Subgroups — #28744235

Order 15104.

526 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21434661	C₂	1	2	yes	maximal, complemented
2	#21426373	C₂	1	1	yes	maximal, characteristic, complemented
3	#21426354	C₂	1	1	yes	maximal, characteristic, complemented
4	#21426326	C₂	1	2	yes	maximal, complemented
5	#21426190	C₂	1	2	yes	maximal, complemented
6	#21426187	C₂	1	1	yes	maximal, characteristic, complemented
7	#21425954	C₂	1	1	yes	maximal, characteristic, complemented
8	#21425951	C₂	1	2	yes	maximal, complemented
9	D₁₆×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₆×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₉×#2215	C₂	1	1	yes	maximal, characteristic
12	#16837933	—	2	2	no	—
13	C₂×Dic₄₇₂	C₂²	1	1	yes	characteristic
14	C₂×#14273671	C₂²	1	2	yes	—
15	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
16	#16837803	C₂²	1	2	yes	—
17	#16837785	C₂²	1	2	yes	—
18	#16837755	—	2	2	no	—
19	#16837643	C₂²	1	2	yes	—
20	#16837629	—	2	2	no	—
21	#16837496	C₂²	1	2	yes	—
22	#16837493	—	2	2	no	—
23	D₈×Dic₅₉	C₂²	1	2	yes	—
24	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
25	#16837229	C₂²	1	2	yes	—
26	#16837228	C₂²	1	1	yes	characteristic
27	#16837207	C₂²	1	2	yes	—
28	#16837205	C₂²	1	2	yes	—
29	#16837203	C₂²	1	2	yes	—
30	C₁₆×Dic₅₉	C₂²	1	1	yes	characteristic
31	C₁₆×Dic₅₉	C₂²	1	1	yes	characteristic
32	C₁₆×Dic₅₉	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₅₉	C₂²	1	1	yes	characteristic
34	C₂×D₁₆×C₅₉	C₂²	1	1	yes	characteristic
35	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
36	C₅₉×#466	C₂²	1	2	yes	—
37	C₅₉×#366	C₂²	1	1	yes	characteristic
38	C₅₉×#357	C₂²	1	2	yes	—
39	C₂×Dic₂₃₆	C₂³	1	2	yes	—
40	C₂×#11745670	—	2	2	no	—
41	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
42	#14273720	—	4	2	no	—
43	#14273718	—	2	2	no	—
44	#14273714	—	2	2	no	—
45	#14273708	—	4	2	no	—
46	#14273706	—	2	2	no	—
47	#14273692	—	2	2	no	—
48	D₄×Dic₅₉	—	2	2	no	—
49	D₄×Dic₅₉	—	4	2	no	—
50	#14273683	—	4	2	no	—
51	C₂×C₄×Dic₅₉	D₄	1	1	yes	characteristic
52	Dic₄₇₂	—	2	2	no	—
53	#14273671	—	2	4	no	—
54	C₂×C₅₉⋊C₁₆	—	2	2	no	—
55	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
56	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
57	#14273661	C₂³	1	2	yes	—
58	#14273658	—	2	2	no	—
59	#14273652	—	2	2	no	—
60	#14273650	—	2	2	no	—
61	#14273648	—	2	2	no	—
62	C₈×Dic₅₉	D₄	1	1	yes	characteristic
63	C₈×Dic₅₉	D₄	1	1	yes	characteristic
64	C₈×Dic₅₉	C₂³	1	1	yes	characteristic
65	C₈×Dic₅₉	C₂³	1	1	yes	characteristic
66	C₂×D₈×C₅₉	C₂³	1	2	yes	—
67	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
68	#121×C₅₉	—	2	2	no	—
69	D₁₆×C₅₉	—	2	2	no	—
70	C₂×C₁₆×C₅₉	—	2	2	no	—
