Subgroups — #28481902

Order 15104.

434 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21432807	C₂	1	1	yes	maximal, characteristic
2	#21424258	C₂	1	1	yes	maximal, characteristic, complemented
3	#21422842	C₂	1	1	yes	maximal, characteristic, complemented
4	#21422657	C₂	1	1	yes	maximal, characteristic
5	#21422655	C₂	1	1	yes	maximal, characteristic
6	#21421956	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#1286	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14273735	—	2	1	no	—
9	#16837952	C₂²	1	1	yes	characteristic
10	#16837479	—	2	1	no	—
11	#16837479	C₄	1	2	yes	complemented
12	#16837467	—	2	1	no	—
13	#16837467	C₄	1	2	yes	complemented
14	#16837455	C₂²	1	1	yes	characteristic
15	C₂×#14273660	—	2	1	no	—
16	C₂×#14273658	—	2	1	no	—
17	#16837370	—	2	1	no	—
18	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
19	#16837268	C₂²	1	1	yes	characteristic
20	#16837154	—	2	1	no	—
21	#16837152	—	2	1	no	—
22	#16837137	—	2	2	no	—
23	#16837135	—	2	2	no	—
24	C₅₉×#431	C₂²	1	1	yes	characteristic
25	C₅₉×#423	C₂²	1	1	yes	characteristic
26	C₅₉×#382	C₂²	1	1	yes	characteristic
27	C₂×#11745685	—	2	1	no	—
28	#14273735	—	4	2	no	—
29	#14273735	—	4	2	no	—
30	C₂×#11745659	—	2	2	no	—
31	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
32	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
33	C₂×C₈×D₅₉	—	2	2	no	—
34	#14273727	Q₈	1	1	yes	characteristic
35	#14273724	Q₈	1	1	yes	characteristic
36	#14273715	—	2	1	no	—
37	#14273701	—	2	1	no	—
38	#14273701	D₄	1	2	yes	—
39	C₄×D₂₃₆	—	2	1	no	—
40	C₄×D₂₃₆	C₂×C₄	1	2	yes	—
41	C₂×C₄⋊Dic₅₉	—	2	1	no	—
42	#14273660	—	4	2	no	—
43	#14273658	—	4	2	no	—
44	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
45	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
46	(C₂×D₅₉)⋊C₈	—	2	2	no	—
47	(C₂×D₅₉)⋊C₈	—	2	2	no	—
48	C₄×C₅₉⋊C₈	—	2	1	no	—
49	C₄×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
50	C₄×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
51	C₄×C₅₉⋊C₈	D₄	1	1	yes	characteristic
52	C₄×C₅₉⋊C₈	D₄	1	1	yes	characteristic
53	C₅₉⋊#95	—	2	1	no	—
54	C₅₉⋊#95	—	2	1	no	—
55	C₂×M₁₆×C₅₉	—	2	1	no	—
56	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
57	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₅₉	—	2	1	no	—
59	C₄⋊C₈×C₅₉	—	2	1	no	—
60	C₄⋊C₈×C₅₉	C₂×C₄	1	2	yes	—
61	C₂²⋊C₈×C₅₉	—	2	1	no	—
62	C₄×C₈×C₅₉	—	2	1	no	—
63	#95×C₅₉	—	2	1	no	—
64	#95×C₅₉	—	2	1	no	—
65	C₂²×C₄×C₅₉	—	4	1	no	—
66	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
69	C₂×C₂²⋊D₅₉	—	2	2	no	—
70	C₂²×Dic₅₉	—	4	1	no	—
71	#11745685	—	4	1	no	—
72	#11745685	—	4	1	no	—
73	#11745685	—	4	1	no	—
