Subgroups — #28482003

Order 15104.

384 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14273659	C₂	1	1	yes	maximal, characteristic, complemented
2	#21422859	C₂	1	1	yes	maximal, characteristic, complemented
3	#21422621	C₂	1	2	yes	maximal
4	#21421953	C₂	1	2	yes	maximal, complemented
5	C₅₉×#2966	C₂	1	1	yes	maximal, characteristic
6	#16837459	C₂²	1	2	yes	—
7	C₂×#14273659	—	2	1	no	—
8	C₄×C₄⋊Dic₅₉	C₂²	1	2	yes	—
9	#16837165	—	2	1	no	—
10	#16837155	—	2	1	no	—
11	#16837153	—	2	1	no	—
12	#16837139	—	2	2	no	—
13	#16837136	—	2	2	no	—
14	C₅₉×#529	C₂²	1	1	yes	characteristic
15	C₅₉×#446	C₄	1	1	yes	characteristic, complemented
16	C₈×Q₈×C₅₉	C₄	1	1	yes	characteristic, complemented
17	C₅₉×#436	C₄	1	1	yes	characteristic, complemented
18	C₅₉×#435	—	2	1	no	—
19	C₈×D₄×C₅₉	—	2	1	no	—
20	C₈×D₄×C₅₉	C₄	1	1	yes	characteristic, complemented
21	C₅₉×#433	—	2	1	no	—
22	C₅₉×#432	C₂²	1	1	yes	characteristic
23	C₅₉×#405	—	2	1	no	—
24	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Dic₅₉	—	2	2	no	—
26	C₂×C₄×Dic₅₉	—	2	2	no	—
27	#14273659	—	2	4	no	—
28	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
29	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
30	#14273638	—	2	2	no	—
31	C₅₉⋊#95	—	2	2	no	—
32	C₂×M₁₆×C₅₉	—	2	1	no	—
33	C₂²×C₈×C₅₉	—	2	1	no	—
34	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
36	#126×C₅₉	—	2	1	no	—
37	#125×C₅₉	Q₈	1	1	yes	characteristic
38	#124×C₅₉	Q₈	1	1	yes	characteristic
39	#123×C₅₉	—	2	1	no	—
40	C₂²⋊Q₈×C₅₉	D₄	1	1	yes	characteristic
41	#121×C₅₉	D₄	1	1	yes	characteristic
42	C₄×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
43	C₄×D₄×C₅₉	—	2	1	no	—
44	C₄×D₄×C₅₉	C₂×C₄	1	1	yes	characteristic
45	#117×C₅₉	—	2	1	no	—
46	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₅₉	—	2	1	no	—
48	C₄⋊C₈×C₅₉	—	2	1	no	—
49	C₄⋊C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
50	C₄⋊C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
51	C₂²⋊C₈×C₅₉	—	2	1	no	—
52	C₂²⋊C₈×C₅₉	—	2	1	no	—
53	C₂²⋊C₈×C₅₉	—	2	1	no	—
54	#97×C₅₉	—	2	1	no	—
55	#97×C₅₉	—	2	1	no	—
56	C₄×C₈×C₅₉	—	2	1	no	—
57	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
60	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
61	C₂×Q₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
62	C₂×D₄×C₅₉	—	2	1	no	—
63	C₂×D₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₉	—	2	1	no	—
65	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
68	C₂²×Dic₅₉	—	4	2	no	—
69	M₁₆×C₅₉	—	4	2	no	—
70	C₂×C₈×C₅₉	—	2	2	no	—
71	C₂×C₈×C₅₉	—	2	1	no	—
72	C₂×C₈×C₅₉	—	2	1	no	—
73	C₂×C₈×C₅₉	—	4	1	no	—
74	C₂×C₈×C₅₉	—	4	2	no	—
75	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₄⋊C₄×C₅₉	—	2	1	no	—
80	C₄⋊C₄×C₅₉	—	2	1	no	—
81	C₄⋊C₄×C₅₉	—	2	1	no	—
82	C₄⋊C₄×C₅₉	—	2	1	no	—
83	C₄⋊C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
84	C₄⋊C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
85	C₂²⋊C₄×C₅₉	—	2	1	no	—
86	C₂²⋊C₄×C₅₉	—	2	1	no	—
87	C₂²⋊C₄×C₅₉	—	2	1	no	—
88	C₂²⋊C₄×C₅₉	—	2	1	no	—
89	C₂²⋊C₄×C₅₉	—	2	1	no	—
90	C₄²×C₅₉	—	2	1	no	—
91	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
94	C₄²×C₅₉	#41	1	1	yes	characteristic
95	C₄⋊Dic₅₉	—	2	4	no	—
96	C₄⋊Dic₅₉	—	4	2	no	—
97	C₄×Dic₅₉	—	8	2	no	—
98	C₂×C₅₉⋊C₈	—	4	2	no	—
99	C₂³×C₅₉	—	2	1	no	—
100	C₂³×C₅₉	#121	1	1	yes	characteristic
