no	—
206	C₄⋊C₈	—	118	2	no	—
207	C₄⋊C₈	C₄×D₅₉	1	2	yes	—
208	#104	—	59	8	no	—
209	C₂²⋊C₈	—	2	1	no	—
210	C₂²⋊C₈	—	118	2	no	—
211	C₂²⋊C₈	—	118	1	no	—
212	#97	—	2	1	no	—
213	C₄×C₈	—	2	1	no	—
214	C₄×C₈	—	2	1	no	—
215	C₄×C₈	—	2	1	no	—
216	C₄×C₈	—	59	2	no	—
217	C₄×C₈	—	59	1	no	—
218	C₄×C₈	—	59	1	no	—
219	C₄×C₈	—	118	2	no	—
220	C₄×C₈	—	118	8	no	—
221	C₄×C₈	—	118	1	no	—
222	C₄×C₈	C₄×D₅₉	1	2	yes	—
223	#41	—	59	4	no	—
224	#41	—	118	2	no	—
225	C₂×Q₈	—	59	1	no	—
226	C₂×D₄	—	59	1	no	—
227	C₂×D₄	—	118	1	no	—
228	C₂²×C₄	—	118	2	no	—
229	C₂²×C₄	—	118	1	no	—
230	C₂²×C₄	—	118	1	no	—
231	C₂²×C₄	—	118	1	no	—
232	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
233	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
234	C₂²×C₄	#11745689	1	1	yes	characteristic
235	M₁₆	—	2	2	no	—
236	M₁₆	C₈×D₅₉	1	4	yes	complemented
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	59	2	no	—
242	C₂×C₈	—	59	2	no	—
243	C₂×C₈	—	118	4	no	—
244	C₂×C₈	—	118	2	no	—
245	C₂×C₈	—	236	4	no	—
246	C₄⋊C₄	—	59	1	no	—
247	C₄⋊C₄	—	59	1	no	—
248	C₄⋊C₄	—	118	1	no	—
249	C₄⋊C₄	—	118	1	no	—
250	C₂²⋊C₄	—	118	1	no	—
251	C₂²⋊C₄	—	118	1	no	—
252	C₂²⋊C₄	—	118	1	no	—
253	C₄²	—	59	4	no	—
254	C₄²	—	118	4	no	—
255	C₄²	—	118	4	no	—
256	C₄²	—	118	1	no	—
257	C₄²	—	118	1	no	—
258	C₄²	—	118	1	no	—
259	C₄²	—	118	2	no	—
260	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
261	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
262	C₄²	D₄×D₅₉	1	1	yes	characteristic
263	C₄²	#11745687	1	1	yes	characteristic
264	C₂³	—	118	1	no	—
265	C₂³	#14273689	1	1	yes	characteristic
266	Q₈	—	59	2	no	—
267	Q₈	—	118	1	no	—
268	D₄	—	59	2	no	—
269	D₄	—	118	4	no	—
270	D₄	—	118	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	118	4	no	—
274	C₂×C₄	—	118	4	no	—
275	C₂×C₄	—	118	4	no	—
276	C₂×C₄	—	118	1	no	—
277	C₂×C₄	—	118	4	no	—
278	C₂×C₄	—	118	1	no	—
279	C₂×C₄	—	118	1	no	—
280	C₂×C₄	—	236	2	no	—
281	C₂×C₄	—	236	2	no	—
282	C₂×C₄	C₄²×D₅₉	1	1	yes	characteristic
283	C₂×C₄	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
284	C₂×C₄	#14273685	1	1	yes	characteristic
285	C₂×C₄	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
286	C₂×C₄	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
287	C₂×C₄	#14273689	1	1	yes	characteristic
288	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
289	C₂×C₄	#14273735	1	2	yes	—
290	C₈	—	2	4	no	—
291	C₈	—	2	4	no	—
292	C₈	—	4	2	no	—
293	C₈	—	59	4	no	—
294	C₈	—	118	2	no	—
295	C₈	—	236	2	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	118	2	no	—
299	C₂²	—	236	1	no	—
300	C₂²	—	236	1	no	—
301	C₂²	#16837249	1	1	yes	characteristic
302	C₂²	#16837467	1	2	yes	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	118	8	no	—
306	C₄	—	118	2	no	—
307	C₄	—	236	1	no	—
308	C₄	—	236	1	no	—
309	C₄	C₄×C₈×D₅₉	1	1	yes	characteristic
310	C₄	#16837359	1	1	yes	characteristic
311	C₄	#104×D₅₉	1	2	yes	—
312	C₄	(C₂²⋊D₅₉)⋊C₈	1	2	yes	—
313	C₄	#16837518	1	2	yes	—
314	C₂	—	2	2	no	—
315	C₂	—	236	2	no	—
316	C₂	#21423866	1	1	yes	characteristic
317	C₂	#21423900	1	1	yes	characteristic
318	C₂	#21423932	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₅₉
commutator	C₄×C₅₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1278	C₂
socle	C₂²×C₅₉	C₄×C₂²⋊C₄
prime radical	C₂×#11745685	C₂×C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1278	D₅₉
hypocenter	C₅₉	C₈×#104
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1278

Complements

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

N	H ≅ G/N	type
3. C₄×C₈×Dic₅₉	C₂	semidirect
5. M₁₆×C₅₉⋊C₈	C₂	semidirect
6. #21421928	C₂	semidirect
7. C₅₉×#1278	C₂	semidirect
9. #16837526	C₄	semidirect
10. #16837467	C₄	semidirect
20. #16837134	C₄	semidirect
21. #16837131	C₄	semidirect
44. #14273653	C₈	semidirect
76. M₁₆×C₅₉	C₂×C₈	semidirect
98. C₄×Dic₅₉	C₂×C₈	semidirect
124. D₂₃₆	C₄×C₈	semidirect
129. Dic₁₁₈	C₄×C₈	semidirect
135. C₅₉⋊C₈	#104	semidirect
160. #1278	D₅₉	semidirect
186. C₅₉	C₈×#104	semidirect
236. M₁₆	C₈×D₅₉	semidirect

Maximal subgroups

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

subgroup	index
#21433052	2
C₄×#14273653	2
C₄×C₈×Dic₅₉	2
#21422479	2
M₁₆×C₅₉⋊C₈	2
#21421928	2
C₅₉×#1278	2
C₈×#104	59