Subgroups — #28479739

Order 15104.

436 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21433065	C₂	1	1	yes	maximal, characteristic
2	#21423783	C₂	1	1	yes	maximal, characteristic, complemented
3	#21422685	C₂	1	1	yes	maximal, characteristic
4	#21422580	C₂	1	1	yes	maximal, characteristic
5	#21422383	C₂	1	1	yes	maximal, characteristic, complemented
6	#21421994	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#1300	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837936	C₂²	1	1	yes	characteristic
9	#16837529	C₄	1	1	yes	characteristic, complemented
10	#16837526	C₄	1	1	yes	characteristic, complemented
11	#16837479	C₄	1	1	yes	characteristic, complemented
12	#16837470	—	2	1	no	—
13	#16837467	C₄	1	1	yes	characteristic, complemented
14	(C₂²⋊D₅₉)⋊C₈	—	2	1	no	—
15	#16837464	—	2	1	no	—
16	#16837455	C₂²	1	1	yes	characteristic
17	#16837359	—	2	1	no	—
18	C₂×#14273638	—	2	1	no	—
19	C₂×#14273638	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₅₉⋊C₈	—	2	1	no	—
21	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
22	#16837143	C₄	1	2	yes	complemented
23	#16837141	C₄	1	2	yes	complemented
24	#16837138	C₄	1	2	yes	—
25	#16837135	C₄	1	2	yes	—
26	#16837124	—	2	1	no	—
27	#16837124	—	2	1	no	—
28	C₅₉×#432	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
30	C₅₉×#376	C₂²	1	1	yes	characteristic
31	C₂×#11745659	—	2	1	no	—
32	C₂²×C₅₉⋊C₈	—	2	1	no	—
33	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
34	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
35	C₂×C₈×D₅₉	—	2	1	no	—
36	#14273721	—	2	1	no	—
37	#14273719	C₂×C₄	1	1	yes	characteristic
38	#14273715	C₂×C₄	1	1	yes	characteristic
39	#14273710	—	2	1	no	—
40	#14273706	C₂×C₄	1	1	yes	characteristic
41	#14273701	C₂×C₄	1	1	yes	characteristic
42	C₄×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
43	C₄×D₂₃₆	C₂×C₄	1	1	yes	characteristic
44	C₄×C₂²⋊D₅₉	—	2	1	no	—
45	#14273684	—	2	1	no	—
46	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
47	Dic₅₉⋊C₈	—	2	1	no	—
48	Dic₅₉⋊C₈	—	2	1	no	—
49	C₂²⋊(C₅₉⋊C₈)	C₂×C₄	1	2	yes	—
50	(C₂×D₅₉)⋊C₈	—	2	1	no	—
51	(C₂×D₅₉)⋊C₈	—	2	1	no	—
52	#14273638	—	2	4	no	—
53	#14273638	—	2	1	no	—
54	#14273638	C₂×C₄	1	2	yes	—
55	#14273637	—	2	1	no	—
56	C₄×C₅₉⋊C₈	—	2	4	no	—
57	C₄×C₅₉⋊C₈	—	2	1	no	—
58	C₄×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
59	C₄×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
60	C₄×C₅₉⋊C₈	D₄	1	1	yes	characteristic
61	C₄×C₅₉⋊C₈	D₄	1	1	yes	characteristic
62	C₈×Dic₅₉	—	2	1	no	—
63	C₂×M₁₆×C₅₉	—	2	1	no	—
64	C₂×C₄²×C₅₉	—	2	1	no	—
65	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
67	C₄⋊C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
