Subgroups — #28494417

Order 15104.

894 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837165	C₂	1	2	yes	maximal
2	C₂×#16837165	C₂	1	2	yes	maximal
3	#21422326	C₂	1	2	yes	maximal
4	C₂²×C₄×C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₅₉	C₂²	1	4	yes	—
6	C₂×C₅₉⋊#95	—	2	2	no	—
7	C₂×C₅₉⋊#95	—	2	2	no	—
8	#16837165	—	2	4	no	—
9	#16837165	C₄	1	8	yes	complemented
10	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₉	—	2	4	no	—
14	C₂×C₄×C₈×C₅₉	C₄	1	4	yes	complemented
15	C₂³×Dic₅₉	—	2	4	no	—
16	C₂×C₄⋊Dic₅₉	—	2	4	no	—
17	C₂×C₄⋊Dic₅₉	—	2	4	no	—
18	C₂×C₄⋊Dic₅₉	—	2	4	no	—
19	C₂×C₄⋊Dic₅₉	—	2	4	no	—
20	C₂×C₄⋊Dic₅₉	—	2	4	no	—
21	C₂×C₄⋊Dic₅₉	C₂×C₄	1	8	yes	—
22	C₅₉⋊#95	—	4	4	no	—
23	C₅₉⋊#95	—	4	4	no	—
24	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₅₉	—	2	2	no	—
28	C₂²×C₈×C₅₉	—	2	2	no	—
29	C₂²×C₈×C₅₉	—	2	2	no	—
30	C₂²×C₈×C₅₉	—	2	2	no	—
31	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₅₉	D₄	1	4	yes	—
35	C₂×C₄²×C₅₉	—	2	2	no	—
36	C₂×C₄²×C₅₉	—	2	2	no	—
37	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₅₉	Q₈	1	2	yes	—
39	C₄×C₈×C₅₉	—	2	8	no	—
40	C₂⁴×C₅₉	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₉	—	2	2	no	—
42	C₂²×C₄×C₅₉	—	2	2	no	—
43	C₂²×C₄×C₅₉	—	2	2	no	—
44	C₂²×C₄×C₅₉	—	2	2	no	—
45	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
46	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₅₉	D₈	1	2	yes	—
48	C₂²×C₄×C₅₉	SD₁₆	1	4	yes	—
49	C₂²×C₄×C₅₉	Q₁₆	1	2	yes	—
50	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₉	#41	1	2	yes	—
55	C₂²×C₄×C₅₉	#41	1	2	yes	—
56	C₂²×C₄×C₅₉	#41	1	2	yes	—
57	C₂²×C₄×C₅₉	#41	1	2	yes	—
58	C₂²×Dic₅₉	—	2	8	no	—
59	C₂²×Dic₅₉	—	4	4	no	—
60	C₂²×Dic₅₉	—	4	4	no	—
61	C₂²×Dic₅₉	—	4	4	no	—
62	C₂²×Dic₅₉	—	4	4	no	—
63	C₂²×Dic₅₉	—	4	4	no	—
64	C₂²×Dic₅₉	—	4	4	no	—
65	C₂×C₈×C₅₉	—	2	2	no	—
66	C₂×C₈×C₅₉	—	2	2	no	—
67	C₂×C₈×C₅₉	—	2	2	no	—
68	C₂×C₈×C₅₉	—	2	2	no	—
69	C₂×C₈×C₅₉	—	2	8	no	—
70	C₂×C₈×C₅₉	—	2	8	no	—
71	C₂×C₈×C₅₉	C₄²	1	4	yes	—
72	C₂×C₈×C₅₉	C₂²⋊C₄	1	4	yes	—
73	C₄²×C₅₉	—	2	2	no	—
74	C₄²×C₅₉	—	2	2	no	—
75	C₄²×C₅₉	—	2	4	no	—
76	C₄⋊Dic₅₉	—	2	8	no	—
77	C₄⋊Dic₅₉	—	2	8	no	—
78	C₄⋊Dic₅₉	—	4	4	no	—
79	C₄⋊Dic₅₉	—	4	4	no	—
80	C₂³×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂³×C₅₉	C₄×D₄	1	2	yes	—
82	C₂³×C₅₉	#121	1	2	yes	—
83	C₂³×C₅₉	#124	1	2	yes	—
84	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
85	C₂³×C₅₉	C₂×SD₁₆	1	2	yes	—
86	C₂³×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