74	#11745685	—	4	1	no	—
75	C₂×D₂₃₆	—	2	1	no	—
76	C₂×D₂₃₆	—	2	1	no	—
77	C₂×D₂₃₆	C₄⋊C₄	1	2	yes	—
78	C₂×C₄×D₅₉	—	2	2	no	—
79	C₂×Dic₁₁₈	—	2	1	no	—
80	M₁₆×C₅₉	—	4	1	no	—
81	M₁₆×C₅₉	—	4	1	no	—
82	C₂×C₈×C₅₉	—	2	2	no	—
83	C₂×C₈×C₅₉	—	4	1	no	—
84	C₄⋊C₄×C₅₉	—	4	1	no	—
85	C₄⋊C₄×C₅₉	—	4	1	no	—
86	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
89	C₄²×C₅₉	#41	1	1	yes	characteristic
90	(C₂×D₅₉)⋊C₄	—	2	2	no	—
91	(C₂×D₅₉)⋊C₄	—	2	2	no	—
92	C₄⋊Dic₅₉	—	2	1	no	—
93	C₄⋊Dic₅₉	—	4	1	no	—
94	C₄⋊Dic₅₉	—	4	1	no	—
95	C₄⋊Dic₅₉	C₄⋊C₄	1	2	yes	—
96	C₂×C₅₉⋊C₈	—	2	1	no	—
97	C₂×C₅₉⋊C₈	—	2	1	no	—
98	C₂×C₅₉⋊C₈	—	4	1	no	—
99	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₅₉⋊C₈	D₈	1	2	yes	—
104	C₂×C₅₉⋊C₈	SD₁₆	1	2	yes	—
105	#11745659	—	4	4	no	—
106	C₈×D₅₉	—	4	4	no	—
107	C₂³×C₅₉	#121	1	1	yes	characteristic
108	C₂²×D₅₉	—	2	2	no	—
109	C₂×C₄×C₅₉	—	4	1	no	—
110	C₂×C₄×C₅₉	—	4	1	no	—
111	C₂×C₄×C₅₉	—	8	1	no	—
112	C₂×C₄×C₅₉	—	8	1	no	—
113	C₂×C₄×C₅₉	—	8	1	no	—
114	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
117	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
118	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
122	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
124	C₂²⋊D₅₉	—	4	2	no	—
125	C₂²⋊D₅₉	—	4	2	no	—
126	C₂×Dic₅₉	—	2	2	no	—
127	C₂×Dic₅₉	—	8	1	no	—
128	C₂×Dic₅₉	—	8	1	no	—
129	C₂×Dic₅₉	—	8	1	no	—
130	D₂₃₆	—	4	2	no	—
131	D₂₃₆	—	4	2	no	—
132	D₂₃₆	—	4	1	no	—
133	D₂₃₆	—	4	1	no	—
134	C₄×D₅₉	—	4	2	no	—
135	C₄×D₅₉	—	4	2	no	—
136	Dic₁₁₈	—	4	1	no	—
137	Dic₁₁₈	—	4	1	no	—
138	C₈×C₅₉	—	4	2	no	—
139	C₈×C₅₉	—	8	1	no	—
140	C₅₉⋊C₈	—	2	1	no	—
141	C₅₉⋊C₈	—	2	1	no	—
142	C₅₉⋊C₈	—	8	1	no	—
143	C₅₉⋊C₈	#102	1	2	yes	—
144	C₅₉⋊C₈	#102	1	2	yes	—
145	#8676	—	59	1	no	maximal, Sylow
146	C₂²×C₅₉	—	2	1	no	—
147	C₂²×C₅₉	—	2	1	no	—
148	C₂²×C₅₉	#380	1	1	yes	characteristic
149	C₂²×C₅₉	#459	1	1	yes	characteristic
150	C₂²×C₅₉	#461	1	1	yes	characteristic
151	C₂×D₅₉	—	4	2	no	—
152	C₂×D₅₉	—	4	2	no	—
153	C₂×D₅₉	—	4	2	no	—
154	C₄×C₅₉	—	2	1	no	—
155	C₄×C₅₉	—	2	1	no	—
156	C₄×C₅₉	—	2	1	no	—
157	C₄×C₅₉	—	2	1	no	—
158	C₄×C₅₉	—	16	1	no	—
159	C₄×C₅₉	C₂×#102	1	1	yes	characteristic
160	C₄×C₅₉	#417	1	1	yes	characteristic
161	C₄×C₅₉	#474	1	1	yes	characteristic
162	C₄×C₅₉	#476	1	1	yes	characteristic
163	Dic₅₉	—	4	2	no	—
164	Dic₅₉	—	16	1	no	—
165	#2931	—	59	1	no	—
166	#1846	—	59	1	no	—
167	#1564	—	59	1	no	—
168	#1517	—	59	1	no	—
169	#1517	—	59	1	no	—
170	#1286	—	59	1	no	—
171	#1286	D₅₉	1	1	yes	characteristic, complemented
172	C₂×C₅₉	—	2	1	no	—
173	C₂×C₅₉	—	2	1	no	—
174	C₂×C₅₉	#1802	1	1	yes	characteristic
175	C₂×C₅₉	#1812	1	1	yes	characteristic
176	C₂×C₅₉	#1814	1	1	yes	characteristic