101	Q₈×C₅₉	—	4	1	no	—
102	D₄×C₅₉	—	4	1	no	—
103	D₄×C₅₉	—	4	2	no	—
104	C₂×C₄×C₅₉	—	2	2	no	—
105	C₂×C₄×C₅₉	—	2	1	no	—
106	C₂×C₄×C₅₉	—	2	1	no	—
107	C₂×C₄×C₅₉	—	2	1	no	—
108	C₂×C₄×C₅₉	—	2	1	no	—
109	C₂×C₄×C₅₉	—	2	1	no	—
110	C₂×C₄×C₅₉	—	4	1	no	—
111	C₂×C₄×C₅₉	—	4	1	no	—
112	C₂×C₄×C₅₉	—	4	1	no	—
113	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
116	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
117	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×Dic₅₉	—	4	4	no	—
120	C₂×Dic₅₉	—	8	2	no	—
121	C₂×Dic₅₉	—	8	2	no	—
122	C₈×C₅₉	—	2	1	no	—
123	C₈×C₅₉	—	2	1	no	—
124	C₈×C₅₉	—	2	1	no	—
125	C₈×C₅₉	—	4	1	no	—
126	C₈×C₅₉	—	4	1	no	—
127	C₈×C₅₉	#104	1	2	yes	—
128	C₅₉⋊C₈	—	8	2	no	—
129	#8696	—	59	1	no	maximal, Sylow
130	C₂²×C₅₉	—	2	1	no	—
131	C₂²×C₅₉	—	2	1	no	—
132	C₂²×C₅₉	—	4	1	no	—
133	C₂²×C₅₉	—	4	1	no	—
134	C₂²×C₅₉	—	4	1	no	—
135	C₂²×C₅₉	#380	1	1	yes	characteristic
136	C₂²×C₅₉	#471	1	2	yes	—
137	C₄×C₅₉	—	2	1	no	—
138	C₄×C₅₉	—	2	1	no	—
139	C₄×C₅₉	—	2	1	no	—
140	C₄×C₅₉	—	2	1	no	—
141	C₄×C₅₉	—	2	1	no	—
142	C₄×C₅₉	—	4	1	no	—
143	C₄×C₅₉	—	4	1	no	—
144	C₄×C₅₉	—	4	1	no	—
145	C₄×C₅₉	C₂×#104	1	1	yes	characteristic
146	C₄×C₅₉	#421	1	1	yes	characteristic
147	Dic₅₉	—	8	4	no	—
148	#2966	D₅₉	1	1	yes	characteristic
149	C₄×#106	—	59	1	no	—
150	#1568	—	59	1	no	—
151	#1509	—	59	2	no	—
152	#1285	—	59	2	no	—
153	C₂×C₅₉	—	2	2	no	—
154	C₂×C₅₉	—	8	1	no	—
155	C₂×C₅₉	#1806	1	1	yes	characteristic
156	C₂×C₅₉	#1812	1	1	yes	characteristic
157	C₂×C₅₉	#1815	1	1	yes	characteristic
158	#529	C₂×D₅₉	1	1	yes	characteristic
159	#446	Dic₅₉	1	1	yes	characteristic, complemented
160	C₈×Q₈	Dic₅₉	1	1	yes	characteristic, complemented
161	#436	Dic₅₉	1	1	yes	characteristic, complemented
162	#435	—	2	1	no	—
163	C₈×D₄	—	2	1	no	—
164	C₈×D₄	Dic₅₉	1	1	yes	characteristic, complemented
165	#433	—	2	1	no	—
166	#432	—	59	2	no	—
167	#432	C₂×D₅₉	1	1	yes	characteristic
168	C₂×#106	—	118	1	no	—
169	#405	—	2	1	no	—
170	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
171	C₄×C₄⋊C₄	—	59	2	no	—
172	#340	—	118	1	no	—
173	#335	—	118	1	no	—
174	#334	—	118	1	no	—
175	#328	—	118	2	no	—
176	#327	—	118	2	no	—
177	C₅₉	#8696	1	1	yes	characteristic, complemented, Sylow
178	C₂×M₁₆	—	2	1	no	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
181	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
182	#126	—	2	1	no	—
183	#125	Dic₁₁₈	1	1	yes	characteristic
184	#124	Dic₁₁₈	1	1	yes	characteristic
185	#123	—	2	1	no	—
186	C₂²⋊Q₈	D₂₃₆	1	1	yes	characteristic
187	#121	D₂₃₆	1	1	yes	characteristic
188	C₄×Q₈	C₂×Dic₅₉	1	1	yes	characteristic
189	C₄×D₄	—	2	1	no	—
190	C₄×D₄	C₂×Dic₅₉	1	1	yes	characteristic
191	#117	—	2	1	no	—
192	C₂×C₄⋊C₄	—	118	2	no	—
193	C₂×C₄²	—	118	2	no	—
194	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
195	#106	—	118	4	no	—
196	C₄⋊C₈	—	2	1	no	—
197	C₄⋊C₈	—	2	1	no	—
198	C₄⋊C₈	—	118	2	no	—
199	C₄⋊C₈	C₂×Dic₅₉	1	1	yes	characteristic
200	C₄⋊C₈	C₂×Dic₅₉	1	1	yes	characteristic
201	C₂²⋊C₈	—	2	1	no	—
202	C₂²⋊C₈	—	2	1	no	—
203	C₂²⋊C₈	—	2	1	no	—
204	C₂²⋊C₈	—	118	2	no	—
205	#97	—	2	1	no	—
206	#97	—	2	1	no	—
207	#97	—	118	2	no	—
208	C₄×C₈	—	2	1	no	—
209	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
210	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