87	C₂³×C₅₉	#135	1	2	yes	—
88	C₂³×C₅₉	#135	1	2	yes	—
89	C₂×C₄×C₅₉	—	2	2	no	—
90	C₂×C₄×C₅₉	—	2	2	no	—
91	C₂×C₄×C₅₉	—	2	2	no	—
92	C₂×C₄×C₅₉	—	2	2	no	—
93	C₂×C₄×C₅₉	—	2	8	no	—
94	C₂×C₄×C₅₉	—	2	4	no	—
95	C₂×C₄×C₅₉	—	2	4	no	—
96	C₂×C₄×C₅₉	#106	1	4	yes	—
97	C₂×C₄×C₅₉	#107	1	4	yes	—
98	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
99	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₉	#117	1	4	yes	—
101	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
102	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
103	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
104	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
105	C₂×C₄×C₅₉	C₄×D₄	1	4	yes	—
106	C₂×C₄×C₅₉	C₄×Q₈	1	2	yes	—
107	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₅₉	#123	1	4	yes	—
109	C₂×C₄×C₅₉	#125	1	2	yes	—
110	C₂×C₄×C₅₉	#126	1	4	yes	—
111	C₂×Dic₅₉	—	4	8	no	—
112	C₂×Dic₅₉	—	4	8	no	—
113	C₂×Dic₅₉	—	4	8	no	—
114	C₂×Dic₅₉	—	8	4	no	—
115	C₂×Dic₅₉	—	8	4	no	—
116	C₂×Dic₅₉	—	8	4	no	—
117	C₂×Dic₅₉	—	8	4	no	—
118	C₈×C₅₉	—	2	4	no	—
119	C₈×C₅₉	—	2	4	no	—
120	#10820	—	59	1	no	maximal, Sylow
121	C₂²×C₅₉	C₄×C₄⋊C₄	1	1	yes	characteristic
122	C₂²×C₅₉	#380	1	1	yes	characteristic
123	C₂²×C₅₉	#382	1	1	yes	characteristic
124	C₂²×C₅₉	#388	1	4	yes	—
125	C₂²×C₅₉	C₂×#106	1	1	yes	characteristic
126	C₂²×C₅₉	C₂×#107	1	1	yes	characteristic
127	C₂²×C₅₉	#427	1	1	yes	characteristic
128	C₂²×C₅₉	#427	1	1	yes	characteristic
129	C₂²×C₅₉	C₄×D₈	1	2	yes	—
130	C₂²×C₅₉	C₄×SD₁₆	1	4	yes	—
131	C₂²×C₅₉	C₄×Q₁₆	1	2	yes	—
132	C₂²×C₅₉	#465	1	4	yes	—
133	C₂²×C₅₉	#466	1	2	yes	—
134	C₂²×C₅₉	#467	1	2	yes	—
135	C₂²×C₅₉	#486	1	2	yes	—
136	C₂²×C₅₉	#487	1	2	yes	—
137	C₂²×C₅₉	#488	1	4	yes	—
138	C₄×C₅₉	—	2	4	no	—
139	C₄×C₅₉	—	2	4	no	—
140	C₄×C₅₉	#376	1	1	yes	characteristic
141	C₄×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
142	C₄×C₅₉	#381	1	1	yes	characteristic
143	C₄×C₅₉	#383	1	1	yes	characteristic
144	C₄×C₅₉	#387	1	4	yes	—
145	Dic₅₉	—	8	8	no	—
146	C₂²×C₄×C₈	D₅₉	1	1	yes	characteristic
147	C₂×#340	—	59	2	no	—
148	C₂×#340	—	59	2	no	—
149	C₄⋊#95	—	59	2	no	—
150	C₂×C₅₉	#1434	1	1	yes	characteristic
151	C₂×C₅₉	C₄×#106	1	1	yes	characteristic
152	C₂×C₅₉	C₄×#107	1	1	yes	characteristic
153	C₂×C₅₉	#1807	1	1	yes	characteristic
154	C₂×C₅₉	#1808	1	1	yes	characteristic
155	C₂×C₅₉	#1834	1	1	yes	characteristic
156	C₂×C₅₉	#1835	1	1	yes	characteristic
157	C₂×C₅₉	#1933	1	4	yes	—