68	C₂²⋊C₈×C₅₉	—	2	1	no	—
69	#97×C₅₉	—	2	1	no	—
70	#97×C₅₉	C₂×C₄	1	2	yes	—
71	C₄×C₈×C₅₉	—	2	1	no	—
72	#95×C₅₉	—	2	1	no	—
73	#95×C₅₉	C₂×C₄	1	2	yes	—
74	C₂²×C₄×C₅₉	—	2	2	no	—
75	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
78	C₂×C₂²⋊D₅₉	—	2	1	no	—
79	C₂×D₂₃₆	C₄²	1	1	yes	characteristic
80	C₂×C₄×D₅₉	—	2	1	no	—
81	C₂×Dic₁₁₈	C₄²	1	1	yes	characteristic
82	M₁₆×C₅₉	—	4	2	no	—
83	C₂×C₈×C₅₉	—	2	1	no	—
84	C₂×C₈×C₅₉	—	2	1	no	—
85	C₂×C₈×C₅₉	—	2	2	no	—
86	C₄²×C₅₉	—	2	2	no	—
87	C₄²×C₅₉	—	2	2	no	—
88	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₅₉	#41	1	1	yes	characteristic
92	C₂²⋊Dic₅₉	—	2	1	no	—
93	(C₂×D₅₉)⋊C₄	—	2	1	no	—
94	(C₂×D₅₉)⋊C₄	—	2	1	no	—
95	(C₂×D₅₉)⋊C₄	—	2	1	no	—
96	(C₂×D₅₉)⋊C₄	—	2	1	no	—
97	C₄⋊Dic₅₉	C₄²	1	1	yes	characteristic
98	C₄⋊Dic₅₉	C₄²	1	1	yes	characteristic
99	Dic₅₉⋊C₄	—	2	1	no	—
100	Dic₅₉⋊C₄	—	2	1	no	—
101	Dic₅₉⋊C₄	—	2	1	no	—
102	Dic₅₉⋊C₄	—	2	1	no	—
103	C₄×Dic₅₉	—	2	1	no	—
104	C₂×C₅₉⋊C₈	—	2	2	no	—
105	C₂×C₅₉⋊C₈	—	2	2	no	—
106	C₂×C₅₉⋊C₈	—	2	2	no	—
107	C₂×C₅₉⋊C₈	—	2	2	no	—
108	C₂×C₅₉⋊C₈	—	2	2	no	—
109	C₂×C₅₉⋊C₈	—	2	2	no	—
110	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
114	#11745659	—	4	2	no	—
115	C₈×D₅₉	—	4	2	no	—
116	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
117	C₂²×D₅₉	—	2	1	no	—
118	C₂×C₄×C₅₉	—	2	2	no	—
119	C₂×C₄×C₅₉	—	2	1	no	—
120	C₂×C₄×C₅₉	—	2	1	no	—
121	C₂×C₄×C₅₉	—	2	4	no	—
122	C₂×C₄×C₅₉	—	4	1	no	—
123	C₂×C₄×C₅₉	—	4	2	no	—
124	C₂×C₄×C₅₉	—	4	2	no	—
125	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
126	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
128	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
131	C₂²⋊D₅₉	—	4	2	no	—
132	C₂×Dic₅₉	—	2	1	no	—
133	C₂×Dic₅₉	—	2	1	no	—
134	C₂×Dic₅₉	—	2	1	no	—
135	D₂₃₆	—	4	1	no	—
136	C₄×D₅₉	—	4	1	no	—
137	C₄×D₅₉	—	4	1	no	—
138	Dic₁₁₈	—	4	1	no	—
139	C₈×C₅₉	—	4	2	no	—
140	C₈×C₅₉	—	4	1	no	—
141	C₈×C₅₉	—	4	1	no	—
142	C₅₉⋊C₈	—	2	4	no	—
143	C₅₉⋊C₈	—	2	1	no	—
144	C₅₉⋊C₈	—	2	1	no	—
145	C₅₉⋊C₈	—	4	2	no	—
146	C₅₉⋊C₈	#104	1	2	yes	—
147	C₅₉⋊C₈	#104	1	2	yes	—
148	#8249	—	59	1	no	maximal, Sylow
149	C₂²×C₅₉	—	2	1	no	—
150	C₂²×C₅₉	—	2	1	no	—
151	C₂²×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₅₉	#444	1	2	yes	—
153	C₂×D₅₉	—	4	1	no	—
154	C₂×D₅₉	—	4	1	no	—
155	C₂×D₅₉	—	4	1	no	—
156	C₄×C₅₉	—	2	1	no	—
157	C₄×C₅₉	—	2	1	no	—
158	C₄×C₅₉	—	2	1	no	—
159	C₄×C₅₉	—	2	1	no	—
160	C₄×C₅₉	—	2	1	no	—
161	C₄×C₅₉	—	4	4	no	—
162	C₄×C₅₉	C₂×#104	1	1	yes	characteristic
163	C₄×C₅₉	C₂×#104	1	1	yes	characteristic
164	Dic₅₉	—	4	1	no	—
165	Dic₅₉	—	4	1	no	—
166	Dic₅₉	—	4	1	no	—
167	#2966	—	59	1	no	—
168	#1726	—	59	1	no	—
169	#1524	—	59	1	no	—
170	#1500	—	59	1	no	—
