Subgroups — #28744239

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	#21434663	C₂	1	2	yes	maximal, complemented
2	#21426375	C₂	1	1	yes	maximal, characteristic, complemented
3	#21426327	C₂	1	2	yes	maximal, complemented
4	#21426189	C₂	1	2	yes	maximal, complemented
5	#21426184	C₂	1	1	yes	maximal, characteristic, complemented
6	#21426179	C₂	1	1	yes	maximal, characteristic
7	Q₃₂×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
8	SD₃₂×Dic₅₉	C₂	1	2	yes	maximal, complemented
9	D₁₆×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₁₆×D₅₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₉×#2249	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837931	—	2	2	no	—
13	C₂×C₁₆×D₅₉	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₅₉	C₂²	1	1	yes	characteristic
15	#16837804	C₂²	1	2	yes	—
16	#16837756	—	2	2	no	—
17	#16837640	C₂²	1	2	yes	—
18	#16837635	C₂²	1	2	yes	—
19	#16837628	—	2	2	no	—
20	Q₁₆×Dic₅₉	C₂²	1	2	yes	—
21	SD₁₆×Dic₅₉	—	2	2	no	—
22	D₈×Dic₅₉	C₂²	1	2	yes	—
23	C₄×C₈×D₅₉	C₂²	1	1	yes	characteristic
24	#16837230	C₂²	1	2	yes	—
25	#16837228	C₂²	1	1	yes	characteristic
26	#16837228	C₂²	1	1	yes	characteristic
27	#16837206	C₂²	1	2	yes	—
28	#16837206	C₂²	1	2	yes	—
29	#16837203	C₂²	1	2	yes	—
30	#16837203	C₂²	1	2	yes	—
31	C₄×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
32	C₁₆×Dic₅₉	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₅₉	C₂²	1	1	yes	characteristic
34	C₂×Q₃₂×C₅₉	C₂²	1	1	yes	characteristic
35	C₂×SD₃₂×C₅₉	C₂²	1	2	yes	—
36	C₂×D₁₆×C₅₉	C₂²	1	1	yes	characteristic
37	C₅₉×#495	C₂²	1	2	yes	—
38	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
39	C₂×C₈×D₅₉	D₄	1	1	yes	characteristic
40	C₂×C₈×D₅₉	C₂³	1	1	yes	characteristic
41	#14273720	—	4	2	no	—
42	#14273719	—	2	2	no	—
43	#14273709	—	4	2	no	—
44	#14273703	—	2	2	no	—
45	#14273698	—	2	2	no	—
46	Q₈×Dic₅₉	—	2	2	no	—
47	D₄×Dic₅₉	—	2	2	no	—
48	(C₂²⋊D₅₉)⋊C₄	—	4	2	no	—
49	#14273683	—	4	2	no	—
50	C₄²×D₅₉	D₄	1	1	yes	characteristic
51	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
52	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
53	C₁₆×D₅₉	—	2	2	no	—
54	C₁₆×D₅₉	—	2	2	no	—
55	#14273661	C₂³	1	2	yes	—
56	#14273661	C₂³	1	2	yes	—
57	#14273659	—	2	2	no	—
58	#14273651	—	2	2	no	—
59	#14273651	—	2	2	no	—
60	#14273648	—	2	2	no	—
61	#14273648	—	2	2	no	—
62	C₄×C₅₉⋊C₈	D₄	1	1	yes	characteristic
63	C₈×Dic₅₉	D₄	1	1	yes	characteristic
64	C₈×Dic₅₉	C₂³	1	1	yes	characteristic
65	C₂×Q₁₆×C₅₉	C₂³	1	2	yes	—
66	C₂×SD₁₆×C₅₉	—	2	2	no	—
67	C₂×D₈×C₅₉	C₂³	1	2	yes	—
68	#124×C₅₉	—	2	2	no	—
69	Q₃₂×C₅₉	—	2	2	no	—
70	SD₃₂×C₅₉	—	2	4	no	—
71	D₁₆×C₅₉	—	2	2	no	—
72	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
75	C₂×Q₈×C₅₉	—	2	2	no	—
76	C₂×D₄×C₅₉	—	2	2	no	—
77	C₂×C₂²⋊D₅₉	—	4	2	no	—
78	C₂²×Dic₅₉	—	4	2	no	—
79	C₂×C₄×D₅₉	D₈	1	1	yes	characteristic
80	C₂×C₄×D₅₉	D₈	1	1	yes	characteristic
81	C₂×C₄×D₅₉	C₂×D₄	1	1	yes	characteristic
82	Q₁₆×C₅₉	—	2	2	no	—
83	Q₁₆×C₅₉	—	2	2	no	—
84	SD₁₆×C₅₉	—	4	4	no	—
85	D₈×C₅₉	—	2	2	no	—
86	D₈×C₅₉	—	2	2	no	—
87	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
89	C₂²⋊C₄×C₅₉	—	4	2	no	—
90	C₂²⋊C₄×C₅₉	—	4	2	no	—
91	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