71	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
72	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
73	#107×C₅₉	C₂³	1	2	yes	—
74	#102×C₅₉	—	2	2	no	—
75	C₂×D₄×C₅₉	—	2	2	no	—
76	C₂×D₄×C₅₉	—	4	2	no	—
77	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×Dic₅₉	—	4	2	no	—
79	C₂²×Dic₅₉	D₈	1	1	yes	characteristic
80	C₂²×Dic₅₉	D₈	1	1	yes	characteristic
81	C₂×Dic₁₁₈	—	2	2	no	—
82	D₈×C₅₉	—	2	2	no	—
83	D₈×C₅₉	—	2	2	no	—
84	C₂×C₈×C₅₉	—	2	1	no	—
85	C₂×C₈×C₅₉	—	2	1	no	—
86	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
88	C₄⋊C₄×C₅₉	—	2	2	no	—
89	C₂²⋊C₄×C₅₉	—	4	2	no	—
90	C₂²⋊Dic₅₉	—	4	2	no	—
91	C₂²⋊Dic₅₉	—	4	2	no	—
92	C₂²⋊Dic₅₉	—	4	2	no	—
93	C₂²⋊Dic₅₉	—	4	2	no	—
94	#11745670	—	4	4	no	—
95	C₄⋊Dic₅₉	—	2	2	no	—
96	C₄⋊Dic₅₉	—	4	2	no	—
97	Dic₅₉⋊C₄	—	4	2	no	—
98	Dic₅₉⋊C₄	—	4	2	no	—
99	Dic₅₉⋊C₄	—	4	2	no	—
100	C₄×Dic₅₉	—	4	2	no	—
101	C₄×Dic₅₉	D₈	1	1	yes	characteristic
102	C₄×Dic₅₉	D₈	1	1	yes	characteristic
103	C₄×Dic₅₉	C₂×D₄	1	1	yes	characteristic
104	C₄×Dic₅₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₅₉⋊C₈	—	2	2	no	—
106	C₂×C₅₉⋊C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₅₉⋊C₈	C₂×D₄	1	1	yes	characteristic
108	Dic₂₃₆	—	2	2	no	—
109	Dic₂₃₆	—	2	2	no	—
110	C₁₆×C₅₉	—	2	1	no	—
111	C₁₆×C₅₉	#41	1	2	yes	—
112	C₅₉⋊C₁₆	—	2	1	no	—
113	C₅₉⋊C₁₆	#41	1	2	yes	—
114	C₂³×C₅₉	—	4	2	no	—
115	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
116	D₄×C₅₉	—	4	2	no	—
117	D₄×C₅₉	—	4	2	no	—
118	D₄×C₅₉	—	8	4	no	—
119	C₂×C₄×C₅₉	—	2	1	no	—
120	C₂×C₄×C₅₉	—	2	1	no	—
121	C₂×C₄×C₅₉	—	4	2	no	—
122	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
124	C₂×Dic₅₉	—	2	2	no	—
125	C₂×Dic₅₉	—	2	2	no	—
126	C₂×Dic₅₉	—	4	2	no	—
127	C₂×Dic₅₉	—	4	2	no	—
128	C₂×Dic₅₉	—	8	4	no	—
129	C₂×Dic₅₉	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₅₉	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₅₉	C₂×D₈	1	1	yes	characteristic
132	C₂×Dic₅₉	C₂×D₈	1	1	yes	characteristic
133	Dic₁₁₈	—	4	2	no	—
134	Dic₁₁₈	—	4	2	no	—
135	C₈×C₅₉	—	2	1	no	—
136	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
137	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
138	C₅₉⋊C₈	—	2	1	no	—
139	C₅₉⋊C₈	#135	1	2	yes	—
140	#33735	—	59	1	no	maximal, Sylow
141	C₂²×C₅₉	—	2	1	no	—
142	C₂²×C₅₉	—	2	1	no	—
143	C₂²×C₅₉	—	8	2	no	—
144	C₂²×C₅₉	—	8	2	no	—
145	C₂²×C₅₉	—	8	2	no	—
146	C₂²×C₅₉	#508	1	2	yes	—
147	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
148	C₄×C₅₉	—	2	1	no	—
149	C₄×C₅₉	—	8	2	no	—
150	C₄×C₅₉	#532	1	1	yes	characteristic
151	C₄×C₅₉	C₂×#135	1	1	yes	characteristic
152	Dic₅₉	—	2	1	no	—
153	Dic₅₉	—	2	1	no	—
154	Dic₅₉	—	2	1	no	—
155	Dic₅₉	—	8	2	no	—