171	#1451	—	59	1	no	—
172	#1302	—	59	1	no	—
173	#1300	D₅₉	1	1	yes	characteristic, complemented
174	C₂×C₅₉	—	2	2	no	—
175	C₂×C₅₉	#1757	1	1	yes	characteristic
176	C₂×C₅₉	C₄×#104	1	1	yes	characteristic
177	C₂×C₅₉	#1765	1	1	yes	characteristic
178	D₅₉	—	8	1	no	—
179	#529	—	59	1	no	—
180	#446	—	59	1	no	—
181	C₈×Q₈	—	59	1	no	—
182	#436	—	59	1	no	—
183	#435	—	118	1	no	—
184	C₈×D₄	—	59	1	no	—
185	C₈×D₄	—	118	1	no	—
186	#433	—	118	1	no	—
187	#432	C₂×D₅₉	1	1	yes	characteristic
188	#431	—	59	1	no	—
189	#405	—	118	1	no	—
190	C₄×M₁₆	C₂×D₅₉	1	1	yes	characteristic
191	C₂×#97	—	59	1	no	—
192	C₂×#97	—	118	1	no	—
193	C₂×C₄×C₈	—	59	1	no	—
194	C₂×C₄×C₈	—	118	1	no	—
195	#376	C₂×D₅₉	1	1	yes	characteristic
196	#330	—	59	2	no	—
197	#329	—	59	2	no	—
198	#328	—	59	2	no	—
199	#327	—	59	2	no	—
200	#322	—	118	1	no	—
201	#322	—	118	1	no	—
202	C₅₉	#8249	1	1	yes	characteristic, complemented, Sylow
203	C₂×M₁₆	—	2	1	no	—
204	C₂×M₁₆	—	118	1	no	—
205	C₂²×C₈	—	59	1	no	—
206	C₂²×C₈	—	59	1	no	—
207	C₂²×C₈	—	118	1	no	—
208	C₂²×C₈	—	118	1	no	—
209	#126	—	118	1	no	—
210	#125	—	59	1	no	—
211	#124	—	59	1	no	—
212	#123	—	118	1	no	—
213	C₂²⋊Q₈	—	59	1	no	—
214	#121	—	59	1	no	—
215	C₄×Q₈	—	59	1	no	—
216	C₄×D₄	—	59	1	no	—
217	C₄×D₄	—	118	1	no	—
218	#117	—	118	1	no	—
219	C₂×C₄²	—	2	1	no	—
220	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
221	C₄⋊C₈	—	118	1	no	—
222	C₄⋊C₈	—	118	1	no	—
223	C₄⋊C₈	—	118	1	no	—
224	C₄⋊C₈	C₄×D₅₉	1	1	yes	characteristic
225	C₄⋊C₈	C₄×D₅₉	1	1	yes	characteristic
226	C₂²⋊C₈	—	2	1	no	—
227	C₂²⋊C₈	—	59	2	no	—
228	C₂²⋊C₈	—	118	1	no	—
229	C₂²⋊C₈	—	118	1	no	—
230	#97	—	2	1	no	—
231	#97	—	59	2	no	—
232	#97	—	118	4	no	—
233	#97	—	118	1	no	—
234	#97	—	118	1	no	—
235	#97	C₄×D₅₉	1	2	yes	—
236	C₄×C₈	—	2	1	no	—
237	C₄×C₈	—	59	1	no	—
238	C₄×C₈	—	59	1	no	—
239	C₄×C₈	—	59	1	no	—
240	C₄×C₈	—	59	1	no	—
241	C₄×C₈	—	118	4	no	—
242	C₄×C₈	—	118	1	no	—
243	C₄×C₈	—	118	1	no	—
244	#95	—	2	1	no	—
245	#95	C₄×D₅₉	1	2	yes	—
246	C₂×Q₈	—	59	1	no	—
247	C₂×D₄	—	59	1	no	—
248	C₂×D₄	—	118	1	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	118	1	no	—
251	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
252	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
253	C₂²×C₄	#11745687	1	1	yes	characteristic
254	M₁₆	—	4	2	no	—
255	M₁₆	—	236	2	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	59	1	no	—
260	C₂×C₈	—	59	1	no	—
261	C₂×C₈	—	59	1	no	—
262	C₂×C₈	—	59	1	no	—
263	C₂×C₈	—	118	2	no	—
264	C₂×C₈	—	118	2	no	—
265	C₂×C₈	—	118	2	no	—
266	C₂×C₈	—	118	2	no	—
267	C₂×C₈	—	118	2	no	—
268	C₂×C₈	—	118	2	no	—
269	C₂×C₈	—	236	2	no	—
270	C₄⋊C₄	—	59	1	no	—
271	C₄⋊C₄	—	59	1	no	—