211	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
212	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
213	#95	—	118	2	no	—
214	C₂×Q₈	C₄⋊Dic₅₉	1	1	yes	characteristic
215	C₂×D₄	—	2	1	no	—
216	C₂×D₄	C₄⋊Dic₅₉	1	1	yes	characteristic
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	236	2	no	—
219	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
220	C₂²×C₄	#11745685	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
222	M₁₆	—	4	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	236	2	no	—
229	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
230	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
231	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
232	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	118	4	no	—
238	C₄⋊C₄	—	236	2	no	—
239	C₄⋊C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
240	C₄⋊C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
241	C₂²⋊C₄	—	2	1	no	—
242	C₂²⋊C₄	—	2	1	no	—
243	C₂²⋊C₄	—	2	1	no	—
244	C₂²⋊C₄	—	2	1	no	—
245	C₂²⋊C₄	—	2	1	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	472	2	no	—
248	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
249	C₄²	#11745685	1	1	yes	characteristic
250	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
251	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
252	C₂³	—	2	1	no	—
253	C₂³	#14273701	1	1	yes	characteristic
254	Q₈	—	4	1	no	—
255	D₄	—	4	1	no	—
256	D₄	—	4	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	472	2	no	—
267	C₂×C₄	—	472	2	no	—
268	C₂×C₄	—	236	4	no	—
269	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
270	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
271	C₂×C₄	#14273686	1	1	yes	characteristic
272	C₂×C₄	#14273701	1	1	yes	characteristic
273	C₂×C₄	#14273706	1	1	yes	characteristic
274	C₂×C₄	#14273706	1	1	yes	characteristic
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	2	1	no	—
278	C₈	—	4	1	no	—
279	C₈	—	4	1	no	—
280	C₈	—	472	2	no	—
281	C₈	#14273654	1	2	yes	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	1	no	—
287	C₂²	#16837262	1	1	yes	characteristic
288	C₂²	#16837665	1	2	yes	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	—	4	1	no	—
297	C₄	—	472	4	no	—
298	C₄	C₂×#14273654	1	1	yes	characteristic
299	C₄	#16837419	1	1	yes	characteristic
300	C₂	—	2	2	no	—
301	C₂	—	8	1	no	—
302	C₂	#21424116	1	1	yes	characteristic
303	C₂	#21424142	1	1	yes	characteristic
304	C₂	#21424155	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16837262
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#2966	C₂
socle	C₂²×C₅₉	#380
prime radical	#121×C₅₉	D₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#2966	D₅₉
hypocenter	C₅₉	#8696
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966

Complements

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

N	H ≅ G/N	type
1. C₄×#14273659	C₂	semidirect
2. #21422859	C₂	semidirect
4. #21421953	C₂	semidirect
15. C₅₉×#446	C₄	semidirect
16. C₈×Q₈×C₅₉	C₄	semidirect
17. C₅₉×#436	C₄	semidirect
20. C₈×D₄×C₅₉	C₄	semidirect
159. #446	Dic₅₉	semidirect
160. C₈×Q₈	Dic₅₉	semidirect
161. #436	Dic₅₉	semidirect
164. C₈×D₄	Dic₅₉	semidirect
177. C₅₉	#8696	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14273659	2
#21422859	2
#21422621	2
#21421953	2
C₅₉×#2966	2
#8696	59