156	Dic₅₉	—	8	2	no	—
157	Dic₅₉	#508	1	2	yes	—
158	#3168	—	59	2	no	—
159	#2253	—	59	1	no	—
160	#2249	—	59	1	no	—
161	#2242	—	59	2	no	—
162	#2217	—	59	2	no	—
163	#2216	—	59	1	no	—
164	#2215	D₅₉	1	1	yes	characteristic
165	C₄×Q₃₂	—	59	1	no	—
166	C₄×SD₃₂	—	59	2	no	—
167	C₄×D₁₆	—	59	1	no	—
168	C₂×C₄×C₁₆	—	59	1	no	—
169	C₂×C₅₉	—	2	2	no	—
170	C₂×C₅₉	—	16	2	no	—
171	C₂×C₅₉	#3169	1	1	yes	characteristic
172	C₂×C₅₉	C₂×#508	1	1	yes	characteristic
173	C₂×C₅₉	C₂×#508	1	1	yes	characteristic
174	#529	—	118	2	no	—
175	C₂×Q₃₂	—	59	1	no	—
176	C₂×SD₃₂	—	59	2	no	—
177	C₂×D₁₆	C₂×D₅₉	1	1	yes	characteristic
178	C₂²×C₁₆	—	59	1	no	—
179	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
180	#499	—	59	2	no	—
181	#495	—	59	2	no	—
182	#488	—	118	2	no	—
183	#467	—	59	2	no	—
184	#466	C₂×D₅₉	1	2	yes	—
185	#465	—	118	2	no	—
186	C₄×Q₁₆	—	59	2	no	—
187	C₄×SD₁₆	—	118	2	no	—
188	C₄×D₈	—	59	2	no	—
189	C₂×C₄×C₈	—	59	1	no	—
190	#367	—	59	2	no	—
191	#366	—	59	1	no	—
192	#366	C₂×D₅₉	1	1	yes	characteristic
193	#358	—	59	2	no	—
194	#358	—	59	2	no	—
195	#357	—	59	2	no	—
196	#357	C₂×D₅₉	1	2	yes	—
197	C₄×C₁₆	—	59	1	no	—
198	C₄×C₁₆	—	59	1	no	—
199	C₄×C₁₆	—	59	1	no	—
200	C₄×C₁₆	—	59	1	no	—
201	C₅₉	#33735	1	1	yes	characteristic, complemented, Sylow
202	C₂×Q₁₆	—	59	2	no	—
203	C₂×SD₁₆	—	118	2	no	—
204	C₂×D₈	C₂²×D₅₉	1	2	yes	—
205	C₂²×C₈	—	59	1	no	—
206	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
207	#126	—	236	2	no	—
208	#125	—	118	2	no	—
209	#124	—	118	2	no	—
210	#123	—	236	2	no	—
211	C₂²⋊Q₈	—	118	2	no	—
212	#121	—	2	2	no	—
213	C₄×Q₈	—	118	2	no	—
214	C₄×D₄	—	118	2	no	—
215	C₄×D₄	—	236	2	no	—
216	#117	—	236	2	no	—
217	C₂×C₄²	—	59	1	no	—
218	Q₃₂	—	118	2	no	—
219	SD₃₂	—	118	4	no	—
220	D₁₆	—	2	2	no	—
221	C₂×C₁₆	—	2	2	no	—
222	C₂×C₁₆	—	59	1	no	—
223	C₂×C₁₆	—	59	1	no	—
224	C₂×C₁₆	—	118	2	no	—
225	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
226	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
227	#107	—	59	2	no	—
228	#107	C₂²×D₅₉	1	2	yes	—
229	#106	—	118	2	no	—
230	#103	—	118	2	no	—
231	#103	—	118	2	no	—
232	#102	—	2	2	no	—
233	#102	—	118	2	no	—
234	C₄×C₈	—	59	1	no	—
235	C₄×C₈	—	59	1	no	—
236	C₄×C₈	—	59	1	no	—
237	C₄×C₈	—	59	1	no	—
238	C₂×Q₈	—	118	2	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	4	2	no	—
241	C₂²×C₄	—	59	1	no	—
242	C₂²×C₄	—	59	1	no	—
243	C₂²×C₄	—	236	2	no	—
244	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
245	Q₁₆	—	118	2	no	—
246	Q₁₆	—	118	2	no	—
247	SD₁₆	—	236	4	no	—
248	D₈	—	2	2	no	—
249	D₈	—	2	2	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	59	1	no	—
253	C₂×C₈	—	59	1	no	—
254	C₂×C₈	—	118	2	no	—