272	C₄⋊C₄	—	118	1	no	—
273	C₄⋊C₄	—	118	1	no	—
274	C₄⋊C₄	—	118	1	no	—
275	C₄⋊C₄	—	118	1	no	—
276	C₂²⋊C₄	—	118	1	no	—
277	C₂²⋊C₄	—	118	1	no	—
278	C₂²⋊C₄	—	118	1	no	—
279	C₂²⋊C₄	—	118	1	no	—
280	C₂²⋊C₄	—	118	1	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	118	1	no	—
284	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
285	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
286	C₄²	D₄×D₅₉	1	1	yes	characteristic
287	C₄²	#11745689	1	1	yes	characteristic
288	C₂³	—	118	1	no	—
289	C₂³	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
290	Q₈	—	236	1	no	—
291	D₄	—	236	1	no	—
292	D₄	—	236	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	118	1	no	—
301	C₂×C₄	—	118	1	no	—
302	C₂×C₄	—	118	1	no	—
303	C₂×C₄	—	236	1	no	—
304	C₂×C₄	—	236	1	no	—
305	C₂×C₄	C₄²×D₅₉	1	1	yes	characteristic
306	C₂×C₄	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
307	C₂×C₄	#14273685	1	1	yes	characteristic
308	C₂×C₄	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
309	C₂×C₄	#14273689	1	1	yes	characteristic
310	C₂×C₄	#14273689	1	1	yes	characteristic
311	C₈	—	4	2	no	—
312	C₈	—	4	1	no	—
313	C₈	—	4	1	no	—
314	C₈	—	59	2	no	—
315	C₈	—	59	2	no	—
316	C₈	—	118	4	no	—
317	C₈	—	118	1	no	—
318	C₈	—	118	1	no	—
319	C₈	—	236	2	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	236	1	no	—
323	C₂²	—	236	1	no	—
324	C₂²	—	236	1	no	—
325	C₂²	#16837249	1	1	yes	characteristic
326	C₂²	#16837521	1	2	yes	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	4	4	no	—
333	C₄	—	236	1	no	—
334	C₄	—	236	1	no	—
335	C₄	—	236	1	no	—
336	C₄	#104×D₅₉	1	1	yes	characteristic
337	C₄	#104×D₅₉	1	1	yes	characteristic
338	C₂	—	2	2	no	—
339	C₂	—	472	1	no	—
340	C₂	#21423895	1	1	yes	characteristic
341	C₂	#21423900	1	1	yes	characteristic
342	C₂	#21423937	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1300	C₂
socle	C₂²×C₅₉	C₄×C₂²⋊C₄
prime radical	#14273701	C₂×C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1300	D₅₉
hypocenter	C₅₉	#8249
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1300

Complements

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

N	H ≅ G/N	type
2. #21423783	C₂	semidirect
5. #21422383	C₂	semidirect
6. #21421994	C₂	semidirect
7. C₅₉×#1300	C₂	semidirect
9. #16837529	C₄	semidirect
10. #16837526	C₄	semidirect
11. #16837479	C₄	semidirect
13. #16837467	C₄	semidirect
22. #16837143	C₄	semidirect
23. #16837141	C₄	semidirect
173. #1300	D₅₉	semidirect
202. C₅₉	#8249	semidirect

Maximal subgroups

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

subgroup	index
#21433065	2
#21423783	2
#21422685	2
#21422580	2
#21422383	2
#21421994	2
C₅₉×#1300	2
#8249	59