158	C₂×C₅₉	#1934	1	4	yes	—
159	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
160	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
161	C₂²×C₄⋊C₄	—	59	4	no	—
162	C₂²×C₄²	C₂×D₅₉	1	1	yes	characteristic
163	C₂×C₄×C₈	—	2	4	no	—
164	C₂×C₄×C₈	Dic₅₉	1	4	yes	complemented
165	C₂×#95	—	118	2	no	—
166	C₂×#95	—	118	2	no	—
167	#340	—	59	8	no	—
168	#340	—	118	4	no	—
169	C₅₉	#10820	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₄	—	118	4	no	—
171	C₂³×C₄	D₂₃₆	1	1	yes	characteristic
172	C₂³×C₄	D₂₃₆	1	1	yes	characteristic
173	C₂³×C₄	C₂²×D₅₉	1	1	yes	characteristic
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	C₄×D₅₉	1	4	yes	—
179	C₂²×C₈	C₂×Dic₅₉	1	2	yes	—
180	C₂²×C₈	C₂×Dic₅₉	1	2	yes	—
181	C₂²×C₈	C₂²⋊D₅₉	1	4	yes	—
182	C₂×C₄⋊C₄	—	59	8	no	—
183	C₂×C₄⋊C₄	—	118	4	no	—
184	C₂×C₄⋊C₄	—	118	4	no	—
185	C₂×C₄⋊C₄	—	118	4	no	—
186	C₂×C₄⋊C₄	—	118	4	no	—
187	C₂×C₄⋊C₄	—	118	4	no	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	—	2	2	no	—
190	C₂×C₄²	Dic₁₁₈	1	2	yes	—
191	C₂×C₄²	C₂×Dic₅₉	1	2	yes	—
192	C₄×C₈	—	2	8	no	—
193	#95	—	236	4	no	—
194	#95	—	236	4	no	—
195	C₂⁴	C₂×D₂₃₆	1	1	yes	characteristic
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	118	8	no	—
201	C₂²×C₄	—	236	4	no	—
202	C₂²×C₄	—	236	4	no	—
203	C₂²×C₄	—	236	4	no	—
204	C₂²×C₄	—	236	4	no	—
205	C₂²×C₄	—	236	4	no	—
206	C₂²×C₄	—	236	4	no	—
207	C₂²×C₄	#11745660	1	4	yes	—
208	C₂²×C₄	D₄₇₂	1	2	yes	—
209	C₂²×C₄	Dic₂₃₆	1	2	yes	—
210	C₂²×C₄	C₄⋊Dic₅₉	1	2	yes	—
211	C₂²×C₄	C₄⋊Dic₅₉	1	2	yes	—
212	C₂²×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₄×D₅₉	1	2	yes	—
214	C₂²×C₄	#11745685	1	2	yes	—
215	C₂²×C₄	#11745685	1	2	yes	—
216	C₂²×C₄	#11745685	1	2	yes	—
217	C₂²×C₄	#11745685	1	2	yes	—
218	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	8	no	—
225	C₂×C₈	—	2	8	no	—
226	C₂×C₈	C₄×Dic₅₉	1	4	yes	—
227	C₂×C₈	C₂²⋊Dic₅₉	1	4	yes	—
228	C₄⋊C₄	—	118	8	no	—
229	C₄⋊C₄	—	118	8	no	—
230	C₄⋊C₄	—	236	4	no	—
231	C₄⋊C₄	—	236	4	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	2	no	—
234	C₄²	—	2	4	no	—
235	C₂³	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
236	C₂³	C₄×D₂₃₆	1	2	yes	—
237	C₂³	#14273701	1	2	yes	—
238	C₂³	#14273715	1	2	yes	—
239	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
240	C₂³	C₂×#11745660	1	2	yes	—
241	C₂³	C₂×Dic₂₃₆	1	1	yes	characteristic
242	C₂³	#14273749	1	2	yes	—
243	C₂³	#14273749	1	2	yes	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	472	4	no	—