92	C₂²⋊Dic₅₉	—	4	2	no	—
93	C₂²⋊Dic₅₉	—	4	2	no	—
94	(C₂×D₅₉)⋊C₄	—	4	2	no	—
95	C₄⋊Dic₅₉	—	2	2	no	—
96	C₄⋊Dic₅₉	—	2	2	no	—
97	C₄⋊Dic₅₉	—	4	2	no	—
98	Dic₅₉⋊C₄	—	4	2	no	—
99	Dic₅₉⋊C₄	—	4	2	no	—
100	Dic₅₉⋊C₄	—	4	2	no	—
101	C₄×Dic₅₉	—	4	2	no	—
102	C₄×Dic₅₉	D₈	1	1	yes	characteristic
103	C₄×Dic₅₉	D₈	1	1	yes	characteristic
104	C₄×Dic₅₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₅₉⋊C₈	C₂×D₄	1	1	yes	characteristic
106	C₂×C₅₉⋊C₈	C₂×D₄	1	1	yes	characteristic
107	C₈×D₅₉	—	2	2	no	—
108	C₈×D₅₉	—	2	1	no	—
109	C₈×D₅₉	—	2	1	no	—
110	C₁₆×C₅₉	#41	1	2	yes	—
111	C₁₆×C₅₉	#41	1	2	yes	—
112	C₅₉⋊C₁₆	—	2	1	no	—
113	C₅₉⋊C₁₆	—	2	1	no	—
114	C₂³×C₅₉	—	4	2	no	—
115	C₂²×D₅₉	C₂×D₈	1	1	yes	characteristic
116	Q₈×C₅₉	—	4	2	no	—
117	Q₈×C₅₉	—	4	2	no	—
118	D₄×C₅₉	—	4	2	no	—
119	D₄×C₅₉	—	4	2	no	—
120	C₂×C₄×C₅₉	—	4	2	no	—
121	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
124	C₂²⋊D₅₉	—	8	4	no	—
125	C₂×Dic₅₉	—	4	2	no	—
126	C₂×Dic₅₉	—	4	2	no	—
127	C₂×Dic₅₉	—	8	4	no	—
128	C₂×Dic₅₉	C₂×D₈	1	1	yes	characteristic
129	C₂×Dic₅₉	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₅₉	C₂×D₈	1	1	yes	characteristic
131	C₄×D₅₉	—	2	2	no	—
132	C₄×D₅₉	—	2	1	no	—
133	C₄×D₅₉	—	2	1	no	—
134	C₄×D₅₉	—	2	2	no	—
135	C₈×C₅₉	#135	1	2	yes	—
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	—	2	1	no	—
140	#33735	—	59	1	no	maximal, Sylow
141	C₂²×C₅₉	—	8	2	no	—
142	C₂²×C₅₉	—	8	2	no	—
143	C₂²×C₅₉	—	8	2	no	—
144	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
145	C₂×D₅₉	—	2	1	no	—
146	C₂×D₅₉	—	2	1	no	—
147	C₂×D₅₉	#508	1	2	yes	—
148	C₄×C₅₉	—	2	1	no	—
149	C₄×C₅₉	—	2	1	no	—
150	C₄×C₅₉	—	8	2	no	—
151	C₄×C₅₉	#532	1	1	yes	characteristic
152	C₄×C₅₉	C₂×#135	1	1	yes	characteristic
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	D₅₉	1	1	yes	characteristic, complemented
161	#2242	—	59	2	no	—
162	#2217	—	59	2	no	—
163	#2216	—	59	1	no	—
164	#2215	—	59	1	no	—
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	—	16	2	no	—
170	C₂×C₅₉	#3169	1	1	yes	characteristic
171	C₂×C₅₉	C₂×#508	1	1	yes	characteristic
172	C₂×C₅₉	C₂×#508	1	1	yes	characteristic
173	D₅₉	—	2	2	no	—
174	#529	—	118	2	no	—
175	C₂×Q₃₂	C₂×D₅₉	1	1	yes	characteristic
176	C₂×SD₃₂	C₂×D₅₉	1	2	yes	—
177	C₂×D₁₆	C₂×D₅₉	1	1	yes	characteristic
178	C₂²×C₁₆	—	59	1	no	—
179	C₂²×C₁₆	—	59	1	no	—
180	#499	—	59	2	no	—
181	#495	C₂×D₅₉	1	2	yes	—
182	#488	—	118	2	no	—
183	#467	—	59	2	no	—
184	#466	—	59	2	no	—
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	—	59	1	no	—
193	#358	—	59	2	no	—
194	#358	—	59	2	no	—
195	#357	—	59	2	no	—
196	#357	—	59	2	no	—
197	C₄×C₁₆	—	59	1	no	—
198	C₄×C₁₆	—	59	1	no	—
199	C₄×C₁₆	—	59	1	no	—
200	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
201	C₅₉	#33735	1	1	yes	characteristic, complemented, Sylow
202	C₂×Q₁₆	C₂²×D₅₉	1	2	yes	—
203	C₂×SD₁₆	—	2	2	no	—
204	C₂×D₈	C₂²×D₅₉	1	2	yes	—
205	C₂²×C₈	—	59	1	no	—
206	C₂²×C₈	—	59	1	no	—
207	#126	—	236	2	no	—
208	#125	—	118	2	no	—
209	#124	—	2	2	no	—
210	#123	—	236	2	no	—
211	C₂²⋊Q₈	—	118	2	no	—
212	#121	—	118	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	—	2	2	no	—
219	SD₃₂	—	2	4	no	—
220	D₁₆	—	2	2	no	—
221	C₂×C₁₆	—	59	1	no	—
222	C₂×C₁₆	—	59	1	no	—