177	D₅₉	—	8	2	no	—
178	C₂×#131	—	118	1	no	—
179	#532	—	59	1	no	—
180	#436	—	59	2	no	—
181	#436	—	118	1	no	—
182	C₈×D₄	—	59	2	no	—
183	C₈×D₄	—	118	1	no	—
184	#431	—	59	1	no	—
185	#431	C₂×D₅₉	1	1	yes	characteristic
186	C₂×#107	—	118	1	no	—
187	C₂×#106	—	118	1	no	—
188	#423	C₂×D₅₉	1	1	yes	characteristic
189	#408	—	118	1	no	—
190	C₂×C₄×C₈	—	59	1	no	—
191	#382	—	59	1	no	—
192	#382	C₂×D₅₉	1	1	yes	characteristic
193	#335	—	118	1	no	—
194	#334	—	118	1	no	—
195	#327	—	118	2	no	—
196	#327	—	118	2	no	—
197	C₅₉	#8676	1	1	yes	characteristic, complemented, Sylow
198	C₂×#41	—	118	1	no	—
199	#131	—	236	2	no	—
200	#131	—	236	2	no	—
201	C₂×M₁₆	—	2	1	no	—
202	C₂×M₁₆	—	118	2	no	—
203	C₂²×C₈	—	59	1	no	—
204	C₂²×C₈	—	59	1	no	—
205	C₂²×C₈	—	118	2	no	—
206	C₄⋊Q₈	—	59	1	no	—
207	#127	—	59	1	no	—
208	#124	—	118	1	no	—
209	#121	—	59	2	no	—
210	#121	—	118	1	no	—
211	C₄×D₄	—	59	2	no	—
212	C₄×D₄	—	118	1	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄⋊C₄	—	118	1	no	—
215	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
216	#107	—	236	2	no	—
217	#106	—	236	2	no	—
218	C₄⋊C₈	—	2	1	no	—
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	118	1	no	—
221	C₄⋊C₈	C₄×D₅₉	1	2	yes	—
222	C₂²⋊C₈	—	2	1	no	—
223	C₂²⋊C₈	—	118	2	no	—
224	C₂²⋊C₈	—	118	2	no	—
225	C₂²⋊C₈	—	118	1	no	—
226	C₄×C₈	—	2	1	no	—
227	C₄×C₈	—	59	1	no	—
228	C₄×C₈	—	59	1	no	—
229	C₄×C₈	—	59	1	no	—
230	C₄×C₈	—	59	1	no	—
231	C₄×C₈	—	118	1	no	—
232	#95	—	2	1	no	—
233	#95	—	2	1	no	—
234	#95	—	118	1	no	—
235	#95	—	118	1	no	—
236	#41	—	236	1	no	—
237	#41	—	236	1	no	—
238	#41	—	236	1	no	—
239	#41	—	236	1	no	—
240	C₂×Q₈	—	118	1	no	—
241	C₂×D₄	—	59	2	no	—
242	C₂×D₄	—	118	2	no	—
243	C₂×D₄	—	118	1	no	—
244	C₂×D₄	—	118	1	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	—	118	2	no	—
247	C₂²×C₄	—	236	1	no	—
248	C₂²×C₄	#11745685	1	1	yes	characteristic
249	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
250	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
251	M₁₆	—	4	1	no	—
252	M₁₆	—	4	1	no	—
253	M₁₆	—	236	4	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	59	1	no	—
257	C₂×C₈	—	59	1	no	—
258	C₂×C₈	—	59	2	no	—
259	C₂×C₈	—	59	2	no	—
260	C₂×C₈	—	59	1	no	—
261	C₂×C₈	—	59	1	no	—
262	C₂×C₈	—	118	1	no	—
263	C₂×C₈	—	118	1	no	—
264	C₂×C₈	—	236	4	no	—
265	C₂×C₈	—	236	1	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	59	2	no	—
269	C₄⋊C₄	—	118	1	no	—
270	C₄⋊C₄	—	236	1	no	—
271	C₄⋊C₄	—	236	1	no	—
272	C₂²⋊C₄	—	118	2	no	—
273	C₂²⋊C₄	—	118	2	no	—
274	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
275	C₄²	#11745685	1	1	yes	characteristic