255	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
256	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	118	2	no	—
259	C₄⋊C₄	—	236	2	no	—
260	C₄⋊C₄	—	236	2	no	—
261	C₄⋊C₄	—	236	2	no	—
262	C₄⋊C₄	—	236	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	—	236	2	no	—
265	C₂²⋊C₄	—	236	2	no	—
266	C₂²⋊C₄	—	236	2	no	—
267	C₂²⋊C₄	—	236	2	no	—
268	C₄²	—	59	1	no	—
269	C₄²	—	59	1	no	—
270	C₄²	—	59	1	no	—
271	C₄²	—	59	1	no	—
272	C₄²	—	236	2	no	—
273	C₁₆	—	2	1	no	—
274	C₁₆	—	59	2	no	—
275	C₁₆	—	118	1	no	—
276	C₁₆	#11745687	1	2	yes	—
277	C₂³	—	4	2	no	—
278	C₂³	D₈×D₅₉	1	1	yes	characteristic
279	Q₈	—	236	2	no	—
280	Q₈	—	236	2	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	8	4	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	59	1	no	—
288	C₂×C₄	—	59	1	no	—
289	C₂×C₄	—	59	1	no	—
290	C₂×C₄	—	59	1	no	—
291	C₂×C₄	—	118	2	no	—
292	C₂×C₄	—	118	2	no	—
293	C₂×C₄	—	472	4	no	—
294	C₂×C₄	—	236	2	no	—
295	C₂×C₄	—	236	2	no	—
296	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
297	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	59	2	no	—
300	C₈	—	118	1	no	—
301	C₈	#41×D₅₉	1	1	yes	characteristic
302	C₈	C₂×#11745687	1	1	yes	characteristic
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	8	2	no	—
306	C₂²	—	8	2	no	—
307	C₂²	—	8	2	no	—
308	C₂²	#16837831	1	2	yes	—
309	C₂²	C₂×D₈×D₅₉	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	8	2	no	—
312	C₄	—	59	2	no	—
313	C₄	—	118	1	no	—
314	C₄	—	118	1	no	—
315	C₄	—	118	1	no	—
316	C₄	—	472	2	no	—
317	C₄	—	472	2	no	—
318	C₄	#16837947	1	1	yes	characteristic
319	C₄	#135×D₅₉	1	1	yes	characteristic
320	C₂	—	2	2	no	—
321	C₂	—	16	2	no	—
322	C₂	#21434670	1	1	yes	characteristic
323	C₂	#508×D₅₉	1	1	yes	characteristic
324	C₂	C₂×#16837831	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₉
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting, prime radical	C₅₉×#2215	C₂
socle	C₂²×C₅₉	C₂²×D₈
hypercenter	#2215	D₅₉
hypocenter	C₅₉	#33735
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2215

Complements

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

N	H ≅ G/N	type
1. #21434661	C₂	semidirect
2. #21426373	C₂	semidirect
3. #21426354	C₂	semidirect
4. #21426326	C₂	semidirect
5. #21426190	C₂	semidirect
6. #21426187	C₂	semidirect
7. #21425954	C₂	semidirect
8. #21425951	C₂	semidirect
9. D₁₆×Dic₅₉	C₂	semidirect
10. C₂×C₁₆×Dic₅₉	C₂	semidirect
201. C₅₉	#33735	semidirect

Maximal subgroups

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

subgroup	index
#21434661	2
#21426373	2
#21426354	2
#21426326	2
#21426190	2
#21426187	2
#21425954	2
#21425951	2
D₁₆×Dic₅₉	2
C₂×C₁₆×Dic₅₉	2
C₅₉×#2215	2
#33735	59