223	C₂×C₁₆	—	118	2	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	—	59	2	no	—
229	#106	—	118	2	no	—
230	#103	—	118	2	no	—
231	#103	—	118	2	no	—
232	#102	—	118	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	C₂²×D₅₉	1	1	yes	characteristic
238	C₂×Q₈	—	2	2	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	236	2	no	—
241	C₂²×C₄	—	59	1	no	—
242	C₂²×C₄	—	59	1	no	—
243	C₂²×C₄	—	59	1	no	—
244	C₂²×C₄	—	236	2	no	—
245	Q₁₆	—	2	2	no	—
246	Q₁₆	—	2	2	no	—
247	SD₁₆	—	4	4	no	—
248	D₈	—	2	2	no	—
249	D₈	—	2	2	no	—
250	C₂×C₈	—	59	1	no	—
251	C₂×C₈	—	59	1	no	—
252	C₂×C₈	—	118	2	no	—
253	C₂×C₈	—	118	1	no	—
254	C₂×C₈	—	118	1	no	—
255	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
256	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
257	C₄⋊C₄	—	118	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	—	4	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	—	236	2	no	—
272	C₄²	D₄×D₅₉	1	1	yes	characteristic
273	C₁₆	—	118	1	no	—
274	C₁₆	—	118	1	no	—
275	C₁₆	#11745687	1	2	yes	—
276	C₁₆	#11745687	1	2	yes	—
277	C₂³	—	4	2	no	—
278	C₂³	—	59	1	no	—
279	Q₈	—	4	2	no	—
280	Q₈	—	4	2	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	472	4	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	59	1	no	—
286	C₂×C₄	—	59	1	no	—
287	C₂×C₄	—	59	1	no	—
288	C₂×C₄	—	118	2	no	—
289	C₂×C₄	—	118	2	no	—
290	C₂×C₄	—	118	1	no	—
291	C₂×C₄	—	118	1	no	—
292	C₂×C₄	—	472	4	no	—
293	C₂×C₄	—	236	2	no	—
294	C₂×C₄	—	236	2	no	—
295	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
296	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
297	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
298	C₈	—	118	1	no	—
299	C₈	—	118	1	no	—
300	C₈	#14273746	1	2	yes	—
301	C₈	C₂×#11745687	1	1	yes	characteristic
302	C₈	C₂×#11745687	1	1	yes	characteristic
303	C₂²	—	8	2	no	—
304	C₂²	—	8	2	no	—
305	C₂²	—	8	2	no	—
306	C₂²	—	59	2	no	—
307	C₂²	—	118	1	no	—
308	C₂²	—	118	1	no	—
309	C₂²	C₂×D₈×D₅₉	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	8	2	no	—
313	C₄	—	59	2	no	—
314	C₄	—	118	1	no	—
315	C₄	—	118	1	no	—
316	C₄	—	472	2	no	—
317	C₄	—	472	2	no	—
318	C₄	#16837950	1	1	yes	characteristic
319	C₄	C₂×#14273746	1	1	yes	characteristic
320	C₂	—	16	2	no	—
321	C₂	—	118	2	no	—
322	C₂	#21434673	1	1	yes	characteristic
323	C₂	#508×D₅₉	1	1	yes	characteristic
324	C₂	#508×D₅₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₉
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₅₉×#2249	C₂
socle	C₂²×C₅₉	C₂²×D₈
prime radical	#21426179	C₂
hypercenter	#2249	D₅₉
hypocenter	C₅₉	#33735
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2249

Complements

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

N	H ≅ G/N	type
1. #21434663	C₂	semidirect
2. #21426375	C₂	semidirect
3. #21426327	C₂	semidirect
4. #21426189	C₂	semidirect
5. #21426184	C₂	semidirect
7. Q₃₂×Dic₅₉	C₂	semidirect
8. SD₃₂×Dic₅₉	C₂	semidirect
9. D₁₆×Dic₅₉	C₂	semidirect
10. C₄×C₁₆×D₅₉	C₂	semidirect
11. C₅₉×#2249	C₂	semidirect
160. #2249	D₅₉	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
#21434663	2
#21426375	2
#21426327	2
#21426189	2
#21426184	2
#21426179	2
Q₃₂×Dic₅₉	2
SD₃₂×Dic₅₉	2
D₁₆×Dic₅₉	2
C₄×C₁₆×D₅₉	2
C₅₉×#2249	2
#33735	59