252	C₂×C₄	—	472	4	no	—
253	C₂×C₄	—	472	4	no	—
254	C₂×C₄	—	472	4	no	—
255	C₂×C₄	—	236	8	no	—
256	C₂×C₄	—	236	8	no	—
257	C₂×C₄	—	236	8	no	—
258	C₂×C₄	#14273659	1	4	yes	—
259	C₂×C₄	#14273661	1	4	yes	—
260	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
261	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
262	C₂×C₄	#14273684	1	4	yes	—
263	C₂×C₄	#14273686	1	1	yes	characteristic
264	C₂×C₄	#14273686	1	1	yes	characteristic
265	C₂×C₄	#14273686	1	1	yes	characteristic
266	C₂×C₄	#14273686	1	1	yes	characteristic
267	C₂×C₄	C₄×C₂²⋊D₅₉	1	4	yes	—
268	C₂×C₄	C₄×Dic₁₁₈	1	2	yes	—
269	C₂×C₄	#14273706	1	2	yes	—
270	C₂×C₄	#14273710	1	4	yes	—
271	C₂×C₄	#14273719	1	2	yes	—
272	C₂×C₄	#14273721	1	4	yes	—
273	C₈	—	2	4	no	—
274	C₈	—	2	4	no	—
275	C₂²	C₄×C₄⋊Dic₅₉	1	1	yes	characteristic
276	C₂²	#16837262	1	1	yes	characteristic
277	C₂²	#16837268	1	1	yes	characteristic
278	C₂²	#16837293	1	4	yes	—
279	C₂²	C₂×#14273659	1	1	yes	characteristic
280	C₂²	C₂×#14273661	1	1	yes	characteristic
281	C₂²	#16837440	1	1	yes	characteristic
282	C₂²	#16837440	1	1	yes	characteristic
283	C₂²	C₄×D₄₇₂	1	2	yes	—
284	C₂²	C₄×#11745660	1	4	yes	—
285	C₂²	C₄×Dic₂₃₆	1	2	yes	—
286	C₂²	#16837631	1	4	yes	—
287	C₂²	#16837638	1	2	yes	—
288	C₂²	#16837643	1	2	yes	—
289	C₂²	#16837749	1	2	yes	—
290	C₂²	#16837753	1	2	yes	—
291	C₂²	#16837757	1	4	yes	—
292	C₄	—	2	4	no	—
293	C₄	—	2	4	no	—
294	C₄	—	472	8	no	—
295	C₄	#16837248	1	1	yes	characteristic
296	C₄	C₄×C₂²⋊Dic₅₉	1	1	yes	characteristic
297	C₄	#16837265	1	1	yes	characteristic
298	C₄	#16837270	1	1	yes	characteristic
299	C₄	#16837286	1	4	yes	—
300	C₂	#21422323	1	1	yes	characteristic
301	C₂	C₄×#14273659	1	1	yes	characteristic
302	C₂	C₄×#14273661	1	1	yes	characteristic
303	C₂	#21424121	1	1	yes	characteristic
304	C₂	#21424125	1	1	yes	characteristic
305	C₂	#21424224	1	1	yes	characteristic
306	C₂	#21424227	1	1	yes	characteristic
307	C₂	#21424772	1	4	yes	—
308	C₂	#21424779	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₉
Fitting	C₂²×C₄×C₈×C₅₉	C₂
socle, prime radical	C₂⁴×C₅₉	C₂×D₄
cosocle	C₂³×C₄×C₅₉	C₂³
hypercenter	C₂²×C₄×C₈	D₅₉
hypocenter	C₅₉	#10820
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
9. #16837165	C₄	semidirect
14. C₂×C₄×C₈×C₅₉	C₄	semidirect
164. C₂×C₄×C₈	Dic₅₉	semidirect
169. C₅₉	#10820	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837165	2
#21422326	2
C₂²×C₄×C₈×C₅₉	2
#10820	59