276	C₄²	D₄×D₅₉	1	1	yes	characteristic
277	C₄²	Q₈×D₅₉	1	1	yes	characteristic
278	C₂³	—	118	2	no	—
279	C₂³	#14273700	1	1	yes	characteristic
280	Q₈	—	236	1	no	—
281	Q₈	—	236	1	no	—
282	D₄	—	236	2	no	—
283	D₄	—	236	2	no	—
284	D₄	—	236	2	no	—
285	D₄	—	236	2	no	—
286	D₄	—	236	1	no	—
287	D₄	—	236	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	8	1	no	—
291	C₂×C₄	—	8	1	no	—
292	C₂×C₄	—	8	1	no	—
293	C₂×C₄	—	118	2	no	—
294	C₂×C₄	—	472	1	no	—
295	C₂×C₄	—	472	1	no	—
296	C₂×C₄	—	472	1	no	—
297	C₂×C₄	—	236	2	no	—
298	C₂×C₄	—	236	2	no	—
299	C₂×C₄	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
300	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
301	C₂×C₄	#14273684	1	1	yes	characteristic
302	C₂×C₄	#14273700	1	1	yes	characteristic
303	C₂×C₄	#14273705	1	1	yes	characteristic
304	C₂×C₄	#14273705	1	1	yes	characteristic
305	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
306	C₂×C₄	SD₁₆×D₅₉	1	1	yes	characteristic
307	C₂×C₄	#14273754	1	1	yes	characteristic
308	C₂×C₄	#14273759	1	1	yes	characteristic
309	C₈	—	4	2	no	—
310	C₈	—	8	1	no	—
311	C₈	—	59	2	no	—
312	C₈	—	59	2	no	—
313	C₈	—	118	1	no	—
314	C₈	—	118	1	no	—
315	C₈	—	472	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	236	2	no	—
319	C₂²	—	236	2	no	—
320	C₂²	—	236	2	no	—
321	C₂²	#16837261	1	1	yes	characteristic
322	C₂²	#16837596	1	1	yes	characteristic
323	C₂²	#16837606	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	16	1	no	—
329	C₄	—	944	1	no	—
330	C₄	—	236	2	no	—
331	C₄	#102×D₅₉	1	1	yes	characteristic
332	C₄	#16837402	1	1	yes	characteristic
333	C₄	#16837681	1	1	yes	characteristic
334	C₄	#16837691	1	1	yes	characteristic
335	C₂	—	2	1	no	—
336	C₂	—	2	1	no	—
337	C₂	—	472	2	no	—
338	C₂	#21424095	1	1	yes	characteristic
339	C₂	#21424141	1	1	yes	characteristic
340	C₂	#21424149	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16837261
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1286	C₂
socle	C₂²×C₅₉	#380
prime radical	#16837952	C₂²
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1286	D₅₉
hypocenter	C₅₉	#8676
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1286

Complements

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

N	H ≅ G/N	type
2. #21424258	C₂	semidirect
3. #21422842	C₂	semidirect
6. #21421956	C₂	semidirect
7. C₅₉×#1286	C₂	semidirect
11. #16837479	C₄	semidirect
13. #16837467	C₄	semidirect
171. #1286	D₅₉	semidirect
197. C₅₉	#8676	semidirect

Maximal subgroups

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

subgroup	index
#21432807	2
#21424258	2
#21422842	2
#21422657	2
#21422655	2
#21421956	2
C₅₉×#1286	2
#8676	59